Real Analysis with Economic Applications - Chapter H pot

Chapter H Economic Applications Even the limited extent of convex analys is we co ve re d in Chap t er G endows on e with surprisingly powerful methods. Unfortuna tely, in practice, it is not always easy to recognize t he s itu ation s in w h ich th e se methods are applicab le. To g et a fe elin g for in which sort o f economic models conv ex analysis may turn o ut to p ro vide the righ t mode of attack, one surely need s a lot of p rac tic e. Our o bjective in this chapte r is thus to presen t a smorgas bord of economic applications that illustrate the mu ltifariou s use of con v ex analysis in general, and the basic separation-b y-h yperplane and linear- extension argumen ts in particular. In our first app lication we revisit expected utility the ory, but this tim e using preferences that are poten tially incomp lete. Our objective is to extend both the clas- sical and the Anscombe-A u m an n Expecte d Utility The ore ms (Section F.3) into the realm of incomplete p references, and introduce the recently popular multi-prior de- cision making models. We then turn to welfare economics. In particular, we prove the Second Welfare Theorem, obtain a useful characterization o f Pareto op tima in pure distribution problems, a nd talk about Harsanyi’s Utilitarianis m Theorem . As an application to information theory, we prov ide a sim ple proof of the celebrated Blackwell’s Theorem on comparin g the valu e of information service s, and as an ap- plication to financial economics, w e pro v ide various formulations of the N o-A rb itrage Theorem. Finally, in the con text of cooperativ e game theory, we c haracterize the Nash barga inin g solution, and examin e some basic applicatio ns to coalitional games without side payments. While the contents of these applicat ions are fairly d iver se (and hence they can be read independently of each other), the methods with which they are studied here all stem from basic con vex analysis. 1 A p plic ations to E xpecte d Ut ili ty Theor y This section continues the investigation o f expected utility the ory we sta rted in S e c- tion F.3. We adopt here the notation and definition s introduced in that section, so it ma y be a good idea to hav e a quick review of Section F.3 before commencing to the analysis pr ovid ed belo w. 1.1 The Expected Mu lti-U tility Theo rem Let X be a nonempty fin ite set, which is in terpr eted as a set of (monetary or nonm one- tary) prizes/alternatives, a nd recall tha t a lo ttery (or a probability distribution) on X is a map p ∈ R X + suc h that S x∈X p(x)=1. As in Section F.3, w e den ote the set of all lotteries on X by L X , which is a compact and conv ex subset of R X . 372 The Expected Utilit y Theorem states that any comp lete preference relation  on L X that satisfies the Independence and C on tinuity Axioms (Section F.3.1) admits an expected u tility repres entat ion. That is, for an y such preference relation there is a utility funct ion u ∈ R X such that p  q if and only if E p (u) ≥ E q (u) (1) for an y p, q ∈ L X . 1 Our goal here is to extend this result to the realm of inc omplete prefere n ce s. Th e disc u ss ion presented in Sectio n B .4, an d a swift comparison of Propositions B.9 a n d B.10 suggest th at one’s objective in this regard should be to obtain a multi- utility an alogue of this theore m. And ind eed , we have: The Expected Multi-Utility Theorem. (Dubra-Macc heroni-Ok) Let X be any none mpty finite set, and  apreferencerelationonL X .Then satisfies the Inde- pende nc e a nd Continuity Axioms if, and only if, t h e re exis ts a non empty set U ⊆ R