By Alex Talbot Coram (auth.)
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Extra resources for State, Anarchy and Collective Decisions
4. Imagine that there are two individuals and the pay-offs for individual one and two are written v1 and v2 . Assume that there is some good that is in®nitely divisible and that the pay-offs can be represented by a continuous function. It might be though that the good is the bene®t from buying and distributing a pizza or from some joint venture. Let the amount they can get in the original position be c1 for individual 1 and c2 for individual 2. It will be observed that any move to the right of c1 makes player 1 better off and any move above c2 makes player 2 better off.
In other cases the selection of strategies will be more dif®cult. The problem of ®nding a solution to a game with two players is illustrated with the following example. To simplify the illustration assume that a strategy is the choice of a point on some closed interval, say a; b. This might occur for example where the strategy is some amount of a resource to devote to an activity and the resource can be treated as if it is in®nitely divisible. In this case ski is a choice of some amount, xk , where xk P a; b.
Condition [Cii] says that the players will only co-operate if ri a. That is, after players in the ®rst other two groups have withdrawn it must be the case that defection by one player in group Ri would cause conditional co-operation to collapse. To see the necessity of condition [Cii], suppose that ri > a. In this case the optimal strategy for each player in group Ri is to defect. Similarly if ri < a no member of group Ri ®nds it worthwhile to co-operate. 42 State, Anarchy and Collective Decisions It follows immediately that, for different values of m H , only one group of conditional co-operators co-operate.