Real Analysis with Economic Applications - Chapter H pot

The Expected Utility Theorem states that any complete preference relation onLX that satisfies the Independence and Continuity Axioms Section F.3.1 admits anexpected utility representati

Chapter H

Economic Applications

Even the limited extent of convex analysis we covered in Chapter G endows one withsurprisingly powerful methods Unfortunately, in practice, it is not always easy torecognize the situations in which these methods are applicable To get a feeling for inwhich sort of economic models convex analysis may turn out to provide the right mode

of attack, one surely needs a lot of practice Our objective in this chapter is thus

to present a smorgasbord of economic applications that illustrate the multifarioususe of convex analysis in general, and the basic separation-by-hyperplane and linear-extension arguments in particular

In our first application we revisit expected utility theory, but this time usingpreferences that are potentially incomplete Our objective is to extend both the clas-sical and the Anscombe-Aumann Expected Utility Theorems (Section F.3) into therealm of incomplete preferences, and introduce the recently popular multi-prior de-cision making models We then turn to welfare economics In particular, we provethe Second Welfare Theorem, obtain a useful characterization of Pareto optima inpure distribution problems, and talk about Harsanyi’s Utilitarianism Theorem As

an application to information theory, we provide a simple proof of the celebratedBlackwell’s Theorem on comparing the value of information services, and as an ap-plication to financial economics, we provide various formulations of the No-ArbitrageTheorem Finally, in the context of cooperative game theory, we characterize theNash bargaining solution, and examine some basic applications to coalitional gameswithout side payments While the contents of these applications are fairly diverse(and hence they can be read independently of each other), the methods with whichthey are studied here all stem from basic convex analysis

This section continues the investigation of expected utility theory we started in tion F.3 We adopt here the notation and definitions introduced in that section, so itmay be a good idea to have a quick review of Section F.3 before commencing to theanalysis provided below

Let X be a nonempty finite set, which is interpreted as a set of (monetary or tary) prizes/alternatives, and recall that a lottery (or a probability distribution)

nonmone-on X is a map p ∈ RX

+ such that S

x∈Xp(x) = 1 As in Section F.3, we denotethe set of all lotteries on X by LX, which is a compact and convex subset of RX

The Expected Utility Theorem states that any complete preference relation  on

LX that satisfies the Independence and Continuity Axioms (Section F.3.1) admits anexpected utility representation That is, for any such preference relation there is autility function u ∈ RX such that

for any p, q ∈ LX.1 Our goal here is to extend this result to the realm of incompletepreferences The discussion presented in Section B.4, and a swift comparison ofPropositions B.9 and B.10 suggest that one’s objective in this regard should be toobtain a multi-utility analogue of this theorem And indeed, we have:

The Expected Multi-Utility Theorem (Dubra-Maccheroni-Ok) Let X be anynonempty finite set, and  a preference relation on LX Then  satisfies the Inde-pendence and Continuity Axioms if, and only if, there exists a nonempty set U ⊆ RX

(of utility functions) such that, for any p, q ∈ LX,

p q if and only if Ep(u)≥ Eq(u) for all u ∈ U (2)

This result shows that one can think of an agent whose (possibly incomplete)preferences over lotteries inLX satisfy the Independence and Continuity Axioms “asif” this agent distinguishes between the prizes in X with respect to multiple objectives(each objective being captured by one u ∈ U) This agent prefers lottery p over q ifthe expected value of each of her objectives with respect to p is greater than thosewith respect to q If such a domination does not take place, that is, p yields a strictlyhigher expectation with respect to some objective, and q with respect to some otherobjective, then the agent remains indecisive between p and q (and settles her choiceproblem by means we do not model here)

Before we move on to the proof of the Expected Multi-Utility Theorem, let usobserve that the classical Expected Utility Theorem is an easy consequence of thisresult All we need is the following elementary observation

Lemma 1 Let n ∈ N, and take any a, u ∈ Rn such that a = 0, Sn

ai = 0, and

u1 >· · · > un.If Sn

aiui ≥ 0, then we must have ai > 0 and aj < 0 for some i < j

Proof We must have n ≥ 2 under the hypotheses of the assertion Let αi :=

ui− ui+1 for each i = 1, , n − 1 We have

So, by the hypothesesSn

ai ≥SK

Another Proof for the Expected Utility Theorem Let X be a nonempty finite setand be a complete preorder on LX that satisfies the Independence and ContinuityAxioms Let us first assume that there is no indifference between the degeneratelotteries, that is, either δx δy or δy δx for any distinct x, y ∈ X

By the Expected Multi-Utility Theorem, there exists a nonempty set U ⊆ RX

such that (2) holds for any p, q ∈ LX Define X := {(x, y) ∈ X2 : δx δy}, andfor any (x, y) ∈ X , let u(x,y) be an arbitrary member of U with u(x,y)(x) > u(x,y)(y).(That there is such a u(x,y) follows from (2).) Define u :=S

(x,y)∈Xu(x,y), and noticethat δz δw iff u(z) > u(w), for any z, w ∈ X (Why?) In fact, u is a von Neumann-Morgenstern utility function for, as we show next

