TY - JOUR

T1 - The dynamics of self-referential paradoxical games

AU - Nicolis, J. S.

AU - Bountis, T.

AU - Togias, K.

N1 - Funding Information:
The authors tnhk thae referees for useful sguoeng, swshicth iimproved the qutyaof li this presentati. Onoe ofnus (T.B.) acknowledges the ®nancial support of the General Secretatrof iRaeh ansdTecaehrynoctoefhHloiceMgllyeiof nnDevelopmeistnt r under the PENED 996-gr1taan4dnthe University of Prsaafor ta `Krthaae’ odory research grant. He is also grateful to the Dmeentpof Mathemarattsiocfteh University of Cypus forr its hospitality, while pat ofrths worik was being completed. K.T. also tnhs MrkaDimitris Tassoulis for many discussions on the results of this paper.
Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

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

AN - SCOPUS:0035783640

VL - 16

SP - 319

EP - 332

JO - Dynamical Systems

JF - Dynamical Systems

SN - 1468-9367

IS - 4

ER -