Question

Monitoring: An employee (player 1) who works for a boss (player 2) can either work (W) or shirk (S), while his boss can either monitor the employee (M) or ignore him (I). As in many employee-boss relationships, if the em- ployee is working then the boss prefers not to monitor, but if the boss is not monitoring then the employee prefers to shirk. The game is represented by the following matrix: Player 2 M 1 1,1 1, 2 w Player 1 s 0,2 2,1 a. Draw the best-response function of each player. b. Find the Nash equilibrium of this game. What kind of game does this game remind you of?

Answer 1

Solution: (a) First notice that there is no pure strategy Nash equilibrium. For mixed-strategies, do as follows: Let p be the probability that player 1 chooses W and q the probability that player 2 chooses M. It follows that 2 (W.9) > 2(5,9) if and only if 1 > 2(1-9), or q> and 12(P, M) > 1,(,1) if and only if p +2(1-P) > 2p+(1-P), or p < It follows that for player 1, 1 p=0 BR19) PE [0,1] if q= if q> 1 p=1 9=1 1 9=0 and for player 2, ifp BR2(P) = 39 € [0,1] if p = 3 if p > (b) From the two best response correspondences the unique (mixed) Nash equilibrium is (p,q) = 6,2). Notice that we could also calculate the mixed-strategy Nash equilib- rium without solving for best response correspondences, and instead just finding probability p that makes player 2 indifferent between M and l; and probability q that makes player 1 indifferent between W and S.

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