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Games, Volume 13, Issue 6 (December 2022) – 15 articles

Cover Story (view full-size image): Games have been successfully applied in security, such as for optimal resource allocations, anticipation of attack scenarios and corresponding attacks, and many more. The security context, however, has some particularities that have motivated the design of games whose payoffs are probability distributions, with stochastic orders to determine best strategies. These games were found to exhibit many interesting and partly pathological phenomena, ranging from the non-convergence of fictitious play in some zero-sum games, to the non-existence of Nash equilibria despite continuous, yet only vector-valued, payoffs. However, for the same reason, their study also leads to a variety of new concepts and possibilities, such as lexicographic Nash equilibria or the account for “disappointment” about equilibria. View this paper
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11 pages, 303 KiB  
Article
Rent Dissipation in Simple Tullock Contests
by Alex Dickson, Ian A. MacKenzie and Petros G. Sekeris
Games 2022, 13(6), 83; https://doi.org/10.3390/g13060083 - 13 Dec 2022
Cited by 3 | Viewed by 1849
Abstract
We investigate observed rent dissipation—the ratio of the total costs of rent seeking to the monetary value of the rent—in winner-take-all and share contests, where preferences are more general than usually assumed in the literature. With concave valuation of the rent, we find [...] Read more.
We investigate observed rent dissipation—the ratio of the total costs of rent seeking to the monetary value of the rent—in winner-take-all and share contests, where preferences are more general than usually assumed in the literature. With concave valuation of the rent, we find that contests can exhibit observed over-dissipation if the contested rent is below a threshold and yet observed under-dissipation with large rents: the nature of preferences implies contestants are relatively effortful in contesting small rents. Considering more general preferences in contests thus allows us to reconcile the Tullock paradox—where rent-seeking levels are relatively small despite the contested rent being sizeable—with observed over-dissipation of rents in experimental settings, where contested rents are arguably small. Full article
(This article belongs to the Special Issue Advances in the Theory and Applications of Contests and Tournaments)
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17 pages, 1636 KiB  
Article
Cournot’s Oligopoly Equilibrium under Different Expectations and Differentiated Production
by Nora Grisáková and Peter Štetka
Games 2022, 13(6), 82; https://doi.org/10.3390/g13060082 - 05 Dec 2022
Cited by 2 | Viewed by 2457
Abstract
The subject of this study is an oligopolistic market in which three firms operate in an environment of quantitative competition known as the Cournot oligopoly model. Firms and their production are differentiated, which brings the theoretical model closer to real market conditions. The [...] Read more.
The subject of this study is an oligopolistic market in which three firms operate in an environment of quantitative competition known as the Cournot oligopoly model. Firms and their production are differentiated, which brings the theoretical model closer to real market conditions. The main objective was to expand the Cournot duopoly and add another firm, resulting in an oligopolistic market structure assuming a partially differentiated production and coalition strategy between two firms. This article contains an oligopolistic model specifically designed for three different types of expectations, and has been applied to find and verify the stability of the net equilibrium of oligopolists. The market of telecommunication operators in Slovakia was selected as a real market case with accessible data on an oligopoly with three companies and partial differentiation. There are studies in which the authors limit their considerations to a certain number of repetitions of oligopolistic games. An infinite time interval is considered here. Three types of future expectations were considered: a simple dynamic model (or naïve expectations) in which the oligopolist assumes that its competitors will behave in the future based on their response functions, an adaptive expectations model in which the oligopolist considers a weighted average of the quantities offered by its competitors, and real expectations in which firms behave as rational players and do not have complete information about demand and offer output based on expected marginal profit. While the presented model proved to be stable under naïve and adaptive expectations, no stable equilibrium was found under real expectations and further results indicate a chaotic behavior. Full article
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24 pages, 3023 KiB  
Article
A Complete Analysis on the Risk of Using Quantal Response: When Attacker Maliciously Changes Behavior under Uncertainty
by Thanh Hong Nguyen and Amulya Yadav
Games 2022, 13(6), 81; https://doi.org/10.3390/g13060081 - 02 Dec 2022
Viewed by 1258
Abstract
In security games, the defender often has to predict the attacker’s behavior based on some observed attack data. However, a clever attacker can intentionally change its behavior to mislead the defender’s learning, leading to an ineffective defense strategy. This paper investigates the attacker’s [...] Read more.
