Preference identification

Transcription

1 Preference identification C. Chambers F. Echenique N. Lambert UC San Diego Caltech Stanford April

2 This paper An experimenter and a subject. Subject makes choices according to some on set X. Experimenter conducts a finite choice experiment of size k. Rationalizing preference: k. When does k?

3 Example Subject chooses among alternatives: X = R n +. Choices come from, a continuous preference. B k = {x k, y k }. A finite experiment: choose from B 1,..., B k. B = k=1 B k is dense.

4 Example y x y x

5 Example y V U x y U V x

6 Example y y V x y U V U x U and y V s.t y k x for all rationalizing k x x

7 Example y y V x x U x y U V x U and y V s.t y k x for all rationalizing k But x y. s.t. is cont. and B = B.

8 Example: a discontinuity. Infinite data (X ): recover ; x y Limiting infinite data (B = k=1 B k): x y s.t. B = B. Finite data: can t rule out y k x, no matter how large k.

9 Complete indifference. Let X = R n +. X X is the complete indifference preference. Fix a continuous preference on X. Proposition (informal) There exists rationalizing k for each k s.t complete indifference = lim k k

10 We need some theory to discipline the rationalizations.

11 Findings Purely a-theoretical, non-parametric estimation is misguided. Theory is needed. General conditions for convergence of preferences. Utility functions are more complicated. thm. (bijection) not enough.

12 Theorem (informal) X = ([a, b]) Ω, set of Anscombe-Aumann acts; or X = R n. Objective monotonicity. Connection between order and topology on X.

13 Theorem (informal) Theorem Let be monotone and cont.; k strongly rationalize the kth finite choice experiment generated by.

14 Theorem (informal) Theorem Let be monotone and cont.; k strongly rationalize the kth finite choice experiment generated by. Then, k For any utility u for u k for k s.t. u k u

15 Theorem (informal) Theorem Let be monotone and cont.; k strongly rationalize the kth finite choice experiment generated by. Then, k (in the topology of closed convergence). For any utility u for u k for k s.t. u k u (in the topology of compact convergence).

16 Discussion. Monotonicity. Convergence of any arbitrary nonparametric preference estimate. Choose k to conform to desired theory. Utility can t be arbitrary. Only get convergence of selected utility estimates. Require an identification theorem for each specific theory.

17 Monotone rationalizations. y y y V x y U V Let (x, y ) U V. U x x x

18 Monotone rationalizations. y y y V x x x U x y U V Let (x, y ) U V. = x, y B x x y y = x x k y y

19 Dependence on existence of. 1/2 1/2

20 Dependence on existence of. 1/2 1/2

21 Dependence on existence of. 1/2 1/2

22 Dependence on existence of. 1/2 1/2

23 Dependence on existence of. Problem is mitigated (in some cases) by diameter of set of rationalizing preferences shrinking as k.

24 Convergence of selected utility estimates. Let X = [0, 1], = and u (x) = x. If u n = x n, then 0 = lim n u n. But n = for all n.

25 Convergence of selected utility estimates. Let X = [0, 1], = and u (x) = x. If u n = x n, then 0 = lim n u n. But n = for all n. (For ε > 0, can choose u n with u n = 1 or u n 1 = 1 and 0 = lim n u n (x) for all x [0, 1 ε].)

26 m n x x n u n (x) = u (m n x) = u n u

27 Dated rewards Model of intertemporal choice. X = R 2 +. Interpret (t, x) X has a monetary quantity x delivered on date t. (x, t) (x, t ) iff x x and t t.

28 Dated rewards. x m n x x n 1 t u n (x) = u (m n x) = u n u

29 Model The set of alternatives is X. Let X be Polish, locally compact, partially ordered by. Let <.

30 Definitions, a binary relation on X, is a preference relation if it is complete and transitive; locally strict if x y and V a nbd of (x, y) implies (x, y ) V with x y ; weakly monotone if x y implies x y. strictly monotonic if x y implies x y, and x > y implies x y. continuous if {y X : y x} and {y X : x y} are closed.

