Optimal voting schemes with costly information acquisition

TL;DR: It is shown that, of all mechanisms, a sequential one is optimal and works as follows: one agent at a time is selected to acquire information and report the resulting signal and the restriction to ex-post efficiency is shown to be without loss when the available signals are sufficiently imprecise.

About: This article is published in Journal of Economic Theory.The article was published on 2009-01-01 and is currently open access. It has received 96 citations till now. The article focuses on the topics: Optimal decision.

Summary

1 Introduction

• In many environments, groups rather than individuals make decisions.
• The committee must analyze the proposed road's costs and benefits, which are complex.
• In all of these cases, even though the committee members have the same objective, decisions may be inefficient because members may prefer to rely on others' opinions instead of acquiring information themselves.
• The total amount of information aggregated in the optimal mechanism is an endogenous object of their interest.

2 The Model

• There are two possible states of the world: H and L. Each of these states occurs with probability one-half.
• (We also implicitly assume that information acquisition takes no time.)the authors.the authors.
• There is a Social Planner (SP) who wants to maximize expected sum of the individuals' utilities, net of the expected total cost of information acquisition: EQUATION where L is the expected number of voters who collect information.
• The utilities are not transferable; that is, the SP cannot use a transfer scheme to induce the voters to acquire information.
• The lemma above states that the posteriors about the state of the world and about the next signal after observing a sequence of signals depend only on the difference between the numbers of signals H and L observed so far.

3 The First-Best Mechanism

• This section characterizes the first-best voting scheme.
• The following proposition characterizes the first-best mechanism.
• Notice that not all of the states can be reached.
• Hence, given a certain d, the more voters have been already asked, the more likely it is that the SP makes a decision.

4 Preliminaries Canonical Mechanisms

• A voting mechanism is an extensive-form game with imperfect information where the players are the voters.
• Chance may randomize when taking an action.
• When a voter is asked to collect information, she does not know her position in the sequence, and she does not know what other voters reported to the SP.
• (b) If, at some information set, a certain voter does not acquire information but takes an action in e, then modify the game such that chance moves at that information set and takes the same action as the voter took in (G, e). 2 Myerson (1986) claims a similar result for a more general class of multistage games with communication.
• Let us assume that the optimal mechanism is asymmetric.

Incentive Compatibility

• The goal of this subsection is to explicitly characterize the incentive compatibility constraint, that is, the constraint that guarantees the voters indeed have incentive to collect information when asked instead of just reporting something.
• Hence, when she computes the probability of a sequence, she takes away a signal H and computes the likelihood of the remainder of the sequence.
• For each mechanism, there exists another one that operates as follows:.
• The probability mixture of incentive compatible mechanisms is also incentive compatible.

Continuation Mechanisms

• The arguments of most proofs regarding optimality of mechanisms involve modifying the mechanism at some states.
• When the scheme must be incentive compatible, there is an interaction between different continuation mechanisms.
• Furthermore, the continuation mechanism M never specifies that the SP asks more voters than the number available after reaching V (l, d), that is N − l.
• The right side of the inequality is the SP's posterior at V (l, d) about the true state of the world.
• This lemma is essential for characterizing the optimal ex-post efficient mechanism.

5 Optimal Mechanisms

• The authors first characterize the optimal ex-post efficient mechanism.
• The authors show that it has very similar properties to the first-best voting scheme.
• Then the authors discuss some attributes of this mechanism.
• Finally, the authors show that the ex-ante optimal mechanism sometimes involves ex-post inefficient decisions.

5.1 Optimal Ex-post Efficient Mechanism

• The authors are ready to characterize the optimal mechanism in the class of ex-post efficient mechanisms, that is, the class of mechanisms where the SP always makes a majority decision.
• The SP keeps asking the voters sequentially to collect information and report it to him.
• Since the function f is decreasing, the more voters the SP has already asked, the less precise a posterior induces the SP to stop asking voters and take an action.
• Furthermore, the function f never jumps down by more than one.
• As the authors pointed out earlier, from Lemma 3, it follows that the optimal mechanism generically involves randomization.

