the strategic inconsistency of paretian liberalism

The Strategic Inconsistency of Paretian LiberalismAuthor(s): Roy GardnerSource: Public Choice, Vol. 35, No. 2 (1980), pp. 241-252Published by: SpringerStable URL: http://www.jstor.org/stable/30023799 .

Accessed: 10/06/2014 14:30

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Springer is collaborating with JSTOR to digitize, preserve and extend access to Public Choice.

http://www.jstor.org

This content downloaded from 195.34.79.103 on Tue, 10 Jun 2014 14:30:20 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=springer

http://www.jstor.org/stable/30023799?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


The strategic inconsistency of Paretian liberalism

ROY GARDNER* Iowa State University

Abstract

In this paper, the problem of the Paretian liberal is cast as a preference revelation game whose outcome function satisfies Gibbard's libertarian condition and strong Pareto optimality. Strategic consistency requires that the equilibrium of the game agree with the sincere outcome. It is shown that, whether viewed in a cooperative or non-cooperative context, the liberal social choice function is strategically inconsistent. This result suggests that, from a strategic standpoint, a different resolution of the liberal paradox is desirable.

Introduction

As one of the major institutions within democracy, individual rights belong in any complete social choice theory. Formalizing such rights, however, has proved to be a difficult matter. Sen (1970: 78-88) argued that a person's right consisted in a pair of social states (x, y) such that if the person pre- ferred x to y or y to x so did society. Let f be a social choice function whose choice set from among the available social possibilities S is given by {x in S: for all y in X, - yPx }, where P is the social preference relation. Sen then imposed the following three conditions on f: (1) universal domain (U) - any profile of individual preference orderings is possible; (2) liberalism (L) - each person in society has at least one right; (3) weak Pareto (WP) - if everyone in society prefers x to y, then xPy.1 Thus arises the liberal paradox: there is no social choice function satisfying conditions (U) (L) and (WP). An example will serve to illustrate the technical problem. Let there be two agents and three social states, x, y, and z. Agent 1 has the right (x, y) and preferences xyz.2 Agent 2 has the right (x, z) and the preferences yxz. Then xPy and zPx by applying (L), while yPz follows from (WP). There is

* The author wishes to thank Alan Gibbard, Nicholas Miller, Prasanta Pattanaik, Ken- neth Shepsle, Philip Straffin and an anonymous referee for their helpful comments.

Public Choice 35 (1980) 241-252. All rights reserved. Copyright m 1980 Martinus Ni/hoffPublishers by, The Hague/Boston/London.



242 R. Gardner

no social choice, each social state being inferior to some other. Among the many resolutions to this problem proposed in the literature,

that of Gibbard (1974) has attracted perhaps the most attention. Gibbard proposes to break a cycle of the above sort at its weakest link, as follows. Again suppose individual 1 has the right (x, y). If individual 1 prefers x to y, and there is no social state z such that

(1) individual 1 prefers y to z or is indifferent between them, and (2) another individual claims his right to z over x,

then xPy. However, if there is such a z, then individual l's right is waived, and xPy holds only in case of (WP). Gibbard's (1974, see especially pp. 399- 402 for a precise statement) condition that each agent have an alienable right in this sense will be denoted (L"). Thus, in the example just cited, individual l's right to x over y is waived because individual 2 claims his right to z over x and individual 1 prefers y to z. On the other hand, individual 2's right to z over x is not waived, since x is the worst alternative for 2. Thus, the social choice should be y in this case. In general, there are social choice functions satisfying conditions (U), (L"), and (WP).

Gibbard's suggestion has evoked considerable response. Sen (1976, for further discussion) for instance questions its moral appripriateness. Kelly (1976, for details) argues that Gibbard's social choice function requires irra- tional expectations on the part of agents. Aldrich argues that there are social choices, in particular at the constitutional level, where conditions like (L) and (L") do not apply. Alrich goes on to argue that the liberal paradox has significant ramifications for game theory (see Aldrich, 1977, as well as the exchange between Aldrich and Miller, 1977).

This paper abstracts from the moral issues but does attempt to join these strands of criticism in the positive sphere. Thus, the liberal social choice function is assumed to be the outcome function for a 2-person game of strategy. Each player in this game has as his strategic variable the preferences he reveals. A social choice function is strategically consistent3 if the equilibrium of the game agrees with the sincere outcome. It is shown that, whether viewed as a cooperative or noncooperative game, the liberal social choice function is strategically inconsistent. There is no general strategic reason to expect the sincere outcome of a game played according to Gibbard's rules.

