The core of a cooperative game

One of the main issues in cooperative game theory is to find an allocation which splits v(N) but which is stable against deviations by individual parties and groups of parties. The main idea of the core is to view the values v(S) of each coalition as the opportunity cost of that group entering into the grand coalition. For an allocation to be in the core, the benefits that every coalition jointly receive must exceed the opportunity cost - the benefit they would receive if they deviated from the grand coalition.

Thus, we say that an allocation x is in the core if it cannot be blocked by any coalition S. In other words, the set of feasible allocations in the core is the set:

Note that the core may be empty. As an example of how cooperative game theory works, consider the following three-player majority game. Suppose that there are three players, A, B and C. When they all act together, they can obtain a payoff of $1. Any two of them acting together can obtain a payoff of в, where 0 <0< 1. The characteristic function of this cooperative game is:

Suppose that the payments xA, xB, xC are in the core. Then these payments must be non-negative, and we must have xA + xB + xC = 1 and xA + xB, xA + xC, xB + xC >0. But if this second set of inequalities holds, they must also hold if we sum them together, so we must have:

However, since xA + xB + xC = 1, this implies that we need Q< 2/3 for the core to be non-empty.

We will use this simple majority game together with the following 0-1 normalisation of a cooperative game, to understand the conditions under which the Coase Theorem does or does not hold when there are three or more players. The 0-1 normalisation works as follows. Suppose that we have cooperative game with a characteristic function v(S), and which has v({i}) Ф 0 for some i, and v(N) Ф 1. We seek to transform the payoffs of the game such that it is strategically equivalent to the original game (that is, it has the same structure as the original game), but which has v({i}) = 0 for all i, and v(N) = 1.

This can be done by adding (not necessarily positive) numbers Zj to the individual payoff of each individual to give them v({i}) = 0. In other words, set zt = -v({i}). This number must be added to every coalition of which i is a member. In particular, the new value of the grand coalition must be:

Now, divide the new value of every coalition by K. The resulting characteristic function is strategically equivalent to the original game, since each individual's payoff has been scaled up by zt and divided by a constant K. Moreover, we have v({i}) = 0 for all i, and v( N) = 1. This normalisation will prove extremely useful in the analysis that follows.

 
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