ANRV344-PL11-16 ARI 17 April 2008 13:11

CoalitionsMacartan HumphreysDepartment of Political Science, Columbia University, New York, New York 10027;email: mh2245@columbia.edu

Annu. Rev. Polit. Sci. 2008. 11:351–86

First published online as a Review in Advance onFebruary 15, 2008

The Annual Review of Political Science is online athttp://polisci.annualreviews.org

This article’s doi:10.1146/annurev.polisci.11.062206.091849

Copyright c© 2008 by Annual Reviews.All rights reserved

1094-2939/08/0615-0351$20.00

Key Words

commitments, cooperative, noncooperative, contracts, Nashprogram

AbstractThe game theoretic study of coalitions focuses on settings in whichcommitment technologies are available to allow groups to coordinatetheir actions. Analyses of such settings focus on two questions. First,what are the implications of the ability to make commitments andform coalitions for how games are played? Second, given that coali-tions can form, which coalitions should we expect to see forming? Iexamine classic cooperative and new noncooperative game theoreticapproaches to answering these questions. Classic approaches havefocused especially on the first question and have produced powerfulresults. However, these approaches suffer from a number of weak-nesses. New work attempts to address these shortcomings by mod-eling coalition formation as an explicitly noncooperative process.This new research reintroduces the problem of coalitional instabil-ity characteristic of cooperative approaches, but in a dynamic setting.Although in some settings, classic solutions are recovered, in othersthis new work shows that outcomes are highly sensitive, not only tobargaining protocols, but also to the forms of commitment that canbe externally enforced. This point of variation is largely ignored inempirical research on coalition formation. I close by describing newagendas in coalitional analysis that are being opened up by this newapproach.

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INTRODUCTION

Coalitions seem to be inescapable in poli-tics. Although much of contemporary polit-ical theory holds to the principle that indi-viduals are the fundamental unit of analysis,in actual models the units are often coali-tions: trading blocs, nations, states, govern-ments, houses, ethnic groups, parties, classes,households.

In most instances, coalitions are givens; wesimply assume that they exist and that theybehave as unitary actors, bracketing the ques-tions about how they persist and how actionsare coordinated within them.

There are two instances, however, in whichcoalitional possibilities play a more prominentrole in political analyses. The first arises whenthe possibility of coalitional action poses athreat to analysis that is based on individualmaximization. Olson (1965) warns that it isnaive to predict actions of individuals basedon group interests in settings where agree-ments cannot be enforced. The converse isalso true, however: It is a mistake to predictthe actions of groups from the preferences ofindividuals in settings in which agreementscan be enforced without duly taking accountof that fact. This is what von Neumann &Morgenstern (1944) call “the crucial prob-lem” in modeling games and economic behav-ior. The basic question is: Given that coali-tions can form, what outcomes should weexpect?

The second instance arises when groupsthemselves are an important subject of analy-sis. The concern here is with the creation andmaintenance of group structures—the ori-gins of nations (Alesina & Spolaore 1997),the formation of classes (Roemer 1982) orcoalitions between classes (Rogowski 1989,Iversen & Soskice 2006), and the study of en-dogenous ethnicity (Posner 2005), migration(Tiebout 1956), and government formation(Laver & Shepsle 1996, Snyder et al. 2005).The question here is: Given that coalitionscan form, which coalitions should we expectto see forming?

To address these questions we need somenotion of what it means to be a coalition. Thisdefinitional question is surprisingly thornyand goes to the heart of the project of coali-tional analysis. The basic difficulty is that it isinsufficient to define a coalition as a group ofindividuals who act together or act somehowin each other’s interests. As Krehbiel (1993)notes in the case of political parties, even ifthe actions of individuals in a group are cor-related, that implies nothing about the agencyof groups. Krehbiel suggests that evidence forthe agency of political parties is found whenindividuals vote contrary to their preferencesbut consistent with the group’s goals. This ap-proach is compelling, but it leaves any the-ory of coalitions that purports to be based onmethodological individualism and utility max-imization in a quandary. On the one hand, ifthe seemingly distinctive actions of individu-als in a coalition can be adequately accountedfor simply through a standard examination ofindividual incentives, then there is no specialplace for coalitional analysis. On the otherhand, if these actions cannot be accounted forby individual incentives, then it would seemthat either the assumption of methodologicalindividualism or the utility maximization as-sumption must be dropped.

Coalitional analysis could indeed beundertaken by dropping either of these as-sumptions. But for the purposes of this review,I take a middle approach and let a coalitiondenote a set of two or more agents that haveengaged in an externally enforced agreementto undertake a set of actions subject to theterms of their agreement. To emphasize theexternal enforcement, I use the metaphor ofplayers signing contracts. The insistence onexternal enforcement is important because“self-enforcing” agreements can be fullyanalyzed without reference to coalitions.As the enforcement technology is taken tobe exogenous, coalitional analysis is in thissense always partial equilibrium analysis. Forgenerality, the definition does not requirethat actions taken are necessarily in the

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interest of the group, nor that they are notin the immediate interests of the individualsconcerned. The definition excludes somesubjects of interest in the analysis of coalitionsmore broadly defined—such as self-enforcingcoordination—but it has the merit of placingthe analytic emphasis on the availability ofcommitment technologies, and, as I discussbelow, variation in the types of commitmentsthat are available to players is key to under-standing the types of coalitional structuresthat we expect to observe. Using this defini-tion, the two questions under study can berephrased as follows: In situations in which in-dividuals can make commitments (from someuniverse of enforceable commitments) toeach other, how do we expect them to behave,and who do we expect to coordinate withwhom?

The remainder of this review considersgame theoretic approaches to answering thesetwo questions. Traditionally, the main at-tempts to respond to the first question usedthe tools of cooperative game theory to iden-tify outcomes that are in some sense immuneto coalitional deviations. By assuming thatgains from cooperation are indeed maximizedin such settings, this approach has placed lit-tle emphasis on the second question of whichgroups are in fact likely to form in equilib-rium. The next section motivates the prob-lem through the examination of two games,Prisoners’ Dilemma and Divide the Dollar. Ithen describe some of the major solution con-cepts that derive from a cooperative approachto studying such problems. These solutions,though developed without explicit referenceto a particular game form, correspond to so-lutions that do arise from more fully specifiednoncooperative analyses.

Although it has significant appeal, coop-erative game theory has been criticized bypolitical scientists on a number of grounds.Many of these criticisms miss the mark, per-haps because of a lack of clarity regarding whatmodeling features are specific to the coop-erative approach. But for some applicationsthe cooperative approach does suffer from

PD: Prisoners’Dilemma

DD: Divide theDollar

two shortcomings: its inability to take accountof all strategically relevant information anda demonstrable sensitivity of negotiated out-comes to bargaining protocols.

Recent work uses noncooperative coalitiontheory to respond to some of these shortcom-ings and to address both questions of substan-tive interest: what coalitions will form andwhat strategy profile will ensue. A commonframework for doing this is to add a non-cooperative pregame in which coalitions areformed and then analyze the base game non-cooperatively with coalitions as the unit ofanalysis. In such cases, players predict the out-comes of the game between coalitions and se-lect coalitional membership on the basis ofthis. Some of these games use open and closedmembership rules; others employ sequentialgame forms or dynamic processes that al-low for continual coalitional evolution. Al-though it is well known that bargaining out-comes are sensitive to protocols, I highlightthe extreme sensitivity of outcomes to detailsof the legal environment—that is, to implicitassumptions regarding not just how contractsare formed but what types of contracts canbe formed. I examine the implications of onesuch assumption for models of governmentformation, the assumption that parties cancommit to bargaining only with a subset ofother parties. This assumption, though rarelyif ever defended, is shown to have substantiveimport.

I close by recommending three fruitful di-rections for future work: a reassessment of theconditions for coalitional stability; a more ex-plicit focus on the types of commitment de-vices available to potential coalition partners,and a better linking of these to applications;and an examination of coalitional possibilitiesin settings in which commitment devices areendogenous.

MOTIVATING CASES

We can motivate the problem of coalitionformation through a discussion of the Pris-oners’ Dilemma (PD) and Divide the Dollar

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(DD) games. In each case, I note how thegames change once coalitional possibilities areconsidered. Later, I return to these gameswhen discussing different approaches to mod-eling coalitional processes.

A Prisoners’ Dilemma with Contracts

Assume that two individuals each need to de-cide whether to provide evidence against theother for some joint crime. Payoffs for thedifferent strategy combinations are as givenin Figure 1.

This is a standard PD game, and its so-lution is well known. The inescapable fact isthat rational, utility-maximizing players de-fect (they play D), even though they wouldboth be better off if both kept quiet (playedC). This is a sad result, and the sadness of ithas provoked an enormous literature to findconditions under which rational individualsmight instead cooperate. One approach is toask how the prisoners would play if they couldsign contracts with each other.

The traditional approach for answeringthis question is the cooperative approach. Im-portantly, the cooperative approach “solves”the PD not by bringing a different solutionconcept to the same game, but by solving adifferent game: one with contracting possibil-ities. To emphasize the point, I refer to the PDas the base game and the PD with contracts asthe expanded game.

How does play in the expanded game differfrom play in the base game? The answer de-

BC D

A C13

13

2−1

D−1

20

0

Figure 1A prisoners’ dilemma.

pends on what kind of contracts can be signed.The universe of possible contracts is not welldefined, a feature I return to later. I focushere, however, on three prominent types ofcontract that can be used to form expandedgames.

� Contracts over pure strategies. Onepossibility is that contracts are signedwith respect to pure strategies: “If bothof us have signed this contract, then Iagree to play C, and if I fail to do so,some terrible thing will happen to me.”In the case where contracts can only besigned with respect to pure strategies,there are three efficient contracts, (C,D), (D, C), and (C, C), although one ofthese, (C, C), is individually rational inthe sense of being an improvement overno contract for both players.

� Contracts over correlated strategies.Contracts could also invoke jointlymixed strategies of the following form:“If both of us have signed this contract,then I, Player A, will condition my strat-egy on some publicly observable ran-dom event x, which occurs with prob-ability q. If x occurs, I will play C (andyou will play D); otherwise, I will playD (and you will play C).” This familyof contracts is larger and allows play-ers to do better than they would if theyonly relied on pure strategies. In thiscase, the individually rational and effi-cient contracts can be parameterized by{q : q ∈ [ 1

3 , 23 ]}; and these different con-

tracts correspond exactly to divisions ofa dollar.

These two types of contract spec-ify particular strategies to be played byplayers, but contracts need not be lim-ited to this. A common expansion of thespace of contracts is to allow contractsto stipulate side payments to be madebetween players.

� Contracts with side payments. If in-dividuals can commit both to a particu-lar course of action in the game and tobilateral financial transfers, then even if

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contracts over correlated strategies arenot feasible, another set of contracts canbe implemented that returns the samevalues. In each of these, one player playsD while the other plays C; the playerwho plays D then makes a side paymentof 1 + α to the other, where α ∈ [0, 1]represents the division of the surplus.In this case (though not always), the setof possible payoffs from the game withtransferable utility and pure strategiesis equivalent to those from the gamewith nontransferable utility and corre-lated strategies.

In the case of the simple contract, thereappears to be just one plausible candidate—allcooperate. This could be supported by manybargaining protocols. For example, we couldimagine that player 1 makes a take-it-or-leave-it offer to player 2. If players sign, the con-tract is implemented; if not, then players playthe PD (base game) noncooperatively. How-ever, in the latter two cases, with multiple con-tracting possibilities, the contracting problemseems considerably more difficult, and the so-lution may plausibly depend on the type ofbargaining allowed.

Finding a solution to such a game is of greattheoretical interest for the following reason.We see that, for a special case, the PD withcontracts reduces to a problem of dividinga dollar. It turns out, however, that this isnot unique to this PD or even to two-playerPDs more generally; rather, for a very largeclass of two-player games—excluding zero-sum games—it is possible to normalize util-ities such that the set of efficient contractscorresponds precisely to the segment [0, 1].The introduction of contracts thus vastly re-duces the complexity of the universe of gamesthat has to be analyzed. Thus, if we can findone reliable solution to this problem, then wehave a solution for a vast array of games thatcan be characterized by the same simple util-ity space. That is the tremendous attractionof finding solutions to games with contracts:the promise of generality.

A Divide-the-Dollar Gamewith Contracts

Contracting becomes considerably morecomplicated once we move beyond two play-ers. To see why, consider the following normalform (noncooperative) game. There are threeplayers, N = {1, 2, 3}. The strategy space foreach player is given by a two-dimensional unitsimplex, �, with generic element s j ∈ �, rep-resenting the set of all possible three-way di-visions of a dollar. If two or more people agreeon an allocation, then each of the three play-ers gets his share in the agreed-on allocation;otherwise, all receive 0.

What are the Nash equilibria of this game?Unlike the PD case, where the Nash equilib-rium was unique, here there are many equi-libria. Any profile in which all three playersplay the same strategy is a Nash equilibrium.Any profile in which two players make thesame offers is also a Nash equilibrium, pro-vided that the third does not offer more toeither of the other two than they offer them-selves. These are all efficient equilibria. Thereis also an inefficient equilibrium, in whicheach player proposes taking the full dollar forhimself.

