Abstract
This thesis consists of three separate essays in Game Theory. Each essay is contained in one chapter and can be read as a standalone piece. In Chapter 1, we extend the Imagined-continuum model
of Kalai and Shmaya (2018) by allowing the number of players to change in each stage of the game.
We introduce this modification to the model of Kalai and Shmaya (2018) by assuming that there
is an infinite population of players and by defining a random variable that, in each stage of the
game, selects from this population a finite subset of players who are allowed to play the game.
Although the number of players is finite, they see the set of players as a continuum. This simplifies the solution to the game but introduces some errors that players make in evaluating random
variables in the game. Kalai and Shmaya (2018) derive a bound on this error, but this bound increases with the time period and so it vanishes at infinity. Our Theorem 1 is a refinement of Kalai
and Shmaya (2018)’s bound since, by making the right assumption on the number of players in
each period, we can make this bound arbitrarily small.
In Chapter 2, we explore the continuity-compactness trade-off in topological spaces and show
how this is related to the existence of a pure strategy Nash equilibrium in normal-form games. We
show that in compact simple games it is always possible to change the topology on the joint strategy space to maintain its compactness while obtaining the continuity of players’ payoff functions.
However, we show that for games that have a discontinuity on the diagonal of the joint strategy
space, this is not enough to derive existence of a pure strategy Nash equilibrium because other
topological properties fail to hold. In particular, if we specify the topology on the joint strategy
space in such a way that makes this space compact and the payoff functions continuous, we find
that: the joint strategy space of this class of games cannot be a locally convex topological vector
space; the topology on this space cannot be the product topology.
In Chapter 3, we study the relationship between expost Nash equilibrium and Bayesian Nash
equilibrium. In particular, we show that in Bayesian games if a strategy vector is an approximate
expost stable Nash equilibrium then this strategy vector is also an approximate Bayesian Nash
equilibrium. This result builds upon the work of Kalai (2004) who has shown a stronger version of
the converse of the above statement for semianonymous large games. In general, the existence of
an approximately expost Nash equilibrium does not necessarily imply the existence of a Bayesian
Nash equilibrium. This means that an equivalence result between approximately expost Nash
and Bayesian Nash equilibrium cannot be established. In the future, we intend to find additional
conditions that are needed to obtain existence of a Bayesian Nash equilibrium.