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Channel Coordination Under Competition a Strategic View Channel Coordination under Competition A Strategic View Guillermo Gallego Masoud Talebian Columbia University University of Newcastle CARMA Retreat Newcastle Aug 18, 2012 Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Competition Suppliers Market Competition Suppliers Market Bertrand model considers oligopolistic firms which compete with price. Gallego and Hu [2007] consider the same setting, but firms are allowed to dynamically change their prices. Coordination Market Supplier Retailer Coordination Market Supplier Retailer A monopolistic supply chain is not coordinated with simple wholesale pricing contracts, due to several factors including: Double marginalization: Spengler [1950], ... Not enough sales effort: Cachon and Lariviere [2005], Taylor [2002], Krishnan et al. [2004], ... Competition and Coordination Suppliers Market Retailer Competition and Coordination Suppliers Market Retailer Elmaghraby [2000]: “If suppliers submit curves that reflect volume discount, then solving for the least-cost (set of) suppliers given the submitted curves is not guaranteed to be a simple task.” Cachon [2003]” “More research is needed on how multiple suppliers compete for the affection of multiple retailers, i.e., additional emphasis is needed on many-to-one or many-to-many supply chain structures.” Model Suppliers Market Retailer Contracts: general format quantity discounts Pricing: price taker or price setting retailer Method: non-cooperative non-zero-sum game theory ci : capacity of supplier i. P c = ci c−i = c − ci r(c) = maxfπ(x): x 2 [0; c]g. s(c) = arg maxfπ(x): x 2 [0; c]g. ri (c) = r(c) − r(c−i ) Definitions P π(x) = i xi pi (x) − e(x) I pi (x): the price of product i when selling x units. I e(x): the sales cost when selling x units. Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly r(c) = maxfπ(x): x 2 [0; c]g. s(c) = arg maxfπ(x): x 2 [0; c]g. ri (c) = r(c) − r(c−i ) Definitions P π(x) = i xi pi (x) − e(x) I pi (x): the price of product i when selling x units. I e(x): the sales cost when selling x units. ci : capacity of supplier i. P c = ci c−i = c − ci Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly ri (c) = r(c) − r(c−i ) Definitions P π(x) = i xi pi (x) − e(x) I pi (x): the price of product i when selling x units. I e(x): the sales cost when selling x units. ci : capacity of supplier i. P c = ci c−i = c − ci r(c) = maxfπ(x): x 2 [0; c]g. s(c) = arg maxfπ(x): x 2 [0; c]g. Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Definitions P π(x) = i xi pi (x) − e(x) I pi (x): the price of product i when selling x units. I e(x): the sales cost when selling x units. ci : capacity of supplier i. P c = ci c−i = c − ci r(c) = maxfπ(x): x 2 [0; c]g. s(c) = arg maxfπ(x): x 2 [0; c]g. ri (c) = r(c) − r(c−i ) Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly P π(xjT ) = π(x) − Ti (xi ). r(cjT ) = maxfπ(xjT ): x 2 [0; c]g. s(cjT ) = arg maxfπ(xjT ): x 2 [0; c]g. Definitions Ti : the contract between supplier i and the retailer. Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Definitions Ti : the contract between supplier i and the retailer. P π(xjT ) = π(x) − Ti (xi ). r(cjT ) = maxfπ(xjT ): x 2 [0; c]g. s(cjT ) = arg maxfπ(xjT ): x 2 [0; c]g. Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly While supermodularity is preserved under maximization (Topkis [1998]), submodularity is not generally preserved under maximization. Maximizing a submodular function in general is NP hard. Sub-modularity Assumption r is submodular, i.e., has decreasing difference. (r(x _ y) − r(x) ≤ r(y) − r(x ^ y)). Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Sub-modularity Assumption r is submodular, i.e., has decreasing difference. (r(x _ y) − r(x) ≤ r(y) − r(x ^ y)). While supermodularity is preserved under maximization (Topkis [1998]), submodularity is not generally preserved under maximization. Maximizing a submodular function in general is NP hard. Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Multi-modularity Lemma r(c) is submodular if: For all i = 1; :::; m: pi (x) is anti-multimodular, For all i = 1; :::; m: pi (x) is decreasing on xj for all j = 1; :::; m, e(x) is multimodular. Multimodularity can be considered as a stronger assumption than submodularity. Multimodularity results in component-wise concavity. For Multimodularity definition and properties refer to Hajek [1985] and Li and Yu [2012]. Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Equilibrium Theorem In a competitive setting with sophisticated contracts, a set of contracts are Nash equilibrium if and only if they are in the following format: 8 >0 for x = 0 > <>≥ r (c + x) for 0 < x < s (c) T ∗(x) = i −i i i r (c) for s (c) ≤ x ≤ min s (c ) > i i j6=i i −j > :≥ ri (c) for minj6=i si (c−j ) < x ≤ ci In addition, all Nash equilibria result in a coordinated chain with unique profit split as: ∗ ∗ Ti (si (cjT )) = ri (c) = r(c) − r(c−i ) ∗ P r(cjT ) = r(c−i ) − (m − 1)r(c) Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Franchise contracts Proposition A franchise contract is equilibrium if and only if it is in the following format: ∗ Ti (x) = r(c) − r(c−i ) for 0 < x Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Marginal-units discount contracts Proposition A marginal units discount contract is equilibrium if and only if it charges price p for units less than threshold l, and 0 for units over the threshold, where: @r r(c)−r(c−i ) (c− ) ≤ p and l = @ci i p Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly All-units discount contracts Proposition An all units discount contract is equilibrium if and only if it charges price p if order is less than threshold l, and, r(c)−r(c−i ) if order is more si (ci ) than or equal to threshold, where: @r (c− ) ≤ p and l = min 6= s (c− ) @ci i j i i j Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Example m = 2 c = [5; 10] Fully substitutable products Market of size 10 Fixed price of 20 2 2 π(x) = 20(x1 + x2) − (x1 + x2) − :01x1 − x2 Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Example: Contracts Supplier 2 Contracts ($) Supplier 1 Capacity (units) Comparative statics Proposition As a supplier’s capacity increases, her profit increases, the competitor suppliers’ profits decrease, and the retailer’s profit increases. Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Example: Sensitivity to first supplier’s capacity Total Chain Supplier 2 Retailer Profit($) Supplier 1 Capacity (units) Example: Sensitivity to second supplier’s capacity Total Chain Retailer Supplier 1 Profit($) Supplier 2 Capacity (units) Capacities Proposition Under sophisticated contracts capacity buildings are coordinated, i.e., ∗ ∗ ∗ the equilibrium capacities are given by ci (c−i ) = arg maxci fr(ci + c−i )g Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Wholesale Pricing Theorem In competitive setting with wholesale prices, there exists a Nash equilibrium, in which the chain is uncoordinated unless suppliers’ capacities are less than a threshold. Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly Example: Win-lose analysis Supplier 1 Supplier 2 Retailer G. P. Cachon. Supply chain coordination with contracts. In S. Graves and T. de Kok, editors, Handbook in OR/MS: Supply Chain Management. North-Holland, 2003. G. P. Cachon and M. A. Lariviere. Supply chain coordination with revenue-sharing contracts: Strenghts and limitations. Management Science, 51(1):30–44, 2005. W. J. Elmaghraby. Supply contract competition and sourcing policies. Manufacturing & Service Operations Management, 2:350–371, 2000. G. Gallego and M. Hu. Dynamic pricing of perishable assets under competition. Working Paper, IEOR Department, Columbia University, NY, 2007. B. Hajek. Extremal splittings of point processes. Mathematics of Operations Research, 10(4), 1985. H. Krishnan, R. Kapuscinski, and D. Butz. Coordinating contracts for decentralized supply chains with retailer promotional effort. Management Science, 50(1):48–63, 2004. Q. Li and P. Yu. Multimodularity and structural properties of stochastic dynamic programs. working paper, 2012. Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly J. J. Spengler. Vertical integration and anti-trust policy. The Journal of Political Economy, 58(4):347–352, 1950. T. Taylor. Supply chain coordination under channel rebates with sales effort effects. Management Science, 48(8):992–1007, 2002. D. M. Topkis. Supermodularity and complementary. Princeton University Press, 1998. Guillermo Gallego and Masoud Talebian Channel Coordination in Oligopoly.
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