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Superstition and Rational Learning
Abstract
We argue that some, but not all, superstitions can persist when learning is rational and players are patient, and illustrate our argument with an example inspired by the Code of Hammurabi. The code specified an "appeal by surviving in the river" as a way of deciding whether an accusation was true. According to our theory, a mechanism that uses superstitions two or more steps off the equilibrium path, such as "appeal by surviving in the river," is more likely to persist than a superstition where the false beliefs are only one step off the equilibrium path. (JEL C72, C73, D83, D84)
... In the case of clientelism, this might be done through the party machine (Stokes 2005;Nichter 2008;Larreguy, Marshall and Querubín 2016;Rueda 2017;Gosnell 1937;Gosnell 1933); for social norms, the in-group's density that enforces the behavior might function as the monitor (Cruz 2019;White and Laird 2020). For religious groups, the religious norms, religiosity, or a belief in the enforcement of supernatural incentives may be akin to monitoring-at least from the perspective of the religious voter (Boas and Smith 2015;Fudenberg and Levine 2006). ...
... Moreover, these immaterial incentives have the potential to be very persuasive as they do not necessarily rely on a budget, even if they are large. In a way, they can result in superstitious equilibria like those described byFudenberg and Levine (2006).It is worth mentioning that this theoretical model does not pretend to explain all scenarios where a voter faces exogenous incentives. Voting behavior is a complex subject, and it would be unwise to ignore that there are other mechanisms at work when voters that have received a promise of an exogenous incentive cast their votes. ...
Why would an individual yield to exogenous incentives and vote against their intrinsic preferences? I aim at answering this question by presenting a game that models the choices of voters who believe that they might receive an incentive associated with their behavior. This game considers the size of the incentive offered, the voter's choice as a strategic best-response to other voters' behavior, and the competitiveness of the election. I find equilibria that show that yielding to exogenous incentives is a best strategy under not very stringent conditions. Finally, I illustrate the predictions from the model using three cases where it has been observed that material or immaterial incentives affected-or still affect-the behavior of voters: vote-buying and voter coercion during the United States Gilded Age, the electoral behavior of African-Americans, and religious voting.
¿Por qué alguien cedería ante incentivos exógenos para votar en contra de sus preferencias intrínsecas? Busco responder esta pregunta presentando un juego de votación que modela las decisiones de votantes que creen que podrían recibir un incentivo asociado a su comportamiento. Este juego considera el tamaño del incentivo ofertado, la decisión del votante como una mejor respuesta estratégica al comportamiento de otros votantes, y la competitividad de la elección. Encuentro equilibrios que muestran que ceder ante incentivos exógenos es la mejor estrategia bajo condiciones no muy rigurosas. Finalmente, ilustro las predicciones del modelo usando tres casos en donde se ha observado que incentivos materiales o inmateriales afectaron-o siguen afectando-el comportamiento del votante: compra de votos y la coerción a votantes durante la Edad Dorada de los Estados Unidos, el comportamiento electoral de los Afro-Americanos, y el voto religioso.
... In practice, such biases may arise for a variety of reasons ranging from a lack of role models (Nguyen 2008; Porter and Serra 2020), imperfect information about available financial aid (Roderick et al. 2009;Johnson et al. 2011;Dynarski et al. 2018), its administrative complexity (Bettinger et al. 2012), poverty generating aspiration failures (Ray 2006;Dalton et al. 2016;Genicot and Ray 2017), to simply being generated by media reports of students burdened by their loans (Advisory Committee on Student Financial Assistance 2001). Finally, findings in the literature studying effects of perceived returns to effort support the modeling assumption of human capital accumulation depending on students' beliefs (Azmat and ...
... However, one could extend it into a repeated decision problem (for with multiple stages of education or overlapping generations) and introduce motives for experimenting with off-equilibrium path decisions. Fudenberg and Levine (2006) show that in such a framework beliefs of biased agents (or their future generations) could indeed converge to the objective ones under two conditions: (1) if agents were sufficiently patient, and (2) if the educational decision was taken sufficiently often. Unfortunately, in reality the educational decisions are taken rather rarely and as such the biased beliefs can well survive in the long-run. ...
Investigating interaction of the lumpy nature of educational investments and informational frictions on returns to and costs of education, I show that pessimistic beliefs can be self-confirmed in equilibrium. Among some of its consequences, I argue that the commonly pursued research methods may not always identify the true underlying skill distributions.
... The parents use this superstition to make the children do not waste their food. This finding were relevant with the research done by Fudenberg andLevine in 2006. Fudenberg andLevine (2006:632) have done research about how superstition can survive if human have rational think and also the type of superstition. ...
... the effect caused; there is irrational thought happen, when people leave the leftover food. It shows that KubuangTigoBalehsociety belief this as great sumbang and it is not ethic if we leftover the food. The parents use this superstition to make the children do not waste their food. This finding were relevant with the research done by Fudenberg andLevine in 2006. Fudenberg andLevine (2006:632) have done research about how superstition can survive if human have rational think and also the type of superstition. Fudenberg and Levine made a game theoretic model using rational learning, so that it elevate superstition beyond the events. Besides, Fudenberg and Levine characterized the condition of misbelieve that s ...
... We start with a simple, but instructive result that requires a preliminary de…nition. We say that ( ; f ) has own-strategy independent feedback if what each player observes ex post about the strategies of other players is independent of his own strategy, that is, for each 32 This is a restrictive assumption that is violated by all of our examples. 33 The next result illustrates its strength. ...
... The intuition is as follows: as an ambiguity averse agent comes close to learning the probabilities of payo¤s associated to strategies played many times, the strategies tried only a few times and hence involving higher uncertainty become relatively less appealing (see Anderson, 2012). Fudenberg and Levine (2006) obtain a re…nement of SCE based on the idea of steady-state learning with extremely patient and long-lived agents and illustrate it with an application. Similar ideas are worth exploring in the current framework. ...
We propose to bring together two conceptually complementary ideas: (1) selfconfi?rming equilibrium (SCE): at a rest point of learning dynamics in a game played recurrently, agents best respond to confi?rmed beliefs, i.e. beliefs consistent with the evidence they accumulate, and (2) ambiguity aversion: agents, coeteris paribus, prefer to bet on events with known rather than unknown probabilities, more generally, agents distinguish objective from subjective uncertainty, which is captured by their ambiguity attitudes. Using as a workhorse the ?smooth ambiguity model of Klibanoff, Marinacci and Mukerji (2005), we provide a de?nition of "smooth SCE" which generalizes the traditional concept of Fudenberg and Levine (1993a,b), called Bayesian SCE, and admits Waldean (maxmin) SCE as a limit case. We show that the set of equilibria expands as ambiguity aversion increases. The intuition is simple: by playing the same "status-quo" strategy in a stable state an agent learns the implied objective probabilities of payoffs, but alternative strategies yield payoffs with unknown probabilities; keeping beliefs ?fixed, increased aversion to ambiguity makes such strategies less appealing. We rely on this core intuition to show that different notions of equilibrium are nested in a simple way, from ?ner to coarser: Nash, Bayesian SCE, Smooth SCE and Waldean SCE. We also prove some equivalence results, under special assumptions about the information structure.
... In these cases, the superstitious belief is insulated from falsification due to the unobservability of the cause itself. A detailed explanation of this type is provided by Fudenberg and Levine (2006) who argue that, ". . . a mechanism that uses superstitions two or more steps off the equilibrium path . . . is more likely to persist (p. ...
... Observations weighted using sampling weights. *** p<0.01, ** p<0.05, * p<0.1 Data are drawn from the2000, 2002, 2004, and 2006 ...
