A one participant extensive-form sport with good data. The participant’s identify is Zeno. The sport has an infinite horizon. For each non-negative integer N there’s a node at which Zeno chooses both to Proceed or Stop. If he Quits at node N his payoff is -N/(N+1). If he continues he strikes on to node N+1.
There’s additionally a terminal node at infinity. This node is reached if and provided that Zeno Continues in any respect finite nodes. The payoff at infinity is 1.
Right here is the story: Zeno stands on the beginning line of a Marathon. After operating the space he decides whether or not to stop, and so on. His objective is to complete the Marathon and run into the arms of his proud and adoring followers. If he doesn’t get to the end line (the node at infinity) then all he has achieved is make himself drained with no compensating adoration. The farther he runs earlier than quitting the extra drained he’s.
The sport has a singular subgame good equilibrium: Zeno completes the Marathon. Nevertheless it has one other technique which is unimprovable by a one-stage deviation: Zeno quits at each alternative.
This latter technique has a pleasant behavioral interpretation. Zeno lacks confidence in his dedication to finish the Marathon. Particularly he desires to finish it however he expects that if he runs one other half of the space he’ll wind up quitting as soon as he will get there so why hassle. And certainly the explanation he is aware of he’ll stop after operating half the space is that he is aware of that when he will get there he’ll know that after operating one other half the space he’ll nonetheless stop so why hassle.
This can be a nice instance for educating the One Stage Deviation Precept, which asserts that methods which can be unimprovable are additionally SPE. The OSDP requires the sport to be steady at infinity. The Marathon sport isn’t steady at infinity.
To make it steady at infinity, assume that Zeno’s followers might be virtually as happy with him if he runs 26.1999 miles as they’d be if he ran the remaining one-millionth of a mile. If that’s the case, then Zeno’s payoff from almost finishing the Marathon is optimistic and the Quitting technique turns into improvable.
