We aim at characterizing the functions that could be explained (recoverable) as a best reply of payoff-maximizing players in contests for a fixed prize. We show that recoverability strongly differs between Decisive Contests, where the prize is allocated with certainty, and Possibly Indecisive Contests, where the prize might not be awarded. In the latter, any arbitrary set of best reply functions is recoverable, thus "anything goes." In the former, best reply functions have to satisfy strong conditions in some cases. We provide an outline of possible applications of our results to R & D and labor markets.
{"title":"Properties of contests: Constructing contest success functions from best-responses","authors":"Luis C. Corchón, Marco Serena","doi":"10.3934/jdg.2022001","DOIUrl":"https://doi.org/10.3934/jdg.2022001","url":null,"abstract":"We aim at characterizing the functions that could be explained (recoverable) as a best reply of payoff-maximizing players in contests for a fixed prize. We show that recoverability strongly differs between Decisive Contests, where the prize is allocated with certainty, and Possibly Indecisive Contests, where the prize might not be awarded. In the latter, any arbitrary set of best reply functions is recoverable, thus \"anything goes.\" In the former, best reply functions have to satisfy strong conditions in some cases. We provide an outline of possible applications of our results to R & D and labor markets.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70034459","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}
E. Accinelli, Armando García, Laura Policardo, Edgar J. Sanchez-Carrera
{"title":"A predator-prey economic system of tax evasion and corrupt behavior","authors":"E. Accinelli, Armando García, Laura Policardo, Edgar J. Sanchez-Carrera","doi":"10.3934/jdg.2022025","DOIUrl":"https://doi.org/10.3934/jdg.2022025","url":null,"abstract":"","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"20 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81353543","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}
Making software open source can have substantial positive effects on the quality and diffusion of a software and strengthen the sales of complementary products. However, it is a large concern of firms that a competitor might use the very same source code to start its own competitive project, a so-called fork. This paper analyzes whether the threat of forking prevents a firm to open its source code. We consider three different regimes: In the first regime a firm develops and sells software under a proprietary license, in the second regime, it uses an open source business model. The third regime is characterized by the competition of two related open source projects. The switching times between the regimes are optimally determined. We find that the optimal strategy substantially depends on the initial state value and the extent to which a competitor can make use of the firm's software quality. A small initial software quality can prevent a firm to open the code when it cannot afford competition, only with a competitive advantage open source is attractive then. For a large initial software quality, a firm would never open the code immediately, it would either wait or keep it proprietary forever.
{"title":"Opening the source code: The threat of forking","authors":"A. Seidl, S. Wrzaczek","doi":"10.3934/jdg.2022010","DOIUrl":"https://doi.org/10.3934/jdg.2022010","url":null,"abstract":"Making software open source can have substantial positive effects on the quality and diffusion of a software and strengthen the sales of complementary products. However, it is a large concern of firms that a competitor might use the very same source code to start its own competitive project, a so-called fork. This paper analyzes whether the threat of forking prevents a firm to open its source code. We consider three different regimes: In the first regime a firm develops and sells software under a proprietary license, in the second regime, it uses an open source business model. The third regime is characterized by the competition of two related open source projects. The switching times between the regimes are optimally determined. We find that the optimal strategy substantially depends on the initial state value and the extent to which a competitor can make use of the firm's software quality. A small initial software quality can prevent a firm to open the code when it cannot afford competition, only with a competitive advantage open source is attractive then. For a large initial software quality, a firm would never open the code immediately, it would either wait or keep it proprietary forever.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"16 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78801695","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}
Motivated by an optimal visiting problem, we study a switching mean-field game on a network, where both a decisional and a switching time-variable are at disposal of the agents for what concerns, respectively, the instant to decide and the instant to perform the switch. Every switch between the nodes of the network represents a switch from begin{document}$ 0 $end{document} to begin{document}$ 1 $end{document} of one component of the string begin{document}$ p = (p_1, ldots, p_n) $end{document} which, in the optimal visiting interpretation, gives information on the visited targets, being the targets labeled by begin{document}$ i = 1, ldots, n $end{document}. The goal is to reach the final string begin{document}$ (1, ldots, 1) $end{document} in the final time begin{document}$ T $end{document}, minimizing a switching cost also depending on the congestion on the nodes. We prove the existence of a suitable definition of an approximated begin{document}$ varepsilon $end{document}-mean-field equilibrium and then address the passage to the limit when begin{document}$ varepsilon $end{document} goes to begin{document}$ 0 $end{document}.