2. Model and assumptions

There are two individuals, denoted 1 and 2, and a set S of social states, S = {x, y, z }. Although this seems restrictive, as far as individual rights are con- cerned, no generality is lost.4 Player 1 has the right (x, y); player 2, the right (x, z). These rights are fixed. Each player i has a complete, reflective, transitive ordering Ri of S belonging to the set of all such orderings, R6. The



243

Table

1. Outcome

function,

fL (R 1,

R 2)

Player

2

Player

1

xzy

x(yz)

xyz

(xy)z

yxz

y(xz)

yzx

(yz)x

zyx

z(yx)

zxy

(zx)y

(xyz)

xyz

x

x

x

x

x

x

y

y

y, z

z

z

x

x

x(yz)

x

x

x

x

x

x

y

y, z

z

z

z

x

x

xzy

x

x

x

x

x

x

z

z

z

z

z

x

x

(xz)y

x

x

x

x

x

x, z

z

z

z

z

z

x, z

x, z

zxy

x

x

x

x

x

z

z

z

z

z

z

z

z

z(xy)

x

x

x

x, y

y

y, z

y, z

z

z

z

z

z

z

zyx

z

z

y

y

y

y, z

y, z

z

z

z

z

z

z

(yz)x

z

y, z

y

y

Y

Y

y

y, z

z

z

z

z

y, z

yzx

y, z

y

y

Y

Y

Y

Y

y

y, z

y, z

y, z

y, z

y

y(zx)

y

y

Y

Y

Y

Y

Y

y

y, z

y, z

y, z

y, z

y

yxz

y

y

y

y

y

y

y

y

y, z

y, z

y, z

y

y

(yx)z

x

x

x

x, y

y

y

Y

y

y, z

y, z

z

x

x, y

(xyz)

x

x

x

x, y

y

y

y

y, z

z

z

z

x, z

x, y, z



244 R. Gardner

Table

2. Admissible

strategies,

player

1

Sincere

preferences

Admissible

strategies

Remarks

xyz

xyz,

yxz,

(yx)z

xzy

xzy

straightforward

zxy

zxy,

zyx

zyx

zxy,

zyx

yzx

yxz

straightforward,

sincerity

inadmissible

yxz

yxz

straightforward

x(yz)

xyz,

x(yz),

xzy

essentially

straightforward

y(xz)

yxz

straightforward,

sincerity

inadmissible

z(xy)

zxy,

zyx

sincerity

inadmissible

(xy)z

yxz

straightforward,

sincerity

inadmissible

(xz)y

xzy,

(xz)y,

zxy

essentially

straightforward

(yz)x

zyx,

(yz)x,

yzx,

y(zx),

yxz

essentially

straightforward

(xyz)

6R

essentially

straightforward



The strategic inconsistency of Pareto liberalism 245

social choice function fL is a map from R X 6 - 2s - 0, satisfying conditions (U), (L"), and (SP).s The outcomes fL (R 1, R2) are computed in Table 1. Note that fL is not particularly decisive: in 33/169 cases, there is more than one social choice.

Table 1 has another interpretation, namely as the matrix for the 2-person game in strategic form. Player 1 has the row strategies; player 2, the column strategies. Regardless of what player i's actual preference ordering is - here- after denoted Ri* - he can announce any possible order to fL. This interpretation makes fL a voting scheme.6 For instance, agent 1 is free to waive his right unconditionally by announcing indifference between x and y. Given sincere preferences (R 1 *, R2 *), the sincere outcome is fL(R1 *,R2 *). This is to be compared with the game outcome.

To play the game, the various players must be able to compare the results of various strategies. Here once faces the inevitable complication of indecisiveness: not just single states but sets of states must be compared. To meet this complication, the following set preference relation is introduced (Gardenfors, 1978, is the source of this ordering).

Definition. Let A and B be non-empty subsets of S and Ri a preference ordering of S. Then A GiB if and only if one of the following conditions holds:

(1) A C B and for all x in A and y in B - A, xR y (2) B CA and for all x in A - B and y in B, xRiy (3) Neither A C B nor B C A, and for all x in A - B and y in B - A,

xRiy.

If xRiy in (1)-(3) is in fact xPiy, then A Gi B. What the relation A Gi B. says is that set A is at least as good for i as set

B; A Gi B says that set A is better for i than set B. To illustrate, a player with preferences xyz has the G ordering.