The Nash equilibrium solution conceptclearly has difficulties making fine predictionsfor this game, since any division and no di-vision can all arise as outcomes. In consid-ering a closely related game, von Neumann& Morgenstern (1944, p. 223) concluded that“there seems to be no escape from the ne-cessity of considering agreements concludedoutside the game . . . if we do not allow themthen it is hard to see what if anything will gov-ern the conduct of a player.” In other words,the authors suggest a form of coalitional anal-ysis in this case not because contracting pos-sibilities are likely to be part of player strate-gies, but rather in order to select a solution tothe base game. In fact, however, it is not clearthat allowing agreements solves the problemin this case. If we allow for the possibility thatplayers can write binding contracts over thestrategies that they will adopt, the question

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then becomes: What contracts will be writ-ten, with what resulting payoffs?

As with the PD example, we can envisiona setting in which, prior to the base game de-scribed above, DD players can sign a set ofcontracts with each other. And as with the PDgame, these contracts may specify explicitly orimplicitly the actions that will be taken by allthe signing partners in the base game and mayor may not specify other actions such as themaking of side payments. Again, as with thePD case, the set of efficient contracts corre-sponds exactly to divisions of a pie.1

In this case, however, matters are morecomplicated because players have to select notjust the contract but also their contractingpartners. The problem is further complicatedby the fact that for any initial contract and re-sulting set of payoffs, some rival contract maybe available to a set of players that would makethem all better off. This raises new concerns:If a contract is offered, does the offer remainopen while a player examines other offers? If acontract is signed, can it be superseded by an-other contract? With more than three players,the contracting possibilities become still morecomplex; in this case, the optimal actions of acontracting pair could reasonably depend onwhether the other players sign contracts witheach other, and what kinds of contracts theysign.

THE COOPERATIVE APPROACH

The classic game theoretic approach to ad-dressing problems in which contracts can besigned uses cooperative game theory. In manyaccounts, cooperative game theory is the studyof games in which contracts can be signed.This is misleading because noncooperativeapproaches can just as easily be used for gameswith contracts and commitments, as long as

1Note that the pie being divided in the expanded game isnot the same as that being divided in the base game. To seethe difference, imagine that players had utility functionsof ui = ((s )i )2. In this case, a 50/50 division between twoplayers would yield payoffs of 0.25 each, whereas a con-tract that specified a 50% probability of a 100/0 or a 0/100division would yield expected utility of 0.5 each.

the strategies for signing contracts and mak-ing commitments are specified. Rather, coop-erative game theory is an approach to solvinggames.

Cooperative Modeling Assumptions

The cooperative approach is characterized bytwo assumptions: a representation assumptionand a solution assumption.

The representation assumption is that, inthe presence of contract signing possibili-ties, the only strategically relevant informa-tion from the base game is the set of profilesof utilities that can be attained by coalitionsof players, possibly as a function of the full setof coalitions that form. The important dif-ference between different contracts is in theutilities they assign to different players, not inthe details of the particular strategies that theyspecify to achieve these utilities. In practice,the representation assumption takes form inthe use of a “characteristic function” (or oneof its relatives) to characterize the base game.

The solution assumption is that generalprotocol-free outcomes should be the objectof study. This assumption is logically distinctfrom the representation assumption. The ideais that the details through which contracts aredeveloped are in practice likely to be unknow-able, and predictions should therefore dependon what coalitions are capable of attaining byworking together rather than on the particularprocedures of bargaining that are used.

Cooperative solutions thus select payoffprofiles and do not directly describe either thestrategies that may be used to sign contractsor the strategies that are employed in the basegame once contracts are (or are not) signed.

I describe the approach in detail for theclassic case of “coalitional games with trans-ferable utility” or “cooperative games withside-payments,” although more general ap-proaches exist.2 For a population N of players,the characteristic function v assigns a worth to

2Two generalizations are worth emphasizing. First, the the-ory can allow for cases in which coalitions can jointly ensuresome set of outcomes that may be valued differently by

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each coalition, v : 2N → R1.3 In practice in

most cooperative work, v is a primitive. How-ever, it should be possible to derive v from basegames. To do this, we need to make two typesof assumptions: effectivity assumptions abouthow much value a coalition can gain and trans-ferability assumptions about how this worthcan be divided among coalition members.

Effectivity assumptions are used to treatthe worth of the coalition as the maximumamount that members of a coalition can “guar-antee” for themselves (the amount for whichthe coalition is “effective”) (Aumann 1961). Asshown in our discussion of the PD, if playerscan collude, then the set of strategies that areavailable to them jointly is the correspond-ing set of correlated strategies. The questionthen is, given these strategy spaces, what cana coalition guarantee itself? Two notions ofcoalitional effectivity are prominent in the lit-erature, corresponding to the minimax andmaximin derivations of v. Intuitively we thinkof S and N − S choosing strategies sequen-tially. S is α-effective for x if it can guaranteeitself the payoffs of x even if it goes first. S isβ-effective if it can guarantee itself these ben-efits if it goes last. Other alternatives are pos-sible. For example, v(S ) may denote the sumof payoffs to members of S from Nash equilib-rium play or from Nash bargaining betweenS and N− S. For Chandler & Tulkens (1995),S is γ-effective for x if x corresponds to theequilibrium payoffs to S in a game betweenS and the remaining players acting as single-tons. The key feature to note here is that inorder to derive a cooperative game from a basegame, we need to already accept some suppo-

different members without the possibility for side-payments among them [nontransferable utility (NTU)games]. See Aumann (1961) for a discussion of the core forNTU games and Maschler & Owen (1989) on the “con-sistent Shapley value” for NTU games. Second, greatergenerality can be achieved by replacing the characteristicfunction with a partition function. A game in “partitionfunction form” (Thrall & Lucas 1963) specifies a payofffor every coalition, S, conditional on a particular partition,P, of N.3Implicitly, single-member coalitions are included in thisset.

TU: transferableutility

sitions about how coalitions are likely to reactto each other.

The transferability assumption is the as-sumption that utility can be directly trans-ferred from one player to another. It can bedefended without reference to interpersonalutility comparisons if players have sufficientaccess to a divisible and transferable good, y,that enters each player’s utility linearly.

With these two assumptions in hand, anumber of minor assumptions allow a usefulnormalization. First we assume that v is “su-peradditive,” i.e., that v(A∪B) ≥ v(A )+v(B).[In fact, superadditivity is a result of both theminimax (Luce & Raiffa 8.2) and maximin(Aumann 1961, Sec. 9) derivations of v.] Thesuperadditivity assumption is based on theidea that whatever underlying strategies areplayed by A and B to generate v(A )+v(B) canalso be employed by A and B working in con-cert (although see Hyndman & Ray 2007 fora counterargument). It is considerably weakerthan the assumption of “strict superadditiv-ity” (the requirement that this inequality bestrict), which is not satisfied, for example,in the DD game we considered above. Sec-ond, we assume that v is essential: v(N ) >∑

i∈N v(i ). Note that under superadditivity,essential games are the interesting ones fromthe perspective of potential for cooperation.For essential supperaditive games, we can al-ways normalize utilities to define a character-istic function with v(N ) = 1 and v(i ) = 0 forall i. This is termed a (0 − 1) normalizationof the game v. Henceforth, we assume that allTU games are (0 − 1)-normalized.

Consider now the cooperative representa-tion of the PD and DD games with contract-ing discussed above. Under the assumptionthat players play Nash in the case of bargain-ing failure, a characteristic function for thePD is given by v : 2{A,B} → R

1, v(C) = (|C | ≥2). For the DD game, there are many Nashequilibria in the game between two-personcoalitions and the remaining singleton, butthey are all payoff-equivalent (for the coali-tions): the coalition claims the pie. For sin-gletons, we can consider the game between

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themselves and some pair acting as a coali-tion, or else consider the three-player gamein which an inefficient Nash equilibrium isplayed. In this case, each can expect a payoffof 0. This yields the 0–1 characteristic func-tion: v : 2{1,2,3} → R

1, v(C) = (|C | ≥ 2).We now consider some of the prominent

cooperative solution concepts for games ofthis form.

Cooperative Solutions

There exists a very wide class of solutions togames in cooperative form, with different ap-proaches emphasizing different principles ofbargaining behavior. Here I describe what areperhaps the three most prominent solutions—the Nash bargaining solution, the core, andthe Shapley value—and show how each han-dles the PD and DD games with contracts.4

The Nash bargaining solution. The Nashbargaining solution, used for settings with orwithout transfers, is designed for what can becalled (e.g., Hart & Mas-Colell 1996) “purebargaining models,” in which either a com-prehensive agreement is reached or all betsare off. The solution is not designed to handlecases with partial breakdowns, where agree-ments are in fact made only between sub-groups.

The Nash bargaining solution is defined asfollows.

Definition 1. The Nash bargaining so-lution for a (0 − 1) TU game is the payoffvector x that maximizes

∏i∈N xi subject to∑

i∈N xi = 1.

Nash (1953) justified this solution usinga particular noncooperative game in whichplayers made threats in a context with a risk ofbargaining failure. However, the real power

4For a sampling of other cooperative solutions, see thetop cycle set (Ward 1961), the bargaining set (Aumann &Maschler 1964), the nucleolus (Schmeidler 1969), the un-covered set (Miller 1980), and the stability set (Rubinstein1980). For more general solution concepts that can be ap-plied under conditions of cooperative or noncooperativebehavior, see Greenberg (1980) and Chwe (1994).

of the result is that Nash also showed that thebargaining solution is the unique outcomefrom any bargaining procedure, includingany noncooperative process, that is efficient,symmetric, and scale-invariant, and whoseoutcomes are “independent of irrelevant alter-natives” (that is, if a bargaining solution pickssome point z, and if some unchosen point x isremoved from the set of acceptable solutions,then z is still chosen). This axiomatic founda-tion provides strong support to the bargainingsolution. Scale invariance is presupposedin all games employing von Neumann–Morgenstern utility; Pareto efficiency isattained in a wide class of bargaining games;symmetry simply requires that all relevantfeatures of the game are captured by the utilitypossibility set. The most difficult to defend isin fact independence of irrelevant alternatives(Rubinstein 1982), although even this arisesnaturally in many bargaining settings.

For the PD example, the Nash bargain-ing solution predicts an even split of the pie.For the DD problem, the Nash bargaining so-lution, using only information on coalitionsof size 1 and 3, yields division ( 1

3 , 13 , 1

3 ). Forthe PD case, this solution is intuitive, and aswe will see, it corresponds to noncooperativesolutions to the same game. For the DD ex-ample, the solution seems less plausible—whywould the players split the pie when any twocould do better by excluding the third? Thereason is that the Nash bargaining solutionis appropriate only if a constrained class ofcontracts—comprehensive contracts—are al-lowed. This feature obtains in some multilat-eral bargaining settings (such as Diermeier &Merlo 2004), but not all.

The core. Perhaps the most prominent so-lution concept for cooperative games is the“core,” defined by Gillies (1959). The basicidea can be traced back to von Neumann &Morgenstern’s Theory of Games.

Definition 2. The core for a (0 − 1) TUgame is the set of payoff vectors {x} for which∑

i∈N xi = 1 and∑

i∈C xi ≥ v(C) for allC ⊂ N.

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Informally, a payoff vector is in the core ifno coalition can, by acting alone, achieve anoutcome that every member prefers to thatvector. There is a clear rationale for the coreconcept in cases where players can sign con-tracts. The idea is closely analogous to theidea of Nash equilibrium except that, to be inthe core, we require not just that no individ-ual deviations benefit any individual but alsothat no group deviations benefit any group. Inthis regard, we may expect it to be more diffi-cult to find points in the core than it is to findNash equilibria. Unlike in the Nash equilib-rium notion, however, we emphasize that thedeviations from an outcome x by C in the defi-nition of the core imply changes in coalitionalstructures and thus, possibly, changes in thestrategies employed by all players rather thanjust those employed by C.5

Returning to our motivating examples, wefind that whereas the Nash bargaining solu-tion corresponded closely to the noncooper-ative solutions, the core solution looks verydifferent for both of these examples. In thePD case, the core is set-valued: The entire setof (individually rational) efficient contracts isin the core, reflecting the fact that for each ofthese, some bargaining procedure could pro-duce them in equilibrium. In all cases, how-ever, the substantive outcome is the same: Oneplayer cooperates and the other does not, anda compensating side payment is made. For theDD game, the core is empty. No outcome ispredicted at all, since for any candidate solu-tion, an argument can be made for why thatoutcome would not obtain. The emptiness ofthe core, as I argue below, reflects a fundamen-

5The core has been defined here for cooperative games. Aclosely analogous definition exists for many social choicesettings; in this case, the core is a set of outcomes ratherthan a set of payoff profiles. Consider any preference ag-gregation rule f that can be represented by a set of “decisivecoalitions” D(a, b), such that b is “socially preferred” to aif and only if there is a coalition, C ∈ D(a, b), whose mem-bers all strictly prefer b to a. Then a is in the core of f for agiven preference profile if there is no b that is socially pre-ferred to a. This is equivalent to the condition that there isno coalition that can and wants to induce a change from ato b.

tal difficulty of generating a prediction for thesubstantive outcome of this game when non-comprehensive agreements can be signed.

