Selling to Cournot oligopolists: pricing under uncertainty & generalized mean residual life Stefanos Leonardos 1 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. 1/36
Overview Introduction 1 The Model 2 Market equilibrium: existence & uniqueness 3 Supplier’s optimal pricing decision MRL and GMRL functions The DGMRL class 4 Examples – Future Research 5 2/36
Outline Introduction 1 The Model 2 Market equilibrium: existence & uniqueness 3 Supplier’s optimal pricing decision MRL and GMRL functions The DGMRL class 4 Examples – Future Research 5 3/36
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. 4/36
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. 5/36
Outline Introduction 1 The Model 2 Market equilibrium: existence & uniqueness 3 Supplier’s optimal pricing decision MRL and GMRL functions The DGMRL class 4 Examples – Future Research 5 6/36
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. 7/36
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. 8/36
Formal setting � Extensive, 2-stage game. � Strategy sets: retailers (followers): q i : R ≥ 0 → R ≥ 0 , order-quantity for price r . supplier (Stackelberg leader): wholesale price r > 0. � Payoff functions: retailers: u i ( q | r ) = q i ( α − q ) − rq i = q i ( α − r − q ) supplier: u s ( r ) = E u s ( r | α ) = r · E q ( r | α ) � Demand uncertainty: α is realized from a: continuous cdf F , F := 1 − F , with finite expec- tation E α < + ∞ , and nonnegative support contained 0 ≤ α L ≤ α H ≤ + ∞ . 9/36
Outline Introduction 1 The Model 2 Market equilibrium: existence & uniqueness 3 Supplier’s optimal pricing decision MRL and GMRL functions The DGMRL class 4 Examples – Future Research 5 10/36
Outline Introduction 1 The Model 2 Market equilibrium: existence & uniqueness 3 Supplier’s optimal pricing decision MRL and GMRL functions The DGMRL class 4 Examples – Future Research 5 11/36
Base case: deterministic demand � The supplier knows the demand parameter α : α L = α H . 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 3 ( α − r ) + . i ( r ) = 1 and each retailer orders quantity q ∗ � Proof: Second stage Given r , α the retailers play a classic Cournot duopoly. Hence, unique equilibrium strategies: i ( r ) = 1 3 ( α − r ) + q ∗ Uniqueness implies that the supplier correctly predicts the retailers’ order-quantities. 12/36
Base case: deterministic demand � The supplier knows the demand parameter α : α L = α H . 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 3 ( α − r ) + . i ( r ) = 1 and each retailer orders quantity q ∗ � Proof: First stage Consider only subgame perfect equlibria On the equilibrium path, the supplier’s payoff is u s ( r | α ) = rq ∗ ( r | α ) = 2 3 r ( α − r ) + , for 0 ≤ r . here: two second stage retailers. 12/36
General case: stochastic demand � The supplier has incomplete information about α : α L < α H . On the equilibrium path, the supplier’s payoff is u s ( r ) = 2 3 r E ( α − r ) + , for 0 ≤ r ≤ α H . Maximization with respect to r is not straightforward u s ( 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 u s ( r ) � ∞ u s ( r ) = 2 3 r E ( α − r ) + = 2 F ( u ) d u 3 r r 13/36
Outline Introduction 1 The Model 2 Market equilibrium: existence & uniqueness 3 Supplier’s optimal pricing decision MRL and GMRL functions The DGMRL class 4 Examples – Future Research 5 14/36
Mean residual life (MRL) function Definition The MRL function m ( · ) of a nonnegative random variable α with cdf F and E α < + ∞ , is defined as � ∞ 1 m ( r ) := E ( α − r | α > r ) = F ( u ) d u , for r < α H F ( r ) r and m ( r ) := 0, otherwise. � Express u s ( r ) in terms of m ( r ) and differentiate ( as if with product rule) u s ( r ) = 2 3 r m ( r ) F ( r ) d u s d r ( r ) = 2 3 ( m ( r ) − r ) F ( r ) = 2 � m ( r ) � − 1 F ( r ) ← FOC? 3 r r for 0 < r < α H . 15/36
Generalized mean residual life (GMRL) function Definition The GMRL function m ( · ) of a nonnegative random variable α with cdf F and E α < + ∞ , is defined as ℓ ( r ) := m ( r ) , 0 < r < α H r and ℓ ( r ) := 0, otherwise. � Interpretation: expected additional demand as a percentage of the given. 16/36
Generalized mean residual life (GMRL) function Definition The GMRL function m ( · ) of a nonnegative random variable α with cdf F and E α < + ∞ , is defined as ℓ ( r ) := m ( r ) , 0 < r < α H r and ℓ ( r ) := 0, otherwise. � From the previous slide: d u s d r ( r ) = 2 3 ( m ( r ) − r ) F ( r ) = 2 � m ( r ) � 3 r − 1 F ( r ) r 1 m ( r ) is weakly decreasing: F is (DMRL) or 2 (i) m ( r ) / r is strictly decreasing and (ii) eventually < 1. 16/36
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 E α 2 is finite, then r ∗ exists and is the unique solution of (1). In this case, if 1 2 E α < α L , then r ∗ = 1 2 E α . Otherwise, r ∗ ∈ [ α L , α H ). � Remarks: Two technicalities lim r → + ∞ ℓ ( r ) < 1 iff E α 2 < + ∞ . m ( r ) = r over an interval iff α ∼ Pareto (2) over this interval. 17/36
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). 18/36
Outline Introduction 1 The Model 2 Market equilibrium: existence & uniqueness 3 Supplier’s optimal pricing decision MRL and GMRL functions The DGMRL class 4 Examples – Future Research 5 19/36
The DGMRL & IGFR classes I � The (GMRL) function is related to the (GFR) function introduced by Lariviere & Porteus (2001) g ( r ) := r h ( 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 orderings. 20/36
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