Core of a Game

The core of a game is a central concept in cooperative game theory. Core of a game refers to situations in which individuals can gain from cooperating in groups, but there is conflict over the allocation of the gains from cooperation. An allocation is in the core if no subgroup of individuals can unilaterally improve the position of all its members. An allocation in the core is presumably stable: once achieved, there is no incentive to move away from it. It is also efficient in the sense that no other allocation makes all individuals better off. The power of an individual can be measured by how favorably the individual is treated by the allocations in the core in comparison with other feasible allocations.

Allocations in the core can be very extreme. For example, suppose a group of three voters must decide on the division of a resource. Three votes are needed to pass a proposal, with one voter controlling two votes and two voters controlling one vote each. The first voter has veto power because no proposal can be passed without that person's votes. The core allocates the entire resource to the veto voter. Given any other allocation (say, one that gives a positive share to the second voter), the first and third voter together can pass a proposal that divides the second voter's share between them.

The core does not consider the possibility of forming voting blocs. In this example, the two nonveto voters together have as many votes as the veto voter. If these two voters are able to act together as a single unit, there is no compelling reason why the veto voter must obtain the whole resource.

There may be more than one outcome in the core. If there are two voters that must agree on the division of a resource, any division that exhausts the resource is in the core because no voter can change the division unilaterally and it is not possible to make both voters better off.

The core may also be empty. Suppose a voting body with three voters must divide a resource by simple majority. Given any allocation, there are always two voters that would do better by dividing the third voter's share between them. If the core is empty, the situation is potentially unstable because given any outcome, there is a subgroup that has the ability and motivation to enforce a different one.

More generally, assuming that voters decide on the division of a fixed resource, the core is either empty (in the absence of veto voters) or distributes everything between the veto voters. Thus, the veto voters have all the power in the group according to the core. When voters are voting over a choice of policy or any other issue, the core is defined in the same way: a policy is in the core if no group of voters prefers another policy and can enforce it. If policies can be unambiguously ranked along some dimension, the core corresponds to the outcome preferred by the median voter.

...