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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.
The triangle of partitions and coalitions for games involving n = 1 to 9 players and k coalitions in which a member's position within the coalition is unimportant is summarized in Table 3.
Example of a 3-Firm Coalitional Game with No Dominant Members
Consider the three-player, pure-strategy, static game depicted in Table 4. (1) This onetime, non-cooperative static game involves an industry that consists of three identical profit-maximizing firms. The market-clearing price is assumed to a function of total industry output. Each profit-maximizing firm is free to produce as much output as it wants subject to capacity limitations. In the standard model of imperfect competition (see, for example, Sweezy 1939), total industry profits are maximized when firms collectively behave as a monopolist. This is analogous to the situation depicted in Table 4 where any one firm produces and the other two firms do not produce. While output is constrained, it will be assumed that each firm has enough capacity to behave as a profit-maximizing monopolist. As the number of producing firms in the industry increases, total industry and per firm profits decline. The non-cooperative Nash equilibrium strategy profile for this game is {Produce, Produce, Produce}.
Suppose that the firms in this industry are able to form coalitions by entering into explicit or tacit collusive arrangements. Payoffs reflect each member's relative dominance in the coalition. The greater the payoff, the more influence a member has when formulating coalition strategy. The symmetry of the payoffs in Table 4 indicates that all firms have equal Shapely (1953) values. This suggests that there are no dominant coalition members. Each producing firm receives an equal share of total industry profits. In the extreme case of a coalition consisting of a single firm, the partition [{[P.sub.1]}{[P.sub.2]}{[P.sub.3]}] results in the same strategy profile {Produce, Produce, Produce} as in the non-cooperative static game depicted in Table 4. Each coalition member earns a maximum profit of 10 and total industry profit is 30. If only two firms produce, total industry profits increase to 60 with each member earning a profit of 30. Finally, monopoly profits increase to 90.
Consider the partition [{[P.sub.1], [P.sub.2]} {[P.sub.3]}] in greater detail. While coalition members cooperate by formulating a common production strategy, strategic interaction between coalitions is non-cooperative. This will be referred to as the Cournot coalition game. In the first stage, members of the {[P.sub.1], [P.sub.2]} coalition agree to set production quotas. In the second stage, the {[P.sub.1], [P.sub.2]} coalition plays a non-cooperative, one-time, static game with the {[P.sub.3]} coalition. This Cournot coalition game is depicted in Table 5.
The Nash equilibrium strategy profiles for the game depicted in Table 5 are {[Produce, Don't produce], Produce} and {[Don't produce, Produce], Produce}. Since the payoffs are symmetrical, the payoffs in Table 5 apply to any partition involving 2-1 coalitions. Table 6 summarizes the coalition payoffs for the five possible partitions in this game. The payoff to each firm in a 1-member coalition is identical to the Nash equilibrium strategy profile {Produce, Produce, Produce} in the non-cooperative, one-time, static game depicted in Table 4.
The 2-member coalitions in Table 6 are stable since the payoff to each firm (30/2 = 15) is greater than by not cooperating (10). In fact, Table 6 suggests that another stable and more profitable coalition exists. A {[P.sub.1], [P.sub.2], [P.sub.3]} coalition that mimics monopoly output strictly dominates every other partition provided that all members have identical production quotas since the payoff to each firm is 90/3 = 30 > 15 > 10. Although it is possible for a coalition consisting of two or three firms to have a Nash equilibrium strategy profile, the only rational coalition is {[P.sub.1], [P.sub.2], [P.sub.3]}. In general, a necessary condition for stability in the Cournot coalition game is for each member to be at least as well off than by not cooperating.
The Nash equilibrium payoffs to members of the {[P.sub.1], [P.sub.2], [P.sub.3]} Cournot coalition in which there are no dominant players are summarized in Table 7. (2) The absence of dominant coalition members is denoted [P.sub.1](1/3) [??] [P.sub.2](1/3) [??] [P.sub.3](1/3). The numbers in parentheses represent each member's share of the total coalition payoff. Table 7 modifies Table 4 to reflect the payoffs to individual members of the {[P.sub.1], [P.sub.2], [P.sub.3]} coalition. Each firm has a choice in the first stage of the game either to form a coalition with either or both rivals, or to play a non-cooperative game. Thus, the strategy profile {Produce, Produce, Produce} may be interpreted either as the firms' decisions not to cooperate in the first stage, that is, a [{[P.sub.1]} {[P.sub.2],} {[P.sub.3]}] partition, or to form a coalition in which production quotas are the same as if the firms were playing a non-cooperative game. Thus, the strategy profile {Produce, Produce, Produce} must constitute at least one Nash equilibrium, although there are others.
The Nash equilibrium strategy profiles for the [{[P.sub.1], [P.sub.2], [P.sub.3]}] partition in Table 7 are {[Produce, Don't produce, Don't produce]}, {[Don't produce, Produce, Don't produce]} and {[Don't produce, Don't produce, Produce]}. That is, the coalition produces an output level equivalent to the output of a single firm in a non-cooperative game. The difference is that all firms produce and have equal production quotas. The Shapely payoffs to each member of the [{[P.sub.1], [P.sub.2], [P.sub.3]}] partition is 30, while the payoffs to each member of the [{[P.sub.1]} {[P.sub.2]} {[P.sub.3]}] partition is 10. Clearly, the strategy profiles {[Produce, Don't produce, Don't produce]}, {[Don't produce, Produce, Don't produce]} and {[Don't produce, Don't produce, Produce]} dominate the strategy profile {Produce, Produce, Produce} since no member can improve its payoff by abandoning the coalition.
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