31 Model B 1, B 2,... is a collection of subsets of X of cardinality two. Interpretation: B i = {x i, y i }; experimenter asks a subject to choose between x i and y i.

32 Model Let Σ k = {B 1,..., B k }. A finite experiment of order k is a function c : Σ k 2 X s.t c(b i ) B i

33 Model Let B = k=1 B k X. Assume: Any subset of B of cardinality two is in some B k. B is dense in X.

34 Model A choice sequence is an increasing collection of experiments: Denote by C k the set of all finite experiments of order k. Consider c : N k Ck s.t. k, c k C k for all k < l, c l Σk = c k The set of choice sequences is denoted C. c c if ( k)( B i Σ k )c k (B i ) c k (B i ).

35 Model The choice function of order k generated by is defined by c (B i ) = arg max B i = {x B i : x y for all y B i }

36 Model: Rationalizability weakly rationalizes a finite experiment c of order k if c(b i ) c (B i ) for all B i Σ k. weakly rationalizes a choice sequence c if c c. strongly rationalizes a finite experiment c of order k if c(b i ) = c (B i ) for all B i Σ k.

37 Definitions Order > has open intervals if is an open set. {(x, y) : x > y}

38 Convergence 1 Theorem Suppose that 1. is continuous and strictly monotone, 2. > has open intervals, 3. every continuous and st. mon. preference relation is locally strict. Let c c be a choice sequence, and let k be a continuous and strictly monotone preference that weakly rationalizes c k. Then, k in the closed convergence topology.

39 Basically the previous theorem applies to X R n. For X = ([0, 1]) does not have open intervals. (For example let x = δ 1 and x be the uniform distribution. Then if z ε = 1+ε 2 δ 1/2 + 1 ε 2 δ 1 we have z ε / [x, x] for any ε > 0 while z 0 (x, x).)

40 For a choice sequence c, let P k (c) = { : is cont. st. mon. weakly rationalizing c k }. For a set of binary relations S, define diam(s) = sup (, ) S 2 δ C (, ) to be the diameter of S according to the metric δ C which generates the topology on preferences.

41 Theorem Suppose that < has open intervals. Let c be a choice sequence, and suppose that each strictly monotone continuous preference is also locally strict. Then lim k diam(pk (c)) 0.

42 Definitions (X, B) has the countable order property if x X and V nbd of x x, x B V with x x x. X has the squeezing property if for any sequence {x n }, if x n x then there is an increasing sequence {x n}, and an a decreasing sequence {x n }, such that x n x n x n, and lim n x n = x = lim n x n.

43 Definitions Proposition If X is R n, or ([a, b]), then X has the countable order and squeezing properties.

44 Convergence 2 Theorem Suppose 1. is cont. and weakly monotone, 2. (X, Σ ) has the countable order property, and X the squeezing property. Let k be a continuous and weakly monotone preference that strongly rationalizes the choice function of order k generated by. Then, k in the closed convergence topology.

45 proof ideas; X = R n k (cpctness) WTS: =.

46 proof ideas; X = R n k (cpctness) WTS: =. Let x y and x U V y. Arbitrary convergent seq. (x n, y n ) (x, y); squeeze the sequence by (x n, y n). x n = inf{x m : m n} and y n = sup{y m : m n} (x n, y n) (x, y) x n and y n.

47 proof ideas y x y; U V x

48 proof ideas y V y N U x y; U V x N U and y N V x N x

49 proof ideas y y V y N U x y; U V x N U and y N V x U B, x x N. y V B y N y x x N x

50 proof ideas y y V y N x y; U V x N U and y N V x U B, x x N. y V B y N y x x N x U Since k, N N s.t. ( n N )x n y.

51 proof ideas Then: x n x n x N x n y y N y n y n n N weak monotonicity hence avoids any sequence to have the wrong preference; so x y by completeness.

