Choice Function
We consider an agent’s “behavior” as a hypothetical response to the following questionnaire, one for each A ⊆ X:
Q(A) Assume you must choose from a set of alternatives A. Which alternative do you choose?
◮ A choice function C assigns to each set A ⊆ X a unique element of A with the interpretation that C(A) is the chosen element from the set A.
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Rm Here are a couple of remarks on choice functions:✆
1. We assume that the agent selects a unique element in A for every question Q(A).
2. The choice function C does not need to be observable. 3. The agent behaving in accordance with C will choose C(A) if
she has to make a choice from a set A.
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Rational Choice
We (Economics) assume that when the agent has in mind a preference relation % on X, then given any choice problem Q(A) for A ⊆ X, she chooses an element in A which is “% optimal.”
Def An induced choice function C% is the function that assigns every nonempty set A ⊆ X the %-best element of A.
Def A choice function C can be rationalized if there is a preference relation % on X so that C = C%, i.e., C(A) = C%(A) for any A in the domain of C.
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Q Under what conditions any choice functions can be presented✆
“as if” derived from some preference relation?
Def Choice function C satisfies (Sen’s) condition α if for any A⊂ B, C(B) ∈ A implies C(A) = C(B).
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Choice ⇐⇒ Preference in General Decision Problem
Thm Assume C is a choice function with a domain containing at least all subsets of X of size 2 or 3. If C satisfies condition α, then there is a preference relation % on X so that C = C%.
Proof Define % by x % y if x = C({x, y}). Let us first show that
%satisfies completeness and transitivity.
Completeness: Follows from that C = ({x, y}) is well-defined. Transitivity: If x % y and y % z, then by definition of % we have C({x, y}) = x and C({y, z}) = y. If C({x, z}) = z, then, by condition α, C({x, y, z}) 6= x. Similarly, by C({x, y}) = x and condition α, C({x, y, z}) 6= y, and by C({y, z}) = y and condition α, C({x, y, z}) 6= z. A contradiction to C({x, y, z}) ∈ {x, y, z}. Next we show that C(A) = C%(A) for all A ⊆ X. Suppose on contrary C(A) 6= C%(A). That is, C(A) = x and
C%(A) = y(6= x). By definition of % and y % x, this means C({x, y}) = y, contradicting condition α.
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Satisficing Procedure
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Q Is there any interesting and general choice procedure that is✆ different from “rational man” yet can be described as if the decision maker maximizes a preference relation?
Def Satisficing (by Herbert Simon)
Let v : X → R be a valuation of the elements in X, and let v∗ ∈ R be a threshold of satisfaction. Let O be an ordering of the alternatives in X.
Given a set A(⊂ X), the decision maker arranges the elements of this set in a list L(A, O) according to the ordering O, then chooses the first element in L(A, O) that has a v-value at least as large as v∗. If there is no such element in A, she chooses the last element.
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Q Is this procedure rationalized by some preference relation?✆
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A Yes, since it satisfies condition α! (verify it by yourself)✆
→ See the discussion in lecture 3 of Rubinstein.
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