D I S C O N T I N U O U S DEMAND FUNCTIONS: ESTIMATION AND PRICING. Rotterdam May 24, 2018

Transcription

1 D I S C O N T I N U O U S DEMAND FUNCTIONS: ESTIMATION AND PRICING Arnoud V. den Boer University of Amsterdam N. Bora Keskin Duke University Rotterdam May 24, 2018

2 Dynamic pricing and learning: Learning optimal selling price from accumulating sales data

3 Dynamic pricing and learning: Learning optimal selling price from accumulating sales data Cont. armed MAB, observing demand d(p) and reward r(p) = p d(p)

4 Dynamic pricing and learning: Learning optimal selling price from accumulating sales data Cont. armed MAB, observing demand d(p) and reward r(p) = p d(p) Standard assumption: d( ) is continuous

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6 Nous admettons que la fonction F (p) qui exprime la loi de la demande ou du débit est une fonction continue...

7 Motivation Is the assumption of continuous demand functions reasonable?

8 Motivation Is the assumption of continuous demand functions reasonable? Price comparison websites: Substantial empirical evidence that seller s rank heavily influences its demand. Ignoring these discontinuities may distort parameter estimates by 50 to 100 percent. (Baye et al., J. Econ. Manag. Strategy 2009)

9 Motivation Is the assumption of continuous demand functions reasonable? Price comparison websites: Substantial empirical evidence that seller s rank heavily influences its demand. Ignoring these discontinuities may distort parameter estimates by 50 to 100 percent. (Baye et al., J. Econ. Manag. Strategy 2009) Rankings in online marketplaces (e.g. Amazon s BuyBox)

10 Motivation

11 Motivation

12 Motivation Is the assumption of continuous demand functions reasonable? Price comparison websites: Substantial empirical evidence that seller s rank heavily influences its demand. Ignoring these discontinuities may distort parameter estimates by 50 to 100 percent. (Baye et al., J. Econ. Manag. Strategy 2009) Rankings in online marketplaces (e.g. Amazon s BuyBox)

13 Motivation Is the assumption of continuous demand functions reasonable? Price comparison websites: Substantial empirical evidence that seller s rank heavily influences its demand. Ignoring these discontinuities may distort parameter estimates by 50 to 100 percent. (Baye et al., J. Econ. Manag. Strategy 2009) Rankings in online marketplaces (e.g. Amazon s BuyBox) Product search with price thresholds

14 Motivation

15 Motivation

16 Motivation Many online applications challenge Cournot s continuity assumption

17 Motivation Many online applications challenge Cournot s continuity assumption Not treated in dynamic pricing or MAB literature

18 Central questions 1 Is there a substantial cost of neglecting demand discontinuities in dynamic pricing and learning?

19 Central questions 1 Is there a substantial cost of neglecting demand discontinuities in dynamic pricing and learning? 2 If yes, how to implement estimation and pricing in the presence of demand discontinuities?

20 Model Price-setting monopolist: decision variable p t [p, p]

21 Model Price-setting monopolist: decision variable p t [p, p] Consumer demand: Poisson random variable with mean d(p t ) d(p) = { e α 0+β 0p if κ 0 p κ 1 e αn+βnp if κ n < p κ n+1 (n = 1,..., N)

22 Model Price-setting monopolist: decision variable p t [p, p] Consumer demand: Poisson random variable with mean d(p t ) d(p) = { e α 0+β 0p if κ 0 p κ 1 e αn+βnp if κ n < p κ n+1 (n = 1,..., N) Model uncertainty: unknown demand parameters θ n = (α n, β n ) (n = 0, 1,..., N) unknown discontinuity points κ n (n = 1,..., N) θ = (θ 0, θ 1,..., θ N ) Θ κ = (κ 1,..., κ N ) K

23 Model Price-setting monopolist: decision variable p t [p, p] Consumer demand: Poisson random variable with mean d(p t ) d(p) = { e α 0+β 0p if κ 0 p κ 1 e αn+βnp if κ n < p κ n+1 (n = 1,..., N) Model uncertainty: unknown demand parameters θ n = (α n, β n ) (n = 0, 1,..., N) unknown discontinuity points κ n (n = 1,..., N) θ = (θ 0, θ 1,..., θ N ) Θ κ = (κ 1,..., κ N ) K Pricing policy: π = (p 1, p 2,...) non-anticipating