Why do seemingly irrational superstitions persist? We analyze the widely held belief among Asians that children born in the Year of the dragon are superior. We use pooled cross section data from the U.S. Current Population Survey to show that Asian immigrants to the United States born in the 1976 year of the dragon are more educated than comparable immigrants from non-dragon years. In contrast, no such educational effect is noticeable for dragon-year children in the general U.S. population. We also provide evidence that Asian mothers of dragon year babies are more educated, richer, and slightly older than Asian mothers of non-dragon year children. This suggests that belief in the greater superiority of dragon-year children is self-fulfilling since the demographic characteristics associated with parents who are more willing and able to adjust their birthing strategies to have dragon children are also correlated with greater investment in their human capital.
... One exception is Fudenberg and Levine (2006). The authors ask how superstitions can persist if we are ever rational, and if so, which types of superstitions would be more likely to persist. ...
... However, some "clever" practices can get around this restriction. 9 This would have implications on our estimation strategy. Personalized plates allows vehicle owners to, subject to certain restrictions, personalize their plate numbers up to eight digits (e.g., 1 LOVE U, WWW, RELAX, etc.). ...
This paper estimates the value of superstitions by studying the auctions of vehicle license plates. We show that the value of superstitions is economically significant, which justifies their persistence in human civilization. We also document the changes of the value of superstitions across different types of plates, across different policy regimes, and across different macroeconomic environments. Interestingly, some of the changes are rather consistent with economic intuition.
... In summary, the ban on lending leads to an outcome that is opposite to the values that the books are believed to promote. Further, these credit constraints reduce welfare, increase corruption because it limits the ability of would-be-entrepreneurs to open new businesses and thus increase the influence of wealthy wizards (Paldam, 2002;Fudenberg and Levine, 2006). ...
Recent studies in psychology and neuroscience offer systematic evidence that fictional works exert a surprisingly strong influence on readers and have the power to shape their opinions and worldviews. Building on these findings, we study what we term Potterian economics, the economic ideas, insights, and structure, found in Harry Potter books, to assess how the books might affect economic literacy. A conservative estimate suggests that more than 7.3 percent of the world population has read the Harry Potter books, and millions more have seen their movie adaptations. These extraordinary figures underscore the importance of the messages the books convey. We explore the Potterian economic model and compare it to professional economic models to assess the consistency of the Potterian economic principles with the existing economic models. We find that some of the principles of Potterian economics are consistent with economists models. Many other principles, however, are distorted and contain numerous inaccuracies, contradicting professional economists views and insights. We conclude that Potterian economics can teach us about the formation and dissemination of folk economics, the intuitive notions of naive individuals who see market transactions as a zero-sum game, who care about distribution but fail to understand incentives and efficiency, and who think of prices as allocating wealth but not resources or their efficient use.
... Since beliefs or even superstitions (see, e.g., Fudenberg & Levine, 2006, and the experimental "god games" in Hajikhameneh & Iannaccone, 2023) about luck are common, perhaps people form beliefs about the dice. Still, the direction is unclear; some subjects might expect positive serial correlation (the "hot hand") and others might expect negative serial correlation (the "law of averages"); this was confirmed in the comments made. ...
Do people know their own risk preferences, or do risk choices change with experience and observation? We provide a straightforward test in the laboratory. People make an initial decision concerning a lottery choice and then experience 24 unpaid practice periods in which they roll the dice, record the outcome, and record the would-be payoff. They then make a final decision for the lottery choice; one of the first and last periods is randomly chosen for payment. Our primary hypothesis is that people will become less risk-averse by having made and experienced the practice rolls. We do find that people are significantly more likely to become less risk-averse than more risk-averse over time. We note that this move towards assuming increased risk goes in the opposite direction from what is at least arguably predicted by loss aversion and reference dependence. We find that women’s preferences change much less during a session than men’s preferences change. We feel that our literally hands-on approach ensures a degree of engagement that helps to accelerate the learning process. We argue that measures obtained after people have had experience with a mechanism are more meaningful, and that this principle might well extend more generally to other elicitation tasks.
... Superstitions are one of many types of partially oral folklore that are often found around society [19]. The term superstition refers to beliefs or influences that the rational mind cannot match [21]. Superstitions are irrational beliefs in which an object, action, or goal is associated with an event that affects its outcome [30][33] [34]. ...
... Superstitious tendencies are defined as irrational thoughts or practices involving supernatural ideation that are objectively unscientific (Fudenberg & Levine, 2006;Vyse, 1997). Although superstitious tendencies actively attribute the outcomes of events to fate or magic, which often diverge from the truth, they ubiquitously exist across cultures. ...
Whether and how interpersonal experiences predispose people to show superstitious tendencies have been largely unexamined by past studies. By adopting a multimethod approach, three studies tested (a) whether ostracism increases superstitious tendencies through thwarted perceived control, (b) whether the dispositional need for closure moderates the effect of ostracism on superstitious tendencies and (c) whether restoring ostracized people's thwarted control weakens their superstitious tendencies. The results revealed that ostracized participants had higher superstitious tendencies than nonostracized participants did (Studies 1-3). Moreover, thwarted control mediated the effect of ostracism on superstitious tendencies (Study 2). In addition, the dispositional need for closure moderated the effect of ostracism on superstitious tendencies, such that the effect was stronger among participants with a high need for closure (Studies 1-2). Finally, restoring ostracized participants' perceived control weakened the effect of ostracism on superstitious tendencies (Study 3). Altogether, these findings feature the essential role of thwarted perceived control in understanding the link between ostracism and superstitious tendencies and the implication of control restoration in weakening the link. They also highlight the importance of dispositional characteristics in moderating people's responses to superstitions following ostracism and related forms of interpersonal maltreatment.
... In summary, the ban on lending leads to an outcome that is opposite to the values that the books are believed to promote. Further, these credit constraints reduce welfare, increase corruption because it limits the ability of would-be-entrepreneurs to open new businesses and thus increase the influence of wealthy wizards (Paldam, 2002;Fudenberg and Levine, 2006). ...
Recent studies in psychology and neuroscience offer systematic evidence that fictional works exert a surprisingly strong influence on readers and have the power to shape their opinions and worldviews. Building on these findings, we study 'Potterian economics', the economic ideas, insights, and structure, found in Harry Potter books, to assess how the books might affect economic literacy. A conservative estimate suggests that more than 7.3% of the world's population has read the Harry Potter books, and millions more have seen their movie adaptations. These extraordinary figures underscore the importance of the messages the books convey. We explore the Potterian economic model and compare it to professional economic models to assess the consistency of the Potterian economic principles with the existing economic models. We find that some of the principles of Potterian economics are consistent with economists' models. Many other principles, however, are distorted and contain numerous inaccuracies, contradicting professional economists' views and insights. We conclude that Potterian economics can teach us about the formation and dissemination of folk economics-the intuitive notions of naïve individuals who see market transactions as a zero-sum game, who care about distribution but fail to understand incentives and efficiency, and who think of prices as allocating wealth but not resources or their efficient use.
... In summary, the ban on lending leads to an outcome that is opposite to the values that the books are believed to promote. Further, these credit constraints reduce welfare, increase corruption because it limits the ability of would-be-entrepreneurs to open new businesses and thus increase the influence of wealthy wizards (Paldam, 2002;Fudenberg and Levine, 2006). ...
