Rational Choice and the Condorcet Jury Theorem

GAMES AND ECONOMIC BEHAVIOR ARTICLE NO.

22, 364]376 Ž1998.

GA970596

Rational Choice and the Condorcet Jury Theorem* Jorgen Wit ¨ CREED, Department of Economics, Uni¨ ersity of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands Received May 10, 1996

In a recent paper, Austen-Smith and Banks Ž1996, Amer. Polit. Sci. Re¨ . 90, 34]45., criticize the current literature on the Condorcet jury theorem as neglecting the behavioral underpinnings of decision-making. They leave open the question whether allowing mixed strategies would sustain the conclusions of the Condorcet jury theorem. In this paper, it is shown that these conclusions can hold in equilibrium. In other words, ‘‘a rational choice foundation for the claim that majorities invariably ‘do better’ than individuals’’ is derived. Ž Journal of Economic Literature Classification Number: D72. Q 1998 Academic Press

I. INTRODUCTION In voting theory, a famous result is due to Condorcet. His Condorcet jury theorem applies in those situations where voters have a common goal, but do not know how to obtain this goal. Voters are informed differently about the performance of alternative ways of reaching it. The Condorcet jury theorem states that a group of voters using majority rule is more likely to choose the right action than an arbitrary single voter is. The strength of this statement has been enlarged by several studies weakening the statistical assumptions of the model Že.g., Ladha, 1992, 1993; Berg, 1993.. In a recent paper, Austen-Smith and Banks Ž1996., however, criticize the current literature on the Condorcet jury theorem. They argue that it neglects the behavioral underpinnings of decision-making. An implicit assumption necessary to prove the theorem is that individuals in a group act identically to individuals making decisions in isolation. Austen-Smith and Banks demonstrate that this sincere ¨ oting assumption is ‘‘inconsistent *I am indebted to Richard McKelvey for suggesting this problem to me and for his helpful comments. The contributions of two anonymous referees, Arthur Schram and Randolph Sloof, are also greatly acknowledged. I also thank the California Institute of Technology for showing great hospitality during the completion of an earlier version of this paper. 364 0899-8256r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

CONDORCET JURY THEOREM

365

with a game-theoretic view of collective behavior.’’ That is, they show that the pure strategies played by group members in the Condorcet analysis do not constitute an equilibrium in general. However, Austen-Smith and Banks leave open the question whether allowing mixed strategies would sustain the conclusions of the Condorcet jury theorem. Myerson Ž1994. shows that such an approach can be successful. Under fairly general assumptions, he concludes that there exists a sequence of equilibria such that the probability of choosing the right action by the group converges to 1 as the group size goes to infinity. In this paper, we derive the Bayesian Nash equilibria in the simple game of Austen-Smith and Banks Ž1996.. Although the game suffers from a multiplicity of equilibria, equilibrium selection arguments are provided to select the most focal one for every parameter configuration. Then it is shown that Condorcet’s conclusions hold in the selected equilibrium for every group of voters. In other words, ‘‘a rational choice foundation for the claim that majorities invariably ‘do better’ than individuals’’ is derived for the Austen-Smith and Banks model.

II. MODEL The alternative space X s  A, B4 consists of two proposals, A and B. One of these proposals has to be chosen by a group of n voters, labeled as N s  1, 2, . . . , n4 , where n is odd. The group uses majority rule in order to arrive at a collective decision, i.e., the proposal that receives more than d ' Ž n y 1.r2 votes will be implemented. Voters simultaneously vote for one of the two proposals. The model is further characterized by a so-called state Žof the world.. This state is an element of the state space S s  A, B4 . The state s is unknown to all voters, but it is assumed that they have a common prior probability p that the true state is A, p s P w s s A x s 1 y P w s s B x. Individual preferences are identical. They depend on the proposal chosen and the state. They can be described by the utility function u i : X = S ª R Ž i g N .: u i Ž A, A. s u i Ž B, B . s 1, u i Ž A, B . s u i Ž B, A. s 0. In words, every voter wants to implement proposal A if the state is A, and B if the state is B. The so-called correct proposal is the one with the same label as the state. Before voters make their decision, they receive a private signal t i Ž i g N ., which is an element from the signal space T s  A, B4 . Voters will be characterized by the private signal they receive, i.e., private signals will be considered as voters’ types. Types are independent draws from the conditional distribution P w?< s x which is described by P w t i s A < s s A x s q A , P w t i s B < s s B x s q B , where 12 - q A - 1 and 12 - q B - 1. Finally, we will

JORGEN WIT ¨

366

also sometimes use the following definitions to reduce the notational burden: P'

p 1yp

q'

1 y qB qA

-1

q'

qB 1 y qA

) 1.