X (of utility funct io n s) suc h tha t, for an y p, q ∈ L X , p  q if and only if E p (u) ≥ E q (u) for all u ∈ U. (2) This result sho w s that one can think of an agent whose (possibly incomplete) preferences over lotteries in L X satisfy the Independence and C ontinuity Axioms “as if” t his agen t distinguishes bet ween the prizes i n X with respect t o multiple objectives (each objective being captured by one u ∈ U). This agent prefers lottery p over q if the expected value of each of her objectives with respect to p is gr eater than those with respect to q. If suc h a domination does not take place, that is, p yields a strictly higher expectation with respect to some objective, and q with respect to some other objective, then the a gent remains indecisive between p and q (and settles her choice problem b y means w e do not m odel here). Before we m ove on to the proof of the Expected M ulti-U tility Theorem , let us observe that the classical Expected Utility Theorem is an easy conseque nce of this result. All w e need is the follow ing elemen tary observat ion. Lem ma 1. Let n ∈ N, and take any a, u ∈ R n such that a = 0, S n a i =0, an d u 1 > ···>u n . If S n a i u i ≥ 0, then we m ust ha v e a i > 0 and a j < 0 for som e i<j. Proo f. We must have n ≥ 2 under the hypotheses of the assertion. Let α i := u i − u i+1 for each i =1, , n − 1. We have α 1 a 1 + α 2 (a 1 + a 2 )+···+ α n−1 n−1 S i=1 a i + u n n S i=1 a i = n S i=1 a i u i . 1 Reminder. E p (u):= S x∈X p(x)u(x). 373 So, by the hy pothese s S n a i =0and S n a i u i ≥ 0, we have α 1 a 1 + α 2 (a 1 + a 2 )+···+ α n−1 n−1 S i=1 a i ≥ 0. (3) If th e claim was false , t h ere w ou ld exist a k ∈ {1, ,n−1} such that a 1 , ,a k ≤ 0 and a k+1 , , a n ≥ 0 with S k a i < 0.Ifk = n − 1, this read ily con tra dicts (3), so conside r the case where n ≥ 3 and k ∈ {1, , n − 2}. In that case, (3) implies α K S K a i > 0 for some K ∈ {k +1, , n − 1}, and w e find 0= S n a i ≥ S K a i > 0.  Another P roof for the Expected Ut ility Theorem . Let X be a nonempty finite se t and  be a comp lete preord er on L X that satisfies the Independence and Contin uit y Axiom s. Let us first assume that there is no indifference betwe en the degenerate lotteries, that is, either δ x  δ y or δ y  δ x for any d i stin c t x, y ∈ X. By the Expected Multi-Utilit y Th eorem, there exists a nonempty set U ⊆ R X suc h that (2) holds for any p, q ∈ L X . Define X := {(x, y) ∈ X 2 : δ x  δ y }, and for 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 suc h a u (x,y) follows from (2).) Define u := S (x,y)∈X u (x,y) , and notice that δ z  δ w iff u(z) >u(w), for any z,w ∈ X. (Why?) In fact, u is a v on Neumann- Morgens t ern ut ility functi o n for ,asweshownext. Take any p, q ∈ L X . Note first that p  q implies E p (u) ≥ E q (u) by (2), so given that  is complete, if we c an show th at p  q implies E p (u) > E q (u), it will follow that (1) holds. To deriv e a con tradiction, suppose p  q but E p (u)=E q (u). Then, by Lemma 1, w e can find two prizes x and y in X such that δ x  δ y ,p(x) >q(x) and p(y) <q(y). (Clearly, q(y) > 0.) For any 0 < ε ≤ q(y), define the lottery r ε ∈ L X as r ε (x):=q(x)+ε,r ε (y):=q(y) − ε and r ε (z):=q(z) for all z ∈ X\{x, y }. Since δ x  δ y we have u(x) >u(y), so E p (u)=E q (u) < E r ε (u) for any 0 < ε ≤ q(y). Since  is comple te, (2) th e n implies that r ε  p for all 0 < ε ≤ q(y). But r 1 m → q, so this contradicts the Con tin uity Axiom, and we are done. It remains to relax the assumption that there is no indifference between any two degenera te lotteries on X. Let Y be a ⊇-ma ximal subset of X such that δ x is not indifferent