The Shapley value. The Shapley value triesto relate the size of a share of the pie that ac-crues to an individual in a cooperative settingto that individual’s contribution to the size ofthe pie. Again there is an underlying bargain-ing logic. Shapley asks whether any allocationhas the following property: Whenever one in-dividual, i, demands more of another, j, on thegrounds that if i left the coalition j would losesome benefits, then j can object to i’s demandon the grounds that if j left i would lose atleast as much as j would if i left. The difficultyis in calculating the marginal contributions ofdifferent players, since these may depend onother players. Perhaps a player’s contributionis large if some other player is in the group butsmall otherwise. If so, then perhaps the mea-surement of that other player’s contributionshould take account of his impact on the firstplayer’s contribution. And so on. The idea ofthe Shapley value (for cooperative TU games)is that a value can be associated with each in-dividual if we average over the whole set ofmarginal contributions that she could maketo every coalition that does not contain heralready. Letting R denote the set of all per-mutations of N, with typical element R, andletting Si (R) denote the set of players that pre-cede player i in the permutation R, the Shapleyvalue is defined as follows.

Definition 3. The Shapley value for a (0−1) TU game is the payoff vector x with:

xi (N, v) = 1|N|!

∑R∈R

[v(Si (R) ∪ {i}) − v(Si (R))].

With a little calculation, one can check thatunder the 0 − 1 normalization xi ∈ [0, 1]and

∑xi = 1. The Shapley value can be

uniquely derived using axioms of symmetry,efficiency, additivity [an axiom on the addi-tion of subgames; if w = v + u, xi (N, w) =xi (N, v)+xi (N, u))], and the dummy axiom—if a player’s actions never increase the util-ity possibilities of any set, then that player

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gains no share of the surplus (Osborne &Rubinstein 1994, 14.4.2).

There are a number of known relations be-tween the Shapley value and the core. Oneis that if a game is convex (in the sense thatthe larger the coalitions are, the larger themarginal contributions of one coalition to an-other are), then the core is nonempty and x isin the core (see, e.g., Mas-Colell et al. 1995,18.AA.1).

In both the PD and the DD games wehave been considering, the Shapley value cor-responds to the Nash bargaining solution.

Hybrid Solutions

There are also solutions that, though cooper-ative in spirit, are closely associated with non-cooperative solutions. Two of these are thestrong equilibrium and coalition-proof Nashequilibrium.

Strong Nash equilibrium. The “strongequilibrium” solution concept, due toAumann (1959), lies right at the intersectionbetween cooperative and noncooperative ap-proaches. It is defined for explicitly noncoop-erative games and acts as a refinement of Nashequilibrium that excludes Nash equilibriafrom which some joint deviation is beneficialto some coalition. In terms of the discussionabove, the notion drops the representationassumption of cooperative approaches butmaintains the solution assumption. We cansee immediately from the definition that if anequilibrium is strong, then it is a Nash equi-librium and it is also Pareto-efficient. Theseproperties indicate how strong the equilib-rium concept is. In fact it is so strong that itoften fails to exist. A strong equilibrium doesnot exist for either the PD or the DD examplesstudied above. Although sometimes related tothe core, the two notions are distinct, as canbe seen from the PD example. The core existsin the PD we studied above because it is as-sumed that if the two individuals do not forma coalition, then they play the Nash equilib-rium. Strong equilibrium, however, assumes

that if one player deviates from the coalition,then the other’s strategy is unaffected.

Coalition-proof Nash equilibrium. Bern-heim et al.’s (1987) notion of coalition-proofNash equilibrium also drops the represen-tation assumption and substantially weakensthe solution assumption. They also drop theassumption invoked in the previous solutionsthat agreements are binding and examineonly the possibility that (unmodeled) commu-nication between players can give rise to self-enforcing agreements. Thus, the coalitionsthey consider do not in fact satisfy the defi-nition of coalitions we have been using so far.Unlike strong equilibrium, Bernheim et al.consider a group deviation to be admissible ifand only if it is not itself subject to deviationsby subgroups of the deviating coalition (for atreatment of linkages between the two con-cepts, see Konishi et al. 1997). All strong equi-libria are coalition-proof, but not all coalition-proof equilibria are strong. Like strongequilibrium, coalition-proof equilibrium canbe thought of as a refinement of Nash equilib-rium; and like strong equilibrium, coalition-proof Nash equilibrium may often fail to exist.

In the PD game we have been examin-ing, all-defect is a coalition-proof equilibriumeven though it is not strong. The reason is thatalthough the individual prisoners may wellagree not to talk, once they are separated theyhave incentives to ignore the agreement. Thisis the first solution we have seen that predictsall-defect as an outcome in the PD—a featurethat follows directly from the inadmissibilityof externally binding agreements. However,in the DD example, none of the original Nashequilibria are coalition-proof. Consider anyNash equilibrium in which player A receivesa positive amount; then consider a discussionin which players B and C agree on a deviationthat splits A’s share positively among them.Then both players B and C have an incentiveto implement this deviation, even in the ab-sence of formal contracts. This result againpoints to the fragility of the Nash equilibriumsolution for distributive games.

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Cooperative-Noncooperative Links

A common concern is that cooperative so-lutions, such as those described above, arenot “supported” by a well-defined game thatleads to these outcomes through standard pro-cesses of individual utility maximization. Inresponse, a large body of research seeks explic-itly to characterize the linkages between co-operative and noncooperative concepts. Themain aim of the so-called Nash program is tofind families of noncooperative games whoseoutcomes correspond to major cooperativesolutions.6 This extensive literature has foundlinks between noncooperative games and awide class of cooperative solutions.

Nash bargaining solution. Nash(1953) pro-vided his own contract-signing game thatimplemented his solution with a refinementof Nash equilibrium. In addition, perhaps themost prominent noncooperative bargainingmodel, the alternating-offers model due toStahl (1972) and Rubinstein (1982), yields theNash bargaining solution as the payoffs fromsubgame perfect equilibrium in the limit asδ tends to unity (Binmore et al. 1986,Binmore 1987). Krishna & Serrano (1996)provide a related result for n-person games.

The core. Bergin & Duggan (1999) demon-strate that the core (whether defined with re-spect to α- or β-effectivity) is subgame per-fect Nash implementable (see also Serrano &Vohra 1994 for subgame perfect Nash imple-mentation of the core and Maskin 1999 forNash implementation). Perry & Reny (1994)model a situation in which bargaining takesplace in continuous time; proposers proposedivisions subject to ratification by concernedparties. If a contract is accepted, the contract-ing parties leave the game. The game contin-ues among remaining players. Perry & Renydemonstrate that an allocation can be sup-ported as a stationary subgame perfect equi-librium outcome only if it is in the core. Im-

6See Bergin & Duggan (1999) for a treatment of differentapproaches to the Nash program.

portantly, in this game, if the core is emptythen a stationary subgame perfect equilib-rium does not exist. In this case, the empti-ness of a core predicts the absence of an effi-cient stationary subgame perfect equilibrium.Chatterjee et al. (1993) provide a model ofn-person coalitional bargaining with sequen-tial offers, transferable utility, and time dis-counting. Focusing on strictly supperadditivegames, they find that efficient stationary equi-librium payoffs converge to a point in the core,as δ converges to 1.

There are also strong results for simplegames of the form commonly of interest to po-litical scientists. For one-dimensional spacesand noncollegial rules, Banks & Duggan(2000) show that if players are perfectly pa-tient and have strictly concave utility, thenoutcomes associated with (no-delay, station-ary) equilibria are always in the core (andif players are nearly perfectly patient, out-comes are close to the core). Thus, for ex-ample, with simple majority rule, the median(core) is implementable in stationary strate-gies through a standard bargaining protocol(see also Jackson & Moselle 2002). Indeed,Banks & Duggan (2000) provide conditionsunder which every core point can be selectedin equilibrium. In addition, recent work thatbetter incorporates the dynamic nature of pol-icy choice and change demonstrates the stabil-ity properties of the core in a noncooperativesetting. Banks & Duggan (2006) show for ageneral bargaining model that if a status quois in the core then it will stay there; indeed,a “static” stationary equilibrium exists if andonly if the status quo is in the core. This resultholds with no assumptions on discount rates.Surprisingly, however, the core, though reten-tive, is not attractive. Banks & Duggan (2006)also show that if a status quo is not in the corein a one-dimensional bargaining game, thenno legislator will ever propose it. Neverthe-less, consistent with Banks & Duggan (2000),with sufficiently patient legislators, outcomesare arbitrarily close to the core.

These results demonstrate the usefulnessof the core when it is nonempty. Can anything

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be learned about the outcome of noncooper-ative games when the core is empty? Maskin2003 shows that, in a class of superadditivegames in partition form, core emptiness is anecessary condition for a grand coalition notto form. Below, I describe more results thatrelate core emptiness to the absence of ab-sorbing states in dynamic processes.

The Shapley value. Bargaining models thatproduce the Shapley value include those ofHarsanyi (1981), Hart & Moore (1990), Win-ter (1994), and Gul (1989). One simple proto-col described by Hart & Mas-Colell (1996) isthe following. A proposer is randomly drawnto propose a division of the pie. If the pro-posal is accepted by all players, it is imple-mented; if not, then with some probabilityq < 1, the proposer is removed from the gameand receives a payoff of 0, and bargaining con-tinues among the remaining players over thevalue obtainable by them. This protocol has aunique subgame perfect solution, which cor-responds to the Shapley value for the TU case(for pure bargaining problems it implementsthe Nash solution). Even though this proce-dure implements the Shapley value, it does notreadily correspond to a likely contracting pro-cedure in many circumstances. For example,in a majority-rule DD game, as studied above,there is no reason to expect that players shouldseek unanimity in contracts that they propose.Gul’s (1989) analysis is perhaps more convinc-ing in this respect. In Gul’s model, there isa pairwise meeting technology. When a pairmeets, one player offers to buy out the other,making a transfer in exchange for the indi-vidual’s resources (for example, her vote). Ifthe offer is rejected, the union dissolves; if ac-cepted, the seller leaves the market and thebuyer continues to negotiate with remainingplayers. Expected payoffs from efficient sta-tionary subgame perfect solutions converge tothe Shapley values as players become patient.This provides a strong noncooperative ratio-nale for the Shapley value, although, as notedby Hart & Mas-Colell (1996), the result holdsonly for efficient solutions, not all solutions.

Other work, not covered here owing tospace limitations, has found noncooperativerationales for a range of other cooperative so-lutions, including the Kalai-Smorodinsky so-lution (Haake 2000, Moulin 1984, Trockel1999) and the uncovered set (Coughlan & leBreton 1999). The model studied by Hart &Mas-Colell (1996), described above, imple-ments the Maschler-Owen (1989) solution fornontransferable utility games. The model hasbeen extended by Vidal-Puga (2005) to allowfor preexisting coalitional structures.

Criticisms of theCooperative Approach

The cooperative approach has been criticizedin recent writing in political theory and, in-creasingly, published political theory relies onnoncooperative methods. The criticisms ofthe cooperative approach have, however, beendiffuse. A clearer sense of the strengths andweaknesses of the approach is important forunderstanding when the predictions of coop-erative approaches should or should not beinvoked. I consider five prominent criticismsto which cooperative theorists can readily re-spond, followed by two criticisms that I be-lieve are more fundamental.

1. Cooperative approaches select outcomesbased on normative rather than positivecriteria. This critique is more salientfor some solutions, such as the Shapleyvalue or the Nash bargaining solution,than for others. The core, for example,has little to offer as a normative solution.In addition, as seen above, a success ofthe Nash program has been its ability todemonstrate how normatively desirableoutcomes can arise from decentralizedbargaining.

2. Cooperative solutions do not exist (enough).Diermeier & Krehbiel (2003) suggestthat the emptiness of the core for manypolitical applications is a reason not touse it. Their rationale emanates in partfrom a desire to use a single solutionconcept for all purposes. One response

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is that existence is neither general of,nor unique to, cooperative solutions;many cooperative solutions generallyexist (or are nonempty), whereas Nashequilibria can fail to exist for reasonsas simple as not having a closed strat-egy set. A deeper response is that mul-tiple solution concepts are not inconsis-tent with theoretical coherence, and theemptiness or nonexistence of some canitself contain relevant information.7 Inthe case of the core, emptiness reflectsa feature of the underlying strategic sit-uation (Hart & Kurz 1983). Thus, inthe party platform selection problem, agame has a (noncooperative) Nash equi-librium if and only if there is a nonemptycore (Duggan 2005); and, as shown be-low, in many games with nonemptycores it is easy to construct plausibledynamic noncooperative games that ex-hibit cycles.