III. STRATEGIES AND EQUILIBRIUM A voting strategy ¨ i is an element from the space of functions V: T ª w0, 1x. It describes i’s probability of voting for proposal t g  A, B4 , given that i’s type is t. As in Feddersen and Pesendorfer Ž1995., a symmetry assumption is made. It entails that agents of the same type use the same strategy. Hence, one can define ¨ i Ž A . s pA

¨ i Ž B . s pB .

Together, these voting strategies constitute a voting strategy profile ¨ s Ž ¨ 1 , ¨ 2 , . . . , ¨ n .. The equilibrium concept used is Bayesian Nash equilibrium. In voting games Žwith n odd., this equilibrium condition can be simplified by observing that i’s strategy only matters if the other agents tie, i.e. when i is pi¨ otal. Define the event Pivi Ž ¨ yi . '  voter i is pivotal given that the others use strategy ¨ yi 4 . DEFINITION 1. A Bayesian Nash equilibrium in the voting game is a strategy profile ¨ ) such that U E u i < ¨ iU , Pivi Ž ¨ yi . , ti U G E u i < ¨ i , Pivi Ž ¨ yi . , ti

; i g N, ¨ i g V , t i g T .

The equilibrium condition is trivially satisfied if others use strategies ¨ yi that never make i pivotal. Within V ny 1 there are only two profiles ¨ yi that establish this event, namely, all other voters always vote for A, independent of their type, and all other voters always vote for B, independent of their type. Applying the symmetry condition, the two strategy profiles where all voters vote for the same proposal, irrespective of their private signal, constitute Bayesian Nash equilibria of the game. We say that voters vote uninformati¨ ely in these equilibria, as their votes do not reveal their private information. What remains to be shown is whether there are any other strategy profiles that satisfy the equilibrium conditions Žand make the probability that an arbitrary voter is pivotal greater than zero..

367

CONDORCET JURY THEOREM

In comparing the differences in expected utility between voting for A and B, it suffices to consider the function U U w Ž t i . s E u i Ž A, s . y u i Ž B, s .
t i g  A, B 4 .

This function is proportional to i’s expected difference in utility between voting for A and voting for B, given that i’s type is t i and i is pivotal. By U ., t i < s x can be written as the prodindependence, the expression P wPivi Ž ¨ yi U uct of P wPivi Ž ¨ yi .< s x and P w t i < s x. The former, i.e. the conditional probabilU .< s x, is given by ity of being pivotal given s, P wPivi Ž ¨ yi ny1 min ŽŽ ny1 .r2, k .

Ý

Ý

ks0

js0

=

ž

ny 1yk k ny1 w P t s A< s x Ž 1 y P w t s A< s x . k

/

k kyj p j 1 y pA . j AŽ

ž/



ny1yk Ž . ny1 Ž 1 y pB . ny1 r2yj pBŽ ny1.r2ykqj . yj 2

0

For notational simplicity this function will be denoted by F Ž P w t s A < s x, pA , pB .. Now, it is possible to simplify the equilibrium conditions by the following lemmas. The first lemma formalizes the intuition that the information contained in the signals should be reflected in the equilibrium strategies. LEMMA 1. In equilibrium, a type-A equilibrium probability of ¨ oting for A is strictly greater than a type-B equilibrium probability of ¨ oting for A Ž pA ) 1 y pB .. Moreo¨ er, in equilibrium there is at most one type using a mixed strategy. Proof. It is sufficient to show that w Ž A. ) w Ž B .. In words, i’s expected utility difference between voting for A and voting for B, is larger if i receives an A signal than if i receives a B signal: w Ž A. y w Ž B . s p q A F Ž q A , pA , pB . y Ž 1 y p . Ž 1 y q B . F Ž 1 y q B , pA , pB . y p Ž 1 y q A . F Ž q A , pA , pB . q Ž 1 y p . q B F Ž 1 y q B , pA , pB . s p Ž 2 q A y 1 . F Ž q A , pA , pB . q Ž 1 y p . Ž 2 q B y 1 . F Ž 1 y q B , pA , pB . )0. B

