Beliefs in Network Games (Revised version of CentER DP 2007-46)

Networks can have an important effect on economic outcomes. Given the complexity of many of these networks, agents will generally not know their structure. We study the sensitivity of game-theoretic predictions to the specification of players’ (common) prior on the network in a setting where players play a fixed game with their neighbors and only have local information on the network structure. We show that two priors are close in a strategic sense if and only if (i) the priors assign similar probabilities to all events that involve a player and his neighbors, and (ii) with high probability, a player believes, given his type, that his neighbors’ conditional beliefs are close under the two priors, and that his neighbors believe, given their type, that. . . the conditional beliefs of their neighbors are close, for any number of iterations.Network games;incomplete information;higher order beliefs;continuity;random networks;population uncertainty

Beliefs in Network Games (Replaced by CentER DP 2008-05)

Networks can have an important effect on economic outcomes. Given the complexity of many of these networks, agents will generally not know their structure. We study the sensitivity of game-theoretical predictions to the specification of players’ (common) prior on the network in a setting where players play a fixed game with their neighbors and only have local information on the network structure. We show that two priors are close in a strategic sense if and only if (1) the priors assign similar probabilities to all events that involve a player and his neighbors, and (2) with high probability, a player believes, given his type, that his neighbors’ conditional beliefs are similar, and that his neighbors believe, given their type, that. . . the conditional beliefs of their neighbors are similar, for any number of iterations. Also, we show that the common but unrealistic assumptions that the size of the network is common knowledge or that the types of players are independent are far from innocuous: if these assumptions are violated, small probability events can have a large effect on outcomes through players’ conditional beliefs.Network games;incomplete information;higher order beliefs;continuity;random networks;population uncertainty

The Minority Game: An Economics Perspective

This paper gives a critical account of the minority game literature. The minority game is a simple congestion game: players need to choose between two options, and those who have selected the option chosen by the minority win. The learning model proposed in this literature seems to differ markedly from the learning models commonly used in economics. We relate the learning model from the minority game literature to standard game-theoretic learning models, and show that in fact it shares many features with these models. However, the predictions of the learning model differ considerably from the predictions of most other learning models. We discuss the main predictions of the learning model proposed in the minority game literature, and compare these to experimental findings on congestion games.Learning;congestion games;experiments.

Congestion, equilibrium and learning: The minority game

The minority game is a simple congestion game in which the players' main goal is to choose among two options the one that is adopted by the smallest number of players. We characterize the set of Nash equilibria and the limiting behavior of several well-known learning processes in the minority game with an arbitrary odd number of players. Interestingly, different learning processes provide considerably different predictions

The minority game: An economics perspective

This paper gives a critical account of the minority game literature. The minority game is a simple congestion game: players need to choose between two options, and those who have selected the option chosen by the minority win. The learning model proposed in this literature seems to differ markedly from the learning models commonly used in economics. We relate the learning model from the minority game literature to standard game-theoretic learning models, and show that in fact it shares many features with these models. However, the predictions of the learning model differ considerably from the predictions of most other learning models. We discuss the main predictions of the learning model proposed in the minority game literature, and compare these to experimental findings on congestion games.Comment: 30 pages, 4 figure

A logic for reasoning about ambiguity

Standard models of multi-agent modal logic do not capture the fact that information is often \emph{ambiguous}, and may be interpreted in different ways by different agents. We propose a framework that can model this, and consider different semantics that capture different assumptions about the agents' beliefs regarding whether or not there is ambiguity. We examine the expressive power of logics of ambiguity compared to logics that cannot model ambiguity, with respect to the different semantics that we propose.Comment: Some of the material in this paper appeared in preliminary form in "Ambiguous langage and differences of belief" (see arXiv:1203.0699

Learning to be prepared

Behavioral economics provides several motivations for the common observation that agents appear somewhat unwilling to deviate from recent choices: salience, inertia, the formation of habits, the use of rules of thumb, or the locking in on certain modes of behavior due to learning by doing. This paper provides discrete-time adjustment processes for strategic games in which players display precisely such a bias towards recent choices. In addition, players choose best replies to beliefs supported by observed play in the recent past, in line with much of the literature on learning. These processes eventually settle down in the minimal prep sets of Voorneveld (2004, 2005).adjustment; learning; minimal prep sets; behavioral bias; salience

Sectoral TFP developments in the OECD

This note describes the sectoral total factor productivity (TFP) developments in the OECD between 1970-1990. Based on the ISDB data of the OECD, we confirm the stylised fact that TFP growth is relatively high in agriculture and relatively low in services. Within manufacturing, the TFP growth in chemicals and in capital goods is high whereas it is low in food processing, paper and publishing and metals. The TFP growth in services sectors like construction, financial services and other (government) services seems to be zero or even negative, while it is relatively high in transport and communication. These sectoral pictures are not universal. Differences between countries are rather large. Also, the TFP growth per year appears to be non constant over time. We use the results from this study in our dynamic CGE model WorldScan to model differences in productivity growth between the sectors. In particular, we employ this mechanism in the European long term scenarios.

Congestion, Equilibrium and Learning: The Minority Game

The minority game is a simple congestion game in which the players’ main goal is to choose among two options the one that is adopted by the smallest number of players. We characterize the set of Nash equilibria and the limiting behavior of several well-known learning processes in the minority game with an arbitrary odd number of players. Interestingly, different learning processes provide considerably different predictions.Learning;congestion games;replicator dynamic;perturbed best response dynamics;quantal response equilibria;best-reply learning

Finite depth of reasoning and equilibrium play in games with incomplete information

The standard framework for analyzing games with incomplete information models players as if they have an infinite depth of reasoning, which is not always consistent with experimental evidence. This paper generalizes the type spaces of Harsanyi (1967-1968) so that players can have a finite depth of reasoning. We do this restricting the set of events that a player of a finite depth can reason about. This approach allows us to extend the Bayesian-Nash equilibrium concept to environments with players with a finite depth of reasoning. We demonstrate that the standard approach of modeling beliefs with Harsanyi type spaces fails to capture the equilibrium behavior of players with a finite depth, at least in some games. Consequently, the standard approach cannot be used to describe the equilibrium behavior of players with a finite depth in general