Recent studies in psychology and neuroscience offer systematic evidence that fictional works exert a surprisingly strong influence on readers and have the power to shape their opinions and worldviews. Building on these findings, we study ‘Potterian economics’, the economic ideas, insights and structure, found in Harry Potter books, to assess how the books might affect economic literacy. A conservative estimate suggests that more than 7.3% of the world’s population has read the Harry Potter books, and millions more have seen their movie adaptations. These extraordinary figures underscore the importance of the messages the books convey. We explore the Potterian economic model and compare it to professional economic models to assess the consistency of the Potterian economic principles with the existing economic models. We find that some of the principles of Potterian economics are consistent with economists’ models. Many other principles, however, are distorted and contain numerous inaccuracies, contradicting professional economists’ views and insights. We conclude that Potterian economics can teach us about the formation and dissemination of folk economics—the intuitive notions of naïve individuals who see market transactions as a zero-sum game, who care about distribution but fail to understand incentives and efficiency and who think of prices as allocating wealth but not resources or their efficient use.
‘I think the writers [of popular literature] are not particularly sympathetic to or don’t understand how a market works. It’s not easy to convey that to a child. It’s not always easy to convey it to grown-ups.’
Gary Becker (New York Times, August 21, 2011, p. SR5).
‘With all due respect to Richard Posner, Cass Sunstein, or Peter Schuck [reference to the books these scholars published in 2005], no book released in 2005 will have more influence on what kids and adults around the world think about government than [Rowling’s] The Half-Blood Prince.’
Benjamin Barton (Michigan Law Review, 2006, p. 1525).
‘As economic theorists, we organize our thoughts using what we call models. The word “model” sounds more scientific than “fable” or “fairy tale” although I do not see much difference between them. The author of a fable draws a parallel to a situation in real life. He has some moral he wishes to impart to the reader. The fable is an imaginary situation that is somewhere between fantasy and reality. Any fable can be dismissed as being unrealistic or simplistic, but this is also the fable’s advantage. Being something between fantasy and reality, a fable is free of extraneous details and annoying diversions. In this unencumbered state, we can clearly discern what cannot always be seen in the real world. On our return to reality, we are in possession of some sound advice or a relevant argument that can be used in the real world.”
Ariel Rubinstein (Econometrica, 2006, p. 881).
‘An investigation of novels and [economic] models…may help us better understand how the public thinks about economic issues.’
Tyler Cowen (The Street Porter and the Philosopher: Conversations on Analytical Egalitarianism, 2008, p. 321).
... persist not due to the pressure of vertical cultural transmission or ignorance, but due to the fact that individual conformity to the status quo institutions and cultural norms may simply be a mutual best response. Fudenberg and Levine (2006) show that false beliefs about off-equilibrium actions, which they call "superstitions," can persist for a long time if they prevent behavior that could reveal the information needed to disconfirm those false beliefs. 43 See also Bisin and Verdier (2015) for a more general theoretical model studying the interaction between culture and institutions and its effect on economic activity. ...
This chapter talks about three major intertwined themes that emerge in the new cultural economics: causal effects of culture on economic outcomes and institutions, the origins of culture, and the issues of cultural transmission, persistence, and change. It emphasizes the studies related to the field of long-run economic growth and development which explores culture as one of the key “deep determinants” of economic performance. The chapter reviews some of the recent empirical studies attempting to identify the causal effects of culture, and examines the research on the origins of culture, its social benefits, and costs. It also considers the evidence on cultural persistence and discusses the mechanisms of cultural transmission and change.
... set inclusion) sequentially -supporting set in Γ(u) coincides with the largest perfectly -supporting set in Γ(u). Remark Fudenberg and Levine (1993) introduced the notion of self-confirming equilibrium that is a generalization of Nash equilibrium for extensive-form games, in which players correctly predict the actions their opponents take along the equilibrium path, but may have misconceptions about what their opponents would do at information sets that are never reached when the equilibrium is played [see also Fudenberg and Levine (2006, 2009), Esponda (2013, Battigalli et al. (2015), and Fudenberg and Kamada (2015)]. 12 Dekel et al. (1999Dekel et al. ( , 2002) developed a refinement of sequentially rationalizable self-confirming equilibrium (SRSCE), which is related to assuming sequential rationality and almost common certainty of payoffs in an epistemic model with independent beliefs [cf. ...
We extend the generic equivalence result of Blume and Zame (Econometrica 62:783–794, 1994) to a broader context of perfectly and sequentially rational strategic behavior (including equilibrium and nonequilibrium behavior) through a unifying solution concept of “mutually acceptable course of action” (MACA) proposed by Greenberg et al. (Econ Theory 40:91–112, 2009. https://doi.org/10.1007/s00199-008-0349-5). As a by-product, we show, in the affirmative, Dekel et al.’s (J Econ Theory 89:165–185, 1999) conjecture on the generic equivalence between the sequential and perfect versions of rationalizable self-confirming equilibrium.
... Sebagian besar peneliti dari beragam disiplin ilmu telah membahas hal mengenai superstisi. Fudenberg & Levine, (2006) Urang hamil indak bulieh makan jo piriang rompeang beko rompeang anaknyo. ...
... Any Bpresent bias ( Frederick et al. 2002) will lead them to weight this aspect of the deal substantially less than immediate benefits and costs. This tendency may be strengthened by the negative effect of cognitive load on self-control (Baumeister et al. 2007;Fudenberg and Levine 2006;Shiv and Fedorikhin 1999;Ward and Mann 2000). Lower self-control is linked to higher use of easy-toaccess credit products (Gathergood 2012). ...
Personal contract purchase (PCP) plans are innovative and increasingly popular forms of car finance. PCPs are inherently more complex than established financing options. The present study used experimental behavioural science to explore consumers’ comprehension of PCP plans and scope for beneficial interventions. Choice tasks, product rating tasks, and multiple choice comprehension questions were deployed to measure the consistency of decision-making and explicit comprehension of the product. Disclosures and advice were varied across conditions. A representative sample (n = 100) of consumers was initially given information on PCP deals as typically disclosed by car dealers. Results revealed that understanding was poor. One quarter of participants performed below chance on multiple-choice comprehension questions. Participants were prone to inconsistencies and objective mistakes when deciding between and rating offers. Disclosures designed to improve processing of mileage and cost information had ambiguous effects. Consumer advice sheets improved comprehension and reduced mistakes, with advice containing a diagram outperforming advice containing only text. The findings raise consumer protection concerns and support improved advice and stronger regulation. © 2018 Springer Science+Business Media, LLC, part of Springer Nature
... Note that there are other stronger notions of equilibria, for example, Markov Perfect equilibrium(Maskin & Tirole, 2001) requires policies to be best responses at all states. It is known that such equilibria are harder to find by learning(Fudenberg & Levine, 2006;Fudenberg & Peysakhovich, 2016). It is interesting to consider whether different kinds of choices (e.g. ...
Social conventions - arbitrary ways to organize group behavior - are an important part of social life. Any agent that wants to enter an existing society must be able to learn its conventions (e.g. which side of the road to drive on, which language to speak) from relatively few observations or risk being unable to coordinate with everyone else. We consider the game theoretic framework of David Lewis which views the selection of a social convention as the selection of an equilibrium in a coordination game. We ask how to construct reinforcement learning based agents that can solve the convention learning task in the self-play paradigm: at training time the agent has access to a good model of the environment and a small amount of observations about how individuals in society act. The agent then has to construct a policy that is compatible with the test-time social convention. We study three environments from the literature which have multiple conventions: traffic, communication, and risky coordination. In each of these we observe that adding a small amount of imitation learning during self-play training greatly increases the probability that the strategy found by self-play fits well with the social convention the agent will face at test time. We show that this works even in an environment where standard independent multi-agent RL very rarely finds the correct test-time equilibrium.
... The purpose of this section is to delineate the key hypotheses of Chinese numerological belief as discussed in the introduction section. We specifically refer to Fudenberg and Levine's (2006) theoretical model in forming our superstition hypotheses. This paper theorizes that superstitious beliefs can persist if their probability of being exposed as untrue is sufficiently low. ...
