Question

Two people have $10 to divide between themselves. They use the following procedure. Each person names a number of dollars (a nonnegative integer), at most equal to 10. If the sum of the amounts that the people name is at most 10, then each person receives the amount of money she named (and the remainder is destroyed). If the sum of the amounts that the people name exceeds 10 and the amounts named are different, then the person who named the smaller amount receives that amount and the other person receives the remaining money. If the sum of the amounts that the people name exceeds 10 and the amounts named are the same, then each person receives $5. Determine the best response of each player to each of the other player’s actions; plot them in a diagram and thus find the Nash equilibria of the game.

Answer 1

The best response for player 2 can be stated as:
(where X1 equals the dollar that a person names and Y2(X1) being the amount the person receives)

X1	Y2(X1)
0	10
1	9,10
2	8,9,10
3	7,8,9,10
4	6,7,8,9,10
5	5,6,7,8,9,10
6	5,6
7	6
8	7
9	8
10	9

Th best responses for player 1 would be the same.

Nash equilibria is the set of strategies that every person forms given no person has any incentive to change. Hence, we can say that there are 4 Nash equilibria: (5,5) , (5,6) , (6,5) , (6,6)