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Volume 11, June

Games, Volume 11, Issue 3 (September 2020) – 16 articles

Cover Story (view full-size image): A long-standing concern in game theory is to determine which multiple pure Nash equilibria is, in some sense, the “best”. Measures, such as payoff dominance, risk dominance, and others, have been proposed, but comparisons remain complicated because each measure reflects different previously obscure game theoretic attributes. Insight and answers are gained by using a newly developed way to separate games into their individual and group behavioral aspects. An added advantage of this decomposition is how it uncovers new intuitive structures of 2 × 2 potential games. View this paper
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Article
Cost of Reasoning and Strategic Sophistication
Games 2020, 11(3), 40; https://doi.org/10.3390/g11030040 - 22 Sep 2020
Cited by 4 | Viewed by 1108
Abstract
I designed an experiment to study the persistence of the prevailing levels of reasoning across games. Instead of directly comparing the k-level(s) of reasoning for each game, I used cognitive load to manipulate the strategic environment by imposing variations on the subject’s [...] Read more.
I designed an experiment to study the persistence of the prevailing levels of reasoning across games. Instead of directly comparing the k-level(s) of reasoning for each game, I used cognitive load to manipulate the strategic environment by imposing variations on the subject’s cost of reasoning and their first- and second-order beliefs. Subjects have systematic changes in k-level(s) of reasoning across games. That finding suggests that subjects are responsive to changes in the strategic environment. Changes in k-level(s) of reasoning are mostly consistent with the endogenous depth of reasoning model when subjects are more cognitively capable or facing less cognitively capable opponents. Subjects have cognitive bounds, but often choose a lower-type action due to their beliefs about their opponents. Finally, cognitive ability plays a significant role in subjects making strategic adjustments when facing different strategic environments. Full article
(This article belongs to the Special Issue Behavioral Game Theory)
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Article
Optimism Bias during the Covid-19 Pandemic: Empirical Evidence from Romania and Italy
Games 2020, 11(3), 39; https://doi.org/10.3390/g11030039 - 17 Sep 2020
Cited by 19 | Viewed by 3814
Abstract
Given the importance of perceived susceptibility to a disease in adopting preventive behaviors, and the negative impact of optimism bias on prevention, this paper aimed to explore to what extent comparative optimism bias (understood as the tendency to assess a lower probability for [...] Read more.
Given the importance of perceived susceptibility to a disease in adopting preventive behaviors, and the negative impact of optimism bias on prevention, this paper aimed to explore to what extent comparative optimism bias (understood as the tendency to assess a lower probability for oneself to experience negative health events compared to others) is present in the specific context of the Covid-19 pandemic, in two countries with different profiles in terms of the spread of the disease: Italy and Romania. After identifying optimism bias in both countries, we tested whether it depends on respondents’ characteristics like gender, age, education, health status and whether or not they have the opportunity to work from home. We surveyed 1126 Romanians and 742 Italians, and found that optimism bias depends on self-reported health status, and that optimism bias increases with age. Inconclusive evidences were found regarding gender and education level, as well as the option to work from home. Full article
(This article belongs to the Special Issue Impact of Overconfidence and Optimism on Individual Decisions)
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Article
Estimating Case-Based Learning
Games 2020, 11(3), 38; https://doi.org/10.3390/g11030038 - 15 Sep 2020
Cited by 2 | Viewed by 1326
Abstract
We propose a framework in order to econometrically estimate case-based learning and apply it to empirical data from twelve 2 × 2 mixed strategy equilibria experiments. Case-based learning allows agents to explicitly incorporate information available to the experimental subjects in a simple, compact, [...] Read more.