The purpose of this paper is to examine Malaysian motorists’ willingness to pay for Chinese lucky plate numbers. Regression analyses reveal that the Chinese bidders are more likely to win and pay significantly more for the Chinese lucky numbers than their non-Chinese counterparts. Consistent with resale motive considerations, the non-Chinese bidders impose the same discount for unlucky numbers as their Chinese counterparts but do not pay a premium for the lucky numbers. We also document the influence of peer effects on the winning bid price. Specifically, Chinese bidders tend to pay more for the same plate numbers won by their peers in the previous auction, whereas both Chinese and non-Chinese bidders are less averse to the unlucky numbers if there are precedents of such ownership among their peers. Overall, our findings suggest that policymakers and marketers need to be sensitive to the subtle differences in the cultural cues of consumers in a multiracial society such as Malaysia.
... In summary, the ban on lending leads to an outcome that is opposite to the values that the books are believed to promote. Further, these credit constraints reduce welfare, increase corruption because it limits the ability of would-be-entrepreneurs to open new businesses and thus increase the influence of wealthy wizards (Paldam, 2002;Fudenberg and Levine, 2006). ...
Recent studies in psychology and neuroscience find that fictional works exert strong influence on readers and shape their opinions and worldviews. We study the Potterian economy, which we compare to economic models, to assess how Harry Potter books affect economic literacy. We find that some principles of Potterian economics are consistent with economists’ models. Many others, however, are distorted and contain numerous inaccuracies, which contradict professional economists’ views and insights, and contribute to the general public’s biases, ignorance, and lack of understanding of economics.
... Thus, the ban on lending leads to an outcome that is opposite to the values that the books are believed to promote. Further, these credit constraints reduce welfare, increase corruption because it limits the ability of would-beentrepreneurs to open new businesses, and increase the influence of wealthy wizards (Paldam 2002, Fudenberg andLevine 2006). ...
... Registrars of each municipality in which marriages are celebrated are required by law to fill and send to the Italian Central Office of Statistics (Istat) a form for each marriage. 6 This form provides both information on the marriage itself (the date and province where the marriage has been celebrated, 5 See also Fudenberg and Levine (2006) for a formal game-theoretic model explaining why superstitious beliefs may survive even if the players are rational. 6 In addition, religious marriages are accounted for in this data collection. ...
In Italy, it is believe d that Tuesdays and Fridays are particularly un luckydays for weddings as well as the 17th day of each month. Previous studies realizedin the aftermath of the Second World War have shown the strong influence that these superstitions had in determining the wedding dates in the entire country. We have used exhaustive data collection of all marriages celebrated in Italy in the years 2007–2009 to investigate whether superstitions are still able to influence the choicesof spouses. We find that this influence is still present after the great economic, socialand demographic transformation of Italian society. We also show that a wife’s education reduces the influence of superstition on the choice of the date of marriage while those who opt for a religious rite are also those who are more careful in avoiding inauspicious days.
... persist not due to the pressure of vertical cultural transmission or ignorance, but due to the fact that individual conformity to the status quo institutions and cultural norms may simply be a mutual best response. Fudenberg and Levine (2006) show that false beliefs about off-equilibrium actions, which they call "superstitions," can persist for a long time if they prevent behavior that could reveal the information needed to disconfirm those false beliefs. 43 See also Bisin and Verdier (2015) for a more general theoretical model studying the interaction between culture and institutions and its effect on economic activity. ...
This paper reviews recent economics literature on culture, with an emphasis on its relation to the field of long-run growth and development. It examines the key issues debated in the new cultural economics: causal effects of culture on economic outcomes, the origins and social costs of culture, as well as cultural transmission, persistence, and change. Some of these topics are illustrated in application to the economic analysis of envy-related culture.
... In that case, players may remain holding differing beliefs about the other players' and nature's moves in those contingencies even after arbitrarily long interactions. In particular, in games against nature, the players may disagree about the consequences in counterfactual situations, and some superstitions may persist (Fudenberg and Levine 2006). Note that, as in our paper, the main reason for lack of learning and asymptotic agreement is the lack of identification: one cannot identify the strategies from the observable moves. ...
Under the assumption that individuals know the conditional distributions of signals given the payoff-relevant parameters, existing results conclude that as individuals observe infinitely many signals, their beliefs about the parameters will eventually merge. We first show that these results are fragile when individuals are uncertain about the signal distributions: given any such model, vanishingly small individual uncertainty about the signal distributions can lead to substantial (nonvanishing) differences in asymptotic beliefs. Under a uniform convergence assumption, we then characterize the conditions under which a small amount of uncertainty leads to significant asymptotic disagreement.
... The same happened in 1293 in Pistoia, where the coeval chronicles reported that after an earthquake, which repeatedly struck the city for eight days, all the citizens participated in a number of processions (Adrasto Barbi 1927). 5. The belief that God was the ultimate cause of natural events and the associated rituals prescribed by the Catholic religion can be interpreted in a framework in which superstition is consistent with rational learning (Fudenberg and Levine 2006). ...
This paper presents a unique historical experiment to explore the dynamics of institutional change in the Middle Ages. We have assembled a novel dataset, where information on political institutions for northern-central Italian cities between 1000 and 1300 is matched with detailed information on the earthquakes that occurred in the area and period of interest. Exploiting the panel structure of the data, we document that the occurrence of an earthquake retarded institutional transition from autocratic regimes to self-government (the commune) in cities where the political and the religious leaders were one and the same person (Episcopal see cities), but not in cities where political and religious powers were distinct (non-Episcopal see cities). Such differential effect holds both for destructive seismic episodes and for events that were felt by the population but did not cause any material damage to persons or objects. Ancillary results show that seismic events provoked a positive and statistically significant differential effect on the construction and further ornamentation of religious buildings between Episcopal and non-Episcopal see cities. Our findings are consistent with the idea that earthquakes, interpreted in the Middle Ages as manifestation of the will and outrage of God, represented a shock to people’s religious beliefs and, as a consequence, enhanced the ability of political-religious leaders to restore social order after a crisis relative to the emerging communal institutions. This interpretation is supported by historical evidence.
... On the other hand, finding a horseshoe or a four-leaf clover or carrying a specific talisman or token in one's wallet or purse can supposedly bring luck and ward off evil. 1 Associating some events with luck, or the absence thereof, may have rational or practical origins. 2 However, believing in lucky and unlucky numbers or days, despite being very wide-spread, seems devoid of any rational basis. For example, the number 13 is considered almost universally unlucky in the European culture and Friday the 13th, is seen as a particularly inauspicious day. 3 Fudenberg and Levine (2006) formulate a game-theoretic model to show that some superstition can emerge even if people are rational. In particular, they argue that superstitions are more likely to be believed if they are further removed from the game equilibrium path. ...
We use the UK Labor Force survey to investigate whether the socio-economic outcomes of people born on the 13th day of the month, and of those born on Friday the 13th, differ from the outcomes of people born on more auspicious days. In many European countries, including the UK, number 13 is considered unlucky and Friday the 13th is seen as an especially unlucky day. We find little evidence that people born on the 13th or those born on Friday the 13th are significantly less likely to be employed, earn lower wages or that they are more likely to stay unmarried compared to people born on other days.
... In Nigeria, traditional ceremonies still accompany yam production and consumption and belief in certain sociocultural practices (Olayide, 1980, Orkwor et al. 1998, Diop, 1998, Stifel et al., 2009). On the relationship between taboos and agriculture and household consumption, Benabou and Triole (2007) and Fudenberg and Livine (2006) developed theoretical model to explain the persistence of superstitions and taboos involved in yam production. In addition, Anyanwu et al. (2003) and Simpa (2011) confirmed that some of the socio-cultural practices includes designating some lands as evil hence not cultivated; designating some work taboo days in which crops are not planted and animals are not grazed; and prohibition of female gender from some farm operations or from production of certain crops; and that this situation is affected by farmers' level of education. ...