In security games, the defender often has to predict the attacker’s behavior based on some observed attack data. However, a clever attacker can intentionally change its behavior to mislead the defender’s learning, leading to an ineffective defense strategy. This paper investigates the attacker’s imitative behavior deception under uncertainty, in which the attacker mimics a (deceptive) Quantal Response behavior model by consistently playing according to a certain parameter value of that model, given that it is uncertain about the defender’s actual learning outcome. We have three main contributions. First, we introduce a new maximin-based algorithm to compute a robust attacker deception decision under uncertainty, given the defender is unaware of the attacker deception. Our polynomial algorithm is built via characterizing the decomposability of the attacker deception space as well optimal deception behavior of the attacker against the worst case of uncertainty. Second, we propose a new counter-deception algorithm to tackle the attacker’s deception. We theoretically show that there is a universal optimal defense solution, regardless of any private knowledge the defender has about the relation between their learning outcome and the attacker deception choice. Third, we conduct extensive experiments in various security game settings, demonstrating the effectiveness of our proposed counter-deception algorithms to handle the attacker manipulation. Full article
(This article belongs to the Special Issue Game-Theoretic Analysis of Network Security and Privacy)
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26 pages, 500 KiB  
Article
Games over Probability Distributions Revisited: New Equilibrium Models and Refinements
by Stefan Rass, Sandra König and Stefan Schauer
Games 2022, 13(6), 80; https://doi.org/10.3390/g13060080 - 01 Dec 2022
Viewed by 1743
Abstract
This article is an overview of recent progress on a theory of games, whose payoffs are probability distributions rather than real numbers, and which have their equilibria defined and computed over a (suitably restricted yet dense) set of distributions. While the classical method [...] Read more.
This article is an overview of recent progress on a theory of games, whose payoffs are probability distributions rather than real numbers, and which have their equilibria defined and computed over a (suitably restricted yet dense) set of distributions. While the classical method of defining game models with real-valued utility functions has proven strikingly successful in many domains, some use cases from the security area revealed shortcomings of the classical real-valued game models. These issues motivated the use of probability distributions as a more complex object to express revenues. The resulting class of games displays a variety of phenomena not encountered in classical games, such as games that have continuous payoff functions but still no equilibrium, or games that are zero-sum but for which fictitious play does not converge. We discuss suitable restrictions of how such games should be defined to allow the definition of equilibria, and show the notion of a lexicographic Nash equilibrium, as a proposed solution concept in this generalized class of games. Full article
(This article belongs to the Section Applied Game Theory)
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11 pages, 247 KiB  
Article
An Experimental Investigation of Trusting Behaviour
by Clelia Mazzoni and Patrizia Sbriglia
Games 2022, 13(6), 79; https://doi.org/10.3390/g13060079 - 22 Nov 2022
Cited by 1 | Viewed by 1302
Abstract
In this paper, we present the results of an experiment conducted in Italy on trusting behaviour. Our subjects participated in a trust game and filled in a questionnaire on trust and trustworthiness based on the attitudinal questions reported in the European Value Survey. [...] Read more.
In this paper, we present the results of an experiment conducted in Italy on trusting behaviour. Our subjects participated in a trust game and filled in a questionnaire on trust and trustworthiness based on the attitudinal questions reported in the European Value Survey. The aims of the research are twofold. Firstly, using the experimental dataset, we construct two measures of trustworthiness among all recipients in the experiment, one based on the questionnaires’ answers and another based on the strategy method. We then compare the ex-ante behavioural decision to trust (before participants are allocated to a group) with the ex-post decision to trust (after participants are allocated to a group and trustors are informed of the level of trustworthiness of all the recipients who have been randomly allocated to each group). Our main finding is that trust strongly varies once the information is disclosed to trustors. The effect on trust is greater when the strategy method is used. Secondly, we compare the behavioural measure of trust with the attitudinal measures of trust and trustworthiness and find that there is only a weak correlation between the two measures, thus confirming, for the Italian case, similar findings in country-based studies on trust. Full article
(This article belongs to the Special Issue A Yin and Yang Perspective on the Trust Game: Trust and Reciprocity)
19 pages, 452 KiB  
Article
A Note on Binary Strategy-Proof Social Choice Functions
by Achille Basile, Anna De Simone and Ciro Tarantino
Games 2022, 13(6), 78; https://doi.org/10.3390/g13060078 - 18 Nov 2022
Viewed by 1306
Abstract
Let Φn be the set of the binary strategy-proof social choice functions referred to a group of n voters who are allowed to declare indifference between the alternatives. We provide a recursive way to obtain the set Φn+1 from [...] Read more.