Motivated by an optimal visiting problem, we study a switching mean-field game on a network, where both a decisional and a switching time-variable are at disposal of the agents for what concerns, respectively, the instant to decide and the instant to perform the switch. Every switch between the nodes of the network represents a switch from begin{document}$ 0 $end{document} to begin{document}$ 1 $end{document} of one component of the string begin{document}$ p = (p_1, ldots, p_n) $end{document} which, in the optimal visiting interpretation, gives information on the visited targets, being the targets labeled by begin{document}$ i = 1, ldots, n $end{document}. The goal is to reach the final string begin{document}$ (1, ldots, 1) $end{document} in the final time begin{document}$ T $end{document}, minimizing a switching cost also depending on the congestion on the nodes. We prove the existence of a suitable definition of an approximated begin{document}$ varepsilon $end{document}-mean-field equilibrium and then address the passage to the limit when begin{document}$ varepsilon $end{document} goes to begin{document}$ 0 $end{document}.
{"title":"A time-dependent switching mean-field game on networks motivated by optimal visiting problems","authors":"Fabio Bagagiolo, Luciano Marzufero","doi":"10.3934/jdg.2022019","DOIUrl":"https://doi.org/10.3934/jdg.2022019","url":null,"abstract":"<p style='text-indent:20px;'>Motivated by an optimal visiting problem, we study a switching mean-field game on a network, where both a decisional and a switching time-variable are at disposal of the agents for what concerns, respectively, the instant to decide and the instant to perform the switch. Every switch between the nodes of the network represents a switch from <inline-formula><tex-math id=\"M1\">begin{document}$ 0 $end{document}</tex-math></inline-formula> to <inline-formula><tex-math id=\"M2\">begin{document}$ 1 $end{document}</tex-math></inline-formula> of one component of the string <inline-formula><tex-math id=\"M3\">begin{document}$ p = (p_1, ldots, p_n) $end{document}</tex-math></inline-formula> which, in the optimal visiting interpretation, gives information on the visited targets, being the targets labeled by <inline-formula><tex-math id=\"M4\">begin{document}$ i = 1, ldots, n $end{document}</tex-math></inline-formula>. The goal is to reach the final string <inline-formula><tex-math id=\"M5\">begin{document}$ (1, ldots, 1) $end{document}</tex-math></inline-formula> in the final time <inline-formula><tex-math id=\"M6\">begin{document}$ T $end{document}</tex-math></inline-formula>, minimizing a switching cost also depending on the congestion on the nodes. We prove the existence of a suitable definition of an approximated <inline-formula><tex-math id=\"M7\">begin{document}$ varepsilon $end{document}</tex-math></inline-formula>-mean-field equilibrium and then address the passage to the limit when <inline-formula><tex-math id=\"M8\">begin{document}$ varepsilon $end{document}</tex-math></inline-formula> goes to <inline-formula><tex-math id=\"M9\">begin{document}$ 0 $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"78 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76784876","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 presence of big data may adversely affect obtaining classification accuracy in many life applications, such as genes dataset, which can contain many unnecessary data in the classification process. In this study, a two-stage mathematical model is proposed through which the features are selected. The first stage relies on the Fuzzy Statistical Dependence (FSD) technique, which is one of the filter techniques, and in the second stage, the Binary Bat Algorithm (BBA) is used, which depends on an appropriate fitness function to select important parameters. The experimental results proved that the proposed algorithm, which we refer to as FSD-BBA, excels over other methods in terms of classification accuracy and the number of influencing genes selected.
