Question

Q21. A game theorist is walking down the street in his neighborhood and finds $20. Just as he picks it up, two neighborhood kids, Mark and Nancy, run up to him, asking if they can have it. Because game theorists are generous in nature, he says he’s willing to let them have the $20, but only according to the following procedure: Mark and Nancy are each to (independently) submit a written request as to their share of the $20. Let m denote the amount that Mark requests for himself and n be the amount that Nancy requests for herself. m and n are required to be chosen from the interval [0, 20]. If m + n ≤ 20, then the two receive what they requested, and the remainder (20 - m - n) is split equally between them. If, however, m + n > 20, then they get nothing, and the game theorist keeps the $20. Mark and Nancy are the players in this simultaneous-move game. Assume that each of them has a payoff equal to the amount of money that he or she receives. Find all Nash equilibria for this game.

Answer 1

Player 2
		I	A
Player 1	I	2,1 0,0 0,0 1,2
	A

To find the Nash equilibria, we examine each action profile in turn.

(I,I)

Neither player can increase her payoff by choosing an action different from her current one. Thus this action profile is a Nash equilibrium.

(I,A)

By choosing A rather than I, player 1 obtains a payoff of 1 rather than 0, given player 2's action. Thus this action profile is not a Nash equilibrium. [Also, player 2 can increase her payoff by choosing I rather than A.]

(A,I)

By choosing I rather than A, player 1 obtains a payoff of 2 rather than 0, given player 2's action. Thus this action profile is not a Nash equilibrium. [Also, player 2 can increase her payoff by choosing A rather than I.]

(A,A)

Neither player can increase her payoff by choosing an action different from her current one. Thus this action profile is a Nash equilibrium.

We conclude that the game has two Nash equilibria, (I,I) and (A,A).