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Games, Volume 9, Issue 3 (September 2018)

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Open AccessFeature PaperArticle Preference Based Subjective Beliefs
Games 2018, 9(3), 50; https://doi.org/10.3390/g9030050
Received: 22 May 2018 / Revised: 26 June 2018 / Accepted: 5 July 2018 / Published: 16 July 2018
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Abstract
We test the empirical content of the assumption of preference dependent beliefs using a behavioral model of strategic decision making in which the rankings of individuals over final outcomes in simple games influence their beliefs over the opponent’s behavior. This approach— by analogy
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We test the empirical content of the assumption of preference dependent beliefs using a behavioral model of strategic decision making in which the rankings of individuals over final outcomes in simple games influence their beliefs over the opponent’s behavior. This approach— by analogy with Psychological Game Theory—allows for interdependence between preferences and beliefs but reverses the order of causality. We use existing evidence from a multi-stage experiment in which we first elicit distributional preferences in a Random Dictator Game, then estimate beliefs in a related 2×2 effort game conditional on these preferences. Our structural estimations confirm our working hypothesis on how social preferences shape beliefs: subjects with higher guilt (envy) expect others to put less (more) effort, which reduces the expected difference in payoffs. Full article
(This article belongs to the Special Issue Economic Behavior and Game Theory)
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Open AccessCommunication Cultural Transmission and Extortion
Games 2018, 9(3), 49; https://doi.org/10.3390/g9030049
Received: 23 May 2018 / Revised: 17 June 2018 / Accepted: 10 July 2018 / Published: 15 July 2018
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Abstract
This paper explores the role of cultural transmission in extortion. Extortion is modeled as an asymmetric contest where individuals from one group attempt to take from individuals in another group. Using a model of cultural transmission, this paper finds the existence of a
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This paper explores the role of cultural transmission in extortion. Extortion is modeled as an asymmetric contest where individuals from one group attempt to take from individuals in another group. Using a model of cultural transmission, this paper finds the existence of a unique asymptotically stable equilibrium where there are a fraction of people who defend against extortion and a fraction of people who take from others. The degree of extortion is decreased when: (1) extortion is less effective; (2) socialization efforts of parents who resist are more effective; and (3) socialization efforts of parents who abstain from extortion are more effective. A key finding is the existence of a complementarity between the effectiveness of socialization. When socialization is more effective for resisting extortion and choosing not to extort, there is a much larger reduction of extortion than either increase could achieve alone. This provides a potential explanation for why some countries like Singapore and Hong Kong were able to greatly reduce corruption and suggests potential policy applications. Full article
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Open AccessArticle On the Stability of Couples
Games 2018, 9(3), 48; https://doi.org/10.3390/g9030048
Received: 22 May 2018 / Revised: 28 June 2018 / Accepted: 2 July 2018 / Published: 11 July 2018
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Abstract
In this article, we analyze the stability of couples on the marriage market. Recent developments in cooperative game theory allow a new model that uses team games which make it possible to model the marriage market. Coalition structures can model couples. We analyze
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In this article, we analyze the stability of couples on the marriage market. Recent developments in cooperative game theory allow a new model that uses team games which make it possible to model the marriage market. Coalition structures can model couples. We analyze two cases: a symmetrical one with only one type of men and one type of women; and one with several types of women and men. Full article
Open AccessArticle Critical Discount Factor Values in Discounted Supergames
Games 2018, 9(3), 47; https://doi.org/10.3390/g9030047
Received: 5 June 2018 / Revised: 27 June 2018 / Accepted: 6 July 2018 / Published: 10 July 2018
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Abstract
This paper examines the subgame-perfect equilibria in symmetric 2×2 supergames. We solve the smallest discount factor value for which the players obtain all the feasible and individually rational payoffs as equilibrium payoffs. We show that the critical discount factor values are
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This paper examines the subgame-perfect equilibria in symmetric 2×2 supergames. We solve the smallest discount factor value for which the players obtain all the feasible and individually rational payoffs as equilibrium payoffs. We show that the critical discount factor values are not that high in many games and they generally depend on how large the payoff set is compared to the set of feasible payoffs. We analyze how the stage-game payoffs affect the required level of patience and organize the games into groups based on similar behavior. We study how the different strategies affect the set of equilibria by comparing pure, mixed and correlated strategies. This helps us understand better how discounting affects the set of equilibria and we can identify the games where extreme patience is required and the type of payoffs that are difficult to obtain. We also observe discontinuities in the critical values, which means that small changes in the stage-game payoffs may affect dramatically the equilibrium payoffs. Full article
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Open AccessArticle Computation of Sparse and Dense Equilibrium Strategies of Evolutionary Games
Games 2018, 9(3), 46; https://doi.org/10.3390/g9030046
Received: 8 May 2018 / Revised: 1 July 2018 / Accepted: 4 July 2018 / Published: 7 July 2018
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Abstract
The evolution of social or biological species can be modeled as an evolutionary game with the equilibrium strategies of the game as prediction for the ultimate distributions of species in population, when some species may survive with positive proportions, while others become extinct.
