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Psychological Review Psychological Review The Dual Accumulator Model of Strategic Deliberation and Decision Making Russell Golman, Sudeep Bhatia, and Patrick Bodilly Kane Online First Publication, December 23, 2019. http://dx.doi.org/10.1037/rev0000176 CITATION Golman, R., Bhatia, S., & Kane, P. B. (2019, December 23). The Dual Accumulator Model of Strategic Deliberation and Decision Making. Psychological Review. Advance online publication. http://dx.doi.org/10.1037/rev0000176 Psychological Review © 2019 American Psychological Association 2019, Vol. 1, No. 999, 000 ISSN: 0033-295X http://dx.doi.org/10.1037/rev0000176 The Dual Accumulator Model of Strategic Deliberation and Decision Making Russell Golman Sudeep Bhatia Carnegie Mellon University University of Pennsylvania Patrick Bodilly Kane McGill University What are the mental operations involved in game theoretic decision making? How do players deliberate (intelligently, but perhaps imperfectly) about strategic interdependencies and ultimately decide on a strategy? We address these questions using an evidence accumulation model, with bidirectional connections between preferences for the strategies available to the decision maker and beliefs regarding the opponent’s choices. Our dual accumulator model accounts for a variety of behavioral patterns, including limited iterated reasoning, payoff sensitivity, consideration of risk- reward tradeoffs, and salient label effects, and it provides a good quantitative fit to existing behavioral data. In a comparison with other popular behavioral game theoretic models fit at the individual subject level to choices across a set of games, the dual accumulator model makes the most accurate out-of-sample predictions. Additionally, as a cognitive-process model, it can also be used to make predictions about response time patterns, time pressure effects, and attention during deliberation. Stochastic sampling and dynamic accumulation, cognitive mechanisms foundational to decision making, play a critical role in explaining well-known behavioral patterns as well as in generating novel predictions. Keywords: behavioral game theory, sequential sampling, preference accumulation, bidirectional processing, cognitive modeling Supplemental materials: http://dx.doi.org/10.1037/rev0000176.supp Strategic decision making is an important feature of human dependent strategic settings—in which outcomes depend on the behavior. From cooperating on collaborative projects, to nego- choices of two or more decision makers—are ubiquitous in tiating agreements, to competing over limited resources, inter- everyday life. Game theory provides a mathematical framework with which decision making in strategic settings can be for- mally represented and analyzed (Hart, 1992; Luce & Raiffa, 1957; von Neumann & Morgenstern, 1944). It is an important area of research in economics, political science, biology, com- X Russell Golman, Department of Social and Decision Sciences, Car- puter science, philosophy, and psychology, where it is used to negie Mellon University; Sudeep Bhatia, Department of Psychology, Uni- describe the types of coordination, cooperation, and conflict versity of Pennsylvania; Patrick Bodilly Kane, Biomedical Ethics Unit, that can be observed in groups of decision makers. McGill University. Traditional game theory is concerned with studying the be- This document is copyrighted by the American Psychological Association or one of its allied publishers. Russell Golman and Sudeep Bhatia developed the model, ran the sim- havior of idealized decision makers, who deliberate rationally This article is intended solely for the personal use ofulations the individual user and is not to be disseminated broadly. generating predicted behavior across a variety of games, and wrote an initial draft of the article. Patrick Bodilly Kane performed the model and who find themselves in an equilibrium in which they can 1 fitting analysis and helped revise the article along with Russell Golman and form accurate expectations about each other’s choices. Not Sudeep Bhatia. We thank Sally Pan and Andrew Chang for superb research surprisingly, humans display pervasive and systematic depar- assistance. Earlier versions of this article were presented at the Foundations tures from rationality, which cannot be accounted for by the of Utility and Risk Conference, at meetings of the Society for Mathemat- traditional Nash equilibrium prediction or its various refine- ical Psychology and the Cognitive Science Society, and at the Network for Integrated Behavioral Sciences. Some of the results presented in this article have been published in the Proceedings of the 39th Annual Conference of 1 The traditional Nash equilibrium solution concept abstracts away from the Cognitive Science Society (2017). Funding for Sudeep Bhatia was any particular dynamical process that would presumably converge to it. received from the National Science Foundation Grant SES-1626825. The implicit assumption is that any reasonable process would converge to Correspondence concerning this article should be addressed to Rus- a Nash equilibrium. This is not so (see, e.g., Hofbauer & Sigmund, 2003), sell Golman, Department of Social and Decision Sciences, Carnegie and actually long-run behavior can be quite sensitive to initial conditions as Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213. E-mail: well as the specific assumptions about the dynamics describing how [email protected] players adjust their strategies over time (Golman, 2011). 1 2 GOLMAN, BHATIA, AND KANE ments. This has led to the growth of behavioral game theory, maker’s preferences, and these emerging preferences in turn in- which acknowledges bounded rationality (see Camerer, 2003a, fluence beliefs about the opponent’s choices. Decision makers 2003b; Colman, 2003).2 build up a sense of how much they like their own strategies as well Models of limited iterated reasoning or error-prone reasoning as how much the opponent might like his, but need not form relax the traditional assumptions of perfect rationality and sophis- probabilistic beliefs. They each choose the strategy they like best tication to better predict the behavior of boundedly rational deci- (which is not necessarily a best response to what they believe the sion makers. According to these models, boundedly rational deci- opponent will choose). Still, as decision makers are able to repre- sion makers may have simplistic beliefs about the behavior of their sent an opponent’s actions, they can respond in an intelligent opponents (or partners), but retain the capability to intelligently manner to what they think the opponent might do and also revise choose strategies that maximize their own rewards given these these beliefs as they deliberate (Goodie, Doshi, & Young, 2012; simplistic beliefs, as with Level-k reasoning (Nagel, 1995; Stahl & Hedden & Zhang, 2002). Wilson, 1994, 1995) or cognitive hierarchy theory (Camerer, Ho, We show that our dual accumulator model predicts a variety of & Chong, 2004). Alternatively, they may have sophisticated be- behavioral patterns, including limited iterated reasoning, payoff liefs about their opponents, but occasionally err in their choices of sensitivity, and consideration of risk-reward tradeoffs (Capra, Go- strategies, as with quantal response equilibrium (McKelvey & eree, Gomez, & Holt, 1999; Crawford, Costa-Gomes, & Iriberri, Palfrey, 1995). Bringing both elements together, decision makers 2013; Goeree & Holt, 2001). Additionally, it provides a compel- may hold simplistic beliefs as well as make occasional mistakes, as ling, cognitively grounded account of the effects of strategy sa- with noisy introspection (Goeree & Holt, 1999, 2004). lience on strategic choice (Crawford, Gneezy, & Rottenstreich, Although these behavioral game theoretic models have been 2008; Hargreaves Heap, Rojo Arjona, & Sugden, 2014; Mehta, developed by relaxing one or more of the unrealistically strong Starmer, & Sugden, 1994). Existing behavioral game theory mod- assumptions of Nash equilibrium, they nonetheless require the els cannot account for all of these patterns (with a parsimonious, formation of probabilistic beliefs and the calculation of expected consistent specification across games), and some of these effects values. Decision makers often struggle with these tasks in simple lie entirely outside the scope of many of these models. Finally, we choice settings, suggesting that they may not be employing them in fit our dual accumulator model to individual subject-level data complex strategic settings either (Gigerenzer & Todd, 1999). In- (assuming that each individual has stable parameter values across deed, Agranov, Potamites, Schotter, and Tergiman (2012) find that games) using an existing, publicly available data set (Stahl & approximately half of subjects are unable to best respond to Wilson, 1995) and compare our model’s fit against Level-k rea- computer opponents known to be playing a uniform mixed strategy soning, cognitive hierarchy theory, logit quantal response equilib- in the p-beauty contest game. Our approach is to begin with rium, and (telescoping) noisy introspection. In a hold-one-out plausible cognitive mechanisms—the same cognitive mechanisms analysis, we find that the dual accumulator model makes the most commonly thought to be at play in other domains of preferential accurate out-of-sample predictions. decision making—rather than with the traditional Nash equilib- Our dual accumulator model can be seen as a cognitive imple- rium benchmark. We aim to characterize
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