Let Φn be the set of the binary strategy-proof social choice functions referred to a group of n voters who are allowed to declare indifference between the alternatives. We provide a recursive way to obtain the set Φn+1 from the set Φn. Computing the cardinalities |Φn| presents difficulties as the computation of the Dedekind numbers. The latter give the analogous number of social choice functions when only strict preferences are admitted. A comparison is given for the known values. Based on our results, we present a graphical description of the binary strategy-proof social choice functions in the case of three voters. Full article
(This article belongs to the Special Issue Strategy-Proof Mechanism Design)
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8 pages, 285 KiB  
Article
The Unanimity Rule under a Two-Agent Fixed Sequential Order Voting
by Marina Bánnikova and José-Manuel Giménez-Gómez
Games 2022, 13(6), 77; https://doi.org/10.3390/g13060077 - 17 Nov 2022
Viewed by 1442
Abstract
This paper studies how the cost of delay and voting order affect agents’ decisions in a unanimity voting mechanism. Specifically, we consider two-voter conclaves with commonly known preferences over two alternatives, the cost of delay, and the following novelty: each voter has a [...] Read more.
This paper studies how the cost of delay and voting order affect agents’ decisions in a unanimity voting mechanism. Specifically, we consider two-voter conclaves with commonly known preferences over two alternatives, the cost of delay, and the following novelty: each voter has a subjective deadline—a moment in time when he/she prefers immediate agreement on any alternative, rather than future agreement on his/her most-preferred alternative. Our key finding shows that patience is not necessarily a main attribute of strategic advantage. When the first voter is the same at every stage, this voter will obtain his/her preferred alternative, even if he/she is the least patient one. However, this first movement advantage disappears when agents alternate as the first voter of each stage: in this case, the most patient voter always wins. Full article
(This article belongs to the Special Issue Game Theory with Applications to Economics)
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15 pages, 1197 KiB  
Article
The Black Box as a Control for Payoff-Based Learning in Economic Games
by Maxwell N. Burton-Chellew and Stuart A. West
Games 2022, 13(6), 76; https://doi.org/10.3390/g13060076 - 16 Nov 2022
Cited by 2 | Viewed by 1982
Abstract
The black box method was developed as an “asocial control” to allow for payoff-based learning while eliminating social responses in repeated public goods games. Players are told they must decide how many virtual coins they want to input into a virtual black box [...] Read more.
The black box method was developed as an “asocial control” to allow for payoff-based learning while eliminating social responses in repeated public goods games. Players are told they must decide how many virtual coins they want to input into a virtual black box that will provide uncertain returns. However, in truth, they are playing with each other in a repeated social game. By “black boxing” the game’s social aspects and payoff structure, the method creates a population of self-interested but ignorant or confused individuals that must learn the game’s payoffs. This low-information environment, stripped of social concerns, provides an alternative, empirically derived null hypothesis for testing social behaviours, as opposed to the theoretical predictions of rational self-interested agents (Homo economicus). However, a potential problem is that participants can unwittingly affect the learning of other participants. Here, we test a solution to this problem in a range of public goods games by making participants interact, unknowingly, with simulated players (“computerised black box”). We find no significant differences in rates of learning between the original and the computerised black box, therefore either method can be used to investigate learning in games. These results, along with the fact that simulated agents can be programmed to behave in different ways, mean that the computerised black box has great potential for complementing studies of how individuals and groups learn under different environments in social dilemmas. Full article
(This article belongs to the Special Issue Learning and Evolution in Games, 1st Edition)
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24 pages, 369 KiB  
Article
Egalitarian-Equivalence and Strategy-Proofness in the Object Allocation Problem with Non-Quasi-Linear Preferences
by Hiroki Shinozaki
Games 2022, 13(6), 75; https://doi.org/10.3390/g13060075 - 15 Nov 2022
Viewed by 1195
Abstract
We consider the problem of allocating heterogeneous objects to agents with money, where the number of agents exceeds that of objects. Each agent can receive at most one object, and some objects may remain unallocated. A bundle is a pair consisting of an [...] Read more.
