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Continuous-time Repeated Games with Imperfect Information: Folk Theorems and Explicit Results Open Access


Other title
Equilibrium characterization
Repeated games
Continuous-time games
Folk theorem
Imperfect observability
Game theory
Intertemporal incentives
Type of item
Degree grantor
University of Alberta
Author or creator
Bernard, Benjamin
Supervisor and department
Frei, Christoph (Mathematical and Statistical Sciences)
Examining committee member and department
Brown, David (Economics)
Klumpp, Tilman (Economics)
Melnikov, Alexander (Mathematical and Statistical Sciences)
Frei, Christoph (Mathematical and Statistical Sciences)
Choulli, Tahir (Mathematical and Statistical Sciences)
Aguerrevere, Felipe (Business)
Department of Mathematical and Statistical Sciences
Mathematical Finance
Date accepted
Graduation date
Doctor of Philosophy
Degree level
This thesis treats continuous-time models of repeated interactions with imperfect public monitoring. In such models, players do not directly observe each other's actions and instead see only the impacts of the chosen actions on the distribution of a random signal. Often, there are two reasons why this signal imperfectly reflects the chosen actions: (a) information is continuously available but it is noisy, or (b) events are observable but occur only at intermittent occasions. In a continuous-time setting, these two different types of information can be cleanly distinguished, where Brownian motion is used to model noise in the continuous information and Poisson processes indicate the arrival of informative events with an intensity that depends on players' actions. The first major result of this thesis is a folk theorem for continuous-time repeated games even when players receive only noisy information about past play. The folk theorem gives sufficient conditions such that players achieve asymptotic efficiency as they get arbitrarily patient. Because more outcomes are sustainable in equilibrium when more information is observed, this result also applies when players receive both aforementioned types of imperfect information. In the proof, we restrict ourselves to strategies that are adjusted only at identical copies of certain stopping times. This has two important implications: (1) despite the possibility of switching actions infinitesimally fast, players do not need to do so to attain asymptotic efficiency, and (2) continuous-time equilibria can be attained as limits of equilibria in discrete-time repeated games where the length of the time period is random, rather than fixed. The other main result of this thesis is a characterization of all payoffs that are attainable in equilibrium in such games with two finitely patient players. Relating optimal actions and incentives to the boundary of the equilibrium payoff set, we obtain a differential equation describing the curvature of the set at almost every point. The equilibrium payoff set is obtained from an iterative procedure, which is similar to that known for discrete-time repeated games but leads to an explicit characterization in our setting. Our result shows that the two types of information have drastically different impacts on the equilibrium payoff set. This is due to the fundamental difference in which the two types of information are used to provide incentives: while the continuous information can be used only to transfer value between players, the discontinuous information may be used to transfer or destroy value upon the arrival of an infrequent event. The quantitative nature of the result makes it possible to precisely measure the impact of abrupt information on the efficiency of players' payoffs in equilibrium. Thus, one can compare the value of additional information to the cost of procuring or providing it, which may lead to interesting applications in mechanism design and information disclosure.
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
Citation for previous publication
Benjamin Bernard and Christoph Frei: The folk theorem with imperfect public information in continuous time, Theoretical Economics, forthcoming

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File title: Introduction
File title: Continuous-time Repeated Games with Imperfect Information
File author: Benjamin Bernard
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