Notation for the canonical setting

As we have already mentioned, in what follows, with the exception of the first part of chapter 3 and chapter 4, we will work with a canonical context space that is the set of probability distributions over a set of states and for each context, define a preference relation over the set of alternatives. Thus the canonical model applies to decisions under risk or uncertainty.

Let S :“ ts, t, u, . . .u, 1 ă |S| “ n, denote an abstract set of future states of nature, and ∆”∆pSq the set of probability distributions over S with ele-

ments denoted byp,q, andr. For allpP∆, letA:“ ta, b, c, . . .u, 1ă|A|“m, be the set of alternatives on which the agent’s strict preference relation,ą_p, is defined.8

For a givenp,ą_p is a binary relation over A. The following notation for a family of preference relations

tą_pĂAˆA:pP∆u, tpA,ą_pq:pP∆u, and pA,ą_pqp_P∆,

will refer to the same object: the context preferences or, simply, preferences of the individual agent.

Each strict preference relation is therefore a subset of the collection ordered pairs of elements of A satisfying: at context p, for any alternatives a and b in

A, a is strictly preferred to b if and only if a ą_p b. When no strict preference holds in either direction for some pairpa, bq(that is neitheraą_p bnorb ą_p a), we write a „p b. Note that by definition, „p is reflexive and symmetric, but

not necessarily an equivalence relation as transitivity (a „p b and b „p c to-

gether imply a„p c)may fail to hold.

This notation (for context-free preferences) is used by both Fishburn (1970) and Kreps (1988) in their classic textbooks on decision theory. Our choice of strict as opposed to weak preference as primitive is partly due the appeal of the asymmetry condition discussed in section (1.1.2) and partly due to the fact that a statement of strict preference by an agent is more concrete and readily observable than its weak counterpart.

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Unless otherwise stated, the sets A and S will be understood to be countable. Also, note that the set of alternatives are not state-dependent because they are available to the decision-maker prior to any particular state obtaining.

This approach also allows us to distinguish transitivity of ą_. from that of

„.. The latter is widely recognized as failing to hold in simple examples: con-

sider an agent who strictly prefers no sugar at all to a single sugar cube in her glass of tea, but who is unable to distinguish between two glasses of tea, one of which contains a single granule more than the other. Transitivity of strict preference appears much more natural and mathematically useful than than that of „. which may also capture more general kinds of incomparability as

the following example shows.

Example 1.2. Let A be a collection of points in R2

and let ą. denote a

preference relation over A corresponding to some context for which we sup- press notation. If ą_. coincides with the strict order on R2

, then x ą_. y iff both x1 ą y1 and x2 ą y2 are true (where ą is the standard order on R and

the subscripts on x and y denote the first and second axes).That is to say, c

is strictly preferred to a if and only if it lies strictly to the north-east of a. Clearly, by transitivity of ą on R, ą_. is transitive.

✻ ✲ x2 x1 r c r a r b

Any pair of points that lie to the south-east or north-west of one another are incomparable under this order. Thus in the above diagram,a andb are incom-

parable under ą., as are b and c. In such cases, as no strict preference holds

in either direction, and we may write a „. b „. c. However since c ą. a, we

see that „. fails to be transitive.

Although we do not relax the condition that „. be transitive in the present

thesis, our approach is the natural starting point from which to do so.

Set notation for context space

For any subset E of A, the set of p in ∆ satisfying a ą_p b for every b P E is denoted by BapEq, thus a is (strictly) better than every element of E for all

p P BapEq. Similarly, for any subset E of A, the set of p in ∆ for which b is

strictly preferred to a for all b P E is denoted WapEq, so that a is (strictly)

worse than every element ofE for eachpPWpEq. We will make extensive use of the shorthand Bab for Baptbuq and similarly Wab :“Waptbuq ” Bbptauq “:

Bba. Finally, the set of pP∆ for which a„p b will be called Nab, and the set

Nabc is defined astpP∆ : a„p b„p c„p au.

We make use of the fact that S has n elements to identify ∆ with the n´1 dimensional simplex inRn_{, tpP R}n

` :p1 ` ¨ ¨ ¨ `pn“1u, whereRn<sub>`</sub> is the set

of vectors inRn _{for which every element of the vector is strictly positive, and}

s

Rn

` will denote the closure of this set in Rn, that is

s

Rn

`:“ tpP Rn:pi ě0 for all i“1, . . . , nu.9

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We prefer this notation on the grounds that it is more in line with modern notation used in the mathematical community outside economics. The standard notation in economics beingRn

`` for the strictly positive vectors andR

n

Where useful, we will consider ∆ as a subset of Rsn´1

` , that is we will identify

each probability distribution in ∆ with a point in the set tp P Rsn´1

` : p1 ` ¨ ¨ ¨ `pn´1 ď1u. In appendix (1.B) of this chapter, we provide some essential

results that justify the relationships between these spaces.