the universal-instability theorem

The Universal-Instability TheoremAuthor(s): Thomas SchwartzSource: Public Choice, Vol. 37, No. 3 (1981), pp. 487-501Published by: SpringerStable URL: http://www.jstor.org/stable/30023509 .

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The universal-instability theorem*

THOMAS SCHWARTZ**

University of Texas at Austin

1. Introduction: The concept of generalized exchange

When individual or corporate creatures with partly conflicting preferences cooperate or compromise, form an alliance or make a deal, they exchange support across issues: they support a package of positions, one on each issue involved in the exchange; although none of them favors every position in the package, they agree (in effect) to all the positions because they prefer the package as a whole to the alternative package that would prevail without the exchange. Vote-traders obviously exchange support across issues. So do coalition-government partners; for them the issues concern legislation, port- folio assignments, or both. So, implicitly, do the supporters of a political candidate or legislative leader who builds a winning platform from planks that have too little support to be enacted singly. The participants in an ordinary economic trade also exchange support across issues: if I trade you a banana for a coconut, I have supported your favorite position on the issue of who gets the banana in return for your support of my favorite position on the issue of who gets the coconut. The exchange of support across issues - generalized exchange, for short - characterizes all political, economic, and other interpersonal activity involving partial conflict of interest.

Kadane (1972), Oppenheimer (1972), and Bernholz (1973), building on an insight of Downs (1957), have independently proved that whenever a majority of voters with separable preferences support a package of

* Research sponsored by a Sid Richardson Foundation grant to the Institute for Constructive Capitalism, University of Texas. ** I presented an earlier version at the Center for the Study of American Business, Washington University, St. Louis, 15 February 1979. I have generalized the result presented in that version thanks to an important suggestion by Kenneth Shepsle and Barry Weingast.

I thank Peter Bernholz and Joe Oppenheimer for many helpful discussions over the years and for comments on much earlier drafts. Their brilliant pioneer papers, cited below, occasioned my thoughts in this subject. I owe some of the interpretive ideas in section 5 to Bernholz.

Public Choice 37: 487-501 (1981) 0048-5829/81/0373-0487 $02.25. ( 1981 Martinus Nijhoff Publishers, The Hague. Printed in the Netherlands.



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minority positions (positions that would each be opposed by a majority if considered singly), there must exist an alternative package preferred by some (other) majority to the given package. Downs stated his result in terms of party platforms and electoral competition, Kadane in terms of referendum writing, and Oppenheimer and Bernholz in terms of vote trading. Here is a statement in terms of generalized exchange: Assum- ing majority rule and separability of preferences, an outcome (a position package) for which generalized exchange is essential - one obtainable only through some generalized exchange - must be unstable in this sense: some actors have the joint power to overturn it in favor of an alternative outcome that they like more.

The majority-rule assumption severely limits the generality of this theorem. So does the separability assumption, which says that each actor's preference among the positions on any issue is independent of the positions adopted on other issues - meaning, for example, that an actor who prefers red wine to white with meat must also prefer red wine to white with fish. Bernholz (1974) and Schwartz (1977) generalized the theorem, Bernholz by weakening the majority-rule assumption, Schwartz by further weakening this assumption and also weakening the separability assumption somewhat. Miller (1976) and Enelow and Koehler (1979) varied the theorem another way: Retaining the majority-rule assumption, they replaced separability by the assumption, also quite restrictive, that every actor pursues a sophisti- cated voting strategy in the sense of Farquharson (1969).

I shall prove a generalization of all these results that rests on no empiri- cally questionable assumptions about actors' preferences or the way issues are decided or anything else. All I assume is the following: If a position on some issue would be rejected in the absence of any generalized exchange involving that issue, then some group of actors with the power to reject the position in question prefer that it be rejected, except possibly as part of a generalized-exchange package. From this I shall deduce that any outcome x for which generalized exchange is essential must be unstable: an alternative outcome, y, is preferred to x by some group with the power to overturn x in favor of y.

