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Game Theory in Supply Chain Analysis∗ 评分

Game theory has become an essential tool in the analysis of supply chains with multiple agents, often with conflicting objectives. This chapter surveys the applications of game theory to supply chain analysis and outlines game-theoretic concepts that have potential for future application. We discuss
games have not yet found application in SCM, so we avoid these as well (e. g, zero-sum games and games in extensive form) The material in this chapter was collected predominantly from Moulin(1986), Friedman(1986) Fudenberg and Tirole(1991),Vives(1999) and Myerson(1997). Some previous surveys of G C models in management science include Lucas's(1971)survey of mathematical theory of games, Feichtinger and Jorgensen's (1983)survey of differential games and Wang and Parlar's(1989 survey of static models. A recent survey by Li and Whang(2001)focuses on application of G-T tools in five specific OR/MS models 1.2 Game setup To break the ground for our next section on non-cooperative games, we conclude this section by introducing basic GT notation. A warning to the reader: to achieve brevity, we intentionally sac rifice some precision in our presentation. See texts like Friedman(1986)and Fudenberg and Tirole (1991) if more precision is required Throughout this chapter we represent games in the normal form. A game in the normal form consists of (1) players(indexed by i=1,.m),(2) strategies(or more generally a set, of strategies denoted by x2,i=1,…,m) available to each player and(3) payoffs(π;(x1,2,…,xn),i=1,…,m) received by each player. Each strategy is defined on a set Xi, i Xi, so we call the Cartesian prod- uctX1×X2×…,× Xn the strategy space( typically the strategy space is?"). Each player may have a unidimensional strategy or a multi-dimensional strategy. However, in simultaneous-move games each players set of feasible strategies is independent of the strategies chosen by the other players, i. e, the strategy choice by one player is not allowed to limit the feasible strategies of another player a player's strategy can be thought of as the complete instruction for which actions to take in a game. For example. a player can give his or her strategy to a person that has absolutely no knowl- edge of the player s payoff or preferences and that person should be able to use the instructions contained in the strategy to choose the actions the player desires. Because each player's strategy is a complete guide to the actions that are to be taken, in the normal form the players choose their strategies simultaneously. Actions are adopted after strategies are chosen and those actions correspond to the chosen strategies As an alternative to the "one-shot"selection of strategies in the normal form, a game can also be represented in extensive form. Here players choose actions only as needed, i.e., they do not make an a priori commitment to actions for any possible sample path. Extensive form games have not been studied in SCM, so we focus only on normal forIm games. The normal form can a. so be described static game, in contrast to the extensive form which is a dynamic game If the strategy has no randomly determined choices, it is called a pure strategy; otherwise it is called a miced strategy. There are situations in economics and marketing that have applied mixed strategies: e.g., search models(Varian 1980) and promotion models(Lal 1990). Ho lowever. mixed strategies have not been applied in SCM, in part because it is not clear how a manager would actu ally implement a mixed strategy. (For example, it seems unreasonable to suggest that a manager should"Hip a coin"when choosing capacity. Fortunately, mixed strategy equilibria do not exist in games with a unique pure strategy equilibrium. Hence, in those games attention can be restricted to pure strategies without loss of generality. Therefore, in the remainder of this chapter we consider only pure strategies In a non-cooperative game the players are unable to make binding commitments regarding which strategy they will choose before they actually choose their strategies. In a cooperative game players are able to make these binding commitments. Hence, in a cooperative game players can make side-payments and form coalitions. We begin our analysis with non-cooperative static games. In all sections except the last one we work with the games of complete information, i.e., the players strategies and payoffs are common knowledge to all players As a practical example throughout this chapter. we utilize the classic newsvendor problem trans formed into a game. In the absence of competition each newsvendor buys Q units of a single product at the beginning of a single selling