to δ y for any distinct x, y ∈ Y. For any p ∈ L Y , define p  ∈ L X with p  | Y = p and p  (x):=0for all x ∈ X\Y. Define   on L Y by p   q iff p   q  . By what we ha ve established abov e, t here is a u ∈ R Y suc h that p   q iff E p (u) ≥ E q (u), for an y p, q ∈ L Y . We extend u to X intheobviouswaybyletting,foranyx ∈ X\Y, u(x):=u(y x ) where y x is an y element of Y with δ x ∼ δ y x . It is easy to show (by using the Independence Axiom) that u ∈ R X is a von Neuman n -M o rgen stern u tility for . We leave estab lish ing this finalstepasanexercise.  The rest of the present subsection is dev oted to pro ving the “only if” part of the Expected Multi- Utility Theorem . (The “if” part is straightfor wa rd .) The main 374 argument is based on the external characterization of closed and convex sets in a Euclidean space, and is con tained in the follo wing result whic h is a bit more general tha n we need a t present. Proposition 1 . Let Z be any nonempty finite set, S ⊆ R Z , and Y := span(S). Assume that S is a compact and conv ex set wi th a l-in t Y (S) = ∅, and  is a continuous affine preference relation on S. Then, there exists a nonempty L ⊆ L(R Z , R) suc h that 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 tak en here to be nonzero. This is mo re than we need to e s tab lish the Expect ed Mu lti-U tility The ore m. Note firstthatif in that theorem was degenerate in the sense that it declared all lotteries indifferent to eac h other, then we would be done by using a constant utilit y function. Let us then assume that  is not degenerate . Th en, applying Proposition 1 with X and L X playing the roles of Z and S, respectively, we find a nonempty subset L of nonzero linear functionals on R Z suc h th at, for any p, q ∈ L X , p  q iff L(p) ≥ L(q) for all L ∈ L. But by Examp le F.6, for each L ∈ L there exists a u L ∈ R X \{0} such that L(σ)= S x∈X σ(x)u L (x)=E σ (u L ) for all σ ∈ R X . L ettin g U := {u L : L ∈ L}, therefore, comp le tes th e pr oof of the Expected M u lti-Utility The ore m. It rem ains to pro ve Proposition 1 . This is exa ctly where con vex analysis turns out to be the essen tial tool of analysis. ProofofProposition1. Define A := {s − t : s, t ∈ S and s  t}, and check that A is a con vex sub set of R Z .Wedefine next C := V {λA : λ ≥ 0} which is easily v er ified to be a convex cone in R Z . 2 The follow ing claim, which is a restatemen t of Lemma F.2, show s why this cone is importan t 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 s ubset of R Z . 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 d 2 (s, t )=  S z∈Z |s(z) − t(z)| 2  1 2 , and the ε-neighborhood of s ∈ S in Y is N ε,Y (s)={r ∈ Y : d 2 (s, r) < ε}. 375 The plan of attack is forming! Note first that if  = ∅, then ch oosing L as consisting only of the zero functional completes the proof. Assume then that  = ∅, whic h ensures that C is a proper subset of R Z . (Why ?) Thus, since we now know that C is a closed convex cone in R Z , we may apply Corollary G.6 to conclu d e that C must be the intersection of the closed half spaces which contain it, a nd wh ich are defined by its supporting hyperplanes. (Where did the second part come from?) Let L denote the set of all nonzero linear functionals on R Z that correspond to these h y perplanes (Corollary F.4). Observ e that if L ∈ L, then there m ust exist a real number α such that {r ∈ R Z : L(r)=α} supports C. Since this hy perplane contains som e 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 2 L(σ ∗ )=L( 1 2 σ ∗ ) ≥ α and 2α =2L(σ ∗ )=L(2σ ∗ ) ≥ α, whic h i s possible only if α =0. Consequen tly, we have σ ∈ C iff L(σ) ≥ 0 fo r all L ∈ L. By C laim 1, therefore, we have ( 4) for a ny s, t ∈ S. It remains to prove Claim 2, which requires some care. Here goes the argum e nt. 4 Take any (λ m ) ∈ R ∞ + ,and(s m ), (t m ) ∈ S ∞ suc h that σ := lim m→∞ λ m (s m − t m ) ∈ Y and s m  t m for all m =1, 2, We wish to sho w that σ ∈ C.Ofcourse,ifs m = t m for infinitely man y m, then we would trivially have σ ∈ C, so it is withou t loss of gen era lity to let s m = t m for each m. Now pick an y s ∗ ∈ al-int Y (S).SinceS is con vex, al-int Y (S) equals the inte r ior of S in Y, so there ex ists an ε > 0 suc h that N ε,Y (s ∗ ) ⊆ S. 5 Take any 0 < δ < ε, and define T := {r ∈ S : r  s ∗ and d 2 (r, s ∗ ) ≥ δ}. Since  is cont inu ous, T is clo sed subset of S.(Yes?)SinceS is com pact, therefore, T is comp ac t as well. Now let d m := d 2 (s m ,t m ), and notice that d m > 0 and λ m (s m − t m )= λ m d m δ δ d m (s m − t m ) ,m=1, 2, So, letting γ m := λ m δ d m and r m := s ∗ + δ d m (s m − t m ) , we get λ m (s m − t m )=γ m (r m − s ∗ ) ,m=1, 2, (5) It is easily verified that d 2 (r m ,s ∗ )=δ < ε, so r m ∈ N ε,Y (s ∗ ) for each m. Since N ε,Y (s ∗ ) ⊆ S, t herefore, we have r m ∈ S for each m. Moreover, r m − s ∗ ∈ C, so by Claim 1, r m  s ∗ . It follows that r m ∈ T for each m. But s ince λ m (s m − t m ) → σ, we have γ m (r m − s ∗ ) → σ. Since d 2 (r m ,s ∗ ) ≥ δ for all m, therefore, (γ m ) must be a bounded sequence. (Why?) 4 The proof I h ave given in an earlier version of this text was somewhat clumsy. The elegant argum ent given below is due to Juan Dubra. 5 Note. It is in this step tha t we invok e the finiteness of Z. (Recall Ex ample G.5.) 376 We no w know that there exist subsequences of (γ m ) and (r m ) that conv erge in R + and T, say to γ and r, respectively. (Wh y?) Since (λ m (s m − t m )) conver ges t o σ,(5) implies that σ = γ(r − s ∗ ), so σ ∈ C.  Exercise 1. H (Throughout this exercise we use the notation adopted in Proposition 1 and its proof.) We s ay t hat  is weakly continuous if {α ∈ [0, 1] : αs+(1−α)t  αs  +(1− α)t  } isaclosedsetforanys, s  ,t,t  ∈ S. In the statement of Proposition 1, we ma y replace the word “contin u ous” with “weakly continuous.” To prov e this, all you need is to verify that the cone C defined in the proof o f Proposition 1 remains closed in R Z with weak continuity. Assume that  = ∅ (otherwise Proposition 1 is vacuous), and proceed as follows. (a) Show that s ∗ − t ∗ ∈ al-int Y (A) for some s ∗ ,t ∗ ∈ al-int Y (S). (b)Prove:Ifσ ∈ al-cl(C), then σ = λ(s − t) for some λ ≥ 0 and 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 < 1 and s, t ∈ S. Finally, u se this fact and weak continuity to show that C is algebraically closed. By Observation G.3, C is thus a closed subset of R Z . Exercise 2. (Aumann) Let X b e any nonempty finite set. Prove: If the preference relation  on L X satisfies the Independence and Continuity Axioms, then there exists a u ∈ R X such that, for any p, q ∈ L X , p  q implies E p (u) > E q (u) and p ∼ q implies E p (u)=E q (u). Exercise 3. Let X be a nonempty finite set. Show that if the subsets U and V of R X represent a preference relation  a s in the Expected Multi-Utility Theorem, then V must belong to the closure of the con vex cone generated by U and all constant functions on X. 1.2 K nightian U ncertain ty Let us no w turn to ex pected utility theory under uncertain ty, and recall the Anscombe- Auman n framework (Section F.3.2). H ere Ω stands for a nonempty finite se t of states, and X that of p rizes. A h orse r ace l ottery is any map from Ω in to L X —wedenotethe setofallhorseracelotteriesbyH Ω,X . 