Econ 710 - Lec1 Preference

Econ 710Lecture Slides 1

Decision Theory Foundations

Fei Li

University of North Carolina, Chapel Hill

August 26, 2013

Fei Li (UNC) Lecture 1 August 26, 2013 1 / 32

Introduction

Decision theory (DT) is the study of decision making of a decisionmaker (DM).

DT is the basis of the predictions, restrictions, and welfareimplications obtained in economic study.

In general, a DM can be

a single person (consumer)an organization (government, �rm, NGO).

Normally, we assume the choices made by the DM in di¤erentsituations are coherent.

This coherency may be a time-and-environment in-varying decisionrule.

If we know the decision rule, we can understand (and predict) theDM�s choice in a di¤erent environment.


Introduction

Much of DT is concerned with formulating a notion of �rational�decision making, and �nding conditions under which observed choicesare consistent with rationality.

Two and a half approaches to DT:

Choice behavior as the primitivePreference (relation) as the primitive(derived approach) Utility maximization as the primitive


Alternatives

All approaches start with a set of alternatives, X .

Each x 2 X is supposed to be a complete description of everythingthe DM might �care about�.

Example

X = {economics, political science}

X = {(x1, ...xK ) 2 RK+ : xk = consumption of goods k 2 K}

X = {(x1, x2) 2 R2+ : x1 = $ today, x2 = $ tmr}

X = {(x1, x2...) 2 R∞+ : xt = consumption in period t 2 N}


Preference-Based Approach

The primitive datum is a binary (�preference�) relation � on X . Ifx � y , we say �x is weakly preferred to y�or �x is at least as goodas y�.

� is meant to be a psychological object, a description of how the DMfeels about one alternative versus another.

Choice is derived from the preference relation. When the feasible setis B � X , the set of possibly chosen alternatives is the set of mostpreferred feasible alternatives:

C � (B,�) := fx 2 B : x � y for all y 2 Bg .

Note: without making further assumption, C � (B,�) may be emptyor contain multiple alternatives.

The actual choice is one alternative in C � (B,�) when it isnon-empty.


Preference-Feasibility Separation

An important aspect of the formulation C � (B,�) is that the DM�spreferences do not depend on the set of feasible alternatives.In reality, it may not be true.

ExampleMenu A: {steak, chicken},DM chooses chicken.

Menu B: {steak, chicken, frog legs},DM chooses steak.


Preference-Feasibility Separation

Example (by Tversky and Kahneman)Flu outbreak in town of 600 people. The DM must choose one of twotreatment strategies:

Treatment A: 400 will die.Treatment B: w.p. 1/3 nobody dies; w.p. 2/3 they all die.

the DM chooses B.

Or we can ask a sophisticated politician to phase alternativesdi¤erently:Treatment A�: 200 people will be saved.Treatment B�: w.p. 1/3 all are saved; w.p. 2/3 non is saved.

the DM chooses A�.


Rationality in the Preference-Based Approach

In the preference-based approach, the DM�s �rationality� is embodied bytwo axioms about the preference relation �, which are

1 completeness (for all x , y 2 X , x � y or y � x), and2 transitiveness (for all x , y , z 2 X , if x � y and y � z , then x � z).

De�nitionIf a preference relation � is complete and transitive, we say that it is arational preference relation.



TheoremSuppose � is rational. Then, for every �nite non-empty set B,C � (B;�) 6= ?.

Proof.We prove the statement by mathematical induction. First, supposeB = fxg is a singleton. Since x � x (completeness), C � (B;�) = x ,which is non-empty. Second, �x n � 1 and suppose that whenB = fx1, x2, ...xng, C � (B;�) 6= ?. Also suppose xm 2 C � (B;�). Nowlet A = B [ fxn+1g, there are two possibilities:

1 xm � xn+1, then xm 2 C � (A;�), or2 xn+1 � xm but not xm � xn+1, then xn+1 2 C � (A;�).



When B is in�nite, C � (B;�) might be empty.

Example

Suppose B =�xn jxn = 1� 1

n ,where n 2 Nand the DM feels that

bigger is better.

Example

Suppose B = fx 2 [0, 1)g and the DM feels that bigger is better.

In both examples, C � (B;�) is empty.


Implications of Rational Preference

Given a binary relation � on X , de�ne two others:

strict preference: x � y i¤ x � y and not y � x ,indi¤erence: x � y i¤ x � y and y � x .

LemmaIf � is rational, then

1 both � and � are transitive.2 for any x , y , z 2 X ,

x � y � z =) x � z , andx � y � z =) x � z .

3 � is irre�exive (x � x never holds) but � is re�exive (x � x for allx).


Implications of Rational Preference

We prove the �rst statement only: � is transitive.

Proof.Suppose x � y and y � z , we want to show that x � z . By the de�nitionof � and transitivity of �, we know that x � y � z . We only need toshow � not z � x�part.Suppose z � x . By transitivity of �, we must have z � y , which is acontradiction!


