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Király B, Varga T, Szabó G, Garay J. Evolutionarily stable payoff matrix in hawk-dove games. BMC Ecol Evol 2024; 24:65. [PMID: 38769504 PMCID: PMC11107024 DOI: 10.1186/s12862-024-02257-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2024] [Accepted: 05/15/2024] [Indexed: 05/22/2024] Open
Abstract
BACKGROUND Classical matrix game models aim to find the endpoint of behavioural evolution for a set of fixed possible interaction outcomes. Here, we introduce an evolutionary model in which not only the players' strategies but also the payoff matrix evolves according to natural selection. RESULTS We start out from the hawk-dove matrix game and, in a way that is consistent with the monomorphic model setup of Maynard Smith and Price, introduce an evolving phenotypic trait that quantifies fighting ability and determines the probability of winning and the cost of losing escalated hawk-hawk fights. We define evolutionarily stable phenotypes as consisting of an evolutionarily stable strategy and an evolutionarily stable trait, which in turn describes a corresponding evolutionarily stable payoff matrix. CONCLUSIONS We find that the maximal possible cost of escalating fights remains constant during evolution assuming a separation in the time scales of fast behavioural and slow trait selection, despite the fact that the final evolutionarily stable phenotype maximizes the payoff of hawk-hawk fights. Our results mirror the dual nature of Darwinian evolution whereby the criteria of evolutionary success, as well as the successful phenotypes themselves, are a product of natural selection.
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Affiliation(s)
- Balázs Király
- Institute of Technical Physics and Materials Science, HUN-REN Centre for Energy Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary.
| | - Tamás Varga
- Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., Szeged, H-6720, Hungary
| | - György Szabó
- Institute of Technical Physics and Materials Science, HUN-REN Centre for Energy Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary
- Institute of Evolution, HUN-REN Centre for Ecological Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary
| | - József Garay
- Institute of Evolution, HUN-REN Centre for Ecological Research, Konkoly-Thege Miklós út 29-33., Budapest, H-1121, Hungary
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Zhang M, Huang Y, Jin Y, Bao Y. Government regulation strategy, leading firms' innovation strategy, and following firms imitation strategy: An analysis based on evolutionary game theory. PLoS One 2023; 18:e0286730. [PMID: 37289768 PMCID: PMC10249874 DOI: 10.1371/journal.pone.0286730] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Accepted: 05/22/2023] [Indexed: 06/10/2023] Open
Abstract
In the innovation ecosystem, the knowledge-based game behavior of each subject not only pertains to its own survival and development but also affects evolution of the innovation ecosystem. The present study investigates the choice of government's regulation strategy, leading firms' innovation protection strategy and following firms' imitation strategy from the perspective of group evolutionary game. Based on the cost-benefit perspective, an asymmetric tripartite evolutionary game model and a simulation model are constructed to analyze the strategies and stability of the evolutionary equilibrium of each subject. We focus mainly on the protection intensity of innovation achievements by leading enterprises and the difficulty of imitation and substitution by following enterprises. The cost of patent operation and maintenance, government subsidies, and the relative difficulty of technology substitution and imitation were identified as the key factors affecting the evolutionary equilibrium of the system. Based on different scenarios resulting from the aforementioned factors, four equilibrium states are observed in the system, namely {no government regulation, technology secrecy, substitution}, {no government regulation, technology secrecy, imitation}, {no government regulation, patent application, imitation}, and {government regulation, patent application, imitation}. Finally, the study suggests corresponding recommendations for the three parties, which can help governments as well as the leading and following firms to choose appropriate behavioral strategies. At the same time, this study offers positive insights to participants in the global innovation ecosystem.
