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.

Appendix

In this Appendix we reference a software program, written in Python 2.7.6, for the generation of the TG-solution. Information concerning computation times, and the number of found vectors for the solution, are given further below.

On the Computation Time

We carried out the calculation of the solution with different values of n and δ. The considered games were the following:

G2: n = 2, v({1}) = v({2}) = 0, v({1,2}) = 1;

G3: n = 3, v(S) = the same of G2, and v({3}) = 1, v({1,3}) = 1, v({2,3}) = 2, v({1,2,3}) = 3 (it is the example of Sect. 4);

G4: n = 4, v(S) = the same of G3, and v({4}) = 0, v({1,4}) = v({2,4}) = 0, v({3,4}) = 1, v({1,2,4}) = v({1,3,4}) = v({2,3,4}) = 1, v({1,2,3,4}) = 3

We used the same affinities reported in Table 1; for all other cases we set $a_{i}^{S} = 1$.

The computation has been undertaken using an ACER Aspire 3935 computer, with 4 GB DDR3 Memory, Intel CoreTM 2 Duo processor P7450 (2,13 GHz, 1066 MHz FSB).

OS: Ubuntu 14.04.

The numbers of found vectors for the TG-solution and the computation times are shown in Tables 5 and 6.

Table 5. Number of found vectors for the TG-solution.

Table 6. Computation times (in seconds).