52 proof ideas Conversely, show x y = x y.

53 proof ideas Conversely, show x y = x y. Let x N x (1/k) B with x x, and y N y (1/k) B with y y; So x x y y Now, n = x n y n large enough. n k n k 1 such that x nk y ; and let x = x nk and y = y nk. Then (x n k, y n k ) (x, y) and x nk nk y nk. Thus x y.

54 Utility functions

55 Compact set of utilities Let V R X be a compact set of cont. utility functions. Φ(u) preference relation represented by u. Theorem Suppose V is compact, and that all Φ(V) are locally strict. Let c be a choice sequence, and let k V weakly rationalize c k. Then, there exists V such that k in the closed convergence topology. Furthermore, if k also weakly rationalizes c k, then k.

56 Example: Expected utility Let Π be finite set of prizes and (Π) set of lotteries. nonconstant expected utility preference are locally strict. V is (homeomorphic to) S = {u R X : x u x = 0, u = 1}; so compact.

57 Utility representations We need a canonical utility representation. Here we use the equal coordinates idea. For X = R n it s literally the ray of equal coordinates. For X = ([a, b]) it s [a, b].

58 Model Let M X s.t M is connected and totally ordered by <. m M and nbd U of m in X m, m M, with m [m, m] U. (If m is not the largest element of M we can choose m such that m < m, and if m is not the smallest element we can choose m such that m < m.) Any bd seq in X is bounded by elements of M.

59 Homeomorphism Let Φ : U R mon such that Φ(u) is the preference represented by u U. Equivalence relation on U; ˆΦ : U/ R is defined in the natural way. Theorem ˆΦ is a homeomorphism.

60 in the limit = identification Theorem Suppose that and are two complete and continuous binary relations. Suppose that is locally strict, and let B X be dense. If B B B B, then =.

61 in the limit = identification Theorem Suppose that and are two continuous and complete binary relations. Suppose X is connected, and let B X be dense. If B B = B B, then =.

62 in the limit = identification Example Let X = [0, 1] [2, 3] and B = ([0, 1) (2, 3]), and observe that the rankings x y iff x y and x y iff x y or (x, y) = (1, 2) have the same restriction to B. e.g. can be represented by u(x) = x and by u(x) = x on [0, 1] and u(x) = x 1 on [2, 3].

63 Topology on preferences Basic requirement for a topology on preferences is that preferences that are close should have similar choice behavior. If (x n, y n ) (x, y), n, x y, then x n n y n for all n large enough. Close preference have the same choice behavior for alternatives that are close to each other.

64 Topology on preferences Let X be Polish and locally compact. Then the topology of closed convergence is the smallest topology for which the sets are open. {(x, y, ) : x y}

65 Topology on preferences Let X be metrizable. Let F n be a sequence of closed sets in X. Li(F n ) and Ls(F n ) are closed subsets of X defined by: x Li(F n ) iff for all nbd V of x there is N N such that F n V for all n N. x Ls(F n ) iff for all nbd V of x and all N N there is n N s.t F n V. Note: Li(F n ) Ls(F n ). Definition F n converges to F in the topology of closed convergence if Li(F n ) = F = Ls(F n ).

66 Topology on preferences Lemma Let (X, d) be a locally compact separable metric space. The set of all closed subsets of X, endowed with the topology of closed convergence, is a compact metrizable space.

67 Related literature With demand theory primitives: Mas-Colell (Restud 78); closest to us. Reny (Ecma 2015) Kübler and Polemarchaskis (Ecma forth) Polemarchakis, Selden and Song (2017) Utility homeomorphisms. Mas-Colell (JET 74) Border and Segal (JET 92) Topologies on preferences: Literature in the 70s (papers by Debreu, Kannai, Grodal and Hildenbrand).

68 Conclusion Convergence of finite-experiment rationalizing preferences. Some pathological examples. Sufficient conditions for convergence of preferences. Convergence of utilities is more subtle.