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 June 2, 2018 1Supported by a fellowship of the Alexander S. Onassis Public Benefit Foundation. 1/25
Overview The Model 1 Market equilibrium: existence & uniqueness 2 Market efficiency 3 Comparative statics 4 2/25
Outline The Model 1 Market equilibrium: existence & uniqueness 2 Market efficiency 3 Comparative statics 4 3/25
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. 4/25
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. 5/25
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 ex- pectation E α < + ∞ , and nonnegative support contained in 0 ≤ α L ≤ α H ≤ + ∞ . 6/25
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. 7/25
Outline The Model 1 Market equilibrium: existence & uniqueness 2 Market efficiency 3 Comparative statics 4 8/25
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 ) + . and each retailer orders quantity q ∗ i ( r ) = 1 � Proof: standard. 9/25
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 10/25
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 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 . 11/25
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. � Inverse of price elasticity of expected demand: � − 1 dr E q ∗ ( r | α ) � − 1 � � d − F ( r ) = e − 1 ℓ ( r ) = m ( r ) F ( r ) · r = − r · E q ∗ ( r | α ) < 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 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 ) � − 1 F ( r ) 3 r r 1 m ( r ) is weakly decreasing: F is (DMRL) or 2 (i) m ( r ) / r is strictly decreasing and (ii) eventually < 1. 12/25
The DGMRL & IGFR classes � The (GMRL) function is related to the (GFR) (increasing generalized failure rate) function introduced by Lariviere & Porteus (2001) g ( r ) := r h ( 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). 13/25
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 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 ). � Generalization to any n ≥ 1 identical retailers: statement of main Theo- rem independent of n (due to second-stage equilibrium uniqueness). 14/25
Outline The Model 1 Market equilibrium: existence & uniqueness 2 Market efficiency 3 Comparative statics 4 15/25
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. 16/25
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 � Π D � � n − 2 � . In Agg The Price of Uncertainty equals: PoU := inf α ∈ S = 1 − O Π U Agg particular, there exist demand levels for which the stochastic outperforms the deterministic market. 17/25
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: � E Π U � � n − 1 � Int PoA := sup ≤ 1 + O E Π U F ∈DG Agg � n − 2 � . If we restrict to DMRL distributions the bound improves to 1 + O � Caution: integrating the market has non-trivial implications on the infor- mation structure. � Problem: PoA is not lower-bounded by 1. 18/25
Outline The Model 1 Market equilibrium: existence & uniqueness 2 Market efficiency 3 Comparative statics 4 19/25
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 X 1 ∼ F 1 , X 2 ∼ F 2 be two nonnegative and continuous, DGMRL demand distributions with finite second moment. If X 1 � mrl X 2 , then r ∗ 1 ≤ r ∗ 2 . � In words: if market X 1 is "less than" market X 2 in a concrete stochastic sense, then wholesale prices in market X 1 are lower than in market X 2 . � Upshot: conclusions depend on the notion of market size – variability that we will employ. challenge established intuitions derived under specific instruments. 20/25
Market Size � Compare markets via their size Theorem Let X 1 ∼ F 1 , X 2 ∼ F 2 be two nonnegative, DGMRL demand distributions, with finite second moments, such that X 1 � mrl X 2 . If Z satisfies some mild conditions, then r ∗ X 1 + Z ≤ r ∗ X 2 + Z . If φ is an increasing, convex function, then r ∗ φ ( X 1 ) ≤ r ∗ φ ( X 2 ) . � But: stochastically larger market does not imply a higher wholesale price. 21/25
Demand Variability � Compare markets via their variability Theorem Let X 1 ∼ F 1 , X 2 ∼ F 2 be two nonnegative, DGMRL demand distributions with finite second moments and α L 1 ≤ α L 2 . If either X 1 , X 2 or both are DMRL and X 1 � ew X 2 , then r ∗ 1 ≤ r ∗ 2 . If either X 1 , X 2 or both are IFR and X 1 � disp X 2 , then r ∗ 1 ≤ r ∗ 2 . � But: a more variable market does not imply a higher wholesale price. 22/25
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