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368

The lemma restricts the possible nature of the equilibrium strategies. That is, equilibrium strategy profiles are either pure Ž pA s 1 and pB s 1. or hybrid Ž pA s 1 and 0 - pB - 1, or 0 - pA - 1 and pB s 0.. The next lemma provides necessary and sufficient conditions for the informati¨ e equilibrium pA s 1 and pB s 1. The proof can be found in Austen-Smith and Banks Ž1996.. The lemma entails that informative voting under majority rule only constitutes an equilibrium in those situations where the impact of A and B signals is more or less equally informative. That is, only for those parameter configurations where voters would agree that the proposal that receives a majority of signals should be chosen. LEMMA 2. pA s 1 and pB s 1 is an equilibrium

m q dq 1 q d G P G q dq1 q d . Ž 1 .

The next lemma establishes a more detailed characterization of the partially informative hybrid equilibria. The interpretation of this lemma is quite intuitive. It shows that the nature of the equilibria depends on the voting rule that would be unanimously chosen if voters have complete information about other voters’ signals. More precisely, the equilibrium where voters with a B signal mix between A and B occurs in cases where a small number of A types suffices to vote for A under complete information. In other words, B types try to offset the bias created by the information structure: A types possess ‘‘better’’ information than B types. Equivalently, the equilibrium where A types use a mixed strategy occurs in situations where a small number of B types is sufficient to favor B. Now, the A types are the ones trying to compensate the bias. LEMMA 3. pA s 1 and 0 - pB - 1 is an equilibrium 0 - pA - 1 and pB s 1 is an equilibrium

m m

q dq 1 ) P ) q dq1 q d Ž 2 . q dq 1 q d ) P ) q dq1 . Ž 3 .

Proof. Given that others use the strategy pA s 1 and 0 - pB - 1, an arbitrary voter correctly votes for A in state A with probability c A ' q A q Ž1 y q A .Ž1 y pB ., and correctly votes for B in state B with probability c B ' q B pB . Using Lemma 1, the equilibrium condition for pA s 1 and

369

CONDORCET JURY THEOREM

0 - pB - 1 is w Ž B . s 0. That is, a B-type is indifferent between voting for A and voting for B, i.e.,

p Ž1 y qA .

ny1 ny1

 0  0

c AŽ ny1.r2 Ž 1 y c A .

Ž ny1 .r2

2

s Ž 1 y p . qB

ny1 ny1

c BŽ ny1.r2 Ž 1 y c B .

Ž ny1 .r2

2

p

m

s

1yp

ž

qB 1 y qA

s

qB 1 y qA

Ž nq1 .r2

/

ž

ž

cB Ž 1 y cB . cA Ž 1 y cA .

Ž ny1 .r2

/

1 y q B pB 1 y Ž 1 y q A . pB

Ž ny1 .r2

/

.

Using q A q q B ) 1, the right-hand side of this expression is strictly decreasing in pB . For pB s 1, the condition is P s q dq 1 q d and for pB s 0 it reduces to P s q dq 1 , which proves the first part of the lemma. Using the symmetry of the model, the second part follows straightforwardly. B We now have a full characterization of the equilibria in the game. All voters uninformatively voting for either proposal constitutes two equilibria that hold for all parameter configurations Žp , q A , q B .. In addition, there is a third equilibrium for certain parameter configurations. These configurations are given by inequalities Ž1. ] Ž3.. They are illustrated in Fig. 1, where q dq1 G P is Ž4. and P G q dq1 is Ž5..

IV. EQUILIBRIUM SELECTION The existence of multiple equilibria raises the issue of equilibrium selection. That is, are there refinements of Bayesian Nash equilibrium that exclude some of these equilibria, or do any of the equilibria possess focal

FIG. 1. Parameter configurations.

