Selling to Cournot oligopolists: pricing under uncertainty & - - PowerPoint PPT Presentation

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Selling to Cournot oligopolists: pricing under uncertainty & generalized mean residual life

Stefanos Leonardos1 and Costis Melolidakis

National and Kapodistrian University of Athens Department of Mathematics, Division of Statistics & Operations Research

January 9, 2018

1Supported by the Alexander S. Onassis Public Benefit Foundation.

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Overview

1

Introduction

2

The Model

3

Market equilibrium: existence & uniqueness Supplier’s optimal pricing decision MRL and GMRL functions

4

The DGMRL class

5

Examples – Future Research

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Outline

1

Introduction

2

The Model

3

Market equilibrium: existence & uniqueness Supplier’s optimal pricing decision MRL and GMRL functions

4

The DGMRL class

5

Examples – Future Research

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Motivation

Pricing decisions

1 often made under incomplete information. 2 common problem: conditions on the distribution of the parameter of

uncertainty that ensure a unimodal and hence “well behaved” revenue function. In the present paper

1 a supplier sets wholesale price, uncertain abou the exact retailers’ val-

uation of his product.

2 having no uncertainty about the market demand, the retailers engage in

a classic Cournot competition with marginal cost equal to the wholesale price set by the supplier. Objective: optimize the supplier’s pricing-decision.

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The Upshot

Implications of our study are twofold

1 economic-modelling perspective: endogenize the cost-formation in the

classic Cournot model in a way that is both realistic and mathematically tractable.

2 technical perspective: identify and study a mild unimodality condition

that ensures a well-behaved stochastic revenue function. Condition: decreasing generalized mean residual life (DGMRL)

1 suitable generalization of the mean residual life (MRL) function, 2 arises naturally in the general context of Cournot competition with

endogenous cost formation under stochastic demand and

3 generalizes the widely used increasing generalized failure rate (IGFR)

condition.

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Outline

1

Introduction

2

The Model

3

Market equilibrium: existence & uniqueness Supplier’s optimal pricing decision MRL and GMRL functions

4

The DGMRL class

5

Examples – Future Research

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The Model I

We consider the following two-stage game (supply chain) Upstream supplier:

1 enough capacity to cover any possible demand from the retailers 2 constant marginal cost, normalized to zero (not trivial) 3 decision variable: wholesale price r.

Downstream retailers:

1 fixed number of n ≥ 1 competing Cournot firms. 2 decision variable: order quantity to the supplier. 3 marginal cost r: wholesale price determined strategically.

Cost is endogenized in the classic Cournot competition.

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The Model II

Market structure:

1 single homogeneous good. 2 affine inverse demand function

p = α − q (r | α) where α is the demand parameter and q (r | α) the total quantity. Demand uncertainty:

1 demand parameter α is realized after supplier’s pricing decision but

prior to retailers’ quantity decisions.

2 supplier is uncertain about the retailers’ willingness-to-pay for his price.

All the above are common knowledge among supplier & retailers.

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Formal setting

Extensive, 2-stage game. Strategy sets: retailers (followers): qi : R≥0 → R≥0, order-quantity for price r. supplier (Stackelberg leader): wholesale price r > 0. Payoff functions: retailers: ui (q | r) = qi (α − q) − rqi = qi (α − r − q) supplier: us (r) = Eus (r | α) = r · Eq (r | α) Demand uncertainty: α is realized from a: continuous cdf F, F := 1 − F, with finite expec- tation Eα < +∞, and nonnegative support contained 0 ≤ αL ≤ αH ≤ +∞.

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Outline

1

Introduction

2

The Model

3

Market equilibrium: existence & uniqueness Supplier’s optimal pricing decision MRL and GMRL functions

4

The DGMRL class

5

Examples – Future Research

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Outline

1

Introduction

2

The Model

3

Market equilibrium: existence & uniqueness Supplier’s optimal pricing decision MRL and GMRL functions

4

The DGMRL class

5

Examples – Future Research

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Base case: deterministic demand

The supplier knows the demand parameter α: αL = αH. Proposition The complete information two-stage game has a unique subgame perfect Nash equilibrium, under which the supplier’s optimal price is r ∗ (α) = α/2 and each retailer orders quantity q∗

i (r) = 1 3 (α − r)+.

