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). 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). 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):
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):
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).
- ↑ Drechsel, J. (2010). Cooperative Lot Sizing Games in Supply Chains. Lecture Notes in Economics and Mathematical Systems 644. Springer-Verlag, Berlin Heidelberg.
- ↑ 2.0 2.1 2.2 Young, H.P. (1985). Methods and principles of cost allocation. Cost Allocation: Methods, Principles, Applications (H.P. Young Ed.). Amsterdam: Elseviers Science Publishers B.V. pp. 3–29.