season. Demand during the season is a random variable D with distribution function FD and density function /D. Each unit is purchased for c and sold on the market for r>c. The newsvendor solves the following optimization problem max T= max Ep[r min(D, Q)-cQI with the unique solution Goodwill penalty costs and salvage revenues can easily be incorporated into the analysis, but for our needs we normalized them out. Now consider the g-t version of the newsvendor problem with two retailers competing on product availability. (See Parlar 1988 for the first analysis of this problem, which is also one of the first articles modeling inventory management in a G-T framework). It is useful to consider only the two-player version of this game because only then are graphical analysis and interpretations feasi- ble. Denote the two players by subscripts i, =1, 2, itj, their strategies(in this case stocking quantities )by Qi and their payoffs by Ti We introduce interdependence of the players' payoffs by assuming the two newsvendors sell the same product. As a result, if retailer i is out of stock, all unsatisfied customers try to buy the product at retailer j instead. Hence, retailer i's total demand is Di+(D-Q;): the sum of his own demand and the demand from customers not satisfied by retailer i. Payoffs to the two players are ther ( Q, Q,)=Ep/ i min(D,+(D,-Q,+, Qi)-cQl,i,j=1,2 2 Non-cooperative static games In non-cooperative static games the players choose strategies simultaneously and are thereafter committed to their chosen strategies. Non-cooperative GT seeks a rational prediction of how the game will be played in practice. The solution concept for these games was formally introduced by John Nash(1950) although some instances of using similar concepts date back a couple of centuries The concept is best described through best response functions DEFINITIoN 1. Given the n-player game, plager i's best response (function)to the strategies a of the other players is the strategy a* that maximizes plager i's payoff Ti(i, I-i) (-a)= arg max丌(x2,x-) If Ti is quasi-concave in Ti the best response is uniquely defined by the first-order conditions. In the context of our competing newsvendors example, the best response functions can be found by optimizing each player"s payoff functions w r t. the player's own decision variable Q i while taking the competitor's strategy Qi as given. The resultin Le resulting best response lunctions are Q* (Q)=Fnl (D3-Q)+ 乙,7 Soine Inlay argue that gt should be a tool for choosing how a manager should play a gaIne, which Illay involve playing against rational or semi-rational players. In some sense there is no conflict between these descriptive and normative roles for GT, but this philisophical issue surely requires more in-depth treatment than can be afforded lere Taken together, the two best response functions form a best response mapping R2-R(or in the more general case Rn-R"). Clearly, the best response is the best player i can hope for given the decisions of other players. Naturally, an outcome in which all players choose their best, responses is a candidate for the non-cooperative solution. Such an outcome is called a Nash equilibrium (hereafter NE)of the game DEFINITION 2. An outcome(ai, r,,,*)2s a Nash equilibrium of the game if c i is a best response to m for alli=1, 2 Going back to competing newsvendors NE is characterized by solving a sgstem of best responses that translates into the system of first-order conditions Q*(Q2) C D1+(D2-Q T Q2(Q#) F When analyzing games with two players it is often instrumental to graph the best response functions to gain intuition. Best responses are typically defined implicitly through the first-order conditions which makes analysis difficult. Nevertheless, we can gain intuition by finding out how each player reacts to an increase in the stocking quantity by the other player(i.e,aQi (Q,/aQ) through employing implicit differentiation as follows 0Q(Q) dQ: dQ rifD: +(D-Q,)+D, >Q;(Qi)Pr(D,>@j) 2 <0. fD: +(D-Q)(Qi) The expression says that the slopes of the best response functions are negative, which implies an intuitive result that each player's best response is monotonic.ly decreasing in the other player's strategy. Figure 1 presents this result for the symmetric newsvendor game. The equilibrium is located on the intersection of the best responses and we also see that the best responses are, indeed ecreasing One way to think about a ne is as a fired point of the best response mapping Rn-> Rn. Indeed according to the definition, Ne must satisfy the system of equations aT; axi=0, all i. Recall that a fixed point a of mapping f(r), Rn-Rm is any x such that f(c)=.. Define f; (21, ..,n) aTi aTi+ i. By the definition of a fixed point