Harsanyi transformation

Harsanyi transformation
Name	Harsanyi transformation

Harsanyi transformation

Introduction

The Harsanyi transformation is a method introduced by John Harsanyi to convert games with incomplete information into games with imperfect information, facilitating analysis by players and analysts. It relates to concepts articulated by John Nash, John Harsanyi, Reinhard Selten, Lloyd Shapley, and engages with frameworks employed by Thomas Schelling, Robert Aumann, Herman Rubin, and Kenneth Arrow. The transformation connects to institutions and results studied at Princeton University, Stanford University, University of California, Berkeley, Harvard University, and influenced work by Nobel Memorial Prize in Economic Sciences recipients and researchers associated with Cowles Foundation, RAND Corporation, and Center for Advanced Study in the Behavioral Sciences.

Formal definition

Formally, the transformation introduces Nature as an initial mover that selects types according to a common prior and informs players privately, converting the original game to an extensive-form representation. This construction echoes methods used in analyses by Leonid Hurwicz, John von Neumann, Oskar Morgenstern, Paul Samuelson, and Kenneth Arrow while incorporating probability measures like those in work by Andrey Kolmogorov, Thomas Bayes, Ronald Fisher, and Jerzy Neyman. The definition specifies a type space, action spaces, payoff functions, and beliefs consistent with Bayesian updating refined by criteria discussed by Robert Aumann, David Lewis, Hillel Yaari, and Milton Friedman.

Applications in game theory

Researchers apply the transformation to derive solution concepts such as Bayesian Nash equilibrium and sequential equilibrium, building on foundations laid by John Nash, Reinhard Selten, Roger Myerson, Eric Maskin, and Lloyd Shapley. It is used in mechanism design problems investigated by Jules Dupuit, William Vickrey, Leonid Hurwicz, Eric Maskin, and Roger Myerson and in auction theory studied by Paul Milgrom, Robert Wilson, Vickrey, and William Vickrey. The technique underpins signaling and screening models developed by Michael Spence, Joseph Stiglitz, George Akerlof, Kenneth Arrow, and Bengt Holmström', and informs bargaining analyses referencing John Nash, Robert Aumann, Herbert Simon, and Thomas Schelling.

Relationship to Bayesian games

The transformation provides the canonical link between extensive-form games with chance moves and static representations of games of incomplete information exemplified by Bayesian games described by John Harsanyi, John Nash, Reinhard Selten, and elaborated upon by Steven Brams, Kenneth Arrow, Robert Aumann, and David Kreps. It formalizes the common prior assumption discussed by Robert Aumann, David Lewis, Milton Friedman, and Kenneth Arrow and interfaces with belief hierarchies explored by Mertens, Partha Dasgupta, Dov Samet, and Hillel Yaari. Bayesian updating rules used in these contexts draw on principles from Thomas Bayes, Andrey Kolmogorov, Jerzy Neyman, and are applied in work at Harvard University, Stanford University, and MIT.

Examples

Canonical examples include auctions where bidders have private valuations as in analyses by Paul Milgrom, Robert Wilson, William Vickrey, and Roger Myerson; signaling games inspired by Michael Spence and George Akerlof; and entry deterrence problems studied by Thomas Schelling and John Nash. Classic textbook cases invoke scenarios from studies at Princeton University, Yale University, London School of Economics, and University of Chicago, and model applications in regulatory settings referenced by Kenneth Arrow, Joseph Stiglitz, and Bengt Holmström.

Extensions and generalizations

Extensions generalize the transformation to infinite type spaces, correlated priors, and dynamic type evolution as in work by Robert Aumann, Mertens, Borgers, Geanakoplos, and Fudenberg. Generalizations connect to epistemic game theory frameworks by Aumann, Roger Myerson, David Lewis, Brandenburger, and Dekel and to refinements like perfect Bayesian equilibrium developed by David Kreps, Robert Wilson, Eric Maskin, and Jean Tirole. Applications intersect with stochastic process theories from Andrey Kolmogorov, Norbert Wiener, and Paul Lévy when modeling evolving types.

Criticisms and limitations

Critics point to reliance on the common prior assumption criticized in debates involving Robert Aumann, David Lewis, John Harsanyi, and Amartya Sen and to challenges modeling higher-order beliefs discussed by Brandenburger, Dekel, Mertens, and Samet. Practical limitations arise in large or continuous type spaces as noted by John Nash, Roger Myerson, Paul Milgrom, and Robert Wilson, and in empirical identification problems debated at institutions like NBER, CEPR, and IZA.

Category:Game theory