24 Performance Revenue loss in T periods relative to a clairvoyant

25 Performance Revenue loss in T periods relative to a clairvoyant Single-period revenue function R(p, κ, θ) = p d(p, κ, θ)

26 Performance Revenue loss in T periods relative to a clairvoyant Single-period revenue function R(p, κ, θ) = p d(p, κ, θ) Regret or revenue loss due to demand model uncertainty π (T, κ, θ) = T { E π sup t=1 p [p,p] } {R(p, κ, θ)} R(p t, κ, θ)

27 Performance Revenue loss in T periods relative to a clairvoyant Single-period revenue function R(p, κ, θ) = p d(p, κ, θ) Regret or revenue loss due to demand model uncertainty π (T, κ, θ) = T { E π sup t=1 Objective: choose π to minimize p [p,p] } {R(p, κ, θ)} R(p t, κ, θ) R π (T ) = sup { π (T, κ, θ) : κ K, θ Θ }

28 Central questions 1 Is there a substantial cost of neglecting demand discontinuities in dynamic pricing and learning? 2 If yes, how to implement estimation and pricing in the presence of demand discontinuities?

29 Cost of ignoring demand discontinuities

30 Cost of ignoring demand discontinuities

31 Cost of ignoring demand discontinuities

32 Cost of ignoring demand discontinuities

33 Cost of ignoring demand discontinuities No discontinuity Loss T 1/2

34 Cost of ignoring demand discontinuities Ignored discontinuity Loss T No discontinuity Loss T 1/2

35 Central questions 1 Is there a substantial cost of neglecting demand discontinuities in dynamic pricing and learning? 2 If yes, how to implement estimation and pricing in the presence of demand discontinuities?

36 Estimating a discontinuous demand function Two-step maximum likelihood estimation: Log-likelihood function t N ( L t : (κ, ϑ) ds ϑ n (1, p s ) e ϑn (1,ps)) I{κ n < p s κ n+1 } s=1 n=0 ˆθ t (κ) = arg max ϑ {L t (κ, ϑ)}

37 Estimating a discontinuous demand function Two-step maximum likelihood estimation: Log-likelihood function t N ( L t : (κ, ϑ) ds ϑ n (1, p s ) e ϑn (1,ps)) I{κ n < p s κ n+1 } s=1 n=0 ˆθ t (κ) = arg max ϑ {L t (κ, ϑ)} Step 1 (discontinuity estimation) { ( ˆκ t = arg max Lt κ, ˆθt (κ) )} κ

38 Estimating a discontinuous demand function Two-step maximum likelihood estimation: Log-likelihood function t N ( L t : (κ, ϑ) ds ϑ n (1, p s ) e ϑn (1,ps)) I{κ n < p s κ n+1 } s=1 n=0 ˆθ t (κ) = arg max ϑ {L t (κ, ϑ)} Step 1 (discontinuity estimation) { ( ˆκ t = arg max Lt κ, ˆθt (κ) )} κ Step 2 (demand parameter estimation) ˆθ t = ˆθ t (ˆκ t )

39 Estimating a discontinuous demand function

40 Estimating a discontinuous demand function

41 Estimating a discontinuous demand function p (2) ˆκ 1 < p (3)

42 Estimating a discontinuous demand function p (3) ˆκ 1 < p (4)

43 Estimating a discontinuous demand function p (4) ˆκ 1 < p (5)

44 Estimating a discontinuous demand function p (5) ˆκ 1 < p (6)

45 Estimating a discontinuous demand function p (6) ˆκ 1 < p (7)

46 Estimating a discontinuous demand function p (7) ˆκ 1 < p (8)

47 Estimating a discontinuous demand function p (8) ˆκ 1 < p (9)

48 Estimating a discontinuous demand function Highest likelihood if p (4) ˆκ 1 < p (5).

49 Designing a near-optimal policy Time horizon {1,..., T }. Discontinuity estimation and pricing policy π

50 Designing a near-optimal policy Time horizon {1,..., T }. Discontinuity estimation and pricing policy π (1) [Explore] Use M equidistant prices p = p 1 < < p M = p.

