Value-based distance between information structures

Fabien Gensbittel, Marcin Peski, and Jérôme Renault


We define the distance between two information structures as the largest possible difference in value across all zero-sum games. We provide a tractable characteri- zation of distance and use it to discuss the relation between the value of informa- tion in games versus single-agent problems, the value of additional information, informational substitutes, complements, or joint information. The convergence to a countable information structure under value-based distance is equivalent to the weak convergence of belief hierarchies, implying, among other things, that for zero-sum games, approximate knowledge is equivalent to common knowledge. At the same time, the space of information structures under the value-based dis- tance is large: there exists a sequence of information structures where players acquire increasingly more information, and ε > 0 such that any two elements of the sequence have distance of at least ε. This result answers by the negative the second (and last unsolved) of the three problems posed by Mertens in his paper “Repeated Games” (1986).


Value of information; universal type space;

JEL codes

  • C7: Game Theory and Bargaining Theory


Fabien Gensbittel, Jérôme Renault, and Marcin Peski, Large space of information structures, TSE Working Paper, n. 19-1006, April 2019.


Fabien Gensbittel, Marcin Peski, and Jérôme Renault, Value-based distance between information structures, Theoretical Economics, vol. 17, n. 3, July 2022, pp. 1225–1267.

Published in

Theoretical Economics, vol. 17, n. 3, July 2022, pp. 1225–1267