This universal-instability theorem has myriad applications. It helps explain the continual shifting of successful legislative alliances, the tendency of ruling parties eventually to lose elections, the instability of coalition governments, and the fugacity of dominant international alliances (com- pared with the greater durability of dominant single actors, such as Rome). It also shows the instabilities that have exercised game theorists and social- choice theorists to be ubiquitous and inescapable: far from being peculiar to special institutions, preference-configurations, or conflict-of-interest situations, they afflict almost all political, economic, and other interpersonal activity almost all the time; in particular, unstable outcomes under majority voting are the rule rather the exception.



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Those of us who theorize under the commodious umbrella of 'public choice' have long been concerned with two phenomena, seemingly uncon- nected, both styled 'collective irrationality.' One is the instability of majority-voting and other collective-choice outcomes, as exemplified by the classical voting paradox. The other is the failure of actors with partly conflicting preferences to cooperate in mutually advantageous ways, as in prisoners'-dilemma games and free-rider problems. My theorem subsumes the second variety of 'collective irrationality' under the first, in this sense: If, to achieve a particular outcome, it is necessary to solve the cooperation problem, then doing so automatically gives rise to the stability problem. In other words, if cooperation by actors with partly conflicting preferences is essential to an outcome, that outcome must be unstable - which helps explain the difficulty of securing cooperation.

A noteworthy feature of my theorem is that it is utterly trivial.

2. Notation: The formal setting

Formally, the theorem is about a positive integer n, sets c1,. ...,on and 92, a binary relation D, and a function F. I begin by interpreting these items and defining two others - a set II and a function r - in terms of them.

Two or more actors, individual or corporate, are to decide n issues, l,. . .,con, in some collective fashion. Each owi is a set, finite or infinite,

of feasible alternation position. If wi is to be decided before w, and if the choice from oy depends on how wi was decided, then i <. (10o the extent that issue indices are not determined by this rule, they are purely arbitrary.)

E2 is the set of feasible outcomes, each an ordered n-tuple (xl ,...,xn) for which xl E co1, x2 E co2, etc. Although a feasible outcome must consist of feasible positions, not every combination of feasible positions, one on each issue, need be a feasible outcome: feasible positions on difference issues might combine in nonfeasible ways. That is why I do not define E2 as the cartesian product of w1

,.n .,Wn.

D is the dominance relation on E2 - the relation of one feasible outcome to another when the former is preferred to the latter by some group of actors with the power to block or overturn the latter in favor of the former. In other words, xDy holds if, and only if, x, y E U2, and for every y-strategy - every joint strategy, by any group of actors, leading to y - some actors who prefer x to y have a joint x-strategy that would be effective against the given y-strategy. If, for example, majorities and they alone always can get their way (which I do not assume), then D is the majority-preference relation on U. Note that I allow coalition efficacy to depend on ordered pairs of outcomes. And by holding E2 and actors' preferences fixed in this discussion, I allow the possibility that coalition efficacy depends on these parameters, too.



490

Whereas a feasible outcome is a feasible combination of positions on all n issues, a partial feasible outcome is a feasible combination of positions on just the first k issues for some k < n. Let H be the set of partial feasible outcomes.

Definition: IH = {(x ,.. .,k)

I either k = n and (x I,.. .,xk) E 2, or else 0 <<k < n and (xl ,...,xk, xk+1, . .,Xn) E E2 for some Xk+l1,.

x., fxn

Note that every feasible outcome is a partial feasible outcome: f C IH. Note also that the empty vector, 0, is a partial feasible outcome: an element (x , ... Xk) of H is 0 if k = 0.

The collective-choice process under consideration determines a choice set from each &i and from 02. The choice set from some issue or from E2 may have more than one member, in which case the process in question has not determined a unique outcome: it has narrowed the range of possible outcomes, leaving the final selection to some random or other arbitrary or exogenous mechanism. (Possibly all choice sets are singletons; I just do not assume this.) The positions on oi that do not belong to the choice set from oi are the rejected positions. Assuming that the first k - 1 issues have already been decided, the choice sets from ok and from 02 may depend on which positions were in fact chosen from c