51 Designing a near-optimal policy Discontinuity estimation and pricing policy π Time horizon {1,..., T }. (1) [Explore] Use M equidistant prices p = p 1 < < p M = p. (2) [Estimate] Compute ˆκ and ˆθ.

52 Designing a near-optimal policy Discontinuity estimation and pricing policy π Time horizon {1,..., T }. (1) [Explore] Use M equidistant prices p = p 1 < < p M = p. (2) [Estimate] Compute ˆκ and ˆθ. (3) [Exploit] Based on ˆκ and ˆθ, use the estimated optimal price in the remaining T M periods,

53 Designing a near-optimal policy Discontinuity estimation and pricing policy π Time horizon {1,..., T }. (1) [Explore] Use M equidistant prices p = p 1 < < p M = p. (2) [Estimate] Compute ˆκ and ˆθ. (3) [Exploit] Based on ˆκ and ˆθ, use the estimated optimal price in the remaining T M periods, but a factor log(m)/m away from the estimated discontinuities.

54 Analysis of estimation errors Theorem (discontinuity estimation error) There exist constants M 1, z 1, γ 1 > 0 such that, if M M 1, then { P π ˆκ n κ n > z } 1 log M for some n = 1,..., N γ 1 M M.

55 Analysis of estimation errors Theorem (discontinuity estimation error) There exist constants M 1, z 1, γ 1 > 0 such that, if M M 1, then { P π ˆκ n κ n > z } 1 log M for some n = 1,..., N γ 1 M M. Theorem (parameter estimation error) There exist constants M 2, z 2, γ 2 > 0 such that, if M M 2, then { P π ˆθ n θ n 2 > z } 2 log M for some n = 0, 1,..., N γ 2 M M.

56 Sufficient condition for good performance Theorem (upper bound on regret) There exists a constant C > 0 such that, if M = T, then R π (T ) C T log T for all T 4(N + 1) 2.

57 Summary of results Ignored discontinuity Loss T No discontinuity Loss T 1/2

58 Summary of results Ignored discontinuity Loss T No discontinuity Loss T 1/2 Discontinuity estimation Loss T 1/2 logt

59 Some intuition

60 Extensions What if discontinuities vary over time?

61 Extensions What if discontinuities vary over time? Include change-point detection module in policy Retains O( T log T ) regret

62 Extensions What if discontinuities vary over time? Include change-point detection module in policy Retains O( T log T ) regret What if there are inventory constraints?

63 Extensions What if discontinuities vary over time? Include change-point detection module in policy Retains O( T log T ) regret What if there are inventory constraints? Asymptotic regime, inventory ξ T, T.

64 Extensions What if discontinuities vary over time? Include change-point detection module in policy Retains O( T log T ) regret What if there are inventory constraints? Asymptotic regime, inventory ξ T, T. Include stochastic-approximation module in policy

65 Extensions What if discontinuities vary over time? Include change-point detection module in policy Retains O( T log T ) regret What if there are inventory constraints? Asymptotic regime, inventory ξ T, T. Include stochastic-approximation module in policy Retains O( T log T ) regret

66 Message of this talk

67 Message of this talk Neglecting discontinuities can cost a lot (linear regret)

68 Message of this talk Neglecting discontinuities can cost a lot (linear regret) Taking it into account retains asymptotic optimality

69 Message of this talk Neglecting discontinuities can cost a lot (linear regret) Taking it into account retains asymptotic optimality Extensions in the paper: changing discontinuity points, inventory constraints

70 Message of this talk Neglecting discontinuities can cost a lot (linear regret) Taking it into account retains asymptotic optimality Extensions in the paper: changing discontinuity points, inventory constraints Interesting research problems:

71 Message of this talk Neglecting discontinuities can cost a lot (linear regret) Taking it into account retains asymptotic optimality Extensions in the paper: changing discontinuity points, inventory constraints Interesting research problems: - Rank-induced discontinuities in other problems?

72 Message of this talk Neglecting discontinuities can cost a lot (linear regret) Taking it into account retains asymptotic optimality Extensions in the paper: changing discontinuity points, inventory constraints Interesting research problems: - Rank-induced discontinuities in other problems? - Nonparametric approach to discontinuous MABs.

73 THE END