Question

Say that T and C each have 60 million dollars to spend in 3 primaries: Nevada, Hawaii, and California. Nevada has 10 delegates, Hawaii has 20, and California has 30. If a candidate spends more money in a state than his opponent, then he gets all the delegates. If the candidates spend equal amounts of money in a state, then they split the state’s delegates. For example, if T spends 5 million in Nevada, 15 million in Hawaii, and 40 million in California, which we can write as (5,15,40), and C spends (15,25,20), then T wins California and gets 30 delegates while C wins Nevada and Hawaii and gets 10 + 20 = 30 delegates.

a. Are there pure strategy Nash equilibria in this game? If so, find them.
b. Now say that T has 80 million dollars while C still has 60. Are there pure strategy Nash equilibria in this game? If so, find them.

Answer 1

(a) T spends 5 million in Nevada, 15 million in Hawaii, and 40 million in California, which we can write as (5,15,40), and C spends (15,25,20), then T wins California and gets 30 delegates while C wins Nevada and Hawaii and gets 10 + 20 = 30 delegates. Then there is no pure strategy Nash equilibria in this game.

(b) Now say that T has 80 million dollars while C still has 60 million dollars. Then T spends 20 million in Nevada, 30 million in Hawaii, and 30 million in California, which we can write as (20,30,30) and C spends (15,25,20). Then T wins California, Nevada and Hawaii and gets all the delegates and C loss. On the other hand, T spends 25 million in Nevada, 30 million in Hawaii, and 25 million in California, which we can write as (25,30,25). Here is also T wins and C loss.

Then there are two pure strategy Nash equilibria in this game, i.e., (20,30,30) and (25,30,25)