370

JORGEN WIT ¨

properties? We shall now provide three arguments that select a unique equilibrium for every parameter vector Žp , q A , q B .. In voting games, elimination of weakly dominated strategies is a natural refinement ŽFeddersen and Pesendorfer, 1995.. Applying this criterium eliminates the uninformative strategy ‘‘always vote for A’’ Ž pA s 1 and pB s 0. for q n ) P, leaving pA s 0 and pB s 1 as the selected equilibrium. The equilibrium ‘‘always vote for B’’ Ž pA s 0 and pB s 1. is eliminated for P ) q n, leaving pA s 1 and pB s 0 as the selected equilibrium. In both cases, each voter, irrespective of the signal t s Ž t 1 , t 2 , . . . , t n ., has a strictly dominant strategy to vote for one of the proposals. Next, for parameters Žp , q A , q B . satisfying Ž1., Ž2., or Ž3., three equilibria exist. The uninformative equilibria constitute two symmetric counterparts. Hence, the Žpartially. informative equilibrium, described in Lemmas 2 and 3, possesses a focal property. It ‘‘conspicuously distinguishes it from the other equilibria’’ ŽMyerson, 1991.. Finally, in the intermediate intervals q dq 1 G P G q n and q n G P G q dq1 again only the uninformative voting equilibria exist. For these parameters, one equilibrium can now be justified by a continuity argument. This is the equilibrium, where all voters always vote for B Ž pA s 0 and pB s 1. in the former, and always vote for A in the latter. Then, for any parameter vector Žp , q A , q B ., the equilibrium strategy profile Ž pA , pB . is continuous in the parameters. Summarizing, the equilibrium strategy profile is characterized by Fig. 2 for the selection criteria used. Note that this figure also unambiguously characterizes the optimal strategies for a single individual Ž n s 1.. The next section will examine what kind of outcomes are supported by the equilibrium strategies.

V. CONDORCET JURY THEOREM The question raised in the Introduction is whether the conclusions of the Condorcet jury theorem remain valid in equilibrium. A first issue is whether it is true that groups do better than individuals in equilibrium, i.e.,

FIG. 2. Equilibrium strategies.

371

CONDORCET JURY THEOREM

is it the case that a group Ž n ) 1. is more likely to choose the correct proposal than an individual Ž n s 1. facing the same decision problem with parameters Žp , q A , q B .? Another issue is whether a group will almost certainly choose the correct proposal as the group size n goes to infinity? To answer these questions, let Pn be defined as the probability that a group of size n votes for the correct proposal in the selected equilibrium. In case of finite n, it needs to be established whether in equilibrium the group is more likely to choose the correct proposal than a single individual. Three cases are considered, depending on the parameter configurations. First, in case the equilibrium strategy is informative wŽ1.x, the voters act according to the behavioral assumption of the Condorcet jury theorem. Hence, the Condorcet jury theorem applies Žsee, e.g., Ladha, 1992.. Next, for those cases in which the group votes uninformatively wŽ4. and Ž5.x, the single individual will also vote uninformatively Žcf. Fig. 2.. In those extreme situations, Pn is equal to P1. Remains to be considered are those situations where the group plays the game according to the hybrid equilibrium wŽ2. and Ž3.x. The following theorem shows that Pn is strictly greater than P1 in this case. THEOREM 1.

If Žp , q A , q B . satisfies Ž2. or Ž3., then Pn ) P1.

Proof. If Žp , q A , q B . satisfies Ž2., then Fig. 2 shows that the selected equilibrium is described by pA s 1 and 0 - pB - 1. The proof of Lemma 3 shows that an arbitrary voter correctly votes for A in state A with probability 1 y Ž1 y q A . pB , and correctly votes for B in state B with probability q B pB . Thus, the probability Pn can be written as n

Pn s p

Ý

ks Ž nq1 .r2

n k

n

qŽ 1 y p .

k

1 y Ž 1 y q A . pB

ž /

Ý

ks Ž nq1 .r2

n k

ž /w

q B pB x w 1 y q B pB x k

nyk

Ž 1 y q A . pB nyk

.

We will show that this expression is a strictly concave function in pB with a maximum in the equilibrium. First, let n

f Ž x. s

Ý

ks Ž nq1 .r2

ny k n k x Ž1 y x. k

ž /

0 F x F 1.

JORGEN WIT ¨

372

Then it is straightforward to show that the derivative df Ž x .rdx is given by ny1 sn ny1 2

df Ž x .

 0  0

dx

x Ž ny1.r2 Ž 1 y x .

Ž ny1 .r2

0 F x F 1.

Using this identity, the derivative of Pn with respect to pB is given by ny1 s yp Ž 1 y q A . n n y 1 dpB 2 dPn

= Ž 1 y q A . pB

1 y Ž 1 y q A . pB

Ž ny1 .r2

Ž ny1 .r2

ny1 q Ž 1 y p . qB n n y 1 2

 0

w q B pB x Ž ny1.r2 w 1 y q B pB x Ž ny1.r2 .