Take any p, q ∈ LX Note first that p  q implies Ep(u) ≥ Eq(u) by (2), so giventhat  is complete, if we can show that p q implies Ep(u) > Eq(u), it will followthat (1) holds To derive a contradiction, suppose p q but Ep(u) = Eq(u).Then, byLemma 1, we can find two prizes x and y in X such that δx δy, p(x) > q(x) andp(y) < q(y) (Clearly, q(y) > 0.) For any 0 < ε ≤ q(y), define the lottery rε∈ LX as

rε(x) := q(x) + ε, rε(y) := q(y)− ε and rε(z) := q(z) for all z ∈ X\{x, y}.Since δx δy we have u(x) > u(y), so Ep(u) = Eq(u) < Er ε(u) for any 0 < ε ≤ q(y).Since  is complete, (2) then implies that rε  p for all 0 < ε ≤ q(y) But r1

m → q,

so this contradicts the Continuity Axiom, and we are done

It remains to relax the assumption that there is no indifference between any twodegenerate lotteries on X Let Y be a ⊇-maximal subset of X such that δx is notindifferent to δy for any distinct x, y ∈ Y For any p ∈ LY, define p ∈ LX with

p|Y = p and p (x) := 0 for all x ∈ X\Y Define  on LY by p  q iff p  q Bywhat we have established above, there is a u ∈ RY such that p  q iff Ep(u)≥ Eq(u),for any p, q ∈ LY.We extend u to X in the obvious way by letting, for any x ∈ X\Y,u(x) := u(yx)where yx is any element of Y with δx ∼ δy x.It is easy to show (by usingthe Independence Axiom) that u ∈ RX is a von Neumann-Morgenstern utility for

The rest of the present subsection is devoted to proving the “only if” part ofthe Expected Multi-Utility Theorem (The “if” part is straightforward.) The main

argument is based on the external characterization of closed and convex sets in aEuclidean space, and is contained in the following result which is a bit more generalthan we need at present.

Proposition 1 Let Z be any nonempty finite set, S ⊆ RZ, and Y := span(S).Assume that S is a compact and convex set with al-intY(S) =∅, and  is a continuousaffine preference relation on S Then, there exists a nonempty L ⊆ L(RZ

, R) suchthat

s  t if and only if L(s) ≥ L(t) for all L ∈ L, (4)for any s, t ∈ S If =∅, then each L ∈ L can be taken here to be nonzero

This is more than we need to establish the Expected Multi-Utility Theorem Notefirst that if in that theorem was degenerate in the sense that it declared all lotteriesindifferent to each other, then we would be done by using a constant utility function.Let us then assume that  is not degenerate Then, applying Proposition 1 with Xand LX playing the roles of Z and S, respectively, we find a nonempty subset L ofnonzero linear functionals on RZ

such that, for any p, q ∈ LX,

p q iff L(p) ≥ L(q) for all L ∈ L

But by Example F.6, for each L ∈ L there exists a uL

∈ RX

\{0} such that L(σ) =S

x∈Xσ(x)uL(x) = Eσ(uL) for all σ ∈ RX

Letting U := {uL : L ∈ L}, therefore,completes the proof of the Expected Multi-Utility Theorem

It remains to prove Proposition 1 This is exactly where convex analysis turnsout to be the essential tool of analysis

Proof of Proposition 1 Define A := {s − t : s, t ∈ S and s  t}, and check that

A is a convex subset of RZ We define next

C := V

{λA : λ ≥ 0}

which is easily verified to be a convex cone in RZ.2 The following claim, which is arestatement of Lemma F.2, shows why this cone is important for us

Claim 1 For any s, t ∈ S, we have s  t iff s − t ∈ C

Now comes the hardest step in the proof

Claim 2 C is a closed subset of RZ.3

2 C is the positive cone of a preordered linear space Which space is this?

3 Reminder R Z is just the Euclidean space R |Z| , thanks to the finiteness of Z For instance, for any s, t ∈ R Z and ε > 0, the distance between s and t is d2(s, t) = S

z ∈Z |s(z) − t(z)| 2  1

, and the ε-neighborhood of s ∈ S in Y is N ε,Y (s) = {r ∈ Y : d 2 (s, r) < ε}.

The plan of attack is forming! Note first that if = ∅, then choosing L asconsisting only of the zero functional completes the proof Assume then that =∅,which ensures that C is a proper subset of RZ (Why?) Thus, since we now knowthat C is a closed convex cone in RZ, we may apply Corollary G.6 to conclude that

C must be the intersection of the closed half spaces which contain it, and which aredefined by its supporting hyperplanes (Where did the second part come from?) Let

L denote the set of all nonzero linear functionals on RZ that correspond to thesehyperplanes (Corollary F.4) Observe that if L ∈ L, then there must exist a realnumber α such that {r ∈ RZ : L(r) = α} supports C Since this hyperplane containssome point of C, we have L(σ) ≥ α for all σ ∈ C and L(σ∗) = α for some σ∗ ∈ C.But, C is a cone, so α

2 = 1

2L(σ∗) = L(1

2σ∗) ≥ α and 2α = 2L(σ∗) = L(2σ∗) ≥ α,which is possible only if α = 0 Consequently, we have σ ∈ C iff L(σ) ≥ 0 for all

L∈ L By Claim 1, therefore, we have (4) for any s, t ∈ S

It remains to prove Claim 2, which requires some care Here goes the argument.4

Take any (λm)∈ R∞

+, and (sm), (tm)∈ S∞ such that

σ := lim

m→∞λm(sm− tm)∈ Y and sm  tm for all m = 1, 2,

We wish to show that σ ∈ C Of course, if sm = tm for infinitely many m, then wewould trivially have σ ∈ C, so it is without loss of generality to let sm= tm for eachm

