Voting: a game theory look

[Poofler]

Why do they think no one vote matters?

If it's because your one vote won't decide an election, you esentially are describing the current state of the world where we vote out of civic duty. And without civic duty, the cheating incentive would run up the numbers past just Bob and his neighbor.

If it's because they do not recognize the legitamacy of the process, that specific reason why it is illegitimate matters. If the culprit is rampant corruption of results, invalidating the votes, there really is no incentive, and I don't see how anyone votes. The action would be meaningless. If all voters believe no candidate was different from another candidate, then I see no incentive to vote. But they would still abide by however the existing government decided the election, as they abstained out of indifference and utility, not out of disgust for government.

[Dan.]

Your post is better directed at Borodog's post. I'm just trying to explain, through game theory, why people vote.

[JaredL]

OP,

The voting problem is a different one from the prisoners' dilemma, which is often used to describe cartel behavior with incentives to cheat. The difference is that in the case of the cartel, you always have the incentive to cehat. If we have an agreement to keep the price high and quantity low, you have the incentive to undercut me if our prices are above equilibrium.

In the voting game, your best action depends on the actions of others. In a standard voting game, where voting is costly though the incentives are such that you would always vote if you knew that you would be the swing voter, there is no pure strategy equilibrium (in other words no situation where everyone is voting or not voting with certainty) except in 1 special case.

To see this, suppose there is an equilibrium such that everyone is either voting or not voting with certainty and there is at least one person not voting for candidate A (who is running against B). There are these cases:
- A wins by more than 1
- A wins by 1
- tie
- B wins by 1
- B wins by more than 1

Clearly any candidate winning by more than 1 can't happen in a pure strategy equilibrium. Similar reasoning indicates that no candidate can win by 1 - anyone voting for the loser would rather not vote given what everyone else is doing. In the case of a tie it is assumed that an A nonvoter would rather vote if they increase their candidate's chance from .5 to 1 of winning.

So the only possible equilibrium in pure strategies would have all A voters voting and a perfect tie. Obviously this is symmetric. So the only way a pure strategy equilibrium in this game exists is if the number of voters supporting each candidate is the same. The equilibrium is for everyone to vote. I'm assuming here that ties are broken via coin toss. If it's known ahead of time which candidate wins in a tie, there is no pure strategy equilibrium no matter how much support there is on each side.

There is obviously a mixed strategy equilibrium. Everyone on side A votes with the same probability that is a function of the number of A voters. Reverse for B. As usual, the probabilities are setup so that every voter is indifferent between voting and not voting. I'm assuming everyone faces the same cost here. It gets a bit more complicated with heterogeneous costs.

In equilibrium the probability of voting is a decreasing function of the number of people on your side. This goes to zero as the number of people gets large.

Not sure if this is what you were looking for.

[JaredL]

[ QUOTE ]
What's the original incentive NOT to vote? So that no one votes, as a possibility. I didn't see that.

[/ QUOTE ]

In economics or political science models the standard assumption is that voting is costly. For example, you have to go to the place. It takes time, and can actually cost money if you have to leave your job to do it, though this isn't usually the case.

I suspect it's some sort of protest nonsense in his case. Not voting has some protest value to him so by voting he would lose that.

[Dan.]

Definately an excellent analysis. I only disagree with this:[ QUOTE ]
Similar reasoning indicates that no candidate can win by 1 - anyone voting for the loser would rather not vote given what everyone else is doing.

[/ QUOTE ]
I think a certain percentage of the population would still vote for a loser even if they knew they would lose (ie, voting for a third party that has no chance), since voting, even for the loser, can alter future games.

[JaredL]

[ QUOTE ]
Definately an excellent analysis. I only disagree with this:[ QUOTE ]
Similar reasoning indicates that no candidate can win by 1 - anyone voting for the loser would rather not vote given what everyone else is doing.

[/ QUOTE ]
I think a certain percentage of the population would still vote for a loser even if they knew they would lose (ie, voting for a third party that has no chance), since voting, even for the loser, can alter future games.

[/ QUOTE ]

Sure, everything in there is assuming a one-shot game. Obviously that's not realistic. There are other things that aren't realistic. For example, I had no clue whatsoever how many people there even are in the state of Pennsylvania much less knowing how many like or dislike Santorum.

You can take care of the problem you mention by simply getting rid of those people. Only consider those who would rather not vote for a losing candidate. The election won't start out 0-0 but that doesn't affect anything.

[Poofler]

Jared, just to clarify, the equilibrium scenarios suppose every voter knows how many voters are in the game, and for which side?

[JaredL]

[ QUOTE ]
Jared, just to clarify, the equilibrium scenarios suppose every voter knows how many voters are in the game, and for which side?

[/ QUOTE ]

Yes.

Assumptions:

1. Players know number of voters on each side.
2. Each voter would rather vote if that would make their candidate win or tie. They would rather stay home otherwise.
3. Each voter faces the same cost to voting.
4. The above is common knowledge among voters.

[valenzuela]

Im sure Borodog means something like this.

+EV of voting: 0,001(because the chances that your vote will matter is really small)
-EV of voting: -4 ( you have to go the voting place, you usually have to wait, etc.)

0,001 -4= -3,999.



Thereby voting is -EV.

Btw I agree with him, but Im more into pvn argument that voting is telling others what to do.

[Butso]

virtually all the game theory arguments made in this thread are wrong.