Choice with Menu-Dependent Rankings (Presentation Slides)

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1 Choice with Menu-Dependent Rankings (Presentation Slides) Paulo Natenzon October 22nd,

2 References that appear on the slides are [1], [2], [3], [4], [5], [6], [7]. References [1] Markus K. Brunnermeier and Jonathan A. Parker. Optimal expectations. American Economic Review, 95(4): , September [2] M.K. Brunnermeier, C. Gollier, and J.A. Parker. Beliefs in the utility function. Optimal beliefs, asset prices, and the preference for skewed returns. The American Economic Review, 97(2): , [3] Jon Elster.. Cambridge University Press, [4] Peter C. Fishburn. Utility Theory for Decision-Making. John Wiley and Sons, New York, [5] Christian Gollier. Optimal illusions and decisions under risk. Working paper, University of Toulouse I, January [6] Leonard J. Savage. The Foundations of Statistics. John Wiley, New York, [7] Ran Spiegler. On two points of view regarding revealed preferences and behavioral economics. Technical report, University College London, September

3 Choice with Menu-dependent Rankings Paulo Natenzon October 22, 2008 Standard Theory s (Berger and Smith 1997) You get a letter asking for a donation... X = {$0, $100, $500} c : P(X ) X c({$0, $100}) =$100 c(x )=$0 Standard Theory Standard Theory s X finite set of possible choice objects P(X )=2 X \ { } all choice situations B P(X ) collection of choice situations c : B P(X ) choice function if c(b) B for all B B. Axiom (Weak axiom of Revealed Preference WARP) If x and y are both in A and B and if x c(a) and y c(b), then x c(b). Theorem The choice function c : P(X ) P(X ) satisfies WARP if and only if it maximizes a rational preference relation X X. Fable: The fox and the grapes Standard Theory s I am sure they are sour.

4 Standard Theory s Two interpretations of : Sore looser Jon Elster s (1982) interpretation preferences underlying a choice may be shaped by the constraints Adaptive preferences Counteradaptive preferences Second interpretation suggests model with: A mapping A A; Choice c(a) =c(a, A) ={x A : x A y, y A} The m function Preference over menus Preliminary results Definition Let m : A [0, 1] n by ( ) m(a) := max p(x), max p(y),...,max p(z) p A p A p A For example, m({p}) =p m( ) = (1, 1,...,1) Also, m(a) 1 =1 A = {p}. Menus of lotteries Preference over menus Preliminary results X Prizes: finite set with elements x, y,...,z. Lotteries: simplex (X ) of probability measures over X with elements p, q, r. A Menus: closed subsets of (X ) denoted A, B, C. ℵ Pairs (A, p) A such that p A. A representation Preference over menus Preliminary results Definition A sour grapes representation is a pair (v, a) that consists of a vnm utility index v R n and a constant a 0 such that the preference on A is represented by the function W : A R defined by W (A) :=max p A v + a ( m(a) 1 1) m(a), p

5 Uniqueness Preference over menus Preliminary results Proposition (uniqueness) Let (v, a) be a sour grapes representation for the preference over menus in A. Then(w, b) is another sour grapes representation for the same preference if and only if there exist α>0 and β R such that w = αv + β b = αa. The Given a menu of consumption streams A, the agent chooses among the options with the following procedure: For each probability measure µ P find a consumption stream c µ A that maximizes Eµ[U(c)]. Choose the measure µ that maximizes a well-being index given by [ 1 W(µ) :=E Eµ[ U(c µ ) s1,...,st] ]. T t T The Brunnermeier & Parker, Gollier (2005, 2005, 2007) Time is T = {1, 2,...,T }; Underlying probability space (Ω, A, P); State of the world s :Ω T {ω1,ω2,...,ωs}; st obtained from s by fixing t T; Consumption streams c :Ω T R+; Utility index U : R T + R increasing, strictly quasiconcave. : consumption example T =2 Ω={ω1,ω2,...,ωS} Allocate savings in t =1andconsumeint =2 One unit endowment Two assets: safe with return R > 0, risky with return R + Z Z1 < Z2 < < ZS and Z1 < 0 < ZS Choose fraction α to invest in safe asset Solvency constraint: R + Z 0 α [ R/ZS, R/Z1]

6 Given a belief µ (Ω), agent chooses portfolio α to maximize Eµu(R + αz). Well-being function simplifies: [ 1 W(µ) =E Eµ[ U(c µ ) s 1,...,st] ]. T t T = E [ 1 2 E µ[u(c µ ) s1]+ 1 2 E µ[u(c µ ) s1, s2] ] = 1 2 E µ[u(c µ )] E[U(cµ )] : properties in terms of choice functions: (Ω, A, P) a probability space; (Z, d) a metric space of prizes; A is a set of acts X :Ω Z; P(A) =2 A \ { } all choice situations B P(A) collection of choice situations Choice functions c : B P(A) Nice properties: Excess optimism and risk taking; Preferences for skewness; General equilibrium with heterogeneous beliefs and gambling; Undersaving and overconfidence. : properties Definition c : B A is an optimal expectations choice function if there are α (0, 1), a utility index u : Z R, and a space M of measures on (Ω, A) including P such that B B, µ M,!X µ B that maximizes Eµu X µ ; for all B Bwe have c(b) =X µ where µ arg max µ M {αe µu X µ +(1 α)eu X µ }. In this case we say that (α, u, M) represents c.

