## Abstract

We study diffusion of cooperation in an n-population game in continuous time. At each instant, the game involves n random individuals, one from each population. The game has the structure of a prisoner's dilemma, where each player can choose a continuous decision variable associated with the probability of cooperating or defecting. We turn the game into a positive dynamical system. Then, we propose a novel strategy that is the saturation of a polynomial function. The strategy requires to each player exclusively the knowledge of her/his own current average payoff, along with her/his own payoffs in the cooperative and noncooperative equilibria; no information about other players' payoffs is required. The proposed strategy guarantees local stability of the cooperative equilibrium if the degree p of the polynomial is greater than or equal to 2. Conversely, the noncooperative equilibrium becomes unstable, for p large enough, if and only if a certain Metzler matrix depending on the payoffs has a positive Frobenius eigenvalue. We prove that the n-dimensional box of all payoffs between the noncooperative and the cooperative ones is positively invariant. Finally we show that, for p large, the domain of attraction of the cooperative equilibrium inside this box becomes arbitrarily close to the box itself.

Original language | English |
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Pages (from-to) | 3478-3498 |

Journal | SIAM Journal on Control and Optimization |

Volume | 56 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2018 |

## Keywords

- Dynamic games
- Equilibria
- Invariant sets
- Prisoner's dilemma
- Stability