44 1 SUPPLY CHAIN OPERATIONS MANAGEMENT Dantzig developed the simplex algorithm in linear programming. A few years later, a relationship between certain types of games (explicitly, zero- sum games) and their solution by linear programming was pointed out. Here we are concerned with two-persons zero-sum games. Situations where there may be more than one player, potential coalitions, cooperation, asymmetry of information (where one player may know something the other does not) etc. are practically important but are not within our scope of study. Two-Persons Zero-Sum Games Two-persons zero-sum games involve two players. Each has only one move (decision) to take and both make their moves simultaneously. Each player has a set of alternatives, say A =( n AAAA , ,,, 321 ) for the first player and B=( m BBBB , ,,, 321 ) for the second player. When both players make their moves (i.e. they select a decision alternative) an outcome ij O follows, corresponding to the pair of moves ),( ji BA which was selected by each of the players respectively. In two-persons zero-sum games, addi- tional assumptions are made: (1) n AAAA , ,,, 321 as well as m BBBB , ,,, 321 and ij O are known to both players. (2) Players do not know with what probabilities the opponent’s alternatives will be selected. (3) Each player has a preference that can be ordered in a rational and con- sistent manner. In strictly competitive games, or zero-sum games, the players have directly opposing preferences, so that a gain by a player is a loss to its opponent. That is; The Gain to Player 1 = The Loss of Player 2 The concepts of pure and mixed strategies, minimax and maximin strategies, saddle points, dominance etc. are also defined and elaborated. For example, two rival companies, A and B, are the only ones. Company A has three alternatives 321 ,, AAA expressing different strategic while B has four alternatives 4321 ,,, BBBB . The payoff matrix to A (a loss to B) is given by: APPENDIX: ESSENTIALS OF GAME THEORY 45 This problem has a solution, called a saddle-point, because the least greatest loss to B is equal to the greatest minimum gain to A. When this is the case, the game is said to be stable, and the pay-off table is said to have a saddle-point. This saddle-point is also called the value of the game, which is the least entry in its row, and the greatest entry in the column. Not all games can have a pure, single strategy, saddle-point solution for each player. When a game has no saddle point, a solution to the game can be devised by adopting a mixed strategy. Such strategies result from the com- bination of pure strategies, each selected with some probability. Such a mixed strategy will then result in a solution which is stable, in the sense that player 1's maximin strategy will equal player 2's minimax strategy. Mixed strategies therefore induce another source of uncertainty. Non-Zero Sum Games Consider the bimatrix game (A,B) = ( ) ijij ba , . Let x and y be the vector of i x j 10 ,1 1 ≤≤= ∑ = j m j j yy TT xByxAy == ba VV , and an equilibrium is defined for each strategy if the following conditions hold ba VV ≤≤ xBAy , . For example, consider the 2*2 bimatrix game. We see that () ( ) ( ) ()()() 222221221222211211 222221221222211211 bybbxbbxybbbbV ayaaxaaxyaaaaV b a +−+−++−−= +−+−++−−= Then, for an admissible solution for the first player, we require that ),(),0();,(),1( yxVyVyxVyV aaaa ≤ ≤ , 1 B 2 B 3 B 4 B 1 A .6 3 1.5 -1.1 2 A -7 .1 .9 .5 3 A 3 0 5 .8 mixed strategies with elements and y , and such that 0 ,1 1 ≤= ∑ = i n i i xx ≤1, . The value of the game for each of the players is given by: which is equivalent to 0 ;0)1()1( ≥ − ≤ − − − axAxyxayxA , where, () ( ) 122222211211 ; aaaaaaaA − = + −− = . That is when, ⎪ ⎩ ⎪ ⎨ ⎧ =−<< ≥−= ≤−= 010 01 00 aAythenx aAythenx aAythenx . In this sense there can be three solutions (0,y), (x,y) and (1,y). We can similarly obtain a solution for the second player using parameters B and b. Say that 0 ≠ A and 0≠B , then a solution for x and y satisfies the follow- ing conditions: ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ <≥ >≤ <≥ >≤ 0/ 0/ 0/ 0/ BifBbx BifBbx AifAay AifAay As a result, a simultaneous solution leads to the following equations for (x,y), which we have used in the text: ( ) () 22 12 * 11 21 12 22 aa a y A aaaa − == −−+ ( ) () 22 12 * 11 12 21 22 aa a x A aaaa − == −−+ . In this case, the value of the game is: ( ) ( ) ( ) ()()() () () () () 11 12 21 22 12 22 21 22 22 11 12 21 22 12 22 21 22 22 22 12 * 11 12 21 22 22 12 * 11 12 21 22 a b V a a a axya axa aya V b b b bxyb bxb byb aa a x Aaaaa bb b y Bbbbb =−−+ +− +− + =−−+ +− +− + − == −−+ − == −−+ For further study of games and related problems we refer to Moulin 1981; Nash 1950; Von Neumann and Morgenstern 1944; Thomas 1986. 