# A Mutation Threshold for Cooperative Takeover

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Prisoner’s Dilemma

## 3. Simulation Design and Methods

## 4. Results

#### 4.1. Spatiality and Population Dynamics

#### 4.2. Influence of the Error Rate

#### 4.3. Heritability and Mutations of the Error Rate

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PD | Prisoner’s dilemma |

IPD | Iterated prisoner’s dilemma |

ALLC | Always cooperate |

ALLD | Always defect |

TFT | Tit-for-Tat |

RND | Random strategy |

RNA | Ribonucleic acid |

## Appendix A

#### Appendix A.1. Details of the Implementation

**Figure A1.**Distribution of error rates. (

**A**) Probability density functions for three distributions of the error rate with most probable value $\widehat{p}={10}^{-3}$ and shape parameter ${log}_{10}s=-0.75,-0.50,-0.25$ from which the error rates are drawn, where the logarithm of the error rates follows a normal distribution. Initial distributions of error rates are drawn from the distribution, while mutations correspond to corrections that are parametrized from a similar distribution with $m=0$. The lognormal distribution was chosen so that mutations increase or decrease the error rates with the same distribution probability, regardless of the value of the error rates (note the logarithmic horizontal axis while the vertical axis is linear). The value of m thus specifies the mode $\widehat{p}$ of the distribution, and the shape parameter s the extent of mutations. (

**B**) Probability density functions for the same three distributions of error rates, shown this time with axis scaling reversed with respect to panel (

**A**), highlighting the effect of varying parameter s. While increasing the value of the shape parameter has a minimal effect on the mode of the distribution, it extends the tail of the distribution to include higher values of the error rate.

Algorithm 1: Strategy replacement procedure. |

#### Appendix A.2. Evaluation of Quantitative Data

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**Figure 1.**Evolution algorithm. (

**A**) At the beginning of each simulation, strategies ALLC (“always cooperate”, green), ALLD (“always defect”, red), and TFT (“mutual reciprocation”, blue) are placed randomly on the lattice. Then the following process is repeated at each lattice site: strategies play M PD games against every neighbor in their neighborhood, and the scores of each player are recorded. (

**B**) In a second step, the score of each strategy is compared against their neighbor’s score, and (

**C**) the highest scoring strategy (boxed in yellow) is assigned to the site being examined. This process is repeated for each of the T iterations of the model.

**Figure 2.**First fourteen iterations of the model for a deterministic simulation (i.e., $p=0$): strategies ALLC (green), ALLD (red), TFT (blue), and RND (“play randomly”, pink) are initially placed randomly on the lattice, then IPD games are played and the highest scoring strategies are propagated. For such a simulation where no stochasticity is included—each move is determined by the assigned strategy and the players make no mistake playing the PD—a very brief invasion by ALLD precedes an eventual predominance of TFT. RND is quickly eliminated from the lattice, as in most of the simulations over the parameter space of the model. The majority of such deterministic simulations become stationary after only a few tens iterations.

**Figure 3.**Population evolution and final lattice state for two simulations with $\widehat{p}\simeq 3.16\times {10}^{-5}$ (

**A**,

**B**) and $\widehat{p}\simeq 3.16\times {10}^{-2}$ (

**C**,

**D**). Simulations were carried out on a Cartesian lattice of size $L=128$ with periodic boundaries over $T=500$ iterations of the model, with iterated prisoner’s dilemma (IPD) games of $M=2000$ moves. In the second lattice (

**D**), formations of ALLC cooperating together successfully survive while being surrounded by defectors. An animation of the two simulations is available in the Supplementary Material of the online version of this article.

**Figure 4.**Final populations fractions for error rates that are immutable, heritable, and subject to mutations. Relative population frequencies with respect to the total population of strategies following either a strategy of unconditional cooperation (ALLC), unconditional defection (ALLD), or mutual reciprocity (TFT), as a function of the most probable error rate of the initial distribution $\widehat{p}$. Final population fractions are averaged over samples of 20 simulations with random initial conditions, with shaded areas proportional to the standard deviation of final populations. Simulations were all carried out on a 2D periodic lattice of side length $L=128$ sites over $T=500$ iterations of the model, with IPD games of $M=2000$ moves. Error rates were initially set according to a lognormal distribution with the most probable values in abscissa and a fixed shape parameter $s=0.5$. (

**A**) Error rates are immutable over the duration of the simulation. (

**B**) Error rates become heritable (i.e., the player losing the game adopts its opponent’s error rate in addition to its strategy). (

**C**) Introduction of a mutation probability $\mu ={10}^{-4}$. While heritability readily contributes to an increase in cooperative behavior—mostly at lower (≲${10}^{-3}$) error rates—the introduction of mutations (

**C**) is the primary driver of the dominance of cooperative behavior, allowing TFT to invade the lattice on a much larger region of the system’s parameter space.

