Cost game

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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

  1. Drechsel, J. (2010). Cooperative Lot Sizing Games in Supply Chains. Lecture Notes in Economics and Mathematical Systems 644. Springer-Verlag, Berlin Heidelberg.
  2. 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.
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