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Voter Rationality

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  1. Voter Rationality Oana Carja Daniel Kluesing Sang Won Lee

  2. a rational individual will not vote • Economically, reward for voting is small. • There is almost no chance that one voter will swing the entire election. • Using probability model we will show • Irrational behavior of voters • Probability that my vote is decisive • Relation of turnout and swing state

  3. Expected Utility of voting • Assumption • People will vote only when they expect payoff greater than the payoff of not voting. • Variables • Gain : What you get when your candidate wins election. • Loss: Time/Effort spent on voting • p : P(My Candidate wins | I go to vote) • q : P(My Candidate wins | I don’t go to vote)

  4. Expected Utility of voting Expected Utility • Given that I go to vote. • Given that I don’t go to vote • To vote, one should expect more. p(Gain – Loss ) + (1-p)( – Loss) ≥ q(Gain)  (p-q) Gain ≥ Loss p My Candidate Win (Gain – Loss) p(Gain – Loss ) + (1-p) ( – Loss) My Candidate Lose ( – Loss) 1-p ≥ q My Candidate Win (Gain) q(Gain) My Candidate Lose (0) 1-q

  5. Ex) an individual in a swing state, Missouri • Assumption • Poll Result on Nov. 3rd will be the actual fraction of voters. • Undecided voter works as independent 50-50 coin flip. • The target voter supports for Obama. • In this situation, Obama need to get 43,203 from undecided voter to make tie. The number of undecided voter who actually voted for Obama, denoted X, follows B(73139, 0.5) • By central limit theorem, normal approximation is used. B(73139,0.5) ≈ N( 36569.5 , 135.22)

  6. Ex) an individual in a swing state, Missouri • p : P(Obama wins | I go to vote) = P( at least tie without my vote) = P( X ≥ 43,023 ) • q : P(Obama wins | I don’ go to vote) = P( Obama win without my vote) = P (X ≥ 43,204 ) • With normal approximation, P( X ≥ 43,023 ) = 0 P( X ≥ 43,024 ) = 0 • In the original equation, (p-q) Gain ≥ Loss since (p-q) is ZERO, LHS of the equation is zero regardless of personal gain. • It’s irrational to go to vote, even in the swing state like Missouri.

  7. Probability of Casting the Decisive Vote • The probability of having a decisive vote in the election equals the probability that your state is necessary for an Electoral College win, times the probability that your vote is decisive in your home-state P(decisive vote in the entire election) = P(your home state is decisive ) × P(your vote is decisive in your home state) • P( the state is decisive) • The probability that your home-state’s electoral votes are necessary for your candidate winning is: • P(|Oev - Mev|<E) +1/2P((|Oev - Mev|=E) • (P for some states are shown in the table.)

  8. Probability of Casting the Decisive Vote • P(your vote is decisive in your home state) = P(tie in your home state | you do not vote) • Two method in calculation of P(tie). (CASE 1) Binomial distribution for undecided voters, with a free parameter p of voting for Obama (CASE 2) Binomial distribution for all voters, with parameter equal to the fraction of Obama voters • Even with a relatively small election of 1000 voters, the probability of casting a decisive vote is small.

  9. Probability of Casting the Decisive Vote • P(your vote is decisive vote in the entire election) • Combining previous two probability • Even for a voter in a swing state like Florida, probability of casting a decisive vote on the national scale is essentially zero for any reasonable national election. Even in close elections like in 2000

  10. Implications for turnout • From the equation, a voter turn out when (p-q) Gain ≥ Loss • (p-q) represents “How much my vote matters?” • My vote matters more in swing states than safe states. Equation says an individual in swing states more likely to go to vote than one in the safe states. • In the 2008 presidential election, the safer the state was, the less people turn out to vote. It showed negative dependence. • ρ(X,Y) = - 0.25 (2008 U.S. Presidential Election Data)

  11. Summary and Conclusion • A rational voter should not turn out to vote as their expected reward is negligible • One voter cannot swing the election even in the swing states. • Those who lives in swing states are more likely to turn out. • Require other method to explain voter behavior • Considering civic duty, a sense of patriotism • Finding Equilibrium – If no one turn out because it’s irrational, I have chance to swing the election.

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