In this paper, we focus on the equal surplus sharing interval solutions for cooperative games, where the set of players are finite and the coalition values are interval numbers. We consider the properties of a class of equal surplus sharing interval solutions consisting of all convex combinations of them. Moreover, an application based on transportation interval situations is given. Finally, we propose three solution concepts, namely the interval Shapley value, ICIS-value and IENSC-value, for this application and these solution concepts are compared.
{"title":"On the equal surplus sharing interval solutions and an application","authors":"O. Palanci, M. Eki̇ci̇, S. Z. A. Gök","doi":"10.3934/jdg.2020023","DOIUrl":"https://doi.org/10.3934/jdg.2020023","url":null,"abstract":"In this paper, we focus on the equal surplus sharing interval solutions for cooperative games, where the set of players are finite and the coalition values are interval numbers. We consider the properties of a class of equal surplus sharing interval solutions consisting of all convex combinations of them. Moreover, an application based on transportation interval situations is given. Finally, we propose three solution concepts, namely the interval Shapley value, ICIS-value and IENSC-value, for this application and these solution concepts are compared.","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":"70034155","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 paper we study zero-sum stochastic games for pure jump processes on a general state space with risk sensitive discounted criteria. We establish a saddle point equilibrium in Markov strategies for bounded cost function. We achieve our results by studying relevant Hamilton-Jacobi-Isaacs equations.
{"title":"Zero-sum games for pure jump processes with risk-sensitive discounted cost criteria","authors":"Chandan Pal, Somnath Pradhan","doi":"10.3934/jdg.2021020","DOIUrl":"https://doi.org/10.3934/jdg.2021020","url":null,"abstract":"In this paper we study zero-sum stochastic games for pure jump processes on a general state space with risk sensitive discounted criteria. We establish a saddle point equilibrium in Markov strategies for bounded cost function. We achieve our results by studying relevant Hamilton-Jacobi-Isaacs equations.","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":"70034691","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}
Laura Aquilanti, S. Cacace, F. Camilli, Raul De Maio
Finite mixture models are an important tool in the statistical analysis of data, for example in data clustering. The optimal parameters of a mixture model are usually computed by maximizing the log-likelihood functional via the Expectation-Maximization algorithm. We propose an alternative approach based on the theory of Mean Field Games, a class of differential games with an infinite number of agents. We show that the solution of a finite state space multi-population Mean Field Games system characterizes the critical points of the log-likelihood functional for a Bernoulli mixture. The approach is then generalized to mixture models of categorical distributions. Hence, the Mean Field Games approach provides a method to compute the parameters of the mixture model, and we show its application to some standard examples in cluster analysis.
{"title":"A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions","authors":"Laura Aquilanti, S. Cacace, F. Camilli, Raul De Maio","doi":"10.3934/jdg.2020033","DOIUrl":"https://doi.org/10.3934/jdg.2020033","url":null,"abstract":"Finite mixture models are an important tool in the statistical analysis of data, for example in data clustering. The optimal parameters of a mixture model are usually computed by maximizing the log-likelihood functional via the Expectation-Maximization algorithm. We propose an alternative approach based on the theory of Mean Field Games, a class of differential games with an infinite number of agents. We show that the solution of a finite state space multi-population Mean Field Games system characterizes the critical points of the log-likelihood functional for a Bernoulli mixture. The approach is then generalized to mixture models of categorical distributions. Hence, the Mean Field Games approach provides a method to compute the parameters of the mixture model, and we show its application to some standard examples in cluster analysis.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48983101","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 extend the methods from Nurbekyan, Saude "Fourier approximation methods for first-order nonlocal mean-field games" [Port. Math. 75 (2018), no. 3-4] and Liu, Jacobs, Li, Nurbekyan, Osher "Computational methods for nonlocal mean field games with applications" [arXiv:2004.12210] to a class of non-potential mean-field game (MFG) systems with mixed couplings. Up to now, splitting methods have been applied to potential MFG systems that can be cast as convex-concave saddle-point problems. Here, we show that a class of non-potential MFG can be cast as primal-dual pairs of monotone inclusions and solved via extensions of convex optimization algorithms such as the primal-dual hybrid gradient (PDHG) algorithm. A critical feature of our approach is in considering dual variables of nonlocal couplings in Fourier or feature spaces.