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Affiliation(s)
- Mengke Zhang
- School of Management, Beijing Union University, Beijing, China
| | - Yan Huang
- School of Management, Beijing Union University, Beijing, China
| | - Yifan Jin
- School of Management, Beijing Union University, Beijing, China
| | - Yuan Bao
- School of Management, Beijing Union University, Beijing, China
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3
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State-controlled epidemic in a game against a novel pathogen. Sci Rep 2022; 12:15716. [PMID: 36127449 PMCID: PMC9488893 DOI: 10.1038/s41598-022-19691-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2022] [Accepted: 09/01/2022] [Indexed: 11/13/2022] Open
Abstract
The pandemic reminded us that the pathogen evolution still has a serious effect on human societies. States, however, can prepare themselves for the emergence of a novel pathogen with unknown characteristics by analysing potential scenarios. Game theory offers such an appropriate tool. In our game-theoretical framework, the state is playing against a pathogen by introducing non-pharmaceutical interventions to fulfil its socio-political goals, such as guaranteeing hospital care to all needed patients, keeping the country functioning, while the applied social restrictions should be as soft as possible. With the inclusion of activity and economic sector dependent transmission rate, optimal control of lockdowns and health care capacity management is calculated. We identify the presence and length of a pre-symptomatic infectious stage of the disease to have the greatest effect on the probability to cause a pandemic. Here we show that contrary to intuition, the state should not strive for the great expansion of its health care capacities even if its goal is to provide care for all requiring it and minimize the cost of lockdowns.
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Křivan V, Cressman R. The asymmetric Hawk-Dove game with costs measured as time lost. J Theor Biol 2022; 547:111162. [DOI: 10.1016/j.jtbi.2022.111162] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2022] [Revised: 05/06/2022] [Accepted: 05/10/2022] [Indexed: 10/18/2022]
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Broom M, Křivan V. Two-strategy games with time constraints on regular graphs. J Theor Biol 2020; 506:110426. [PMID: 32777217 DOI: 10.1016/j.jtbi.2020.110426] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2019] [Revised: 07/24/2020] [Accepted: 07/27/2020] [Indexed: 11/19/2022]
Abstract
Evolutionary game theory is a powerful method for modelling animal conflicts. The original evolutionary game models were used to explain specific biological features of interest, such as the existence of ritualised contests, and were necessarily simple models that ignored many properties of real populations, including the duration of events and spatial and related structural effects. Both of these areas have subsequently received much attention. Spatial and structural effects have been considered in evolutionary graph theory, and a significant body of literature has been built up to deal with situations where the population is not homogeneous. More recently a theory of time constraints has been developed to take account of the fact that different events can take different times, and that interaction times can explicitly depend upon selected strategies, which can, in turn, influence the distribution of different opponent types within the population. Here, for the first time, we build a model of time constraint games which explicitly considers a spatial population, by considering a population evolving on an underlying graph, using two graph dynamics, birth-death and death-birth. We consider one short time scale along which frequencies of pairs and singles change as individuals interact with their neighbours, and another, evolutionary time scale, along which frequencies of strategies change in the population. We show that for graphs with large degree, both dynamics reproduce recent results from well-mixed time constraint models, including two ESSs being common in Hawk-Dove and Prisoner's Dilemma games, but for low degree there can be marked differences. For birth-death processes the effect of the graph degree is small, whereas for death-birth dynamics there is a large effect. The general prediction for both Hawk-Dove and Prisoner's dilemma games is that as the graph degree decreases, i.e., as the number of neighbours decreases, mixed ESS do appear. In particular, for the Prisoner's dilemma game this means that cooperation is easier to establish in situations where individuals have low number of neighbours. We thus see that solutions depend non-trivially on the combination of graph degree, dynamics and game.
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Affiliation(s)
- Mark Broom
- Department of Mathematics, City, University of London, London, UK.
| | - Vlastimil Křivan
- Centre for Mathematical Biology, Department of Mathematics, Faculty of Science, University of South Bohemia, Branišovská 1760, 370 05 České Budějovice, Czech Republic; Czech Academy of Sciences, Biology Centre, Institute of Entomology, Branišovská 31, 370 05 České Budějovice, Czech Republic.