,...,-Wk-1. These choice sets

also may depend on whether there is to be any generalized exchange across

Ck* , g*,C>On* When (xl,. . .,k) E H, F(x ,. ., k) is the choice set from U2, given that

x,.- - -.,k have been chosen from ot,. . .-k and that there is to be no generalized exchange across Ok+1 ,- .,con; in short, F(x ,.. .-,xk) is the choice set from 92 in the presence of (x1 ,. . ,Xk) and the absence of subsequent exchange. So interpreted, F(x1,.. .,xk) is a subset of 02, e'very member of F(xl,.. .,Xk) has the form (xl,...,xk,

Xk+1.... *,Xn), and F(x1,

S. .,Xk) is simply the no-exchange choice set from 02 when k = 0 - when (xI,... .Xk) = 0, that is.

If (x),. . .,Xk-I)E I - 2, let r(x1,.. .,Xk-l) be the set of rejected positions on Wk in the presence of (x1 ,. .,Xk-1) and the absence of subsequent exchange.

Definition: r is the function on H - U2 defined by the condition:

r(xl,... xk-1) = Ok - {x Ix is the kth coordinate of some vector in

F(xl,...,Xk- 1), (The reason r is defined on H - U2 rather than H as a whole is that if x E ,2 then r(x) would have to be a subset of the (k + 1)th issue, which does not exist.)



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3. Assumptions

The theorems cited in section 1 assume, in effect, that 92 is the Cartesian product of wo ,.-. .,cn,, hence that every combination of feasible positions, one on each issue, is itself feasible. That is preposterous. Among other things, it rules out budget constraints: single affordable (feasible) positions on different issues need not be jointly affordable. I assume only that R is a subset of the Cartesian product of ol ,. . .,n-

n

_eCX oi. (1) i=1

This assumption follows from my interpretation of 92 and Co,...,cn. It does not restrict the underlying collective-choice process or situation in any way.

The following two assumptions are similarly nonrestrictive: each just follows from my interpretation of F.

x E F(x) whenever x E C2. (2)

If Xk+1 is the (k + 1)th coordinate of some vector in F(x,,.. .,xk), then F(x ,.. .,xk, Xk+l) C F(x ,. . .,xk). (3)

The connection between collective choices and the dominance relation, D, is controversial. I make one modest assumption about this connection - the assumption advertised in section 1.

If Xk Er(x ,.. .,k -1) and x E F(xl,...xk-_X ,Xk), then yDx for some y. (4)

Suppose the first k - 1 issues have already been decided, and a position xk E Wk would be rejected in the absence of any generalized exchange across WCk,. - .,on. Surely, then, some actors with the joint power to reject xk prefer that it be rejected, except possibly as part of a generalized-exchange package. As captured by (4), this means that if an outcome x con- tains xk plus whatever has already been chosen (xx,.. - -,k-1, say) and is not based on any generalized exchange across COk,. . .,Cn (ifx E F(x1,..., Xk- 1 ,Xk), that is), then x is dispreferred to some feasible outcome y by some actors with the joint power to block or overturn x in favor of y.

Example: Three two-sides issues are to be decided by majority voting. So D is the relation of majority preference on 12, and every generalized exchange is, in effect, a vote trade. x1 has been chosen from w1 - never mind how or why. In the absence of subsequent vote trading, x2 would be chosen from

wC2 and Y2 rejected. In the presence of x2 and the absence of subsequent



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vote trading, x3 would be chosen from 03 and Y3 rejected, while in the presence of Y2 and the absence of subsequent vote trading, Y3 would be chosen and x3 rejected. Formally:

F(x I)= {(x1, x2, x3)},

so that F(xl,,x2)= {(x ,x2,x3)},r(x1)= {y2}, and r(xl, x2) = {y3}.

But F(xi,y2) = {(xl,y2,Y3)},

so that r(x, Y2) = {X)}.

The situation is depicted by the following decision tree, in which trade-free choices are circled:

X3

x1 S2

x3

According to (4), because y2 Er(xl) and (xl, y2, y3)EF(xl, y2), some outcome dominates (x1, Y2, Y3). This means that a majority prefer some outcome, presumably (xl, x2, x3), to (x1, Y2, Y3).

