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• Both parties have an equal bargaining power.
• All players weigh their payoffs by the same measure. In other words, the payoff
of every coalition can be distributed to participants using a metric that is
recognized by all parties, money for example.
• No income effects on the choice behavior of the players, entailing that the players
are not faced with a boundary condition.
III.2.2.1Transferable Utility and Side Payments
1. Cooperative Framework
A n-person cooperative game with transferable utilities is a pair 95 [N, v(S)], where N
={1,2, ..., n} is a finite, non-empty, set of players, and where each non-empty subset of
players S N is called a coalition.
In this cooperative game, every coalition S is assigned its worth by a characteristic
function, denoted as v(S) and where v: 2 N $ , where 2 N is the set of all subsets of N.
The function v allocates to each coalition S in 2 N the maximum payoff that can be
achieved if the players in the cooperation cooperate. So an allocation of v(S) to parties in
the coalition S is defined by a payoff vector allocates outcome to player
For a vector payoff and a coalition S, the total payoff to players is:
such that
The valuation function v can be characterized by the following axioms:
• By convention, v(ø) = 0 where ø is the empty set.
• The characteristic function is super-additive, in other words, the union of any two
disjoint groups of players never diminishes the total benefits: v(S T) " v(S) +
v(T), holds all coalitions S and T, with S T=ø. For any two disjoint coalitions S
and T of N, superadditivity implies that the coalition is at least any beneficial to its
members as the non-cooperative payoffs they would have gained if they did not
join the coalition. In this regard, the coalition is optimal.
• v is convex:
95 We generally follow the notation given in Wooders and Kaneko, 2004.
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