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

1/25

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

June 2, 2018

1Supported by a fellowship of the Alexander S. Onassis Public Benefit Foundation.

2/25

Overview

1

The Model

2

Market equilibrium: existence & uniqueness

3

Market efficiency

4

Comparative statics

3/25

Outline

1

The Model

2

Market equilibrium: existence & uniqueness

3

Market efficiency

4

Comparative statics

4/25

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.

5/25

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.

6/25

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 ex- pectation Eα < +∞, and nonnegative support contained in 0 ≤ αL ≤ αH ≤ +∞.

7/25

The Outcome

Implications of our study

1 technical perspective: identify and study a mild unimodality condition

that characterizes the equilibrium price and ensures a “well-behaved” non-deterministic revenue function.

2 economic perspective: obtain a mathematically tractable pricing model

and perform comparative statics and market efficiency analysis via prob- abilistic tools.

8/25

Outline

1

The Model

2

Market equilibrium: existence & uniqueness

3

Market efficiency

4

Comparative statics

9/25

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: standard.

10/25

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

11/25

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 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.

12/25

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. Inverse of price elasticity of expected demand: ℓ (r) =

• −F (r)

m (r)F (r) · r

−1

=

• −r ·

d dr Eq∗ (r | α)

Eq∗ (r | α)

−1

= e−1

<r>

Realistic assumption: decreasing

12/25

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.

13/25

The DGMRL & IGFR classes

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

F (r) is the failure rate function.

GFR gives the percentage decrease in the probability of a stock out from increasing the stocking quantity by 1%. Widely used unimodality condition: g (r) is increasing (IGFR). Satisfied by most commonly used probability distributions. IGFR ⊂ DGMRL (and more results on moments and limiting behavior).

14/25

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).

Generalization to any n ≥ 1 identical retailers: statement of main Theo- rem independent of n (due to second-stage equilibrium uniqueness).

15/25

Outline

1

The Model

2

Market equilibrium: existence & uniqueness

3

Market efficiency

4

Comparative statics

16/25

On a silver platter

Probability of stockout: Theorem For any DMRL distribution F, the probability F (r ∗) of an immediate stockout (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 via MRL representation of r ∗. Necessity: does not extend to the class of DGMRL distributions.

17/25

Price of Uncertainty

Question: for any realized demand value α how does the performance of the stochastic market compare to that of the deterministic market? Theorem The Price of Uncertainty equals: PoU := infα∈S

• ΠD

Agg

ΠU

Agg

• = 1 − O

n−2. In

particular, there exist demand levels for which the stochastic outperforms the deterministic market.

18/25

Price of Anarchy

Question: for any realized demand value α how does the performance of the competitive market compare to that of the integrated market? Theorem The Price of Anarchy is upper-bounded: PoA := sup

F∈DG

• EΠU

Int

EΠU

Agg

• ≤ 1 + O
• n−1

If we restrict to DMRL distributions the bound improves to 1 + O

n−2.

Caution: integrating the market has non-trivial implications on the infor- mation structure. Problem: PoA is not lower-bounded by 1.

19/25

Outline

1

The Model

2

Market equilibrium: existence & uniqueness

3

Market efficiency

4

Comparative statics

20/25

Comparative Statics

Equation r ∗ = m (r ∗) allows comparative statics via stochastic orderings (partial orders with respect to various criteria on the space of all demand distributions), Shaked and Shanthikumar (2007). Lemma Let X1 ∼ F1, X2 ∼ F2 be two nonnegative and continuous, DGMRL demand distributions with finite second moment. If X1 mrl X2, then r ∗

1 ≤ r ∗ 2 .

In words: if market X1 is "less than" market X2 in a concrete stochastic sense, then wholesale prices in market X1 are lower than in market X2. Upshot: conclusions depend on the notion of market size – variability that we will employ. challenge established intuitions derived under specific instruments.

21/25

Market Size

Compare markets via their size Theorem Let X1 ∼ F1, X2 ∼ F2 be two nonnegative, DGMRL demand distributions, with finite second moments, such that X1 mrl X2. If Z satisfies some mild conditions, then r∗

X1+Z ≤ r∗ X2+Z.

If φ is an increasing, convex function, then r ∗

φ(X1) ≤ r ∗ φ(X2).

But: stochastically larger market does not imply a higher wholesale price.

22/25

Demand Variability

Compare markets via their variability Theorem Let X1 ∼ F1, X2 ∼ F2 be two nonnegative, DGMRL demand distributions with finite second moments and αL1 ≤ αL2. If either X1, X2 or both are DMRL and X1 ew X2, then r ∗

1 ≤ r ∗ 2 .

If either X1, X2 or both are IFR and X1 disp X2, then r ∗

1 ≤ r ∗ 2 .

But: a more variable market does not imply a higher wholesale price.

23/25

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.

24/25

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.

25/25

Thank you