Since Y2 would be defeated in the presence of xt and the absence of subsequent vote trading, some majority - M, let us say - prefer x2 to Y2 in the presence of x1 . So long as each member of M realizes that x2 would lead to x3 and Y2 to y3 in the absence of vote trading, M will prefer (xi, x2, x3) to (x , y2, Y3), as required.

But what if those who support x2 do so only because they mistakenly believe that x2 will lead to Ya and Y2 to X3? Then, it seems, no majority will prefer (xI, x2, x3) to (xl; Y2, Y3) after all. Thus, (4) apparently pre- supposes, for every issue, that sufficiently many actors have either accurate beliefs about choices on subsequent issues, or preferences that do not depend on subsequent choices, or some combination of the two.

I say 'apparently,' because whether (4) really is thus restrictive is partly a semantic question. Suppose that, in the presence of x1, a majority prefer x2 to y2 only because they mistakenly believe x2 will lead to y3 and y2 to



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x3. Then while they do not prefer (xl, x2,x3) to (xl ,Y2,Y3) under the descriptions '(x, , x, x3)' and '(x1, y2, y3),' they do prefer (x1,x, x, x3)to (x1, Y2, y3) under a different pair of descriptions, to wit:

(dl) 'the outcome that would prevail if xl and x2 were chosen and there were no vote trading involving W3,'

(d2) 'the outcome that would prevail if xl and Y2 were chosen and there were no vote trading involving W3.'

The majority in question fail to realize that '(x , x2, x3)' and (d1) describe the same outcome; similarly for '(xl, Y2, Y3)' and (d2). It is common for someone to prefer one thing to another under one pair of descriptions ('the martini with the olive' and 'the martini with the twist') but not under another pair of descriptions ('the poisoned martini' and 'the unpoisoned martini'), failing to realize that both pairs of descriptions describe the same pair of things (that the martini with the olive is poisoned while the martini with the twist is not).

In the case at hand, (4) requires that a majority prefer some outcome, presumably (xl, x2, x3), to (xl, Y2, Y3). If this means that a majority prefer (x1, x2, x3) to (x1, Y2, Y3) under some pair of descriptions or other - restricted, perhaps, to a specified range that includes (d1) and (d2) as

well as '(xI, x2, x3)' and '(xI, Y2, Y3)' - then (4) is satisfied even in the troublesome case lately cited. But if it means that a majority prefer (x 1, x2, x3) to (x , Y2, Y3) under the specific descriptions '(x1, x2, x3)' and

'(xl, Y2, ya),' then (4) might not be satisfied. I favor the former, more liberal interpretation, because a preference for an outcome x to an outcome y under any of a range of descriptions that includes (dl) and (d2) is a sufficient incentive to block or overturn y in favor of x.

But whichever way one interprets (4), this assumption is far less restrictive than those used in any of the publications cited earlier. And it is the only of my assumptions that is in any way restrictive: (1)-(3) are utterly devoid of empirical content.

4. The theorem

Assuming (1)-(4), the theorem says that every feasible outcome foreign to F(O), the no-exchange choice set from 92, is unstable. That, of course, includes every feasible outcome for which generalized exchange is essential.

Theorem: Assuming (1)-(4), if x E U - F(4) then yDx for some y. n

Proof: x E X i by (1); say x = (x ,...,). But x E F(x) i= 1



494

by (2), while x 9 F(O) by hypothesis. That is, x E F(x ,.. .,xi) when i = n but not when i = 0. For some k = 1, 2,.. .,n - 1, then,

x E F(xl ,.. .ykXk+1) but x 9- F(xl .... xk),

whence F(xl,...,Xk, Xk+ 1) q F(xx ... ,-Xk),

and thus, by (3), Xk+1 is not the (k + 1)th coordinate of any vector in

F(xl,.. ..Xk). By (1), however,xk+l1 E k+1. Therefore,

Xk+ lE r(x1, *...xk)

But xEF(xl,.. .,k,Xk+l).

So yDx for some y by (4).