7 : properties Proposition (Ran Spiegler, 2008) Fix Ω={ω1,...,ωS}, the objective measure P and α (0, 1). Let X0 be a safe action, i.e., u(x0(ω)) = 0 for every ω Ω. Then there exist a material payoff function u and a pair of actions X1, X2 such that X1 c({x0, X1}) and c({x0, X1, X2}) ={X0}. : properties Proof. Construct the following payoff function Act / state ω1 ω2 ωs X X1 1 k k X2 m n n where k = 1+P({ω 1}) 1 P({ω1}) ε for ε>0 small, m > 1andn > (k + ε)m. : properties Proposition (Ran Spiegler, 2008 extended) Fix Ω={ω1,...,ωS}, the objective measure P and α (0, 1). Let X0 be a safe action, i.e., u(x0(ω)) = 0 for every ω Ω. Then there exist a material payoff function u and a pair of actions X1, X2 such that c({x0, X1}) ={X1} and c({x0, X1, X2}) ={X0}. (Ran Spiegler s extended) Three assets and two states of nature P =(q, 1 q) =(1/2, 1/2) Utility payoffs given by payoff in ω1 payoff in ω2 Asset X0 0 0 X1 1 k X2 m n

8 When q =1/2, taking m =2,n =7,k =2.9: The Savage Set of states of nature Ω Prize space Z Preference on the set of acts X :Ω Z satisfying (A1) weak order (A2) non-degeneracy (A3) eventwise monotonicity (A4) weak comparable probability (A5) small event continuity (A6) uniform monotonicity (A7) sure-thing principle For details see Fishburn (1970). Μ Definition We say that M preserves independence if for all µ M (and that includes P) the set of random variables A is an independency. Proposition 1 An optimal expectations choice function represented by (u, M) where M preserves independence maximizes a rational preference relation R A A. Moreover, R admits a Savage subjective utility representation with the state independent utility index given by u and the subjective belief given by a convex combination of the objective measure P and a special product measure µ. The Savage Theorem (Savage) Axioms A1 A7 imply that there exists a unique, finitely additive, non-atomic probability measure µ and a state-independent utility function u : Z R, such that the individual ranks finite-outcome acts f :Ω Z on the basis of Eµu(f )= u(f (ω))dµ(ω) = x u(x)µ(f 1 (x)) Proof. See Fishburn (1970).

9 Proof of Proposition 1 1 For each act X :Ω Z in A choose a measure µx M that solves max E µu X (1) µ M Project µx into the σ-algebra generated by X and build the product measure µ of these projections for all X A. 2 Take any menu of acts B Band consider the optimal belief µ M and the induced choice X B. Note that Eµ u X = Eµ u X (2) since µ maximizes expression (1) for all X and, in particular, for X. Proof of Proposition 1 (continued) 5 Pick any X B that cannot be induced for any belief µ M. Now consider the measure µ M where the projection of µ into the sigma algebra generated by any X B is equal to the projection of P, except for the projection into the sigma algebra generated by X, for which it is the same as the projection of µ. Since X cannot be induced with µ, there is a X B such that E µu X > E µu X for all X B and E µu X > E µu X Note that Eu X Eu X for all X B. Proof of Proposition 1 (continued) 3 Now we claim that X solves max {E µ u X + Eu X } (3) X B and we show this in two steps. 4 Take any X B that can be induced for some belief µx M, i.e., where EµX u X E u X for all X B. µx Since M preserves independence, we can take µx such that EµX u X = E µ u X. Hence Eµ u X + Eu X = Eµ u X + Eu X EµX u X + Eu X = Eµ u X + Eu X Proof of Proposition 1 (continued) 5 (continued) Therefore Eµ u X + Eu X = Eµ u X + Eu X E µu X + Eu X > E µu X + Eu X E µu X + Eu X = Eµ u X + Eu X so X in fact solves problem (3).

10 Proof of Proposition 1 (continued) 6 This means that c maximizes the utility index U(X )=Eµ u X + Eu X and of course it will also maximize 1 2 U(X )=1 2 [E µ u X + Eu X ] = E ( 1 2 µ + 1 P)u X 2 and this is just an expected subjective utility representation, where the subjective measure is a convex combination of µ and P. Discussion and next steps Relation between SG and OE? and comonotonicity? in mixture space? Theories of weakening of WARP? Complete and transitive preference over menus? A Theory of Summary for relaxations of WARP; suggests model A A; ; violates WARP; OE + Independence = Savage;