# Transforming Games with Affinities from Characteristic into Normal Form

## Abstract

von Neumann and Morgenstern, while introducing games in extensive form in their book [1], also supplied a method for transforming such games into normal form. Once more in their book [1], the same authors provided a method for transforming games from characteristic function form into normal form, although limited to constant-sum games. In his paper [2], Gambarelli proposed a generalization of this method to variable-sum games. In this generalization, the strategies are the requests made by players to join any coalition, with each player making the same request to all coalitions. Each player’s payment consists of the player’s request multiplied by the probability that the player is part of a coalition really formed. Gambarelli introduced a solution for games in characteristic function form, made up of the set of Pareto-Optimal payoffs generated by Nash Equilibria of the transformed game.

In this paper, the above transformation method is generalized to the case in which each player’s requests vary according to the coalition being addressed. Propositions regarding the existence of a solution are proved. Software for the automatic generation of the solution is supplied.

## Keywords

Game theory Characteristic function Nash equilibria Pareto optimality Transformation## Notes

### Acknowledgements

This paper is sponsored by MIUR, by research grants from the University of Bergamo, by the Group GNAMPA of INDAM and the statutory funds (no. 11/11.200.322) of the AGH University of Science and Technology. The authors thank Angelo Uristani for his useful suggestions.

Finally, the authors would like to thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.

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