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The evolution of social or biological species can be modeled as an evolutionary game with the equilibrium strategies of the game as prediction for the ultimate distributions of species in population, when some species may survive with positive proportions, while others become extinct. We say a strategy is dense if it contains a large and diverse number of positive species, and is sparse if it has only a few dominant ones. Sparse equilibrium strategies can be found relatively easily, while dense ones are more computationally costly. Here we show that by formulating a “complementary” problem for the computation of equilibrium strategies, we are able to reduce the cost for computing dense equilibrium strategies much more efficiently. We describe the primary and complementary algorithms for computing dense as well as sparse equilibrium strategies, and present test results on randomly generated games as well as a more biologically related one. In particular, we demonstrate that the complementary algorithm is about an order of magnitude faster than the primary algorithm to obtain the dense equilibrium strategies for all our test cases. Full article
(This article belongs to the Special Issue Mathematical Biology and Game Theory)
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Open AccessArticle Cooperation and Strategic Complementarity: An Experiment with Two Voluntary Contribution Mechanism Games with Interior Equilibria
Games 2018, 9(3), 45; https://doi.org/10.3390/g9030045
Received: 25 May 2018 / Revised: 21 June 2018 / Accepted: 3 July 2018 / Published: 7 July 2018
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Abstract
In public goods game experiments, designs implementing non-linearities in the production are less common than the standard linear setting, especially so under the assumption that the private goods production and public goods aggregation function are both non-linear. We study a voluntary contribution game
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In public goods game experiments, designs implementing non-linearities in the production are less common than the standard linear setting, especially so under the assumption that the private goods production and public goods aggregation function are both non-linear. We study a voluntary contribution game (VCM) in which returns from the private project have diminishing marginal benefits and the contributions to the joint project exhibit pairwise strategic complementarities. As a control, we use a public goods game with an identical private production technology, but with the standard linear public goods aggregation. In addition to the aggregation technology, we manipulate the group size variable: In both treatments, the subjects will first play a VCM game in groups of five for 20 rounds, after which the group size is reduced to two, and the game is played for another 20 rounds. A significant over-contribution is observed in both settings when the group size is five. The rate of over-contribution is much higher under the complementary technology, but as predicted by theory, the contributions drop drastically when the group size is reduced from n = 5 to n = 2 within this treatment. Our experiment also provides empirical evidence that the so-called group size effect is present in both treatments, but it is much weaker under the standard aggregation technology. Full article
(This article belongs to the Special Issue Public Good Games)
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Open AccessArticle An Automated Method for Building Cognitive Models for Turn-Based Games from a Strategy Logic
Games 2018, 9(3), 44; https://doi.org/10.3390/g9030044
Received: 18 May 2018 / Revised: 2 July 2018 / Accepted: 4 July 2018 / Published: 6 July 2018
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Abstract
Whereas game theorists and logicians use formal methods to investigate ideal strategic behavior, many cognitive scientists use computational cognitive models of the human mind to predict and simulate human behavior. In this paper, we aim to bring these fields closer together by creating
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Whereas game theorists and logicians use formal methods to investigate ideal strategic behavior, many cognitive scientists use computational cognitive models of the human mind to predict and simulate human behavior. In this paper, we aim to bring these fields closer together by creating a generic translation system which, starting from a strategy for a turn-based game represented in formal logic, automatically generates a computational model in the Primitive Information Processing Elements (PRIMs) cognitive architecture, which has been validated on various experiments in cognitive psychology. The PRIMs models can be run and fitted to participants’ data in terms of decisions, response times, and answers to questions. As a proof of concept, we run computational modeling experiments on the basis of a game-theoretic experiment about the turn-based game “Marble Drop with Surprising Opponent”, in which the opponent often starts with a seemingly irrational move. We run such models starting from logical representations of several strategies, such as backward induction and extensive-form rationalizability, as well as different player types according to stance towards risk and level of theory of mind. Hereby, response times and decisions for such centipede-like games are generated, which in turn leads to concrete predictions for future experiments with human participants. Such precise predictions about different aspects, including reaction times, eye movements and active brain areas, cannot be derived on the basis of a strategy logic by itself: the computational cognitive models play a vital role and our generic translation system makes their construction more efficient and systematic than before. Full article
