Voting: a game theory look
 

Borodog's thread inspired me, but I decided to create a new thread to discuss it.

Basically, it's possible that no one vote at all. However, if no one votes, a single person has incentive to "cheat" the no-one-vote agreement, since his or her vote would be the only one counted, so whatever he or she says goes. Now, if one person cheats, another then also has incentive to cheat, and the result is a landslide where a whole slew of people go out and vote, since they all are given incentive to vote. It's a very simple game theory problem relating to cartel agreements, quite common in microeconomics.

Re: Voting: a game theory look
 

Such an argument might hold, but in only holds for a most a small number of voters.

Additionally, this argument ignores issues exterior to the game rules, such as whether or not people are not voting because they do not recognize the legitimacy of the process itself. If literally nobody is voting because they don't believe in the legitimacy of the voting system, one person really does not have an incentive to vote and decide everything, because nobody will abide by the results anyway.

Re: Voting: a game theory look
 

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Such an argument might hold, but in only holds for a most a small number of voters.

Additionally, this argument ignores issues exterior to the game rules, such as whether or not people are not voting because they do not recognize the legitimacy of the process itself. If literally nobody is voting because they don't believe in the legitimacy of the voting system, one person really does not have an incentive to vote and decide everything, because nobody will abide by the results anyway.

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Possibly, though every non-voter who lacks faith in the system must be 100% certain that no one will abide by the results. If they perceive there to be even a 1% chance the results hold and the decision of the few voters is enforced, there becomes incentive.

Re: Voting: a game theory look
 

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Such an argument might hold, but in only holds for a most a small number of voters.

Additionally, this argument ignores issues exterior to the game rules, such as whether or not people are not voting because they do not recognize the legitimacy of the process itself. If literally nobody is voting because they don't believe in the legitimacy of the voting system, one person really does not have an incentive to vote and decide everything, because nobody will abide by the results anyway.

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Possibly, though every non-voter who lacks faith in the system must be 100% certain that no one will abide by the results. If they perceive there to be even a 1% chance the results hold and the decision of the few voters is enforced, there becomes incentive.

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All you need is for most people to be fairly certain that most people will not abide by the results. People do communicate, you know.

Re: Voting: a game theory look
 

I don't see how knowing AND TRUSTING (which is another assumption of yours) that most people will not abide erradicates incentive in the event that...

Re: Voting: a game theory look
 

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I don't see how knowing AND TRUSTING (which is another assumption of yours) that most people will not abide erradicates incentive in the event that...

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Your friend Bob elects himself King. Do you genuflect and start paying him taxes?

Re: Voting: a game theory look
 

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I don't see how knowing AND TRUSTING (which is another assumption of yours) that most people will not abide erradicates incentive in the event that...

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Your friend Bob elects himself King. Do you genuflect and start paying him taxes?

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Can you define the current state of this world? I took OP's post to assume we had a couple of candidates, known countrywide, and a complete and sudden disdain/lack of faith in the election process. From this I presume a small, but sizable population actually vote for the candidates. I don't vote for Bob, I vote for one of the candidates. I don't presume I will be the only one to doubt with certainty the committment of the voters to abstain, or the ability of the elected with exsiting infrastracture to force the will, no matter how small, upon the people.

Are you imagining the desert island economics example?

Re: Voting: a game theory look
 

No, I'm imagining the scenario in the OP, i.e. where it is likely that only a single person is voting. Such a scenario does not exist in a vacuum. The only reason I can think of for nobody voting is that nobody believes in voting, or that the results of the vote are legitimate.

Re: Voting: a game theory look
 

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

Re: Voting: a game theory look
 

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

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The assumption Borodog set forth is that no one person's vote matters, so if everyone beleives this, then no one votes. And so this experiment just begins with the assumption that no one votes because the see no value in it.

Re: Voting: a game theory look
 

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.

Re: Voting: a game theory look
 

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

Re: Voting: a game theory look
 

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.

Re: Voting: a game theory look
 

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

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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.

Re: Voting: a game theory look
 

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.

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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.

Re: Voting: a game theory look
 

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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.

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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.

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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.

Re: Voting: a game theory look
 

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

Re: Voting: a game theory look
 

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

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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.

Re: Voting: a game theory look
 

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.

Re: Voting: a game theory look
 

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

Re: Voting: a game theory look
 

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.

Re: Voting: a game theory look
 

lmao

Re: Voting: a game theory look
 

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

Re: Voting: a game theory look
 

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).

Re: Voting: a game theory look
 

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

Re: Voting: a game theory look
 

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Butso, would you explain yourself?

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Re: Voting: a game theory look
 

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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...

Re: Voting: a game theory look
 

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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.

Re: Voting: a game theory look
 

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.