We consider the problem of allocating heterogeneous objects to agents with money, where the number of agents exceeds that of objects. Each agent can receive at most one object, and some objects may remain unallocated. A bundle is a pair consisting of an object and a payment. An agent’s preference over bundles may not be quasi-linear, which exhibits income effects or reflects borrowing costs. We investigate the class of rules satisfying one of the central properties of fairness in the literature, egalitarian-equivalence, together with the other desirable properties. We propose (i) a novel class of rules that we call the independent second-prices rules with variable constraints and (ii) a novel condition on constraints that we call respecting the valuation coincidence. Then, we establish that the independent second-prices rule with variable constraints that respects the valuation coincidence is the only rule satisfying egalitarian-equivalence, strategy-proofness, individual rationality, and no subsidy for losers. Our characterization result implies that in the case of three or more agents, there are few opportunities for agents to receive objects under a rule satisfying egalitarian-equivalence and the other desirable properties, which highlights the strong tension between egalitarian-equivalence and efficiency. In contrast, in the case of two agents and a single object, egalitarian-equivalence is compatible with efficiency. Full article
(This article belongs to the Special Issue Strategy-Proof Mechanism Design)
32 pages, 2389 KiB  
Article
Endogenous Game Choice and Giving Behavior in Distribution Games
by Emin Karagözoğlu and Elif Tosun
Games 2022, 13(6), 74; https://doi.org/10.3390/g13060074 - 03 Nov 2022
Viewed by 1731
Abstract
We experimentally investigated the effects of the possibility of taking in the dictator game and the choices of passive players between the dictator game and the taking game on the distribution decisions of active players. Our main findings support our hypothesis: when the [...] Read more.
We experimentally investigated the effects of the possibility of taking in the dictator game and the choices of passive players between the dictator game and the taking game on the distribution decisions of active players. Our main findings support our hypothesis: when the dictator game is not exogenously given but chosen by the receivers (or passive players), this makes them accountable, which leads to less giving by dictators. We also conducted an online survey to gain further insights about our experimental results. Survey participants predicted most of the observed behavior in the experiment and explained the factors that might have driven the predicted behavior using reasoning similar to ours. Our results provide a new perspective for the dependence of giving in the dictator game on contextual factors. Full article
(This article belongs to the Section Behavioral and Experimental Game Theory)
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13 pages, 807 KiB  
Article
Characterizing Agent Behavior in Revision Games with Uncertain Deadline
by Zhuohan Wang and Dong Hao
Games 2022, 13(6), 73; https://doi.org/10.3390/g13060073 - 01 Nov 2022
Viewed by 1230
Abstract
Revision game is a very recent advance in dynamic game theory and it can be used to analyze the trading in the pre-opening stock market. In such games, players prepare actions that will be implemented at a given deadline, before which they may [...] Read more.
Revision game is a very recent advance in dynamic game theory and it can be used to analyze the trading in the pre-opening stock market. In such games, players prepare actions that will be implemented at a given deadline, before which they may have opportunities to revise actions. For the first time, we study the role of the deadline in revision games, which is the core component that distinguishes revision games from classic games. We introduce the deadline distribution into revision game model and characterize the sufficient and necessary condition for players’ strategies to constitute an equilibrium. The equilibrium strategy with respect to the deadline uncertainty is given by a simple differential equation set. Governed by this differential equation set, players initially fully cooperate, and the cooperation level decreases as time progresses. The uncertainty has a great impact on players’ behavior. As the uncertainty increases, players become more risk averse, in the sense that they prefer lower mutual cooperation rate rather than higher payoff with higher uncertainty. Specifically, they will not stay in full cooperation for a long time, while after they deviate from the full cooperation, they adjust their plans more slowly and cautiously. The deadline uncertainty can improve the competition and avoid collusion in games, which could be utilized for auction design and pre-opening stock market regulations. Full article
(This article belongs to the Section Algorithmic and Computational Game Theory)
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20 pages, 378 KiB  
Article
Rational Play in Extensive-Form Games
by Giacomo Bonanno
Games 2022, 13(6), 72; https://doi.org/10.3390/g13060072 - 30 Oct 2022
Viewed by 2004
Abstract
We argue in favor of a departure from the equilibrium approach in game theory towards the less ambitious goal of describing only the actual behavior of rational players. The notions of Nash equilibrium and its refinements require a specification of the players’ choices [...] Read more.
We argue in favor of a departure from the equilibrium approach in game theory towards the less ambitious goal of describing only the actual behavior of rational players. The notions of Nash equilibrium and its refinements require a specification of the players’ choices and beliefs not only along the equilibrium play but also at counterfactual histories. We discuss an alternative—counterfactual-free—approach that focuses on choices and beliefs along the actual play, while being silent on choices and beliefs at unreached histories. Such an approach was introduced in an earlier paper that considered only perfect-information games. Here we extend the analysis to general extensive-form games (allowing for imperfect information) and put forward a behavioral notion of self-confirming play, which is close in spirit to the literature on self-confirming equilibrium. We also extend, to general extensive-form games, the characterization of rational play that is compatible with pure-strategy Nash equilibrium. Full article
(This article belongs to the Topic Game Theory and Applications)
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15 pages, 291 KiB  
Article
Misperception and Cognition in Markets
by Benjamin Young
Games 2022, 13(6), 71; https://doi.org/10.3390/g13060071 - 28 Oct 2022
Cited by 2 | Viewed by 1274
Abstract
We consider a market setting where a consumer holds either a naive or sophisticated perception of their preference over products. We introduce the concept of a cognitive equilibrium, in which the consumer can transition between the cognitive states of naiveté and sophistication depending [...] Read more.