Now pick any s∗ ∈ al-intY(S) Since S is convex, al-intY(S)equals the interior of

S in Y, so there exists an ε > 0 such that Nε,Y(s∗) ⊆ S.5 Take any 0 < δ < ε, anddefine

It is easily verified that d2(rm, s∗) = δ < ε, so rm

∈ Nε,Y(s∗) for each m Since

Nε,Y(s∗)⊆ S, therefore, we have rm

∈ S for each m Moreover, rm

− s∗ ∈ C, so byClaim 1, rm  s∗ It follows that rm ∈ T for each m But since λm(sm − tm) → σ,

We now know that there exist subsequences of (γm)and (rm)that converge in R+

and T, say to γ and r, respectively (Why?) Since (λm(sm

− tm)) converges to σ, (5)

Exercise 1.H (Throughout this exercise we use the notation adopted in Proposition 1 and its proof.) We say thatis weakly continuous if{α ∈ [0, 1] : αs+(1−α)t 

αs + (1− α)t }is a closed set for anys, s , t, t ∈ S.In the statement of Proposition

1, we may replace the word “continuous” with “weakly continuous.” To prove this, all you need is to verify that the coneC defined in the proof of Proposition 1 remains closed in RZ with weak continuity Assume that =∅(otherwise Proposition 1 is vacuous), and proceed as follows.

(a) Show thats∗− t∗ ∈ al-intY(A)for somes∗, t∗ ∈ al-intY(S)

(b) Prove: Ifσ ∈ al-cl(C),then σ = λ(s− t) for someλ≥ 0and s, t∈ S

(c) Using the Claim proved in Example G.10, conclude that(1− λ)(s∗− t∗) + λ(s−t)∈ C for all 0≤ λ < 1and s, t∈ S Finally, use this fact and weak continuity to show that C is algebraically closed By Observation G.3, C is thus a closed subset

ofRZ

Exercise 2 (Aumann) Let X be any nonempty finite set Prove: If the preference relationonLX satisfies the Independence and Continuity Axioms, then there exists

au∈ RX such that, for anyp, q∈ LX,

p q implies Ep(u) > Eq(u) and p∼ q implies Ep(u) = Eq(u)

Exercise 3 Let X be a nonempty finite set Show that if the subsets U and V of

RX represent a preference relationas in the Expected Multi-Utility Theorem, then

V must belong to the closure of the convex cone generated by U and all constant functions onX

Let us now turn to expected utility theory under uncertainty, and recall the Aumann framework (Section F.3.2) Here Ω stands for a nonempty finite set of states,and X that of prizes A horse race lottery is any map from Ω intoLX — we denote theset of all horse race lotteries by HΩ,X.6 In the Anscombe-Aumann setup, the prefer-ence relation of an individual is defined on HΩ,X If this preference relation is com-plete and it satisfies the Independence, Continuity, No-Triviality, and MonotonicityAxioms∗ (Section F.3.2), then there exist a utility function u ∈ RX and a probabilitydistribution μ ∈ LΩ such that

6 Reminder For any h ∈ H Ω,X and ω ∈ Ω, we write h ω for h(ω), that is, hω(x) is the probability

of getting prize x in state ω.

for any f, g ∈ HΩ,X.

We now ask the following question: How would this result modify if  was notknown to be complete? A natural conjecture in this regard is that  would thenadmit an expected multi-utility representation with multiple prior beliefs, that is,there would exist a nonempty set U ⊆ RX of utility functions and a nonempty set

M ⊆ LΩ of prior beliefs such that f  g iff

S

ω∈Ω

μ(ω)Ef ω(u)≥ S

ω∈Ω

μ(ω)Eg ω(u) for all (μ, u) ∈ M × U

for any f, g ∈ HΩ,X Unfortunately, to the best of the knowledge of this author,whether this conjecture is true or not is not known at present What is known isthat if the incompleteness of  stems only from one’s inability to compare the horserace lotteries that differ across states, then the conjecture is true (with U being asingleton)

To make things precise, let us recall that a preference relation  on HΩ,X induces

a preference relation∗ on LX in the following manner:

p∗ q if and only if p  q ,where p stands for the constant horse race lottery that equals p at every state, andsimilarly for q Obviously, if  is complete, so is ∗ Converse is, however, false.Indeed, the property that  is complete enough to ensure the completeness of ∗ ismuch weaker than assuming outright that  is complete It is this former propertythat the Knightian uncertainty theory is built on

The Partial Completeness Axiom∗ The (induced) preference relation ∗ iscomplete, and ∗=∅

When combined with the Monotonicity Axiom∗, this property makes sure that,given any h ∈ HΩ,X,

(p, h−ω) (q, h−ω) if and only if (p, h−ω) (q, h−ω ) (7)for any lotteries p, q ∈ LX and any two states ω, ω ∈ Ω.7 (Why?) Thus, if an in-dividual cannot compare two horse race lotteries f and g, then this is because theydiffer in at least two states For a theory that wishes to “blame” one’s indecisive-ness on uncertainty, and not on risk, the Partial Completeness Axiom∗ is thus quiteappealing.8

8 Well, I can’t pass this point without pointing out that I fail to see why one should be expected

to have complete preferences over lotteries, but not on acts While many authors in the field seem

to take this position, it seems to me that any justification for worrying about incomplete preferences over acts would also apply to the case of lotteries.