We propose a framework in order to econometrically estimate case-based learning and apply it to empirical data from twelve 2 × 2 mixed strategy equilibria experiments. Case-based learning allows agents to explicitly incorporate information available to the experimental subjects in a simple, compact, and arguably natural way. We compare the estimates of case-based learning to other learning models (reinforcement learning and self-tuned experience weighted attraction learning) while using in-sample and out-of-sample measures. We find evidence that case-based learning explains these data better than the other models based on both in-sample and out-of-sample measures. Additionally, the case-based specification estimates how factors determine the salience of past experiences for the agents. We find that, in constant sum games, opposing players’ behavior is more important than recency and, in non-constant sum games, the reverse is true. Full article
(This article belongs to the Special Issue Behavioral Game Theory)
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Article
Information Disclosure through Technology Licensing
Games 2020, 11(3), 37; https://doi.org/10.3390/g11030037 - 04 Sep 2020
Viewed by 1408
Abstract
We show that even if information transmission through an honest outside agency is not possible due to the possibility of collusion between the firms and the outside agency, information transmission is still possible through technology licensing. However, unlike the case of a cost-free [...] Read more.
We show that even if information transmission through an honest outside agency is not possible due to the possibility of collusion between the firms and the outside agency, information transmission is still possible through technology licensing. However, unlike the case of a cost-free honest outside agency, where information transmission always occurs under a quantity setting oligopoly, information transmission through licensing does not always occur. Full article
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Article
Shame and Theory-of-Mind Predicts Rule-Following Behavior
Games 2020, 11(3), 36; https://doi.org/10.3390/g11030036 - 01 Sep 2020
Cited by 1 | Viewed by 1932
Abstract
This paper examines the idea that adherence to social rules is in part driven by moral emotions and the ability to recognize the emotions of others. Moral emotions like shame and guilt produce negative feelings when social rules are transgressed. The ability to [...] Read more.
This paper examines the idea that adherence to social rules is in part driven by moral emotions and the ability to recognize the emotions of others. Moral emotions like shame and guilt produce negative feelings when social rules are transgressed. The ability to recognize and understand the emotions of others is known as affective theory of mind (ToM). ToM is necessary for people to understand how others are affected by the violations of social rules. Using a laboratory experiment, individuals participated in a rule-following task designed to capture the propensity to follow costly social rules and completed psychometric measures of guilt, shame, and ToM. The results show that individuals who feel more shame and have higher ToM are more likely to follow the rules. The results from this experiment suggest that both shame and ToM are important in understanding rule-following. Full article
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Article
A Note on Stabilizing Cooperation in the Centipede Game
Games 2020, 11(3), 35; https://doi.org/10.3390/g11030035 - 20 Aug 2020
Cited by 1 | Viewed by 1770
Abstract
In the much-studied Centipede Game, which resembles the Iterated Prisoners’ Dilemma, two players successively choose between (1) cooperating, by continuing play, or (2) defecting and terminating play. The subgame-perfect Nash equilibrium implies that play terminates on the first move, even though continuing play [...] Read more.
In the much-studied Centipede Game, which resembles the Iterated Prisoners’ Dilemma, two players successively choose between (1) cooperating, by continuing play, or (2) defecting and terminating play. The subgame-perfect Nash equilibrium implies that play terminates on the first move, even though continuing play can benefit both players—but not if the rival defects immediately, which it has an incentive to do. We show that, without changing the structure of the game, interchanging the payoffs of the two players provides each with an incentive to cooperate whenever its turn comes up. The Nash equilibrium in the transformed Centipede Game, called the Reciprocity Game, is unique—unlike the Centipede Game, wherein there are several Nash equilibria. The Reciprocity Game can be implemented noncooperatively by adding, at the start of the Centipede Game, a move to exchange payoffs, which it is rational for the players to choose. What this interchange signifies, and its application to transforming an arms race into an arms-control treaty, are discussed. Full article
(This article belongs to the Special Issue Pro-sociality and Cooperation)
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Article
Valuable Cheap Talk and Equilibrium Selection
Games 2020, 11(3), 34; https://doi.org/10.3390/g11030034 - 20 Aug 2020
Cited by 2 | Viewed by 1436
Abstract
Intuitively, we expect that players who are allowed to engage in costless communication before playing a game would be foolish to agree on an inefficient outcome amongst the set of equilibria. At the same time, however, such preplay communication has been suggested as [...] Read more.
