Raymond Hayden Economics

Player 2

L'

R'

L

2,1

0,0

M

0,2

0,1

R

1,3

then chooses between two actions, L' and R', after which the game ends. Payoffs are given in the extensive form in Figure 4.1.1.

Using the normal-form representation of this game given in Figure 4.1.2, we see that there are two pure-strategy Nash equilibria—(L,I/) and (R, R'). To determine whether these Nash equilibria are subgame-perfect, we use the extensive-form representation to define the game's subgames. Because a subgame is defined to begin at a decision node that is a singleton information set (but is not the game's first decision node), the game in Figure 4.1.1 has no subgames. If a game has no subgames then the requirement of subgame-perfection (namely, that the players' strategies constitute a Nash equilibrium on every subgame) is trivially satisfied. Thus, in any game that has no subgames, the definition of subgame-perfect Nash equilibrium is equivalent to the definition of Nash equilibrium, so in Figure 4.1.1 both (L, I/) and (R, R') are subgame-perfect Nash equilibria. Nonetheless, (R,R') clearly depends on a noncredible threat: if player 2 gets the move, then playing L' dominates playing R', so player 1 should not be induced to play R by 2's threat to play R' if given the move.

One way to strengthen the equilibrium concept so as to rule out the subgame-perfect Nash equilibrium (R,R') in Figure 4.1.1 is to impose the following two requirements.

Requirement 1 At each information set, the player with the move must have a belief about which node in the information set has been reached by the play of the game. For a nonsingleton information set, a belief is a probability distribution over the nodes in the information set; for a singleton information set, the player's belief puts probability one on the single decision node.

Requirement 2 Given their beliefs, the players' strategies must be sequentially rational. That is, at each information set the action taken by the player with the move (and the player's subsequent strategy) must be optimal given the player's belief at that information set and the other players' subsequent strategies (where a "subsequent strategy" is a complete plan of action covering every contingency that might arise after the given information set has been reached).

In Figure 4.1.1, Requirement 1 implies that if the play of the game reaches player 2's nonsingleton information set then player 2 must have a belief about which node has been reached (or, equivalently, about whether player 1 has played L or M). This belief is represented by the probabilities p and 1 — p attached to the relevant nodes in the tree, as shown in Figure 4.1.3.

Given player 2's belief, the expected payoff from playing R' is p ■ 0 + (1 — p) • 1 — 1 —p, while the expected payoff from playing V is p ■ 1 + (1 — p) ■ 2 = 2 — p. Since 2 — p > 1 — p for any value of p, Requirement 2 prevents player 2 from choosing R'. Thus, simply requiring that each player have a belief and act optimally given this belief suffices to eliminate the implausible equilibrium (R,R') in this example.

Requirements 1 and 2 insist that the players have beliefs and act optimally given these beliefs, but not that these beliefs be rea

1 R

sonable. In order to impose further requirements on the players' beliefs, we distinguish between information sets that are on the equilibrium path and those that are off the equilibrium path.

Definition For a given equilibrium in a given extensive-form game, an information set is on the equilibrium path if it will be reached with positive probability if the game is played according to the equilibrium strategies, and is off the equilibrium path if it is certain not to be reached if the game is played according to the equilibrium strategies (where "equilibrium" can mean Nash, sub game-perfect, Bayesian, or perfect Bayesian equilibrium).

Requirement 3 At information sets on the equilibrium path, beliefs are determined by Bayes' rule and the players' equilibrium strategies.

In the subgame-perfect Nash equilibrium (L, L') in Figure 4.1.3, for example, player 2's belief must be p = 1: given player l's equilibrium strategy (namely, L), player 2 knows which node in the information set has been reached. As a second (hypothetical) illustration of Requirement 3, suppose that in Figure 4.1.3 there were a mixed-strategy equilibrium in which player 1 plays L with probability q\, M with probability q2, and R with probability 1 —

qi — q2. Then Requirement 3 would force player 2's belief to be