How Plausible Are Rational Preference Relations?

One view: who cares? DT is the theory of perfectly rational decisionmaking, meant to be a benchmark towards which imperfectly rationalreal-world people should strive. (normative view of DT)

Of those who do care, many view completeness as merely a technicalrequirement.

Transitivity, on the other hand, is violated in at least two plausiblescenarios.


A Plausible Scenario for Intransitivity

Imperceptible Di¤erences

Consider a DM who likes sugar in her co¤ee.

Let xi = a cup of co¤ee with i grains of sugar.

Since the DM cannot taste the di¤erent that one grain of sugarmakes xi � xi+1,But x50 � x1 !

This means, in some situations, the DM may be neither � nor � istransitive.


Another Plausible Scenario for Intransitivity

Aggregation.

Note that a DM could be an organization containing multiplepersons. In such a case, the DM�s preference should re�ect itsmembers�preferences.Family F = fDad ,Mom,Childg . Each family member i 2 F has arational preference relation �i . The family�s prevalences are based onmajority rule:

x � y i¤ ji : x �i y j � 2.Suppose the family members�preferences are:

x �D y �D z , y �M z �M x , and z �C x �C y .

The family preference is:

x � y , y � z , and z � x !

We have the classic Condorcet paradox cyclical preference pro�le.Then � is intransitive.


Utility Function

If transitivity is a very strong assumption, why not dump it?

De�nitionA preference relation � on X is represented by a utility functionu : X ! R if

x � y , u(x) � u(y).

If u represents �,

C �(B,�) = fx jx 2 argmaxy2Bu(y)g

Two immediate results:

1 Any representable � is rational.2 If u represents � and f : R ! R is strictly increasing, thenV = f � u also represents �.


Utility Function

TheoremA preference can be represented by a utility function only if it is rational.

Proof.Suppose � on X is represented by u.

For any x , y 2 X , we have either u(x) � u(y) or u(x) � u(y), so �must be complete.

For x , y , z 2 X , suppose x � y and y � z , we must have

u(x) � u(y) � u(z)

So x � y .

Oops! To have a utility function representation, we have to buy atransitive preference.


Utility Function

Utility functions are easier to work with than preferences, so having autility representation for preferences is convenient.

The question is, how restrictive is it to assume that the DM�spreferences are representable?

Unfortunately, it is not true that any rational preference can berepresented by a utility function. Certain conditions are necessary.

TheoremA rational preference on a �nite alternative set can be represented by autility function.


Utility Function

Proof.Let n = jX j. When n = 0, trivial.Suppose n > 0. Construct fXig as follows.

1 X0 = X ,2 X1 = X0 � C �(X0,�)....3 Xi+1 = Xi � C �(Xi ,�),

Xm+1 = ? or Xm = C �(Xm ;�) for some m � n.Let u(x) = m� i if x 2 C �(Xi ,�).Suppose x � y , and x 2 C �(Xi ;�), then u(x) = m� i , y 2 Xi , andu(y) � m� i = u(x).Suppose u(x) � u(y). Straightforwardly, x � y .


Utility Function

What if the alternative set is not �nite?

countable set: �ne. See (Kreps, 2012)uncountable set: in general, no! More conditions are needed.

Example (Lexicographic preferences)

X = [0, 1]2, (x1, y1) � (x2, y2) whenever1 either x1 > x2, or2 x1 = x2 and y1 > y2.

Proof.There is a family of intervals f[u(x , 0), u(x , 1)]gx2[0,1]. For each interval,pick a rational number ρ(x) s.t. u(x , 0) < ρ(x) < u(x , 1). For x > x 0,u (x , 0) > u (x 0, 1), so ρ (x) is strictly increasing, and a one-to-one mapbetween a subset of rational numbers (countable)and [0, 1] (uncountable)is constructed. Contradiction!


Choice-Based Approach

The primitive datum in this approach is a choice structure,(B,C ),where

B � 2X /? is the set of possible feasible sets (nonempty),C : B ! 2X is the choice correspondence,

and by de�nition,

? 6= C (B) � B for any B 2 B.

Example

X = fx , y , zg and let B = ffx , yg, fy , zg, fx , zgg: all size-two sets. Onepossible choice structure is (B,C1), where C1(fx , yg) = x ,C1(fy , zg) = y and C1(fx , zg) = x .

This approach is based on observable, measurable objects (feasiblesets, choice behavior), as opposed to unobservable psychologicalobjects (preference).


Rationality in the Choice-Based Approach

In this approach, the DM�s rationality assumption is embodied by thefollowing axiom.

De�nition (WARP)

A choice structure (B,C ) satis�es the Weak Axiom of RevealedPreference (WARP) i¤

for all sets A,B 2 B, x 2 C (A) and y 2 A,if y 2 C (B) and x 2 B, then x 2 C (B).

WARP simply means that if x is chosen when y is available, thenwhenever (in a di¤erent situation, with a di¤erent feasible set) y is chosenwhen x is feasible, x is also chosen.

