(Voter participation) Two candidates, A and B, compete in an elec- tion. Of the n citizens, k support candidate A and m (= n - k) support candidate B.
Each citizen decides whether to vote, at a cost, for the candidate she supports, or to abstain. A citizen who abstains receives the payoff of 2 if the candidate she supports wins, 1 if this candidate ties for first place, and 0 if this candidate loses.
A citizen who votes receives the payoffs 2 - c, 1 - c, and -c in these three cases, where 0 c 1.
a. For k = m = 1, is the game the same (except for the names of the actions) as any considered so far in this chapter?
b. For k = m, find the set of Nash equilibria. (Is the action profile in which everyone votes a Nash equilibrium? Is there any Nash equilibrium in which the candidates tie and not everyone votes? Is there any Nash equilibrium in which one of the candidates wins by one vote? Is there any Nash equilibrium in which one of the candidates wins by two or more votes?)
c. What is the set of Nash equilibria for k m?
If, when sitting in a traffic jam, you have ever thought about the time you might save if another road were built, the next exercise may lead you to think again.