Intuitively, we expect that players who are allowed to engage in costless communication before playing a game would be foolish to agree on an inefficient outcome amongst the set of equilibria. At the same time, however, such preplay communication has been suggested as a rationale for expecting Nash equilibria in general. This paper presents a plausible formal model of cheap talk that distinguishes and resolves these possibilities. Players are assumed to have an unlimited opportunity to send messages before playing an arbitrary game. Using an extension of fictitious play beliefs, minimal assumptions are made concerning which messages about future actions are credible and hence contribute to final beliefs. In this environment, it is shown that meaningful communication among players leads to a Nash equilibrium (NE) of the action game. Within the set of NE, efficiency then turns out to be a consequence of imposing optimality on the cheap talk portion of the extended game. This finding contrasts with previous “babbling” results. Full article
(This article belongs to the Special Issue Behavioral Game Theory)
Article
With Potential Games, Which Outcome Is Better?
Games 2020, 11(3), 33; https://doi.org/10.3390/g11030033 - 17 Aug 2020
Cited by 3 | Viewed by 1471
Abstract
Lower one- or two-dimensional coordination, or potential games, are popularly used to model interactive behavior, such as innovation diffusion and cultural evolution. Typically, this involves determining the “better” of competing solutions. However, examples have demonstrated that different measures of a “good” choice can [...] Read more.
Lower one- or two-dimensional coordination, or potential games, are popularly used to model interactive behavior, such as innovation diffusion and cultural evolution. Typically, this involves determining the “better” of competing solutions. However, examples have demonstrated that different measures of a “good” choice can lead to conflicting conclusions; a fact that reflects the history of game theory in equilibrium selection. This behavior is totally explained while extending the analysis to the full seven-dimensional class of potential games, which includes coordination games. Full article
(This article belongs to the Special Issue Behavioral Game Theory)
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Article
Promotion of Green Technology under Different Environmental Policies
Games 2020, 11(3), 32; https://doi.org/10.3390/g11030032 - 09 Aug 2020
Cited by 2 | Viewed by 1692
Abstract
In this paper, I develop a two-stage game of pollution abatement technology adoption in a Cournot oligopoly to investigate a firm’s decision to adopt pollution abatement technology. In particular, I study the adoption incentives and welfare implications of popular environmental policies, namely emission [...] Read more.
In this paper, I develop a two-stage game of pollution abatement technology adoption in a Cournot oligopoly to investigate a firm’s decision to adopt pollution abatement technology. In particular, I study the adoption incentives and welfare implications of popular environmental policies, namely emission fees and quotas. Tradeable permits result in identical outcomes to emission fees. Within each policy regime, the conditions for Nash equilibria are identified where both firms invest in the green technology, neither firm invests in the technology, or only one firm invests. The following extensions are also analyzed: asymmetric adoption costs, increase in the marginal cost of production from adoption, and a type-dependent fee where adoption reduces the emission fee. Social welfare under an emission fee is identical to that under a quota. However, when policy is (not) stringent, firms are more willing to adopt expensive technology under a fee (quota) than under a quota (fee, respectively). Full article
(This article belongs to the Special Issue Environmental Economics and Game Theory)
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Article
Against All Odds: Tentative Steps toward Efficient Information Sharing in Groups
Games 2020, 11(3), 31; https://doi.org/10.3390/g11030031 - 07 Aug 2020
Cited by 1 | Viewed by 1519
Abstract
When groups face difficult problems, the voices of experts may be lost in the noise of others’ contributions. We present results from a “naturally noisy” setting, a large first-year undergraduate class, in which the expert’s voice was “lost” to such a degree that [...] Read more.