Example

X = fx , y , zg and let B = ffx , yg, fx , y , zg, fx , zgg. The choicestructure is (B,C2), where C2(fx , yg) = x , y 2 C2(fx , y , zg). Ifx 2 C2(fx , y , zg), the choice structure satis�es WARP; otherwise, no.


Relationship between rational preference and WARP

The preference-based approach is more intuitive and it allows us touse utility-maximaztion analysis.

However, the choice-based approach is more rooted on observablebehavior.

Are they related? Or1 If a DM has a rational preference �, do his decision rule C �(B;�)satis�es WARP?

2 If a DM�s choice satis�es WARP, is there any rational preference that isconsistent with these choices?



TheoremSuppose � is rational. Then the choice structure generated by �,C �(.,�) satis�es WARP.

Proof.Suppose we have x , y 2 B and x 2 C �(B,�). So x � y . Now consideranother feasible set B 0 such that x , y 2 B 0. Suppose y 2 C �(B 0,�). Sofor any z 2 B 0, y � z . Since x � y and � is transitive,x 2 C �(B 0,�).



What about the other direction?

De�nitionA choice structure (B,C ) is rationalized by a rational preference � i¤

C (B) = C �(B,�) for all B 2 B.

In other words,

� generates the choice structure(B,C ),if a DM�s choice is consistent with (B,C ), we can interpret his choiceas if he has preference � and makes his choice following thepreference.

We want to �nd conditions under which (B,C ) is rationalized by arational preference. (complete and transitive) Is WARP itself enough?



Example (Non-existence)

X = x , y , z and B = ffx , yg , fy , zg , fx , zgg, C (fx , yg) = fxg,C (fy , zg) = fyg and C (fx , zg) = fzg .

The choice structure satis�es WARP, but it cannot be rationalized.

we must have x � y , y � z and z � x , which violates transitivity!

Key:

1 fx , y , zg is not feasible,2 WARP only imposes pairwise restrictions on C (�) ,3 so it lost its bite.



Example (Non-uniqueness)

X = x , y , z and B = ffx , yg, fx , zg, fx , y , zg, g.C (fx , yg) = C (fx , zg) = Cfx , y , zg = fxg

The choice structure satis�es WARP,

we must have x � y , x � z , but don�t know his preference over yand z .

Key:

1 fy , zg is not feasible,2 so WARP does not impose restriction on the choice (preference) overthem,

3 the preference can be arbitrary or non-comparable,4 we do not have the completeness.


The Revealed Preference Relation

The natural candidate for a rationalizing preference is

De�nitionGiven (B,C ), the corresponding revealed preference relation, ��, isde�ned by x �� y for all x 2 X , and for all x 6= y by

x �� y i¤ for some B 2 B, x 2 C (B) and y 2 B.

We say x is revealed at least as good as y . Note that

the condition requires some B not all or x , y it self,

the de�nition does not say that (B,C ) satis�es WARP,�� may not be rational,�� is complete if B contains all size-two subsets.


The Revealed Preference Theorem

TheoremSuppose (B,C ) is a choice structure such that

1 WARP is satis�ed,2 B includes all size-two subsets,3 B includes all size-three subsets,

then

1 �� is rational, and2 it is the unique rational preference that rationalizes (B,C ).


Proof: �� is rational

Since B contains all size-two subsets, �� is complete. We show thetransitivity here.

Suppose that for some x , y , z , we have x �� y and y �� z . We onlyneed to show x 2 C (fx , y , zg) so that x �� y , z .Since C (fx , y , zg is non-empty, there three cases:

1 x 2 C (fx , y , zg), done.2 y 2 C (fx , y , zg). x �� y ) 9B 2 B s.t. x , y 2 B andx 2 C (B) WARP) x 2 C (fx , y , zg).

3 z 2 C (fx , y , zg). y �� z ) 9B 0 2 B s.t. y , z 2 B 0 andy 2 C (B 0) WARP) y 2 C (fx , y , zg). The argument in case 2 impliesx 2 C (fx , y , zg).

Hence, x �� z .


Proof: �� rationalizes (B,C)

We want to show that C (B) = C �(B,��). First, we show that

C (B) � C �(B,��) for all B 2 B.

By the de�nition of ��, for any x 2 C (B), we have x �� y for anyy 2 B. So x 2 C � (B,��).


Proof: �� rationalizes (B,C)

It remains to show the reverse inclusion.

Let B 2 B and x 2 C �(B,��), then x �� y for all y 2 BPick y � 2 C (B).Since x �� y �, there exists a By � 2 B s.t. x 2 C (By �) and y � 2 By � .(the de�nition of revealed preference)

Since y � 2 C (B) and x 2 B, WARP implies that x 2 C (B). (thede�nition of revealed preference again)

So C �(B,��) � C (B) for all B 2 B.


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Econ 710 - Lec1 Preference

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