Let us illustrate this point by an example. Suppose that the decision maker discounts any prospect delayed by more than six months at 18 percent, but that his rate of discount for prospects delayed six months or less is zero. Now the decision maker must choose at t0 between two alternatives. Alternative A1 is a payment of $5000 at t0 + 1 year. Alternative A2 is a payment of $10 000 at t0 + 5 years, but A2 has a cancellation clause: at any time during the first year, for a cancellation fee of $100, the decision maker can cancel his decision for A2 and receive the payment of $5000 at t0 + 1 year.
At t0, the discounted present values are:
Alternative A1 $4237
Alternative A2 $4371
Accordingly, the decision maker chooses alternative A2. However, at t1 = t0 + 6 months and one day, the payoff for alternative A1 is less than six months away, and so is not discounted, and is valued at $5000. To obtain this payment, however, the decision maker must pay the cancellation fee of $100. The net values discounted to t1 are
Alternative A1 $4900
Alternative A2 $4748
Therefore, the rational decision maker reverses his decision.
This is a one-person game. Suppose we express these decisions as plans of action for the successive stages like the pure strategies as understood by von Neumann and Morgenstern. The decision maker has three pure strategies:
- (1) Choose A1.
- (2) Choose A2, then do not cancel.
- (3) Choose A2, then cancel.
The payoffs of these strategies, discounted to t0, are:
- (1) $4237
- (2) $4371
- (3) $4127
Why, then, does our rational decision maker not simply choose strategy 2 and stick with it? Suppose that the decision maker has a weak will, in
Schelling’s sense, and knows that he does. Then he can anticipate that if he chooses A2, he will indeed cancel it after six months and in fact carry out strategy 3. Because of his weakness of will, strategy 2 simply is not available to him. That being so, in the spirit of Ulysses and the Sirens, (note Elster, 1977) the rational but weak-willed decision maker will choose strategy 1 and alternative A1.
This is not to say that intertemporal inconsistency does not exist. No doubt a strong-willed decision maker, having chosen strategy 2, will feel some subjective tension in the nature of regret or temptation during the time interval t1 to t2 = t0 + one year. Does rationality require him to act on the temptation? Well - perhaps it does.