In other words, the first order condition, dPnrdpB s 0, can be written as

p 1yp

s

ž

Ž nq1 .r2

qB 1 y qA

ž

/

Ž ny1 .r2

1 y q B pB 1 y Ž 1 y q A . pB

/

,

which is equal to the equilibium condition Žcf. Lemma 3.. Next, the second derivative d 2 Pn dpB2

ny1 2 s yp Ž 1 y q A . n n y 1 2

 0

ny1 2

= Ž 1 y q A . PB Ž 1 y Ž 1 y q A . pB . ny1 q Ž 1 y p . q B2 n n y 1 2

 0

1 y 2 Ž 1 y q A . pB

ny1 2

Ž ny3 .r2

= q B pB Ž 1 y q B pB .

Ž ny3 .r2

w 1 y 2 q B pB x ,

which is strictly less than zero if

p 1yp

Ž 1 y 2 Ž 1 y q A . pB . -

ž

Ž nq1 .r2

qB 1 y qA

=

ž

/

1 y q B pB 1 y Ž 1 y q A . pB

Ž ny3 .r2

/

Ž 1 y 2 q B pB . .

CONDORCET JURY THEOREM

373

In the local optimum, identified by the first order conditions, this expression holds if

Ž 1 y q B pB . Ž 1 y 2 Ž 1 y q A . pB . ) Ž 1 y 2 q B pB . Ž 1 y Ž 1 y q A . pB . m

qB ) 1 y q A ,

which always holds. As the function is continuously differentiable in pB on w0, 1x, this implies that the local maximum is also the global maximum on w0, 1x. In other words, in equilibrium the probability of making a correct decision is strictly greater than both the non-Nash outcome where everybody votes informatively Ž pB s 1. and where everybody votes uninformatively for A Ž pB s 0.. For the range of parameters that support this hybrid equilibrium, the single individual Ž n s 1. either votes informatively or votes always for A. In the first situation, the Condorcet jury theorem applies, where the group voting informatively always does better than the single individual voting informatively. Hence, Pn ) P1. In case the individual always votes for A, the probability of choosing the correct proposal is equal to the probability of making a correct decision by a group voting uninformatively for A. We have seen above, that Pn is strictly greater than this probability. The second part of the theorem can be proved in a similar way by using symmetry. B From the proof of the theorem it becomes clear that the equilibrium probability Pn is also greater than the probability that a group chooses the correct proposal when all voters use non-Nash informative strategies. This implicitly establishes limit behavior. The Condorcet jury theorem shows that a group with members who vote informatively, votes with probability 1 for the correct proposal as the group size n goes to infinity ŽLadha, 1992.. More formally, and for all parameter configurations, the next theorem establishes the asymptotic properties. THEOREM 2. n ª `.

For any Žp ,q A ,q B ., the probability Pn con¨ erges to 1 as

Proof. We will consider three cases, Ži. q A - q B , Žii. q A ) q B , and Žiii. q A s q B . Figure 2 shows that in the limit the selected equilibrium strategies do not depend on the value of p in cases Ži. and Žii.. In case Ži., there is an n9 such that for any n ) n9, pAn s 1 and 0 - pBn - 1, where superscripts n indicate the dependence on n. In this equilibrium, the asymptotic expression for Pn Žcf. Theorem 1. is given by lim Pn s p q Ž 1 y p . 1q B p Bn ) 1r24 .

nª`

JORGEN WIT ¨

374

In Lemma 3, it was shown that the mixing probability pBn is implicitly defined by

p 1yp

s

ž

Ž nq1 .r2

qB

ž

/

1 y qA

Ž ny1 .r2

1 y q B pB

/

1 y Ž 1 y q A . pB

.

Solving this expression for pBn gives pBn s

Hn s

Hn y 1

Ž 1 y q A . Hn y q B p 1yp

ž

1 y qA qB

,

where

Ž nq1 .r2 2r Ž ny1 .

/

.

Taking the limit n ª ` results in lim Hn s

nª`

1 y qA qB

and

lim pBn s

nª`

1 1 y q A q qB

.