(This article belongs to the Special Issue Logic and Game Theory)
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Open AccessFeature PaperArticle Dynamic Contributions to a Public Project: The Impact of Rising Marginal Benefit and Completion Benefits
Games 2018, 9(3), 43; https://doi.org/10.3390/g9030043
Received: 24 May 2018 / Revised: 21 June 2018 / Accepted: 23 June 2018 / Published: 1 July 2018
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Abstract
Many public projects are funded in a dynamic manner in which contributors are able to make gradual increases in contributions and condition additional contributions on the cooperation of others. This study presents results from experiments in which subjects with an initial fixed endowment
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Many public projects are funded in a dynamic manner in which contributors are able to make gradual increases in contributions and condition additional contributions on the cooperation of others. This study presents results from experiments in which subjects with an initial fixed endowment make contributions to a public project gradually over a series of rounds. A 2 × 2 experimental design is used to examine the effectiveness of multiple thresholds that once crossed increase the marginal benefit of a contribution to the public project and the existence of a completion benefit upon project completion. Results reveal that when the multiple threshold design is combined with a completion benefit, overall contributions and project completion rates increase relative to other treatments. Without the presence of a completion benefit, contributions in the multiple threshold design are not significantly different from a constant marginal benefit design. In addition, completion benefits are shown to strongly encourage additional cooperation and project completion. Finally, projects are more likely to be completed when substantial contributions occur in the early rounds. Full article
(This article belongs to the Special Issue Public Good Games)
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Open AccessArticle The Optimal Strategy in the Minimum Effort Game
Games 2018, 9(3), 42; https://doi.org/10.3390/g9030042
Received: 21 May 2018 / Revised: 20 June 2018 / Accepted: 27 June 2018 / Published: 29 June 2018
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Abstract
A simple expression is derived for the optimal strategy in the minimum effort game. This maps from player beliefs to an optimal effort level. From this expression the set of Nash equilibria in the game is fully characterized. All Nash equilibria are symmetric
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A simple expression is derived for the optimal strategy in the minimum effort game. This maps from player beliefs to an optimal effort level. From this expression the set of Nash equilibria in the game is fully characterized. All Nash equilibria are symmetric and involve at most two actions being played with positive probability. We discuss how our expression for the optimal strategy can help inform on the comparative statics of a change in the number of players and effort cost benefit ratio. Full article
(This article belongs to the Special Issue Public Good Games)
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Open AccessArticle To Tender or Not to Tender? Deliberate and Exogenous Sunk Costs in a Public Good Game
Games 2018, 9(3), 41; https://doi.org/10.3390/g9030041
Received: 25 May 2018 / Revised: 15 June 2018 / Accepted: 22 June 2018 / Published: 26 June 2018
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Abstract
In an experimental study, we compare individual willingness to cooperate in a public good game after an initial team contest phase. While players in the treatment setup make a conscious decision on how much to invest in the contest, this decision is exogenously
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In an experimental study, we compare individual willingness to cooperate in a public good game after an initial team contest phase. While players in the treatment setup make a conscious decision on how much to invest in the contest, this decision is exogenously imposed on players in the control setup. As such, both groups of players incur sunk costs and enter the public good game with different wealth levels. Our results indicate that the way these sunk costs have been accrued matters especially for groups on the losing side of the contest: Given the same level of sunk costs, contributions to the public good are lower for groups which failed to be successful in the preceding between-group contest. Furthermore, this detrimental effect is more pronounced for individuals who play a contest with deliberate contributions before. Full article
(This article belongs to the Special Issue Public Good Games)
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Open AccessArticle Buying Optimal Payoffs in Bi-Matrix Games
Games 2018, 9(3), 40; https://doi.org/10.3390/g9030040
Received: 23 May 2018 / Revised: 20 June 2018 / Accepted: 22 June 2018 / Published: 26 June 2018
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Abstract
