Two-population replicator dynamics and number of Nash equilibria in matrix games

This paper deals with the approximation of Nash equilibria in m-player games. We present conditions under which an approximating sequence of games admits near-equilibria that approximate near-equilibria in the limit game. We apply the results to two classes of games: (i) a duopoly game approximated by a sequence of matrix games, and (ii) a stochastic game played under the S-adapted information structure approximated by games played over a sampled event tree. Numerical illustrations show the usefulness of this approximation theory.

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IsoMaTrix: a framework to visualize the isoclines of matrix games and quantify uncertainty in structured populations

10.1101/2020.06.24.170183 ◽

2020 ◽

Author(s):

Jeffrey West ◽

Yongqian Ma ◽

Artem Kaznatcheev ◽

Alexander R. A. Anderson

Keyword(s):

Spatial Configuration ◽

Replicator Dynamics ◽

Evolutionary Game ◽

Payoff Matrix ◽

Structured Populations ◽

Three Dimensions ◽

Continuous Treatment ◽

Hybrid Automata ◽

Matrix Games ◽

Structured Matrix

AbstractSummaryEvolutionary game theory describes frequency-dependent selection for fixed, heritable strategies in a population of competing individuals using a payoff matrix, typically described using well-mixed assumptions (replicator dynamics). IsoMaTrix is an open-source package which computes the isoclines (lines of zero growth) of matrix games, and facilitates direct comparison of well-mixed dynamics to structured populations in two or three dimensions. IsoMaTrix is coupled with a Hybrid Automata Library module to simulate structured matrix games on-lattice. IsoMaTrix can also compute fixed points, phase flow, trajectories, velocities (and subvelocities), delineated “region plots” of positive/negative strategy velocity, and uncertainty quantification for stochastic effects in structured matrix games. We describe a result obtained via IsoMaTrix’s spatial games functionality, which shows that the timing of competitive release in a cancer model (under continuous treatment) critically depends on the initial spatial configuration of the tumor.Availability and implementationThe code is available at: https://github.com/mathonco/isomatrix.

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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

Journal of Mathematical Biology ◽

10.1007/s00285-019-01440-6 ◽

2019 ◽

Vol 80 (3) ◽

pp. 743-774 ◽

Cited By ~ 3

Author(s):

Tamás Varga ◽

Tamás F. Móri ◽

József Garay

Keyword(s):

Replicator Dynamics ◽

Time Constraints ◽

Asymptotically Stable ◽

Matrix Games ◽

Rest Point

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Pure Nash equilibria of coordination matrix games

Economics Letters ◽

10.1016/j.econlet.2005.01.024 ◽

2005 ◽

Vol 89 (1) ◽

pp. 7-11 ◽

Cited By ~ 4

Author(s):

David P. Roberts

Keyword(s):

Nash Equilibria ◽

Matrix Games

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Predicting Nash equilibria for microbial metabolic interactions

Bioinformatics ◽

10.1093/bioinformatics/btaa1014 ◽

2020 ◽

Author(s):

Jingyi Cai ◽

Tianwei Tan ◽

S H Joshua Chan

Keyword(s):

Nash Equilibria ◽

Metabolic Flux ◽

Evolutionary Game ◽

Growth Conditions ◽

Supplementary Information ◽

Community Interactions ◽

Simulation Code ◽

Matrix Games ◽

Strategy Space ◽

Metabolic Interactions

Abstract Motivation Microbial metabolic interactions impact ecosystems, human health and biotechnology profoundly. However, their determination remains elusive, invoking an urgent need for predictive models seamlessly integrating metabolism with evolutionary principles that shape community interactions. Results Inspired by the evolutionary game theory, we formulated a bi-level optimization framework termed NECom for which any feasible solutions are Nash equilibria of microbial community metabolic models with/without an outer-level (community) objective function. Distinct from discrete matrix games, NECom models the continuous interdependent strategy space of metabolic fluxes. We showed that NECom successfully predicted several classical games in the context of metabolic interactions that were falsely or incompletely predicted by existing methods, including prisoner’s dilemma, snowdrift and cooperation. The improved capability originates from the novel formulation to prevent ‘forced altruism’ hidden in previous static algorithms while allowing for sensing all potential metabolite exchanges to determine evolutionarily favorable interactions between members, a feature missing in dynamic methods. The results provided insights into why mutualism is favorable despite seemingly costly cross-feeding metabolites and demonstrated similarities and differences between games in the continuous metabolic flux space and matrix games. NECom was then applied to a reported algae-yeast co-culture system that shares typical cross-feeding features of lichen, a model system of mutualism. 488 growth conditions corresponding to 3,221 experimental data points were simulated. Without training any parameters using the data, NECom is more predictive of species’ growth rates given uptake rates compared with flux balance analysis with an overall 63.5% and 81.7% reduction in root-mean-square error for the two species. Availability Simulation code and data are available at https://github.com/Jingyi-Cai/NECom.git Supplementary information Supplementary data are available at Bioinformatics online.

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