{"title":"Gene subset selection using fuzzy statistical dependence technique and binary bat algorithm","authors":"M. Mahmoud, Fatima Mahmood Hasan, O. Qasim","doi":"10.3934/jdg.2022011","DOIUrl":"https://doi.org/10.3934/jdg.2022011","url":null,"abstract":"The presence of big data may adversely affect obtaining classification accuracy in many life applications, such as genes dataset, which can contain many unnecessary data in the classification process. In this study, a two-stage mathematical model is proposed through which the features are selected. The first stage relies on the Fuzzy Statistical Dependence (FSD) technique, which is one of the filter techniques, and in the second stage, the Binary Bat Algorithm (BBA) is used, which depends on an appropriate fitness function to select important parameters. The experimental results proved that the proposed algorithm, which we refer to as FSD-BBA, excels over other methods in terms of classification accuracy and the number of influencing genes selected.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"108 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81179669","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 model of the dynamics of distributions of individual wealth, or of individual Darwinian fitness, is here developed. Pairs of individuals are recurrently and randomly matched to play a game over a resource. In addition, all individuals have random access to a constant background source, and their fitness or wealth depreciates over time. For brevity, we focus on the well-known Hawk-Dove game. In the base-line model, the probability of winning a fight over a resource is the same for both parties. In an extended version, the individual with higher current fitness or wealth has a higher probability of winning. Analytical results are given for the fitness/wealth distribution at any given time, for the evolution of average fitness/wealth over time, and for the asymptotics with respect to both time and population size. Long-run average fitness/wealth is non-monotonic in the value of the resource, thus providing a potential explanation of the so-called curse of the riches.
{"title":"The dynamics of fitness and wealth distributions — a stochastic game-theoretic model","authors":"Sylvain Gibaud, J. Weibull","doi":"10.3934/jdg.2022016","DOIUrl":"https://doi.org/10.3934/jdg.2022016","url":null,"abstract":"A model of the dynamics of distributions of individual wealth, or of individual Darwinian fitness, is here developed. Pairs of individuals are recurrently and randomly matched to play a game over a resource. In addition, all individuals have random access to a constant background source, and their fitness or wealth depreciates over time. For brevity, we focus on the well-known Hawk-Dove game. In the base-line model, the probability of winning a fight over a resource is the same for both parties. In an extended version, the individual with higher current fitness or wealth has a higher probability of winning. Analytical results are given for the fitness/wealth distribution at any given time, for the evolution of average fitness/wealth over time, and for the asymptotics with respect to both time and population size. Long-run average fitness/wealth is non-monotonic in the value of the resource, thus providing a potential explanation of the so-called curse of the riches.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"32 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85385114","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 focus on stationary (ergodic) mean-field games (MFGs). These games arise in the study of the long-time behavior of finite-horizon MFGs. Motivated by a prior scheme for Hamilton–Jacobi equations introduced in Aubry–Mather's theory, we introduce a discrete approximation to stationary MFGs. Relying on Kakutani's fixed-point theorem, we prove the existence and uniqueness (up to additive constant) of solutions to the discrete problem. Moreover, we show that the solutions to the discrete problem converge, uniformly in the nonlocal case and weakly in the local case, to the classical solutions of the stationary problem.