In 1986 Truman Bewley proved the following extension of the Anscombe-AumannTheorem for incomplete preferences on HΩ,X that satisfy the Partial CompletenessAxiom.9

Bewley’s Expected Utility Theorem Let Ω and X be any nonempty finite sets,and a preference relation on HΩ,X Then satisfies the Independence, Continuity,No-Triviality, Monotonicity, and Partial Completeness Axioms∗ if, and only if, thereexist a (utility function) u ∈ RX and a nonempty set M ⊆ LΩ such that, for any

μ(ω)Eg ω(u) for all μ ∈ M

An individual whose preference relation over horse race lotteries satisfies the ioms of Bewley’s Expected Utility Theorem holds initial beliefs about the true state

ax-of the world, but her beliefs are imprecise in the sense that she does not hold onebut many beliefs In ranking two horse race lotteries, she computes the expectedutility of each horse race lottery using each of her prior beliefs (and hence attaches toevery act a multitude of expected utilities) If f yields higher expected utility than

g for every prior belief that the agent holds, then she prefers f over g If f yieldsstrictly higher expected utility than g for some prior belief, and the opposite holdsfor another, then she remains indecisive as to the ranking of f and g

This model is called the Knightian uncertainty model, and has recently beenapplied in various economic contexts ranging from financial economics to contracttheory and political economy.10 This is not the place to get into these matters atlength, but let us note that all of these applications are based on certain behavioralassumptions about how an agent would make her choices when she cannot comparesome (undominated) feasible acts, and hence introduces a different (behavioral) di-mension to the associated decision analysis.11

We now turn to the proof of Bewley’s Expected Utility Theorem The structure

of this result is reminiscent of that of the Expected Multi-Utility Theorem, so you

9 While the importance of Bewley’s related work was recognized widely, his original papers mained as working papers for a long time (mainly by Bewley’s own choice) The first of the three papers that contain his seminal analysis has appeared in print only in 2002 while the rest of his papers remain unpublished.

re-10 The choice of terminology is due to Bewley But I should say that reading Frank Knight’s 1921 treatise did not clarify for me why he chose this terminology At any rate, it is widely used in the literature, so I will stick to it As for applications of the theory, see, for instance, Billot, et al (2000), Dow and Werlang (1992), Epstein and Wang (1994), Mukerji (1998), Ghirardato and Katz (2000), and Rigotti and Shannon (2005).

11 For instance, Bewley’s original work presupposes that, when there is a status quo act in the choice problem of an agent, then she would stick to her status quo if she could not find a better feasible alternative according to her incomplete preferences An axiomatic foundation for this behavioral postulate is recently provided by Masatlioglu and Ok (2005).

may sense that convex analysis (by way of Proposition 1) could be of use here This

is exactly the case

Proof of Bewley’s Expected Utility Theorem We only need to prove the “onlyif” part of the assertion Assume that  satisfies the Independence, Continuity, No-Triviality, Monotonicity, and Partial Completeness Axioms∗.By the Expected UtilityTheorem, there exists a u ∈ RX such that p∗ q iff Ep(u)≥ Eq(u), for any p, q ∈ LX.Let T := {Ep(u) : p ∈ LX} and S := TΩ.The proof of the following claim is relativelyeasy — we leave it as an exercise

Claim 1 S is a compact and convex subset of RΩ, and al-intspan(S)(S) = ∅

Now define the binary relation  on S by

F  G if and only if f  gfor any f, g ∈ LΩ such that F (ω) = Ef ω(u)and G(ω) = Eg ω(u) for all ω ∈ Ω

Claim 2  is well-defined Moreover,  is affine and continuous

Proof of Claim 2 We will only prove the first assertion, leaving the proofs of theremaining two as easy exercises And for this, it is clearly enough to show that, forany f, f ∈ LΩ, we have f ∼ f whenever Ef ω(u) = Efω(u) for all ω ∈ Ω (Is it?)Take any h ∈ HΩ,X By the Monotonicity Axiom∗, if Ef ω(u) = Ef 

ω(u) for all ω ∈ Ω,then (fω, h−ω)∼ (fω, h−ω)for all ω ∈ Ω Thus, by the Independence Axiom∗ (applied

so applying the Independence Axiom∗ one more time, we get f ∼ f

By the Partial Comparability Axiom∗, we have = ∅, which implies that thestrict part of  is nonempty as well Then, Claims 1 and 2 show that we may applyProposition 1 (with Ω playing the role of Z) to find a nonempty subset L of nonzerolinear functionals on RZ

such that, for any F, G ∈ S,

F  G iff L(F) ≥ L(G) for all L ∈ L

Clearly, there exists a σL

(q, h−ω) holds for all h ∈ HΩ,X and ω ∈ Ω By definition of  and (8), therefore, wehave σL(ω)(Ep(u)− Eq(u))≥ 0 for each L ∈ L It follows that σL

∈ RΩ +\{0} for each

∗∗ Exercise 6 (An open problem) Determine how Bewley’s Expected Utility Theorem would modify if we dropped the Partial Completeness Axiom∗ from its statement.