When groups face difficult problems, the voices of experts may be lost in the noise of others’ contributions. We present results from a “naturally noisy” setting, a large first-year undergraduate class, in which the expert’s voice was “lost” to such a degree that bringing forward even more inferior information was optimal. A single individual had little chance to improve the outcome and coordinating with the whole group was impossible. In this setting, we examined the change in behavior before and after people could talk to their neighbors. We found that the number of people who reduced noise by holding back their information strongly and significantly increased. Full article
(This article belongs to the Special Issue Experiments on Communication in Games)
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Article
Hierarchical Structures and Leadership Design in Mean-Field-Type Games with Polynomial Cost
Games 2020, 11(3), 30; https://doi.org/10.3390/g11030030 - 06 Aug 2020
Cited by 1 | Viewed by 1677
Abstract
This article presents a class of hierarchical mean-field-type games with multiple layers and non-quadratic polynomial costs. The decision-makers act in sequential order with informational differences. We first examine the single-layer case where each decision-maker does not have the information about the other control [...] Read more.
This article presents a class of hierarchical mean-field-type games with multiple layers and non-quadratic polynomial costs. The decision-makers act in sequential order with informational differences. We first examine the single-layer case where each decision-maker does not have the information about the other control strategies. We derive the Nash mean-field-type equilibrium and cost in a linear state-and-mean-field feedback form by using a partial integro-differential system. Then, we examine the Stackelberg two-layer problem with multiple leaders and multiple followers. Numerical illustrations show that, in the symmetric case, having only one leader is not necessarily optimal for the total sum cost. Having too many leaders may also be suboptimal for the total sum cost. The methodology is extended to multi-level hierarchical systems. It is shown that the order of the play plays a key role in the total performance of the system. We also identify a specific range of parameters for which the Nash equilibrium coincides with the hierarchical solution independently of the number of layers and the order of play. In the heterogeneous case, it is shown that the total cost is significantly affected by the design of the hierarchical structure of the problem. Full article
(This article belongs to the Special Issue Mean-Field-Type Game Theory)
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Article
Signalling, Information and Consumer Fraud
Games 2020, 11(3), 29; https://doi.org/10.3390/g11030029 - 30 Jul 2020
Viewed by 1628
Abstract
In a two-sided asymmetric information market, the role of the accuracy of consumers’ imperfect and private information on the level of fraud, incidence of fraud and trade under price rigidity is examined. Consumers receive a costless but noisy private signal of quality. The [...] Read more.
In a two-sided asymmetric information market, the role of the accuracy of consumers’ imperfect and private information on the level of fraud, incidence of fraud and trade under price rigidity is examined. Consumers receive a costless but noisy private signal of quality. The product offered in the market can be of two exogenously given qualities and it is common knowledge that the consumer is not willing to pay a high price for a low quality product. A low quality seller chooses to be either honest (by charging the lower market price) or dishonest (by charging the higher price). We show that equilibria involving fraud exist for all parameter values. Furthermore, for some parameter values, we find that -in equilibrium- a higher precision of consumers’ private information leads to higher levels of fraud and incidence of fraud, reducing consumers’ welfare. We provide conditions for the public revelation of consumers’ private information to be a Pareto improvement. Full article
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Article
Numerical Solution of Open-Loop Nash Differential Games Based on the Legendre Tau Method
Games 2020, 11(3), 28; https://doi.org/10.3390/g11030028 - 23 Jul 2020
Viewed by 1484
Abstract
In this paper, an efficient implementation of the Tau method is presented for finding the open-loop Nash equilibrium of noncooperative nonzero-sum two-player differential game problems with a finite-time horizon. Regarding this approach, the two-point boundary value problem derived from Pontryagin’s maximum principle is [...] Read more.
