0 such that any two elements of the sequence have distance at least ε : having more and more information may lead nowhere. As a consequence, the completion of (Z * , d) is not compact, hence not homeomorphic to the set of consistent probabilities over the states of the worldàworldà la Mertens and Zamir. This example answers by the negative the second (and last unsolved) of the three problems posed by J.F. Mertens in his paper "Repeated Games", ICM 1986.
(This abstract was borrowed from another version of this item.)"> 0 such that any two elements of the sequence have distance at least ε : having more and more information may lead nowhere. As a consequence, the completion of (Z * , d) is not compact, hence not homeomorphic to the set of consistent probabilities over the states of the worldàworldà la Mertens and Zamir. This example answers by the negative the second (and last unsolved) of the three problems posed by J.F. Mertens in his paper "Repeated Games", ICM 1986.
(This abstract was borrowed from another version of this item.)">
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The large space of information structures

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  • Gensbittel, Fabien
  • Renault, Jérôme
  • Peski, Marcin
Abstract
We revisit the question of modeling incomplete information among 2 Bayesian players, following an ex-ante approach based on values of zero-sum games. K being the finite set of possible parameters, an information structure is defined as a probability distribution u with finite support over K × N × N with the interpretation that: u is publicly known by the players, (k, c, d) is selected according to u, then c (resp. d) is announced to player 1 (resp. player 2). Given a payoff structure g, composed of matrix games indexed by the state, the value of the incomplete information game defined by u and g is denoted val(u, g). We evaluate the pseudo-distance d(u, v) between 2 information structures u and v by the supremum of |val(u, g) − val(v, g)| for all g with payoffs in [−1, 1], and study the metric space Z * of equivalent information structures. We first provide a tractable characterization of d(u, v), as the minimal distance between 2 polytopes, and recover the characterization of Peski (2008) for u v, generalizing to 2 players Blackwell's comparison of experiments via garblings. We then show that Z * , endowed with a weak distance d W , is homeomorphic to the set of consistent probabilities with finite support over the universal belief space of Mertens and Zamir. Finally we show the existence of a sequence of information structures, where players acquire more and more information, and of ε > 0 such that any two elements of the sequence have distance at least ε : having more and more information may lead nowhere. As a consequence, the completion of (Z * , d) is not compact, hence not homeomorphic to the set of consistent probabilities over the states of the worldàworldà la Mertens and Zamir. This example answers by the negative the second (and last unsolved) of the three problems posed by J.F. Mertens in his paper "Repeated Games", ICM 1986.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Gensbittel, Fabien & Renault, Jérôme & Peski, Marcin, 2019. "The large space of information structures," TSE Working Papers 19-1006, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:122929
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    References listed on IDEAS

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    1. Lehrer Ehud & Monderer Dov, 1994. "Discounting versus Averaging in Dynamic Programming," Games and Economic Behavior, Elsevier, vol. 6(1), pages 97-113, January.
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    3. Renault, Jérôme & Venel, Xavier, 2017. "A distance for probability spaces, and long-term values in Markov Decision Processes and Repeated Games," TSE Working Papers 17-748, Toulouse School of Economics (TSE).
    4. Peski, Marcin, 2008. "Comparison of information structures in zero-sum games," Games and Economic Behavior, Elsevier, vol. 62(2), pages 732-735, March.
    5. Jérôme Renault & Xavier Venel, 2017. "Long-Term Values in Markov Decision Processes and Repeated Games, and a New Distance for Probability Spaces," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 349-376, May.
    6. Dinah Rosenberg & Bernard de Meyer & Ehud Lehrer, 2010. "Evaluating Information in Zero-Sum Games with Incomplete Information on Both Sides," Post-Print hal-00586037, HAL.
    7. Bernard De Meyer & Ehud Lehrer & Dinah Rosenberg, 2009. "Evaluating information in zero-sum games with incomplete information on both sides," Documents de travail du Centre d'Economie de la Sorbonne 09035, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
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    Cited by:

    1. Olivier GOSSNER & Jean-François MERTENS, 2020. "The Value of Information in Zero-Sum Games," Working Papers 2020-19, Center for Research in Economics and Statistics.

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    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General

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