46 1 SUPPLY CHAIN OPERATIONS MANAGEMENT REFERENCES 47 REFERENCES Agrawal V, Seshadri S (2000) Risk intermediation in supply chains. IIE Transactions 32: 819-831. Akerlof G (1970) The Market for Lemons: Quality Uncertainty and the Market Mechanism. Quarterly Journal of Economics, 84: 488-500. Bank D (1996) Middlemen find ways to survive cyberspace shopping. Wall Street Journal, 12 December, p. B6. Barzel Y (1982) Measurement cost and the organization of markets. Journal of Law and Economics 25: 27-47 Bowersox DJ (1990) The strategic benefits of logistics alliances. Harvard Business Review, July-august 68: 36-45. Cachon G (2003) Supply chain coordination with contracts. In: De Kok, AG, Graves S (Eds.), Handbooks in Operations Research and Manage- ment Science. Elsevier, Amsterdam. Caves RE, Murphy WE (1976) Franchising firms, markets and intangible assets. Southern Economic Journal 42: 572-586. Christopher M (1992) Logistics and Supply Chain Management, Pitman, London. Christopher M (2004) Creating resilient supply chains. Logistics Europe, 14-21. Friedman JW (1986) Game theory with Applications to Economics. Oxford University Press. Fudenberg D, Tirole J (1991) Game Theory. MIT Press. Cambridge, Mass. Klaus P (1991) Die Qualitat von Bedienungsinterakionen, in M. Bruhn and B. Strauss,( Eds.), Dienstleistungsqualitat: Konzepte-Methoden-Erfahrungen, Wiebaden Lafontaine F (1992) Contract theory and franchising: some empirical results. Rand Journal of Economics 23(2): 263-283. La Londe B, Cooper M (1989) Partnership in providing customer service: a third-party perspective. Council of Logistics Management, Oak Brook, IL. Lambert D, Stock J (1982) Strategic Physical Distribution Management, R.D. Irwin Inc., Ill., p. 65. McIvor R (2000) A practical framework for understanding the outsourcing process. Supply Chain Management: an International Journal vol.5(1): 22-36. Moulin H (1995) Cooperative Microeconomics: A Game-Theoretic Intro- duction. Princeton University Press. Princeton ,New Jersey Nash F (1950) Equilibrium points in N-person games. Proceedings of the National Academy of Sciences 36: 48-49. 48 1 SUPPLY CHAIN OPERATIONS MANAGEMENT Newman R (1988) The buyer-supplier relationship under just in time, Journal of Production and Inventory Management 4: 45-50. Rao K, Young RR (1994) Global Supply Chain: factors influencing Out- sourcing of logistics functions. International Journal of Physical Distri- bution & Logistics Management 24(6). Rey P (1992) The economics of franchising, ENSAE Paper, February, Paris. Rey P, Tirole J (1986) The logic of vertical restraints. American Economic Review 76: 921-939 Reyniers DJ, Tapiero CS (1995a) Contract design and the control of qual- ity in a conflictual environment. Euro J. of Operations Research, 82(2): 373-382. Reyniers, Diane J, Tapiero CS (1995b) The supply and the control of qual- ity in supplier-producer contracts. Management Science 41: 1581-1589. Riordan M (1984) Uncertainty, asymmetric information and bilateral con- tracts. Review of Economic Studies 51: 83-93. Ritchken P, Tapiero CS (1986) Contingent Claim Contracts and Inventory Control. Operations Research 34: 864-870 Rubin PA, Carter JR (1990) Joint optimality in buyer–supplier negotia- tions. Journal of Purchasing and Materials Management 26(1): 54-68. Tapiero CS (1995) Acceptance sampling in a producer-supplier conflicting environment: Risk neutral case, Applied Stochastic Models and Data Analysis 11: 3-12 Tapiero CS (1996) The Management of Quality and Its Control, Chapman and