{"title":"Discrete approximation of stationary Mean Field Games","authors":"T. Bakaryan, D. Gomes, H'ector S'anchez Morgado","doi":"10.3934/jdg.2022022","DOIUrl":"https://doi.org/10.3934/jdg.2022022","url":null,"abstract":"In this paper, we focus on stationary (ergodic) mean-field games (MFGs). These games arise in the study of the long-time behavior of finite-horizon MFGs. Motivated by a prior scheme for Hamilton–Jacobi equations introduced in Aubry–Mather's theory, we introduce a discrete approximation to stationary MFGs. Relying on Kakutani's fixed-point theorem, we prove the existence and uniqueness (up to additive constant) of solutions to the discrete problem. Moreover, we show that the solutions to the discrete problem converge, uniformly in the nonlocal case and weakly in the local case, to the classical solutions of the stationary problem.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"36 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78650984","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}
Marco Cirant, D. Gomes, Edgard A. Pimentel, H. Sánchez-Morgado
Here, we prove the existence of smooth solutions for mean-field games with a singular mean-field coupling; that is, a coupling in the Hamilton-Jacobi equation of the form begin{document}$ g(m) = -m^{- alpha} $end{document} with begin{document}$ alpha>0 $end{document} . We consider stationary and time-dependent settings. The function begin{document}$ g $end{document} is monotone, but it is not bounded from below. With the exception of the logarithmic coupling, this is the first time that MFGs whose coupling is not bounded from below is examined in the literature. This coupling arises in models where agents have a strong preference for low-density regions. Paradoxically, this causes the agents move towards low-density regions and, thus, prevents the creation of those regions. To prove the existence of solutions, we consider an approximate problem for which the existence of smooth solutions is known. Then, we prove new a priori bounds for the solutions that show that begin{document}$ frac 1 m $end{document} is bounded. Finally, using a limiting argument, we obtain the existence of solutions. The proof in the stationary case relies on a blow-up argument and in the time-dependent case on new bounds for begin{document}$ m^{-1} $end{document} .
{"title":"On some singular mean-field games","authors":"Marco Cirant, D. Gomes, Edgard A. Pimentel, H. Sánchez-Morgado","doi":"10.3934/JDG.2021006","DOIUrl":"https://doi.org/10.3934/JDG.2021006","url":null,"abstract":"Here, we prove the existence of smooth solutions for mean-field games with a singular mean-field coupling; that is, a coupling in the Hamilton-Jacobi equation of the form begin{document}$ g(m) = -m^{- alpha} $end{document} with begin{document}$ alpha>0 $end{document} . We consider stationary and time-dependent settings. The function begin{document}$ g $end{document} is monotone, but it is not bounded from below. With the exception of the logarithmic coupling, this is the first time that MFGs whose coupling is not bounded from below is examined in the literature. This coupling arises in models where agents have a strong preference for low-density regions. Paradoxically, this causes the agents move towards low-density regions and, thus, prevents the creation of those regions. To prove the existence of solutions, we consider an approximate problem for which the existence of smooth solutions is known. Then, we prove new a priori bounds for the solutions that show that begin{document}$ frac 1 m $end{document} is bounded. Finally, using a limiting argument, we obtain the existence of solutions. The proof in the stationary case relies on a blow-up argument and in the time-dependent case on new bounds for begin{document}$ m^{-1} $end{document} .","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45239573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze hypotheses tests using classical results on large deviations to compare two models, each one described by a different Hölder Gibbs probability measure. One main difference to the classical hypothesis tests in Decision Theory is that here the two measures are singular with respect to each other. Among other objectives, we are interested in the decay rate of the wrong decisions probability, when the sample size begin{document}$ n $end{document} goes to infinity. We show a dynamical version of the Neyman-Pearson Lemma displaying the ideal test within a certain class of similar tests. This test becomes exponentially better, compared to other alternative tests, when the sample size goes to infinity. We are able to present the explicit exponential decay rate. We also consider both, the Min-Max and a certain type of Bayesian hypotheses tests. We shall consider these tests in the log likelihood framework by using several tools of Thermodynamic Formalism. Versions of the Stein's Lemma and Chernoff's information are also presented.