{"title":"Splitting methods for a class of non-potential mean field games","authors":"Siting Liu, L. Nurbekyan","doi":"10.3934/JDG.2021014","DOIUrl":"https://doi.org/10.3934/JDG.2021014","url":null,"abstract":"We extend the methods from Nurbekyan, Saude \"Fourier approximation methods for first-order nonlocal mean-field games\" [Port. Math. 75 (2018), no. 3-4] and Liu, Jacobs, Li, Nurbekyan, Osher \"Computational methods for nonlocal mean field games with applications\" [arXiv:2004.12210] to a class of non-potential mean-field game (MFG) systems with mixed couplings. Up to now, splitting methods have been applied to potential MFG systems that can be cast as convex-concave saddle-point problems. Here, we show that a class of non-potential MFG can be cast as primal-dual pairs of monotone inclusions and solved via extensions of convex optimization algorithms such as the primal-dual hybrid gradient (PDHG) algorithm. A critical feature of our approach is in considering dual variables of nonlocal couplings in Fourier or feature spaces.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47055320","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 paper we consider a mean field approach to modeling the agents flow over a transportation network. In particular, beside a standard framework of mean field games, with controlled dynamics by the agents and costs mass-distribution dependent, we also consider a path preferences dynamics obtained as a generalization of the so-called noisy best response dynamics. Such a preferences dynamics says the agents choose their path having access to global information about the network congestion state and based on the observation of the decision of the agents that have preceded. We prove the existence of a mean field equilibrium obtained as a fixed point of a map over a suitable set of time-varying mass-distributions, defined edge by edge in the network. We also address the case where the admissible set of controls is suitably bounded depending on the mass-distribution on the edge itself.
{"title":"Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach","authors":"Fabio Bagagiolo, Rosario Maggistro, R. Pesenti","doi":"10.3934/JDG.2021007","DOIUrl":"https://doi.org/10.3934/JDG.2021007","url":null,"abstract":"In this paper we consider a mean field approach to modeling the agents flow over a transportation network. In particular, beside a standard framework of mean field games, with controlled dynamics by the agents and costs mass-distribution dependent, we also consider a path preferences dynamics obtained as a generalization of the so-called noisy best response dynamics. Such a preferences dynamics says the agents choose their path having access to global information about the network congestion state and based on the observation of the decision of the agents that have preceded. We prove the existence of a mean field equilibrium obtained as a fixed point of a map over a suitable set of time-varying mass-distributions, defined edge by edge in the network. We also address the case where the admissible set of controls is suitably bounded depending on the mass-distribution on the edge itself.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41780636","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 develop a model for the industry dynamics in the electricity market, based on mean-field games of optimal stopping. In our model, there are two types of agents: the renewable producers and the conventional producers. The renewable producers choose the optimal moment to build new renewable plants, and the conventional producers choose the optimal moment to exit the market. The agents interact through the market price, determined by matching the aggregate supply of the two types of producers with an exogenous demand function. Using a relaxed formulation of optimal stopping mean-field games, we prove the existence of a Nash equilibrium and the uniqueness of the equilibrium price process. An empirical example, inspired by the UK electricity market is presented. The example shows that while renewable subsidies clearly lead to higher renewable penetration, this may entail a cost to the consumer in terms of higher peakload prices. In order to avoid rising prices, the renewable subsidies must be combined with mechanisms ensuring that sufficient conventional capacity remains in place to meet the energy demand during peak periods.
{"title":"The entry and exit game in the electricity markets: A mean-field game approach","authors":"R. Aid, Roxana Dumitrescu, P. Tankov","doi":"10.3934/JDG.2021012","DOIUrl":"https://doi.org/10.3934/JDG.2021012","url":null,"abstract":"We develop a model for the industry dynamics in the electricity market, based on mean-field games of optimal stopping. In our model, there are two types of agents: the renewable producers and the conventional producers. The renewable producers choose the optimal moment to build new renewable plants, and the conventional producers choose the optimal moment to exit the market. The agents interact through the market price, determined by matching the aggregate supply of the two types of producers with an exogenous demand function. Using a relaxed formulation of optimal stopping mean-field games, we prove the existence of a Nash equilibrium and the uniqueness of the equilibrium price process. An empirical example, inspired by the UK electricity market is presented. The example shows that while renewable subsidies clearly lead to higher renewable penetration, this may entail a cost to the consumer in terms of higher peakload prices. In order to avoid rising prices, the renewable subsidies must be combined with mechanisms ensuring that sufficient conventional capacity remains in place to meet the energy demand during peak periods.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48965523","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 difference and differential Stackelberg game theoretic models with several followers of opinion control in marketing networks. It is assumed that in the stage of analysis of the network its opinion leaders have already been found and are the only objects of control. The leading player determines the marketing budgets of the followers by resource allocation. In the basic version of the models both the leader and the followers maximize the summary opinions of the network agents. In the second version the leader has a target value of the summary opinion. In all four models we have found the Stackelberg equilibrium and the respective payoffs of the players analytically. It is shown that the hierarchical control system is ideally compatible in all cases.