Proof: Second stage Given r, α the retailers play a classic Cournot duopoly. Hence, unique equilibrium strategies: q∗

i (r) = 1

3 (α − r)+ Uniqueness implies that the supplier correctly predicts the retailers’

• rder-quantities.

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Base case: deterministic demand

The supplier knows the demand parameter α: αL = αH. Proposition The complete information two-stage game has a unique subgame perfect Nash equilibrium, under which the supplier’s optimal price is r ∗ (α) = α/2 and each retailer orders quantity q∗

i (r) = 1 3 (α − r)+.

Proof: First stage Consider only subgame perfect equlibria On the equilibrium path, the supplier’s payoff is us (r | α) = rq∗ (r | α) = 2 3r (α − r)+ , for 0 ≤ r. here: two second stage retailers.

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General case: stochastic demand

The supplier has incomplete information about α: αL < αH. On the equilibrium path, the supplier’s payoff is us (r) = 2 3r E (α − r)+ , for 0 ≤ r ≤ αH. Maximization with respect to r is not straightforward us (r) may not be concave, hence not unimodal. an optimal price may not even exist. requiring concavity: too restrictive. Goal: derive a mild unimodality condition for us (r) us (r) = 2 3r E (α − r)+ = 2 3r

∞

r

F (u) du

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Outline

1

Introduction

2

The Model

3

Market equilibrium: existence & uniqueness Supplier’s optimal pricing decision MRL and GMRL functions

4

The DGMRL class

5

Examples – Future Research

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Mean residual life (MRL) function

Definition The MRL function m (·) of a nonnegative random variable α with cdf F and Eα < +∞, is defined as m (r) := E (α − r | α > r) = 1 F (r)

∞

r

F (u) du, for r < αH and m (r) := 0, otherwise. Express us (r) in terms of m (r) and differentiate (as if with product rule) us (r) = 2 3rm (r)F (r) dus dr (r) = 2 3 (m (r) − r)F (r) = 2 3r

m (r)

r − 1

• F (r)

← FOC? for 0 < r < αH.

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Generalized mean residual life (GMRL) function

Definition The GMRL function m (·) of a nonnegative random variable α with cdf F and Eα < +∞, is defined as ℓ (r) := m (r) r , 0 < r < αH and ℓ (r) := 0, otherwise. Interpretation: expected additional demand as a percentage of the given.

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Generalized mean residual life (GMRL) function

Definition The GMRL function m (·) of a nonnegative random variable α with cdf F and Eα < +∞, is defined as ℓ (r) := m (r) r , 0 < r < αH and ℓ (r) := 0, otherwise. From the previous slide: dus dr (r) = 2 3 (m (r) − r)F (r) = 2 3r

m (r)

r − 1

• F (r)

1 m (r) is weakly decreasing: F is (DMRL) or 2 (i) m (r) /r is strictly decreasing and (ii) eventually < 1.

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Fixed-Point Theorem

Theorem If the supplier has incomplete information α ∼ F, then

1 (necessary) If an optimal price r ∗ of the supplier exists, then it satisfies

the fixed point equation r ∗ = m (r ∗) (1)

2 (sufficient) If F is strictly DGMRL and Eα2 is finite, then r ∗ exists

and is the unique solution of (1). In this case, if 1

2Eα < αL, then

r ∗ = 1

2Eα. Otherwise, r ∗ ∈ [αL, αH).

Remarks: Two technicalities limr→+∞ ℓ (r) < 1 iff Eα2 < +∞. m (r) = r over an interval iff α ∼ Pareto (2) over this interval.

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On a silver platter

Theorem For any DMRL distribution F, the probability F (r ∗) of no transaction in equilibrium satisfies F (r ∗) ≤ 1 − e−1 This bound is robust: independent of the particular distribution F, and tight: attained by the exponential and, asymptotically, by a parametric Beta distribution. Proof: immediate through MRL representation of r ∗. Generalization to any n ≥ 1 identical retailers: statement of main Theo- rem independent of n (due to second-stage equilibrium uniqueness).

