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Suppose one is voting to decide some political proposal or elect some office, and there's not enough information, (or the information is contrary), to make a confident decision. Perhaps the proposal is a law that's hundreds of pages of dense text, and there's not enough time to read it. Perhaps the candidates are untrustworthy, but one might be worse, yet even that's unclear.

When it is it better to pass, (not vote), and when is it better to guess? Also, when is even the choice of whether to pass or guess uncertain?

Some assumptions:

  • The voter is currently not registered in a political party.
  • No communication is possible, but voters can observe the results of the vote.
  • This will be an iterated game (there will be multiple rounds of voting).

Note: This is not a question about specific candidates or proposals, nor a broad question about many candidates and proposals. It's more of a game-theory question relevant to politics.

Some readers may wonder how guesses could possibly be worthwhile. I'd compare it to a wire dilemma, where doing nothing is as bad as cutting the wrong wire -- in which case a guess always offers better odds than doing nothing. Or like in Jan-ken where an expert player can trounce a beginner, but not if the beginner moves randomly. In roguish political contests voters may be expertly read by campaigners who stay several steps ahead, so that (as with Jan-ken) the voters' own strategies and biases will be employed against them, but making no choice leaves the choice to other outmaneuvered voters.

  • @indigochild sorry. I did! I was thinking 'gamification'. Something entirely different. – user1530 Mar 10 '17 at 17:57
  • @indigochild - I don't think so. There's almost never a need for it. – Bobson Mar 10 '17 at 18:07
  • I don't think "how to vote" is on topic but I'm open to opposition on that matter. – user10303 Mar 11 '17 at 14:35
  • @user30031, Not "how to vote", whether to vote ignorantly, the pros and cons of guessing vs. not voting. – agc Mar 11 '17 at 15:23
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    I think the question is on topic, but that game theory doesn't translate to real-life politics very well. – Bobson Mar 12 '17 at 4:19
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The voter would only want to vote when they are the marginal voter. If they don't know when they are the marginal voter, their optimum strategy is to vote in their first election, and then vote based on the turnout in the last election. If either candidate wins by a comfortable margin, there is no reason to vote (their vote wouldn't have mattered). Otherwise their vote may change the outcome. If they are uncertain about the payoffs from any particular candidate, they will need to learn over time by voting and observing how well various candidates do for them.

Utility

What does our theoretical voter's utility look like? The graph below shows how their utility changes based on the percentage of voters who vote for their position (for example, when X=0.05 the voter's candidate received 5% of the vote in an election).

enter image description here

It's a basic step-wise function. When their candidate receives less than half of the total votes, the voter receives a negative utility (the green line below is below the y-axis). When their candidate wins the election, they receive a positive utility. The amount of the utility does not increase based on exactly what share of votes are received: 50%+1 is just as good as 100%. I'm pretty ambivalent about what happens at exactly at 50%, but for the chart I called that a loss (50%+1 is required to win).

Under this function, it should be clear that the voter will want to make sure they receive the positive utility and not the negative utility. So how do they do that?

Marginal Utility

A marginal utility function describes the increase in utility from a certain action. The chart below shows how each additional percentage of votes received increases our voter's utility:

enter image description here

The marginal utility is always 0, except when increasing from 50% to 50%+1. This is derived straight from the first chart (a marginal utility function is the first derivative of utility function).

What does all that mean? Our voter should only vote when they are the marginal voter. That is, when their vote would break a tie in favor of their candidate.

At this point it's worth noting that since our voter is unregistered, there is actually a cost (disutility) to voting: they have to register to vote. They might incur other costs like becoming informed about issues, locating their polling place, meeting party members, etc. Since the cost is a simple constant, it just decrease the utility by some set amount every election.

The Iterated Game

Elections will happen with some frequency. Although in any one election our voter should only vote when they are the marginal voter, they really wish to maximize their returns over a long period of time (not just win a single election). How does this change things?

Normally in this situation the solution would be too coordinate with other voters. However, our voter can't communicate - they can only observe the outcome of elections. In this situation, our voter will have to learn from past elections.

Although game theory would assume that our voter knows exactly when they are the marginal voter, in practice no one would ever know this. The voter would vote or not vote in a given election, and then observe what voter turnout and election results accrue. If their either side wins by a land-slide, they probably don't need to vote next time (their vote wouldn't have changed anything). If the election is reasonably close, the voter may decide to vote in the next election. How close is "reasonably close" will depend on the voter's risk tolerance (how much uncertainty are they comfortable with?).

What's the best strategy? Political scientist Robert Axelrod discussed this in his famous book, The Evolution of Cooperation. Axelrod hosted a large iterated prisoner's dilemma tournament where people could submit different strategies. The winner was tit-for-tat: the voter should vote in their first election. From their choices are based on the last round of voting. If their side did not win by a comfortable margin in the last election, they will show up to vote in this election. Otherwise, they will stay home.

Uncertainty about results

Finally, let's incorporate the possibility that the voter doesn't know exactly how to evaluate candidates. Basically we are saying that the voter doesn't know how high on the y-axis the green lines are. The good scenario could be very high up, or it could be very low (even negative!). The worst scenario could be very close to the best scenario, or they could be worlds apart.