Theorem 2

• The optimal ex-post efficient mechanism described in Theorem 1 is generically unique and involves randomization only at a single state.
• First, the authors show that if, for a certain pair (p, c), there are at least two different optimal mechanisms, then there exists an optimal mechanism that involves randomizations in at least two different states.
• By Lemma 6, it follows that there are at least two continuation mechanisms that have the same efficiency.
• Since there are only finitely many continuation mechanisms, if the optimal mechanism were not generically unique, there would exist two continuation mechanisms with the same efficiency for a positive measure of (p, c).

5.2 Discussion of the Ex-post Optimal Mechanism Infinitely Many Voters

• Recall that the function g was decreasing because, as the number of voters who have not yet collected information decreases, the value of asking more voters also decreases.
• The way to guarantee a large probability of being pivotal to the voters is to make decisions after signal sequences where the difference between the numbers of different signals is small.
• (Otherwise the model is the same as before.).
• This is because as k 0 goes to infinity, the probability of being pivotal, and hence also the benefit from collecting information, goes to zero.
• Suppose that there are infinitely many voters, and the first-best mechanism is not incentive compatible.

Robustness

• A common critique of Bayesian mechanism design is that to design the optimal mechanism, the SP has to have perfect knowledge about the information structure of the environment.
• If the decreasing step function is too large, then this equilibrium is one in which the voters do not collect any information.
• Hence, the value of the objective function of the SP is also close to the value corresponding to the incentive compatible mechanism.
• Then there exists a unique symmetric mixed-strategy equilibrium.

5.3 Ex-ante Optimal Mechanism

• The authors show that the optimal mechanism sometimes involves ex-post inefficient decisions.
• That is, the mechanism characterized in Theorem 1 is not always optimal.
• It will be shown that if the cost of information acquisition is small enough, then the optimal ex-post efficient mechanism can be improved upon by replacing a continuation mechanism with an ex-post inefficient continuation mechanism.
• At V (K, 1), the SP asks an additional voter, and if she confirms his posterior, he makes the majority decision.
• If their reports are the same, he again makes the majority decision.

6 Discussion

• This paper analyzed optimal voting schemes in environments where information acquisition is costly and unobservable.
• The Social Planner stops asking voters if and only if his posterior is more precise than the value corresponding to the number of voters already asked.
• This intuition does not involve any assumption about the distribution of the states of the world and the signals.
• Recall that having the explicit form of the incentive compatibility constraint made it possible to compute the efficiency of continuation mechanisms.
• From ( 26) and ( 27) it follows that the absolute value of the ratio of the coefficients corresponding to the terms with the largest power is 1/l (s).

Frequently asked questions

Q1. What are the contributions in "Optimal voting schemes with costly information acquisition∗" ?

This paper analyzes a voting model where ( i ) there is no conflict of interest among the voters, and ( ii ) information acquisition is costly and unobservable. The social planner asks, at random, one voter at a time to invest in information and to report the resulting signal. Voters are informed of neither their position in the sequence nor the reports of previous voters. Obeying the planner by investing and reporting truthfully is optimal for voters. In this scheme, the social planner stops aggregating information and makes a decision when the precision of his posterior exceeds a cut-off which decreases with each additional report. 

Q2. What is the effect of the ex-ante optimal mechanism?

The authors also show that, if the cost of information acquisition is small, then, surprisingly, the ex-ante optimal mechanism is often ex-post inefficient. 

Q3. What is the way to improve the optimal ex-post efficient mechanism?

It will be shown that if the cost of information acquisition is small enough, then the optimal ex-post efficient mechanism can be improved upon by replacing a continuation mechanism with an ex-post inefficient continuation mechanism. 

Q4. What is the probability that the deviator is asked to report a signal?

Since the SP orders the voters independently of the realizations of the signals,p (A ∩B) = p (s) 2i+ d N ,where (2i+ d) /N is the probability that the deviator is asked to report a signal if a decision is made after a sequence with length 2i+d. 

Q5. What is the argument that the first-best mechanism is not incentive compatible?

The authors argue that for given p and c, there always exists a k0 ∈ N such that the mechanism is not incentive compatible if the SP stops asking voters only if |d| ≥ k0.