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The Spatialized, Continuous-Valued Battle of the Sexes

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Abstract

The dynamics of a spatial, continuous-valued formulation of the iterated battle of the sexes is assessed in this work. The game is played in the cellular automaton manner, i.e., with local and synchronous interaction. The effects of probabilistic updating and memory of past encounters are also taken into account. With deterministic updating, the spatial structure enables the emergence of coordination clusters, leading to the mean payoffs per encounter to values that are accessible only in the cooperative two-person game scenario, which constitutes a notable case of self-organization. With probabilistic updating of choices, both kinds of player tend to reach a full coordination absorbing steady state in the long term. As a general rule, short-term memory of past iterations does not qualitatively alter the ahistoric dynamics. Unlimited trailing memory induces an inertial effect that alters the dynamics to a larger extent, particularly in the probabilistic updating scenario, in which case unlimited trailing memory fully inhibits the dynamics.

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Notes

  1. We have studied such an approach dealing with the spatialized PD game [2, 7].

  2. The probabilistic updating mechanism has been implemented in the binary BOS context in [5].

  3. Which differs from the usual memory implementation in spatial games, consisting of designing strategies which determine the next move of a player given the history (genome) of past moves [14, 15], whereas scores are treated in a Markovian way. Probabilistic simulations with genome-type memory are implemented in [12].

  4. , L=(6y−1)x+1−y+λ(Δp−4(x+y−1)), \(\frac{\partial L}{\partial x}=6y-1-4\lambda\), \(\frac{\partial L}{\partial y}=6x-1-4\lambda\), \(\max L \rightarrow x=y=\frac{1+4\lambda}{6}\rightarrow \Delta p = 4(2\frac{1+4\lambda}{6}-1) \rightarrow \lambda=\frac{3\Delta p +8}{16}\rightarrow x=y = \frac{\Delta p +4}{8}\), .

  5. \(\pi^{(T)}_{i}=\alpha p^{(T-1)}_{i}+p^{(T)}_{i}\), .

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Acknowledgements

This work was carried out during a two-months residence in the University of the West of England (Bristol), supported by EPSRC grant EP/H014381/1.

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  1. ETSI Agrónomos (Estadística, GSC), Universidad Politécnica de Madrid, C.Universitaria, 28040, Madrid, Spain

    Ramón Alonso-Sanz

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Correspondence to Ramón Alonso-Sanz.

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Alonso-Sanz, R. The Spatialized, Continuous-Valued Battle of the Sexes. Dyn Games Appl 2, 177–194 (2012). https://doi.org/10.1007/s13235-012-0042-y

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