Academic journal article Atlantic Economic Journal

# Partitions and Coalitional Stability with Non-Dominant and Dominant Members

Academic journal article Atlantic Economic Journal

# Partitions and Coalitional Stability with Non-Dominant and Dominant Members

## Article excerpt

Introduction

The development of game theory has been characterized by the search for separate equilibria in cooperative and non-cooperative games. But, as Aumann (2003) has commented, game theory should provide a general framework for analyzing strategic behavior regardless of the interaction of players. The analysis of coalitions is a small step in this direction. According to Koczy (2002), coalitional game theory may be analyzed in two-stages. In the first stage, players cooperate by partitioning themselves into coalitions. In the second stage, these coalitions engage in non-cooperative strategic interaction.

This paper applies coalitional game theory to a simple analysis of the stability of coalitions involving imperfectly competitive firms producing a homogeneous product. The next two sections formalize the concept of a coalition in a manner similar to McCain (2004). This is followed by an example of a 3-firm coalitional game in which there are no dominant members. The next section extends the analysis by considering coalitions with "dominant" members. A member is dominant if it is able to extract a larger payoff than other members of the same coalition. This is followed by several numerical examples of coalitional game play involving dominant members and no capacity constraints for subordinate members. The next section considers coalitional game play in which subordinate members are capacity constrained. The final section summarizes the main conclusions of this paper.

Coalitions

The analysis begins by assuming (see, for example, McCain 2004) that game [GAMMA] consists of a set P of N players, that is,

P = {1,2, ..., N}. (1)

The set S of N strategies in this game is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Let v = [{[[sigma].sub.i,j]}.sub.i[member of]P] represent a vector of strategies and [PHI]([GAMMA]) the set of all strategy vectors. From this, there exists a mapping [PHI]([GAMMA]) [right arrow] [R.sup.N] such that the payoff to each player [[pi].sub.i] is

[[pi].sub.i] = [[pi].sub.i] ([{[[sigma].sub.k]}.sup.N.sub.k=1]) (3)

where [[sigma].sub.k] [member of] [{[[sigma].sub.i,j]}.sup.N.sub.j=1] = [S.sub.i].

A coalition of players C is defined as a non-empty subset of P(C[subset]P). The standard condition that a coalition must consist of at least one player is also imposed. Assuming that utility (payoffs) is transferable between members, there exists a mapping [SIGMA](C) [right arrow] R where [SIGMA](C) is the maximum coalition payoff. In much of the literature on cooperative games, this replaces conditions (2) and (3).

Partitions

In the theory of cooperative games, a partition is defined as the set of players P that are divided into various coalitions [C.sub.i] (Lucas and Thrall 1963). The set of partitions is

[PI] = {[C.sub.1], [C.sub.2], ..., [C.sub.N]} (4)

where j [member of] P = [there exists] [C.sub.i] [contains as member] j [member of] [C.sub.i] and [for all]i, k i [not equal to] k [??] [C.sub.i] [intersection] [C.sub.k] = [empty set]. The partition function maps Z([PI], C) [right arrow] R, where C [member of] [PI]. The outcome of a coalitional game is [gamma] = ([PI], [pi]), where [pi] is a vector of member payoffs.

If the position of a coalition member is unimportant, the number of possible partitions of n members is given by the nth Bell number (Bell 1934, Conway and Guy 1995)

B(n) = [n.summation over (k=1)] S(n, k) (5)

In Eq. (5), S(n, k) are Stifling numbers of the second kind (Abramowitz and Stegun 1972; Roman 1984), which may be calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Using Eqs. (5) and (6), a game involving three players can be divided into five partitions, which are summarized in Table 1. Table 2 summarizes the partitions of a 4-player game. In the tables, partitions are identified by square brackets while players belonging to the same coalition are included within curly brackets. …

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