6 In the A nscombe-Aumann setup, the prefer- enc e relation  of an ind iv idu al is defined on H Ω,X . If this preference relation is com- plete and it satisfies the Independence, Cont inuity, No-Tr iv iality, and M on oton icity Axiom s ∗ (Sectio n F.3.2), t h e n ther e exist a u tility funct ion u ∈ R X and a probability distribution μ ∈ L Ω such that f  g if an d on ly i f S ω∈Ω μ(ω)E f ω (u) ≥ S ω∈Ω μ(ω)E g ω (u) (6) 6 Reminder. For any h ∈ H Ω,X and ω ∈ Ω, we write h ω for h(ω), that is, h ω (x) is the probability of getting prize x in state ω. 377 for any f,g ∈ H Ω,X . We now ask the follo wing question: How w ould this result modify if  was not known to be complete? A natural conjecture in this regard is that  would then admit an expected multi-utility representation with multiple prior beliefs, that is, there would exist a nonem p ty set U ⊆ R X of utility function s and a non e mpty set M ⊆ L Ω of prior beliefs such th at f  g iff S ω∈Ω μ(ω)E f ω (u) ≥ S ω∈Ω μ(ω)E g ω (u) for all (μ,u) ∈ M×U for any f,g ∈ H Ω,X . Unfortunately, to the best of the kno wledge of this author, whether this conjecture is true or not is not kno wn at presen t. What is known is that if the incompleteness of  stems on ly from one’s inability t o compar e the horse race lotteries t ha t d i ffer across states, then the conjecture is true (with U being a singleton). To mak e things precise, let us recall that a preference relation  on H Ω,X induces a preference relation  ∗ on L X in the following manner: p  ∗ q if and only if p  q , where p stands for the constant horse race lottery that equals p a t every state, and similarly for q . Obviously, if  is complete, so is  ∗ . Converse is, however, false. Indeed, the propert y that  is complete enough to ensure the completeness of  ∗ is muc h w eaker than assuming outright t hat  is complete. It is this form er propert y that the Knightian u n cer tainty theory is b uilt on. The Partial Com pleteness Axiom ∗ . The (induced) preference relation  ∗ is com plete, an d  ∗ = ∅. When com bined with the Monotonicit y Axiom ∗ , this property m akes s ure that, given any h ∈ H Ω,X , (p, h −ω )  (q, h −ω ) if and only if (p, h −ω  )  (q, h −ω  ) (7) for any lotteries p, q ∈ L X and an y two states ω, ω  ∈ Ω. 7 (Why?) Thus, if an in- dividual cannot compare t wo horse race lotteries f and g, then this is because they differ in at least two states. For a theory that wishes to “blame” one’s indecisive- ness on uncertaint y, and not on risk, the P artial Completeness Axiom ∗ is thu s quite appealing. 8 7 Reminder. For any h ∈ H Ω,X and r ∈ L X , we denote by (r, h −ω ) the horse race lottery that yields the lottery r ∈ L X in state ω and agrees with h in all other states, that is, (r, h −ω ) τ :=  r, if τ = ω h τ , otherwise. 8 Well, I can’t pass this point without pointing out that I fa il to see why one s hould 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 an y justification for worrying about incomplete preferences over acts would also apply to the case of lotteries. 378 In 1986 Truman Bewley pro ved the follow ing extension of the Anscom be-Aumann Theorem for incomplete preferences on H Ω,X thatsatisfythePartialCompleteness Axiom . 