Playing with Information
Jonathan Zvesper
Abstract:
The title of this dissertation is not very informative as to its contents.
We do not look exactly at how information is `played with',
but rather at how information the players have affects the play of the game:
at how they play, with information.
So for example some of the theorems we discuss
are of the form, `if the players have such-and-such information,
then they will make such-and-such choices'.
The contents of this dissertation therefore constitute a series of
contributions to the literature on epistemic game theory. So we study
the connections between beliefs and rational choice in an interactive,
multi-agent setting. We focus, as has been focussed the attention of
epistemic game theorists, only on so-called `non-cooperative' game
theory, i.e.~in which any binding contracts the players can make
between themselves must be explicitly modelled in the game. Indeed,
as much as is possible we try to let each game be the whole story
about the interaction situation, so we generally avoid assuming that
players have exogenous information concerning what other players will
do, based for example on past observation. We call this the
`one-shot' interpretation. It means that the kind of information we
consider is always about the `rationality' of the players, or
information about information of this kind.
In Chapter 1, which also serves to introduce some of the mathematical
models that we use in the dissertation, we add a few minor touches to
a basic but fundamental theorem in epistemic game theory, which
relates the `level' of `mutual belief' to the number of rounds of
iteration of non-optimal strategies. To put a sociological gloss on
that theorem, we could see it as affirming a direct correlation
between in the one hand the extent to which a group of players have
the `same' information and in the other the extent to which those
players' behaviour is co-ordinated by consideration of the preferences
of others. However, you will not find such gloss on the material in
the dissertation, and it's difficult to sum up the issues concisely in
non-specialist terms, though the Introduction does briefly sketch part
of the basic logic of the argument proving the theorem. The `minor
touches' that form our own contribution in Chapter 1 are called there
`generalisations', and one of those is to look at how the logic
extends to the infinitary case, where it turns out that there are some
subtleties that, we argue, call for so-called `neighbourhood', or the
closely-related `topological', models for beliefs, rather than the
`relational' models commonly found in epistemic logic. For those
familiar with formal epistemology: there are hints of a connection
with issues of logical omniscience, as neighbourhood models do not
licence the inference that because you know $\varphi$ and that
$\varphi$ implies $\psi$, then you know $\psi$.
In Chapter 2 we make the distinction, important in logic, between
syntax (language) and semantics (models), and discuss some issues that
arise, like definability. The technical contribution of that Chapter
is to address a foundational question concerning the existence of
belief model that is in a certain sense `complete'.
In Chapter 3 we play with some ideas about dynamics of information,
looking at how epistemic conditions might come about. We introduce
tools from `dynamic epistemic logic', that we adapt to the
neighbourhood model framework that we often use throughout the
dissertation. We also show the importance, for understanding some
game-theoretical predictions, of revisable beliefs: that is, of
modelling situations in which a player might believe something and
later learn that it is not true.
In Chapter 4, we look at a particular game-theoretical situation in
which the players' assumptions can be violated in this way, in which
they can be surprised by the apparent irrationality of a player. So
we turn our attention to games with distinct temporal stages
(so-called `extensive games'). We use tools explained in Chapter 3 in
order to build epistemic models of these situations, in which players
can have beliefs and later find out that they are wrong. (For the
game-theorist: we provide an epistemic foundation, in terms of a
notion of `stable belief in dynamic rationality', for backward
induction.)
Keywords: