Voting: a game theory look

[hmkpoker]

There is also an incentive not to vote (or more accurately, a lack of incentive to vote) for many people, simply because they don't believe that their vote matters enough to be worth the effort. It's a combination of the unlikelihood of their vote mattering, the lack of value even in the unlikely event that their vote does matter (since your choice is between a douche and a turd sandwich), against the costs of voting. For some people the vote is worth it, for others it is not. Undoubtedly, voting becomes more valuable as votes become fewer, so low numbers should be more enticing to voters, and there's an equilibrium of sorts. But as we're seeing, that equilibrium is getting lower all the time.

[Butso]

lmao

[iron81]

Butso, for those of us who aren't well versed in game theory, would you explain yourself?

[NSchandler]

I really don't understand the premise of the problem to be honest, because I don't understand how nobody voting is collectively better than at least 1 person voting. This would make sense only if ALL political undertaking were rent-seeking and ALL politicians were equally adept at rent-seeking. In this case, voting for policy A over policy B is only a transfer from one group to another, and the best collective outcome is for nobody to undertake the costs (assumed to be >0) of voting. However, if this is the case, here is a game theoretic way to approach the problem.

The probability that your vote is decisize is 1/(sqr root (pi*n))*(4p-4p^2)^n, where everybody votes 1 way p percentage of the time and the other way (1-p) percentage of the time. There are 2n+1 voters.

In equilibrium, the chance that your vote is decisive needs to be equal to (costs of voting)/(value of changing the election). So, for example, if you would be willing to pay $1000 to have the election go your way with certainty and it costs $10 to vote, you need to influence the election 1/100 (=$10/$1000) in equilibrium. This makes you indifferent between voting and not voting because 99/100 times you lose $10 and 1/100 times you gain $990, for an EV=0.

I'll assume p=0.5 (this means that people are equally likely to vote the way you want as they are to vote the other way), what n do we need for you to be indifferent between voting and not voting? With these numbers if the number of voters is 6367 you'll be (roughly) indifferent between voting and not voting.

So, say there are 100,000 potential voters. The MSNE would be for each person to vote (roughly) 6.367% of the time (mathematically this not EXACTLY correct for reasons I won't get into, but it's very very close). That is, use a random number generator to come up with a number between 1 and 100,000 and if it comes up 6,367 or less, go vote. Otherwise, stay home. Given these numbers, that's the MSNE. As with all MSNE, you will be indifferent between the two choices here. The only reason you vote 6.367% of the time is to ensure that everybody else remains indifferent between voting and not voting.

But, as others have said, this is different from the prisoner's dilemma. For one, in the prisoner's dilemma you benefit by cheating regardless of the actions of the other players. In this example, you gain by cheating only if your vote is decisive. This tempers the proclivity to cheat. And second, it can only appear to be a prisoner's-dilemma-type payout with rather awkward assumptions (the ones I laid out in the first paragraph).

[xorbie]

if you want to give a "game theory look" you need to actually define a game.

[Dan.]

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Butso, would you explain yourself?

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[Dan.]

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if you want to give a "game theory look" you need to actually define a game.

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Personal attack deleted The game is a person's decision to vote or not. Funny how everyone else understood that no problem...

[xorbie]

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if you want to give a "game theory look" you need to actually define a game.

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Deleted The game is a person's decision to vote or not. Funny how everyone else understood that no problem...

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Pretty sure I'm not retarded, thanks for checking though. "...a person's decision to vote or not" does not a game. I'm not just being nitty for no reason, the results change completely based on the game.

The most basic model has full information and no communication. This is a poor represenation of the real world, and so of course results won't hold. Much more interesting models exist, and many predict non-zero voter turnout.

[xorbie]

To those who are interested in actually thinking about things from a game theory perspective, there are several things to understand. First of all, game theory often results in counter-intuitive results that aren't seen in real life, the reason being that utility is a very hard thing to quantify into a game setting.

The primary example is of course the prisoner's dillemna, and many such issues are resolved by using the concept of repeated play games. However, in one shot games (and you can make a good case for voting being a one shot game), results often run contrary to real life, because people are "irrational".

The reason that non-zero voter turnout is generally an issue is because there is uncertainty as to which side constitutes a majority. If everyone knew that there were 51% democrats, 49% republicans, there would be no incentive for any republicans to vote because they would have no chance of winning the election. Thus only a couple democrats would vote and that would be that.

The problem with saying that this is an equilibrium however is that "equilibrium" doesn't translate into "actually occurs". For example, consider a very simple game which is the model for coordination problems in general:

A guy and girl want to go out on a date. Both would prefer to be where the other one is, but the guy would prefer a sports game while the girl would prefer a concert. Neither is allowed to communicate with the other. In this case, the equilibria are for both to go the sports game or both to go the concert. However it's impossible to actually ensure that either case is brought to fruition because of the case of no communication.

In order to prevent communication from ruining the equilibrium, you would need every 49.1% democrat to 49% republican, so the democrats would have to coordinate for themselves. Or of course just all of them go out and vote, because coordination is more costly than just getting your fatass up to go cast a ballot.

In any case, what uncertainty creates is a situation in which you really don't know which side is going to win, so in this case there is a much bigger chance that you are actually going to pivotal. In this case, you go out and vote with higher frequency (this is actually born out in the fact that closer elections have higher turnout).

Of course since libertarians/green party/communists/whatever else know they have no chance of winning, they are less likely to show up and vote.