Average choice. David Ahn Federico Echenique Kota Saito. Harvard-MIT, March 10, 2016

Transcription

1 David Ahn Federico Echenique Kota Saito Harvard-MIT, March 10, 2016

2 Path independence This paper: An exploration of path independence in stochastic choice.

3 Plott path independence Plott (Ecma 1973) in response to Arrow s impossibility theorem: c(a B) = c(c(a) c(b)) For example: if c(x, y) = x, then c(x, y, z) = c(c(x, y) c(z)) = c(x, z) c = choice

4 Plott path independence Kalai-Megiddo (Ecma 1980) Machina-Parks (Ecma 1981) NO stochastic* choice can be continuous and Plott Path Indep. Restore the impossibility. Primitive: stochastic* choice. (I ll explain what I mean by stochastic*).

5 Path independence We: allow the path to affect choice. Choice from A B is a lottery between choice from A and choice from B. Who A and B are may affect the lottery.

6 Our main result Theorem A stochastic* choice is cont. and path independent iff it is a cont. Luce (or Logit) rule.

7 Our main result Kalai-Megiddo and Machina-Parks Impossibility thm.: NO stochastic* choice can be cont. and PPI. Our paper: tweaking PPI avoids impossibility and characterizes the Luce model.

8 Stochastic choice Stochastic choice: for each A, given prob. of choosing x out of A. Average (= stochastic*) choice: given the average (or mean) stochastic choice from A.

9 Stochastic choice at McDonalds Burger Cheese burger Fries Drink prob Combo Combo Kids menu avg For ex. standard IO models (Berry-Levinsohn-Pakes).

10 Why average choice? Aggregate data can be available when choice frequencies are not. Aggregate data can be more reliably estimated. Allows us to understand how utility depends on object characteristics. This is how economists use the Logit model.

11 Main thm. Luce or Logit: : u(x) ρ(x, A) = y A u(y) ρ (A) = x A xρ(x, A) Luce model iff Path independence Continuity

12 Results - II Characterization of the (ordinally) linear Luce model: ρ(x, A) = u(x) y A u(y) = f (v x) y A f (v y) : ρ (A) = x A xρ(x, A) An avg. choice is cont. PI, and independent iff it is a linear Luce rule.

13 Results - III Characterization of the (cardinally) affine Luce model: : ρ(x, A) = u(x) y A u(y) = y A ρ (A) = x A xρ(x, A) v x + β (v y + β) An avg. choice is cont. PI, independent, and calibrated iff it is an affine Luce rule.

14 Small sample advantage. Luce s IIA ρ(x, {x, y}) ρ(x, {x, y, z}) = ρ(y, {x, y}) ρ(y, {x, y, z}) Theory: ρ is observed. Reality: ρ is estimated.

15 Small sample advantage. Estimating frequencies can require large samples. Luce (1959): need 1000s of observations to test his model. It is clear that rather large sample sizes are required from each subset to obtain sensitive direct tests of axiom 1. avoids the problem.

16 Primitive Let X be a compact and convex subset of R n, with n 2. For ex. X = (P) and P set of prizes

17 Primitive Let X be a compact and convex subset of R n, with n 2. For ex. X = (P) and P set of prizes Let A be the set of all finite subsets of X.

18 Primitive Let X be a compact and convex subset of R n, with n 2. For ex. X = (P) and P set of prizes Let A be the set of all finite subsets of X. An average choice is a function ρ : A X, such that, for all A A, ρ (A) conva.

19 Luce model A stochastic choice is a function ρ : A (X ) s.t. ρ(a) (A). ρ : A (X ) is a continuous Luce rule if a cont. u : X R ++ s.t. u(x) ρ(x, A) = y A u(y).

20 Luce rationalizable ρ is continuous Luce rationalizable if a cont. Luce rule ρ s.t. ρ (A) = x A xρ(x, A).

21 Luce rationalizable ρ is continuous Luce rationalizable if a cont. Luce rule ρ s.t. ρ (A) = x A xρ(x, A). i.e. if cont. u : X R ++ s.t. ρ (A) = ( ) u(x) x A y A u(y) x.

22 Path independence If A B = then for some λ (0, 1). ρ (A B) = λρ (A) + (1 λ)ρ (B),

23 Path independence Contrast with Plott P.I.: ρ (A B) = ρ ({ρ (A), ρ (B)}).

24 Path independence Contrast with Plott P.I.: ρ (A B) = ρ ({ρ (A), ρ (B)}). Let ρ (A) = ρ (A ). Then PPI demands: ρ (A B) = ρ (A B). We allow for the path to matter through the weights on ρ (A) = ρ (A ) and ρ (B).

