[Paper Review] Evolutionary game dynamics with three strategies in finite populations

Jing Wang, Feng Fu+2 more|ArXiv.org|Jan 29, 2007

Evolutionary Game Theory and Cooperation6 references3 citations

TL;DR

This paper proposes a finite-population evolutionary game model with three strategies using a frequency-dependent Moran process, formulated as a system of linear equations to compute fixation probabilities. The key contribution is the introduction of global and local fixation probabilities, showing that Tit-for-Tat (TFT) can invade and dominate populations of All-C and All-D under weak selection and intermediate interaction rounds, explaining the emergence of cooperation in finite populations where it fails in infinite models.

ABSTRACT

We propose a model for evolutionary game dynamics with three strategies $A$, $B$ and $C$ in the framework of Moran process in finite populations. The model can be described as a stochastic process which can be numerically computed from a system of linear equations. Furthermore, to capture the feature of the evolutionary process, we define two essential variables, the {\em global} and the {\em local} fixation probability. If the {\em global} fixation probability of strategy $A$ exceeds the neutral fixation probability, the selection favors $A$ replacing $B$ or $C$ no matter what the initial ratio of $B$ to $C$ is. Similarly, if the {\em local} fixation probability of $A$ exceeds the neutral one, the selection favors $A$ replacing $B$ or $C$ only in some appropriate initial ratios of $B$ to $C$. Besides, using our model, the famous game with AllC, AllD and TFT is analyzed. Meanwhile, we find that a single individual TFT could invade the entire population under proper conditions.

Motivation & Objective

• To model evolutionary game dynamics with three strategies in finite populations, where population size is strictly constant.
• To address the limitations of infinite-population models by incorporating stochastic effects and finite-size dynamics.
• To define and compute global and local fixation probabilities for assessing selection favorability of one strategy over others.
• To analyze the well-known cooperation game involving All-C, All-D, and TFT strategies under finite population conditions.
• To determine under what conditions TFT can successfully invade and replace All-C or All-D in finite populations, despite failing in infinite models.

Proposed method

• Formulates the evolutionary process as a frequency-dependent Moran process with birth-death steps in a finite population of size N.
• Derives fitness functions for each strategy based on payoff matrices and weak selection (w ≪ 1), using payoff averages over interactions.
• Solves for fixation probabilities via a system of linear equations derived from the master equation of the stochastic process.
• Introduces global fixation probability (for any initial ratio of B to C) and local fixation probability (for specific initial ratios) to assess selection favorability.
• Applies the model numerically to the three-strategy game with All-C, All-D, and TFT, computing fixation probabilities for single mutants.
• Uses weak selection approximation to derive conditions under which TFT fixation exceeds neutral fixation, especially for intermediate numbers of rounds in repeated games.

Experimental results

Research questions

• RQ1 Under what conditions can a single Tit-for-Tat (TFT) individual invade and fixate in a finite population of All-C and All-D players?
• RQ2 How do global and local fixation probabilities differ in their ability to predict selection favorability for a strategy in a three-strategy game?
• RQ3 Why does TFT fail to invade in infinite-population models but can succeed in finite populations, and what role does population size play?
• RQ4 How does the number of rounds in a repeated Prisoner’s Dilemma affect the fixation probability of TFT in finite populations?
• RQ5 Can finite population effects explain the emergence of cooperation in games where it is not favored in infinite-population replicator dynamics?

Key findings

• A single TFT individual can achieve a fixation probability greater than 1/N, exceeding that of both All-C and All-D, indicating a higher chance of invading and dominating the population.
• For intermediate numbers of rounds (n), if the equilibrium frequency of TFT is below n/(n+θ), selection favors local replacement of All-C or All-D by TFT.
• Global fixation of TFT over All-C or All-D requires the stricter condition x* < 1/(1+θg) < 1/3 when θg > 2, indicating stronger selection pressure.
• As the number of rounds n increases, the probability that selection favors TFT takeover approaches one, especially when TFT and All-D are nearly matched in payoff over multiple rounds.
• Finite population size effects allow TFT to outperform All-D despite mutual defection in early rounds, due to payoff accumulation in repeated interactions.
• The 1/3 law from infinite-population models still applies in finite populations, but only locally; global replacement requires more stringent conditions.