The study investigated whether socio-cultural factors accentuate technical efficiency of yam
farmers in Kogi State, Nigeria in addition to the socio-economic normally postulated.
Primary data collected from 180 yam farmers randomly selected from three local government
areas, one from each of the socio-cultural group of the State was used. Results indicated that
there is more number of socio-cultural factors that determine the level of technical efficiency
of yam farmers than the socio-economic. The results further show that male farmers are more
affected by socio-cultural factors than female. In addition, the Okuns seems to be more
affected while the Igalas were least affected. However, some of the socio-cultural practices
are shrouded in some form of secrecy and research effort should be geared towards empirical
understanding of their operation. Yam farmers should be provided with more comprehensive
and adequate extension support to manage their farms in line with modern and improved
production technologies, rather than basing their production decisions on factors alien to
modern agricultural production.
... For example, Morris (1996) shows that after an initial public offering, there is room for a speculative premium to emerge before all traders learn about the true distribution of the asset's dividend. In a game-theoretic model with rational learning, Fudenberg and Levine (2006) characterize the conditions under which superstitious beliefs can be expected to persist over time. ...
We provide the first solid evidence that Chinese superstitious beliefs can have significant effects on house prices in a North American market with a large immigrant population. Using real estate data on close to 117,000 house sales, we find that houses with address number ending in “4” are sold at a 2.2% discount and those ending in “8” are sold at a 2.5% premium in comparison to houses with other addresses. These price effects are found either in neighborhoods with a higher than average percentage of Chinese residents, consistent with cultural preferences, or in repeated transactions, consistent with speculative behavior. (JEL D03, R2, Z1)
... 4 The defaults are exactly opposite. If nothing happens, the criminal goes 1 There is a small body of game theory literature with similar undertakings (some tackling fiction, some early legal codes); see Aumann and Maschler's (1985) and Aumann's (2003) studies of the Talmud; Brams' (1983 Brams' ( , 1994 Brams' ( , 2003) studies of the Hebrew Bible and other literary sources; Fudenberg and Levine's (2005) interpretation of the code of Hammurabi; and Mehlmann's (1989, 2000) game theortic studies of Goethe's Faust Brams (1983) argues that superior beings, like the God of the Old Testament, sometimes appear to make mistakes and, thus, appear not to be omniscient. While we could make all the main points in our paper if we relaxed the assumption of omniscience, the exposition is much easier if we stick to the traditional idea of an all-knowing god. ...
In this paper we examine the strategic savviness of the medieval church as portrayed in Richard Wagner's Tannhäuser. We show that the church employed an optimal randomization strategy based on arguments of dominance or trembling-hand perfection. Particular attention is paid to the employed randomization device.
... Fudenberg and Levine argue that this law is based on a superstition and show that Hammurabi has it " exactly right " , in the sense that this law " uses the greatest amount of superstition consistent with patient rational learning " . In a subsequent paper, these economic theorists (Fudenberg and Levine, 2006), discuss several games inspired by the second law in the Code of Hammurabi. The motivation for their paper is to understand why superstitions persist over long periods of time. ...
... Other contributions in that small literature includeAumann and Maschler (1985),Fudenberg and Levine (2005),Harmgart, Huck, and Müller (2006), or Mehlmann (2000). ...
Tannhäuser und der Sängerkrieg auf Wartburg is Richard Wagner's only opera whose title points the audience to a particular scene of the enfolding drama,1 and there can indeed be little doubt that the tournament of song in the last scene of the middle act is the linchpin of its story. It is Tannhäuser's confessional outburst that sets in motion the irresistible chain of events that, ultimately, leads to his and Elisabeth's deaths and redemption. At the same time the scene poses a puzzle that has led some authors to the conclusion that the opera's conception is fundamentally flawed. Here we argue that such a conclusion can be avoided by analysing Tannhäuser's apparently inconsistent behaviour in act two from a rational choice
... The problem, however, is that other stories could be told as well. One could model the same effect by assuming that trust exists by the same mechanisms that superstitions persist—by a lack of willingness to test a hypothesis about potential harms that, if proven correct, could have catastrophic personal consequences (Fudenberg and Levine 2006). Guidance as to which of these routes to explanation is to be the most promising is provided by having a deep understanding of the psychological basis of such effects—something that is the traditional domain of behavioral research. ...
Game-theoretic models of marketing channels typically rely on simplifying assumptions that, from a behavioral perspective, often appear naïve. However, behavioral researchers have produced such an abundance of behavioral regularities that they are impossible to incorporate into game-theoretic models. We believe that a focus on three core findings would benefit both fields; these are: First, beliefs that are held by the various players regarding profit consequences of different actions are incomplete and often biased; second, players’ preferences and optimization objectives are not commonly known; and third, players have insufficient cognitive abilities to achieve optimization objectives. Embracing these three findings shifts the focus from rational decision making to how decision makers learn to improve their decision making skills. Concluding, we believe that greater convergence of game-theoretic modeling and behavioral research in marketing channels would lead to new insights for both fields.
... The problem, however, is that other stories could be told as well. One could model the same effect by assuming that trust exists by the same mechanisms that superstitions persist-by a lack of willingness to test a hypothesis about potential harms that, if proven correct, could have catastrophic personal consequences (Fudenberg and Levine 2006). Guidance as to which of these routes to explanation is to be the most promising is provided by having a deep understanding of the psychological basis of such effects-something that is the traditional domain of behavioral research. ...
This paper reviews the literature that applies behavioral economic models to managerial decisions. It organizes the literature into research that focuses on alternative utility functions and research that focuses on non-equilibrium models. Generally, behavioral models have seen less application to manager decisions than to consumer decisions and therefore there are many opportunities to develop new theoretical models, new laboratory experiments, and new field applications. The application of these models to field data is particularly underdeveloped.
... However, the incentives to experiment depend on how useful the results will be: if an opportunity to experiment arises infrequently, then there is little incentive to actually carry out the experiment. This has implications for backwards induction explored in Fudenberg & Levine (2006), who re-examined the steady-state model of Fudenberg & Levine (1993b) in subclass of games of perfect information where each player moves only once on any path of play. The key observation is that for some prior beliefs experimentation takes place only on the equilibrium path, so a relatively sharp characterization of the limit equilibrium path (the limit of the steady state paths as first the lifetimes go to infinity and then the discount factor goes to 1) is possible. ...
The theory of learning in games explores how, which, and what kind of equilibria might arise as a consequence of a long-run nonequilibrium process of learning, adaptation, and/or imitation. If agents strategies are completely observed at the end of each round (and agents are randomly matched with a series of anonymous opponents), fairly simple rules perform well in terms of the agents worst-case payoffs, and also guarantee that any steady state of the system must correspond to an equilibrium. If players do not observe the strategies chosen by their opponents (as in extensive-form games), then learning is consistent with steady states that are not Nash equilibria because players can maintain incorrect beliefs about off-path play. Beliefs can also be incorrect because of cognitive limitations and systematic inferential errors.
... The problem, however, is that other stories could be told as well. One could model the same effect by assuming that trust exists by the same mechanisms that superstitions persist-by a lack of willingness to test a hypothesis about potential harms that, if proven correct, could have catastrophic personal consequences (Fudenberg and Levine 2006). Guidance as to which of these routes to explanation is to be the most promising is provided by having a deep understanding of the psychological basis of such effects-something that is the traditional domain of behavioral research. ...
