### Abstract

Paradoxical games are non-constant sum, non-negotiable conflicts, in which two contestants (players) blackmail each other, acting as components of a nonlinear dynamical system characterized by time varying probabilities of cooperation. Such games are 'paradoxical' in the sense that both players could win or lose simultaneously and are called 'self-referential' if the parameters of the system depend explicitly on the contestants' probabilities of cooperation. Previously studied two-contestant models with constant parameters were found to be 'conservative', possessing two centres around which the game can oscillate forever. In this paper, we first study the case where all parameters are allowed to vary and find that the dynamics becomes 'dissipative', possessing a single fixed point attractor of moderate equal gains. On this attractor large subsets of initial conditions (strategies) converge as t → ∞ and attain constant cooperation probabilities. If both contestants cooperate 'equally disregarding' the other's tendency to do so, the attractor moves closer to the state of 'maximum pay-off', where both parties cooperate with probability 1. However, in the asymmetric case, where one of the players 'takes less into account' the other's tendency to cooperate, it is the more 'indifferent' player who profits the most! Partially self-referential games, in which only the 'gain' or 'loss' factors due to defection vary by a small parameter ε, lead to the state of full cooperation (and maximum gain), as the stable fixed points of the ε = 0 case either become repellors, or are eliminated via a pitchfork and a saddle-node bifurcation.

Original language | English |
---|---|

Pages (from-to) | 319-332 |

Number of pages | 14 |

Journal | Dynamical Systems |

Volume | 16 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 2001 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mechanics of Materials
- Computational Mechanics
- Applied Mathematics

### Cite this

*Dynamical Systems*,

*16*(4), 319-332. https://doi.org/10.1080/14689360110081741

**The dynamics of self-referential paradoxical games.** / Nicolis, J. S.; Bountis, T.; Togias, K.

Research output: Contribution to journal › Article

*Dynamical Systems*, vol. 16, no. 4, pp. 319-332. https://doi.org/10.1080/14689360110081741

}

TY - JOUR

T1 - The dynamics of self-referential paradoxical games

AU - Nicolis, J. S.

AU - Bountis, T.

AU - Togias, K.

PY - 2001/12

Y1 - 2001/12

N2 - Paradoxical games are non-constant sum, non-negotiable conflicts, in which two contestants (players) blackmail each other, acting as components of a nonlinear dynamical system characterized by time varying probabilities of cooperation. Such games are 'paradoxical' in the sense that both players could win or lose simultaneously and are called 'self-referential' if the parameters of the system depend explicitly on the contestants' probabilities of cooperation. Previously studied two-contestant models with constant parameters were found to be 'conservative', possessing two centres around which the game can oscillate forever. In this paper, we first study the case where all parameters are allowed to vary and find that the dynamics becomes 'dissipative', possessing a single fixed point attractor of moderate equal gains. On this attractor large subsets of initial conditions (strategies) converge as t → ∞ and attain constant cooperation probabilities. If both contestants cooperate 'equally disregarding' the other's tendency to do so, the attractor moves closer to the state of 'maximum pay-off', where both parties cooperate with probability 1. However, in the asymmetric case, where one of the players 'takes less into account' the other's tendency to cooperate, it is the more 'indifferent' player who profits the most! Partially self-referential games, in which only the 'gain' or 'loss' factors due to defection vary by a small parameter ε, lead to the state of full cooperation (and maximum gain), as the stable fixed points of the ε = 0 case either become repellors, or are eliminated via a pitchfork and a saddle-node bifurcation.

AB - Paradoxical games are non-constant sum, non-negotiable conflicts, in which two contestants (players) blackmail each other, acting as components of a nonlinear dynamical system characterized by time varying probabilities of cooperation. Such games are 'paradoxical' in the sense that both players could win or lose simultaneously and are called 'self-referential' if the parameters of the system depend explicitly on the contestants' probabilities of cooperation. Previously studied two-contestant models with constant parameters were found to be 'conservative', possessing two centres around which the game can oscillate forever. In this paper, we first study the case where all parameters are allowed to vary and find that the dynamics becomes 'dissipative', possessing a single fixed point attractor of moderate equal gains. On this attractor large subsets of initial conditions (strategies) converge as t → ∞ and attain constant cooperation probabilities. If both contestants cooperate 'equally disregarding' the other's tendency to do so, the attractor moves closer to the state of 'maximum pay-off', where both parties cooperate with probability 1. However, in the asymmetric case, where one of the players 'takes less into account' the other's tendency to cooperate, it is the more 'indifferent' player who profits the most! Partially self-referential games, in which only the 'gain' or 'loss' factors due to defection vary by a small parameter ε, lead to the state of full cooperation (and maximum gain), as the stable fixed points of the ε = 0 case either become repellors, or are eliminated via a pitchfork and a saddle-node bifurcation.

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U2 - 10.1080/14689360110081741

DO - 10.1080/14689360110081741

M3 - Article

VL - 16

SP - 319

EP - 332

JO - Dynamical Systems

JF - Dynamical Systems

SN - 1468-9367

IS - 4

ER -