3. Cooperative solutions do not predict strate-gies and cannot incorporate institutions.Diermeier & Krehbiel (2003) criti-cize the core for saying nothing aboutstrategies. They argue that noncooper-ative solutions predict behavioral andoutcome-related predictions, whereascooperative solutions only predict out-comes. Garrett & Tsebelis (2001, p. 100)argue that their “most important cri-tique” of power indices follows from thefact that “cooperative game theoreticassumption of enforceable agreementsis institution free.” Both critiques losemuch of their force, however, when-ever the characteristic function is de-rived from a normal form representa-tion. In such cases, although strategiesare not typically modeled explicitly, they

7The use of a solution concept for a given setting reflectsthe information available to the theorist.Thus, if all thatcan be imposed on beliefs is common knowledge of playerrationality,then the appropriate solution corresponds to theset of rationalizable solutions, not the set of Nash equilib-ria.

are implicit, as described above.8 In thesame way, institutional features are notabsent; rather, they are folded into thedefinition of cooperative games.

4. We do not observe the cycling predictedby cooperative theories. Paradoxically, al-though cooperative theory allows forbinding commitments in many politicalsettings in which the core is empty, co-operative solutions are associated withinstability (see, e.g., Johnson & Libecap2003, Shepsle 1986). In fact, however,core emptiness is simply the absence ofa prediction not the prediction of unsta-ble outcomes. At a deeper level, how-ever, as demonstrated below, in somesettings there is a relationship betweencore emptiness and cycling: When con-tracts are of short duration and the coreis empty, cycles can arise in equilibriumand are predicted by dynamic noncoop-erative models (see Konishi & Ray 2003,Kalandrakis 2004). I illustrate this pointwith a discussion of the Laver-Shepslemodel of government formation in the“Markov Processes and Spot Contracts”section below.

5. It is unrealistic to assume that players canarbitrarily make binding commitments toeach other. There are two responses tothis criticism. The first is that such bind-ing agreements are not unique to co-operative approaches and are endemicin noncooperative approaches studyingrelated phenomena. In such cases thebindingness is often hidden in the gametree, which simply declares that if such-and-such a set of actions are taken thegame ends, or the agreement is im-plemented, and so on. Even the abil-ity to make a take-it-or-leave-it offer(as for example in the model of Baron& Diermeier 2001) involves an im-plicit commitment technology; indeed,

8But, as noted above, cooperative solutions will not dis-tinguish between strategy profiles that produce the sameoutcomes.

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LOSS OF STRATEGICALLY RELEVANTINFORMATION

Consider a game between players A and B in which A has nooptions and B can choose between L and R. Payoffs are givenby uA(L) = 0, uA(R) = 1, uB (L) = −1, uB (R) = 0. Usingthe standard minimax derivation, the characteristic functionis given by v(A) = v(B) = 0, v(A, B) = 1. The Shapley andNash solutions suggest an even split; yet clearly if A refused tobargain, the Nash equilibrium would award the entire pie tohim. Why then would he bargain? The problem here is subtleand hangs on a point of law: As part of contracting, can B makea unilateral commitment? Can B credibly commit to play L inthe event of bargaining breakdown and thereby induce A tobargain? If the answer is no, then the characteristic functionloses important information.

in some noncooperative models (suchas that of Diermeier & Merlo 2000),parties can make cash payments to par-ties outside their coalition in exchangefor a commitment to support future ac-tions of the government. The secondresponse is that this concern is aboutscope conditions, not about the basic ap-proach. Cooperative solutions are moti-vated by the idea that binding commit-ments can be made between arbitrarygroups of players. Yet cooperative solu-tions are employed in settings in whichthis assumption cannot be readily de-fended. If this assumption is not jus-tified, then it is not legitimate to ig-nore the details of institutional proce-dures that give rise to social outcomesunless noncooperative foundations thatinclude such features yield the sameoutcomes (as, for example, was claimedfor the case of the uncovered set byMcKelvey 1986).

There remain, however, two deep andlongstanding concerns with the cooperativeapproach.

1. The characteristic function disregardsstrategically relevant information. The charac-teristic function greatly simplifies the repre-

sentation of a game and in doing so may insome cases lose strategically relevant infor-mation. The most obvious problem is thatpayoffs to coalitions are assumed to be inde-pendent of the ways that remaining individ-uals form into coalitions. Thus, positive andnegative externalities across coalitions are ex-cluded. This problem can be resolved by re-placing the characteristic function with a par-tition function, described below.

Other problems persist, however, and ariseeven in the context of two-player games.An illustration—based on one provided byMcKinsey (1952) and referenced by Luce &Raiffa (1957)—of the inconsistencies that mayarise from the minimax derivation of a char-acteristic function is given in the sidebar.

A reasonable response to such problems isthat Nash equilibria payoffs should be usedto assign worths instead of minimax (or max-imin). But, as a general principle, the useof Nash equilibrium to predict play follow-ing contracting presents difficulties for gameswith multiple equilibria. In such cases, it maybe possible to associate a cooperative solutionfor each cooperative game that corresponds toa given Nash equilibrium. But even this doesnot capture some of the subtleties that mayarise. For an indication of some of the morecomplex reasoning that may arise in the caseof multiple equilibria, consider a symmetricBattle of the Sexes game, with a 0 − 1 repre-sentation as given in Figure 2.

BC D

A C.6

.40

0

D0

0.4

.6

Figure 2A Battle of the Sexes game. Refusal to sign acontract could signal expectations about theultimate equilibrium that will be selected.

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In this case, the mixed-strategy noncoop-erative solution is Pareto-inefficient. Bargain-ing, were it possible, could permit players toreach one of the efficient equilibria or even torandomize between them. In this case, as inthe PD that we examined, many cooperativesolutions will assign 0.5 each. However, fol-lowing the logic of forward induction, a rea-sonable inference following a player’s refusalto negotiate is that the player expects an equi-librium in his favor to result in a Nash equi-librium that awards a share greater than 0.5.

2. Protocols matter. The solution assump-tion made by cooperative theorists is that gen-eral bargaining results, if they are attainableat all, can be obtained without specifying pre-cise bargaining protocols. Insofar as it impliesgreater generality, the lack of a detailed de-scription of the game form that implements asolution is a distinct advantage of the cooper-ative approach, not a weakness. Furthermore,consistent with the Nash program, many arti-cles proposing cooperative theories includednoncooperative games that supported thesesolutions (Nash 1953, Harsanyi 1974, Maskin,2003). However, the fact that some bargainingprotocol implements a given solution does notmean that all protocols do, or that all relevantprotocols do. If different plausible contract-signing protocols produce different outcomes(as indeed they do), then cooperative results,despite the advantage of weak assumptions, donot cover the requisite field.

NONCOOPERATIVECOALITION THEORY

New research on coalitions has responded tothese concerns by weakening both the repre-sentation assumption (characteristic functionsare typically replaced with partition functions,or in some cases more general contracts overplayers’ actions) and the solution assump-tion (explicitly noncooperative approaches areused to examine contract selection). Unlikeclassic approaches, in most cases, new non-cooperative coalition theory focuses not juston outcomes but on studying which coali-

COALITIONAL STRUCTURES

A coalitional structure is a partitioning of the set of playersinto a collection of groups, � = (G j )m

j=1, such that everybodyis in some group (∪m

j=1G j = N ) and no one is in two groups(G j ∩ Gk = ∅ for j = k). Typically, we assume that a parti-tioning is defined uniquely up to the reordering of cells and ofindividuals within cells (thus {{1,2},{3,4}} is considered thesame partition as {{4,3},{1,2}}). Let the set of all such parti-tions be given by G. For a population of size n, the cardinalityof G is then given by the Bell number (Rota 1964), a numberthat grows very rapidly in n. The set of partitions for n = 4 is15. For a 100-person senate, the Bell number is on the orderof 10116 (Weisstein 2005).

tional structures (see sidebar) will form inequilibrium.9

As in the discussion of the PD and DDgames above, the approach used in much re-cent work assumes that prior (or in some cases,in parallel) to playing some base game, a coali-tional structure forms, which determines insome manner how individuals will play thebase game. There exist a range of ways of link-ing coalitional structures to the ultimate out-comes and payoffs for individuals. The sim-plest, used for example by Bloch (1996), isto define a utility function for each individualover the universe of partitions, ui : G → R

1 fori ∈ N. Implicitly, once coalitions are formedthere is some conjecture about play betweencoalitions and a corresponding allocation ofbenefits to individual coalition members. Theidea is that, once formed, coalitions play theunderlying game with each other, and eachcoalition maximizes some function reflectingthe welfare of its members, subject, perhaps,to some agreed-on rule for internal distribu-tion of benefits. The most common alterna-tive is to associate with each partition a worthfor each coalition and allow the distribution ofthe value and the partition to be determined

9For older work emphasizing this objective, see Shenoy(1979), Hart & Kurz (1983) and Aumann & Myerson(1988).

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jointly; in this case we have a partition func-tion given by vC : {� ∈ G|C ∈ �} → R

1 forC ∈ 2N and contracts that specify adherenceto the group’s strategy that returns vC . Moregeneral forms can be considered, however. Inprinciple, it is possible that the coalition for-mation stage specifies a set of actions that willbe taken by coalition members as a functionof what other coalitions form, or perhaps as afunction of the set of contracts that are signedby other agents.

For concreteness, in the discussion thatfollows I consider this question for a situationin which the base game is a Baron & Ferejohn–style majority rule bargaining game involv-ing seven players. Coalitional features enter intwo ways: through commitments made amongplayers in some, or through the institutions ofthe base game. We assume that in the basegame, each nondummy coalition has an equalprobability of being recognized to proposean allocation of the pie. The expected pay-offs for each coalition (of patient players), foreach possible coalitional structure, are given

in Table 1, which is based on values providedin Table 1 of Snyder et al. (2005). My tablealso reports the expected average payoff forindividuals within a coalition.

The question then is, knowing what re-turns can be achieved by coalitions, whatcoalitions are likely to form? Or, for the run-ning example we will study, what party struc-tures should we observe in a distributive-politics setting under different commitmentmechanisms? The answer depends sharply onwhat protocols are used and what types ofcommitment are permitted.

Simultaneous Games

Open membership. In the simplest ap-proach, players simultaneously select a coali-tion to join. Yi & Shin (1994) propose amodel in which all players simultaneously an-nounce a message, and those who announcethe same message form a coalition. Utilitiesare assigned as a function of the ensuing parti-tion, perhaps determined by a subgame played

Table 1 Numeric partitions of seven players

(1)Numericpartition

(2)MIWa

(3)Expected coalition payoffb

(4)Expected per capita payoff

a {1,1,1,1,1,1,1} {1,1,1,1,1,1,1} {0.14},{0.14},{0.14},{0.14},{0.14},{0.14},{0.14}

{0.14},{0.14},{0.14},{0.14},{0.14},{0.14},{0.14}

b {2,1,1,1,1,1} {2,1,1,1,1,1} {0.28},{0.14},{0.14},{0.14}, {0.14},{0.14} {0.14},{0.14},{0.14},{0.14}, {0.14},{0.14}c {3,1,1,1,1} {3,1,1,1,1} {0.42},{0.14},{0.14},{0.14},{0.14} {0.14},{0.14},{0.14},{0.14},{0.14}d {2,2,1,1,1} {2,2,1,1,1} {0.28},{0.28},{0.14},{0.14},{0.14} {0.14},{0.14},{0.14},{0.14},{0.14}e {3,2,1,1} {2,1,1,1} {0.4},{0.2},{0.2},{0.2} {0.13},{0.10},{0.2},{0.2}f {4,1,1,1} {1,0,0,0} {1},{0},{0},{0} {0.25},{0},{0},{0}g {4,2,1} {1,0,0} {1},{0},{0} {0.25},{0},{0}h {2,2,2,1} {1,1,1,0} {0.33},{0.33},{0.33},{0} {0.17},{0.17},{0.17},{0}i {3,3,1} {1,1,1} {0.33},{0.33},{0.33} {0.11},{0.11},{0.33}j {3,2,2} {1,1,1} {0.33},{0.33},{0.33} {0.11},{0.17},{0.17}k {4,3} {1,0} {1},{0},{0} {0.25},{0},{0}l {5,2} {1,0} {1},{0} {0.2},{0}m {5,1,1} {1,0,0} {1},{0},{0} {0.2},{0},{0}n {6,1} {1,0} {1},{0} {0.17},{0}o {7} {1} {1} {0.14}

aColumn 2 provides minimum integer weights (MIW) under simple majority rule.bThe expected returns for each coalition (Column 3) are from Snyder et al. (2005).

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within the coalition. If mixed strategies areallowed, then equilibrium existence followsfrom Nash’s theorem. Under such a rule, how-ever, there may be a multiplicity of equilibria.Yi (1997) gives a monotonicity condition (ce-teris paribus, a member of a smaller coalitionbenefits from a shift to a larger coalition) thatensures that the grand coalition is the uniqueequilibrium.

One feature of such open membershiprules is that implicitly players cannot excludeothers from their coalition. They may chooseto leave a coalition they do not like, but theycannot keep other players out (or in). Thus,in the seven-person DD example, we can con-struct an open-rule game in which players whoenter the same coalition receive an equal percapita share of the worth of the coalition inthe coalitional structure. This game clearlydoes not satisfy Yi’s monotonicity condition,and a multiplicity of equilibria exist. The setof (Nash) equilibrium coalitional structures isgiven by the concentrated structure (o) andthe fragmented structures (a and b). For ev-ery other structure, some player has an in-centive to deviate in order to select anothergroup.10 In this case, all equilibria are payoff-equivalent, because players who receive a sub-proportionate share of the pie have an incen-tive to join a group in which they would get atleast a proportionate share, and, importantly,cannot be prevented from doing so.