Hall, London Tapiero CS (2005a) Risk Management, John Wiley Encyclopedia on Actu- arial and Risk Management, Wiley, New York-London Tapiero CS (2005b) Value at Risk and Inventory Control, European Journal of Operations Research 163(3): 769-775. Tapiero CS (2006) Consumers Risk and Quality Control in a Collaborative Supply Chains. European Journal of Operations Research, (available on line, October 18) Thomas LC (1986) Game Theory and Application, Ellis Horwood Ltd, Chichester. Tsay A, Nahmias S, Agrawal N (1998) Modeling supply chain contracts: A review, in Tayur S, Magazine M, Ganeshan R (eds), Quantitative Models of Supply Chain Management, Kluwer International Series Van Damme DA, Ploos van Amstel MJ (1996) Outsourcing Logistics management Activities. The International Journal of Logistics manage- ment 7(2): 85-94. REFERENCES 49 Van Laarhoven P, Berglund M, Peters M (2000) Third-party logistics in Europe – five years later. International Journal of Physical Distribution & Logistics Management 30(5): 425-442. Von Neumann J, Morgenstern O (1944) Theory of Games and Economic Behavior. Princeton University Press. Williamson OE (1985) The Economic Institutions of Capitalism, New York, Free Press. Zeithaml V, Parasuraman A, Berry LL (1990) Delivering Quality Service, Free Press, New York. A supply chain can be defined as “a system of suppliers, manufacturers, distributors, retailers, and consumers where materials flow downstream from suppliers to customers and information flows in both directions” (Geneshan et. al. 1998). The system is typically decentralized which implies that its participants are independent firms each with its own frequently conflicting goals spanning production, service, purchasing, inventory, transportation, marketing and other such functions. Due to these conflicting goals a decen- tralized supply chain is generally much less efficient than the correspond- ing centralized or integrated chain with a single decision maker. Efficiency suffers from both vertical (e.g., buyer-vendor competition) and horizontal (e.g., a number of vendors competing for the same buyer) conflicts of interest. How to manage competition in supply chains is a challenging task which comprises a variety of problems. The overall target is to make, to the extent possible, the decentralized chain operate as efficiently as its benchmark, the corresponding centralized chain. This particular aspect of supply chain management is referred to as coordination. This chapter addresses simple static supply chain models, competition between supply chain members and their coordination. 2.1 STATIC GAMES IN SUPPLY CHAINS In research and management literature where supply chain problems and related game theoretic applications have gained much attention in recent years, we see extensive reviews focusing on such aspects as taxonomy of supply chain management (Geneshan et. al. 1998); integrated inventory models (Goyal and Gupta 1989); game theory in supply chains (Cachon and Netessine 2004); operations management (Li and Whang 2001); price quantity discounts (Wilcox et. al. 1987); and competition and coordination (Leng and Parlar 2005). IN A STATIC FRAMEWORK 2 SUPPLY CHAIN GAMES: MODELING 52 2 SUPPLY CHAIN GAMES: MODELING IN A STATIC FRAMEWORK In the literature, supply chains are distinguished by various features such as: types of decisions; operations; competition and coordination; incentives; objectives; and game theoretic concepts. In this chapter we deal with three essential features of static supply chains, i.e., the supply chains with deci- sions independent of time: customer demand, competition and risk. In this sense we distinguish between • deterministic and random demands; endogenous and exogenous demands • vertical and horizontal competition within supply chains • no risk involved, risk incurred by only one of the parties and risk shared between the parties. In this chapter, supply chain games are combined into three groups. The first group of games represents classical horizontal production and vertical pricing competition under endogenous demands. These games involve decisions about either product prices or