Introduction to Cooperative Game Theory

Published by Mario Oettler on

In cooperative games, participants can forge alliances due to the possibility of external enforcement. This means that participants can negotiate a strategy and then are bound to the negotiation outcome (no cheap talk).

The challenge in cooperative games is to agree on a contract rather than finding a strategy combination. It all comes down to a pay-off function that everybody can agree on. In a cooperative game, players follow two kinds of interest:

  1. Common interest: cooperate to gain a higher pay-off than in an uncooperative situation
  2. Selfish interest: Gain a share as big as possible from the outcome.

Besides that, nobody can be forced to come to an agreement. If someone thinks the result is not good enough, he can leave the negotiation. But once all parties agree on a strategy combination, it can be enforced.

We can distinguish games with transferrable utility or without transferrable utility. Typically, this utility is represented in money.

Let’s consider an example:

In the following game, we see that {Opera, Opera} and {Boxing, Boxing} would be the Nash equilibria.

Without transferrable utility, player B would lose if he accepts going to Boxing. Player A would have no means to compensate him for this conceived loss compared to an evening in the Opera.

With transferrable utility, A could offer B a compensation worth 1. This reduces the pay-off of A to 2 and increases the pay-off of B to 2.