Eguia (2007) studied a model of this formfor cases in which coalitions coordinate onvoting over binary agendas over which play-ers have 0/1 utility. For more general policy-making settings, Caplin & Nalebuff (1997)studied a related model that takes the follow-ing form. There is a fixed number of com-munities (coalitions), M = {1, 2, . . . m}; eachcommunity j implements a policy from some

10Define the binary operator → where b→a means thatsome player in coalition structure b has an incentiveto make a deviation that induces coalition structure a.We then have the following sets of relations over states:

i → e → c → f → md → h → j → g

k

}→ l

⎫⎬⎭ → n → o .

space Xj (note that this space may be distinctfor each community). Let G denote a familyof partitionings of N into these M commu-nities. A policy vector is a choice of policyfor each community and thus is an element ofX ≡ × j∈M Xj . Individuals are distributed ac-cording to a hyperdiffuse distribution μ overa measure space of types, A, and have prefer-ences over the group to which they belong andthe pertinent policy outcomes that are cap-tured by their type. So ui : M × X × A→ R.

Once individuals are grouped into com-munities, they play a game to select anoutcome, and then some political process de-termines the political outcomes Pj in eachcommunity according to Pj : G → Xj .The within-group political process is not de-scribed in any detail in this model, but it is as-sumed that some political decision is reachedin each community and that this decision iscontinuous in the population of types—i.e.,if there is some small change in the popula-tion of types, then there will be a correspond-ingly small change in the policy selected bythe community. In this setting, a social equi-librium is a partition � with a correspondingpolicy outcome (Pj (�))m

j=1, such that, giventhe partition, each player prefers to be in thegroup he is in to being in any other group. Arelatively simple proof establishes equilibriumexistence in the two-community case with anodd number of dimensions. Gomberg (2004)has also found sufficient conditions for exis-tence for an arbitrary number of dimensions.However, Gomberg (2005) shows that with-out further constraints on Pj or ui , an equi-librium can fail to exist in any number of di-mensions if there are just three communities.

Closed membership. In an alternativemodel (game �) studied by von Neumann &Morgenstern (1944) and Hart & Kurz (1983),a player’s strategy set is the set of coalitions ofwhich he is a member. A coalition is formed ifall members of the coalition choose it. Notethat embedded in the structure of the game isthe requirement that an individual can be partof a given group only if all other members

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consent to his membership. Thus, unlike inthe open-membership case, a deviation canbe ruled out if the deviation involves a changeby a player to a coalition in which he is notwelcome.

In the DD game we have been consider-ing, game � produces a wider class of equi-libria. The same extreme structures are againequilibria, but so are the cases with majoritycoalitions. In game �, a coalitional structuremay be sustained as an equilibrium as long asno one person could benefit from destroyingthe coalition of which he is a member. All butstates e, j, and i are equilibria, since in eachcase, members of the trios would prefer thepayoffs they can expect from states b, d, and c,respectively.

In game �, a related game also studiedby Hart & Kurz (1983), a coalition is formedof all members who announce a given coali-tion, whether or not all members of that coali-tion announce it. Thus, if 1 and 2 announce{1,2,3}, while 3 and 4 announce {3,4} theensuing structure is {{1},{2},{3,4}} under �

and {{1,2},{3,4}} under �. Again, a stablecoalitional structure is a coalitional structurethat emerges from a Nash equilibrium in thecoalition-selection game. The interpretationin this case is that players make an open offerto a set of players with whom they are will-ing to work, and that offer can be taken up byany subgroup of players to whom the offer hasbeen extended. The key implication with re-spect to coalitional stability is that, if a playerleaves a coalition, that coalition may continuewithout him. In this case, all but two states areequilibria; unlike in game �, structure j con-tinues to be an equilibrium because a split bya member of a trio would induce state h (notstate d), which would leave the splitter in theworst possible position.

The merit of each of these coalition-formation games depends in part on the plau-sibility of the commitment mechanisms theypresuppose. The open rule depends on acommitment technology that allows playersto commit to hypothetical coalitions withoutknowing the exact make-up of the coalition;

under game � the implicit commitment is toa well-defined set, whereas under game � thecommitment is to a nested set of coalitions.In all cases, the commitment is assumed tobe independent of the choices of membersin other coalitions. More recent studies havemodeled coalition formation as a more dy-namic process. This approach provides a morenatural framework for studying more complexacceptance and rejection rules and coalitionalchoices that depend on the coalitional choicesof other players.

Sequential Contracting Games

The multiplicity of equilibria in the games de-scribed follows in part from the simultaneityof moves. Subsequent work has focused on themore realistic case of sequential contracting.

Chatterjee et al. (1993) provide an earlymodel in which bargaining is built on thecharacteristic function of a TU cooperativegame. A protocol determines an ordering ofproposers. Each proposer selects a coalition,S, and makes an offer to the members of thecoalition; the proposer is committed to thatoffer until all respondents have accepted orrejected it. If an offer is accepted by all mem-bers of S, the players in S leave the game, andbargaining continues among remaining play-ers. If an offer is rejected by any member of S,the rejector becomes the next proposer. Dis-counting occurs between rounds.

Chatterjee et al. (1993) examine the effi-ciency of (stationary subgame perfect Nash)equilibrium for all protocols. They find thatfor equilibrium to be efficient for all proto-cols, we need the condition that for all S,v(S)|S| ≤ v(N)

|N| . This condition is satisfied, for ex-ample, by the PD game with contracts butnot by the DD game with contracts. Theyfind that any efficient stationary equilibriumof a strictly superadditive game is in the core;conversely, if the core is empty, then everystationary equilibrium is inefficient. Theseresults provide another strong link betweenthe core and the efficiency of noncooperativeequilibria; however, these games feature no

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guarantee that efficient equilibria will exist,and indeed in some cases every protocol re-turns inefficient results even if there is a core.

Subsequent work in this tradition hasmoved from games built on characteristicfunctions to games built on partition func-tions. Two models stand out.

The first, studied by Ray & Vohra (1999),employs a protocol of the following form (seealso Bloch 1996 for a closely related model).The game starts with the random recognitionof a proposer. The proposer offers to a coali-tion a contract that specifies an allocation toeach coalition member as a function of theultimate coalitional structure that forms, sub-ject to budget balance. We might think of thecontract as specifying the equilibrium strate-gies that will be played by coalition membersfor each coalitional structure, together witha redistribution among coalition members ofthe value attained when those strategies areplayed. Once such a contract is signed, thesigners leave the game, and contracting con-tinues among the remainder until all contractsare signed. If a contract is rejected, the rejec-tor can propose an alternative contract. Onceall contracting is complete, players play thebase game.

Ray & Vohra (1999) derive their strongestresults for symmetric games and show that,for any partition function, there is an as-sociated numeric partition (a partitioningthat is defined uniquely up to a relabel-ing of players) that results from equilibriumproposal making. In addition, they provideconditions under which no-delay equilibriaobtain.

For the seven-person DD game we havebeen examining, the solution under this pro-tocol has the first player offer an equal di-vision of the pie to four players. Each ofthese accepts and the game ends. Thus, out-come f is selected, although the payoff-equivalent structures g and k could also besustained.

In the second sequential model, studied byGomes (2005; see also Gomes 1999), a ran-dom player is selected who can propose a con-

tract to some set of players that includes a setof actions and an (unrestricted) set of trans-fers. Once a contract is proposed, all mem-bers of the invoked coalition respond withacceptance or rejection following some fixedprotocol. If the signatories of the contract ac-cept the contract, the coalition of these play-ers is formed. Unlike in the Ray & Vohramodel, however, these players remain in thegame and may, by mutual consent, renego-tiate and replace their contract with anothercontract including more players (conditionalon the consent of all players who are hav-ing their contracts rewritten). Also in con-trast to the Ray & Vohra model, the game isdynamic, with players playing the base gamein each period given the contracts in force(I discuss further studies along these lines inthe next section). The space of contracts isthe set of utilities that are awarded to eachplayer in a coalition, conditional on the coali-tional structure and the partition function andsubject to budget balance within the coali-tion. Gomes (2005) shows that if the grandcoalition is efficient, it is reached throughcontracting in finite time. If the grand coali-tion is inefficient, however, then in gameswith positive externalities there are bargain-ing delays, and inefficiencies arise even asplayer patience approaches unity. In thesecases, players delay proposing contracts, hop-ing to free-ride on the cooperative behavior ofothers.

Gomes’ results are consistent with newfindings by Hyndman & Ray (2007) for char-acteristic function games with (long-term)binding agreements, finite states, and purestrategies. They show under general con-ditions that, for arbitrary time-dependentstrategies, every equilibrium path convergesto an efficient outcome. Moreover, undergrand-coalition superadditivity (but not forgeneral partition functions), they show thatevery absorbing payoff limit (steady-state pay-off profile) of every Markovian equilibrium isstatic efficient. These results provide broadsupport for the Coase theorem in settingsin which long-term contracts can be signed

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subject to renegotiation in cases in which thegrand coalition is efficient.

Application of sequential contract forma-tion models to public goods production.To see the logic of the Ray & Vohra so-lution, consider the following public goodsproblem (Ray & Vohra 2001). Each i ∈ N ={1, 2, . . . , n} must choose a level of somepublic good to produce. Payoffs are given byui = ∑n

j = 1 zj − 12 z2

i . The classic Nash equi-librium solution yields zi = 1 for all i, and so∑n

j=1 zj = n and ui = n − 12 .

What strategies would be selected if coali-tions maximized the sum of the welfare oftheir members? For concreteness, we con-sider the case with n = 4. For any groupof size m, it is easy to see that a utility wel-fare maximizer that controls the strategiesof individuals in the group would maximizem

∑mj=1 zj − 1

2

∑mj=1 z2

j , yielding zi = m forall i.

Thus, assuming that coalitions maximizethe sum of their members’ utility but coali-tions play noncooperatively with each other,the average payoffs to members of a coalitionconditional on a coalitional structure are asgiven in Table 2.

The question then is: What coalitionalstructure and set of strategies should we ex-pect to observe in a setting in which con-tracts can be signed? The Nash equilib-rium of the game without contracts produces{3 1

2 ,3 12 ,3 1

2 ,3 12 }, but this is not a very com-

pelling solution, since, as in the PD game, anypair could do better by signing a bilateral con-tract. The grand coalition can achieve payoff

Table 2 Payoffs from different (numeric) coalitionalstructures in a public goods game

Coalitional structure Strategies Payoffs{{1},{2},{3},{4}} {1,1,1,1} {3 1

2 ,3 12 ,3 1

2 ,3 12 }

{{1,2},{3},{4}} {2,2,1,1} {4,4,5 12 ,5 1

2 }{{1,2},{3,4}} {2,2,2,2} {6,6,6,6}{{1},{2,3,4}} {1,3,3,3} {9 1

2 ,5 12 ,5 1

2 ,5 12 }

{{1,2,3,4}} {4,4,4,4} {8,8,8,8}

vector {8,8,8,8}. However, though efficient,this vector is not in the core. Indeed, the high-est payoff that the least well-paid player in thegrand coalition can achieve is 8. But any playerrecieving 8 has an incentive to leave the grandcoalition for a payoff of 9 1

2 . Further splits oraggregations would not, however, produce netgains for all members concerned. This sug-gests that, though inefficient, {{1},{2,3,4}}may nonetheless be a plausible equilibriumstructure.

This is what Ray & Vohra find. Specifically,they show that for all δ above some threshold,there exists a unique equilibrium coalitionalstructure. The equilibrium emerges when thefirst offerer commmits to join no coalition andremains on his own. The second offerer thenproposes a coalition of the remaining threeplayers.

In this example, one player free-rides onthe remaining three. Ray & Vohra (2001)show the conditions under which such inef-ficiencies arise. Strikingly, for the game con-sidered here, inefficiency depends sharply onn. In particular, the unique equilibrium is effi-cient for a sequence of population sizes fromthe set n ∈ {1,2,3,5,8,13,20, . . .}. In all othercases, including the example given above, theoutcome is inefficient. There are, however,upper bounds on the inefficiencies that canresult; in particular, Ray & Vohra show thatthe number of equilibrium coalitions is lessthan log2 n + 1. For example, with one mil-lion players, no more than 20 coalitions formin equilibrium.

The Ray & Vohra model shows that allow-ing for side payments does not prevent inef-ficiencies. However, as the authors note, theinefficient equilibrium appears to depend onthe stipulation that players cannot renegoti-ate. This prohibition rules out the possibility,for example, that a second offerer would pro-pose a contract to all four players, includingthe player already committed, making an offerof the form {10,10,6,6}. For the public goodsgame we have been examining, this stipulationappears especially limiting because the “con-tract” to be renegotiated involves only one

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player. It is a general commitment that theplayer cannot renege on, even though doingso would make all better off.11

However, as we saw in our discussion ofGomes (2005), in settings in which renego-tiation is possible, these inefficiencies do notarise. This is not to say that we can ignorethe possibility of the kinds of inefficienciesidentified by Ray & Vohra, but rather that theinefficiency prediction turns on details of thecontracting, in particular on the possibility forrenegotiation.