quantities with respect to two types of endogenous demands: (i) the quantity demanded for a product as a func- tion of price set for the product and (ii) an inverse demand function with price as a function of the quantity produced or sold. In both cases the de- mands are deterministic, which implies that all produced/supplied products are sold and thus there is no risk involved. Random exogenous demand for products characterizes the second group of games which is related to the classical newsvendor problem. The parties vertically compete by deciding on a price to offer and a quantity to order for a particular price. Since the demand is uncertain, the downstream party, which faces the demand, runs the risk of overestimating or underestimating it. The risk involves costs incurred due to choosing the quantity to order and stock before customer demand is realized. We refer to this group of games as stocking / pricing competition with random demand. The third group of games represents classical risk-sharing interactions between supply chain members. Similar to the second group, the competi- tion is vertical and the demand is exogenous and random. Unlike the sec- ond group, however, incentives to mitigate risk may be offered to a party which faces uncertain customer demands. Since the incentives include buyback and urgent purchase options, some of the uncertainty is trans- ferred from one party to another. In such a case, the risk associated with random demand is shared and the inventories of all involved parties are affected when deciding on what quantities to stock. 2.1 STATIC GAMES IN SUPPLY CHAINS 53 Motivation We describe a few production, pricing and inventory-stock related prob- lems which have been found in various service and industry-related supply chains. Most of these problems have been extensively studied and can be found virtually in every survey devoted to supply chain management including those mentioned above. It is worth noting that, in general, the number of basic supply chain problems is significant and selecting just a few of them for an introductory purpose is not a simple matter. Our selection criterion is based on one of the overall goals of this book– to show how optimal pricing and inventory policies evolve when static operation conditions become dynamic. Under such conditions, we find par- ticularly interesting the static problems which allow for straightforward and, yet natural, dynamic extensions. The problems which we discuss in this chapter will be discussed again in the following chapters to show the effect of production and service dynamics on managerial decisions. The static feature of the problems we select implies that the period of time that the problems encompass is such that no change in system para- meters is observed. Since all products are delivered at once by the end of the period and then instantly sold, these problems ignore the intermediate inventories (and associated costs) before and during the selling season. Due to the focus on stock and pricing policies, shortages as well as left- overs are avoided, as much as possible, by the end of the period. In all the problems that we consider, it is assumed that the information needed for decision-making is available and transparent to the supply chain partici- pants and that the overall order lead-time is smaller than the length of the period so that all deliveries are provided on time. This chapter introduces and discusses basic models of horizontal and vertical competition between supply chain members, the effect of uncertainty and risk sharing as well as basic tools for coping with the competition by coordinating supply chains. The analysis which we employ includes (i) formal statements of problems of each non-cooperative party involved as well as the corresponding centralized formulations where only one deci- sion-maker is responsible for all managerial decisions in the supply chain; (ii) system-wide optimal and equilibria solution for competing parties; (iii) analysis of the effect of competition on supply chain performance and of coordination for improving the performance. In analyzing the problems we use Nash and Stackelberg equilibria which we briefly present next. [...]