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Stackelberg Reasoning in Mixed-Motive Games: an Experimental Investigation

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Stackelberg Reasoning in Mixed-Motive Games: an Experimental Investigation Journal of Economic Psychology 19 (1998) 279±293 Stackelberg reasoning in mixed-motive games: An experimental investigation Andrew M. Colman *, Jonathan A. Stirk Department of Psychology, University of Leicester, Leicester LE1 7RH, UK Received 26 June 1996; accepted 5 May 1997 Abstract The Stackelberg heuristic is a simulation heuristic in which a player optimizes against best- reply counterstrategies, and a game is Stackelberg-soluble if the resulting Stackelberg strate- gies are in equilibrium. To test the hypothesis that players use this heuristic in Stackelberg-sol- uble games, 100 subjects played all 12 ordinally non-equivalent 2 ´ 2 games, nine of which (including Prisoner's Dilemma and Stag Hunt) were Stackelberg-soluble and three (including Battle of the Sexes and Chicken) of which were non-Stackelberg-soluble. Subjects signi®cantly preferred Stackelberg strategies in Stackelberg-soluble games, and a protocol analysis of stated reasons for choice showed that joint payo maximization and strategic dominance were given as reasons signi®cantly more frequently in Stackelberg-soluble than in non-Stackelberg-solu- ble games. Ó 1998 Elsevier Science B.V. All rights reserved. PsycINFO classi®cation: 2340 JEL classi®cation: C72 Keywords: Cognitive hypothesis testing; Decision making; Game theory; Heuristic modeling; Non-zero sum games * Corresponding author. Tel.: 01 16 2522170; fax: 01 16 2522067; e-mail: [email protected]. 0167-4870/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 7 - 4 8 7 0 ( 9 8 ) 0 0 0 0 8 - 7 280 A.M. Colman, J.A. Stirk / Journal of Economic Psychology 19 (1998) 279±293 1. Introduction There are grounds for believing that, in certain classes of games at least, players tend to use a form of reasoning that Colman and Bacharach (1997) called the Stackelberg heuristic. This is a generalization of the ``minorant'' and ``majorant'' models that von Neumann and Morgenstern (1944), Section 14.4.1, pp. 100±104 used to rationalize their solution of strictly competitive games. The assumption underlying the Stackelberg heuristic is that players reason with a type of simulation heuristic (Kahneman and Tversky, 1982) in which both players choose strategies that maximize their own payos on the assumption that any choice will invariably be met by a counterstrategy that maximizes the co-player's payo, as if the co-player could choose second in a perfect-information version of the game with foreknowledge of the ®rst player's choice. For example, a player in a simultaneous-choice game such as the well-known Prisoner's Dilemma game may use the following type of Stackelberg reasoning: Suppose my fellow prisoner were able to move second, with the bene®t of knowing what I had chosen to do. In that case, if I were to cooperate, then my fellow prisoner's sel®sh interest would be best served by respond- ing with a defecting choice, and I would end up with the worst possible payo. On the other hand, if I were to defect, then my fellow prisoner's self-interest would again be served by a defecting choice, but in this case I would get a slightly better payo. It follows that to do the best for my- self in the sequential version of the game I should defect; therefore I will defect in the actual simultaneous-choice version of the game. Formally, in any two-person game C áSi, Hiñ, where Si, i 2 {1, 2} is Play- er i's strategy set and Hi is a real-valued payo function de®ned on the set S S1 ´ S2, Player i applies the Stackelberg heuristic as follows. First, Player i assumes that Player j can anticipate Player i's thinking about the game as if choosing second with perfect information of Player i's choice. This implies that for every strategy si in Player i's strategy set Si, Player j, who is assum- edly rational, responds with a strategy f such that f(si) is invariably a best re- ply of Player j to si, so that Hj si; f si P Hj si; sj 8sj 2 Sj: Second, assuming that both players are fully informed about the rules of the game and the payo functions, Player i expects this to happen and chooses a payo-maximizing strategy based on the assumption that Player j's response will invariably be a best reply f(si). In other words, Player i chooses a count- A.M. Colman, J.A. Stirk / Journal of Economic Psychology 19 (1998) 279±293 281 erstrategy for which maxi Hi(si, f(si)) is attained. This is called Player i's Stackelberg strategy or h strategy. If in any game C the players' h strategies constitute a Nash equilibrium, then C is Stackelberg-soluble (or h-soluble) and the corresponding equilibrium is called a Stackelberg solution (or h so- lution). In a two-person game, a (Nash) equilibrium is an outcome in which each strategy is a payo-maximizing best reply to the co-player's strategy. There are several grounds for believing that the Stackelberg heuristic may in¯uence the thinking of human decision makers. First, the Stackelberg heu- ristic relies on evidential reasoning (Eells, 1985; Jerey, 1983; Nozick, 1969, 1993), a form of reasoning in which decisions are made by maximizing the conditional expected utilities of possible acts rather than the pure expect- ed utilities as in classical decision theory, the utilities of outcomes being weighted by their probabilities given the speci®ed acts, irrespective of wheth- er there is any causal connection between the acts and the outcomes. There is experimental evidence to show that people do, in certain circumstances, use evidential forms of reasoning. For example, Quattrone and Tversky (1984) reported that subjects expressed a greater willingness to vote in an election when they believed that the outcome would depend on the proportion of like-minded voters who voted rather than on the behaviour of non-aligned voters, even though the eect of an individual's vote would be negligible (and equal) in both cases, and they predicted that their preferred candidate would be more likely to win the election if they themselves voted. This is a clear example of evidential reasoning. Second, the Stackelberg heuristic is a form of simulation heuristic, by which people reason about events through an operation resembling the run- ning of a simulation model, and Kahneman and Tversky (1982) have provid- ed empirical evidence that people use simulation heuristics to analyse counterfactual propositions (e.g., ``If only I had driven home by my regular route, I would not have collided with the truck at the intersection'') by men- tally ``undoing'' events that have occurred and then running mental simula- tions of the events with the corresponding parameters of the model altered. The third ground for believing that people use Stackelberg reasoning is the powerful psychological appeal of payo dominance as a criterion for choosing between Nash equilibria in games, and the ability of experimental subjects to exploit payo-dominant equilibrium points by focusing successfully on them in certain pure coordination games (Schelling, 1960, Ch. 3; Mehta et al., 1994a, b) and in other games of common interests (Cooper et al., 1990, 1992a, b). If e and f are any two distinct Nash equilibria in a two-per- son game, then e (strictly) payo-dominates f i Hi(e) > Hi(f) for both 282 A.M. Colman, J.A. Stirk / Journal of Economic Psychology 19 (1998) 279±293 players i 2 {1, 2}. According to the payo-dominance principle, in any game, if one equilibrium point e payo-dominates all others, rational players will choose strategies corresponding to e. The Harsanyi and Selten (1988) general theory of equilibrium selection in games is based on this assumption (togeth- er with a slightly dierent principle of risk dominance), and most game the- orists have accepted its intuitive force (e.g., Crawford and Haller, 1990; Farrell, 1987, 1988; Gauthier, 1975; Lewis, 1969; Sugden, 1995). Games with unique payo-dominant Nash equilibria that are Pareto-opti- mal are called games of common interests (Aumann and Sorin, 1989), and al- though in any such game the payo-dominant equilibrium is the ``obvious'' choice, it has been argued that the payo-dominance principle cannot be jus- ti®ed by conventional game-theoretic reasoning (Gilbert, 1989, 1990). In a two-person common-interest game, for example, Player I has a reason to choose the strategy corresponding to the payo-dominant Nash equilibrium only if there is a reason to expect Player II to do likewise; but Player II has a reason to choose it only if there is a reason to expect Player I to choose it, and we have an in®nite regress that provides neither player with any reason for choosing the payo-dominant equilibrium. Colman and Bacharach (1997) ar- gued that the Stackelberg heuristic provides a clari®cation of the psycholog- ical appeal of payo dominance and an explanation for the otherwise inexplicable ability of experimental subjects to solve certain types of pure co- ordination games in practice. Colman (1997) has shown that all pure coordi- nation games are soluble by the Stackelberg heuristic with the help of the variable-frame theory of Bacharach (1993). Colman and Bacharach (1997) proved that every game of common interest is Stackelberg-soluble or h-soluble, that its h solution is its payo-dominant Nash equilibrium, and that in every game with Pareto-rankable Nash equilib- ria, a Stackelberg solution is a payo-dominant Nash equilibrium. They also proved that there are h-soluble games that are not games of common interests and games with payo-dominant Nash equilibria that are not h solutions.
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