This is another big deviation from normal game theory: typically we would assume that the voter has an unbiased and complete knowledge of everything in the game.

How this is handled would depend on some specifics of our model. Currently the voter has no information about the candidate. However, if we allow the voter to observe some characteristics of the candidate prior to election (say, through public debates, their Twitter feed, etc.) than the voter will be able to make some judgments about the candidate.

This is taking us outside of game theory, but over multiple elections the voter will learn to associate some characteristics of the candidate with the utility they gain. Perhaps instead of a static value the voter has some idea of the distribution of potential benefits, which they update after each election (as in a bayesian game).

Without this prior information, the voter would have no way to discern

  • Thanks, I like game theory too, so +1 there, but this answer seems a bit general. Assumptions of the question I should probably clarify, (or ask another question): Iterated voting is a given, but communication (apart from the vote itself) is not. Let's assume the voter is unregistered and lives in a remote lighthouse, which would simplify the various party-oriented strategems. – agc Mar 10 '17 at 22:56
  • @agc - I added those assumptions to the question and edited my answer. – indigochild Mar 11 '17 at 4:13
  • Thanks for the OP edit, one minor bit, by "unregistered" I'd meant the voter is "not registered in a party", (i.e. not a registered R, D, L, G, etc.), but is registered to vote. Which distinction shouldn't affect your current revision much. – agc Mar 11 '17 at 6:41
  • Hmm, the fact that this question has attracted an in-depth answer on game theory reinforces my hunch that it's off-topic. – user10303 Mar 11 '17 at 14:36
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    @user30031 - Gotcha. A question can be on-topic for multiple SE sites. Voting is on-topic here. Game theory (in this case) is just the analytical tool to answer the question. Other questions use case studies, statistics, or ethical theory. The community will agree or disagree. If it's closed by virtue of being off-topic, so be it. – indigochild Mar 11 '17 at 19:12
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Indigochild's answer supposes the uninformed voter is interested in "winning", i.e. his chosen candidate or proposal wins, and does things that benefit the voter.

This answer is an attempt to sketch some options of uninformed voters who don't so much care about personal benefits, (let's suppose they'll sturdily endure most any ensuing political weather), but who are nevertheless interested in what's best for the nation.

  1. One school of thought is that uninformed voters should never vote. Some radical part of that school would want prospective voters to pass tests showing themselves to be well informed before being permitted to vote. The most radical/criminal fringe of that school would support preemptive disenfranchisement for selected groups that can't pass their tests, (notorious examples: the 1965 Alabama Literacy test & the 1964 Louisiana Literacy test); for them voter suppression seems like virtuous social planning.

  2. Legislators themselves are often poorly informed. Congressmen often are faced with impossibly long bills that are fast tracked, for which there is no time to read before voting upon, (e.g. Patriot Act, et al). Apparently some congressmen evaluate these unread laws based on peer pressure, word of mouth, and by reading summaries, abstracts, and reviews -- similar to how people select books and movies, by reading book covers, the inside flaps, book reviews, or watching movie trailers, reviews, and posters, and so forth.

  3. Not voting allows others who do vote to decide things. If those other voters are known to be well informed and benevolent, the non-voter needn't fret.

  4. Not voting needn't guarantee that those who do vote are well informed or benevolent. Those who do vote might be ignorant or malicious. If the ignorant and malicious are known to reliably vote in their nation's worst interests, and it's known both who is malicious and how they'll vote, this gives other uninformed voters better odds on choosing well -- simply by voting the opposite of the malicious voter's preference.

  5. If a lot of uninformed people vote at random, their votes will tend to cancel each other out.

  6. Random voting is trickier than it seems. Suppose there's three candidates, {A,B,C}, random voter Smith assigns 1/3 odds to each. Another random voter Jones thinks, "that leaves out a choice, that of not voting", which we'll call 'X': {X,A,B,C}. Both X and to not vote are equivalent to voting with the majority. Suppose the other voters favor A, therefore Jones has a random pick between {A,A,B,C} -- since there's two ways to pick A out of four, random voter Jones then has a 50% chance of picking A.

  • About point #6 - In game theory, all actors are rational. There is no possibility that a voter votes randomly, it must be done according to their ranking of preferences. – indigochild Mar 13 '17 at 17:06
  • Second concern - You are incorrectly characterizing my answer. In a game, every actor rationally pursues some goal (given by their utility function and order of preferences). Every actor must do this, or it isn't a game. However, I don't assume that a voter wants things that are best for themselves. They could order their preferences according to some kind of social good and still be rational. – indigochild Mar 13 '17 at 17:11
  • @indigochild, Re #6: It can be rational to guess, e.g. a wire dilemma. GT with random aspects is not novel, e.g. GT informed strategies of Poker. – agc Mar 13 '17 at 18:07
  • A guess is not random. It still conforms to the notion of a ranked-order preference. The idea of people voting randomly in your answer does not, because it suggests (to me, at least) that they do not have a well-ordered set of preferences (because you've assigned them all equal probabilities). – indigochild Mar 13 '17 at 18:14
  • @indigochild, re "every actor must": thanks for the clarification -- would your favored definition of an abstract game require actors' goals to be a single shared utility function and preference order, or does a game permit several ordered rankings, or even several functions? – agc Mar 13 '17 at 18:26

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