Thor, the god of thunder, protects the Nine Realms of Man from his brother Loki, the god of mischief. The game between the brothers takes place over time, with no fixed end date. Loki has a single "Trick" that he can play on the unsuspecting humans, and Thor has a single "Trap" that he can set for Loki. Loki benefits from successfully playing the Trick, while Thor benefits from preventing the trick and trapping Loki. In each period, Loki chooses whether to play the trick, or wait until the next period. Simultaneously, Thor decides whether to set the trap for Loki or wait. Between periods the game continues with probability ? = 2/3 and ends with probability 1/3, giving both players payoff 0.

Because each player can make a move?trap or trick?only once, the game ends once either player does so. If Thor sets the trap and Loki plays the trick in the same period, then the trick fails and Loki is captured. Setting the trap costs Thor c = 3/4. Thus, Thor's payoff is 1 ? 3/4 = 1/4 and Loki's payoff is 0 in this case. If Thor sets the trap in a period and Loki waits, then the trap fails, and Loki can play the trick with no risk in the next period. Thus, Thor's payoff in this case is ?3/4 (Thor gets 0 benefit because Loki avoided the trap, and pays 3/4 cost) and Loki's payoff is ?(1) = 2/3 (he can play the trick in the next round if the game continues). If Thor waits and Loki plays the trick, then Loki's trick succeeds and Thor loses the chance to trap him, payoff 1 for Loki and 0 for Thor. Finally, if both wait, then the current period ends, and the next period begins with probability ? = 2/3, and the with probability 1/3 game terminates with payoffs 0 for both players.

As long as the game continues, every round looks identical. Let us look for an equilibrium in which a player's probability of making a move (trick or trap) does not change over time. Let Thor's payoff in this equilibrium be v_T and Loki's payoff be v_L. The game between the gods can be represented in the following table.

Suppose that in the first round, Thor isn't completely prepared to spring the trap on Loki. In particular, in the first round, Thor cannot set the trap with probability greater than 2/5. In all later rounds Thor is "ready" and can set the trap with any probability t ? [0, 1]. Thus, starting from round 2, the game is identical to the one we solved before. The first round game is therefore

Loki Trick Wait Thor Trap 1 3 Wait UT