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The convexity condition is simply a stronger form of the super-additivity condition. A
transferable utility game is convex if the marginal contribution of each player increases
with the size of the coalition they joins. In other words, it is valuable to join a larger
coalition than a smaller one. Convexity guarantees that the core is non-empty.
2. Transferable Utility
A game [N, v(S)] is a cooperative game with transferable utility is game where within the
coalition each player has the possibility to transfer any amounts of utilitymoney,
including money, to the another player within the same coalition. In what follows, we
adopt the following valuation function (Kaneko & Wooders, 2004)
(1.0)
In this setting, Ui : X % $ is the payoff function for player i, where X is outcome
space for player i and where xi and are, respectively, the outcome and the endowment
of commodities defined on the set X. In using the characteristic function in (1.0), we
focus on transferable utilities where agents can transfer some utility and not necessarily
compare utilities. It is assumed here that V(S) describes the Pareto frontier (For the proof,
see Kaneko & Wooders, 2004.) In this case, the no income effects condition means that
the sum v(S) is independent of the distribution of the money holdings among the
members of S.
3.Transferable Utility and Side Payments
To make up for any inequities that might rise due to the exchange in commodities and
other between members of a coalition, we allow for the transfer of money from one
member of the coalition to another. Transfers of money between members of the same
coalition, or “Side payments”, are a rule of the game and are not a prerequisite of the
utility function. The postulation of “side payments” is independent of the assumption of
transferable utility (see for example, Luce ad Raïffa (1985.)
In the following, we consider side payments in the context of transferable utility. Using
the setting from (1.0), we represent a transferable utility with side payment as
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