Quiz 4. Combining sequential and simultaneous moves and mixed strategies

1. Consider a game in which there are two players, A and B. Player A moves first and chooses either Up or Down. If Up, the game is over, and each player gets a payoff of 2. If A moves Down, then B gets a turn and chooses between Left and Right. If Left, both players get 0; if Right, A gets 3 and B gets 1.

(a) Draw the tree for this game, and find the subgame-perfect equilibrium. [2]

(b) Show this sequential-play game in strategic form, [2] and find all the Nash equilibria. Which is or are subgame perfect and which is or are not? [2]

(c) What method of solution could be used to find the subgame-perfect equilibrium from the strategic form of the game? [1]

2. Show how does the change from simultaneous to sequential change the equilibrium in the battle of sexes? [3]

3. Consider the following game.

		COLUMN
		YES	NO
ROW	YES	x, x	0, 1
	NO	1, 0	1, 1

1. For what values of x does this game have a unique Nash equilibrium? What is that equilibrium? [3]

2. For what values of x does this game have a mixed-strategy Nash equilibrium?[2] With what probability, expresses in terms of x, does each player play YES in this mixed-strategy equilibrium?[3]

3. For what values of x found in part (b), is the game an example of an assurance game, a game of chicken, or a game similar to tennis? Explain. [2]

Answers

1. (a) The game tree is shown at right.

The subgame-perfect equilibrium is (Down, Right).

(b) The strategic form of this game is shown below.

1

		Player B
		If Down,then Left		If Down, then Right
Player A	Up	2, 2	2, 2
	Down	0, 0	3, 1

Here we see that there are two Nash equilibria: (Up, Left) yields a payoff of (2, 2) and (Down, Right) yields a payoff of (3, 1). The (Down, Right) equilibrium is subgame perfect (see answer to part a); it does not rely on either player’s taking an action that is not in her own interest at the time it is taken. The (Up, Left) equilibrium is not subgame perfect. This is a Nash equilibrium only if Player A expects Player B to choose Left, but in the sequential game, Player B would hurt herself (she would get a payoff of 0 rather than of 1) if she chose Left after Player A had already picked Down. Player B’s threat to pick Left in this situation is not credible; it is just this sort of noncredible threat used to support an equi­librium that is eliminated when the requirement of subgame perfectness is imposed.

c) To find the subgame-perfect equilibrium from the strategic form of the game, start with the second-moving player (Player B), and eliminate any of her weakly dominated strategies (If Down, then Left). Once Player B’s weakly dominated strategy is eliminated from consideration, Down dominates Up for Player A, so the unique subgame-perfect (or rollback) equilibrium is (Down; If Down, then Right).

2. There is a first-mover advantage and unique rollback equilibrium.

3. (a) When x < 1 No is a dominant strategy for both players, so (No, No) is the unique Nash equilibrium; there is no equilibrium in mixed strategies.

(b) px = p + 1(1 – p) x = 1/p

qx = q + 1(1 – q) x = 1/q

MSE: Row plays 1/x(Yes) + (1 – 1/x)(No)

Column plays 1/x(Yes) + (1 – 1/x)(No)

There is a mixed-strategy Nash equilibrium when x > 1. In that MSE Yes will be played by both players with probability 1/x.

(c) This is an example of an assurance game because there are two Nash equilibria in pure strategies where the players coordinate on the same strategy.