Game theoretic models of marketing channels typically rely on simplifying assumptions that, from a behavioral perspective,
often appear naïve. However, behavioral researchers have produced such an abundance of behavioral regularities that they are
impossible to incorporate into game theoretic models. We believe that a focus on three core findings would benefit both fields;
these are: first, beliefs that are held by the various players regarding profit consequences of different actions are incomplete
and often biased; second, players’ preferences and optimization objectives are not commonly known; and third, players have
insufficient cognitive abilities to achieve optimization objectives. Embracing these three findings shifts the focus from
rational decision making to how decision makers learn to improve their decision-making skills. Concluding, we believe that
greater convergence of game theoretic modeling and behavioral research in marketing channels would lead to new insights for
both fields.
KeywordsMarketing channels-Game theory-Behavioral decision research
The relevance of special interests lobbying in modern democracies can hardly be questioned. But if large trade associations can overcome the free riding problem and form effective lobbies, why do they not also threaten market competition by forming equally effective cartels? We argue that the key to understanding the difference lies in supply elasticity. The group discipline, which works in the case of lobbying, can be effective in sustaining a cartel only if increasing output is sufficiently costly—otherwise the incentive to deviate is too great. The theory helps organizing a number of stylized facts within a common framework.
The present literature review focused on social and psychological factors associated with superstition. The literature included
conceptual definition of superstition, prevalence and its probable underlying factors. It also sheds light upon the pros and cons
for adherence with superstitious beliefs. Number of online articles published in peer reviewed journals, book reviews, abstracts,
conference proceedings and dissertations were reviewed with a specific focus for psycho-social effects of superstition. The findings revealed that superstitious beliefs are widely spread and there are socio- learning predispositions which lie at backdrop of superstition. Superstition casts negative and positive influences on psychological health of individuals. Overall the paper is a worthy contribution for articulated understanding of phenomenon of superstition and its associated factors which opens horizons for further related and in depth inquiries.
The issue of efficiency of financial markets has always fascinated scientists. It is significant from the point of view of assessing portfolio management effectiveness and behavioural finance. In the first part of this paper, the hypothesis of the unfortunate dates effect was tested upon 29 commodity prices in relation to the following four approaches: close‑close, overnight, open‑open, and open‑close. The rates of return were calculated for the sessions falling on the 13th and 4th day of the month, Friday the 13th and Tuesday the 13th. The study proved the occurrence of seasonal effects on the so‑called unlucky dates.
The key issue in selecting between equilibria in signalling games is determining how receivers will interpret deviations from the path of play. We develop a foundation for these off-path beliefs, and an associated equilibrium refinement, in a model where equilibrium arises from non-equilibrium learning by long-lived senders and receivers. In our model, non-equilibrium signals are sent by young senders as experiments to learn about receivers' behavior, and different types of senders have different incentives for these various experiments. Using the Gittins index (Gittins, 1979), we characterize which sender types use each signal more often, leading to a constraint we call the "compatibility criterion" on the receiver's off-path beliefs and to the concept of a "type-compatible equilibrium." We compare type-compatible equilibria to signalling-game refinements such as the Intuitive Criterion (Cho and Kreps, 1987) and divine equilibrium (Banks and Sobel, 1987).
Glossary
Definition of the Subject
Introduction
Deterministic Learning
Stochastic Learning
Future Directions
Bibliography
A mezuzah is a small case affixed to the doorframe of each room in Jewish homes and workplaces which contains a tiny scroll of parchment inscribed with a prayer. It is customary for religious Jews to touch the mezuzah every time they pass through a door and kiss the fingers that touched it. However, kissing the mezuzah has also become customary for many secular Jews who think of the mezuzah as a good luck charm. In view of a recent revelation that kissing the mezuzah entails a health hazard, the present paper inquires whether it also has some observable benefit. In an experiment conducted among non-religious mezuzah-kissing economics and business students confronted with a logic-problem exam, some were allowed to kiss the mezuzah before taking the exam, whereas the others were asked not to do so or could not do so because it had been removed from the room doorframe. The experiment revealed that participants who did not kiss the mezuzah performed worse than those who kissed it, and that the stronger is one's belief in the mezuzah's luck-enhancing properties, the better he performs when he kisses it but the worse he performs when he does not.
The evil eye belief is a widespread superstition according to which people can cause harm by a mere envious glance at coveted objects or their owners. This paper argues that such belief originated and persisted as a useful heuristic under conditions in which envy was likely to trigger destructive behavior and the avoidance of other people's envy, effectively prescribed by the evil eye belief, was a proper response to that threat. We hypothesize that in weakly institutionalized societies wealth differentiation and vulnerability of productive assets were the key factors enabling envy-induced destructive behavior and contributing to the emergence and spread of the evil eye belief as a cultural defense mechanism. Evidence from small-scale preindustrial societies shows that there is indeed a robust positive association between the incidence of the belief and measures of wealth inequality, controlling for potential confounding factors such as patterns of spatial and cross-cultural diffusion and various dimensions of early economic development. Furthermore, the evil eye belief is more prevalent in agro-pastoral societies that tend to sustain higher levels of inequality and where vulnerable material wealth plays a dominant role in the subsistence economy.
We examine the potential effect of Chinese superstition on the prices of four commodities traded in the US commodity market using daily data from January 1994 to September 2012. We focus on market responses to days that Chinese traders superstitiously deem as either lucky or unlucky. Our results suggest that day 4 in the month (considered unlucky) is associated with significantly lower returns for three commodities (copper, cotton and soybean). The evidence controls for the possible effects of other anomalies and emerges despite the fact that China buys only about half of the US total exports of these commodities. These results seem in conflict with an efficient US commodity market as it opens the possibility for formulating profitable trading rules based on day 4 trading.
The nodes of an extensive-form game are commonly specified as sequences of actions. Rubinstein calls such nodes histories. We find that this sequential notation is superfluous in the sense that nodes can also be specified as sets of actions. The only cost of doing so is to rule out games with absent-minded agents. Further, we present an application. We take an arbitrary assess-ment and define its infinitely-more-likely relation. This relation compares nodes, which are now specified as sets of actions. We find that if the assessment is consistent, then its infinitely-more-likely relation can be additively represented by a density function assigning numbers to actions. This construction is unexpectedly intuitive because it closely resembles that of an ordinary proba-bility density function. Essentially, we have found a clear way to say that consistency embodies the "stochastic independence" of zero-probability actions. Finally, we show that this intuitive result is the driving force behind the algebraic characterizations of consistency in Kreps and Wilson (1982, Appendix) and Perea, Jansen, and Peters (1997). Thus one can comfortably study consistency without reference to sequences of assessments. seminar participants at Auckland, Brigham Young, the London School of Economics, New South Wales, Sydney, and Western Ontario; and conference participants at the Econometric Soci-ety meeting in Sydney, the game theory conference in Stony Brook, and the Midwest mathematical economics meeting in Ann Arbor.
What do we notice and how does this affect what we learn and come to believe? I present a model of an agent who learns to make forecasts on the basis of freely available informa- tion, but is selective in which information he attends to because of limited cognitive resources. I model the agent's choice of whether to attend to and encode information along a dimension as a function of his current beliefs about whether such information is predictive, taking as given that he attends to information along other dimensions. If the agent does not attend to and encode some piece of information, it cannot be recalled at a later date. He uses Bayes' rule to update his beliefs given encoded information, but does not attempt to fill in missing information. I show that, as a consequence of selective attention, the agent may persistently fail to recognize important empirical regularities, make biased forecasts, and hold incorrect beliefs about the statistical relationship be- tween variables. In addition, I identify factors that make such errors more likely or persistent. The model sheds light on a set of systematic biases in inference, including the difficulty people have in recognizing relationships that prior theories do not make plausible and the overattribution of cause to salient event features. The model is applied to help understand the formation and stability of erroneous stereotypes, as well as discrimination based on such stereotypes.
