StrategicVotingModel
ThoughtStorms Wiki
This is a ‘handwaving’ version of a strategic model of discrete public good contribution. BayesianVotingGame is a slightly more technical exposition of the same kind of thing, based on a paper by Palfrey & Rosenthal. Hopefully this version gets across the intuition, though it may not be formally perfect.
(Note to Phil - is it possible to do subscripts in this wiki? Other mathematical notation? And pictures?)
In VotingModel rational individualists contributed to provide a (discrete) public good if they believed there was a big eneough chance their contribution might 'swing' whether the good is provided or not. The most obvious example is the case of 'voting' (if you can consider an election result as a public good - maybe analogy is a better term than example.)
But what determines the probability of being the swinger? Only the actions of other agents, who are all thinking on the same lines as well.
eg. you imagine the voter thinking along these lines - 'There's no point me voting because my vote won't make any difference'. 'Hang on though, if everyone thinks like that no one would vote so my vote would make a difference.' 'But if everyone realises that ....' etc.
setup
As in the basic VotingModel, there are N agents, all thinking like this. Each one will get a benefit ui if the public good is provided, but will suffer a cost ci if they contribute. The good is provided if at least some number n^ agents contribute.
As before, this leads to the conclusion that it is optimal for an agent i to contribute if, and only if:
Ci/ui < prob (n-i = n-1)
Where n-i is the number of other agents who contribute.
To simplify things, we can set ui = 1, and define p = prob (n-i = n-1), and so rewrite this condition as:
Ci < p
Guesstimates
Agents know their own contribution cost levels ci, but they don’t know anyone else’s.
However, they have an expectation of how many other agents are likely to contribute, and this will influence their decision as to whether to contribute themselves.
One way to think of this is to imagine that agents make a ‘guesstimate’ of the number of other agents who will contribute. Agents also have (‘guesstimated’?) knowledge of the distribution of cost values across the population. These facts imply that they can also, at a push, make a guesstimate of p, the probability that they could be the swing contributor.
Using this idea, suppose agents adopt a strategy of contributing if, and only if:
ci < c’
That is, you contribute if your contribution cost is below your guesstimate of the probability p of being the swing contributor. If your guesstimate is correct (c’ = p), then this strategy gets you the best possible result – as we know, it is optimal to contribute iff ci < p.
Equilibrium
To simplify, assume that all agents make the same guesstimate c’. This makes sense if all agents have the same information about the general distribution of cost values in the group. (Assume that agents’ cost values are independently and identically distributed.)
An equilibrium is defined as a situation where all agents get the best possible payoff given the actions of all other agents. I.e., in equilibrium no one has an incentive to do anything different.
We can see that an equilibrium occurs if the agents follow the strategy of contributing if cost is below guesstimate, and if their guesstimate c’ is in fact correct.
I.e.: c’ = p
In this case all those with ci < c’* = p* will contribute, and as c’* = p* it is optimal for them to do so. All those with ci > c’ = p will not contribute, and this too is optimal for them.
Eg. in an election, everyone expects that there will be no more than a couple of percent between two candidates. People base their decision to vote or not on how likely they think it is that their vote could swing the election. Because the expectation is that numbers will be very close, large numbers turn out on both sides, and the result is in fact within one per cent. This situation corresponds to an ‘equilibrium’ in the model – voters’ expectations were justified by the actual turnout, and so they have made the ‘best’ decision according to their preferences.
When will such a situation occur, if at all? Is it meaningful in any way? See: