This paper presents a time-consistent dynamic Shapley value imputation for a class of differential network games. A novel form for measuring the worth of coalitions – named as cooperative-trajectory characteristic function – is developed for the Shapley value imputation. This new class of characteristic functions is evaluated along the cooperative trajectory. It measures the worth of coalitions under the process of cooperation instead of under min-max confrontation or the Nash non-cooperative stance. The resultant dynamic Shapley value imputation yields a new cooperative solution in differential network games.
{"title":"Shapley value for differential network games: Theory and application","authors":"L. Petrosyan, D. Yeung","doi":"10.3934/jdg.2020021","DOIUrl":"https://doi.org/10.3934/jdg.2020021","url":null,"abstract":"This paper presents a time-consistent dynamic Shapley value imputation for a class of differential network games. A novel form for measuring the worth of coalitions – named as cooperative-trajectory characteristic function – is developed for the Shapley value imputation. This new class of characteristic functions is evaluated along the cooperative trajectory. It measures the worth of coalitions under the process of cooperation instead of under min-max confrontation or the Nash non-cooperative stance. The resultant dynamic Shapley value imputation yields a new cooperative solution in differential network games.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70033363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article shows how to apply results of chemical reaction network theory (CRNT) to prove uniqueness and stability of a positive equilibrium for pairs/groups distributional dynamics that arise in game theoretic models. Evolutionary game theory assumes that individuals accrue their fitness through interactions with other individuals. When there are two or more different strategies in the population, this theory assumes that pairs (groups) are formed instantaneously and randomly so that the corresponding pairs (groups) distribution is described by the Hardy–Weinberg (binomial) distribution. If interactions times are phenotype dependent the Hardy-Weinberg distribution does not apply. Even if it becomes impossible to calculate the pairs/groups distribution analytically we show that CRNT is a general tool that is very useful to prove not only existence of the equilibrium, but also its stability. In this article, we apply CRNT to pair formation model that arises in two player games (e.g., Hawk-Dove, Prisoner's Dilemma game), to group formation that arises, e.g., in Public Goods Game, and to distribution of a single population in patchy environments. We also show by generalizing the Battle of the Sexes game that the methodology does not always apply.
{"title":"Using chemical reaction network theory to show stability of distributional dynamics in game theory","authors":"R. Cressman, V. Křivan","doi":"10.3934/jdg.2021030","DOIUrl":"https://doi.org/10.3934/jdg.2021030","url":null,"abstract":"This article shows how to apply results of chemical reaction network theory (CRNT) to prove uniqueness and stability of a positive equilibrium for pairs/groups distributional dynamics that arise in game theoretic models. Evolutionary game theory assumes that individuals accrue their fitness through interactions with other individuals. When there are two or more different strategies in the population, this theory assumes that pairs (groups) are formed instantaneously and randomly so that the corresponding pairs (groups) distribution is described by the Hardy–Weinberg (binomial) distribution. If interactions times are phenotype dependent the Hardy-Weinberg distribution does not apply. Even if it becomes impossible to calculate the pairs/groups distribution analytically we show that CRNT is a general tool that is very useful to prove not only existence of the equilibrium, but also its stability. In this article, we apply CRNT to pair formation model that arises in two player games (e.g., Hawk-Dove, Prisoner's Dilemma game), to group formation that arises, e.g., in Public Goods Game, and to distribution of a single population in patchy environments. We also show by generalizing the Battle of the Sexes game that the methodology does not always apply.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70034448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A novel approach depicting the dynamics of marijuana usage to gauge the effects of peer influence in a school population, is the site of investigation. Consumption of drug is considered as a contagious social epidemic which is spread mainly by peer influences. A relation-based graph-CA (r-GCA) model consisting of 4 states namely, Nonusers (N), Experimental users (E), Recreational users (R) and Addicts (A), is formulated in order to represent the prevalence of the epidemic on a campus. The r-GCA model is set up by local transition rules which delineates the proliferation of marijuana use. Data available in [ 4 ] is opted to verify and validate the r-GCA. Simulations of the r-GCA system are presented and discussed. The numerical results agree quite accurately with the observed data. Using the model, the enactment of campaigns of prevention targeting N, E and R states respectively were conducted and analysed. The results indicate a significant decline in marijuana consumption on the campus when a campaign of prevention targeting the latter three states simultaneously, is enacted.