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6
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Kleptoparasitic interactions modeling varying owner and intruder hunger awareness. Theor Popul Biol 2020; 136:31-40. [DOI: 10.1016/j.tpb.2020.11.002] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2020] [Revised: 10/13/2020] [Accepted: 11/06/2020] [Indexed: 11/21/2022]
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Argasinski K, Rudnicki R. Replicator dynamics for the game theoretic selection models based on state. J Theor Biol 2020; 526:110540. [PMID: 33221278 DOI: 10.1016/j.jtbi.2020.110540] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2019] [Revised: 11/06/2020] [Accepted: 11/10/2020] [Indexed: 11/16/2022]
Abstract
The paper presents an attempt to integrate the classical evolutionary game theory based on replicator dynamics and the state-based approach of Houston and McNamara. In the new approach, individuals have different heritable strategies; however, individuals carrying the same strategy can differ in terms of state, role or the situation in which they act. Thus, the classical replicator dynamics is completed by the additional subsystem of differential equations describing the dynamics of transitions between different states. In effect, the interactions described by game structure, in addition to the demographic payoffs (constituted by births and deaths), can lead to the change in state of the competing individuals. Special cases of reversible and irreversible incremental stage-structured models, where the state changes can describeenergy accumulation, developmental steps or aging, are derived for discrete and continuous versions. The new approach is illustrated using the example of the Owner-Intruder game with explicit dynamics of the role changes. The new model presents a generalization of the demographic version of the Hawk-Dove game,with the difference being that the opponents in the game are drawn from two separate subpopulations consisting of Owners and Intruders. Here, the Intruders check random nest sites and play the Hawk-Dove game with the Owner if they are occupied. Meanwhile, the Owners produce newborns that become Intruders, since they must find a free nest site to reproduce. An interesting feedback mechanism is produced via the fluxes of individuals between the different subpopulations. In addition, the population growth suppression mechanism resulting from the fixation Bourgeois strategy is analyzed.
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Affiliation(s)
- Krzysztof Argasinski
- Institute of Mathematics of Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa, Poland.
| | - Ryszard Rudnicki
- Institute of Mathematics of Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland.
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Defectors' intolerance of others promotes cooperation in the repeated public goods game with opting out. Sci Rep 2020; 10:19511. [PMID: 33177552 PMCID: PMC7659018 DOI: 10.1038/s41598-020-76506-3] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2020] [Accepted: 10/22/2020] [Indexed: 12/21/2022] Open
Abstract
The theoretical and experimental research on opting out (also called conditional dissociation) in social dilemmas has concentrated on the effect this behavior has on the level of cooperation when used against defectors. The intuition behind this emphasis is based on the common property of social dilemmas that individuals are worse off the more their opponents defect. However, this article shows clearly that other opting out mechanisms are better at increasing cooperative behavior. In fact, by analyzing the stable Nash equilibria for the repeated multi-player public goods game with opting out, our results provide a strong argument that the best opting out rule is one whereby the only groups that voluntarily stay together between rounds are those that are homogeneous (i.e., those groups that are either all cooperators or all defectors), when these groups stay together for enough rounds. This outcome emerges when defectors are completely intolerant of individuals who cooperate (e.g., defectors exhibit xenophobic behavior toward cooperators) and so opt out whenever their group has a cooperator in it. The strong preference by defectors to be with like-minded individuals causes all heterogeneous groups to disband after one round.
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Garay J, Cressman R, Xu F, Broom M, Csiszár V, Móri TF. When optimal foragers meet in a game theoretical conflict: A model of kleptoparasitism. J Theor Biol 2020; 502:110306. [DOI: 10.1016/j.jtbi.2020.110306] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/04/2019] [Revised: 03/26/2020] [Accepted: 04/27/2020] [Indexed: 10/24/2022]
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Reducing courtship time promotes marital bliss: The Battle of the Sexes game revisited with costs measured as time lost. J Theor Biol 2020; 503:110382. [DOI: 10.1016/j.jtbi.2020.110382] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2020] [Accepted: 06/15/2020] [Indexed: 11/18/2022]
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A temporal model of territorial defence with antagonistic interactions. Theor Popul Biol 2020; 134:15-35. [DOI: 10.1016/j.tpb.2020.03.005] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2019] [Revised: 03/26/2020] [Accepted: 03/27/2020] [Indexed: 11/18/2022]
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Broom M, Cressman R, Křivan V. Revisiting the "fallacy of averages" in ecology: Expected gain per unit time equals expected gain divided by expected time. J Theor Biol 2019; 483:109993. [PMID: 31491495 DOI: 10.1016/j.jtbi.2019.109993] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2019] [Revised: 07/26/2019] [Accepted: 09/02/2019] [Indexed: 11/16/2022]
Abstract
Fitness is often defined as the average payoff an animal obtains when it is engaged in several activities, each taking some time. We point out that the average can be calculated with respect to either the time distribution, or to the event distribution of these activities. We show that these two averages lead to the same fitness function. We illustrate this result through two examples from foraging theory, Holling II functional response and the diet choice model, and one game-theoretic example of Hamilton's rule applied to the time-constrained Prisoner's dilemma (PD). In particular, we show that in these models, fitness defined as expected gain per unit time equals fitness defined as expected gain divided by expected time. We also show how these fitnesses predict the optimal outcome for diet choice and the prevalence of cooperation in the repeated PD game.