... where q2 q1 p (Q) Q p (q1 q 2 ) q2 (1 ) 0, Q q1 p (q1 q 2 ) 0, q2 ) c q2 Q q 2 ) c q1 q2 p 2 (Q) Q2 2 p (Q) Q q2 p 2 (Q) , Q=q1+q2 Q2 We illustrate this with the following example: Example 2. 6 Let the price be linear in production quantity, p=a-bQ, Q=q1+q2, p(0)=a>c 2 2 p p p p Note that the price requirements, b 0 and 2 2 q1 q2 q1 q2 2 p q1 q 2 0 are met for the selected function Using Proposition 2. 5... q1*=q2*, we obtain the following equation p (2q*) 0 p (2q*) c 2q * (2. 20) Q Define Q' so that p(Q')=c Then it is easy to verify that, Since the two problems are symmetric, Q=q1+q2, 2 J 2 2 q2 q1 J 2 2 J q1 q 2 2 p Q 2 q1 p Q2 q2 p2 Q2 0 This implies that the Hessian of J(q1,q2) is semi-definite negative and thus the function J(q1,q2) is jointly concave in production quantities q1 and q2 for q1 q2 [0,... (2. 22) , p ( 2q n ) c q n p ( 2q n ) Q a 2bq n c q n ( b) 0 76 2 SUPPLY CHAIN GAMES: MODELING IN A STATIC FRAMEWORK 1 2 a c c The payoffs for the , hence, pn= a 3b 3 3 equilibrium are thus identical for both players, J1(q1n,q2n) = J2(q1n,q2n) = (a c) 2 9b Based on (2. 23) we can identify the best response function of the second supplier p (q1 q 2 ) a b(q1 q 2 ) c q 2 ( b) 0 , p (q1 q 2 ) c q 2 q2 and. .. q1[p(q1+ q2R(q1))-c] p (q1 q1 q2 ) c q2 q1 Differentiating this function we find J 1 (q1 ) q1 p(q1 R q2 (q1 )) c q1 p(q1 R q2 (q1 )) (1 Q R q2 (q1 ) ) q1 0 , (2. 24) 2. 2 PRODUCTION/PRICING COMPETITION 75 R where q 2 (q1 ) is determined by differentiating (2. 23) with q2 set equal to q1 q2R(q1) p(Q) (1 Q Thus R q2 (q1 ) q1 R R q 2 (q1 ) ) q1 q 2 (q1 ) p(Q) q1 Q p(Q) p 2 (Q) R q2 (q1 ) Q Q2 p 2 (Q) (1 q 2 (q1... q1[p(q1+q2)-c]+ q2[p(q1+q2)-c] q1 , q2 s.t q1 0, q2 0, p(q1+q2) c (2. 19) 72 2 SUPPLY CHAIN GAMES: MODELING IN A STATIC FRAMEWORK System-wide optimal solution We first study the centralized problem by employing the first-order optimality conditions J (q1 , q 2 ) q1 J (q1 , q 2 ) q2 p (q1 q2 ) c q1 p (q1 q2 ) c q2 p (Q) Q p (Q) Q Q q1 Q q2 q2 q1 p (Q) Q p (Q) Q Q =0, q1 Q =0 q2 p p p , only q1 q2 Q total... ) Q2 R 2 R q 2 (q1 ) ) q1 0 p(Q) p 2 (Q) R q2 (q1 ) (2. 25) Q Q2 R q 2 (q1 ) 0, q1 the greater the production of the first supplier, q1, the lower the production of the second supplier, q2R(q1) Equation (2. 25) implies, Based on (2. 23), (2. 24) and (2. 25) we conclude that the pair (q1s,q2s) constitutes the Stackelberg equilibrium of the production game if there exists a joint solution in q1 and q2 of... J1(q1,q2)= max q1[p(q1+q2)-c] q1 q1 (2. 17) s.t q1 0, p(q1+q2) c The problem of Supplier 2 max J2(q1,q2)= max q2[p(q1+q2)-c] q2 q2 (2. 18) s.t q2 0, p(q1+q2) c, where p(Q) is the price at which the retailer can sell Q product units; q1 and q2 are the quantities produced by suppliers (manufacturers) 1 and 2 2 .2 PRODUCTION/PRICING COMPETITION 71 respectively and sold by the retailer; Q=q1+q2 is the total quantity... p q1 q 2 ab 2 e bQ 0 implying that the equilibrium q1 q2 is not necessarily unique The Nash equilibrium is determined by (2. 22) condition 2 2 ae b 2qn c q n abe b 2qn 0 Setting the left-hand side of this equation as L in Maple >L:=a*exp(-b *2* q)-c-q*a*b*exp(-b *2* q); L := a e ( 2 b q) c qabe ( 2 b q) and substituting specific parameters of the problem a=15,b=0.1,c=1, we have 78 2 SUPPLY CHAIN GAMES:. .. overall supply chain payoff deteriorates under horizontal competition, 3(a c) 2 J1(q1s,q2s)+J2(q1s,q2s)= . game is: ( ) ( ) ( ) ()()() () () () () 11 12 21 22 12 22 21 22 22 11 12 21 22 12 22 21 22 22 22 12 * 11 12 21 22 22 12 * 11 12 21 22 a b V a a a axya axa aya V b b b bxyb bxb byb aa a x Aaaaa bb b y Bbbbb =−−+. ( ) () 22 12 * 11 21 12 22 aa a y A aaaa − == −−+ ( ) () 22 12 * 11 12 21 22 aa a x A aaaa − == −−+ . In this case, the value of the game is: ( ) ( ) ( ) ()()() () () () () 11 12 21 22 12 22 21. and an equilibrium is defined for each strategy if the following conditions hold ba VV ≤≤ xBAy , . For example, consider the 2* 2 bimatrix game. We see that () ( ) ( ) ()()() 22 222 122 122 221 121 1 22 222 122 122 221 121 1 bybbxbbxybbbbV ayaaxaaxyaaaaV b a +−+−++−−= +−+−++−−=