一种新颖的方法描述了大麻使用的动态，以衡量学校人群中同伴影响的影响，这是调查的地点。毒品消费被认为是一种传染性的社会流行病，主要通过同伴影响传播。建立了一个由非使用者(N)、实验使用者(E)、娱乐使用者(R)和成瘾者(A) 4种状态组成的基于关系的图- ca (R - gca)模型，以表示校园流行病的流行程度。r-GCA模型是由描述大麻使用扩散的局部过渡规则建立的。选择[4]中可用的数据来验证和验证r-GCA。对r-GCA系统进行了仿真并进行了讨论。数值计算结果与实测数据吻合较好。利用该模型，分别进行了针对N、E和R州的预防运动的制定并进行了分析。结果表明，当针对后三个州同时实施预防运动时，校园大麻消费量显著下降。
{"title":"A graph cellular automaton with relation-based neighbourhood describing the impact of peer influence on the consumption of marijuana among college-aged youths","authors":"Y. B. Ruhomally, M. Z. Dauhoo, Laurent Dumas","doi":"10.3934/JDG.2021011","DOIUrl":"https://doi.org/10.3934/JDG.2021011","url":null,"abstract":"A novel approach depicting the dynamics of marijuana usage to gauge the effects of peer influence in a school population, is the site of investigation. Consumption of drug is considered as a contagious social epidemic which is spread mainly by peer influences. A relation-based graph-CA (r-GCA) model consisting of 4 states namely, Nonusers (N), Experimental users (E), Recreational users (R) and Addicts (A), is formulated in order to represent the prevalence of the epidemic on a campus. The r-GCA model is set up by local transition rules which delineates the proliferation of marijuana use. Data available in [ 4 ] is opted to verify and validate the r-GCA. Simulations of the r-GCA system are presented and discussed. The numerical results agree quite accurately with the observed data. Using the model, the enactment of campaigns of prevention targeting N, E and R states respectively were conducted and analysed. The results indicate a significant decline in marijuana consumption on the campus when a campaign of prevention targeting the latter three states simultaneously, is enacted.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70034577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a pursuit-evasion differential game problem with countable number pursuers and one evader in the Hilbert space begin{document}$ l_{2}. $end{document} Players' dynamic equations described by certain begin{document}$ n^{th} $end{document} order ordinary differential equations. Control functions of the players subject to integral constraints. The goal of the pursuers is to minimize the distance to the evader and that of the evader is the opposite. The stoppage time of the game is fixed and the game payoff is the distance between evader and closest pursuer when the game is stopped. We study this game problem and find the value of the game. In addition to this, we construct players' optimal strategies.
We consider a pursuit-evasion differential game problem with countable number pursuers and one evader in the Hilbert space begin{document}$ l_{2}. $end{document} Players' dynamic equations described by certain begin{document}$ n^{th} $end{document} order ordinary differential equations. Control functions of the players subject to integral constraints. The goal of the pursuers is to minimize the distance to the evader and that of the evader is the opposite. The stoppage time of the game is fixed and the game payoff is the distance between evader and closest pursuer when the game is stopped. We study this game problem and find the value of the game. In addition to this, we construct players' optimal strategies.
{"title":"Game value for a pursuit-evasion differential game problem in a Hilbert space","authors":"A. J. Badakaya, A. S. Halliru, J. Adamu","doi":"10.3934/JDG.2021019","DOIUrl":"https://doi.org/10.3934/JDG.2021019","url":null,"abstract":"We consider a pursuit-evasion differential game problem with countable number pursuers and one evader in the Hilbert space begin{document}$ l_{2}. $end{document} Players' dynamic equations described by certain begin{document}$ n^{th} $end{document} order ordinary differential equations. Control functions of the players subject to integral constraints. The goal of the pursuers is to minimize the distance to the evader and that of the evader is the opposite. The stoppage time of the game is fixed and the game payoff is the distance between evader and closest pursuer when the game is stopped. We study this game problem and find the value of the game. In addition to this, we construct players' optimal strategies.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70034627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we review a model of stochastic evolution under general noisy best-response protocols, allowing the probabilities of suboptimal choices to depend on their payoff consequences. We survey the methods developed by the authors which allow for a quantitative analysis of these stochastic evolutionary game dynamics. We start with a compact survey of techniques designed to study the long run behavior in the small noise double limit (SNDL). In this regime we let the noise level in agents' decision rules to approach zero, and then the population size is formally taken to infinity. This iterated limit strategy yields a family of deterministic optimal control problems which admit an explicit analysis in many instances. We then move in by describing the main steps to analyze stochastic evolutionary game dynamics in the large population double limit (LPDL). This regime refers to the iterated limit in which first the population size is taken to infinity and then the noise level in agents' decisions vanishes. The mathematical analysis of LPDL relies on a sample-path large deviations principle for a family of Markov chains on compact polyhedra. In this setting we formulate a set of conjectures and open problems which give a clear direction for future research activities.