f(x1,…,xn)--Oπ;(x1,…,xn)/O;+x→bπ(x1,…,xn)/Ox;-0,all lence, a*solves the first-order conditions if and only if it is a fixed point of mapping f(a) de etnea DOV Q2(Q1) Q'(Q2) Figure 1. Best responses in the newsvendor game The concept of ne is intuitively appealing. Indeed, it is a self-fulfilling prophecy. To explain suppose a player were to guess the strategies of the other players. a guess would be consistent with payoff maximization(and therefore reasonable) only if it presumes that strategies are chosen to maximize every player's payoff given the chosen strategies. In other words, with any set of strategies that is not a Ne there exists at least one player that is choosing a non payoff maximizing strategy. Moreover, the ne has a self-enforcing property: no player wants to unilaterally deviate from it since such behavior would lead to lower payoffs. Hence NE seems to be the necessary condition for the prediction of any rational behavior by players While attractive, numerous criticisms of the ne concept exist. Two particularly vexing problems are the non-existence of equilibrium and the multiplicity of equilibria. Without the existence of an equilibrium, little can be said regarding the likely outcome of the game. If there are multiple equilibria, then it is not clear which one will be the outcome. Indeed it is possible the outcome is not even an equilibrium because the players may choose strategies from different equilibria. In some situations it is possible to rationalize away some equilibria via a refinement of the ne concept: e. g trembling hand perfect equilibrium(Selten 1975), sequential equilibrium(Kreps and Wilson 1982) and proper equilibria(Myerson 1997). In fact, it may even be possible to use these refinements to the point that only a unique equilibrium remains. However, these refinements have generally not been applied or needed in the SCM literature. 2 These refinements eliminate equilibria that are based on incredible threats, i. e threats of future actions that would not actually be adopted if the sequence of event in the game led to a point in the game in which those actions could be taken. This issue has not appeared in the SCM literature An interesting feature of the NE concept is that the system optimal solution(a solution that maximizes the total payoff to all players) need not be a NE. Hence, decentralized decision making generally introduces inefficiency in the supply chain.(There are, however, some exceptions: see Mahajan and van ryzin 1999b and Netessine and zhang 2003 for situations in which competition may result in the system-optimal performance). In fact, a Ne may not even be on the Pareto frontier: the set of strategies such that each player can be made better off only if some other player is made worse off. A set of strategies is Pareto optimal if they are on the Pareto frontier otherwise a set of strategies is Pareto inferior. Hence. a e can be pareto inferior. The Prisoner's Dilemma game is the classic example of this: only one pair of strategies is Pareto optimal(both cooperate), and the unique Nash equilibrium(both"defect")is Pareto inferior. A large body of the SCm literature deals with ways to align the incentives of competitors to achieve optimality( see Cachon 2002 for a comprehensive survey and taxonomy In the newsvendor game one could verify that the competitive solution is different from the centralized solution as well, but this issue is not the focus of this chapter 2.1 Existence of equilibrium A NE is a solution to a system of n equations(first-order conditions), so an equilibrium may not exist. Non-existence of an equilibrium is potentially a conceptual problem since in this case it, is not clear what the outcome of the game will be. However, in many games a ne does exist and there are some reasonably simple ways to show that at least one NE exists. As already mentioned, a Ne is a fixed point of the best response mapping. Hence fixed point theorems can be used to estab- lish the existence of an equilibrium. There are three key fixed point theorems, named after their creators: Brouwer, Kakutani and Tarski. (See Border 1999 for details and references. ) However, direct application of fixed point theorems is somewhat inconvenient and hence generally not done (see Lederer and Li 1997 and Majumder and Groenevelt 2001a for existence proofs that are based on Brouwer' s fixed point theorem). Alternative methods, derived from these fixed point theorems have been developed. The simplest(and the most widely used) technique for demonstrating the existence of NE is through verifying concavity of the players' payoffs, which implies continuous best. response functions THEOREM 1(Debreu 1952 ). Suppose that for each player the strategy space is compact and convet and the