16 research outputs found
Interpreting, axiomatising and representing coherent choice functions in terms of desirability
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that appear in imprecise-probabilistic decision making. We provide these choice functions with a clear interpretation in terms of desirability, use this interpretation to derive a set of basic coherence axioms, and show that this notion of coherence leads to a representation in terms of sets of strict preference orders. By imposing additional properties such as totality, the mixing property and Archimedeanity, we obtain representation in terms of sets of strict total orders, lexicographic probability systems, coherent lower previsions or linear previsions
Rationality, decisions and large worlds
Taking Savage's (1954) subjective expected utility theory as a starting point, this thesis distinguishes three types of uncertainty which are incompatible with Savage's theory for small worlds: ambiguity, option uncertainty and state space
uncertainty. Under ambiguity agents cannot form a unique and additive probability function over the state space. Option uncertainty exists when agents cannot assign unique
consequences to every state. Finally, state space uncertainty arises when the state space the agent constructs is not exhaustive, such that unforeseen contingencies
can occur. Chapter 2 explains Savage's notions of small and large worlds, and shows that ambiguity, option and state space uncertainty are incompatible with the small
world representation. The chapter examines whether it is possible to reduce these types of uncertainty to one another. Chapter 3 suggests a definition of objective ambiguity by extending Savage's framework to include an exogenous likelihood ranking over events. The definition
allows for a precise distinction between ambiguity and ambiguity attitude. The chapter argues that under objective ambiguity, ambiguity aversion is normatively permissible.
Chapter 4 gives a model of option uncertainty. Using the two weak assumptions that the status quo is not uncertain, and that agents are option uncertainty averse, we derive status quo bias, the empirical tendency for agents to choose the status quo over other available alternatives. The model can be seen as rationalising status quo bias.
Chapter 5 gives an axiomatic characterisation and corresponding representation theorem for the priority heuristic, a heuristic which predicts binary decisions be-
tween lotteries particularly well. The chapter analyses the normative implications of this descriptive model.
Chapter 6 defends the pluralist view of decision theory this thesis assumes. The chapter discusses possible applications of the types of uncertainty defined in the thesis, and concludes
Coherent and Archimedean choice in general Banach spaces
I introduce and study a new notion of Archimedeanity for binary and
non-binary choice between options that live in an abstract Banach space,
through a very general class of choice models, called sets of desirable option
sets. In order to be able to bring an important diversity of contexts into the
fold, amongst which choice between horse lottery options, I pay special
attention to the case where these linear spaces don't include all `constant'
options.I consider the frameworks of conservative inference associated with
Archimedean (and coherent) choice models, and also pay quite a lot of attention
to representation of general (non-binary) choice models in terms of the
simpler, binary ones.The representation theorems proved here provide an
axiomatic characterisation for, amongst many other choice methods, Levi's
E-admissibility and Walley-Sen maximality.Comment: 34 pages, 7 figure
A theory of desirable things
Inspired by the theory of desirable gambles that is used to model uncertainty
in the field of imprecise probabilities, I present a theory of desirable
things. Its aim is to model a subject's beliefs about which things are
desirable. What the things are is not important, nor is what it means for them
to be desirable. It can be applied to gambles, calling them desirable if a
subject accepts them, but it can just as well be applied to pizzas, calling
them desirable if my friend Arthur likes to eat them. Other useful examples of
things one might apply this theory to are propositions, horse lotteries, or
preferences between any of the above. Regardless of the particular things that
are considered, inference rules are imposed by means of an abstract closure
operator, and models that adhere to these rules are called coherent. I consider
two types of models, each of which can capture a subject's beliefs about which
things are desirable: sets of desirable things and sets of desirable sets of
things. A crucial result is that the latter type can be represented by a set of
the former
Interpreting, axiomatising and representing coherent choice functions in terms of desirability
Choice functions constitute a simple, direct and very general mathematical
framework for modelling choice under uncertainty. In particular, they are able
to represent the set-valued choices that appear in imprecise-probabilistic
decision making. We provide these choice functions with a clear interpretation
in terms of desirability, use this interpretation to derive a set of basic
coherence axioms, and show that this notion of coherence leads to a
representation in terms of sets of strict preference orders. By imposing
additional properties such as totality, the mixing property and Archimedeanity,
we obtain representation in terms of sets of strict total orders, lexicographic
probability systems, coherent lower previsions or linear previsions.Comment: arXiv admin note: text overlap with arXiv:1806.0104
Coherent choice functions without Archimedeanity
We study whether it is possible to generalise Seidenfeld et al.’s representation result for coherent choice functions in terms of sets of probability/utility pairs when we let go of Archimedeanity. We show that the convexity property is necessary but not sufficient for a choice function to be an infimum of a class of lexicographic ones. For the special case of two-dimensional option spaces, we determine the necessary and sufficient conditions by weakening the Archimedean axiom