Question

Consider the following game, which might model the “Split-or-Steal” game show. Two players simultaneously choose whether to split or steal. If they each choose to split, they each get $50. If one chooses steal and the other chooses split, then the stealer gets $100 and the splitter gets $0. If both choose steal, they each get $0.

(a) Assume the players care both about their own monetary earnings and the amount of inequality between their earnings: for a pair of actions (a1, a2) that results in monetary payoffs of $x and $y to players 1 and 2 respectively, player 1’s payoff is 3x − 2|x − y| and player 2’s payoff is 3y − 2|x − y|. For example, when one player chooses split and the other chooses steal, the payoff for the player who chooses split is −200。Represent the game with a payoff matrix.

(b) What is player 1’s best response if player 2 chooses steal? What is player 1’s best response if player 2 chooses split?

Answer 1

This game is very easy to present in the form of a payoff matrix. Payoffs for the players differs from their monetary earnings because they also care about the amount of inequality between their earnings, as mentioned. Thus, let's first calculate the payoffs under different circumstances:

If player 1 chooses to split and player 2 also chooses to split, player 1's payoff = 3(50) - 2|50 - 50| = 150.

If player 1 chooses to steal and player 2 chooses to split, player 1's payoff = 3(100) - 2|100 - 0| = 100

If player 1 chooses to split and player 2 chooses to steal, player 1's payoff = 3(0) - 2|0 - 100| = -200

If player 1 chooses to steal and player 2 chooses to steal, player 1's payoff = 3(0) - 2|0 -0| = 0

The same holds for player 2, as his payoff function is identical to player 1's payoff function. The payoff matrix for this game is given below:

(a)

	Player 2 Splits	Player 2 Steals
Player 1 Splits	(150, 150)	(-200, 100)
Player 1 Steals	(100, -200)	(0, 0)

(b) From the matrix above, we can see that if player 2 chooses to steal, player 1's best response is to play "steal", since this strategy minimizes its loss ( 0 > -200)

If player 2 chooses to split the money, player 1's best response is to play "split", since this strategy maximizes its payoff (150 > 100).