供应链分析中的博弈论应用与概念探讨

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"Game Theory在供应链分析中的应用" 供应链分析是一个涉及多个代理的复杂领域，这些代理通常具有相互冲突的目标。游戏理论已经成为了分析这种环境的关键工具。本章深入探讨了游戏理论在供应链分析中的应用，并预览了未来可能被采用的游戏理论概念。作者Gérard P. Cachon和Serguei Netessine都是来自宾夕法尼亚大学沃顿商学院的专家，他们在2003年2月撰写了这篇论文。文章首先区分了非合作游戏理论与合作游戏理论，并在静态和动态的供应链场景中进行讨论。非合作游戏理论通常涉及各个参与者独立决策，寻求最大化自身的利益，而合作游戏理论则考虑了参与者之间可能通过协作来实现共同利益。在确定非合作游戏的均衡时，研究者特别关注如何证明均衡的存在性和唯一性。一个被广泛用来演示这些理论工具的例子是"新闻贩子游戏"（Newsvendor Game）。新闻贩子面临的是需求不确定性问题，必须决定订购多少商品以满足预期需求，同时最小化过度库存和缺货的成本。这个游戏模型很好地展示了如何应用博弈论来平衡风险和回报，以及如何在信息不对称的情况下做出最优决策。供应链中的博弈问题包括价格战、库存管理、合同设计、产能分配等多个方面。例如，供应商和零售商之间的价格博弈可能导致双方都采取不合作策略，导致低效的市场结果。通过理解并应用游戏理论，可以设计出激励兼容的合同，促进供应链的协调和效率。此外，动态设置下的游戏理论引入了时间序列和连续决策的概念。这在处理如需求预测更新、应急响应或供应链网络重新配置等动态问题时尤其重要。动态博弈可能涉及到参与者对其他方行为的预期和反应，以及随着时间推移的策略调整。游戏理论在供应链分析中发挥着至关重要的作用，它不仅帮助我们理解不同参与者之间的互动，而且为解决实际业务问题提供了理论框架。通过深入研究和应用这些理论，企业可以更好地管理供应链风险，优化资源配置，提高整体绩效。随着电子商务和数字化的不断发展，游戏理论在供应链分析中的应用将更加广泛和深入。

An in teresting feature of the NE concept is that the system optimal solution (a solution that

maximizes the total pa yoﬀ to all players) need not be a NE. Hence, decen tralized decision making

generally introduces ineﬃciency in the supply chain. (There are, however, some exceptions: see

MahajanandvanRyzin1999bandNetessineandZhang 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 s et of strategies such that each player can be made better oﬀ only if some other

pla yer 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 NE 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 Pare to inferior. A large body of

the SCM literature deals with w ays to align the incentives of competitors to achieve optimality (see

Cachon 2002 for a comprehensive survey and taxonomy). In the newsv endor 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 k ey 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 inconv enient 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,

ha ve been developed. The simplest (and the most widely used) technique for demonstrating the

existence of NE is through verifying concavit y 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 convex

and the payoﬀ function is continuous and quasi-concave with respect to each player’s own strategy.

Then there exists at least one pure strategy NE in 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 convex and the payoﬀ function is continuous and quasi-concave with respect to each player’s

own strategy. Then there exists 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 ch an ge 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 x

the global maxim um x

∗

) is on the left (Figure 2) and at x

the global maximum

∗

)isontheright(Figure3). Hence,asmallchangeinx

from x

to x

induces a jump in

the best response of player 2, x

∗

. 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

and Shumsky (2001) for an extension of the newsvendor game to the situation in which product

inve ntory 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 t wo other important settings: two retailers competing with cost functions described

by the Economic Order Quan tity (EOQ) or two service providers competing with service times

described b y the M/M/1 queuing model.

)'(

Figure 2.

)"(

)''(

Figure 3.

x "

Figure 4.

The assumption of a compact strategy space may seem restrictive. For example, in the newsvendor

game the strategy space R

is not bounded from abov e. 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.

To con tinue with the newsvendor game analysis, it is easy to verify that the newsvendor’s objective

function is concave (and hence quasi-concave) w.r.t. the stocking quantity by taking the second

derivative. Hence the conditions of Theorem 1 are satisﬁed and a NE exists. There are virtually

dozens of papers emplo ying 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 pla yer’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).

equilibria

Figure 5. Non-uniqueness of the equlibrium.

If quasi-conca vity of the pla yers’ payoﬀscannotbeveriﬁed, 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 theorem only requires best response map-

pings to be non-decreasing for the existence of equilibrium and does not require quasi-concavity of

the players’ pay o ﬀs (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

x = f (x) on the compact set. It is easy to verify after a few attempts that if f(x) is non-decreasing

(but possibly with jumps up) then it is not possible to derive a situation without an equilibrium.

However, when f(x) 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; play ers’ objectives do not have to be quasi-concave or even con tinuous. Ho wever, 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 ourselv es to the assumption of continuous pa yoﬀ

functions, and in particular, to twice-diﬀerentiable payoﬀ functions.

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