PROBLEM SET #2 – STRATEGIC CHOICE IN TWO-PLAYER GAMES:

ANSWERS AND DISCUSSION

1.         Answer the following questions pertaining to the two-player zero-sum game depicted in the payoff matrix below. Briefly explain each of your answers. (The row Player 1 has four strategies; the column Player 2 has three strategies. The number in each cell is the payoff to Player 1; the payoff to Player 2 is the negative of the number — that is, P1 wants to maximize ands P2 wants to minimize, the payoff.)

 Player 1

                          Player 2

Remember: The game above is zero-sum (as are the next two games), so P1 is maximizing and P2 is minimizing with respect to the payoffs shown in the matrix.

(1)       Does either player have dominated strategies?

            Yes, both.

For P1 (aiming to maximize), s2 is dominated by s1, s3, and s4, and s1 is dominated by s3; s3 and s4 are undominated.

            For P2 (aiming to minimize), c1 is dominated by c2 and c3 is dominated by c2.

(2)       Does either player have a dominant strategy?

            P1 does not (s3 and s4 are both undominated, so nothing can be dominant).

            P2 does have a dominant strategy c2 (as noted above).

(3)       What is Player 1’s maximin strategy?

            s3, which guarantees a payoff to P1 of no less than 3.

(4)       What is Player 2’s minimax strategy?

            c2, which guarantees a payoff to P2 of no more than 3.

(5)       Does the game have an equilibrium outcome?

Yes, (s3, c2) is an equilibrium/saddlepoint (3 is both the maximum in the column and minimum in the row).

(6)       Is this zero-sum game strictly determined?

            Yes, because maximin [for P1] = minimax [for P2] = 3

2.         Answer the following questions pertaining to the two-player zero-sum game depicted in the payoff matrix below. Briefly explain each of your answers. (The row Player 1 has four strategies; the column Player 2 has three strategies. The number in each cell is the payoff to Player 1; the payoff to Player 2 is the negative of the number — that is, P1 wants to maximize ands P2 wants to minimize, the payoff.)

Player 1

                          Player 2

(1)       Does either player have dominated strategies? No

(2)       Does either player have a dominant strategy?

            No [couldn’t be otherwise given the answer to (1)]

(3)       What is Player 1’s maximin strategy?

            s2, which guarantees a payoff to P1 of no less than 2.

(4)       What is Player 2’s minimax strategy?

            c2, which guarantees a payoff to P1 of no more than 2.

(5)       Does the game have an equilibrium outcome?

Yes, (s2, c2) is an equilibrium/saddlepoint (2 is both the maximum in the column and minimum in the row).

(6)       Is this zero-sum game strictly determined?

            Yes, because maximin [for P1] = minimax [for P2] = 2

3.         Answer the following questions pertaining to the two-player zero-sum game depicted in the payoff matrix below. Briefly explain each of your answers. (The row Player 1 has four strategies; the column Player 2 has three strategies. The number in each cell is the payoff to Player 1; the payoff to Player 2 is the negative of the number — that is, P1 wants to maximize ands P2 wants to minimize, the payoff.)

Player 1

                          Player 2

(1)       Does either player have dominated strategies?

            None for P1.

For P2, c1 is dominated by c2.

(2)       Does either player have a dominant strategy?

            No (while c2 dominates c1, c2 does not dominate c3).       

(3)       What is Player 1’s maximin strategy?

            s3, which guarantees a payoff to P1 of no less than 2.

(4)       What is Player 2’s minimax strategy?

            c3, which guarantees a payoff P1 of no more than 3.

5)        Does the game have an equilibrium outcome?

No. In particular, (s3, c3) is not an equilibrium/saddlepoint (3 is not the minimum in the row).

(6)       Is this zero-sum game strictly determined?

No, because maximin [for P1] = 2 < minimax [for P2] = 3. Thus both players should use mixed strategies. P2's optimal mixed strategy will put zero probability on c1 (because it is dominated) and P1's optimal mixed strategy will put zero weight on s1 and s2 (because they are “sequentially dominated,” i.e., they are dominated by s4 and s3 respectively given that the P2 will not use his dominated strategy c1).

4.         Answer the following questions pertaining to the (variable-sum) game depicted in the payoff matrix. Then briefly explain each of your answers. (Each player has just two strategies. The number in lower-left corner of each cell is the payoff to Player 1; the number in the upper-right corner of each cell is the payoff to Player 2. Each player is trying to maximize his payoff.)

Player 1

                        Player 2

Remember. This game is non-zero-sum; each player is maximizing his own payoff (lower left for P1 and upper right for P2). (The players are not attempting to maximize their payoff margin over their opponents; this would turn the game into a zero-sum-game — and would imply that players in Chicken are indifferent between both swerving and both crashing.)

1)        What do you expect the outcome of the game to be if the players must make their strategic choices simultaneously (not knowing what choice the other is making)?

P1 does not have a dominant strategy but P2 does have a dominant strategy in c1. Thus P1 can expect P2 to choose c1; with that expectation, P1’s best reply is s1. Thus outcome should be 5,3. Note: maximin is not in general a reasonable principle of choice in a non-zero-sum game (because your opponent is not necessarily “out to get you”).

(2)       What do you expect the outcome of the game to be if the players make their strategic choices sequentially, with Player 1 moving first and Player 2 second?

P1 needs to look ahead and reason back (Dixit & Nalebuff). P1 thinks: if I choose s1, what will P2 choose? and if I choose s2, what will P2 choose? P2’s best reply to s1 is c1, giving outcome 5,3 -- i.e., a payoff of 5 for P1. P2’s best reply to s2 is also c1, giving outcome 3,5 -- i.e., a payoff of 3 for P1. (c1 is P2's best reply to both of P1's choice precisely because c1 is dominant.) Since P1 can expect a higher payoff from s1 than from s2, P1 chooses s1. P2 replies with c1, so the outcome is 5,3 (as in the simultaneous move game).

(3)       What do you expect the outcome of the game to be if the players make their strategic choices sequentially, with Player 2 moving first and Player 1 second?

Now P2 needs to look ahead and reason back. P2 thinks: if I choose c1, what will P2 choose? and if I choose c2, what will P2 choose? P1’s best reply to c1 is s1, giving outcome 5,3 -- i.e., a payoff of 3 for P2. However, P1’s best reply to c2 is s2, giving outcome 3,4 -- i.e., a payoff of 4 for P2. Since P2 can expect a higher payoff from c2 than c1, P2 chooses c2. P1 replies with s2, so the outcome is 3,4. (So P2, unlike P1, gains by going first. However P1 may be able to neutralize this advantage, as noted in the next question.) Note that c1, though dominant in the simultaneous play game, is no longer dominant in the sequential play game, since P2 (by moving second) can counter it with a contingent strategy (s1 if c1 and s2 if c2).

(4)       Continuing to suppose that the players make their strategic choices sequentially, with Player 2 moving first and Player 1 second, could Player 1 communicate any pre-play message to Player 2 that might improve the outcome for Player 1.

If P1 can communicate to P2 an absolute commitment to choosing s1, this will compel P2 to choose c1 to get a payoff of 3 rather than 2 and will gain P1 5 rather than 3. Note that all that is required of P1 is a simple (unconditional) commitment, not a threat or promise.

Last year, one student suggested a more treacherous bargaining strategy for P1: invite P2 to choose c1 rather than c2, while promising to choose s2, but then double-cross P2. (P2 should be suspicious of P1 intentions, however; if P1 is not going to double-cross P2, why is P1 so interested in getting P2 to switch from c2 to c1 [that alone gains P1 nothing]?)