Preference logic, conditionals and solution concepts in games
Johan van Benthem, Sieuwert van Otterloo, Olivier Roy

Abstract:
Preference is a basic notion in human behaviour, underlying such
varied phenomena as individual rationality in the philosophy of action
and game theory, obligations in deontic logic (we should aim for the
best of all possible worlds), or collective decisions in social choice
theory. Also, in a more abstract sense, preference orderings are used
in conditional logic or non-monotonic reasoning as a way of arranging
worlds into more or less plausible ones. The field of preference logic
studies formal systems that can express and analyze notions of
preference between various sorts of entities: worlds, actions, or
propositions. The art is of course to design a language that combines
perspicuity and low complexity with reasonable expressive power.  In
this paper, we take a particularly simple approach. As preferences are
binary relations between worlds, they naturally support standard unary
modalities. In particular, our key modality \diamond\phi will just say that
\phi is true in some world which is at least as good as the current one. Of
course, this notion can also be indexed to separate agents.  The
essence of this language is already in [4], but our semantics is more
general, and so are our applications and later language
extensions. Our modal language can express a variety of preference
notions between propositions. Moreover, it can "deconstruct" standard
conditionals, providing an embedding of conditional logic into more
standard modal logics. Next, we take the language to the analysis of
games, where some sort of preference logic is evidently needed. We
show how a qualitative unary preference modality suffices for defining
Nash Equilibrium in strategic games, and also the Backward Induction
solution for finite extensive games. Finally, from a technical
perspective, our treatment adds a new twist. Each application
considered in this paper suggests the need for some additional access
to worlds before the preference modality can unfold its true
power. For this purpose, we use various extras from the modern
literature: the global modality, further hybrid logic operators,
action modalities from propositional dynamic logic, and modalities of
individual and distributed knowledge from epistemic logic. The total
package is still modal, but we can now capture a large variety of new
notions. Finally, our emphasis in this paper is wholly on expressive
power. Axiomatic completeness results for our languages can be found
in the follow-up paper.


Keywords: Preference Logic;  Condititional Logic;  Nash Equilibrium;  Backward Induction