9 Bew le y’s Ex pected Utility The o re m. Let Ω an d X be an y nonemp ty finite sets, and  a preference relation on H Ω,X .Then satisfies the Independence, Contin uit y, No-Trivialit y, M on oton ic ity, and Partial Com p lete nes s A x iom s ∗ if, and only if, t he re exist a (utility function) u ∈ R X and a nonempt y set M ⊆ L Ω such that, for an y f,g ∈ H Ω,X , f  g if and on ly if S ω∈Ω μ(ω)E f ω (u) ≥ S ω∈Ω μ(ω)E g ω (u) for all μ ∈ M. An individual whose preference relation over horse race lotteries satisfies the ax- ioms of Bew ley’s Expected Utility Theorem holds initial beliefs about the true state of the w orld, but her beliefs are imprecise in t he sense that she d oes not hold one but many beliefs. In ranking two horse race lotteries, she computes the expected utility of each horse r a ce lotte ry us ing e ach of he r p r ior beliefs (and hen ce attache s to every act a m ultitude of expected utilities). If f yields higher expecte d utility th a n g for every prior belief that the agen t holds, then she prefers f over g. If f yields strictly high er ex pected utility than g for some prior belief, and the opposite holds for another, then she remains indecisive as to the ranking of f and g. Th is model is c alled the Knight ia n unce rtainty mod e l, and h as recently been applied in various econom ic co ntexts rangin g from financial economics to con tract theory and political economy. 10 This is not t he place to g et into th ese ma tters a t length, but let us note th at all of these application s are based on certain behavioral assumptions about how an agen t would m ak e her c hoices when she cannot compare some (undominated) feasible a cts, and hence in troduces a d iffere nt (beha vioral) di- mension to the associated decision analysis. 11 We now turn to th e proof of Bewley’s Expected Utility Theor em. The structu r e of this result is reminiscent of that of th e Expected Multi-Utility Theorem, so you 9 While the importance of Bewley’s related work was recognized widely, his original papers re- mained as working papers for a long time (mainly by Bewley’s own choice). The first of the three papers that co ntain his s eminal analys is has appeared in print only in 2002 while the rest of his papers remain unpublished. 10 The choice of terminology is due to Bewley. But I sho uld say that reading Frank Knight’s 1921 treatise did not clarify f or me wh y he chose this termino logy. 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), a nd Rigotti and Shannon (2005). 11 For i nstance, Bewley’s original work presupposes that, when there is a status quo a ct in th e 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 founda t ion for this behavioral postulate is recently p rovided by Masatlioglu and Ok (2005). 379 may sense th at c onvex analysis (by way of P roposition 1) could be of use here. This is exactly the case. Proo f of Bewley’s Expec ted Utility Theorem. We only need to prov e the “only if” part of the assertion. Assume that  satisfies the Independence, Continuit y, No- Triviality, Mo notonicity, and Partial C ompleten ess Axiom s ∗ . By the Expected Utility Theorem, there exists a u ∈ R X suc h that p  ∗ q iff E p (u) ≥ E q (u), for any p, q ∈ L X . Let T := {E p (u):p ∈ L X } and S := T Ω . The p roof of the following claim i s relatively easy — we lea ve it as an exercise. Claim 1. S is a compact and conv ex subset of R Ω , and al-int span(S) (S) = ∅. Now define the bin ary relation  on S by F  G if a nd only if f  g for any f,g ∈ L Ω suc h that F (ω)=E f ω (u) and G(ω)=E g ω (u) for all ω ∈ Ω. Claim 2.  is well-defined. Mor eover,  is a ffine and continuous. ProofofClaim2.We will only prove the first a ssertion, lea v ing the p roofs of the remain in g t wo as ea sy exer cis es . And for this, it is clearly enough to show that, for an y f,f  ∈ L Ω , we have f ∼ f  whenev er E f ω (u)=E f  ω (u) for all ω ∈ Ω. (Is it?) Take any h ∈ H Ω,X . B y the M onotonicity Axiom ∗ ,ifE f ω (u)=E f  ω (u) for a ll ω ∈ Ω, then (f ω ,h −ω ) ∼ (f  ω ,h −ω ) for all ω ∈ Ω. Th us, by the Independence Axiom ∗ (ap plied |Ω| − 1 man y tim es), 1 |Ω| f +  1 − 1 |Ω|  h = S ω∈Ω 1 |Ω| (f ω ,h −ω ) ∼ S ω∈Ω 1 |Ω| (f  ω ,h −ω )= 1 |Ω| f  +  1 − 1 |Ω|  h so applying the Ind e penden ce A xiom ∗ one more tim e, we get f ∼ f  .  