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Affiliation(s)
- Mark Broom
- Department of Mathematics, City, University of London, London, UK.
| | - Ross Cressman
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada.
| | - Vlastimil Křivan
- Department of Mathematics, Faculty of Science, University of South Bohemia, Branišovská 1760, České Budějovice 370 05, Czech Republic; Czech Academy of Sciences, Biology Centre, Institute of Entomology, Branišovská 31, České Budějovice 370 05, Czech Republic.
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The ESS for evolutionary matrix games under time constraints and its relationship with the asymptotically stable rest point of the replicator dynamics. J Math Biol 2019; 80:743-774. [DOI: 10.1007/s00285-019-01440-6] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2019] [Revised: 09/23/2019] [Indexed: 10/25/2022]
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Garay J. Technical review on derivation methods for behavior dependent functional responses. COMMUNITY ECOL 2019. [DOI: 10.1556/168.2019.20.1.4] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- J. Garay
- MTA Centre for Ecological Research, Evolutionary Systems Research Group, Klebelsberg Kunó u. 3, H-8237 Tihany, Hungary and
- MTA-ELTE Research Group in Theoretical Biology and Evolutionary Ecology and Department of Plant Systematics, Ecology and Theoretical Biology, ELTE Eötvös Loránd University, Pázmány Péter sétány1/c, H-1117 Budapest, Hungary. Phone: , Fax:
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Cressman R, Křivan V. Bimatrix games that include interaction times alter the evolutionary outcome: The Owner–Intruder game. J Theor Biol 2019; 460:262-273. [DOI: 10.1016/j.jtbi.2018.10.033] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2018] [Revised: 10/02/2018] [Accepted: 10/12/2018] [Indexed: 11/16/2022]
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Křivan V, Galanthay TE, Cressman R. Beyond replicator dynamics: From frequency to density dependent models of evolutionary games. J Theor Biol 2018; 455:232-248. [PMID: 29990466 DOI: 10.1016/j.jtbi.2018.07.003] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Game theoretic models of evolution such as the Hawk-Dove game assume that individuals gain fitness (which is a proxy of the per capita population growth rate) in pair-wise contests only. These models assume that the equilibrium distribution of phenotypes involved (e.g., Hawks and Doves) in the population is given by the Hardy-Weinberg law, which is based on instantaneous, random pair formation. On the other hand, models of population dynamics do not consider pairs, newborns are produced by singles, and interactions between phenotypes or species are described by the mass action principle. This article links game theoretic and population approaches. It shows that combining distribution dynamics with population dynamics can lead to stable coexistence of Hawk and Dove population numbers in models that do not assume a priori that fitness is negative density dependent. Our analysis shows clearly that the interior Nash equilibrium of the Hawk and Dove model depends both on population size and on interaction times between different phenotypes in the population. This raises the question of the applicability of classic evolutionary game theory that requires all interactions take the same amount of time and that all single individuals have the same payoff per unit of time, to real populations. Furthermore, by separating individual fitness into birth and death effects on singles and pairs, it is shown that stable coexistence in these models depends on the time-scale of the distribution dynamics relative to the population dynamics. When explicit density-dependent fitness is included through competition over a limited resource, the combined dynamics of the Hawk-Dove model often lead to Dove extinction no matter how costly fighting is for Hawk pairs.
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Affiliation(s)
- Vlastimil Křivan
- Czech Academy of Sciences, Biology Centre, Institute of Entomology, Branišovská 31, České Budějovice 370 05, Czech Republic; Department of Mathematics, Faculty of Sciences, University of South Bohemia, Branišovská 1760, České Budějovice 370 05, Czech Republic.
| | | | - Ross Cressman
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada.
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Garay J, Cressman R, Móri TF, Varga T. The ESS and replicator equation in matrix games under time constraints. J Math Biol 2018; 76:1951-1973. [DOI: 10.1007/s00285-018-1207-0] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2017] [Revised: 12/02/2017] [Indexed: 11/28/2022]
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