The Exact Computational Complexity of Evolutionarily Stable Strategies

Conitzer, Vincent

doi:10.1007/978-3-642-45046-4_9

Cited by 8 publications

(9 citation statements)

References 29 publications

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“…4b). Even computing Evolutionary Stable Strategies (ESS) [13], a refinement of Nash equilibria, is intractable [61,62]. In larger games (e.g., AlphaZero in Section 3.4.2), the reduction in the number of agents that are resistant to mutations is more dramatic (in the sense of the stationary distribution support size being much smaller than the total number of agents) and less obvious (in the sense that more-resistant agents are not always the ones that have been trained for longer).…”

Section: α-Rank and Mccs As A Solution Concept: A Paradigm Shiftmentioning

confidence: 99%

α-Rank: Multi-Agent Evaluation by Evolution

Omidshafiei¹,

Papadimitriou

Piliouras

et al. 2019

Sci Rep

122

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We introduce α - Rank , a principled evolutionary dynamics methodology, for the evaluation and ranking of agents in large-scale multi-agent interactions, grounded in a novel dynamical game-theoretic solution concept called Markov - Conley chains (MCCs). The approach leverages continuous-time and discrete-time evolutionary dynamical systems applied to empirical games, and scales tractably in the number of agents, in the type of interactions (beyond dyadic), and the type of empirical games (symmetric and asymmetric). Current models are fundamentally limited in one or more of these dimensions, and are not guaranteed to converge to the desired game-theoretic solution concept (typically the Nash equilibrium). α -Rank automatically provides a ranking over the set of agents under evaluation and provides insights into their strengths, weaknesses, and long-term dynamics in terms of basins of attraction and sink components. This is a direct consequence of the correspondence we establish to the dynamical MCC solution concept when the underlying evolutionary model’s ranking-intensity parameter, α , is chosen to be large, which exactly forms the basis of α -Rank. In contrast to the Nash equilibrium, which is a static solution concept based solely on fixed points, MCCs are a dynamical solution concept based on the Markov chain formalism, Conley’s Fundamental Theorem of Dynamical Systems, and the core ingredients of dynamical systems: fixed points, recurrent sets, periodic orbits, and limit cycles. Our α -Rank method runs in polynomial time with respect to the total number of pure strategy profiles, whereas computing a Nash equilibrium for a general-sum game is known to be intractable. We introduce mathematical proofs that not only provide an overarching and unifying perspective of existing continuous- and discrete-time evolutionary evaluation models, but also reveal the formal underpinnings of the α -Rank methodology. We illustrate the method in canonical games and empirically validate it in several domains, including AlphaGo, AlphaZero, MuJoCo Soccer, and Poker.

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Section: α-Rank and Mccs As A Solution Concept: A Paradigm Shiftmentioning

confidence: 99%

α-Rank: Multi-Agent Evaluation by Evolution

Omidshafiei¹,

Papadimitriou

Piliouras

et al. 2019

Sci Rep

122

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“…A particularly important difference between Leader and Deadlock dynamics is the existence of an internal fixed point in Leader but not in Deadlock. Fixed points are a property of equilibrium dynamics: in the most general case, even on very long timescales these fixed points might not be realized due to the evolutionary constraints of population size [40] or computation [41,42]. Thus, it is important to check to what extent this qualitative difference can translate to a quantitative difference in finite time horizons.…”

Section: Heterogeneity and Latent Resistancementioning

confidence: 99%

Fibroblasts and Alectinib switch the evolutionary games played by non-small cell lung cancer

Kaznatcheev

Peacock

Basanta

et al. 2017

Preprint

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Heterogeneity in strategies for survival and propagation among the cells that constitute a tumour is a driving force behind the evolution of resistance to cancer treatment. The rules mapping the tumour's strategy distribution to the fitness of individual strategies can be represented as an evolutionary game. We develop a game assay to measure this property in co-culures of alectinib-sensitive and alectinibresistant non-small cell lung cancer. The games are not only quantitatively di↵erent between di↵erent environments, but targeted therapy and cancer associated fibroblasts qualitatively switch the type of game being played from Leader to Deadlock. This provides the first empirical confirmation of a central theoretical postulate of evolutionary game theory in oncology: we can treat not just the player, but also the game. Although we concentrate on measuring games played by cancer cells, the measurement methodology we develop can be used to advance the study of games in other microscopic systems.

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“…However, the question of the exact complexity of ESS existence, given the payoff matrix, remained open. A few years later, Conitzer finally settles this question in [1], showing that ess is actually…”

Section: Previous Workmentioning

confidence: 99%

Existence of Evolutionarily Stable Strategies Remains Hard to Decide for a Wide Range of Payoff Values

Melissourgos

Spirakis

2017

Lecture Notes in Computer Science

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The concept of an evolutionarily stable strategy (ESS), introduced by Smith and Price [4], is a refinement of Nash equilibrium in 2-player symmetric games in order to explain counter-intuitive natural phenomena, whose existence is not guaranteed in every game. The problem of deciding whether a game possesses an ESS has been shown to be Σ P 2 -complete by Conitzer [1] using the preceding important work by Etessami and Lochbihler [2]. The latter, among other results, proved that deciding the existence of ESS is both NP-hard and coNP-hard. In this paper we introduce a reduction robustness notion and we show that deciding the existence of an ESS remains coNP-hard for a wide range of games even if we arbitrarily perturb within some intervals the payoff values of the game under consideration. In contrast, ESS exist almost surely for large games with random and independent payoffs chosen from the same distribution [10].

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The Exact Computational Complexity of Evolutionarily Stable Strategies

Cited by 8 publications

References 29 publications

α-Rank: Multi-Agent Evaluation by Evolution

α-Rank: Multi-Agent Evaluation by Evolution

Fibroblasts and Alectinib switch the evolutionary games played by non-small cell lung cancer

Existence of Evolutionarily Stable Strategies Remains Hard to Decide for a Wide Range of Payoff Values

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