We consider non-zero sum bi-matrix games where one player presumes the role of a leader in the Stackelberg model, while the other player is her follower. We show that the leader can improve her reward if she can incentivise her follower by paying
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We consider non-zero sum bi-matrix games where one player presumes the role of a leader in the Stackelberg model, while the other player is her follower. We show that the leader can improve her reward if she can incentivise her follower by paying some of her own utility to the follower for assigning a particular strategy profile. Besides assuming that the follower is rational in that he tries to maximise his own payoff, we assume that he is also friendly towards his leader in that he chooses, ex aequo, the strategy suggested by her—at least as long as it does not affect his expected payoff. Assuming this friendliness is, however, disputable: one could also assume that, ex aequo, the follower acts adversarially towards his leader. We discuss these different follower behavioural models and their implications. We argue that the friendliness leads to an obligation for the leader to choose, ex aequo, an assignment that provides the highest follower return, resulting in ‘friendly incentive equilibria’. For the antagonistic assumption, the stability requirements for a strategy profile should be strengthened, comparable to the secure Nash equilibria. In general, no optimal incentive equilibrium for this condition exists, and therefore we introduce ε-optimal incentive equilibria for this case. We show that the construction of all of these incentive equilibria (and all the related leader equilibria) is tractable. Full article
(This article belongs to the Special Issue Logic and Game Theory)
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Open AccessArticle An Abstraction-Refinement Methodologyfor Reasoning about Network Games
Games 2018, 9(3), 39; https://doi.org/10.3390/g9030039
Received: 2 May 2018 / Revised: 14 June 2018 / Accepted: 17 June 2018 / Published: 22 June 2018
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Abstract
Network games (NGs) are played on directed graphs and are extensively used in network design and analysis. Search problems for NGs include finding special strategy profiles such as a Nash equilibrium and a globally-optimal solution. The networks modeled by NGs may be huge.
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Network games (NGs) are played on directed graphs and are extensively used in network design and analysis. Search problems for NGs include finding special strategy profiles such as a Nash equilibrium and a globally-optimal solution. The networks modeled by NGs may be huge. In formal verification, abstraction has proven to be an extremely effective technique for reasoning about systems with big and even infinite state spaces. We describe an abstraction-refinement methodology for reasoning about NGs. Our methodology is based on an abstraction function that maps the state space of an NG to a much smaller state space. We search for a global optimum and a Nash equilibrium by reasoning on an under- and an over-approximation defined on top of this smaller state space. When the approximations are too coarse to find such profiles, we refine the abstraction function. We extend the abstraction-refinement methodology to labeled networks, where the objectives of the players are regular languages. Our experimental results demonstrate the effectiveness of the methodology. Full article
(This article belongs to the Special Issue Logic and Game Theory)
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Open AccessArticle A Survey on the Design of Gamified Systems for Energy and Water Sustainability
Games 2018, 9(3), 38; https://doi.org/10.3390/g9030038
Received: 22 May 2018 / Revised: 12 June 2018 / Accepted: 13 June 2018 / Published: 21 June 2018
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Abstract
In a world affected by the constant growth and concentration of the population in urban areas, the problem of preserving natural resources has become a priority. A promising approach to resource conservation is demand management, i.e., the ability to positively influence the behaviour
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In a world affected by the constant growth and concentration of the population in urban areas, the problem of preserving natural resources has become a priority. A promising approach to resource conservation is demand management, i.e., the ability to positively influence the behaviour of the population towards more sustainable consumption. Information and Communication Technology (ICT) tools have shown a great potential in influencing consumers’ behaviour, which could be exploited for the common good. However, the design of so-called persuasive systems for environmental purposes is a challenging task, because it cannot be based solely on the utilitarian motivation of users, but must be able to trigger a broader range of engagement factors deeply rooted in the human psychology. In this paper, we review the main design principles and models at the base of a class of persuasive system that exploits gamification and Games with a Purpose (GWAPs) to engage users towards sustainability; we identify the most commonly used incentive mechanisms for inducing behavioural changes; and present a selection of gamified systems for energy and water conservation. From such a survey, we distill design guidelines to be applied in the design of demand management socio-technical systems. Full article
(This article belongs to the Special Issue Public Good Games)
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