In this paper, an efficient implementation of the Tau method is presented for finding the open-loop Nash equilibrium of noncooperative nonzero-sum two-player differential game problems with a finite-time horizon. Regarding this approach, the two-point boundary value problem derived from Pontryagin’s maximum principle is reduced to a system of algebraic equations that can be solved numerically. Finally, a differential game arising from bioeconomics among firms harvesting a common renewable resource is included to illustrate the accuracy and efficiency of the proposed method and a comparison is made with the result obtained by fourth order Runge–Kutta method. Full article
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Review
A Review of Theoretical Studies on Indirect Reciprocity
Games 2020, 11(3), 27; https://doi.org/10.3390/g11030027 - 20 Jul 2020
Cited by 17 | Viewed by 3325
Abstract
Despite the accumulation of research on indirect reciprocity over the past 30 years and the publication of over 100,000 related papers, there are still many issues to be addressed. Here, we look back on the research that has been done on indirect reciprocity [...] Read more.
Despite the accumulation of research on indirect reciprocity over the past 30 years and the publication of over 100,000 related papers, there are still many issues to be addressed. Here, we look back on the research that has been done on indirect reciprocity and identify the issues that have been resolved and the ones that remain to be resolved. This manuscript introduces indirect reciprocity in the context of the evolution of cooperation, basic models of social dilemma situations, the path taken in the elaboration of mathematical analysis using evolutionary game theory, the discovery of image scoring norms, and the breakthroughs brought about by the analysis of the evolutionary instability of the norms. Moreover, it presents key results obtained by refining the assessment function, resolving the punishment dilemma, and presenting a complete solution to the social dilemma problem. Finally, it discusses the application of indirect reciprocity in various disciplines. Full article
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Article
A Note on Buyers’ Behavior in Auctions with an Outside Option
Games 2020, 11(3), 26; https://doi.org/10.3390/g11030026 - 18 Jul 2020
Cited by 2 | Viewed by 1591
Abstract
In this note I show that the equilibrium in cutoff strategies observed in auctions with a buy-it-now price may also arise in markets where objects are sold simultaneously by auctions and posted prices. However, contrary to auctions with a buy-it-now price where buyers [...] Read more.
In this note I show that the equilibrium in cutoff strategies observed in auctions with a buy-it-now price may also arise in markets where objects are sold simultaneously by auctions and posted prices. However, contrary to auctions with a buy-it-now price where buyers need to know only the total number of players in the market, in the latter environment buyers must also observe the number of active bidders in the auction for the equilibrium to exist in cutoff strategies. Full article
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Article
Switching Competitors Reduces Win-Stay but Not Lose-Shift Behaviour: The Role of Outcome-Action Association Strength on Reinforcement Learning
Games 2020, 11(3), 25; https://doi.org/10.3390/g11030025 - 08 Jul 2020
Cited by 1 | Viewed by 1765
Abstract
Predictability is a hallmark of poor-quality decision-making during competition. One source of predictability is the strong association between current outcome and future action, as dictated by the reinforcement learning principles of win–stay and lose–shift. We tested the idea that predictability could be reduced [...] Read more.
Predictability is a hallmark of poor-quality decision-making during competition. One source of predictability is the strong association between current outcome and future action, as dictated by the reinforcement learning principles of win–stay and lose–shift. We tested the idea that predictability could be reduced during competition by weakening the associations between outcome and action. To do this, participants completed a competitive zero-sum game in which the opponent from the current trial was either replayed (opponent repeat) thereby strengthening the association, or, replaced (opponent change) by a different competitor thereby weakening the association. We observed that win–stay behavior was reduced during opponent change trials but lose–shiftbehavior remained reliably predictable. Consistent with the group data, the number of individuals who exhibited predictable behavior following wins decreased for opponent change relative to opponent repeat trials. Our data show that future actions are more under internal control following positive relative to negative outcomes, and that externally breaking the bonds between outcome and action via opponent association also allows us to become less prone to exploitation. Full article
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