VOTER RATIONALITYOana Carja

Daniel Kluesing

Sang Won Lee

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

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)

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

EXPECTED UTILITY OF VOTING

My Candidate Win (Gain – Loss)

My Candidate Lose ( – Loss)1-p

p

My Candidate Win (Gain)

My Candidate Lose (0)1-q

q

p(Gain – Loss ) + (1-p) ( –

Loss)

q(Gain)

Expected Utility

≥

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)

Obama McCain Undecided(# of eligible voter -1)

Poll result(Nov.3rd, 2008) 49.0% 49.3% 1.7%

Estimate of Actual voter 2,108,127 2,121,034 73,139 4,302,300

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.

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)

state P( the state is decisive)

Fl 0.000309368

MO 0.000841

IN 0.0007669

NC 0.0012406

OH 0.002245

CA 0.306723

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.)

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.

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

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.

(2008 U.S. Presidential Election Data)

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

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.