Application of sequential contract forma-tion models to government formation.Consider now an application of Gomes’ re-sults to a government-formation problem inwhich there is a convex set of policy out-comes, X, and legislators have quasilinearutility of the form ui = yi + vi (x), whereyi is the legislator’s holding of a transfer-able good and vi : X → R

1 is strictly con-cave. In this case, efficiency is achieved bychoosing the (unique) policy outcome thatmaximizes

∑vi (x) and redistributing income

across the grand coalition. Under the commit-ment mechanism studied by Gomes (2005),the grand coalition should form a governmentand implement this policy in finite time. Thisis a classic Coasian solution. However, it runscontrary to the common prediction of min-imal winning coalitions (Riker 1962) and toa more recent prediction of undersized coali-tions (Diermeier & Merlo 2000).

The difference between the predictionfrom Gomes’ model and the more standardpredictions comes down to a subtle differencein commitment technologies. The differenceis not whether commitments may be madebut which commitments can be made. Con-sider the following example. Assume, con-sistent with Diermeier & Merlo (2000), thatplayer i ∈ N = {1,2,3} has preferences over

11See Diamantoudi & Xue (2007) for a setting in which in-efficiencies may obtain even with renegotiation of contractsand transfers between players.

points in R3 given by vi (x) = 2xi − ∑3

k=1 x2k .

The efficient policy is then given by( 1

3 , 13 , 1

3

).

Let the status quo be given by player 1’s ideal,({1,0,0}), with corresponding status quo util-ities of (1,−1,−1).

Assume that player 2 is deterministicallyselected to form a government. What gov-ernment will form? If the formateur is freeto propose a coalition and a set of transfers,as in Gomes’ model, we should expect player2 simply to propose a policy such as (0, 1

2 , 12 )

together with a transfer of 32 from 3 to 2 for

net utilities of (− 12 ,2,−1). This option is not

attractive, however, in the Diermeier & Merlo(2000) model because in that model the deci-sion to enter government and the decision tosplit the pie are separated. If 2 enters govern-ment with 3, then he can expect policy (0, 1

2 , 12 )

but no net transfers between the players forutilities of only (− 1

2 , 12 , 1

2 ). Instead, Diermeier& Merlo (2000) predict that 2 will elect toform a government on his own, implementingpolicy (0,1,0) and receiving no transfers fornet utilities of (−1,1,−1). Thus, Diermeier &Merlo (2000) predict a minority government,whereas in the Gomes model a majority gov-ernment is formed. As noted, however, theGomes model allows for renegotiation sub-ject to the consent of relevant parties, and inthis case this leads to still larger coalitions.In our present example, if player 2 were tooffer an amended contract, he could proposea government policy of ( 1

3 , 13 , 1

3 ) with furthernet transfers of (− 5

6 , 46 , 1

6 ) and net utilities of(− 1

2 ,2 12 ,−1).12

Markov Processesand Spot Contracts

For a given set of membership rules, it is pos-sible to construct dynamic processes in which

12This example is illustrative only. I have not taken accounthere of how expectations of a second-round amendmentwill affect first-round offers. In fact, if player 2 knew hewas a monopoly formateur, he could extract considerablymore from player 1 by implementing even less desirablepolicies in the first round.

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EXAMPLE OF A SIMPLE MARKOV PROCESS

Consider the following process of coalition formation in theseven-player DD examined above. At each stage, an individ-ual is selected at random. The individual can then chooseeither to change groups, conditional on unanimous accep-tance by all members of the receiving group (if there areany), or to nominate one player for expulsion from his owngroup, conditional on the consent of all players other thanthe victim. If these changes occur, then the coalitional struc-ture changes accordingly; in either case, period payoffs areallocated according to the prevailing structure. Payoffs aregiven by the average payoff of a player’s coalition, and sidepayments are not allowed. If players are shortsighted anddrift occurs with positive probability, that is, weak prefer-ence is sufficient to produce a change of groups, then stateh = {2,2,2,1} is the unique stable state; moreover, it is ac-cessible from every other state. Other processes are directly

or indirectly dominated by h, either through expulsion (E→),

strict preferences (S→), or weak preferences (

W→), as follows.

aW→ b

W→ cS→ f

W→ g

oE→ n

E→ mW→ l

E→ g

}W→ k

S→ iS→ e

jS→ e

⎫⎪⎪⎬⎪⎪⎭

S→ dS→ h

Note that states involving minimum winning coalitions arenot stable, since expulsion or drift may result in coalitionalstructure {4,3}. But from this state, a member of the minimumwinning coalition has an incentive to defect to create the three-party system {3,3,1}.

the coalitional partition at each stage is a func-tion of the partition function at the previousstage and the optimizing actions of individu-als. In such processes, payoffs can be receivedin “real time” (Konishi & Ray 2006, Gomes2005, Hyndman & Ray 2007). In many set-tings, this structure is more realistic, althoughit also has the advantage of providing a simpleenvironment in which players can form con-jectures about future (equilibrium) play.

The general structure allows for many dif-ferent ways to model the evolution of groupstructures. Single-step processes allow formoves between coalitional structures that re-quire change in group membership of one per-son only. More general processes allow for the

wholesale change of coalitional structure fromperiod to period. In such cases, the stability ofa state provides a natural equilibrium notion;although different notions of stability obtainfor stochastic processes, the simplest is that astate is stable if it is “absorbing,” in the sensethat once in this state the process will remainin this state. A simple example of such a pro-cess for the seven-player DD game is shownin the sidebar.

Laver & Benoit (2003) study a Markovprocess of coalition formation in which statesare partitions and each member of a coali-tion (party) receives his per capita share of theShapley value of the partition. As in the ex-ample considered above, moves are assumedto occur if they provide immediate benefitsto both the mover and the receiving coalition,although in Laver & Benoit’s examination, ex-pulsions are not permitted. As in the sidebar’sexample of a simple Markov process, it is im-plicitly assumed that players are shortsightedand that once members join coalitions theymake two commitments: to vote with the partyand to receive a per capita share of the sum ofallocations made to their coalition. Laver &Benoit find that although multiple equilibriaexist, larger dominant parties are in generalmore likely than smaller parties to be bothattractive to and accepting of new members,suggesting a tendency of evolution toward asingle-party monopoly coalitional structure.

Konishi & Ray (2003) examine a more gen-eral process in which a (finite) set of statescan be interpreted either as partitions or asstrategy profiles. In their model, players arefarsighted and select moves as a function ofthe expected value given equilibrium strate-gies of all other players at each state. Agree-ments in this model should be thought of asspot contracts—binding, but only for a sin-gle period of play. An equilibrium process ofcoalition formation (EPCF) is one in which, ifa move is made between states, this is alwaysto the (long-term expected) benefit of somecoalition, and if a strictly profitable move ispossible for some coalition from some state,then the state is guaranteed to change and a

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strictly profitable move will be made with pos-itive probability. Konishi & Ray demonstratethat such EPCFs exist for finite (or count-ably infinite) state spaces. Moreover, focus-ing on characteristic functions, they show thatfor sufficiently patient (or sufficiently impa-tient) players, if a deterministic EPCF has aunique limit then that limit is a (weak) corepoint. Further, if an EPCF is absorbing anddeterministic then the absorbing states are inthe “largest consistent set” (Chwe 1994). Thefact that core outcomes emerge in settings inwhich players are farsighted is especially im-portant because the standard justification forthe core, focusing on single deviations, im-plicitly assumes shortsighted players.

Gomes & Jehiel (2005) study finite statespaces, which can be interpreted as out-comes or as coalitional structures, in a relatedgame. In each period, a player can proposea change of states alongside an (unrestricted)set of transfers to other players (in con-trast to the model of Konishi & Ray (2003),where proposals are made by fictitious play-ers and no transfers are allowed). They ex-amine the different implications of long-termand spot contracts. With long-term contracts,they find that efficiency obtains, consistentwith the Coase theorem. A striking result isthat although core points are consistent withequilibria with myopic players, they are notnecessarily equilibria with farsighted players.Gomes & Jehiel provide an extraordinary ex-ample in which there are four states, a, b, c,d, with payoff profiles at each state for eachof three players given by v(a) = (60, 40, 0),v(b) = (40, 0, 60), v(c ) = (0, 60, 40), v(d ) =(64, 64, 64). A move from any state to anyother state can be made with the consentof a majority. Thus, for each of states a, b,and c, there is a division of the dollar, andso with transferable utilities we may expectthe kind of cycling described by Kalandrakis(2004, 2006b). However, unlike in the stan-dard DD game, there is a state d that is a corepoint in the sense that there is no move awayfrom d that produces immediate benefits toany pair, even if transfers among them are al-

lowed. Gomes & Jehiel (2005) show, however,that with sufficient patience, players can electto move away from d, incurring short-termlosses in the expectation of longer-run gains.The key insight is that if selected, player 1 forexample could propose a shift to state a in theexpectation that once in state a, players 1 and2 will both be in strong bargaining positionsshould they have the opportunity to proposean alternative state to the now suffering player3. The key weakness of state d is that it doesnot satisfy the property Gomes & Jehiel term“negative externality–freeness”—that is, fromstate d it is possible for a coalition to induceanother state that hurts a non–coalition part-ner. Strikingly, Gomes & Jehiel show that ifno state exists that is negative externality–free,then the existence of a stable state cannot beguaranteed—there exist some proposal prob-abilities that induce cycles and, possibly, inef-ficiencies.

Application of Markov models to theLaver-Shepsle government formationmodel. As an application of these results togovernment formation, consider the Laver-Shepsle model of government formation(Laver & Shepsle 1996). In each period, aparty is selected to propose a coalition, thatis, an allocation of government portfolios topolitical parties. It is assumed that all partieshave some positive probability of beingselected. Once a government is proposed,it can be vetoed by any of its members. Ifit is not vetoed, it is subjected to a majorityvote. If the motion carries, the governmenttakes office. If the proposal is either vetoedor defeated on the floor, the status quogovernment remains in place. In eithercase, play moves to the next period. Oncea government takes office, policy outcomesare selected via a noncooperative game inwhich each party chooses the optimal policyon the dimensions it controls. Note that theimplicit contracting is very limited. When acoalition is voted in, players accept the policychoices of the coalition members; but playersare not permitted to sign arbitrary contracts,

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a feature that can result in the selection ofPareto-suboptimal policies in equilibrium.The equilibrium concept Laver & Shepsleinvoke is stationary subgame perfect Nashequilibrium.

The Laver-Shepsle model of coalitionformation is unambiguously noncoopera-tive. However, it has been described inthe literature as a non-noncooperative, per-haps cooperative, perhaps structure-induced-equilibrium social choice theoretic model(Diermeier 2006). The reason for this likelyrelates to the way the model is solved. Al-though Laver & Shepsle describe the solu-tion concept as being stationary subgame per-fect Nash equilibrium, in fact they solve forstates, not strategy profiles. Of course, strate-gies and states are closely linked, and so Laver& Shepsle could respond that “equilibriumstates” is shorthand for “states that obtainwhen players play their subgame perfect Nashequilibrium.” There is a second and perhapsmore difficult problem, however: AlthoughLaver & Shepsle provide a game form, they donot provide the universe of player preferencesthat combine with the game form to produce afully specified game. In particular, they do notprovide information regarding how streams ofutility are measured. In the absence of this in-formation, Laver & Shepsle leave some partsof the game unsolved (such as which playerswin a standoff); but more worryingly, the ab-sence of this information makes it difficult toknow whether the solutions proposed indeedcorrespond to subgame perfect equilibria.

In some readings (e.g., Diermeier 2006),the solution provided by Laver & Shepsle isin fact the core and not subgame perfection.In consequence, Diermeier (2006, p. 167) ar-gues that when the core of the (single-period)game is empty, as sometimes it is, “the Laver-Shepsle theory has no empirical content; itdoes not predict anything.” In fact the issueis more subtle. For cases in which the core isempty, Laver & Shepsle indeed do not pre-dict anything. But this is not because of thestructure of their model or the solution con-cept employed; it is because they simply do

not fully identify equilibrium strategies forall preference profiles including those withan empty core. Once sufficient utility infor-mation is provided, the model can yield anempirical prediction resulting from stationarysubgame perfection even in those cases whenthe core is empty; indeed, as we will see be-low, in such cases it can predict governmentinstability.

