The Large Space of Information Structures

We revisit the question of modeling incomplete information among 2 Bayesian\nplayers, following an ex-ante approach based on values of zero-sum games. $K$\nbeing the finite set of possible parameters, an information structure is\ndefined as a probability distribution $u$ with finite support over $K\\times\n\\mathbb{N} \\times \\mathbb{N}$ with the interpretation that: $u$ is publicly\nknown by the players, $(k,c,d)$ is selected according to $u$, then $c$ (resp.\n$d$) is announced to player 1 (resp. player 2). Given a payoff structure $g$,\ncomposed of matrix games indexed by the state, the value of the incomplete\ninformation game defined by $u$ and $g$ is denoted $\\mathrm{val}(u,g)$. We\nevaluate the pseudo-distance $d(u,v)$ between 2 information structures $u$ and\n$v$ by the supremum of $|\\mathrm{val}(u,g)-\\mathrm{val}(v,g)|$ for all $g$ with\npayoffs in $[-1,1]$, and study the metric space $Z^*$ of equivalent information\nstructures.\n We first provide a tractable characterization of $d(u,v)$, as the minimal\ndistance between 2 polytopes, and recover the characterization of Peski (2008)\nfor $u \\succeq v$, generalizing to 2 players Blackwell's comparison of\nexperiments via garblings. We then show that $Z^*$, endowed with a weak\ndistance $d_W$, is homeomorphic to the set of consistent probabilities with\nfinite support over the universal belief space of Mertens and Zamir. Finally we\nshow the existence of a sequence of information structures, where players\nacquire more and more information, and of $\\varepsilon>0$ such that any two\nelements of the sequence have distance at least $\\varepsilon$: having more and\nmore information may lead now here. As a consequence, the completion of\n$(Z^*,d)$ is not compact, hence not homeomorphic to the set of consistent\nprobabilities over the states of the world {\\it \\`a la} Mertens and Zamir. This\nexample answers by the negative the second (and last unsolved) of the three\nproblems posed by J.F. Mertens in his paper ``Repeated Games", ICM 1986.\n

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