By the Partial Comparab ility Axiom ∗ ,wehave = ∅, which implies that the strict part of  is n o n e mpty as we ll. Then, Claims 1 an d 2 s how that we may apply Proposition 1 (with Ω pla ying the role of Z)tofind a nonempt y subset L of nonzero linear functionals o n R Z suc h th at, for any F,G ∈ S, F  G iff L(F ) ≥ L(G) for all L ∈ L. Clearly, there exists a σ L ∈ R Ω \{0} such that L(F )= S ω∈Ω σ L (ω)F (ω) for all F ∈ R Ω (Exam ple F.6). So, F  G iff S ω∈Ω σ L (ω)F (ω) ≥ S ω∈Ω σ L (ω)G(ω) for all L ∈ L, (8) for any F, G ∈ S. By th e Partial Comparability A x iom ∗ ,thereexistp, q ∈ L X such that p  ∗ q so that E p (u) > E q (u). But, by the M onotonicity Ax iom ∗ , (p, h −ω )  380 (q, h −ω ) holds for all h ∈ H Ω,X and ω ∈ Ω. By definition of  and (8), therefor e , we have σ L (ω)(E p (u) − E q (u)) ≥ 0 for each L ∈ L. It follo ws th at σ L ∈ R Ω + \{0} for each L, so letting α L := S ω∈Ω σ L (ω), and defining M := { 1 α L σ L : L ∈ L} completes th e proof.  Exercise 4. Derive the Anscombe-Aumann Expected Utility Theorem from Bewley’s Expected Utility Theorem. Exercise 5. State and prove a uniqueness theorem to supplement Bewley’s Expected Utility Theorem (along the lines of the uniqueness part of the Anscom be-Aumann Expected Utility Theorem). ∗∗ Exerc ise 6.(A n op en pr oblem) Determine ho w Bewley’s Expected Utility T heorem would modify if we dropped the Partial Completeness Axiom ∗ from its statement. 1.3 The Gilboa-Schm eidler Multi-Prior Model In recen t years the theory of individual decision theory underwen t a considerable transformation, because the d escriptive po wer of its most foundational m odel — the one ca pture d by the Anscombe-Au mann Expected Utility Theorem — is found to le ave someth ing to be desired. The most striking em p irical observation that has led to this view stem m e d f rom the 1961 experiments of Da niel E llsberg, which we d iscu s s nex t. Consider tw o urns, eac h con taining 100 balls. The color of an y one of these balls is known to be either red or blac k. It is not kno w n an ything about the distribution of the balls in the first urn, but it is kno wn that exactly 50 balls in the second urn are red. One ball is draw n from each urn at random. Consider the follow ing bets: Bet i α : If the ball drawn from the ith urn is α, then y ou win $10, nothing otherwise, where i ∈ {1, 2} and α ∈ {red, black}. Most people declare in the experiments that they are indifferent bet ween the bets 1 red and 1 black , and between the bets 2 red and 2 black . Giv en the symmetry of the situ ation, this is exactly what one wou ld expect. But how about comparing 1 red versus 2 red ? The answ er is surprising, although you may not think so at first. An overwhelming majority of the subjects in the experiments declar e that they strictly prefer 2 red over 1 red .(Weknowthattheirpreferencesare stric t , becaus e t hey in fact choo se “2 red + a small fee” over “1 red +nofee.”) Itseems the fact that they do not kn ow the r atio of blac k to r ed balls in the first urn — the so-called ambiguity of this urn — bothers the agen ts . This is a serious problem fo r the model en visaged by the Anscombe-Aumann Expected Utilit y Theorem . In deed, no individual w hose preferences can be modeled as in that r esult can behav e this way! (For this rea son, th is situation is commonly c alled the Ellsberg P arad ox.) Let us translate the story at han d to the Anscombe-A uma nn setup. Foremost we need to specify a state space and an outcom e space. Owing to the simplicity of the 381 [...]