In recent years the theory of individual decision theory underwent a considerabletransformation, because the descriptive power of its most foundational model — theone captured by the Anscombe-Aumann Expected Utility Theorem — is found to leavesomething to be desired The most striking empirical observation that has led to thisview stemmed from the 1961 experiments of Daniel Ellsberg, which we discuss next.Consider two urns, each containing 100 balls The color of any one of these balls

is known to be either red or black It is not known anything about the distribution

of the balls in the first urn, but it is known that exactly 50 balls in the second urnare red One ball is drawn from each urn at random Consider the following bets:

Betiα : If the ball drawn from theith urn isα,then you win $10, nothing otherwise,

where i ∈ {1, 2} and α ∈ {red, black} Most people declare in the experiments thatthey are indifferent between the bets 1red and 1black, and between the bets 2red and

2black Given the symmetry of the situation, this is exactly what one would expect.But how about comparing 1redversus 2red?The answer is surprising, although you maynot think so at first An overwhelming majority of the subjects in the experimentsdeclare that they strictly prefer 2red over 1red (We know that their preferences arestrict, because they in fact choose “2red + a small fee” over “1red + no fee.”) It seemsthe fact that they do not know the ratio of black to red balls in the first urn — theso-called ambiguity of this urn — bothers the agents This is a serious problem for themodel envisaged by the Anscombe-Aumann Expected Utility Theorem Indeed, noindividual whose preferences can be modeled as in that result can behave this way!(For this reason, this situation is commonly called the Ellsberg Paradox.)

Let us translate the story at hand to the Anscombe-Aumann setup Foremost weneed to specify a state space and an outcome space Owing to the simplicity of the

situation, the choice is clear:

Ω := {0, 1, , 100} and X := {0, 10}

Here by state ω ∈ Ω we mean the state in which exactly ω of the balls in the firsturn are red In turn, the outcome space X contains all possible payments an agentmay receive through the bets under consideration Given these specifications, then,the bet 1red is modeled by the horse race lottery f : Ω → LX defined by

f (ω) = 100ω δ10+100−ω100 δ0,while the bet 2red is modeled by the horse race lottery f : Ω → LX defined by

f (ω) = 12δ10+ 12δ0.(Notice that f involves risk, but not uncertainty — this is the upshot of the EllsbergParadox.) The horse race lotteries that correspond to the bets 1black and 2black aremodeled similarly; we denote them by g and g , respectively

In this formulation, the data of the Ellsberg Paradox tell us that the preferencerelation of a decision maker on {f, g, f , g } may well exhibit the following structure:

f ∼ g and f f.Now suppose can be represented by means of a prior μ ∈ LΩ and

a utility function u ∈ R{0,10} as in the Anscombe-Aumann Expected Utility Theorem

Of course, without loss of generality, we may assume that u(0) = 0 If u(10) = 0 aswell, then we get f ∼ f in contrast to f f So, assume instead u(10) = 0 Then,

f ∼ g (i.e the indifference between the bets 1red and 1black) means

g Indeed, f = 12f + 12g Consequently, if  satisfies the Independence Axiom∗ and

f ∼ g is true, then f ∼ f must hold per force, we cannot possibly have f f.The Ellsberg Paradox seems to tell us that, to increase the descriptive power of ourmodel, we should break free from the straightjacket of the Independence Axiom∗ atleast so as to allow for the following property in a nontrivial way

The Uncertainty Aversion Axiom∗.12 For any f, g ∈ HΩ,X,

In 1989, Itzhak Gilboa and David Schmeidler offered a brilliant weakening of theIndependence Axiom∗ which goes along with this property very nicely Since then thisweakened version of the Independence Axiom∗ — called the C-Independence Axiom∗

(“C” for “constant”) — became the industry standard

The C-Independence Axiom∗ For any f, g ∈ HΩ,X, p∈ LX,and any 0 < λ ≤ 1,

f  g if and only if λf + (1 − λ) p  λg + (1 − λ) p

The interpretation of this property is identical to that of the usual IndependenceAxiom∗,but now we demand the independence property only with respect to mixingwith constant horse race lotteries Naturally, this is bound to increase the descriptivepower of the decision making model at hand We shall see shortly how it does this

is endowed with a complete preference relation  and chooses from any given ble set the horse race lottery (or act) that is a -maximum in that set, the “use”

feasi-of such a theory lies in its entirety in the structure feasi-of representation it provides for

 And it is exactly at this point that the Gilboa-Schmeidler theory shines bright

It turns out that replacing the Independence Axiom∗ with the C-Independence andUncertainty Aversion Axioms∗ leads us to a beautiful utility representation for theinvolved preferences

The Gilboa-Schmeidler Theorem Let Ω and X be any nonempty finite sets,and a complete preference relation on HΩ,X Then satisfies the C-Independence,Continuity, Monotonicity, and Uncertainty Aversion Axioms∗ if, and only if, there

exist a (utility function) u ∈ RX and a nonempty convex set M ⊆ LΩ such that, forany f, g ∈ HΩ,X,

f  g if and only if min

of each horse race lottery using each of her prior beliefs, and then, relative to herbeliefs, chooses the lottery that yields the highest of the worst possible expectedutility In the literature, this model, which is indeed an interesting way of completingBewley’s model, is often referred to as the maxmin expected utility model withmultiple priors.13

To give a quick example, let us turn back to the Ellsberg Paradox that we sidered at the beginning of this subsection and see how the Gilboa-Schmeidler theoryfares with that To this end, let us adopt the notation of that example, and givenany nonempty M ⊆ LΩ, let us define the map UM :HΩ,X → R by

con-UM(h) := min

S

in the Anscombe-Aumann Theorem.) But what if M = LΩ?Then we have UM(f ) =

0 = UM(g)and UM(f ) = 12 = UM(g ),so we have f ∼ g and f ∼ g while f f and

g g,in full concert with the Ellsberg Paradox In fact, the same result would obtainwith much smaller sets of beliefs as well For instance, if M := {μ ∈ LΩ : μ(ω) > 0only if ω ∈ {49, 50, 51}}, then we have UM(f ) = 4950 = UM(g) and UM(f ) = 12 =

UM(g ),so we remain consistent with the data of the Ellsberg Paradox

The literature on the applications of the Gilboa-Schmeidler model is too large to

be recounted here Moreover, there are now many generalizations and variants of themaxmin expected utility model.14 But going deeper into individual decision theory

13 I have heard many economists being critical of this model because it models a decision maker

as “too pessimistic.” But note that nothing is said about the nature of the set of beliefs in the theorem Depending on the application, this set may be taken to comprise only optimistic beliefs,

so an agent who acts pessimistically relative to her (optimistic) beliefs, may end up behaving not

at all in a way that a pessimistic person would behave (It should be noted that the choice of a particular set of beliefs in an application is a complicated matter which is best discussed within the specifics of that application.)

