Power, Embedded Games, and Coalition Formation

ABSTRACT

Applications of game theory frequently presume but do not show that social structures contain games. This study shows that multiple games are embedded in strong power structures and that power is exercised because 1) the game of those low in power contains a dilemma whereas 2) the game of those high in power does not. As in previous analyses, we find those low in power play the Prisoner's Dilemma game. New to this analysis is the discovery that those high in power play the Privileged game, a game with no dilemma. Also new is the extension of the analysis to the design of coalitions. That extension shows that, when coalition formation succeeds, it eliminates the dilemma of those low in power by transforming their game from Prisoner's Dilemma to Privileged. By contrast, exactly the same coalition structure does not alter the game played by those high in power. Applying well-known game theoretic solution concepts, we predict that low power coalitions will countervail power, but that coalitions of those high in power will not affect power exercise. Experiments testing this theory investigate 1) coalitions of those high in power, 2) low power coalitions organized against multiple high power positions, and 3) opposed coalitions struggling for power against each other. Results strongly support the theory.

Keywords:

• networks
• exchange theory
• rational choice theory
• power
• coalition formation
• ruling class

Authorship is equal. The research reported here was supported by SBR: 9423231 and IIS: 9817518, grants to David Willer from the National Science Foundation.

We wish to thank Marcel van Assen, as a JMS reviewer, and Brent Simpson for helpful comments. (We were so impressed by the quality of the review from Marcel that we asked for the name of the reviewer in order to express our gratitude to him in print.)

Notes

¹According to Rasmusen, “The strategy combination is a Nash equilibrium if no player has incentive to deviate from his strategy given that the other players do not deviate” (1989: 33).

²In network exchange research, a person or collectivity, termed ‘actor,’ is connected, via a ‘relation,’ to one or more other actors, forming a set of connected ‘positions,’ called a ‘network.’ In experiments on exchange networks, actors typically enter into negotiations over resource ‘pools' that are placed between relations. Exchange occurs when actors have agreed how to split the resource pool. In economic terms, the actors play a ‘zero-sum’ game; when one actor gains resources the other loses an equal amount. The actor with the larger part of the resource pool exercises power over the other. (For reviews, see Willer, 1999; Cook & Whitmeyer, 1992.)

³While beyond the scope of this paper, a future paper will extend Strategic analysis to non-bipartite networks.

⁴It matters not that Axelrod's (1984) second criterion for a PD game, 2R>T+S, is not met here in the first few games of the defection chain because no game is iterated. Instead, defection, the dominant strategy, links a series of games.

⁵According to Rasmusen, “Strategy s _i is a weakly dominant strategy if it is a player's best response to any strategies the other players might pick, in the sense that whatever strategies they pick, his payoff is no smaller with s _i than with any other strategy, and is greater in some strategy combination” (1989: 31).

⁶A colleague suggested that Nash subgame perfection is a more efficient way to analyze this structure. Nash, however, is not a procedure for finding games in structures. Nevertheless, once the games are found, Nash asserts that at the 21 – 3 point the equilibrium changes from defection to cooperation.

⁷We adopt the following definition from Rasmusen, “A ‘dominant strategy equilibrium’ is a strategy combination consisting of each player's dominant strategy’ (1989: 28). Although a ‘dominant strategy equilibrium’ is also a ‘Nash equilibrium,’ the converse is not always true. Additionally, a feasible solution x is considered to be ‘Pareto Optimal’ if no other feasible solution y beats or matches x on all objectives and beats x on at least one objective.

⁸The second-and higher order free rider problem exists when the sanctioning of first order defectors is costly. Additionally, as noted by Heckathorn (1989), higher order problems need not exist even when enforcement is costly, so long as the payoff increment (from higher rates of compliance) offsets the cost.

⁹Though favored by sanctioning, in other regards, the powers given to the coalitions were quite limited. For example, unions can strike against corporations one at a time, but our coalition of low power players could not coordinate so as to systematically exclude one high power player while including others.

^a n = 10 for each mean. To avoid repeated measures, the ten data points are from ten separate sessions. The two parts of each session gave one datum point each. Within each part the mean is for the last 6 of 12 rounds of negotiation. For each round, there were maximally three exchanges.

^b All t-tests are 2-tailed.

¹⁰For any t, when (R=T) and (S=P), it is assumed that a player's uncertainty of strategy will produce one further defection.