A one player extensive-form game with perfect information. The player’s name is Zeno. The game has an infinite horizon. For every non-negative integer N there is a node at which Zeno chooses either to Continue or Quit. If he Quits at node N his payoff is -N/(N+1). If he continues he moves on to node N+1.
There is also a terminal node at infinity. This node is reached if and only if Zeno Continues at all finite nodes. The payoff at infinity is 1.
Here is the story: Zeno stands at the starting line of a Marathon. After running the distance he decides whether to quit, etc. His goal is to finish the Marathon and run into the arms of his proud and adoring fans. If he doesn’t get to the finish line (the node at infinity) then all he has done is make himself tired with no compensating adoration. The farther he runs before quitting the more tired he is.
The game has a unique subgame perfect equilibrium: Zeno completes the Marathon. But it has another strategy which is unimprovable by a one-stage deviation: Zeno quits at every opportunity.
This latter strategy has a nice behavioral interpretation. Zeno lacks confidence in his determination to complete the Marathon. In particular he wants to complete it but he expects that if he runs another half of the distance he will wind up quitting once he gets there so why bother. And indeed the reason he knows he will quit after running half the distance is that he knows that when he gets there he will know that after running another half the distance he will still quit so why bother.
This is a great example for teaching the One Stage Deviation Principle, which asserts that strategies that are unimprovable are also SPE. The OSDP requires the game to be continuous at infinity. The Marathon game is not continuous at infinity.
To make it continuous at infinity, assume that Zeno’s fans will be almost as proud of him if he runs 26.1999 miles as they would be if he ran the remaining one-millionth of a mile. If so, then Zeno’s payoff from nearly completing the Marathon is positive and the Quitting strategy becomes improvable.