To see this, consider a version of the modelin which each player is perfectly impatient,placing value only on the policies producedby governments in the present period, and inwhich policy is defined over two policy dimen-sions. Let ij denote a government in whichthe horizontal dimension is controlled by iand the vertical by j. The state space is theset of governments. We now consider two ex-amples (Figure 3). In each case there are fourparties, A, B, C, D, with voting weights givenby 2,1,1,1, respectively. The first case corre-sponds to a situation in which there is a verystrong party; the second case, based on oneprovided by Austen-Smith & Banks (1990),corresponds to a case in which the core isempty.13

We identify Markov perfect equilibria inwhich, in each period, a party is selected topropose a government. This proposal is thensubjected to a veto and a sequential vote and, ifaccepted, becomes the next status quo. Stageutilities are then awarded, and play moves tothe next period. Using backward induction,in equilibrium proposers propose their mostpreferred cabinet among those that will notbe vetoed or invested; if there is no such cab-inet that they prefer to the status quo, theypass. Vetoers veto cabinets if they prefer thestatus quo to the proposal, and all parties votein favor of a coalition if and only if they pre-fer this proposal to the status quo. We assumethat the status quo always lies on the lattice.Given this behavior, the equilibrium propos-als for the first case are provided in the lowerpanels of Figure 3 as a function of the status

13For simplicity we examine cases with nongeneric ideals,but the main result does not depend on this.

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θ σA(θθ) σB(θθ) σC(θ) σD(θθ) θ σA(θθ) σB(θθ) σC(θθ) σD(θθ)

AA, DA, AB,DB SQ SQ SQ SQ AA,DA,DB, AB SQ CC CC CC

BB, BA, CA,CB AA SQ SQ AA BB, BA, CA,CB AA SQ SQ AA

CC, BD, BC, CD AA BB SQ DD CC,BC BB BB SQ SQ

DD, DC, AC,AD AA BB SQ SQ DD,AD AA BB CC SQ

AC,DC AA AA SQ SQ

BD,CD AA BB CC DD

A(2)B(1)

C(1) D(1)

A(2)B(1)

C(1)

D(1)

Figure 3The upper part of each panel shows a collection of four ideal points and possible cabinets from theLaver-Shepsle model. The voting weight of each group is given in parentheses. Black dots signify idealpoints and white dots signify other possible coalition positions. The lower panel shows stationaryproposal strategies for each player conditional on a given status quo government.

quo (θ ). For the first case, it is easy to checkthat there is a single absorbing state at AA.Indeed, given equal proposal probabilities,the probability that AA is reached within 10rounds is well above 99% independent of thestatus quo. With a fixed ordering, state AA isguaranteed within four rounds. This is an ex-ample of a case in which the unique absorbingstate is a core point. In addition, the outcomedoes not depend precisely on features suchas recognition probabilities. We note, how-ever, that from Theorem 7 of Gomes & Jehiel(2005), if transfers can be made between play-ers then we are not guaranteed that this state(or any state) is an equilibrium of this gamefor perfectly patient players for all proposalprobabilities. The reason is that players B, C,and D can induce a move away from AA that

hurts A. Such a move is beneficial in the shortrun for B and C but harmful for D. Player Dmight nevertheless support such a move if hecould subsequently extract a payment from Ato return to AA.

In contrast, in the second case, in which thecore is absent, an equilibrium cycle can occurwith the state passing from AA to CC to BB toAA over time. In this case, the asymptotic dis-tribution does depend on recognition proba-bilities; with equal probabilities, state AA oc-curs with frequency 1

4 and states CC and BBeach occur with probability 3

8 .

Coalitions and Commitments

We have seen that the types of commitmentsthat are permissible have strong implications

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GOVERNMENT FORMATION: EMPIRICALREGULARITIES AND THEORETICALPREDICTIONS

Recent work on government formation has been motivated inpart by two empirical regularities (Ansolabehere et al. 2005).First, the share of the pie (cabinet positions) assigned to aparty in government is proportionate to its seat share (Gamson1961); second, there is little advantage in being a formateur.Depending on the protocols used, both regularities in the em-pirical literature on coalition formation can be supported (orrejected) by theoretical models.

As shown by Baron & Ferejohn (1989), when there are atleast five (patient) players, any distribution can be sustainedas an equilibrium (though not as a stationary equilibrium), nomatter who proposes first. In addition, Kalandrakis (2006a)shows that for monotonic voter rules, including rules with ve-toers and dictators, any (expected) payoff can result from astationary equilibrium of distributive games for an appropri-ate selection of recognition probabilities. For the DD gameconsidered above, any profile of expected payoffs {αi }i∈N withα1 ≥ α2 ≥ α3 can be implemented through an alternating-offers contracting game with patient players and recognitionprobabilities given by ( α1

α1+α2,

α2α1+α2

, 0).In many models that build on the Baron & Ferejohn (1989)

approach with random recognition rules, the formateur enjoysa premium (Snyder et al. 2005). The formateur’s advantagedisappears, however, with patient players under a protocol inwhich the rejector of a proposal becomes the next offerer (Ray& Vohra 1999). In Morelli’s (1999) model of demand com-petition, the formateur advantage is also absent under someconditions (see also Montero & Vidal-Puga 2007).

for the types of coalitional structures we mayexpect. Table 3 highlights the extreme depen-dence of coalitional predictions on the specificcommitment technologies that are deemedfeasible. There is no coalition (party) structurethat is predicted under all the commitmenttechnologies we have examined, and only two({3,2,1,1} and {3,3,1}) are not predicted bysome model of contract writing.

I close this section with a discussion of theimplications of one salient feature of com-mitment technology for models of govern-ment formation: the ability to commit to

negotiate only with a specified set of otherplayers.

The politics of government formation hasbeen a very active area of research in politicalscience during the past 20 years. Much re-cent work ignores policy positions and treatsthe problem of coalition formation as a DDproblem (e.g., Snyder et al. 2005).14

When formulated as a DD game, theoreti-cal predictions clearly depend on the bargain-ing protocols used (see sidebar). But the pre-dictions also depend on the manner in whichcommitment possibilities are modeled. To seehow, we reconsider the DD game under twotreatments. In the first (“majoritarian”) treat-ment, commitments are invoked only uponthe signing of the final deal. A proposal ismade by a randomly selected player on thefloor of the parliament; if it is accepted by amajority, the division is implemented. If re-jected, another player is randomly selected.In the case with three (patient) players, 1

3 isoffered and 2

3 is retained; each player’s ex antetake is 1

3 .Now consider a second (“coalitional”)

treatment. Before going onto the floor of par-liament, players can elect to form a coalition(or in some treatments, a “proto-coalition”).In forming a coalition, players can committo only support motions that they can jointlyagree on. But they cannot simultaneouslycommit to a particular motion. That is, theyagree to reach a substantive agreement witheach other but do not (yet) agree on what thatsubstantive agreement is. In this case, whenselecting partners, individuals calculate whatthey expect to receive from bargaining witheach partner. Given the simple symmetry of

14Important exceptions stand out. Calvert & Dietz (2005)consider externalities in preferences; Jackson & Moselle(2002) and Morelli (1999) allow both ideological positionsand pork allocations. Baron & Diermeier (2001) allow bothtypes of preferences, although, since they assume quasilin-earity and efficient bargaining, within-coalition bargain-ing takes place over divisions of a dollar only. For Stromet al. (1994), ideological differences constrain coalitions;for Laver & Shepsle (1996), preferences are driven by pol-icy concerns only.

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Tab

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the example we are considering, under Rubin-stein bargaining they should expect to receivehalf the pie from either partner. A process isrequired, however, for determining the set ofcoalitions. The simplest approach is one inwhich a formateur is randomly selected andasked to propose a coalition. If his proposalfails, another formateur is drawn.

Although the distinction between majori-tarian and coalitional approaches is not stan-dard in the literature, in fact both of these sys-tems have been used by students of coalitionalgovernment. The majoritarian approach isused, for example, by Snyder et al. (2005)and Laver & Shepsle (1996), and the coali-tional approach is used by Baron & Diermeier(2001), Iversen & Soskice (2006), and Eguia(2007).

Does this modeling choice matter? Un-derlying the distinction is a presupposition ofdifferent commitment technologies. Initially,one might expect it makes little differencewhether one engages in multilateral bargain-ing or selects a coalition and bargains withinit. In the first case, one has a one-third chanceof getting two thirds of the pie and a one-thirdchance of getting one third of the pie; in thesecond case, one has a two-thirds chance ofgaining half the pie. Either way, one expectsone third of the pie. In fact, however, playersmay have different preferences over the twomethods in the presence of asymmetries thatare prominent in models of coalition forma-tion. We consider two such asymmetries.

Size asymmetries. Consider first asymme-tries arising from groups of different sizes orweights of the form studied by Snyder et al.(2005). In this model, a bargain is reached onthe floor to divide up a pie between the mem-bers of some winning majority (a coalition).The question is: Would payoffs differ if al-locational decisions were made within ratherthan across coalitions? The linearity of the re-sults might seem to suggest that, under riskneutrality, benefits would be similar underboth methods. This is not the case, and as Idemonstrate next, different players can have

markedly different preferences over the insti-tutional form in cases like this.

To illustrate, consider the case of bargain-ing over the division of a pie of size 1 betweengroups of size {2,1,1,1}. The expected payoffsunder a majoritarian model are {.4,.2,.2,.2}(Snyder et al. 2005).

What can players expect in a coalitionalversion of this game? Consider first bargain-ing within minimum winning coalitions. Inthis case, there are two types of coalition: smallcoalitions containing the strong player andone weak player, and large coalitions contain-ing the three weak players. Under Rubinsteinbargaining, players in the small coalitions re-ceive 1

2 , while players in the weak coalitionsreceive 1

3 . The following then is an equilib-rium of the coalition game: Each strong playerproposes to bargain with one weak player,each weak player proposes a coalition with thestrong player; all players accept all offers toenter minimum winning coalitions.

In the coalitional game, the strong playeris always sought after as a coalition partner;his expected share of the pie therefore is 1

2 .A weak player has an expected share of just16 . Hence, because of his attractiveness as acoalition partner, the strong player prefers tobargain within coalitions, but the weak play-ers prefer to bargain across coalitions, on thefloor. Since the weak players form a majorityhere, it is clear that in an up-down vote overthe institutional rules, the noncoalitional ap-proach would win.

However, majorities do not always opposecoalitional institutions. Consider the case withgroups of size {2,2,1,1,1}. There continueto be strong benefits to the coalitional sys-tem for the larger parties. Indeed, if pa-tient, large parties will reject any offers tobe in a coalition with small parties, since anysuch winning coalition contains more thantwo players; they only enter government to-gether. Their expected benefits are 1

2 eachcompared to 2

7 under Baron-Ferejohn bar-gaining. In an up-down vote, a majority willfavor a coalitional government commitmentdevice.

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Constraints over admissible divisions.Consider finally a case in which players arerisk-neutral and have uniform voting weightsbut in which constraints on the set of fea-sible allocations generate asymmetries be-tween players. This is the setting examinedby Iversen & Soskice (2006), who focus on re-distribution in the presence of class coalitionssubject to a nonregressivity constraint.

Suppose player j ∈ {1,2,3} has endow-ment to

j = j−13 . The net transfers to j, denoted

p j , are limited by three constraints:

1. Budget balance:∑

j p j = 02. Minimum negative payoffs: p j ≥ −to

j

3. Nonregressivity: p1 ≥ p2 ≥ p3.

Together these constraints determine asubset of the unit simplex that constitutes ac-ceptable distributions. Let an arbitrary pointin the unit simplex be denoted by t, wheret ∈ [0, 1]3,

∑tj = 1. The admissible subset

of offers is then given by the set shown inFigure 4.

The question again is: What will theoutcome look like under each institutionalsetting?

As above, we solve the game among each ofthe pairs using a Rubinstein-style alternating-offers bargaining model. Where tij denotethe triple of payoffs when i and j bargainbilaterally, under risk neutrality in the limitas players tend to perfect patience, equilib-rium outcomes are given by t12 = { 1

2 , 12 ,0},

t23 = {0, 13 , 2

3 }, t13 = { 23 ,0, 1

3 }.The game thus produces sharp asymme-

tries in preferences over coalition partners.Given a choice, 1 would prefer to bargain with3, 3 would prefer to bargain with 2, and 2would prefer to bargain with 1. Under randomappointment of a formateur, the following isan equilibrium: Each player, if selected as for-mateur, proposes his preferred coalition; alloffers of partnering are accepted. Thus, eventhough each coalition will produce markedlydifferent policies, any coalition can form inequilibrium.

Under random appointment of a forma-teur, the expected redistribution vector is

1 3

2

Figure 4Example of a redistribution game of the form studied by Iversen & Soskice(2006). The set of acceptable redistributions is shaded; the “zero nettransfers” case is marked with a star.

{ 718 , 5

18 , 618 } and expected net transfers are

{ 718 ,− 1

18 ,− 618 }.

As in the other cases, we can also study amajoritarian version of this game with equalrecognition probabilities. In the present ex-ample, there is a particularly simple majoritar-ian equilibrium for cases in which players arepatient. When selected, each player i proposes23 for himself and 1

3 for player (i + 2)(mod3);all players accept any offers that accord themat least 1

3 . In this equilibrium, expected bene-fits are 1

3 each, and expected net transfers are{ 1

3 ,0,− 13 }.