... section 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 analysis lies at the very heart of it We begin with introducing the following auxiliary concept Let Ω be a nonempty set If ϕ is a real map on RΩ such that D ϕ (F + α1Ω ) = ϕ(F ) + ϕ (α1Ω ) for any F ∈ RΩ and α ∈ R, then we say that ϕ is C-additive.15... 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... of the decision making process of the agent If we assume that the agent is endowed with a complete preference relation and chooses from any given feasible set the horse race lottery (or act) that is a -maximum in that set, the “use” of such a theory lies in its entirety in the structure of representation it provides for And it is exactly at this point that the Gilboa-Schmeidler theory shines bright... 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 for all h ∈ H ,X p∗ In what follows we assume that L(p∗ ) = 1 and L(p∗ ) = −1.16 Claim 1 For any h ∈ H ,X , there is a unique 0 ≤ h ≤ 1 such that h ∼ h p∗ + (1 − h) p∗ Proof of Claim 1 Define h := inf{α ∈ [0, 1] : α p∗ + (1 − α) p∗ h} , and use the C-Independence... we would have, for any h ∈ H ,X , U (h) = U ( ph ) = L(ph ) = V ( ph ) = V (h) , where ph = h p∗ + (1 − h )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 386 Claim 3 There exists an increasing, superlinear and C-additive real map... 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 H ,X that represents and agrees with L on all constant lotteries Now take any h ∈ H ,X , and replace h( ω) (which is a lottery on LX ) with. .. to the Ellsberg Paradox that we considered at the beginning of this subsection and see how the Gilboa-Schmeidler theory fares with that To this end, let us adopt the notation of that example, and given any nonempty M ⊆ LΩ , let us define the map UM : H ,X → R by UM (h) := min ω∈Ω μ(ω)Ehω (u) : μ ∈ M where u ∈ R{0,10} satisfies 0 = u(0) < u(10) = 1 Assume that is the preference relation on H ,X that is... Harsanyi in 1955, who is best known for his pioneering work on modeling games with incomplete information Harsanyi (192 0-2 000) has shared with John Nash and Reinhard Selten the 1994 Nobel Prize in Economics for his contributions to the analysis of equilibria of noncooperative games 26 There is a considerable social choice literature that revolves around this result If you are interested in this topic, Weymark... race lotteries, she computes the expected utility of each horse race lottery using each of her prior beliefs, and then, relative to her beliefs, chooses the lottery that yields the highest of the worst possible expected utility In the literature, this model, which is indeed an interesting way of completing Bewley’s model, is often referred to as the maxmin expected utility model with multiple priors.13... it Begin by noting that {L ◦ h : h ∈ H ,X } = [−1, 1]Ω 19 We may thus define ψ : [−1, 1]Ω → R as ψ(L ◦ h) := U (h) for all h ∈ H ,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 ∈ H ,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 . games without side payments. While the contents of these applicat ions are fairly d iver se (and hence they can be read independently of each other), the methods with which they are studied here. course here. We thus stop o ur present treatmen t here, and conclude the section with the proof o f the Gilboa-Schm eidler Theorem. T his is a “deep” result, but on e which is not all that hard. ld have, for any h ∈ H Ω,X , U (h) =U(p h )=L(p h )=V (p h )=V (h) , where p h = α h p ∗ +(1− α h )p ∗ .  16 Given that  = ∅, we m ust have p ∗  ∗ p ∗ . (Why?) So, since a von Neumann-Morgenstern