14 Particularly noteworthy, in my opinion, are Ghirardato and Marinacci (2001), Schmeidler (1989), and Maccheroni, Marinacci and Rustichini (2005).

will get us off course here We thus stop our present treatment here, and conclude thesection with the proof of the Gilboa-Schmeidler Theorem This is a “deep” result,but one which is not all that hard to establish once one realizes that convex analysislies at the very heart of it.

We begin with introducing the following auxiliary concept

Dhilqlwlrq Let Ω be a nonempty set If ϕ is a real map on RΩ such that

ϕ (F + α1Ω) = ϕ(F ) + ϕ (α1Ω) for any F ∈ RΩ and α ∈ R,

then we say that ϕ is C-additive.15

The following result is the main step towards the proof of the Gilboa-SchmeidlerTheorem

Lemma 2 Let Ω be a nonempty set, and ϕ : RΩ

→ R an increasing, superlinear and

C-additive map Then, there exists a nonempty convex subset L of positive linearfunctionals on RΩ such that

ϕ(F ) = min{L(F ) : L ∈ L} for all F ∈ RΩ, (9)and

Proof Thanks to the superlinearity of ϕ, Corollary G.5 ensures that, for every

F ∈ RΩ,there is a linear functional LF on RΩ

such that ϕ ≤ LF and ϕ(F ) = LF(F ).Moreover, by C-additivity of ϕ, we have

ϕ(F − 1Ω) + ϕ(1Ω) = ϕ(F ) = LF(F ) = LF(F − 1Ω) + LF(1Ω)

for any F ∈ RΩ Since LF(F − 1Ω) ≥ ϕ(F − 1Ω) and LF(1Ω) ≥ ϕ(1Ω), we mustthen have LF(F − 1Ω) = ϕ(F − 1Ω) and, more to the point, LF(1Ω) = ϕ(1Ω), forany F ∈ RΩ Now let L be the convex hull of {LF : F ∈ RΩ

} By construction, Lsatisfies (9) and (10) Furthermore, because ϕ is superlinear, we have ϕ(0) = 0, so

by monotonicity of ϕ, F ≥ 0 implies L(F ) ≥ ϕ(F ) ≥ ϕ(0) = 0 for all L ∈ L That

Since we know the general representation of a linear functional on a Euclideanspace (Example F.6), the following is an immediate consequence of Lemma 2

15 Reminder 1 Ω is the real function on Ω that equals 1 everywhere.

Corollary 1 Let Ω be a nonempty finite set If ϕ : RΩ → R is an increasing,superlinear and C-additive map with ϕ (1Ω) = 1,then there exists a nonempty convexsubset M of LΩ such that

We are now ready for our main course

Proof of the Gilboa-Schmeidler Theorem We only need to prove the “only if”part of the assertion, and for this we may assume = ∅, for otherwise the claim

is trivial Suppose  satisfies the C-Independence, Continuity, Monotonicity, andUncertainty Aversion Axioms∗ Then, the preference relation ∗ on LX (induced

by ) is complete, and satisfies the Continuity and Independence Axioms (Yes?)Therefore, by the Expected Utility Theorem, there exists an affine map L onLX thatrepresents∗ Since X is finite, there exist (degenerate) lotteries p∗ and p∗ such that

p∗ ∗ p∗ p∗ for all p ∈ LX (Why?) By the Monotonicity Axiom, therefore,

p∗  h  p∗ for all h ∈ HΩ,X

In what follows we assume that L(p∗) = 1 and L(p∗) =−1.16

Claim 1 For any h ∈ HΩ,X, there is a unique 0 ≤ αh ≤ 1 such that

h∼ αh p∗ + (1− αh) p∗ Proof of Claim 1 Define αh := inf{α ∈ [0, 1] : α p∗ + (1− α) p∗  h}, and usethe C-Independence and Continuity Axioms to verify that αh is equal to the task.Claim 2 There exists a unique U :HΩ,X → R that represents  and satisfies

Proof of Claim 2 First define the real map φ on { p : p ∈ LX} by φ( p ) = L(p),and then define U on HΩ,X by U (h) := φ (αh p∗ + (1− αh) p∗ ) , where αh is asfound in Claim 1 Obviously, U represents  and satisfies (11) Besides, if V wasanother such function, we would have, for any h ∈ HΩ,X,

U (h) = U ( ph ) = L(ph) = V ( ph ) = V (h),where ph = αhp∗+ (1− αh)p∗

16 Given that = ∅, we must have p ∗ ∗ p∗ (Why?) So, since a von Neumann-Morgenstern utility function is unique up to strictly increasing affine transformations, the choice of these numbers is without loss of generality.