5. Six things worth noting about this theorem

Thing 1: The theorem does not say that the outcomes in F(O), the no- exchange choice set, are stable, only that no other feasible outcomes are stable. The members of F(O), some or all of them, may or may not be stable, witness these two examples:

Example A: Using majority rule, three voters are to decide two issues: passage vs. defeat of measure a, and passage vs. defeat of measure b. Four outcomes are feasible: both measures pass (ab), a passes but b is defeated (a), a is defeated but b passes (ib), both are defeated (ab). The voters rank the outcomes in order of preterence as follows:

Mr. 1 Mr. 2 Mr. 3

ab ab ab ab ab ab ab ab ab ab ab ab

In the absence of vote trading, both measures would be defeated: F(O)= {ab}. D is the relation of majority preference, depicted on the following page.

An arrow running from the designation of one outcome to that of another indicates that the former outcome dominates the latter; the numerals represent actors who prefer the former to the latter and together have the power to block or overturn the latter in favor of the former. Every



495

ab - { 2,3) --+ ab

{13) ,3)

+b {2,3} {1'2}1 a {2,3}- ab

feasible outcome, including ab, is unstable.

Example B: Five voters rank the same four outcomes this way:

Mr. 1 Mr. 2 Mr. 3 Mr. 4 Mr. 5

ab ab ab ab ab ab ab ab ab ab ab ab ab ab ab ab ab ab ab ab

Once again, issues are decided by_majority rule, D is the relation of majority preference, and F() = {ab}. But this time the four outcomes are related as follows:

(3, 4,5) {1,2,5}

~3a

ab now is stable.

Thing 2: The theorem does not say that every feasible outcome is unstable in the presence of generalized exchange, or even that generalized exchange always produces an unstable outcome. Not only can a generalized exchange coexist with a stable outcome, but the very outcome produced by a



496

generalized exchange can be stable (in which case it must belong to F(0)). The theorem just says every outcome for which generalized exchange is essential - every outcome that can only result from a generalized exchange - is unstable.

In Example B, ab (the sole member of F(0)) is stable. Yet Messrs. 1 and 3 might trade votes, Mr. 1 agreeing to vote for b in return for Mr. 3's agreement to vote for a. Without further trading, the resulting outcome would be ab, which Messrs. 1 and 3 both prefer to ab. To be sure, Messrs. 2 and 4 could block the effect of this trade by trading votes themselves, Mr. 2 agreeing to vote against a in return for Mr. 4's agreement to vote against b, ensur- ing that the actual outcome will be ab, which Messrs. 2 and 4 both prefer to ab - as does Mr. 5. Either way, perfectly rational vote trading (hence generalized exchange) would coexist with the stable ab. And if the second trade occurred, the very result of trading would be ab. This is consistent with the theorem, because vote trading is not essential to ab: ab also is the outcome that would prevail in the absence of any vote trading.

(Assuming majority rule, no ties, and separability of voters' preferences [three assumptions satisfied by Example B], Oppenheimer [1972] and Koehler [1975] argued that [rational] vote trading precludes the existence of any stable outcome. Despite a similar-sounding thesis, Bernholz [1973, 74] did not really commit this error because his term 'logrolling' was intended to apply, not to all cases of vote trading, but only to those supported by majorities [unlike the trade by Messrs. 1 and 3] in order to overturn the no-trade outcome [unlike the trade by Messrs. 2 and 4]).

Thing 3: The theorem does not just apply to voting and vote trading. It also applies, for example, to economic exchange, as illustrated by this example:

Example C: Mr. 1 can exchange a banana for Mr. 2's coconut. Four outcomes are feasible: Mr. 1 gives Mr. 2 a banana and gets a coconut (bc), Mr. 1 gives Mr. 2 a banana but gets no coconut (bc), Mrj:1 gives no banana but still gets a coconut (bc), neither gives anything (bc).Messrs. 1 and 2 rank these outcomes as follows:

Mr. 1 Mr. 2

bc be bc bc bc bc bc bc

If no exchange occurred, the resulting outcome would be bc. That is, F(O) = {bc}. Because bc and bc differ only in Mr. l's action, he has the