{x, y, z }

{x} - {x, y} ------ {y z}{y

} -+ {z}

{x, z }

where arrows denote set preference. The relation Gi is reflexive and transitive, but not complete. This incompleteness, however, will not present any especial problems.

The orderings Gi and Gi are used to compare strategies. Suppose Ri and Ri' are two strategies open to individual i. Then Ri dominates Ri' if and only if



246 R. Gardner

fL (Ri, R1) Gi fL (Ri', Rj) for all R1

and

fL (Ri, R)) i fL(Ri', R1) for some Ri .

A strategy is admissible for i if it is not dominated by any other strategy. Admissible strategies play an important role in the analysis to follow. A game is straightforward for player i if he has but one admissible strategy; and essentially straightforward, if he is indifferent among all admissible strategies.7

3. Two-person liberal games with complete information

The game theoretic analysis of the liberal social choice function begins with an investigation of admissible strategies. Using the set preference relation Gi and Gi defined in the last section, one can show:

Proposition 1. Sincere revelation of preference is not in general admissible.

To see this, consider Table 2, in which are computed the admissible strategies for player 1, the row player, as a function of sincere preference. When player 1 has the preference ordering yzx, his only admissible strategy is yxz. It is straightforward for this player to lie in this situation. It never hurts to lie; indeed, it sometimes even helps. Sincerity is also inadmissible for player 1 when he has the preference y(xz), z(xy) or (xy)z. Similar results can be seen for player 2, the column player, by exploiting the following symmetry of fL, namely

y E fL (Ri, R,) if and only if z E f (Ri, Ri).

Thus, the roles of z and y are switched for player 2, who finds the following sincere preferences inadmissible: zyx, y(xz), z(xy), and (xz)y.8 Farquhar- son (1969: 29-30), in his classic study, held it to be self-evident that sincere voting is admissible and that a straightforward strategy is sincere. The liberal social choice function thus provides a counterexample to both assertions. This strong incentive to lie seems especially regrettable, and it plays a central role in the liberal game.

The cooperative game is considered first. In this setting, both players know each other's preferences. Moreover, the players are free to communi- cate with each other and to make binding agreements before choosing their strategies. The solution concept adopted is the negotiation set (a special case of the core). An outcome is in the negotiation set if neither player can




improve upon it by himself and it is at the same time Pareto optimal. If the cooperative liberal game were strategically consistent, the sincere outcome would be in the negotiation set. However, one has the following:

Proposition 2. The sincere outcome is not in general in the negotiation set.

To see this, consider the situation (RT, R*) = (xyz, zxy). The sincere outcome from Table 1 is {z}. Player 1 however can assure himself of at least the outcome {y, z} by use of the insincere strategy yxz, and {y, z) G_1 {z}. By the same token, player 2 can guarantee himself of the outcome {y, z } by his sincere strategy. The outcomes in the negotiation set therefore are {x} and {x, z}.

The fact that the sincere outcome is not in the negotiation set is again disturbing, since it means that rational players in the cooperative game have both the will and the power to insist on an insincere outcome.

Attention now turns to the noncooperative game. Here the equilibrium concept adopted is that of sophisticated voting (Farquharson, 1969: Ch. 8). Each player begins by ruling out all inadmissible strategies and expects that the other player does likewise. Then, given the reduced game matrix, each player rules out strategies inadmissible at this level. The process continues until no further reductions are possible. A sophisticated outcome is reached if each player ultimately has only one admissible strategy. A sophisticated equilibrium is a Nash equilibrium, although a Nash equilibrium need not be a sophisticated equilibrium. If the noncooperative liberal game were strategically consistent, the sincere outcome would be sophisticated. Instead one has

Proposition 3. The sincere outcome is not in general a sophisticated outcome.

To see this, consider the case where (R1 *, R2*) = (zxy, xzy). After the reduction to admissible strategies, the game matrix is

2 1 xzy zxy (zx)y

zxy x z z zyx z z z

Sincerity is now inadmissible for player 1, as zyx dominates zxy, but sincerity is the only admissible strategy left for player 2. Thus the sophisticated outcome is z = fL (zyx, xzy), whereas the sincere outcome is x.

In the above, the sophisticated outcome is nevertheless Pareto optimal. Even worse then is the reemergence of the liberal paradox in the following



248 R. Gardner

guise:9

Proposition 4. The sophisticated outcome is not in general Pareto optimal.