We consider a market setting where a consumer holds either a naive or sophisticated perception of their preference over products. We introduce the concept of a cognitive equilibrium, in which the consumer can transition between the cognitive states of naiveté and sophistication depending on the degree of exploitation in the market. We compare market outcomes under monopoly and competition. While competition unambiguously improves market outcomes when the consumer’s cognitive state is exogenous, it can strictly lower gains from trade when cognitive states are endogenously determined. Full article
(This article belongs to the Special Issue Economics of Motivated Beliefs)
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26 pages, 616 KiB  
Article
Construction of Equilibria in Strategic Stackelberg Games in Multi-Period Supply Chain Contracts
by Reza Azad Gholami, Leif Kristoffer Sandal and Jan Ubøe
Games 2022, 13(6), 70; https://doi.org/10.3390/g13060070 - 27 Oct 2022
Cited by 1 | Viewed by 1640
Abstract
Almost every supplier faces uncertain and time-varying demand. E-commerce and online shopping have given suppliers unprecedented access to data on customers’ behavior, which sheds light on demand uncertainty. The main purpose of this research project is to provide an analytic tool for decentralized [...] Read more.
Almost every supplier faces uncertain and time-varying demand. E-commerce and online shopping have given suppliers unprecedented access to data on customers’ behavior, which sheds light on demand uncertainty. The main purpose of this research project is to provide an analytic tool for decentralized supply channel members to devise optimal long-term (multi-period) supply, pricing, and timing strategies while catering to stochastic demand in a diverse set of market scenarios. Despite its ubiquity in potential applications, the time-dependent channel optimization problem in its general form has received limited attention in the literature due to its complexity and the highly nested structure of its ensuing equilibrium problems. However, there are many scenarios where a single-period channel optimization solution may turn out to be myopic as it does not consider the after-effects of current pricing on future demand. To remedy this typical shortcoming, using general memory functions, we include the strategic customers’ cognitive bias toward pricing history in the supply channel equilibrium problem. In the form of two constructive theorems, we provide explicit solution algorithms for the ensuing Nash–Stackelberg equilibrium problems. In particular, we prove that our recursive solution algorithm can find equilibria in the multi-periodic variation of many standard supply channel contracts such as wholesale, buyback, and revenue-sharing contracts. Full article
(This article belongs to the Topic Game Theory and Applications)
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10 pages, 1209 KiB  
Article
The Strategy Method Risks Conflating Confusion with a Social Preference for Conditional Cooperation in Public Goods Games
by Maxwell N. Burton-Chellew, Victoire D’Amico and Claire Guérin
Games 2022, 13(6), 69; https://doi.org/10.3390/g13060069 - 25 Oct 2022
Cited by 2 | Viewed by 2079
Abstract
The strategy method is often used in public goods games to measure an individual’s willingness to cooperate depending on the level of cooperation by their groupmates (conditional cooperation). However, while the strategy method is informative, it risks conflating confusion with a desire for [...] Read more.
The strategy method is often used in public goods games to measure an individual’s willingness to cooperate depending on the level of cooperation by their groupmates (conditional cooperation). However, while the strategy method is informative, it risks conflating confusion with a desire for fair outcomes, and its presentation may risk inducing elevated levels of conditional cooperation. This problem was highlighted by two previous studies which found that the strategy method could also detect equivalent levels of cooperation even among those grouped with computerized groupmates, indicative of confusion or irrational responses. However, these studies did not use large samples (n = 40 or 72) and only made participants complete the strategy method one time, with computerized groupmates, preventing within-participant comparisons. Here, in contrast, 845 participants completed the strategy method two times, once with human and once with computerized groupmates. Our research aims were twofold: (1) to check the robustness of previous results with a large sample under various presentation conditions; and (2) to use a within-participant design to categorize participants according to how they behaved across the two scenarios. Ideally, a clean and reliable measure of conditional cooperation would find participants conditionally cooperating with humans and not cooperating with computers. Worryingly, only 7% of participants met this criterion. Overall, 83% of participants cooperated with the computers, and the mean contributions towards computers were 89% as large as those towards humans. These results, robust to the various presentation and order effects, pose serious concerns for the measurement of social preferences and question the idea that human cooperation is motivated by a concern for equal outcomes. Full article
(This article belongs to the Special Issue Learning and Evolution in Games, 1st Edition)
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