Problems

Chapter 9 Problems 1 and 4 (1st Edition)

Chapter 10 Problems 4 and 8 (2nd Edition)


Solutions

9.1 (10.4 2nd edition)

(a)(i) There are two pure-strategy equilibria {Down,Left} and {Up, Right}. There is also a mixed-strategy equilibrium, p(Up) = p(Left) = 2/3, in which Row uses Up with probability 2/3 and Column uses Left with probability 2/3. The payoffs from the mixed-strategy equilibrium are 2/3 to each player.

(ii) If Row commits to Up, she ensures herself a payoff of 2. Similarly, if Column commits to Left, she ensures herself a payoff of 2. First, decide which player has something to gain from an outcome different from the equilibrium. In this game, each player prefers a different equilibrium. Hence, the Row player committing to Up guarantees that the column player responds with Right, earning the Row player her maximum payoff. Similarly, the Column player could commit to Left.

(b)(i) Both players have dominant strategies: Up for Row and Right for Column. Therefore, the unique equilibrium is {Up, Right} with payoffs of (3, 4).

(ii) Since the column player prefers the equilibrium to any other outcome, only the row player has something to gain from a strategic move. However, the only improvement for Row comes from {Up , Left}. Committing to Up will not change the outcome since column will still play Right. Further, no promise can change the outcome since there is no way to redistribute the wealth in {Up, Left} to make both players better off than the equilibrium. Perhaps, row could somehow achieve his best payoff of 4 by using the threat "Down if Right," but this wouldn't be easy.

(c)(i) Both players have a dominant strategy: Up for Row and Right for Column. Therefore, only one equilibrium exists: {Up, Right}, with payoffs of (2,2).

(ii) This is a classic prisoner's dilemma. Both players are playing their dominant strategies, yet another outcome is preferable by both. A number of ways are used to resolve this, especially if the game is repeated. Refer to the lecture notes.

9.4 (10.8 2nd edition)

(a)The Nash equilibrium is {Aggressive, Aggressive }.

(b)(i)

fig. 4

The equilibrium is for the US to be aggressive, and for the USSR to be aggressive regardless of what the US does: {A; A,A}.

(ii) fig. 4c

The equilibrium is for the Soviet Union to be restrained, and for the US to be restrained if the Soviet Union is, and aggressive if the Soviet Union is. {R; R if R, A if A}

(iii) fig. 4d

This game has two decision nodes for the US, and five for the USSR

period 1: USSR moves, period 2: US moves, period 3: USSR moves

Note that in period 3, the USSR moves "aggressive" at all four nodes

Hence, in period 2, the US moves "aggressive" at both nodes

In period 1, USSR's move is irrelevant (it will be reconsidered) so either move is part of an equilibrium strategy.

Hence, there are two equilibria:

{ A, A,A,A,A,A ; A,A }, and
{ R, A,A,A,A,A ; A,A }

(c) Note that the best outcome of those above is in part (ii). Hence, the USSR needs to commit to moving first, and to not being able to reconsider its move later.

Both the United States and the Soviet Union are better off in the situation in which the United States moves last. This is the situation in which the United States retains its flexibility and can respond to the actions of the Soviets. With this order of timing, the Soviet Union's awareness of the United States's ability to respond to its action discourages it from acting aggressively. The Soviets know that if they did so, the United States would match their aggression, and they would suffer as a result. In turn, the Soviet Union knows that restraint will be met with restraint. If the Soviet Union moves last, the United States knows that nothing will restrain the Soviets from acting aggressively. Anticipating this, the United States will also have to act in the same way, and the outcome in which both are aggressive is worse for both countries than the outcome in which both are restrained is. The key point is that the Us must retain its flexibility. If it committed to either action (Restraint or Aggression), it would lose its ability to influence the action of the Soviet Union; if the U.S. action is previously determined, the Soviet Union should choose its dominant strategy of Aggression. In order to retain its influence, the United States should not commit to any actions; it should remain credibly flexible. For example, the U.S. should not state that it will never be aggressive (it should not renounce the use of nuclear weapons) nor should it adopt a unitarily aggressive posture.