Thus, consider the case (R I", R2 *) = (xyz, zxy). This situation is straightforward for player 2, the game matrix of admissible strategies being

2 1 zxy

xyz z yxz y, z (xy)z z

For player 1, strategy yxz now dominates the others since {y, z) _} 1 {z ). The sophisticated outcome therefore is fy, z}. However, {y, z) is Pareto inferior to {x, z }, as {x, z)}G1 {y, z) and {x, z } Q2 {y, z}. Thus, although player 1 does better by lying in this situation, both players could be made even better off than the sophisticated outcome.

It is interesting to note that for the paradigm case motivating Gibbard's (1974: 389-399) analysis, namely (R1 *, R2 *) = (xyz, yzx),"0 the sincere outcome y is the sole outcome in the negotiation set; and the sole sophisticated outcome. This would seem to be the underlying reason for the intui- tive appeal of his solution to the liberal paradox. Unfortunately, such an underlying appeal does not in general obtain.

This discussion of the liberal game with complete information is con- cluded with the following observation:11

Proposition S. The liberal game need not have a sophisticated outcome.

Consider the situation (R 1 *, R2 *) = (zyx, yzx). The reduction to admissible strategies is

2 1 yzx yxz

zyx y, z y zxy z x

No further reduction is possible, and no sophisticated outcome is defined. The strategies (zxy, yzx) and (zyx, yxz) are, however, Nash equilibria. The Nash outcomes z = fL (zxy, yzx) and y = fL (zyx, yxz), are still different from the sincere outcome {y, z}.




4. Two-person liberal games with incomplete information

The last section assumed complete information on the part of the players, including knowledge of each other's preferences.2 This section relaxes that assumption in order to see the effect on the strategic outcome of the liberal game. Each player still knows the game matrix, but now only his own preferences. In a noncooperative context, each player faces a problem of decision making under uncertainty. One attitude that seems reasonable in this context is the adoption of admissible maximin strategies.'3 Such a strategy is the best a player can do in the most unfavorable situation and forms the basis for the initial demands underlying the negotiation set in Proposition 1. If each player adopts admissible maximin strategies, then one has the following:

Proposition 6. Given incomplete information, suppose that each player uses only admissible maximin strategies. Then Pareto inferior outcomes can occur, and with greater frequency than in the case of complete information.

To see this, consider Table 3, which is computed on the basis of Tables 1 and 2. Note that there are five cases which result in Pareto inferior outcomes. Only two of these, (xyz, zxy) and by symmetry (yxz, xzy), are predicted by Proposition 4. These Pareto failures are explained by the fact that for player 1 with preferences xyz (player 2 with preferences xzy), sincerity is not a maximin strategy. Thus, in the case when preferences are (xyz, xzy), the strategic outcome under incomplete information is not x, by y - even though x is best for both. Among the other outcomes there are few surprises. For Gibbard's example (xyz, yzx), the outcome is still y, via the strategies (yxz, yzx). The example in Proposition 5 is resolved in favor of {y, z}.

Now suppose that each player, instead of adopting an admissible maximin strategy, has a prior probability distribution over the other player's strategies. As long as a positive probability is attached to the most unfavorable situation, Proposition 6 shows that Pareto inferior outcomes still occur with positive probability. Again, let the situation become cooperative - say by negotiations - so that the players have the possibility of reaching the negotiation set. Whether they do so or not seems a contingent matter, cru- cially dependent on somehow learning each other's preferences. It seems clear that, at least in the 2-player theory, the hypothesis of incomplete information makes Pareto inferiority of the liberal game outcome a serious problem. In this sense, the liberal paradox persists at the strategic level.



250

Table

3. Outcomes

under

incomplete

information

Player

2

Player

1

xzy

xyz

yxz

yzx

zyx

zxy

xyz

{y,z}a

ya

y

y

y, z

{y, z )a

xzy

za

x

x

z

z

z

zxy

z

x

x

z

z

z

zyx

z

y

y

{y, z }

z

z

yzx

{y,

z}

y

y

y

{y, z}

y, z

yxz

{y,z}a

y

y

y

y,z

{y,z}

a = Pareto

inferior

outcome.



