Weakness of Will in a Game in Extensive Form
Weakness of will may also be a factor in interactive decisions. Consider Game 9.3 in extensive form, shown as Figure 9.2. All decisions are close enough together in time that there is no need to discount payments to present value.
First we note that the perfect equilibrium for this game is for decision maker a to choose behavior strategy Alt1 for a payoff of $4237. However, when we express this game in terms of contingent strategies, we have, for decision maker a,
- (1') Choose Alt1.
- (2') Choose Alt2, then, if b chooses up, choose up.
- (3') Choose Alt2, then, if b chooses up, choose down.
and for decision maker b,
Figure 9.2 Game 9.3: two-person game
(4') If a chooses Alt2 then choose up (5') If a chooses Alt2 then choose down
If decision maker a chooses strategy 3', then decision maker b’s best response is strategy 5', while if decision maker a chooses strategy 2', and b knows this with certainty, then b’s best response is strategy 4'. Taking this into account, the payoffs to be expected from these strategies would seem to be
(1') $4237 (2') $4371 (39) $4137 (4') $3000 (5') $2000
This being so, we ask again, why does decision maker a not simply choose strategy 2'? There are two possibilities: (1) Decision maker b believes that decision maker a has a weak will, and will not carry out strategy 2 but, having arrived at decision-point a2, will choose down. Decision maker b therefore chooses strategy 5'; and this is known to agent a, who then chooses strategy 1' as his best response to strategy 5'. Thus, it seems, the subgame perfect equilibrium can be necessary because of the belief that a has a weak will. (2) The second possibility is that b believes a is dishonest and opportunistic and will choose “down” at decision point A2 regardless of any protestations to the contrary. Thus, the subgame perfect equilibrium can be necessary because of the belief that A is dishonest. But suppose that decision maker A has a strong will, that is, a capability to choose strategy 2’ and stick to it despite the temptation to choose “down” at decision point A2; and suppose that this is known to decision maker b. Suppose decision maker a also is honest, and this, too, is known to decision maker b. Thus, decision maker needs only announce “on my honor, I am choosing strategy 29,” and then b’s rational decision is for strategy 49, and the cooperative solution A1, up, up results.