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Outline

1

Introduction

2

The Model

3

Market equilibrium: existence & uniqueness Supplier’s optimal pricing decision MRL and GMRL functions

4

The DGMRL class

5

Examples – Future Research

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The DGMRL & IGFR classes I

The (GMRL) function is related to the (GFR) function introduced by Lariviere & Porteus (2001) g (r) := rh (r) , where h (r) := f (r)

F (r) is the failure rate function.

Widely used unimodality condition: g (r) is increasing (IGFR). Satisfied by most commonly used probability distributions. For historical accuracy: Singh & Maddala (1976) first to use GFR in income distributions mod- elling under the name proportional failure rate function. Belzunce et.al (1998): define the IGFR & DGMRL classes via stochastic

• rderings.

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The DGMRL & IGFR classes II

Theorem If X is a nonnegative, absolutely continuous random variable, with EX < +∞, then

1 If X is IGFR, then X is DGMRL. 2 If X is DGMRL and m (r) is log-convex, then X is IGFR.

Proof: Part 1: Belzunce et.al (1998). Part 2: log-convexity is restrictive (more general condition open). Question: how much larger is the DGMRL class? IGFR is already partic- ularly inclusive.

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Relations between classes

Relations between the IFR, IGFR, DMRL and DGMRL classes IFR IGFR DMRL DGMRL IGFR property does not imply, nor is implied by the DMRL property. DGMRL class generalizes the IGFR class in the following cases:

1 technical: there exists λ ≥ 1 such that X MRL λX but X HR λX. 2 important for economic modelling: not connected support.

Conceptual difference: instantaneous vs entire behavior of the distribu- tion after point r.

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Properties of the GMRL function

Under proper (but mild) assumptions:

1 GMRL & GFR: inverse limiting behavior (Belzunce et.al 1998)

lim

r→∞ ℓ (r) =

lim

r→+∞

1 g (r) − 1

2 GMRL & regular variation:

ℓ (r) = 1 rF (r)

∞

r

F (u) du =

∞

1

F (ur) F (r) du

3 Derivative formula: a motivation

m′ (r) = m (r) h (r) − 1 = m (r) r · rh (r) − 1 = ℓ (r) g (r) − 1

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Limiting behavior & moments

Theorem Let X be a nonnegative DGMRL random variable, and let β > 0. The following are equivalent limr→+∞ ℓ (r) = c < 1

β

EX β+1 < +∞. In particular, c = 0 if and only if EX β+1 < +∞ for every β > 0. Compare with Theorem (2. Lariviere 2006) Let X be an IGFR random variable with support (α, +∞), and let β > 0. The following are equivalent limr+∞ g (r) = κ > β, with κ possibly infinite. EX β < +∞

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The DGMRL & DMRL classes: some closure properties

DMRL property preserved under

1 strictly increasing, differentiable and concave transformations. 2 αX + β is DMRL. 3 0 < α ≤ 1, X α is DMRL.

DGMRL property

1 preserved: positive scale transformations and left truncations. 2 not clear: right truncations? 3 not preserved: shifting, convolutions (same as IGFR). 4 both possible: mixing.

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Outline

1

Introduction

2

The Model

3

Market equilibrium: existence & uniqueness Supplier’s optimal pricing decision MRL and GMRL functions

4

The DGMRL class

5

Examples – Future Research

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Examples: Convolution of DGMRL

Log-logistic distribution: DGMRL not closed under convolutions.

Figure: The GMRL function for the convolution of two standard log-logistic random variables (k = 2).

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Examples: Not connected support

Mixture of disjoint uniform: DGMRL not necessarily closed under mixing.

Figure: The GMRL function of αλ ∼ λF1 + (1 − λ) F2 for λ = 1/4 (solid) and λ = 3/4 (dotted).

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Examples: Birnbaum-Saunders distribution

DGMRL but not IGFR: Birnbaum-Saunders.

