Abstract
The classical game theoretic resolutions to Selten’s Chain Store game are unsatisfactory; they either alter the game to avoid the paradox or struggle to organize the existing experimental data. This paper applies co-evolutionary algorithms to the Chain Store game and demonstrates that the resulting system’s dynamics are neither intuitively paradoxical nor contradicted by the existing experimental data. Specifically, some parameterizations of evolutionary algorithms promote genetic drift. Such drift can lead the system to transition among the game’s various Nash Equilibria. This has implications for policy makers as well as for computational modelers.
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Notes
Selten (1978)
For a precise game theoretic definition of ‘Common Knowledge,’ see Fudenberg and Tirole (1996), p. 542.
Other structures were tested, such as limiting the Entrants’ memory. However, these structures did not produce significantly different results.
The specification was tested for this paper and produced no describable results. Chen and Ni (1996) tried this approach with an eight-stage Chain Store game for which history was truncated to only include the opponent’s behavior. However, they found that this abbreviated version was also intractable.
These observations are made at the phenotypical level. When an Entrant stays out, the Chain Store does not have the opportunity to respond. Therefore, NE outcomes include all games in which all stages end in either enter-acquiesce or stay out.
For example, if a particular generation resulted in more than 70% of all stages ending in enter-acquiesce, the system would be in the SGP state. If more than 70% of stages 1–4 and stages 6–9 ended in enter-acquiesce and stage 5 ended in stay out more than 70% of the time, the system would be in the SGP-1 state. If more than 70% of stages 1–4 and stages 6–9 ended in enter-acquiesce and stage 5 ended in stay out less than 70% of the time, the system would not be in one of the ten states.
When the system transitions from one state’s neighborhood to another, it can, with very low probability, pass through the neighborhood of a third state. The system only approximates a Markov process, because knowledge that the system has been in a state for several generations rules out the (almost trivially small) possibility that the system is passing through. This technically violates the Markov property.
To balance mutation rates across agents with different lengths of chromosome material, mutation was implemented on a per-bit basis. The per-bit probability of mutation was equal for the CS and Entrant agents. However, this led to a difference in the probability that a parent agent was mutated at least once. In the BC version, the per-bit probability of mutation was 2%. The probability of at least one mutation was 16.6 and 30.5, respectively, for the CS and Entrant agents. In the FSM, the probability of mutation was 5% per 16-bit string. The probability of at least one mutation was 14.3 and 18.5, respectively, for the CS and Entrant agents.
Kreps and Wilson (1982b, p. 254).
This variation decreases considerably when the number of generations is increased. For example, the 0.358 SD reported for the BC model in the first column of Table 8 drops to 0.101 when the number of generations is increased to 10,000. This statistic further drops to 0.068 when the number of generations is increased 20,000.
Repeated attempts to increase the value of this statistic through parameter calibration proved ineffective in raising it above 8%.
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Acknowledgments
This work benefited substantially from conversations with John Miller, Bill McKelvey, Scott Carr, and John Holland. Comments from Computational Economics Editor Hans Amman and anonymous reviewers also significantly improved this paper.
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Tracy, W.M. Paradox Lost: The Evolution of Strategies in Selten’s Chain Store Game. Comput Econ 43, 83–103 (2014). https://doi.org/10.1007/s10614-013-9364-0
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DOI: https://doi.org/10.1007/s10614-013-9364-0