# Cost game

Whenever characteristic function of a game represents costs and players favour lower amounts of allocated goods, this is a cost game (Drechsel, 2010, p.10)[1]. Characteristic function is then noted as c(S). In cooperative game theory, however, many times the focus is on gains - especially in business applications - that can be realized by cooperating players in a coalition set (Young, 1985, p.11)[2]. Given a cost function c the potential gains for a coalition S can then be interpreted as the savings the cooperating player(s) can achieve compared to their non-cooperative approach when they act individually and incur standalone costs c(i) (Young, 1985, p.11)[2]:

v(S)=Σc(i) - c(S) for all S ⊂ N

v is the characteristic function for a cost-savings game and v(S) is the value (or sometimes profit) of coalition S in this game.

When the allocation of net profits or benefits is explicitly sought after and c(S) are for example the costs of a firm providing a subset of outputs S (of N), with r(i) being the revenues of commodity i ∈ N, the characteristic function of net profits from S is defined as (Young, 1985, p.11)[2]:

v(S)=Σr(i) - c(S)

Note: Technically speaking the illustrative characteristic function used in this example should be labelled as a cost function c(S) instead of v(S).

Usually costs and profits (savings) can be handled equivalently as they are determined by one another and c(S) is a dual game of v(S) (Drechsel, 2010, p.11):

c(S) = v(N) - v(N\S) for all S ⊆ N

where v(N\S) denotes the subset value of all players which doesn't include the observed S (value of grouping consisting of grand coalition minus the observed set S).