Consider a contractual setting with two players, where the underlying strategic interaction is given by the following "partnership game":
This matrix describes the technology of the relationship. Suppose that before noncooperative interaction, the players sign an externally enforced contract specifying a transfer t from player 2 to player 1 in the event that either (H, L) or (L, H) is played. The interpretation here is that the court can observe the total output of the relationship (which is either 8, 12, or 16) and is willing to enforce a transfer if output is 12.
However, the court enforces the transfer only if one or both players request enforcement. Each player who requests enforcement must pay a legal fee of c. The court only allows transfers less than or equal to 10 (from one player to the other) because this is the greatest payoff a single player can earn in the partnership. Consider a "grand game" in which the partnership game is one component. Play in the grand game runs as follows. First, the players interact by playing the illustrated partnership game. Then, if either (H, H) or (L, L) is the outcome, the grand game ends. However, if (H, L) or (L, H) is the outcome, then the players interact further by playing the following enforcement game:
The players' payoffs in the enforcement game are added to their payoffs from the partnership game.
(a) Suppose you are a policy maker who determines the fee c and the contract parameter t. Under what conditions on c and t will (H, H) be played in a subgame perfect equilibrium of this game? That is, for what values of c is there a number t that induces cooperation? Remember that t is constrained to be between -10 and 10.
(b) Explain your reasoning for your answer to part (a). In particular, what happens when c = 0?
(c) Suppose that between the partnership game and the enforcement game the players make a joint decision about whether to continue with the enforcement game or "settle out of court"; they can also make a monetary transfer at this time. When the players settle out of court, they avoid the court fee. The default decision is to continue with the enforcement game. Do you think (H, H) can be supported in the negotiation equilibrium of this game? Explain your reasoning without explicitly solving this game.