{"title":"Decision Theory and large deviations for dynamical hypotheses tests:\u0000The Neyman-Pearson Lemma, Min-Max and Bayesian tests","authors":"Hermes H. Ferreira, A. Lopes, S. Lopes","doi":"10.3934/jdg.2021031","DOIUrl":"https://doi.org/10.3934/jdg.2021031","url":null,"abstract":"We analyze hypotheses tests using classical results on large deviations to compare two models, each one described by a different Hölder Gibbs probability measure. One main difference to the classical hypothesis tests in Decision Theory is that here the two measures are singular with respect to each other. Among other objectives, we are interested in the decay rate of the wrong decisions probability, when the sample size begin{document}$ n $end{document} goes to infinity. We show a dynamical version of the Neyman-Pearson Lemma displaying the ideal test within a certain class of similar tests. This test becomes exponentially better, compared to other alternative tests, when the sample size goes to infinity. We are able to present the explicit exponential decay rate. We also consider both, the Min-Max and a certain type of Bayesian hypotheses tests. We shall consider these tests in the log likelihood framework by using several tools of Thermodynamic Formalism. Versions of the Stein's Lemma and Chernoff's information are also presented.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48593571","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}
Pub Date : 2021-01-20DOI: 10.1101/2021.01.18.21250050
E. Alvarez, J. Brida, E. Limas, L. Rosich
This work addresses the spread of the coronavirus through a non-parametric approach, with the aim of identifying communities of countries based on how similar their evolution of the disease is. The analysis focuses on the number of daily new COVID-19 cases per ten thousand people during a period covering at least 250 days after the confirmation of the tenth case. Dynamic analysis is performed by constructing Minimal Spanning Trees (MST) and identifying groups of similarity in contagions evolution in 95 time windows of a 150-day amplitude that moves one day at a time. The number of times countries belonged to a similar performance group in constructed time windows was the intensity measure considered. Groups' composition is not stable, indicating that the COVID-19 evolution needs to be treated as a dynamic problem in the context of complex systems. Three communities were identified by applying the Louvain algorithm. Identified communities analysis according to each country's socioeconomic characteristics and variables related to the disease sheds light on whether there is any suggested course of action. Even when strong testing and tracing cases policies may be related with a more stable dynamic of the disease, results indicate that communities are conformed by countries with diverse characteristics. The best option to counteract the harmful effects of a pandemic may be having strong health systems in place,with contingent capacity to deal with unforeseen events and available resources capable of a rapid expansion of its capacity.
{"title":"Analysis of communities of countries with similar dynamics of the COVID-19 pandemic evolution","authors":"E. Alvarez, J. Brida, E. Limas, L. Rosich","doi":"10.1101/2021.01.18.21250050","DOIUrl":"https://doi.org/10.1101/2021.01.18.21250050","url":null,"abstract":"This work addresses the spread of the coronavirus through a non-parametric approach, with the aim of identifying communities of countries based on how similar their evolution of the disease is. The analysis focuses on the number of daily new COVID-19 cases per ten thousand people during a period covering at least 250 days after the confirmation of the tenth case. Dynamic analysis is performed by constructing Minimal Spanning Trees (MST) and identifying groups of similarity in contagions evolution in 95 time windows of a 150-day amplitude that moves one day at a time. The number of times countries belonged to a similar performance group in constructed time windows was the intensity measure considered. Groups' composition is not stable, indicating that the COVID-19 evolution needs to be treated as a dynamic problem in the context of complex systems. Three communities were identified by applying the Louvain algorithm. Identified communities analysis according to each country's socioeconomic characteristics and variables related to the disease sheds light on whether there is any suggested course of action. Even when strong testing and tracing cases policies may be related with a more stable dynamic of the disease, results indicate that communities are conformed by countries with diverse characteristics. The best option to counteract the harmful effects of a pandemic may be having strong health systems in place,with contingent capacity to deal with unforeseen events and available resources capable of a rapid expansion of its capacity.","PeriodicalId":42722,"journal":{"name":"Journal of Dynamics and Games","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"62320857","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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