497

power to block bcin favor of be, which he prefers to bc, so bcDbc. For the same reason, bcDc. And because be and bc differ only in Mr. 2's section, as do bc and bc, we likewise have bcDbc and bcDbc. But because Messrs. I and 2 both prefer bc to bc and together have the power to trade, bcDbc. So the four outcomes are related as follows:

bc

{1} {2}

{1,2} b

{2}

bc

No outcome is stable. This does not mean there will be no exchange. An agreement to exchange

the banana for the coconut is likely to be enforceable - by conscience if not coercion. And we might regard the enforcement of such an agreement as the cancellation of Mr. l's power to overturn bc in favor of bc and Mr. 2's power to overturn bc in favor of bc: bc would become stable - but only after an enforceable agreement has been made. The dominance structure depicted above, in which bc is unstable, represents the pre-agreement situation, in which every enforceable obligation is either a standing obligation or the creature of an earlier agreement.

An enforceable exchange agreement always alters the rules of the game so as to stabilize an initially unstable outcome. Economic exchange is no different in this respect from majority voting: once some majority success- fully coalesce, enforceably agreeing to support an initially unstable outcome, that outcome can no longer be overturned.

Thing 4: The theorem shows that the instabilities exhibited by the classical voting paradox and similar examples are not mere oddities - that they are not peculiar to certain rare choice processes, feasible sets, or profiles of actors' preferences. On the contrary, every collective-choice outcome to which some form of generalized exchange is essential - and that includes virtually every interesting, realistic collective-choice outcome - is perforce unstable.

Some might contend that elections with small numbers of candidates often have stable outcomes. But that depends on how the feasible alternatives are described. An election that has a stable outcome when the



498

feasible alternatives are taken to be the candidates might lack a stable outcome when the feasible alternatives are taken to be the candidate-platform pairs, each consisting of a candidate plus a platform that he neither did adopt or could have adopted. Looked at this way, even a two-candidate election can have an unstable outcome: the loser might have won had he adopted a different platform.

Thing 5: The theorem helps explain why it is hard to secure cooperation: Because cooperators with partly conflicting preferences exchange support across issues, if cooperation by actors with partly conflicting preferences be needed to achieve a particular outcome, then some actors must have the joint power to block or overturn that outcome in favor of another, which they like more. This is illustrated by the celebrated Prisoners' Dilemma:

Example D: Two prisoners, Messrs. 1 and 2, must each decide whether to confess to a joint crime. The four feasible outcomes and their consequences are displayed in this payoff matrix:

Mr. 2 Does not confess Confesses

Does not Mild penalty for No penalty for Mr. 2. confess for each. Maximum penalty for Mr. 1.

Mr. 1 cc cc

Maximum penalty for Fairly severe penalty for Confesses Mr. 2. each.

No penalty for Mr. 1. cc cc

Seeking to minimize the penalties they will suffer, the prisoners rank the four outcomes as follows:

Mr. 1 Mr. 2

CC CC

CC CC

CC CC

CC CC cc cc

Because cc and cc differ only in Mr. 2's action, he can block or overturn cc in favor of cc, which he prefers to cc. So ccDcc. For similar reasons, ccDcc, ccDcc, and ccDcc. Each prisoner is better off confessing, regardless of



499

what the other does. Therefore, barring cooperation, each will confess: F(O) = {cc}. But both prisoners prefer cc to cc. So if cooperation is possible - if Messrs. 1 and 2 have the joint power to block or overturn cc in favor of cc - then ccDcc, and thus the four outcomes are related this way:

cc

{ 2} {1}

cc (1,cc2) cc { 2) 1

cc, the one outcome for which cooperation is essential, is unstable - as the theorem says it must be. Specifically, either prisoner, acting alone, has the power and incentive to block or overturn cc in favor of another feasible outcome.