{"title":"Optimal resource allocation in the difference and differential Stackelberg games on marketing networks","authors":"A. Korolev, G. Ougolnitsky","doi":"10.3934/jdg.2020009","DOIUrl":"https://doi.org/10.3934/jdg.2020009","url":null,"abstract":"We consider difference and differential Stackelberg game theoretic models with several followers of opinion control in marketing networks. It is assumed that in the stage of analysis of the network its opinion leaders have already been found and are the only objects of control. The leading player determines the marketing budgets of the followers by resource allocation. In the basic version of the models both the leader and the followers maximize the summary opinions of the network agents. In the second version the leader has a target value of the summary opinion. In all four models we have found the Stackelberg equilibrium and the respective payoffs of the players analytically. It is shown that the hierarchical control system is ideally compatible in all cases.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70033324","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 sufficient condition for the uniqueness of a Nash equilibrium in strategic-form games: for any two distinct strategy profiles, there is a player who can obtain a higher payoff by unilaterally changing the strategy from one strategy profile to the other strategy profile. An example of a game that satisfies this condition is the prisoner's dilemma. Viewed as a solution concept, the Nash equilibrium satisfying the condition is stronger than strict Nash Equilibrium and weaker than strict dominant strategy equilibrium.
{"title":"On the uniqueness of Nash equilibrium in strategic-form games","authors":"J. Minagawa","doi":"10.3934/jdg.2020006","DOIUrl":"https://doi.org/10.3934/jdg.2020006","url":null,"abstract":"We consider a sufficient condition for the uniqueness of a Nash equilibrium in strategic-form games: for any two distinct strategy profiles, there is a player who can obtain a higher payoff by unilaterally changing the strategy from one strategy profile to the other strategy profile. An example of a game that satisfies this condition is the prisoner's dilemma. Viewed as a solution concept, the Nash equilibrium satisfying the condition is stronger than strict Nash Equilibrium and weaker than strict dominant strategy equilibrium.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"38 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70033313","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 a complex system model we simulate runs for different strategies of economic agents to study diverse types of fluctuations. The liquidity of financial assets arises as a result of agent's interaction and not as intrinsic properties of the assets. Small differences in the strategic rules adopted by the agents lead to divergent paths of market liquidity. Our simulation also supports the idea that the higher the maximum local allowed fluctuation the higher the path divergence.
{"title":"Financial liquidity: An emergent phenomena","authors":"Alfredo García, M. Szybisz","doi":"10.3934/jdg.2020015","DOIUrl":"https://doi.org/10.3934/jdg.2020015","url":null,"abstract":"In a complex system model we simulate runs for different strategies of economic agents to study diverse types of fluctuations. The liquidity of financial assets arises as a result of agent's interaction and not as intrinsic properties of the assets. Small differences in the strategic rules adopted by the agents lead to divergent paths of market liquidity. Our simulation also supports the idea that the higher the maximum local allowed fluctuation the higher the path divergence.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70033558","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}
Cost sharing problems can arise from situations in which some service is provided to a variety of different customers who differ in the amount or type of service they need. One can think of and airports computers, telephones. This paper studies an airport problem which is concerned with the cost sharing of an airstrip between airplanes assuming that one airstrip is sufficient to serve all airplanes. Each airplane needs an airstrip whose length can be different across airplanes. Also, it is important how should the cost of each airstrip be shared among airplanes. The purpose of the present paper is to give an axiomatic characterization of the Baker-Thompson rule by using grey calculus. Further, it is shown that each of our main axioms (population fairness, smallest-cost consistency and balanced population impact) together with various combina tions of our minor axioms characterizes the best-known rule for the problem, namely the Baker-Thompson rule. Finally, it is demonstrated that the grey Shapley value of airport game and the grey Baker-Thompson rule coincides. Keywords: airport situations, Baker-Thompson rule, grey data, Shapley value.
{"title":"On the grey Baker-Thompson rule","authors":"M. O. Olgun, O. Palanci, S. Z. A. Gök","doi":"10.3934/jdg.2020024","DOIUrl":"https://doi.org/10.3934/jdg.2020024","url":null,"abstract":"Cost sharing problems can arise from situations in which some service is provided to a variety of different customers who differ in the amount or type of service they need. One can think of and airports computers, telephones. This paper studies an airport problem which is concerned with the cost sharing of an airstrip between airplanes assuming that one airstrip is sufficient to serve all airplanes. Each airplane needs an airstrip whose length can be different across airplanes. Also, it is important how should the cost of each airstrip be shared among airplanes. The purpose of the present paper is to give an axiomatic characterization of the Baker-Thompson rule by using grey calculus. Further, it is shown that each of our main axioms (population fairness, smallest-cost consistency and balanced population impact) together with various combina tions of our minor axioms characterizes the best-known rule for the problem, namely the Baker-Thompson rule. Finally, it is demonstrated that the grey Shapley value of airport game and the grey Baker-Thompson rule coincides. Keywords: airport situations, Baker-Thompson rule, grey data, Shapley value.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70033778","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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