Cooperative game theory studies what groups of players can achieve together and how their joint value can be allocated. A transferable utility game assigns a real value to each coalition. The core of a cooperative game asks which allocations no coalition can improve upon; the Shapley value averages marginal contributions across player orderings.
A finite transferable utility game gives each coalition a real value that its members can distribute among themselves. The usual normalization is . An efficient payoff vector satisfies . Simple cooperative games model winning coalitions with values zero and one; convex cooperative games model increasing marginal contributions.
A player's marginal contribution to a coalition is the increase in its value when that player joins. In a convex cooperative game, marginal contributions increase with the preceding coalition. Averaging contributions across player orderings gives the Shapley value.
A coalition is a subset of the players in a transferable utility game, considered as a group that can act together. The game assigns it value . The core of a cooperative game constrains the total payoff assigned to every coalition, while a marginal contribution measures the value gained by adding one player.
A transferable utility game is convex when its coalition-value function is supermodular. Equivalently a player's marginal contribution cannot decrease when the preceding coalition grows. The implication follows by applying the displayed inequality to and with , ; the reverse implication follows by adding successive marginal inequalities. This property concerns coalition values, not geometrical convexity of a strategy space.
Increasing marginal contributions imply for . Summing telescopes to , while efficiency follows by telescoping over the full ordering. Hence every marginal contribution vector is in the core of a cooperative game. The core is a convex set, so their average, the Shapley value, is in it too. This gives an elementary proof of nonemptiness and stability for a convex cooperative game.
The core consists of efficient payoff vectors for which each coalition receives at least its own attainable value. Thus no coalition can improve every member's payoff by leaving. It is a convex set defined by linear constraints, but can be empty. Every convex cooperative game has a nonempty core because each marginal contribution vector belongs to it.
The Shapley value averages each player's marginal contribution over uniformly random player orderings. Exactly orderings have immediately before player , giving the formula. The values sum to by telescoping each ordering. In a simple cooperative game this is the probability of being pivotal. For a convex cooperative game, Shapley value belongs to the core of a convex game guarantees a stable allocation as well.
For a player ordering , is the set before player , and the displayed entries form its marginal contribution vector. Their sum is . The Shapley value is the average of these vectors. In a convex cooperative game, increasing marginal contributions make each such vector satisfy every coalition constraint of the core of a cooperative game.
A weighted voting game assigns player weights and a threshold, and declares coalitions winning when their total weight passes that threshold. The strict convention is equivalent to quota for integer weights; another common convention writes a weak inequality directly. Nontrivial examples are simple cooperative games. Having nonnegative weights does not guarantee a convex cooperative game: two-of-three majority supplies a counterexample.
A simple cooperative game is a monotone transferable utility game in which each coalition either loses, with value zero, or wins, with value one. In the usual nontrivial convention the empty coalition loses and the grand coalition wins. The Shapley value is then the probability that a player changes a losing coalition into a winning one when players enter in uniformly random order.
