# Game theory 100 voters

a. Why is it not a Nash equilibrium for everybody to vote?
b. Why is it not a Nash equilibrium for nobody to vote?
c. Find all the pure strategy Nash equilibria in this game?

If everybody votes, then the liberals have a payoff of $-11\$ $, so they can stay home and get$-10\ instead, improving their payoff. Thus this is not a Nash equilibrium.

If no one votes, then everyone has a payoff of $0\$ $. But then if any person goes to vote, they improve their payoff to$9\, so again this is not a Nash equilibrium.

In fact, no Nash equilibrium in pure strategies exists. Suppose there is one. If the outcome is a tie, then since the villagers are split $51/49$, there must be someone who didn't vote (and thus gets a $0\$ $payoff). If they go vote instead, their candidate wins and they improve their payoff to$9\, so this can't be a Nash equilibrium.

Suppose a candidate wins. If the opposing candidate got at least one vote, then whoever voted for the losing party gets $-11\$ $. If they instead don't vote, they get$-10\, so this is not a Nash equilibrium either.

Finally, suppose a candidate wins, and no one voted for the opposing candidate. If the winning side got at least two votes, then either of the voters could improve their payoff from $9\$ $to$10\ by not voting, so this again is not a Nash equilibrium. If the winning side got a single vote, then any villager of the opposing party could go vote and improve their payoff from $-10\$ $to$-1\, so this can't be a Nash equilibrium either.

The list is exhaustive and so no pure strategy Nash equilibrium exists.

• Thank you for the clear answer