**Figure 5.**Transition towards TFT-mediated cooperation. (

**A**) Relative population frequency for each strategy as a function of the mutation rate $\mu $. Simulation parameters are identical as in Figure 4, and the data presented here refers to simulations initialized with an initial distribution of error rates having $\widehat{p}={10}^{-4}$ and $s=0.5$. Final populations are averaged on ensembles of 20 simulations with random initial distributions of strategies, with shaded areas proportional to the standard deviation in final population fractions. Increasing the mutation probability $\mu $ drives a continuous transition of the final population fractions towards invasion by TFT until a breakdown of cooperative behavior occurs at very high error rates (inset). (

**B**) RGB-coding for relative population frequency of mutual cooperative behavior (TFT) as a function of mutation rate $\mu $ and shape parameter s of the error rate distribution. Measures are derived from 100 samples of s and $\mu $ combinations, and the results for each parameter configuration pair are averaged over 10 simulations with random initial conditions. (

**C**–

**F**) Final lattice state, error rates distributions, and temporal population, stationarity index, and mean error rate evolutions of five representative simulations taken from the statistical ensemble shown on panel (

**A**). The initial distribution of error rates is shown as a dotted curve on (

**D**), grey bars indicate the error rates distribution for all strategies, while colored curves indicate the ones specific for each strategy. Panel (

**E**) shows population evolution with colored curves, while the black curve indicates the stationarity of the simulation. Panel (

**F**) displays the mean of the error rate for each strategy.

**Figure 6.**Temporal snapshots from a simulation including ALLC (green), ALLD (red) and TFT (blue) with initial error rate distribution parameters $\widehat{p}={10}^{-6}$ and $s=1$, and for which $\mu ={10}^{-4}$, taken at regular intervals of 25 iterations each. In this regime ALLC and TFT coexist while ALLD is a predator for ALLC. Traveling wavefronts of ALLC propagate through time on the lattice, although the fragmentation of ALLC populations into spatially distinct colonies acts as a barrier guarding against complete and immediate invasion by ALLD. The dynamics of the three strategies is reminiscent of a forest-fire model where there is an accumulation of trees (ALLC) according to a growth rate dictated by simulation parameters, which is conducive to dramatic reorganizations of the lattice following ignition (ALLD mutants).

**Table 1.**Score matrix of the Prisoner’s dilemma (PD). Numbers refer to the PD’s score matrix as traditionally defined [57], indicating the reward of the player adopting the strategy in the leftmost column. Mutual cooperation leads to the best possible mean outcome, while defection either leads to the absolute maximum or the absolute minimum reward. Letters refer to the general form of the score matrix for the PD: mutual cooperation leads to the reward payoff (R), while mutual defection leads to the punishment (P). Exploitation—with one player defecting while the opponent chooses to cooperate—leads to the temptation payoff (T) and sucker’s payoff (S). The game satisfies the PD constraint when $T>R>P>S$.

Cooperate | Defect | |||
---|---|---|---|---|

Cooperate | R | S | ||

3 | 0 | |||

Defect | T | P | ||

5 | 1 |

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**MDPI and ACS Style**

Champagne-Ruel, A.; Charbonneau, P. A Mutation Threshold for Cooperative Takeover. *Life* **2022**, *12*, 254.
https://doi.org/10.3390/life12020254

**AMA Style**

Champagne-Ruel A, Charbonneau P. A Mutation Threshold for Cooperative Takeover. *Life*. 2022; 12(2):254.
https://doi.org/10.3390/life12020254

**Chicago/Turabian Style**

Champagne-Ruel, Alexandre, and Paul Charbonneau. 2022. "A Mutation Threshold for Cooperative Takeover" *Life* 12, no. 2: 254.
https://doi.org/10.3390/life12020254