Again we find disagreement over the op-timal institutional procedures. The wealthyplayer, 3, is indifferent between systems.Player 1 benefits from the coalitional ap-proach, while 2 prefers taking her chanceson the floor. The key difference in outcomesarises when 2 is selected; if selected as a forma-teur, 2 will form a coalition with 1 and receive12 . If, however, 2 can propose an outcome di-rectly, then she can make better use of heragenda-setting power and propose retaining23 , yielding only 1

3 to player 1.We find therefore that in a wide variety

of settings, the presence of a coalitional com-mitment device can have substantial effectson the outcome of bargaining. In itself, thismakes commitment devices of this form an

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interesting object of study—under what cir-cumstances are we more likely to see coali-tional institutions used?—but beyond that, itimplies that this modeling choice is not in-nocuous, and the utility of a model should bejudged on the reasonableness of this model-ing choice. For majoritarian approaches, thequestion is whether, once formed, members ofa coalition have the ability or incentive to pushtheir partners to renegotiate deals that werepreviously reached in plenary. For coalitionalapproaches, the question is whether coalitionpartners have the ability or incentive to seeksupport for alternative deals from parties out-side their coalition.

CONCLUSIONS

Recent research has greatly expanded our un-derstanding of the kinds of coalitional struc-tures that can, or are likely to, emerge insettings in which players can sign bindingagreements with each other. This work hasmoved beyond an unproductive oppositionthat sometimes obtains between adherents ofcooperative and noncooperative approaches.In this new work, coalition formation is mod-eled as an explicitly noncooperative processthat occurs in settings in which binding agree-ments can be made. The correspondence be-tween Nash equilibrium strategies and coreoutcomes can then be fruitfully examined.

A number of consistent messages haveemerged from this work. First, in many cases,when core outcomes exist, they correspondto predictions that result from noncoopera-tive analysis, both in the more obvious caseswhere players are myopic and in the moresurprising cases where players are farsighted(Banks & Duggan 2000, Konishi & Ray 2003).In settings in which the core is empty, thisfact also provides insights into the types ofequilibria that may obtain, and indeed, consis-tent with older insights, the emptiness of thecore may give rise in noncooperative settingsto cyclical processes induced by noncoopera-tive play. However, there are cases in whichthe core is not a good prediction, even when

it is not empty. For the case in which con-tracts are temporary and large transfers can bemade between players, Gomes & Jehiel (2005)show that although core outcomes may ob-tain for myopic players, these may not obtainif players are sufficiently farsighted. In suchsettings, unless a new condition is satisfied—negative externality–freeness—coalitions maychoose to leave core states in order to extortother players who will make side payments toachieve subsequent gains.

Second, this research has found that in flex-ible negotiation environments (with durablebut modifiable contracts and transferable util-ity), the Coase conjecture that efficient out-comes result from contracting typically holds(under subgame perfection) if grand coalitionscan implement efficient outcomes. However,in cases in which the grand coalition cannotimplement efficient outcomes, or in whichcontracts cannot be renegotiated, inefficientcoalitional structures may result (Hyndman &Ray 2007).

Third, and most important, this new workhighlights the dependence of predicted out-comes not simply on the bargaining protocolbut, more substantively, on details of the spaceof admissible contracts—that is, on the typesof commitments that can be made.

These new developments point to a num-ber of fruitful avenues for new work on coali-tional politics. I emphasize three of these.

The Stability of CoalitionalStructures

We have seen that there has been some confu-sion regarding what lessons to take from theemptiness of the core, particularly in majority-rule settings. Core emptiness does not predictinstability. But scholars have historically beenconscious that in some sense, an empty coremakes the status quo vulnerable; if the ma-jority could simply find the right game form,they could change it. Recent work has clar-ified that the problem of instability is dis-tinct from the problem of the emptiness ofa solution concept. In truly dynamic settings,

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noncooperative solutions can generate stablebehavioral predictions, but in settings wherethe core is empty, such as the DD game, thesedo not translate into stable political outcomes(Kalandrakis 2004, 2006b). Instability thenseems to be inherent in the problem, not inthe solution concept, and it may arise in manysettings of importance to political scientists(Gomes & Jehiel 2005) as a function of pref-erences and protocols rather than simply be-cause of exogenous shocks. There is an openagenda to better understand when such insta-bility is likely to arise in different political andcontractual environments.

The Differential Effects of DifferentCommitment Technologies

We have seen a striking dependence of pre-dicted outcomes on the type of commitmentsthat can be made between players. Whereasmuch work has focused on the impacts of dif-ferent bargaining protocols, less attention hasbeen given to a determination of what kindsof commitments may be made. As we saw inthe example of government formation, somemodels invoke an ability to commit only to ne-gotiate with a subset of players while others donot. This ability has material effect, yet the ra-tionale for such a commitment device is rarelydefended and the impacts of the assumptionare rarely analyzed. Contracting possibilitiesmay vary along a number of dimensions; someof these have been examined, whereas oth-ers remain open questions. In applied work,there has been little effort to match the com-mitment technologies made available in gamedescription to the actual settings under study.

The signatories. In most treatments, con-tracts can either be signed by no individuals orby arbitrary coalitions of individuals. In prin-ciple, however, there may be a rich set of con-straints on who can sign contracts, or who cansign contracts with whom. Plausibly, for ex-ample, contracts may only be signed betweenpairs, or between some pairs and not others.In some settings we may consider contracts

signed by only one person, but in others thesemay be void. In other settings we might con-sider commitments credible only if made bysome share of a group, a majority for exam-ple, or perhaps the grand coalition. Finally,in most work on coalition formation, con-tracts are signed uniquely among membersof a given coalition; yet more general struc-tures are conceivable in which contractingtakes place across networks rather than withincoalitions In such cases, individuals may havea series of contracts with different individualsor groups rather than within the same group.

The domain. Variation in the domain ofcontracts is also relevant, though largely un-explored in the context of coalitional analysis.Considerable work has been done on incom-plete contracts arising where transaction costsmake the specification of all possible even-tualities infeasible. Maskin & Tirole (1999)argue that in such settings, transaction costsneed not prevent optimal contracting; rather,contracts can be written in utility space ratherthan action space. Their results are very gen-eral whenever renegotiation is not permittedand are still general even with renegotiationif players are risk-averse. However, there areother reasons why limited domain may berelevant; in particular, there may be legal orother inhibitions on what is contractible. Inthe model of Iversen & Soskice (2006), con-tracts are not enforceable if they impose costson poorer players. In the Laver & Shepsle(1996) model, parties, although they are inoffice together, cannot make a binding agree-ment on policy choices. Finally, there may belegal restrictions on the ability to make sidepayments. The Gomes & Jehiel (2005) modelis appropriate for analyzing coalition forma-tion only if side payments can be made forfavors rendered.

The costs. All the models examined here as-sume that contracting is essentially costless.In all cooperative models, the payoffs are de-termined by outcomes and not by how thoseoutcomes are reached. Some noncooperative

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approaches include costs in terms of time tocontracting but ignore other features. Thenumber of contracting parties is typically notconsidered a cost to contracting, although itis a plausible concern. Little has been donein this area, although we refer to Gul (1989)for the case of games played on characteristicfunctions and to Macho-Stadler et al. (2006)for a recent examination of bilateral mergersin a Cournot setting.

The duration. A number of the studiesexamined above highlight the importanceof contract duration for equilibrium out-comes. Hyndman & Ray (2007), for exam-ple, usefully distinguish between temporary(or spot) agreements, irreversible agreements(which allow no renegotiation), and perma-nent agreements (which allow renegotiationamong consenting signatories). Many featuresof duration have not been examined, how-ever, such as contracts with endogenous sun-set provisions or contracts whose terminationdepends on actions taken by only a subset ofparties. Further issues arise if the possibility ofreneging is considered, a point I return to be-low. For such cases, further details of contractdesign may become salient, such as whethercontracting provides for joint or severalliability.

Publicity. In present studies, contracts arepublic documents. However, in principle, thepublicity or secrecy of contracts could affectplay in ways not presently allowed for in con-tracting games. This matters especially if con-tracts specify actions to be taken. Such con-tracts could serve not only to commit playerswithin a coalition to work in each other’s inter-ests but also to commit them to a course of ac-tion in the game between coalitions. In othersettings, the publicity of contracts could revealpreviously unavailable information to outsideparties. Finally, as noted above (see Criticismsof the Cooperative Approach), even the re-fusal to sign a contract, if this is known, couldprovide information regarding intended playin a base game and thus alter play.

The process. As is well known, bargain-ing protocols can have a large effect on theoutcome of negotiations. In the context ofcoalition formation, Gul (1989) shows thatpairwise bargaining can lead to Shapley valuepayoffs when all (live) pairs are matched withequal probability. However, he also showsthat in a related partnership game with dif-ferent recognition probabilities, different out-comes obtain. Quite different procedures areproposed by Morelli (1999), Snyder et al.(2005), Diermeier & Merlo (2000), and oth-ers, all with substantive implications. De-termining which approach is “right” is dif-ficult because coalitional bargaining proce-dures are typically not embodied in constitu-tions (Diermeier & Merlo 2004). Moreover,even if formal rules exist, these rules may notreflect the relevant commitment devices avail-able to players. The largest party may well bethe first to be recognized on the floor, buta smaller party might have been the first togather a group and reach a deal in the cor-ridors. The challenge for applied work is toidentify protocols that accord with the actualprocedures used in the cases of interest.

Endogenous CommitmentTechnologies

Throughout this discussion I have treatedcommitment technologies as exogenous. Inmany accounts, these commitment devices areexternal institutional features; compliance isguaranteed by an outside agent or institution.In other accounts (e.g., Jackson & Moselle2002), the commitments that hold parties to-gether are seen less as legal devices and moreas the result of repeated interactions withina larger, unmodeled game. Yet a clear lessonfrom the new work on coalition formation(examined in the previous section, Nonco-operative Coalition Theory) is that differentcommitment technologies produce materiallydifferent outcomes that are valued differentlyby different individuals. Understanding thismapping from commitment technologies towelfare can provide information on the origin

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and institutionalization of those commitmenttechnologies. In turn, a deeper understandingof the origins and the breaking points of com-

mitment technologies can shape our under-standing of when and how political coalitionswill elect to employ them.

ACKNOWLEDGMENTS

I thank David Epstein, Jeff Lax, Lucy Goodhart, and Cyrus Samii for very helpful commentsand suggestions. I am especially indebted to Michael Laver for extremely useful conversationson all parts of this article.

DISCLOSURE STATEMENT

The author is not aware of any biases that might be perceived as affecting the objectivity ofthis review.

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Annual Review ofPolitical Science

Volume 11, 2008Contents

State FailureRobert H. Bates � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �1

The Ups and Downs of Bureaucratic OrganizationJohan P. Olsen � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 13

The Relationships Between Mass Media, Public Opinion, and ForeignPolicy: Toward a Theoretical SynthesisMatthew A. Baum and Philip B.K. Potter � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 39

What the Ancient Greeks Can Tell Us About DemocracyJosiah Ober � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 67

The Judicialization of Mega-Politics and the Rise of Political CourtsRan Hirschl � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 93

Debating the Role of Institutions in Political and EconomicDevelopment: Theory, History, and FindingsStanley L. Engerman and Kenneth L. Sokoloff � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �119

The Role of Politics in Economic DevelopmentPeter Gourevitch � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �137

Does Electoral System Reform Work? Electoral System Lessons fromReforms of the 1990sEthan Scheiner � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �161

The New Empirical BiopoliticsJohn R. Alford and John R. Hibbing � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �183

The Rule of Law and Economic DevelopmentStephan Haggard, Andrew MacIntyre, and Lydia Tiede � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �205

Hiding in Plain Sight: American Politics and the Carceral StateMarie Gottschalk � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �235

Private Global Business RegulationDavid Vogel � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �261

Pitfalls and Prospects in the Peacekeeping LiteratureVirginia Page Fortna and Lise Morje Howard � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �283

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AR344-FM ARI 8 April 2008 18:44

Discursive Institutionalism: The Explanatory Power of Ideasand DiscourseVivien A. Schmidt � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �303

The Mobilization of Opposition to Economic LiberalizationKenneth M. Roberts � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �327

CoalitionsMacartan Humphreys � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �351

The Concept of Representation in Contemporary Democratic TheoryNadia Urbinati and Mark E. Warren � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �387

What Have We Learned About Generalized Trust, If Anything?Peter Nannestad � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �413

Convenience VotingPaul Gronke, Eva Galanes-Rosenbaum, Peter A. Miller, and Daniel Toffey � � � � � � � � �437

Race, Immigration, and the Identity-to-Politics LinkTaeku Lee � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �457

Work and Power: The Connection Between Female Labor ForceParticipation and Female Political RepresentationTorben Iversen and Frances Rosenbluth � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �479

Deliberative Democratic Theory and Empirical Political ScienceDennis F. Thompson � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �497

Is Deliberative Democracy a Falsifiable Theory?Diana C. Mutz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �521

The Social Processes of Civil War: The Wartime Transformation ofSocial NetworksElisabeth Jean Wood � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �539

Political Polarization in the American PublicMorris P. Fiorina and Samuel J. Abrams � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �563

Indexes

Cumulative Index of Contributing Authors, Volumes 7–11 � � � � � � � � � � � � � � � � � � � � � � � � � �589

Cumulative Index of Chapter Titles, Volumes 7–11 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �591

Errata

An online log of corrections to Annual Review of Political Science articles may be foundat http://polisci.annualreviews.org/

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