BayesianVotingGame

(This is based on Palfrey & Rosenthal 'Voter Participation & Strategic Uncertainty' (American Political Science Review 1985). The original model has a vote between two options, but the structure is basically the same - voting preferences are fixed from the start, so the choice is simply whether to vote for your side or abstain.)

The formal model assumes that all agents know F(c), the prior probability distribution of ci's. The model is then a static Bayesian Game. (Note - I don't know how to do subscripts etc. in the wiki.)

A handy way to think about a Bayesian game is to imagine an agent called 'Nature' who acts before the players. This makes the game sequence as follows:

1. nature draws vector c = (c1, .. ­, cN) of players' cost values

2. nature reveals cost ci to player i alone

3. players choose whether to contribute or not

4. players receive payoffs

Strategies

(Loose) Definition: a strategy for player i is a function si(ci) which specifies the action (contribute / don't contribute) that player i would choose if drawn with any cost value ci.

(That is: player i decides on strategy at start of game, before knowing what type he will be drawn as. Alternatively i's strategy specifies what he would do if he was in the position of any player.)

In this particular model it is possible to define all strategies by the following decision rule:

set a critical cost level ci for each possible draw of ci

contribute if the actual draw ci < ci

c' = (c'1, .­. , c'N) is vector of strategies (critical cost levels) chosen by all players

c'-i = (c'1, .. ­, c'i-1, c'i+1, .. ­, cN) is vector of stratgeies of all other players but i

Equilibrium

(Loose) Definition: c' = (c'1, .. , c'*N) is a (pure strategy) Bayesian Nash equilibrium if it maximises payoffs for all players with all cost values.

As in the basic VotingModel, the expected payoffs from contributing / not contributing are:

Uc = pi (ni > n^-1) + pi (ni = n^-1) - ci (setting u = 1)

Un = pi (ni > n^-1)

So i's payoff is maximised by contributing iff:

ci < pi

where pi = prob (ni = n^-1)

But here pi is determined by F(c) and by c'-i, the vector of all other players' strategies. So can write this: pi (F, c'-i).

From the maximisation condition above, the optimal strategy for any c'-i is to set:

c'i = pi (F, c'-i)

Thus the equilibrium condition is:

c'i = pi (F, c'-i)

That is, in equilibrium everyone sets a cutoff c'*i which maximises their payoff given the cutoffs set by all other players.

Symmetric Equilibrium

Simplifying assumption: in equilibrium, everyone sets the same cutoff level c'*.

The equilibrium condition is thus:

c' = p (F, c')

(Note: p = prob (ni = n^-1 given (F, c'-i)) = prob (ni = n^-1 given (F, c')) is identical for all agents as cis are drawn independently.)

Thus: in equilibrium there are n contributing agents whose actual draw ci < c', and (N - n) non-contributors for whom ci >= c'.

Question: Do the assumptions in this model make sense?

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