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Imitation and selective matching in reputational games

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Abstract

This paper investigates imitation and selective matching in reputational games with an outside option. We identify two classes of such games, ultimatum and trust games. By selective matching we mean that short-run players have the possibility of selecting the long-run player they play against. We find that selective matching (unlike random matching) favors the equilibrium associated to reputation in the ultimatum game, but not in the trust game.

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Notes

  1. See Selten (1978), Kreps and Wilson (1982), Milgrom and Roberts (1982), Fudenberg and Levine (1989, 1992).

  2. Notable exceptions are Jackson and Kalai (1999) and Hörner (2002).

  3. See Robson and Vega-Redondo (1996), Vega-Redondo (1997), Tanaka (1999) or Alós-Ferrer (2004) for instance.

  4. See Offerman and Sonnemans (1998) and Pingle and Day (1996).

  5. The other matching process is the so-called playing-the-field. See for instance Vega-Redondo (1997) for a description of the differences between both matching processes.

  6. For more details on this class of matching mechanisms, see Fudenberg and Levine (1998).

  7. Biologists call this class of models Haystack models. See Bergstrom (2002).

  8. The same game is used in price dispersion models. See Salop and Stiglitz (1977).

  9. We keep here the denomination “short-run” and “long-run” players only for convenience, as all players repeatedly play the game.

  10. As will be easy to see, allowing population sizes to be different does not alter the results of the model, but highly complicates their exposition.

  11. See Kreps (1990).

  12. See Binmore et al. (1995).

  13. Alternatively, evolutionary games consider best response dynamics which, relative to imitation dynamics, required much more information (as players need to know the whole structure of the game).

  14. On the justification of the myopic assumption, see Section 5.

  15. Following Oechssler (1997), E and O are called actions since in our model of selective matching short-run players strategy indicates both a long-run player and an action choice.

  16. Most evolutionary game models postulate that agents simply look at the immediate past and use it as a one-point predictor of what will happen next. This is called static expectation. Young (1993) was the first to introduce a process of expectation formation into stochastic evolutionary models. As extensive-form games may present unreached information sets, evolutionary models investigating these games have proposed an extension of the static-expectations approach. See, for instance, Nöldeke and Samuelson (1993), who developed the analysis of evolutionary stability in extensive form games.

  17. As mentioned by Samuelson (1997), mutation is a residual capturing whatever has been excluded when modelling selection.

  18. On this point, see Canning (1989).

  19. See Kandori et al. (1993) and Young (1993).

  20. For more details on this result, see Kandori et al. (1993), Samuelson (1997), and Vega-Redondo (1996).

  21. This follows the Robson and Vega-Redondo (1996) model as well as the biologists formulation of evolutionary dynamics. However, it constitutes a point of departure with Kandori et al. (1993). On this point, see Robson and Vega-Redondo (1996).

  22. This is underlined in the statement of Proposition 1 by specifying for all possible cardinality of subset S I and for all S I .

  23. See Kandori and Rob (1995), Proposition 4.

  24. On the interplay between evolutionary game theory and the equilibrium selection problem, see Samuelson (1997).

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  1. European University Institute, Villa San Paolo, Via della Piazzuola, 43, 50133, Firenze, Italy

    Thierry Vignolo

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Vignolo, T. Imitation and selective matching in reputational games. J Evol Econ 20, 395–412 (2010). https://doi.org/10.1007/s00191-009-0156-6

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