Figure: Birnbaum-Saunders distribution for β = γ = 5: GFR (left), GMRL (right). Behavior depends largely on the selection of the parameters.

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Examples: Cantor distribution

Observations of independent interest.

Figure: Cantor distribution: the unique fixed point at r =

5 12 of the MRL

function (left) and the GMRL function, for values distant from 0 (right).

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Future Research I

Natural extension: retailers may produce limited capacities of their own. Theorem If the supplier has incomplete information α ∼ F and the retailers may produce Ti = T, i ∈ N (symmetric case), then

1 (necessary) If the optimal price r ∗ of the supplier exists, then it satisfies

the translated fixed point equation r ∗ = m (r ∗ + (n + 1) T + c) (2)

2 (sufficient) If F is DMRL, then r ∗ exists and is the unique solution

• f (2).

In this case, if E (α) ≤ 2αL − (n + 1) T − c (= r n

L ), then

r ∗ = 1

2 (E (α) − (n + 1) T − c). Otherwise, r ∗ ∈

• (r n

L )+ , r n H

• .

Sensitivity analysis of the parameters, comparative statics, extensions . . .

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Future Research II

Pareto Regularly varying Subexponential Long-tailed Heavy-tailed DMRL IFR Light-tailed

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Future Research II

Pareto Regularly varying Subexponential Long-tailed Heavy-tailed DMRL IFR Light-tailed IGFR?

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Future Research II

Pareto Regularly varying Subexponential Long-tailed Heavy-tailed DMRL IFR Light-tailed IGFR? DGMRL?

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Future Research II

Pareto Regularly varying Subexponential Long-tailed Heavy-tailed DMRL IFR Light-tailed IGFR? DGMRL? (eventually DGMRL?)

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About

Journal: submitted, under revision. Available online at: https://arxiv.org/abs/1709.09618. https://arxiv.org/abs/1601.07365. Conferences: (earlier version) GAMES 2016, 5th World Congress of the Game Theory Society (2016), The Netherlands. SING12, 12th European Meeting on Game Theory (2016), Denmark contributed talk. GTM16, 10th International Conference on Game Theory and Manage- ment (2016), Russia. M3ST, International Conference on Modern Mathematical Methods, Greece.

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Selected References I

[1]

• M. Banciu & P. Mirchandani (2013), Technical Note – New Results Concerning Prob-

ability Distributions with Increasing Generalized Failure Rates. Operations Research, 61(4): 925–931. [2]

• F. Belzunce, J. Candel & J. M. Ruiz (1995), Ordering of Truncated Distributions

through Concentration Curves. Sankhy¯ a: The Indian Journal of Statistics, Series A (1961-2002), 57(3): 375–383. [3]

• F. Belzunce, J. Candel & J. M. Ruiz (1998), Ordering and Asymptotic Properties of

Residual Income Distributions. Sankhy¯ a: The Indian Journal of Statistics, Series B (1960-2002), 60(2): 331–348. [4] David Bradley & Ramesh Gupta (2003), Limiting behaviour of the mean residual

• life. Annals of the Institute of Statistical Mathematics, 55(1): 217–226.

[5] Martin A. Lariviere, (1999). Supply Chain Contracting and Coordination with Stochastic Demand, Quantitative Models for Supply Chain Management, (eds.) Srid- har Tayur, Ram Ganeshan, Michael Magazine: 233-268. Springer US.

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Selected References II

[6] Martin A. Lariviere & Evan Porteus (2001), Selling to the Newsvendor: An Analysis

• f Price-Only Contracts. Manufacturing & Service Operations Management, 3(4):

293–305. [7] Martin A. Lariviere (2006), A Note on Probability Distributions with Increasing Gen- eralized Failure Rates. Operations Research, 54(3): 602–604. [8] Anand Paul (2005), A Note on Closure Properties of Failure Rate Distributions, Operations Research, 53(4):733–734. [9]

• S. K. Singh & G. S. Maddala (1976), A Function for Size Distribution of Incomes.

Econometrica, 44(5): 963–970.

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Source

Online at: https://arxiv.org/abs/1709.096188.

Thank you for your attention!