The strategic inconsistency ofPareto liberalism 251

5. Conclusion

This paper has investigated the game-theoretic properties of the 2-person liberal game. Several paradoxical conclusions can be drawn. Sincere revelation of preference is neither an admissible nor a maximin strategy in general. The sincere outcome is not always in the negotiation set, nor is it always a sophisticated outcome. Indeed, sophisticated revelation of preference can lead to Pareto inferior outcomes. When agents are unaware of each other's preferences, the possibility of Pareto inferior outcomes is even more pro- nounced. When agents are strategically minded, the liberal paradox has merely been pushed back one level. All of this suggests that, from a game- theoretic point of view, a different resolution of liberal paradox is desirable. Presumably such a resolution must take place at the constitutional level, as the outcome of a certain constitutional game."4 This should be the subject of further research.

NOTES

1. A stronger Pareto condition (SP) could also be imposed: xPy if no one prefers y to x and someone prefers x to y.

2. Throughout this paper, the preference ordering xPy, yPz, xPz will be written xyz. Ties will be denoted by parentheses; thus xPy, ylz, xPz is written x (yz).

3. This idea is introduced in Peleg (1978) and Dutta and Pattanaik (1978). 4. The results can be extended to more than three alternatives by having the remain-

ing social states ranked below x, y, and z by all individuals. However, important questions about coalition rights cannot be raised in the 2-person framework.

5. Gibbard's condition (WP) has been strengthened to (SP) in the interests of deci- siveness, and does not materially affect the results.

6. Gibbard (1973) introduced this notion. It is clear from Table 1 that the liberal voting scheme is manipulable.

7. Farquharson (1969) is the source for sincerity and straightforwardness. Hurwicz (1972) suggested essential-straightforwardness.

8. The author is indebted to an anonymous referee for this observation. 9. In Aldrich's interpretation of the liberal paradox, it is equivalent to the Prisoner's

Dilemma, of which this is reminiscent. 10. To make the identification, let 1 = Edwin, 2 = Angelina, x = wo, y = wE, and z =

wi. 11. The author is obliged to an anonymous referee for pointing this out. 12. Such an assumption is standard in game theory. But, see Harsanyi (1962) and

Satterthwaite. 13. Such strategies need not exist in general, since Gi is a partial order. They do exist

in the present model. 14. Gardenfors (unpublished) contains an analysis along these lines.



252 R. Gardner

REFERENCES

Aldrich, J. (1977a). 'The Dilemma of a Paretian Liberal: Some Consequences of Sen's Theorem.' Public Choice, 30: 1-21.

Aldrich , J. (1977b). 'Liberal Games: Further Comments on Social Choice and Game Theory.' Public Choice, 30: 29-34.

Dutta, B., and Pattanaik, P.K. (1978). 'On Nicely Consistent Voting Systems.' Econo- metrica, 46: 163-170.

Farquharson, R. (1969). Theory of Voting. New Haven: Yale University Press. Gardenfors, F. (1978). 'On Definitions of Manipulation of Social Choice Functions.'

In J.J. Laffont (Ed.), Aggregation and Revelation of Preferences. Amsterdam: North-Holland.

Gardenfors, F. 'Rights, Games and Social Choice.' Unpublished manuscript. Gibbard, A. (1973). 'Manipulation of Voting Schemes: A General Result.' Econo-

metrica, 41: 587-601. Gibbard, A. (1974). 'A Pareto-Consistent Libertarian Claim.' Journal of Economic

Theory, 7: 388-410. Harsanyi, J. (1962). 'Bargaining in Ignorance of the Opponent's Utility Function.'

Journal of Conflict Resolution, 6: 29-38. Hurwicz, L. (1972). 'On Informationally Decentralized Systems.' In C.B. McGuire and

R. Radner (Eds.), Decision and Organization. Amsterdam: North-Holland. Kelly, J.S. (1976). 'Rights Exercising and a Pareto-Consistent Libertarian Claim.'

Journal of Economic Theory, 13: 138-153. Miller, N.R. (1977). ' "Social Preference" and Game Theory: A Comment on "The

Dilemma of a Paretian Liberal".' Public Choice, 30: 23-28. Peleg, B. (1978). 'Consistent Voting Systems.' Econometrica, 46: 153-162. Satterthwaite, M. (Ms.). 'Straightforward Allocation Mechanisms.' Unpublished manu-

script. Sen, A.K. (1970). Collective Choice and Social Welfare. San Francisco: Holden-Day. Sen, A.K. (1976). 'Liberty, Unanimity, and Rights.' Economica, 43: 217-245.



the strategic inconsistency of paretian liberalism

Documents