25 Path independence x ρ ({x, y, z}) ρ ({x, y}) z y

26 Path independence x ρ ({x, y, z}) ρ ({x, y}) z y

27 Path independence x ρ ({x, y, z}) ρ ({x, y}) z y Luce s IIA: ρ(x, {x, y}) ρ(x, {x, y, z}) = ρ(y, {x, y}) ρ(y, {x, y, z})

28 Path independence x y z w

29 Path independence x y z w

30 Violation of path independence x y x y ρ (A \ {x}) ρ (A \ {y}) z w z w

31 Violation of path independence x y ρ (A)? z w

32 Continuity Let x / A. For any sequence x n in X, if x = lim n x n, then ρ (A {x}) = lim n ρ (A {x n }).

33 Theorem An average choice is continuous Luce rationalizable iff it satisfies continuity and path independence.

34 Proof sketch Necessity: ρ (A) = ( ) u(x) x A y A u(y) x

35 Proof sketch Necessity: ρ (A) = ( ) u(x) x A y A u(y) x u(x) + u(x) ρ (A B) = u(x)x + u(x)x x A y B x A x B

36 Proof sketch Necessity: ρ (A) = ( ) u(x) x A y A u(y) x u(x) + u(x) ρ (A B) = u(x)x + u(x)x x A y B x A x B = ρ (A) x A u(x) + ρ (B) x B u(x);

37 Plott path independence If conva convb = then ρ (A B) = ρ ({ρ (A), ρ (B)}).

38 Plott path independence Kalai-Megiddo (Ecma 1980) and Machina-Parks (Ecma 1981): Theorem If ρ is continuous then it cannot satisfy Plott path independence.

39 Plott path independence Proposition If an average choice is continuous Luce rationalizable, then it cannot satisfy Plott path independence.

40 Plott path independence Let x, y, z X be aff. indep.. PPI ρ ({x, y, z}) = ρ (ρ ({x, y}), {z}) = u(ρ ({x, y}))ρ ({x, y}) + u(z)z u(ρ. ({x, y})) + u(z)

41 Plott path independence Let x, y, z X be aff. indep.. PPI ρ ({x, y, z}) = ρ (ρ ({x, y}), {z}) = u(ρ ({x, y}))ρ ({x, y}) + u(z)z u(ρ. ({x, y})) + u(z) But ρ ({x, y, z}) = u(x)x + u(y)y + u(z)z. u(x) + u(y) + u(z)

42 Plott path independence By aff. indep.: u(z) u(x) + u(y) + u(z) = u(z) u(ρ ({x, y})) + u(z). u(ρ ({x, y})) = u(x) + u(y).

43 Plott path independence So: u(x) + u(y) = u(ρ ({x, y})) = u ( ) u(x)x + u(y)y. u(x) + u(y) Choose y arbitrarily close to x while satisfying aff. indep. Then u(x) = 2u(x), a contradiction as u(x) > 0.

44 Plott path independence Proposition No average choice satisfies Plott path independence and (our) path independence.

45 Linear Luce ρ is a linear Luce rule if v R n ; and a monotone and cont. f : R R ++ s.t. ρ(x, A) = f (v x) f (v y). y A

46 Independence u(x) = u(y) iff λ, z ρ ({λx + (1 λ)z, λy + (1 λ)z}) = λρ ({x, y}) + (1 λ)z

47 x ρ (x, y) y ρ (λx + (1 λ)z, λy + (1 λ)z) z

48 x ρ (x, y) y ρ (λx + (1 λ)z, λy + (1 λ)z) z ρ (λ x + (1 λ )z, λ y + (1 λ )z)

49 Linear Luce Theorem An average choice is continuous linear Luce rationalizable iff it satisfies independence, continuity and path independence.

50 Linear Luce Let ρ be cont. Luce rationalizable. Lemma If ρ satisfies independence then u(x) = u(y) iff u(λx + (1 λ)z) = u(λy + (1 λ)z) λ, z

51 Linear Luce Let ρ be cont. Luce rationalizable. Lemma If ρ satisfies independence, then u(x) u(y) iff u(λx + (1 λ)z) u(λy + (1 λ)z) λ, z.

52 Strictly affine Luce ρ is a strictly affine Luce rule if v R n ; and β R s.t. v x + β ρ(x, A) = (v y + β). y A

53 Calibration ρ ({λx + (1 λ)y, λy + (1 λ)x}) = ρ ({x, y}) + 2λ(1 λ) (x + y 2ρ ({x, y}))

54 Calibration Interpret λx + (1 λ)y and λy + (1 λ)x as two perfectly correlated lotteries. ρ ({λx + (1 λ)y, λy + (1 λ)x}) = (λ 2 + (1 λ) 2 u(x)x + u(y)y ) u(x) + u(y) u(y)x + u(x)y + (2λ(1 λ)) u(x) + u(y)

55 Calibration ( x y ) λ ( x y ) ρ λ 1 λ ( y x ) 1 λ λ 1 λ ( y x ) ( x y ) ( y x )

56 Strictly affine Luce Theorem An average choice is strictly affine Luce rationalizable iff it satisfies calibration, independence, continuity and path independence.

