A Nonspeculation Theorem with an Application to Committee Design

Jidong Chen; Mark Fey; W. Ramsay Kristopher

doi:10.1515/bejte-2015-0103

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A Nonspeculation Theorem with an Application to Committee Design

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Published/Copyright: April 1, 2017

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From the journal The B.E. Journal of Theoretical Economics Volume 17 Issue 2

Abstract

Various well known agreement theorems show that if players have common knowledge of actions and a “veto" action is available to every player, then they cannot agree to forgo a Pareto optimal outcome simply because of private information in settings with unique equilibrium. We establish a nonspeculation theorem which is more general than previous results and is applicable to political and economic situations that generate multiple equilibria. We demonstrate an application of our result to the problem of designing an independent committee free of private persuasion.

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Appendix

Proof of Theorem 3:

Let sa(ω) denote the probability with which an action profile a is played when the strategy profile is s(ω), and the state is ω.

Let’s first establish a useful identity. For any strategy profile s(⋅) and any ω′, we have

E[ui(s(ω),ω)|ω∈Pˆ(ω′)]=E[∑a∈Aui(a,ω)sa(ω)|ω∈Pˆ(ω′)]=∑a∈AE[ui(a,ω)sa(ω′)|ω∈Pˆ(ω′)]=∑a∈AE[ui(a,ω)|ω∈Pˆ(ω′)]sa(ω′)

(1) Assume that Condition 3 holds and for i=1,…,n, let vi=∑ω∈Ωπ(ω)vi(ω). For all s and for all i, we have

E[ui(zi,s−i(ω),ω)|Ω]=∑ω′∈Ωπ(ω′)E[ui(zi,s−i(ω),ω)|Pˆ(ω′)]=∑ω′∈Ωπ(ω′)∑a−iE[ui(zi,a−i,ω)|Pˆ(ω′)]s−ia−i(ω′)=∑ω′∈Ωπ(ω′)∑a−ivi(ω′)s−ia−i(ω′)=∑ω′∈Ωπ(ω′)vi(ω′)=vi

This establishes the first part of Condition 2.

(2) For the second part of Condition 2, suppose s satisfies E[ui(s(ω),ω)|Ω]≥vi for all i. This is equivalent to

∑ω′∈Ωπ(ω′)E[ui(s(ω),ω)|Pˆ(ω′)]≥vi

for all i. Summing across individuals, we have

∑ω′∈Ωπ(ω′)∑iE[ui(s(ω),ω)|Pˆ(ω′)]≥∑ivi.

By the second part of Condition 3, we have two types of action profiles: the ones such that E[ui(a,ω)|Pˆ(ω′)]=vi(ω′) for all i and all ω′∈Ω; and the action profiles such that ∑iE[ui(a,ω)|Pˆ(ω′)]<∑ivi(ω′) for all ω′∈Ω.

Given a strategy profile s(⋅), for any state ω′, if the second types of actions are never played with a positive probability, then according to the identity we show above, we get E[ui(s(ω),ω)|Pˆ(ω′)]=∑a∈Avi(ω′)sa(ω′)=vi(ω′), for all i. If this property is satisfied for all states ω′, then we get the result we want.

If, however, under some state ω′, the second types of action are played with a positive probability, we have

∑iE[ui(s(ω),ω)|Pˆ(ω′)]=∑i∑a∈AE[ui(a,ω)|Pˆ(ω′)]sa(ω′)=∑a∈A∑iE[ui(a,ω)|Pˆ(ω′)]sa(ω′)<∑a∈A∑ivi(ω′)sa(ω′)=∑ivi(ω′).

Summing across all possible states, we have ∑ω′∈Ωπ(ω′)∑iE[ui(s(ω),ω)|Pˆ(ω′)]<∑ω′∈Ωπ(ω′)∑ivi(ω′)=∑ivi.

This is a contradiction and thus we have E[uj(s(ω),ω)|Pˆ(ω′)]=E[uj(z,ω)|Pˆ(ω′)] for all j and all ω′. This establishes the second part of Condition 2. ■

Published Online: 2017-4-1

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https://doi.org/10.1515/bejte-2015-0103