{"title":"Large deviations and Stochastic stability in Population Games","authors":"Mathias Staudigl, S. Arigapudi, W. Sandholm","doi":"10.3934/jdg.2021021","DOIUrl":"https://doi.org/10.3934/jdg.2021021","url":null,"abstract":"In this article we review a model of stochastic evolution under general noisy best-response protocols, allowing the probabilities of suboptimal choices to depend on their payoff consequences. We survey the methods developed by the authors which allow for a quantitative analysis of these stochastic evolutionary game dynamics. We start with a compact survey of techniques designed to study the long run behavior in the small noise double limit (SNDL). In this regime we let the noise level in agents' decision rules to approach zero, and then the population size is formally taken to infinity. This iterated limit strategy yields a family of deterministic optimal control problems which admit an explicit analysis in many instances. We then move in by describing the main steps to analyze stochastic evolutionary game dynamics in the large population double limit (LPDL). This regime refers to the iterated limit in which first the population size is taken to infinity and then the noise level in agents' decisions vanishes. The mathematical analysis of LPDL relies on a sample-path large deviations principle for a family of Markov chains on compact polyhedra. In this setting we formulate a set of conjectures and open problems which give a clear direction for future research activities.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70034341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, there has been a revival of interest in cyclic decompositions of stochastic dynamics. These decompositions consider the behavior of dynamics over the short, medium and long run, aggregating cycles of behavior into progressively larger cycles, eventually encompassing the entire state space. We show that these decompositions are equivalent to the aggregative stage of Edmonds' algorithm and that this equivalence can be used to recover well-known results in the literature.
{"title":"Stochastic dynamics and Edmonds' algorithm","authors":"Jonathan Newton, William H. Sandholm","doi":"10.3934/jdg.2021029","DOIUrl":"https://doi.org/10.3934/jdg.2021029","url":null,"abstract":"Recently, there has been a revival of interest in cyclic decompositions of stochastic dynamics. These decompositions consider the behavior of dynamics over the short, medium and long run, aggregating cycles of behavior into progressively larger cycles, eventually encompassing the entire state space. We show that these decompositions are equivalent to the aggregative stage of Edmonds' algorithm and that this equivalence can be used to recover well-known results in the literature.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70034440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Determining the cause of a particular event has been a case of study for several researchers over the years. Finding out why an event happens (its cause) means that, for example, if we remove the cause from the equation, we can stop the effect from happening or if we replicate it, we can create the subsequent effect. Causality can be seen as a mean of predicting the future, based on information about past events, and with that, prevent or alter future outcomes. This temporal notion of past and future is often one of the critical points in discovering the causes of a given event. The purpose of this survey is to present a cross-sectional view of causal discovery domain, with an emphasis in the machine learning/data mining area.
{"title":"Causal discovery in machine learning: Theories and applications","authors":"Ana Rita Nogueira, J. Gama, C. Ferreira","doi":"10.3934/JDG.2021008","DOIUrl":"https://doi.org/10.3934/JDG.2021008","url":null,"abstract":"Determining the cause of a particular event has been a case of study for several researchers over the years. Finding out why an event happens (its cause) means that, for example, if we remove the cause from the equation, we can stop the effect from happening or if we replicate it, we can create the subsequent effect. Causality can be seen as a mean of predicting the future, based on information about past events, and with that, prevent or alter future outcomes. This temporal notion of past and future is often one of the critical points in discovering the causes of a given event. The purpose of this survey is to present a cross-sectional view of causal discovery domain, with an emphasis in the machine learning/data mining area.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70033959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A pricing model for the simplest commodity markets is considered. The model describes the behavior of the Order Book, consisting of orders from producers, consumers and speculators. The paper explores the external impact on this model in the form of large operations by new market participants, who at high speeds begin to push forward their orders, for example, first bids and then asks. Such strategies are called swings. This paper investigates a single cycle of one simple trading strategy of the swing type. Found a particular model case of pricing potentiality price relative to swing operation. An example is given, that shows that the simplest commodity markets with producers and consumers have the internal property that they are potentially vulnerable to external influences. The swinging of prices through large purchases and sales leads to systematic profits of the entrants at the expense of the traditional market participants.