Thing 6: Sen's (1970a and b) celebrated Liberal Paradox, which purports to exhibit an incompatibility between Paretian and liberal prescriptions, also is an instance of my theorem. I illustrate with Sen's own example:

Example E: A single, publicly owned copy of Lady Chatterly 's Lover is to be read by lascivious Mr. El, by prudish Mr. Pee, or by neither; call these feasible outcomes ip, Qp, and 9p. Although Mr. El would enjoy reading the book, the thought of prudish Pee reading it and being shocked or titillated pleases him more. Mr. Pee feels that no one should read the book but if anyone is going to read it then better he than corruptible El. So Messrs. El and Pee rank the three outcomes as follows:

Mr. El Mr. Pee

Qp Vp jZF Qp

Either individual has the right to read or not to read the book as he chooses, so long as the other is not reading it - a modest 'liberal' assumption. Qp



500

would violate Mr. El's right: if Mr. Pee is not reading the book anyway, Mr. El has the right to read it and wants to read it.

Although Zp would violate neither individual's right, it is Pareto in- efficient: both individuals prefer 4p to 4p. But Fp requires cooperation: to secure Qp, Mr. Pee must (in effect) agree to read the book if he gets it, and his agreement must (somehow) be enforceable. In other words, for 4p to prevail, Messrs. El and Pee must exchange support across issues, Mr. El supporting Mr. Pee's favorite position on the issue of whether Mr. El reads the book, in return for Mr. Pee's support of Mr. El's favorite position on the issue of whether Mr. Pee reads the book. So f{p) is the no-exchange choice set (F(0)), and {Qp} is the choice set in the presence of generalized exchange. Besides being Pareto efficient, vp is consistent with individual readership rights: should 4p prevail, Mr. Pee would have chosen to read the book (in order to keep Mr. El from reading it), so Mr. Pee's right not to read it if he chooses would not have been violated.

Suppose Messrs. El and Pee can cooperate to bring about Ep. Then all three outcomes are unstable. Because each individual has the right to read or not to read the book as he chooses, so long as the other is not reading it, Mr. El can block or overturn 9p in favor of Qp, which he prefers to Qp, and Mr. Pee can block or overturn Qp in favor of 4p, which he prefers to 4p. So

/pD'p and QpD9p.

But Messrs. El and Pee together can block or overturn Fp in favor of Ep, which both prefer to 4p. So

Sen finds it paradoxical that all three feasible outcomes are unstable - especially, I suppose, since one of them, Qp, is plainly more reasonable than the others. Although Ido not share this judgment, my immediate point is that the instability of Qp is an instance of a phenomenon more general than Sen suggests: Outcomes, such as Qp, for which generalized exchange is essential are perforce unstable, whether or not they are based on individual rights or liberal principles. If there is a paradox, it is one of generalized exchange, not liberalism or individual rights.

REFERENCES

Bernholz, Peter. (1973). 'Logrolling, arrow paradox and cyclical majorities. Public Choice, 15: 87-95.

Bernholz, Peter. (1974). Logrolling, arrow paradox and decision rules: A generalization. Kyklos, 27: 49-62.



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Downs, Anthony. (1957). An economic theory of democracy. New York: Harper and Row. 55-60.

Enelow, James M., and Koehler, David H. (1979). Vote trading in a legislative context: An analysis of cooperative and noncooperative strategic voting. Public Choice, 34 (2): 157-175.

Farquharson, Robin. (1969). Theory of voting. New Haven: Yale University Press. Kadane, J.B. (1972). On division of the question. Public Choice, 13: 47-54. Koehler, David H. (1975). Vote trading and the voting paradox: A proof of logical

equivalence. American Political Science Review, 69: 954-969. Miller, Nicholas R. (1976). Logrolling, vote trading, and the paradox of voting: Some

game-theoretical comments. Presented to Public Choice Society meeting, Roanoke. May.

Oppenheimer, Joe A. (1972). Relating coalitions of minorities to the voters' paradox, or putting the fly in the democratic pie. Presented to Southwest Political Science Association meeting, 1972.

Schwartz, Thomas (1977). Collective choice, separation of issues and vote trading. American Political Science Review, 71: 999-1010.

Sen, Amartya K. (1970a). The impossibility of a Paretian liberal. Journal of Political Economy, 78.

Sen, Amartya K. (1970b). Collective choice and social welfare. San Francisco: Holden- Day. 79-88.



the universal-instability theorem

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