Claim 3 There exists an increasing, superlinear and C-additive real map ϕ on RΩsuch that

Let us suppose for the moment that Claim 3 is true Then U ( p∗ ) = L(p∗) = 1,

so by (12), we find ϕ(1Ω) = ϕ(L◦ p∗ ) = U ( p∗ ) = 1.17 Therefore, combining Claims

2 and 3 with Corollary 1 completes the proof of the Gilboa-Schmeidler Theorem.18

It remains to prove Claim 3 This is not really difficult, but we still have to break

a sweat for it Begin by noting that {L ◦ h : h ∈ HΩ,X} = [−1, 1]Ω.19 We may thusdefine ψ : [−1, 1]Ω

→ R asψ(L◦ h) := U(h) for all h ∈ HΩ,X

Claim 4 ψ(λF ) = λψ(F ) for any (F, λ) ∈ [−1, 1]Ω

× R++ such that λF ∈ [−1, 1]Ω

Proof of Claim 4 Take any F ∈ [−1, 1]Ω and 0 < λ ≤ 1 Choose any f ∈ HΩ,X

such that L ◦ f = F We wish to show that ψ(λ(L ◦ f)) = λψ(L ◦ f) To this end,take any p ∈ LX with L(p) = 0, and let g := λf + (1 − λ) p Let’s compute ψ(L ◦ g)

in two different ways First, we have

L◦ g = λ (L ◦ f) + (1 − λ) (L ◦ p ) = λ (L ◦ f) ,

so ψ(L ◦ g) = ψ(λ (L ◦ f)) = ψ(λF ) Second, we find a q ∈ LX with q ∼ f (Claim1) and use the C-Independence Axiom∗ to find g ∼ λ q + (1 − λ) p Then, byClaim 2,

U (g) = U (λ q + (1− λ) p ) = L(λq + (1 − λ)p) = λL(q) = λU( q ) = λU(f),

17 Try not to get confused with the notation L ◦ p ∗ is the function that maps any given ω ∈ Ω

to L(p ∗ ) = 1, so L ◦ p ∗ = 1 Ω

18 I hope you see the “big picture” here We know that the Expected Utility Theorem applies to

 ∗ , so we may represent  ∗ with an affine function L But any horse race lottery has an equivalent

in the space of all constant horse race lotteries This allows us to find a U on HΩ,X that represents

 and agrees with L on all constant lotteries Now take any h ∈ H Ω,X , and replace h(ω) (which is

a lottery on LX) with L(h(ω)) (which is a number) — you get L ◦ h It is easy to see that there is a map ϕ on the set of all L ◦ hs such that ϕ(L ◦ h) ≥ ϕ(L ◦ g) iff f  g — this is exactly what (12) says The problem reduces, then, to determine the structure of the function ϕ If we had the full power

of the Independence Axiom ∗ , we would find that ϕ is increasing and affine — this is exactly what

we have done when proving the Anscombe-Aumann Theorem in Section F.3.2 Here all we got is C-Independence, so we will only be able to show that ϕ (actually its (unique) positive homogeneous extension to R Ω ) is increasing and C-additive Adding the Uncertainty Aversion Axiom ∗ into the picture will then give use the superlinearity of ϕ But, thanks to Corollary 1, we know how to represent such functions!

19 The ⊆ part of this equation is obvious To see the ⊇ part, take any F ∈ [−1, 1] Ω , and notice that, for every ω ∈ Ω, there is a p ω ∈ L X such that L(p ω ) = F (ω) by the Intermediate Value Theorem If we define h ∈ H Ω,X by h(ω) := p ω , therefore, we find L ◦ h = F.

so ψ(L ◦ g) = λU(f) = λψ(L ◦ f) = λψ(F ) Conclusion: ψ(λF ) = λψ(F ) for any

F ∈ [−1, 1]Ω

and 0 < λ ≤ 1

To conclude the proof of Claim 4, take any F ∈ [−1, 1]Ω

and λ > 1 with λF ∈[−1, 1]Ω By what we have just established, ψ(F ) = ψ(1λλF ) = 1λψ(λF ), and we aredone

We now extend ψ to the entire RΩby positive homogeneity That is, we define thereal map ϕ on RΩ by

ϕ(αF ) = 1

λψ(λαF ) = 1

λαψ(λF ) = αϕ(F ),where we used Claim 4 to get the second equality here

We are half way through proving Claim 3 We are now in possession of a positivelyhomogeneous real map ϕ on RΩ that satisfies (12) and ϕ(1Ω) = 1.20 We next provethat ϕ is C-additive Since ϕ is positively homogeneous, it is enough to show onlythat ϕ(F + α1Ω) = ϕ(F ) + αϕ(1Ω) for all F ∈ [−1, 1]Ω

and 0 < α ≤ 1.21 Of course,

L◦ f = F for some f ∈ HΩ,X, and L(p) = α for some p ∈ LX Again, find a q ∈ LX

such that f ∼ q , so by the C-Independence Axiom∗, 12f + 12 p ∼ 12 q + 12 p Then U1

21 Suppose we can do this Then for any G ∈ R Ω and β > 0, we take any 0 < λ ≤ β1 with

λG ∈ [−1, 1] Ω , and use the positive homogeneity of ϕ to get

ϕ(G + β1 Ω ) =λ1ϕ(λG + λβ1 Ω ) = 1λ(ϕ(λG) + λβϕ(1 Ω )) = ϕ(G) + βϕ(1 Ω ).

Since ϕ(1Ω) = 1, we may conclude that ϕ is C-additive.

