After comtemplating on Neouto’s post about why rational people vote, I would like to give out some opinions, though I am not sure if it is logically convincing enough. I try to explain voting is a rational decision from the point of view of game thoery.

First of all, let us consider voters as players in a game and they choose from their strategies of “go to vote" or “not go to vote." They make their decisions based on their preferences. Giver players are all rational, they prefer not going to vote to going to vote given the fact that the candidate they support wins. However, voters are fonder of that the candidate they support wins than of that the candidate they support loses. That is,

for every voter, candidate wins without voting > candidates wins with voting > candidate loses without voting > candidate loses with voting.

Note that the preference above implies that players are concerned abouth who is elected. However, this is not often the fact.

In turn, I would like to show that, though it has been very clear, people will go to vote if their vote is critical.

Assume 1)the cost of going to vote is 1. 2) the payoff is 2 if his candidate wins and is -2 if his candidate loses. 3)if the outcome is tie, the court judges who wins and the probability of winning the election for both candidates is 1/2. 4)there are only 2 candidates to be elected. 5) the number of supporters for each candidate is the same, that is, every vote is critical.

(I) supposed the number of players in support of each candidate is 1. (n=1)

Below lies the game in normal form.

** II**

| vote not vote

———————————————————–

**I** vote | **(-1,-1)** ( 1,-2)

not vote | (-2, 1) ( 0, 0)

The payoffs in the matrix can be derived from the assumptions given above. Some remarks are given below:

If playerI goes to vote and the playerII does not, the candidateI wins. Therefore, the payoff for I is 2 (payoff of candidateI winning) – 1 (the cost of voting) =1. The payoff for II is -2 (payoff of candidateII losing).

If both players go to vote, the result is pending and should be judged by the court, giving each candidate probability of 1/2 to win. Thus, the payoff for I is 1/2*2 (the expectation payoff of candidateI winning) + 1/2*(-2) (the expectation payoff of candidateI losing) -1 (the cost of voting) = -1. The same applies to player II.

Evidently, the game is a prisoner’s dilemma. The Nash Equilibrium predicts that the outcome will be both players go to vote.

(II) given the game of k players holds for the argument above, I would like to prove from the premise that the game of k+1 players holds.

Because every player goes to vote if there are k voters in support of each candidate, the (k+1)th vote is critical. Therefore, the (k+1)th player goes to vote as the prisoner’s dilemma predicts.

By mathematical induction (I hope I did not make any mistake in the argument given above), I conclude that everyone will go to vote if the election is intense enough.

Furthermore, I would like to argue that people will regard every election as intense because it raises the probability that a candidate wins by letting the supporters perceive that the election is intense, given that supporters are inclined to believe what candidates say. However, I don’t have enough knowledge to analyze this argument technically. (Perhaps I will refine my argument later when I am equipped with enough tools.)Therefore, I try to argue on basis of what I observe in the elections of Taiwan.

It is broadly observable that in almsot every campaign in Taiwan, no matter how big the difference of poll shows one candidate leads others, candidates cry out “搶救" (rescue) or consider the election 五五波 (fifty to fifty) in the last week of the campaign. Because if the leading candidate does not mobilize the supporters to vote, the second ruuner may win instead when he mobilizes his supporters. Therefore, it resembles a arms competition that each players pretend that he is going to lose if you don’t go to vote.

Although I can not provide a convincing proof that why rational people vote, taking voting as strategic behavior and, in my opinion, trying to explain it from the perspective of game thoery is a good starting point.

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