Glossary
Definition of the Subject
Introduction
Towards a Logic and Geometry of Interaction
Further Directions
Bibliography
We study immigration policy in a small receiving economy under self-selection of migrants. We show that a non-selective immigration
policy choice affects and is affected by the migratory decisions of skilled and unskilled foreign workers. From this interaction
multiple equilibria may arise, which are driven by the natives’ expectations on the skill composition of migrants. In particular,
pessimistic (optimistic) beliefs induce a country to impose higher (lower) barriers to immigration, which worsen (improve)
the skill composition of immigrants and thus confirm initial beliefs. This mechanism induces immigration policy to be self-confirming.
We discuss how the adoption of a skill-selective policy affects this result.
Standard economists explain sin as optimal. If so, why would agents ever experience shame? To explain this anomaly, some economists advance the dynamic inconsistency view, which explains sin as being suboptimal. If so, why would agents experience greater wellbeing if they sin sometimes? The proposed "two-moment decision" theory avoids both anomalies. In the first moment, the agent decides whether to adopt an internal constraint concerning a temptation in all periods. Second, th e agent then decides on whether to suspend the internal constraint in a particular period. The standard and dynamic inconsistency views are special cases of the two-moment decision theory.
The concepts of coordination and cooperation are widely used in economics, and particularly in game theory. They were also at the foundation of development economics at the time of WWII, with Paul Rosenstein-Rodan highlighting the existence of intersectoral spillovers effects, multiple equilibria and underdevelopment traps. These concepts returned to the forefront of development theory in the 1970s with the notions of coordination failure and poverty traps, as well as the research on social norms. One example was Samuel Bowles' seminal concept of 'institutional poverty traps', i.e. highly inegalitarian institutions that persist even though they are inefficient. Membership institutions are of particular relevance in developing countries, and therefore in development economics. The paper explores the cognitive dimensions of coordination failures and institutional traps; it reveals that local institutions in developing countries may be efficient and examines the conditions in which norms create poverty traps, in particular membership norms. Firstly, it is argued that institutions and norms are key causes of the formation and persistence of poverty traps. Institutions and norms are complex cognitive devices, some beliefs and norms appear to be particularly resilient and difficult to revise. Secondly, it is shown that no particular institutional form is ex ante a cause of poverty traps: depending on contexts, the same institutional forms can be efficient or inefficient. It is the combination of multiple elements - economic and political environment, and social norms - that create thresholds effects and entrap groups into low equilibria. Thirdly, it is argued that the norms that organise group membership, because they involve beliefs that are difficult to revise, are typical factors of poverty traps.
I argue that medieval judicial ordeals accurately assigned accused criminals' guilt and innocence. They did this by leveraging a medieval superstition called iudicium Dei. According to that superstition, God condemned the guilty and exonerated the innocent through clergy conducted physical tests. Medieval citizens' belief in iudicium Dei created a separating equilibrium in which only innocent defendants were willing to undergo ordeals. Conditional on observing a defendant's willingness to do so, the administering priest knew he was innocent and manipulated the ordeal to find this. My theory explains the peculiar puzzle of ordeals: trials of fire and water that should've condemned most persons who underwent them did the reverse. They exonerated these persons instead. Boiling water rarely boiled persons who plunged their arms in it. Burning iron rarely burned persons who carried it. Ordeal outcomes were miraculous. But they were miracles of mechanism design.
We estimate a hedonic price model of willingness to pay (WTP) for superstition. The model explains WTP measured by winning bids in Hong Kong license-plate auctions through numeric and alpha-numeric sequences catering to culturally-sensitive superstitions, tempered by self-expression, economic and auction-specific considerations. The paper adds to the extant literature on three dimensions. First, we focus on a commodity that is a necessary accessory with no intrinsic value, known to be available in unlimited supply at a production cost substantially less than the buyer’s WTP. Second, we show differentiated responses by consumers to nuanced cues that exploit well-known local cultural sensitivities. Finally, we incorporate macroeconomic and auction-specific variables to elicit their impact on consumer preferences and behavior within our culturally-sensitive context. Our results suggest quantifiable differences in consumer responses to psychologically-nuanced cultural cues, which management can exploit to its competitive advantage. This evidence is relevant and timely, given the rising corporate interest in serving the vast and growing consumer market in China.
The paper deals with three related issues.1.
It introduces a measure of partial subgame perfection for equilibria of repeated games.
2.
It illustrates that the folk-theorem discontinuity generated by small complexity costs, as exhibited by Abreu and Rubinstein, does not exist in the presence of any level of perfection.
3.
It shows that reactive strategy equilibria, such as tit-for-tat, cannot be subgame perfect, even partially so. As a corollary, this shows a need to use full automata rather than exact automata when studying complexity and perfection in repeated games.
Suppose two players repeatedly meet each other to play a game where1. each uses a learning rule with the property that it is a calibrated forecast of the other's plays, and2. each plays a myopic best response to this forecast distribution.Then, the limit points of the sequence of plays are correlated equilibria. In fact, for each correlated equilibrium there is some calibrated learning rule that the players can use which results in their playing this correlated equilibrium in the limit. Thus, the statistical concept of a calibration is strongly related to the game theoretic concept of correlated equilibrium.Journal of Economic LiteratureClassification Numbers: C72,D83,C44.
If it is common knowledge that the players in a game are Bayesian utility maximizers who treat uncertainty about other players' actions like any other uncertainty, then the outcome is necessarily a correlated equilibrium. Random strategies appear as an expression of each player's uncertainty about what the others will do, not as the result of willful randomization. Use is made of the common prior assumption, according to which differences in probability assessments by different individuals are due to the different information that they have (where "information" may be interpreted broadly, to include experience, upbringing, and genetic makeup). Copyright 1987 by The Econometric Society.
The authors study the steady states of a system in which players learn about the strategies their opponents are playing by updating their Bayesian priors in light of their observations. Players are matched.at random to play a fixed extensive-form game and each player observes the realized actions in his own matches but not the intended off-path play of his opponents or the realized actions in other matches. Because players are assumed to live finite lives, there are steady states in which learning continually takes place. If lifetimes are long and players are very patient, the steady state distribution of actions approximates those of a Nash equilibrium. Copyright 1993 by The Econometric Society.
In a self-confining equilibrium, each player's strategy is a best response to his beliefs about the play of his opponents and each player's beliefs are correct along the equilibrium path of play. Thus, if a self-confirming equilibrium occurs repeatedly, no player ever observes play that contradicts his beliefs, even though beliefs about play at off-path information sets need not be correct. The authors characterize the ways in which self-confirming equilibria and Nash equilibria can differ and provide conditions under which self-confirming equilibria correspond to standard solution concepts. Copyright 1993 by The Econometric Society.
In some experiments rational players who understand the structure of the game could improve their payoff. We hound the size
of the observed losses in several such experiments, lb do this, we suppose that observed play resembles an equilibrium because
players learn about their opponents' play. Consequently, in an extensive-form game, some actions that are not optimal given
the true distribution of opponents' play could be optimal given available information. We find that average losses are small:
0.64 per player with stakes between 30. In one of the three experiments we examine, this also implies a
narrow range of outcome.