Notice that we did not use the Uncertainty Aversion Axiom∗ yet We will use itnow in order to show that ϕ is superadditive, that is,

for any F, G ∈ RΩ (Since ϕ is positively homogeneous, this property entails that ϕ

is superlinear.) Once again, thanks to positive homogeneity, it is enough to establish(13) for (arbitrarily chosen) F and G in [−1, 1]Ω Pick any f, g ∈ HΩ,X such that

L◦ f = F and L ◦ g = G

Let us first consider the case ϕ(F ) = ϕ(G) In this case U (f ) = U (g), so we have

f ∼ g By the Uncertainty Aversion Axiom∗, then, 12f + 12g  f, so, by (12),

ϕ(12F +12G) = ϕ(L◦1

2f + 12g

) = U (12f + 12g)≥ U(f) = ϕ(F ) = 12ϕ(F ) + 12ϕ(G),and (13) follows by positive homogeneity of ϕ

Finally, consider the case ϕ(F ) = ϕ(G), say ϕ(F ) > ϕ(G) Let α := ϕ(F ) − ϕ(G),and define H := G + α1Ω Notice that C-additivity of ϕ and ϕ(1Ω) = 1 entailϕ(H) = ϕ(G) + α = ϕ(F ) So, by what we have shown in the previous paragraph,

where u, v ∈ RX, and M and N are nonempty, closed and convex subsets of LΩ

Prove thatM = N and v is a strictly increasing affine transformation ofu

Exercise 9.H (Nehring) Let Ω and X be two nonempty finite sets, and  a plete preference relation onHΩ,X that satisfies theC-Independence, Continuity and Monotonicity Axioms∗.Define the binary relation on HΩ,X by:

com-f  g iff λf + (1− λ)h  λg + (1 − λ)h for all h∈ HΩ,X and 0 < λ≤ 1

(Whenf  g, one says thatf is unambiguously preferred tog.)

(a) Interpret.

(b) Show that, for any f, g∈ HΩ,X and p, q ∈ LX,

(c) Show thatis a preorder on HΩ,X that satisfies the Independence Axiom∗ (d ) Show that there exists a nonempty convex subsetM ofLΩ and au ∈ RX such that, for any f, g∈ HΩ,X,

ω∈Ω

μ(ω)Ef ω(u)≥ S

ω∈Ω

μ(ω)Eg ω(u) for all μ∈ M

How does this result fare with the interpretation you gave in part (a)?

Exercise 10 (Ambiguity Aversion) LetΩandX be two nonempty finite sets, and

a preference relation onHΩ,X We say thatis ambiguity averse if there exists a preference relation  on HΩ,X which admits a representation as in the Anscombe- Aumann Expected Utility Theorem, and

for any p ∈ LX and f ∈ HΩ,X (Interpretation?) Prove: If  satisfies the C Independence, Continuity, Monotonicity, and Uncertainty Aversion Axioms∗,then

(c) Prove or disprove: If is variational, then it is ambiguity averse.

Exercise 12 (Pessimistic Preferences over Sets of Lotteries) LetX be a nonempty finite set, and denote by c(LX) the set of all nonempty closed subsets of LX We think of c(LX)as a metric space under the Hausdorff metric (Section E.2.5) Prove thatis a continuous and complete preference relation onc(LX)such that, for any

P, Q, R∈ c(LX),

(i) P ⊆ Q impliesP  Q;

(ii)P  R andQ R and P ∩ Q = ∅ implyP ∪ Q  R,

if, and only if, there exists a continuousU :LX → Rsuch that

P  Q iff min{U(p) : p ∈ P } ≥ min{U(p) : p ∈ Q}

for any P, Q∈ c(LX).(Interpretation?)

22 The theory of variational preferences is developed thoroughly in Maccheroni, Marinacci and Rustichini (2005).

2 Applications to Welfare Economics

Throughout this application m and l stand for fixed natural numbers with m ≥ 2

An m-person, l-commodity exchange economy is formally defined as the list

E := ({ωi, ui}i=1, ,m)where ωi

∈ Rl

+ and ui : Rl

+ → R stand for the endowment vector and the utilityfunction of agent i, respectively The following assumptions are often imposed onthese primitives:

Avvxpswlrq (A1) ωi 0 for all i = 1, , m

Avvxpswlrq (A2) ui is continuous and strictly increasing for all i = 1, , m.Avvxpswlrq (A3) ui is strictly quasiconcave for all i = 1, , m

An allocation in an exchange economy E is defined as any vector x = (x1, , xm)∈

ui(xi) < ui(yi) for all i = 1, , m,and strongly Pareto optimal if, for no other allocation y in E, we have

ui(xi)≤ ui(yi) for all i = 1, , m and ui(xi) < ui(yi) for some i = 1, , m.Clearly, these are fundamental efficiency properties Once an allocation fails to satisfyeither of them, one can improve upon this allocation at no welfare cost to the society.Let us take the set of all admissible prices as Rl

++and denote a generic price vector

by p ∈ Rl++.Given an exchange economy E, we define the demand correspondence

of the ith agent on Rl

++× Rl

+ as

di(p, ωi) := arg max{ui(xi) : xi ∈ Rl+ and pxi ≤ pωi}(Example E.4) That is, if xi ∈ di(p, ωi), we understand that xi is one of the mostdesired bundles for person i among all consumption bundles that she could affordgiven the price vector p Of course, if (A2) and (A3) hold, then di can be considered

as a continuous function (Example E.4), a convention which we adopt below Acompetitive equilibrium for E is defined as any (p, x) ∈ Rl