# 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).

## See also

## References

- ↑ 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.