Consequently, lim q B pBn s

nª`

qB 1 y q A q qB

)

qB qB q qB

s

1 2

,

which establishes the asymptotic result for case Ži.. The theorem easily follows in case Žii. being the symmetric counterpart of case Ži.. Finally, case Žiii. can be divided into three subcases, Žiiia. q ) P, Žiiib. q G P G q, and Žiiic. P ) q. In subcases Žiiia. and Žiiic., there exist a hybrid strategy equilibrium. The result follows by the same arguments as in case Ži. and Žii.. Remains to be shown is subcase Žiiib.. For these parameters, the equilibrium strategy is informative. Given that q A s q B ) 12 , the result follows from the original Condorcet jury theorem. B

VI. CONCLUDING DISCUSSION The main result derived in this paper shows that Condorcet’s notion that groups are more likely to make correct decisions than individuals do, holds in equilibrium. Moreover, it is demonstrated that in those situations where the equilibrium strategy is hybrid, the equilibrium outcome is even better than predicted by the original Condorcet jury theorem. This ‘‘positive synergy’’ result might serve as another equilibrium selection argument. It

CONDORCET JURY THEOREM

375

implies that the symmetric hybrid equilibrium strictly payoff-dominates the symmetric Nash equilibria, where every voter votes uninformatively. In a recent paper, Ladha et al. Ž1996. show that asymmetric equilibria also exist in this game. They prove that the same kind of synergy can be realized for those situations where informative voting is not an equilibrium strategy. It can be shown that this synergy is even stronger than the one found in this paper. However, asymmetric equilibria do not have predictive power in a positive theory of voting. These strategies leave the coordination problem unanswered. ŽOf course, coordination can be enhanced if voters can build a history in a repeated game setting.. Subsequent to the writing of this paper, I learned of the paper by McLennan Ž1996., which generalizes the main result of this paper. McLennan proves that whenever sincere voting leads to the conclusions of the Theorem, there are also Nash equilibria that support these conclusions. It is easily shown that sincere voting satisfies this condition in the AustenSmith and Banks model. McLennan does not show the particular form of the equilibrium strategies. In this paper, it is shown that these strategies have intuitive appeal. An interesting parallel can be drawn with Feddersen and Pesendorfer Ž1996.. The strategic voters in their model, the uninformed independents, try to offset the bias created by partisans, in order to let the informed independents decide which mutually beneficial outcome should be chosen. In our mixed strategy equilibria something similar occurs. The ‘‘less’’ informed types strategically compensate the bias created by the information structure by using a mixed strategy. This behavior allows the ‘‘more’’ informed types to put more weight in the collective decision. The current paper and McLennan Ž1996. leave unanswered whether the conclusions of the Condorcet jury theorem continue to hold for those situations where sincere voting does not support these conclusions. Myerson Ž1994. shows that it should be true in the limit as the groupsize goes to infinity, but he provides no general result. This is left for future research.

REFERENCES Austen-Smith, D., and Banks, J. S. Ž1996.. ‘‘Information Aggregation, Rationality and the Condorcet Jury Theorem,’’ Amer. Polit. Sci. Re¨ . 90, 34]45. Berg, S. Ž1993.. ‘‘Condorcet’s Jury Theorem, Dependency Among Jurors,’’ Soc. Choice Welfare 10, 87]95. Feddersen, T., and Pesendorfer, W. Ž1995.. ‘‘Voting Behavior and Information Aggregation in Elections with Private Information,’’ Discussion Paper 1117, The Center For Mathematical Studies in Economics and Management Science. Evanston: Northwestern University.

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Feddersen, T., and Pesendorfer, W. Ž1996.. ‘‘The Swing Voter’s Curse,’’ Amer. Econ. Re¨ . 86, 408]424. Ladha, K. Ž1992.. ‘‘The Condorcet Jury Theorem, Free Speech and Correlated Votes,’’ Amer. J. Polit. Sci. 36, 617]634. Ladha, K. Ž1993.. ‘‘Condorcet’s Jury Theorem in Light of de Finetti’s Theorem,’’ Soc. Choice Welfare 10, 69]85. Ladha, K., Miller, G., and Oppenheimer, J. Ž1996.. ‘‘Democracy: Turbo-charged or Shackled? Information Aggregation by Majority Rule,’’ mimeo. St. Louis: Washington University. McLennan, A. Ž1996.. ‘‘Consequences of the Condorcet Jury Theorem for Beneficial Information Aggregation by Rational Agents,’’ mimeo. Minneapolis: University of Minnesota. Myerson, R. B. Ž1991.. Game Theory: Analysis of Conflict. Cambridge, MA: Harvard Univ. Press. Myerson, R. B. Ž1994.. ‘‘Extended Poisson Games and the Condorcet Jury Theorem,’’ Discussion Paper 1103, The Center For Mathematical Studies in Economics and Management Science. Evanston: Northwestern University.