We examine the role of off-path "superstitions" in macro-economics, and show how a false belief about off-path play is the key element underlying both the Lucas Critique and the game-theoretic concept of self-confirming equilibrium. However, the impact of false beliefs in these two cases is different: In the Lucas case, a policy maker's incorrect beliefs about off-path play can lead to the adoption of mistaken policy innovation. However, the consequences of such an innovation provide evidence that the belief that motivated them was wrong. In contrast, play may never escape an undesirable self-confirming equilibrium, as the action implied by the mistaken belief does not generate data that contradicts it; escape from the self-confirming equilibrium requires that players do a sufficient amount of experimentation with off-path actions.
The concept of a perfect equilibrium point has been introduced in order to exclude the possibility that disequilibrium behavior is prescribed on unreached subgames [Selten 1965 and 1973]. Unfortunately this definition of perfectness does not remove all difficulties which may arise with respect to unreached parts of the game. It is necessary to reexamine the problem of defining a satisfactory non-cooperative equilibrium concept for games in extensive form. Therefore a new concept of a perfect equilibrium point will be introduced in this paper2).
We propose a new criterion for equilibria of extensive games, in the spirit of Selten's perfectness criteria. This criterion requires that players' strategies be sequentially rational: Every decision must be part of an optimal strategy for the remainder of the game. This entails specification of players' beliefs concerning how the game has evolved for each information set, including informaiton sets off the equilibrium path. The properties of sequential equilibria are developed; in particular, we study the topological structure of the set of sequential equilibria. The connections with Selten's trembling-hand perfect equilibria are given.
The concept of a perfect equilibrium point has been introduced in order to exclude the possibility that disequilibrium behavior is prescribed on unreached subgames [SELTEN 1965 and 1973]. Unfortunately this definition of perfectness does not remove all difficulties which may arise with respect to unreached parts of the game. It is necessary to reexamine the problem of defining a satisfactory non-cooperative equilibrium concept for games in extensive form. Therefore a new concept of a perfect equilibrium point will be introduced in this paper2). In retrospect the earlier use of the word "perfect" was premature. Therefore a perfect equilibrium point in the old sense will be called "subgame perfect". The new definition of perfectness has the property that a perfect equilibrium point is always subgame perfect but a subgame perfect equilibrium point may not be perfect. It will be shown that every finite extensive game with perfect recall has at least one perfect equilibrium point. Since subgame perfectness cannot be detected in the normal form, it is clear that for the purpose of the investigation of the problem of perfectness, the normal form is an inadequate representation of the extensive form. It will be convenient to introduce an "agent normal form" as a more adequate representation of games with perfect recall.
In a self-confirming equilibrium, each player correctly forecasts the actions that opponents will take along the equilibrium path, but may be mistaken about the way that opponents would respond to deviations. This paper develops a refinement of self-confirming equilibrium in which players use information about opponents' payoffs in forming beliefs about the way that opponents play off of the equilibrium path. We show that this concept is robust to payoff uncertainty. We also discuss its relationship to other concepts and show that it is closely related to assuming almost common certainty of payoffs in an epistemic model with independent beliefs. Journal of Economic Literature Classification Numbers: C72, D84.
A group of individuals repeatedly plays a fixed extensive-form game, using past play to forecast future actions. Each (asymptotically) maximizes his own immediate expected payoff, believing that others' play corresponds to the historical frequencies of past play. Because players observe only the path of play in each round, they may not learn how others act in parts of the game tree that are not reached infinitely often. Hence, differences and correlations in beliefs about out-of-equilibrium actions may persist indefinitely. The stable points of these learning processes are self-confirming equilibria, a weaker solution concept than Nash equilibria. Journal of Economic Literature Classification Numbers: C72, D83.
The philosophy of equilibrium refinements is that the analyst, if he knows things about the structure of the game, can reject some Nash equilibria as unreasonable. The word “know” in the preceding sentence deserves special emphasis. If in a fixed game the analyst can reject a particular equilibrium outcome, but he cannot do so for games arbitrarily “close by,” then he may have second thoughts about rejecting the outcome. We consider several notions of distance between games, and we characterize their implications for the robustness of equilibrium refinements.
Game theoretic models of learning which are based on the strategic form of the game cannot explain learning in games with large extensive form. We study learning in such games by using valuation of moves. A valuation for a player is a numeric assessment of her moves that purports to reflect their desirability. We consider a myopic player, who chooses moves with the highest valuation. Each time the game is played, the player revises her valuation by assigning the payoff obtained in the play to each of the moves she has made. We show for a repeated win–lose game that if the player has a winning strategy in the stage game, there is almost surely a time after which she always wins. When a player has more than two payoffs, a more elaborate learning procedure is required. We consider one that associates with each move the average payoff in the rounds in which this move was made. When all players adopt this learning procedure, with some perturbations, then, with probability 1 there is a time after which strategies that are close to subgame perfect equilibrium are played. A single player who adopts this procedure can guarantee only her individually rational payoff.
A k-sided die is thrown n times, to estimate the probabilities of landing on the various sides. The MLE of is the vector of empirical proportions . Consider a set of Bayesians that put uniformly positive prior mass on all reasonable subsets of the parameter space. Their posterior distributions will be uniformly concentrated near p. Sharp bounds are given, using entropy. These bounds apply to all sample sequences: There are no exceptional null sets.
Ann. Statist. 1990, Vol. 18, 1317-1324 Diaconis__Persi Freedman__David
For a steady state to be a Nash equilibrium the agents have to perfectly observe the actions of others. This paper suggests a solution concept for cases where players observe only an imperfect signal of what the others' actions are. The model is enriched by specifying the signal that each player has about the actions taken by the others. The solution, which we call rationalizbale conjectural equilibrium (RCE), is a profile of actions such that each player's action is optimal, given the assumption that it is common knowledge that all players maximize their expected utility given their knowledge. The RCE occupies an intermediary position between Nash equilibrium on one hand and Rationalizability style Bernheim-Pearce on the other hand. The concept is demonstrated by several examples in which it refines the rationalizability concept and still is not equivalent to Nash equilibrium.
The paper formulates a simple two-person model of learning with pattern recognition and discusses its implications. In particular, it focuses on the asymptotic behavior of players' beliefs when the game has a mixed-strategy Nash equilibrium.Journal of Economics LiteratureClassification Numbers: C72, D83.
We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.
We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.
We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.
The backward induction (or subgame-perfect) equilibrium of a perfect information game is shown to be the unique evolutionarily stable outcome for dynamic models consisting of selection and mutation, when the mutation rate is low and the populations are large.
Each period, a player must choose an action without knowing the outcome that will be chosen by "Nature," according to an unknown and possibly history-dependent stochastic rule. We discuss have a class of procedures that assign observations to categories, and prescribe a simple randomized variation of fictitious play within each category. These procedures are "conditionally consistent," in the sense of yielding almost as high a time-average payoff as could be obtained if the player chose knowing the conditional distributions of actions given categories. Moreover given any alternative procedure, there is a conditionally consistent procedure whose performance is no more than epsilon worse regardless of the discount factor. Cycles can persist if all players classify histories in the same way; however in an example, where players classify histories differently, the system converges to a Nash equilibrium. We also argue that in the long run the time-average of play should resemble a correlated equilibrium. 1 The authors are grateful for financial support from NSF grants SBR-9223320, SBR-9223175, SBR9409180 and the UCLA Academic Senate. This paper benefited from conversations with Glen Ellison and Dean Foster. We would like to thank participants at the Second International Conference on Economic Theory at Carlos III, Madrid for helpful comments.
Learning by Matching PatternsAn Evolutionary Analysis of Forward and Backward InductionRationalizable Conjectural Equilibrium: Between Nash and RationalizabilityReexamination of the perfectness concept for equilibrium points in extensive games
- V Lambson
- D Noldeke
- L Samuelson
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A Theory of Learning, Experimentation, and Equilibrium in Games
- D Fudenberg
- D M Kreps
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Equilibrium in Games," mimeo.