57 On continuity and Debreu s example Debreu s example: ρ(t, {t, b}) ρ(b, {t, b}) ρ(t, {t, b, b }) ρ(b, {t, b, b })

58 On continuity and Debreu s example Let ρ be continuous Luce rationalizable. z n x then: ρ ({x, y}) 2u(x)x + u(y)y 2u(x) + u(y) = lim n ρ ({x, y, z n }). Thus ρ must be discontinuous.

59 Finite sample test Theory: ρ is observed. Reality: ρ is estimated.

60 Finite sample test Fix A. Estimate ρ(x, A) by sampling from ρ. Luce (1959): It is clear that rather large sample sizes are required from each subset to obtain sensitive direct tests of axiom 1.

61 Finite sample test Fix A. Population choices from A are given by a Luce rule p(a). Observe iid sample X 1,..., X k of choices: X i A for i = 1,..., k. px k = i : X i = x. k

62 Finite sample test Two possibilities to test the Luce model: 1. Use Luce s IIA. Requires: px k py k, for x, y A. 2. Use average choice: µ k = x A xp k x

63 Recall that k ( p k x p k y p ) ( ) x d N 0, 2 p2 x p y py 2. p x = u(x) p y u(y)

64 On the other hand, k(µ k µ) d N(0, Σ), where Σ = (σ l,h ) and σ l,h max{x l x k : x A}.

65 Proposition For any M, a Luce model s.t. asymptotic variance of p k a /p k b relative to max{σ l,h }, the largest element of Σ, is greater than M. The inefficiency in using ratios relative to means can be arbitrarily large.

66 Proof sketch.

67 Recall: Theorem An average choice is continuous Luce rationalizable iff it satisfies continuity and path independence.

68 Proof sketch Necessity: ρ (A) = ( ) u(x) x A y A u(y) x

69 Proof sketch Necessity: ρ (A) = ( ) u(x) x A y A u(y) x u(x) + u(x) ρ (A B) = u(x)x + u(x)x x A y B x A x B

70 Proof sketch Necessity: ρ (A) = ( ) u(x) x A y A u(y) x u(x) + u(x) ρ (A B) = u(x)x + u(x)x x A y B x A x B = ρ (A) x A u(x) + ρ (B) x B u(x);

71 Proof sketch Sufficiency: First determine ρ on A with cardinality 2 and 3. A = {x, y} A = {x, y, z} with x, y, z affinely indep.

72 Proof sketch Sufficiency: Determine ρ(x, {x, y}) and ρ(y, {x, y}) from ρ ({x, y}) = xρ(x, {x, y}) + yρ(y, {x, y}).

73 Proof sketch Sufficiency: Determine ρ(x, {x, y}) and ρ(y, {x, y}) from ρ ({x, y}) = xρ(x, {x, y}) + yρ(y, {x, y}). For affinely indep. x, y, z, ρ(x, {x, y, z}), ρ(y, {x, y, z}) and ρ(z, {x, y, z}) are also determined from ρ ({x, y, z}).

74 Proof sketch By path independence, θ s.t. ρ (A) = θz + (1 θ)ρ (A \ {z}) = θz + (1 θ)[xρ(x, {x, y}) + yρ(y, {x, y})]. x, y and z are affinely indep., ρ(x, A), ρ(y, A) and ρ(z, A) are unique; thus ρ(x, A) = (1 θ)ρ(x, {x, y}) and ρ(y, A) = (1 θ)ρ(y, {x, y}).

75 Proof sketch ρ(x, A) = (1 θ)ρ(x, {x, y}) and ρ(y, A) = (1 θ)ρ(y, {x, y}). Hence ρ(x, A) ρ(x, {x, y}) = ρ(y, A) ρ(y, {x, y}). Luce s IIA!

76 Proof sketch We can define a Luce rule with utility u. Fix x X. Let u(x ) = 1, and Then, so u defines a Luce rule. Let ρ be the implied avg. choice. We need to show ρ = ρ. u(x) = ρ(x, {x, x }) ρ(x, {x, x }). u(x) ρ(x, {x, y}) = u(y) ρ(y, {x, y})

77 Proof sketch The proof is by induction on A. We know that ρ(a) = ρ (A) when A 3.

78 Proof sketch Crucial lemma (roughly stated): Let A A be generic then there is x, y A s.t. is a singleton. conv 0 ({x, ρ (A \ {x})}) conv 0 ({y, ρ (A \ {y})})

79 x z w ρ (A \ {y}) ρ (A) q ρ (A \ {x}) y r

80 Proof sketch ρ(a \ {z}) = ρ (A \ {z}) So conv 0 ({x, ρ(a \ {x})}) = conv 0 ({x, ρ (A \ {x})}) conv 0 ({y, ρ(a \ {y})}) = conv 0 ({y, ρ (A \ {y})}) conv 0 ({x, ρ (A \ {x})}) conv 0 ({y, ρ (A \ {y})}) being a singleton, and path independence Implies ρ(a) = ρ (A).

81 Conclusion Plott s path independent choice leads to an impossibility. A simple modification of path independence, allowing the path to affect the weights of stochastic choice, avoid the impossibility. Our modified path independence and cont. pins down a unique choice: Luce s rule.