payoff function is continous and quasi-concave with respect to each player's oun strategy Then there eri s at least one pure strategy Ne im the game If the game is symmetric (i.e, if the players'strategies and payoffs are identical), one would imagine that a symmetric solution should exist. This is indeed the case, as the next Theorem ascertains THEOREM 2. Suppose that a game is symmetric and for each player the strategy space is compact and conve and the pay off function is continuous and quasi-concave with respect to each player own strateg. Then there erists at least one symmetric pure strategy ne in the game To gain some intuition about why non-quasi-concave payoffs may lead to non-existence of ne, suppose that in a two-player game, player 2 has a bi-modal objective function with two local maxima. Furthermore, suppose that a small change in the strategy of player 1 leads to a shift of the global maximum for player 2 from one local maximum to another To be more specific, let us say that at f the global maximum r%() is on the left(Figure 2) and at i the global ma aximum : 2(2) is on the right(Figure 3). Hence, a small change in a from y to m induces a jump in the best response of player 2, 2. The resulting best response mapping is presented in Figure 4 and there is no NE in pure strategies in this game. As a more specific example, see Netessine d Shumsky(2001)for an extension of the newsvendor game to the situation in which product inventory is sold at two different prices; such game may not have a ne since both players'objectives may be bimodal. Furthermore, Cachon and Harker(2002 ) demonstrate that pure strategy nE may not exist in two other important settings: two retailers competing with cost functions described oy the Economic Order Quantity(EoQ) or two service providers competing with service times described by the M/M/1 queuing model 丌2(x1▲ z2(x") (x1) x2(x1”) Figure 2 F Igure 3 F The assumption of a compact strategy space may seem restrictive. For example, in the newsvendor game the strategy space R+ is not bounded from above. However, we could easily bound it with some large enough finite number to represent the upper bound on the demand distribution. That bound would not impact any of the choices, and therefore the transformed game behaves just as the original game with an unbounded strategy space 9 To continue with the newsvendor game analysis, it is easy to verify that the newsvendor's objective unction is concave(and hence quasi-concave)wr t. the stocking quantity by taking the second derivative. Hence the conditions of Theoren 1 are satisfied and a NE exists. There are virtually dozens of papers employing Theorem 1(see, for example, Lippman and McCardle 1997 for the proof involving quasi-concavity, Mahajan and van Ryzin 1999a and Netessine et al. 2002 for the proofs involving concavity). Clearly, quasi-concavity of each player's objective function only implies uniqueness of the best response but does not imply a unique ne. One can easily envision a situation where unique best response functions cross more than once so that there are multiple equilibria. (see Figure 5) Figure 5. Non-uniqueness of the equlibrium If quasi-concavity of the players' payoffs cannot be verified, there is an alternative existence proof that relies on Tarski's(1955) fixed point theorem and involves the notion of supermodular games The theory of supermodular games is a relatively recent development introduced and advanced by Topkis(1998). Roughly speaking, Tarski's fixed point theoren only requires best response map- pings to be non-decreasing for the existence of equilibrium and does not require quasi-concavity of the players' payoffs(hence, it allows for jumps in best responses). While it may be hard to believe that non-decreasing best responses is the only requirement for the existence of a ne, consider once again the simplest form of a single-dimensional equilibrium as a solution to the fixed point mapping r=f(r )on the compact set. It is easy to verify after a few attempts that, if f( r )is non-decreasing Cbut possibly with jumps up) then it is not possible to derive a situation without an equilibrium However, when f(a)jumps down, non-existence is possible(see Figures 6 and 7) Hence, increasing best response functions is the only (major) requirement for an equilibrium to exist; players'objectives do not have to be quasi-concave or even continuous. However, to describe an existence theorem with non-continuous payoffs requires the introduction of terms and definitions from lattice theory. As a result, we shall restrict ourselves to the assumption of continuous payoff functions, and in particular, to twice-differentiable payoff functions 10

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