## Essays on beliefs and knowledge

##### Abstract

The unifying theme of all three chapters of this dissertation is incomplete information
games. Each chapter investigates two essential components, namely
beliefs and knowledge, of incomplete information games. In particular, the
first two chapter studies an alternative equilibrium notion of Sakovics (2001)-
mirage equilibrium- and the final chapter introduces a new notion of metric to
measure the distance between partitions. All relevant notations and definitions
are defined for each chapter so that any of them can be read independently.
In the first chapter, I restudy the Purification theorem of Harsanyi (1973)
by relaxing the common knowledge assumption on priors for 2 x 2 games.
I show that the limit of the (Mirage) equilibrium points in perturbed games
generically converge to a pure strategy of the original complete information.
This result, unlike the original one in which the limit is a mixed equilibrium
point, is reminiscent of risk dominance criterion of Carlsson and van Damme
(1993). I also study the conditions for different hierarchy levels that yields
risk dominant outcome for coordination games. That is, I give conditions (first
order stochastic dominance and monotone likelihood ratio order) that yield
the risk dominant outcome of a coordination game as the limit of perturbed
game ´a la Harsanyi (1973).
In the second chapter, I attempt to provide a generalization of mirage equilibrium
for dynamic games in the context of Cournot duopoly in which costs
are private information. The task of extending the definition of mirage equilibrium is a nontrivial issue since it is not clear on which level of finite hierarchies
of beliefs the update takes place. I take a short-cut to tackle this problem and
instead of working on beliefs (probability distributions) directly, I work on the
support of them. Broadly speaking, players update their beliefs by eliminating
the support of ”types” that do not explain the opponents’ behavior. I show that
the limit of this update process converges to a Nash equilibrium of the corresponding
complete information game. I also show that the rate of convergence
is linear.
In the third chapter, I define a new metric to measure the distance between
the partitions of a given finite set. I compare the proposed metric with the ones
in the literature through examples.