{"title":"The sensitivity of commodity markets to exchange operations such as swing","authors":"G. Bobrik, P. Bobrik, I. Sukhorukova","doi":"10.3934/jdg.2020022","DOIUrl":"https://doi.org/10.3934/jdg.2020022","url":null,"abstract":"A pricing model for the simplest commodity markets is considered. The model describes the behavior of the Order Book, consisting of orders from producers, consumers and speculators. The paper explores the external impact on this model in the form of large operations by new market participants, who at high speeds begin to push forward their orders, for example, first bids and then asks. Such strategies are called swings. This paper investigates a single cycle of one simple trading strategy of the swing type. Found a particular model case of pricing potentiality price relative to swing operation. An example is given, that shows that the simplest commodity markets with producers and consumers have the internal property that they are potentially vulnerable to external influences. The swinging of prices through large purchases and sales leads to systematic profits of the entrants at the expense of the traditional market participants.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70034067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The allocation problem of rewards/costs is a basic question for players namely individuals and companies that planning cooperation under uncertainty. The involvement of uncertainty in cooperative game theory is motivated by the real world where noise in observation and experimental design, incomplete information and further vagueness in preference structures and decision-making play an important role. In this paper we extend cooperative bubbly games to cooperative fuzzy bubbly games, where the worth of each coalition is a fuzzy bubble instead of an interval. Further, we introduce a set-valued concept called the fuzzy bubbly core. Finally, some results on fuzzy bubbly core are given.
{"title":"On cooperative fuzzy bubbly games","authors":"I. Özcan, S. Z. A. Gök","doi":"10.3934/JDG.2021010","DOIUrl":"https://doi.org/10.3934/JDG.2021010","url":null,"abstract":"The allocation problem of rewards/costs is a basic question for players namely individuals and companies that planning cooperation under uncertainty. The involvement of uncertainty in cooperative game theory is motivated by the real world where noise in observation and experimental design, incomplete information and further vagueness in preference structures and decision-making play an important role. In this paper we extend cooperative bubbly games to cooperative fuzzy bubbly games, where the worth of each coalition is a fuzzy bubble instead of an interval. Further, we introduce a set-valued concept called the fuzzy bubbly core. Finally, some results on fuzzy bubbly core are given.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70034524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Financial crises are an important research topic because of their impact on the economy, businesses, and populations. However, prior research tends to generate reactive systemic risk measures, in the sense that the measure surges after the crisis starts. Few of them succeed in warning of financial crises in advance. In this paper, we first sketch a toy model that produces normal mixture distributions based on a dynamic regime switching model. We derive that the relative concavity among various indices tends to increase before a crisis. Using Polymodel theory, we introduce a measure of concavity as a crisis risk indicator, and test it against known crises observed in the past. We validate this indicator by a trading strategy holding long or short positions on the S & P 500 Index, depending on the indicator value.
{"title":"Crisis risk prediction with concavity from Polymodel","authors":"Yao Kuang, R. Douady","doi":"10.3934/jdg.2021027","DOIUrl":"https://doi.org/10.3934/jdg.2021027","url":null,"abstract":"Financial crises are an important research topic because of their impact on the economy, businesses, and populations. However, prior research tends to generate reactive systemic risk measures, in the sense that the measure surges after the crisis starts. Few of them succeed in warning of financial crises in advance. In this paper, we first sketch a toy model that produces normal mixture distributions based on a dynamic regime switching model. We derive that the relative concavity among various indices tends to increase before a crisis. Using Polymodel theory, we introduce a measure of concavity as a crisis risk indicator, and test it against known crises observed in the past. We validate this indicator by a trading strategy holding long or short positions on the S & P 500 Index, depending on the indicator value.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70034720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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