Understanding Game Theory

Kolokoltsov, Vassili N.; Malafeyev, Oleg

doi:10.1142/7564

Cited by 61 publications

(42 citation statements)

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“…In particular, for the corresponding limit in a game-theoretic setting (replicator dynamics), we can refer to (Benaim & Weibull, 2003) or the last section of (Kolokoltsov & Malafeyev, 2010).…”

Section: Conclusion and Bibliographical Commentsmentioning

confidence: 99%

Nonlinear Markov Games on a Finite State Space (Mean-field and Binary Interactions)

Kolokoltsov¹

2012

IJSP

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Managing large complex stochastic systems, including competitive interests, when one or several players can control the behavior of a large number of particles (agents, mechanisms, vehicles, subsidiaries, species, police units, etc), say N k for a player k, the complexity of the game-theoretical (or Markov decision) analysis can become immense as N k → ∞. However, under rather general assumptions, the limiting problem as all N k → ∞ can be described by a well manageable deterministic evolution. In this paper we analyze some simple situations of this kind proving the convergence of Nashequilibria for finite games to equilibria of a limiting deterministic differential game.

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Section: Conclusion and Bibliographical Commentsmentioning

confidence: 99%

Nonlinear Markov Games on a Finite State Space (Mean-field and Binary Interactions)

Kolokoltsov¹

2012

IJSP

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“…In future work, we will investigate the influence of third parties in inspection games related to the Prisoner's Dilemma ( [54] (Section 10.3)) and on the evolution of market shares in retail duopolies [55]. The case study on which this paper is based examined an alliance that [56] would be described as poorly embedded: in the eyes of some commentators, their alliance reduced rather than enhanced their reputation.…”

Section: Discussion Conclusion and Lessons Learnedmentioning

confidence: 99%

External Pressure on Alliances: What Does the Prisoners’ Dilemma Reveal?

Binner¹,

Fletcher

Kolokoltsov

et al. 2013

Games

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Prompted by a real-life observation in the UK retail market, a two-player Prisoners' Dilemma model of an alliance between two firms is adapted to include the response of a rival firm, resulting in a version of a three-player Prisoners' Dilemma. We use this to analyse the impact on the stability of the alliance of the rival's competition, either with the alliance or with the individual partners. We show that, while strong external pressure on both partners can cause Ally-Ally to become a Nash equilibrium for the two-player Prisoners' Dilemma, weak or asymmetric pressure that plays on the partners' differing objectives can undermine the alliance. As well as providing new insights into how allies should respond if the alliance is to continue, this also illustrates how a third party can most effectively cause the alliance to become unsustainable. We create a new game theoretic framework, adding value to existing theory and the practice of alliance formation and sustainability.

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“…The aim of the present paper is two-folds: 1) To widen the range of applicability of this research by introducing a unified methodology for the analysis of a large class of conflict interactions of social, economic or military character (that turn out to be mathematically similar, but are often discussed in disjoint sets of subject specific journals) describing the pressure executed by a big player (or principal) on a large group of small players that resist the pressure or collaborate, that is the class of games of an agent immersed into a pool of evolutionary and mean-field interacting small players; 2) to build the rigorous mathematical theory of the law of large number limits for the latter conflicts by proving that the controlled deterministic evolutionary equation (kinetic equation) describing the dynamics of interaction can be obtained as the limiting behavior of the controlled Markov models of kth order and/or mean-field interaction (with the number of agents tending to infinity) and thus extending the corresponding theory for the justification of the usual replicator dynamics (see e.g. [16] or Section 11.9 of textbook [56] for the latter). The practical usefulness of this limit is that it provides much more tractable limiting models where carrying out a traditional Markov decision analysis for a large state space is often unfeasible.…”

Section: Objectives and Content Of The Studymentioning

confidence: 99%

The Evolutionary Game of Pressure (or Interference), Resistance and Collaboration

Kolokoltsov¹

2017

Mathematics of OR

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In this paper we extend the framework of evolutionary inspection game put forward recently by the author and coworkers to a large class of conflict interactions dealing with the pressure executed by the major player (or principal) on the large group of small players that can resist this pressure or collaborate with the major player. We prove rigorous results on the convergence of various Markov decision models of interacting small agents (including evolutionary growth), namely pairwise, in groups and by coalition formation, to a deterministic evolution on the distributions of the state spaces of small players paying main attention to situations with an infinite state-space of small players. We supply rather precise rates of convergence. The theoretical results of the paper are applied to the analysis of the processes of inspection, corruption, cyber-security, counter-terrorism, banks and firms merging, strategically enhanced preferential attachment and many other. (2010): 91A22, 91A40, 91A80, 91F99, 60J20 Mathematics Subject Classification

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Understanding Game Theory

Cited by 61 publications

References 0 publications

Nonlinear Markov Games on a Finite State Space (Mean-field and Binary Interactions)

Nonlinear Markov Games on a Finite State Space (Mean-field and Binary Interactions)

External Pressure on Alliances: What Does the Prisoners’ Dilemma Reveal?

The Evolutionary Game of Pressure (or Interference), Resistance and Collaboration

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