Problem Setting:
2 alternative elections
voters' preferences depend on state variable that is not directly observable
each voter receives private signal that is correlated to the state variable
voters may be contingent (state dependent preference), or predetermined (state independent preference-i.e. the same preference in every state)
Aim of paper:
mechanism to elicit and aggregate private signals from voters
output the alternative favoured by the majority
Theoretical Findings:
voters truthfully reporting their signals forms a Bayes Nash Equilibrium
An equilibrium state in which no coalition of voters can deviate and receive a better outcome.
Strong Bayesian Nash Equilibrium
A strategy profile (s1, s2, ..., sT) is said to be a e-strong Bayes Nash Equilibrium if there not not exist any strategy profile (s1', s2', ..., sT') and a subset of agents D such that:
s_{i} = s_{i}' for t \notin D
u_{t}(s_{1}'...,) \geq u_{t}(s_{1}, ..., )
There exists some t\in D sich that u_{t}(s_{1}'...,) \geq u_{t}(s_{1}, ..., ) + eps
In laymans terms, consider 2 strategy profiles, s,s'.
Two alternatives - one is 'correct', each voter votes for correct alternative with probability p.
Theorem Statement: The probability that the majority voting scheme goes to the correct alternative is 1 as the number of voters increases when p>0.5 and (conversely) goes to zero when p<0.5.
Assumptions: voters vote truthfully and do not behave strategically.
Model:
Voters play Nash equilibrium strategy profile.
Voting rule: majority.
Each voter assigned a preference parameter x\in[-1,1] describing alignment to the two alternatives
Unique Baye Nash equilibrium is characterised by two thresholds: x0,x1, x0<x1 such that voters with preference below x0 always vote for one alternative and preference above x1 always vote for the other. Voters with preference between x0 and x1 vote truthfully
Result: as number of voters tends to infinity, the probability of a majority voting scheme outputing the correct alternative tends to 1 if voters play the equilibrium strategy profile.
Pros: Correct alternative output with high probability
Cons: requires sophisticated voters who will perform some computation
Does not consider strategic behaviour.
Introduction
Not all participants have clear preferences over alternatives, even in case of binary decisions.
Those who do not have clear preferences are contingent - have partial information on which alternative is better for them.
Informal Setting
two states (discrete state space), each agent receives a binary signal
imperfectly informed voters who only have partial information regarding which outcome is better
Assumption: distribution of agents is common knowledge (i.e. each voter knows the nature of all other voters)
Goal: select that alternative which would be preferred by the majority if knew the true state of the world.
This is meaningful only when the majority of voters are contingent.
Claim:
truthful strategy profile forms a strong Bayes Nash Equilibrium
"median voter trick"
ensures voters who are "below the median" have conflict of interest to voters who are "above" the median - makes sure less than half the voters have an incentive to deviate and those voters can can only change the outcome in the unfavourable direction by the property of the median.
Implementation:
questionnaire
agents report preference and a prediction of other agents' preferences
Assumptions
Distribution of agents is common knowledge
In case of binary choice, median voter's vote is favoured by majority (median according to sorting on what basis?) preference?
Model Formulation
N agents need to take a binary between alternatives 1 and 2.
There is a set of two possible worlds $W = {1,2}$ where 1 indicates that alternative 1 is better and 2 indicates otherwise.
Agents have common prior belief on likelihood of each world
A mechanism is said to be truthful when the truthful strategy profile forms a strong Bayes Nash Equilibrium.