1 文献综述：博弈论在供应链管理中的应用数 9 艾松 2...

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文献综述：文献综述：博弈论在供应链管理中的应用博弈论在供应链管理中的应用

数数 9 9 艾松艾松

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博弈论在供应链管理中的博弈论在供应链管理中的应用应用

现在还处于探索的阶段现在还处于探索的阶段 ,, 所用的博弈论所用的博弈论理论还比较浅；理论还比较浅；

更多的是用博弈论中的概念、已有的结更多的是用博弈论中的概念、已有的结论等，最常用的就是论等，最常用的就是 NashNash 均衡，均衡， GaGameme 的模型，的模型， StackelbergStackelberg 模型等；模型等；

部分模型用显示原理、部分模型用显示原理、 NashNash 均衡的存均衡的存在性定理来求解均衡结果。在性定理来求解均衡结果。

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文献综述文献综述Huang,Z.M., S.X.Li. 2001.

Co-op advertising models in manufacturer-retailer supply chains:A game theory approach.

European Journal of Operational Research 135,527-544.

Li,S.X., Z.M.Huang, J.Zhu, P.Y.K.Chau. 2002.

Cooperative advertising,game theory and manufacturer-retailer supply chains.

Omega 30,347–357.

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Huang,Z.M., S.X.Li. 2001.Huang,Z.M., S.X.Li. 2001. Co-op advertising models in manufacturer-Co-op advertising models in manufacturer-retailer supply chains:A game theory retailer supply chains:A game theory approach. approach. European Journal of Operational Research European Journal of Operational Research 135,527-544. 135,527-544.

Keyword: Keyword:

Decision analysis; Game theory; Co-op Decision analysis; Game theory; Co-op advertising; Equilibrium; Coordination; advertising; Equilibrium; Coordination; Bargaining problems; Utilities.Bargaining problems; Utilities.

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1.Introduction1.Introduction Vertical co-op advertising is an interactive Vertical co-op advertising is an interactive

relationship between a manufacturer and a relationship between a manufacturer and a retailer in which the retailer initiates and retailer in which the retailer initiates and implements a local advertising and the implements a local advertising and the manufacturer pays part of the cost.manufacturer pays part of the cost.

The main reason for a manufacturer to use co-op advertising is to strengthen the image of the brand and to motivate immediate sales at retailer level.

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1.Introduction1.Introduction Most studies to date on vertical co-op Most studies to date on vertical co-op

advertising have focused on a advertising have focused on a relationship where the manufacturer is a relationship where the manufacturer is a leader and the retailer is a follower.leader and the retailer is a follower.

This paper is intended to discuss the This paper is intended to discuss the relationship between co-op advertising relationship between co-op advertising and efficiency of manufacturer- retailer and efficiency of manufacturer- retailer transactions.transactions.

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1.Introduction1.Introduction

Three co-op advertising model:Three co-op advertising model:

1.a leader-follower noncooperative game:1.a leader-follower noncooperative game:manufacturer is a leader;manufacturer is a leader;

2.a noncooperative simultaneous move ga2.a noncooperative simultaneous move game;me;

3.a cooperative game.3.a cooperative game.

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2.Assumptions2.Assumptions

S—retailer’s sales response volume S—retailer’s sales response volume function of product; function of product;

a —retailer’s local advertisinga —retailer’s local advertising level; level;

qq—manufacturer’s national brand name —manufacturer’s national brand name investmentinvestment

t —fraction of total local advertising t —fraction of total local advertising expenditures which manufacturer sharesexpenditures which manufacturer shares

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One-period sales response volume function:

Expected sales response volume:

profit marginaldollar ser'manufacturm

profit marginaldollar sretailer'r

0)(,0,, ,0

),(~

~

E

qaqaS

qaqaS ),(

10


The manufacturer’s,retailer’s,system’s The manufacturer’s,retailer’s,system’s expected profit functions are as follows:expected profit functions are as follows:

Note: “cq” should be “q”

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3.Stackelberg equilibrium3.Stackelberg equilibrium

We model the relationship between the manWe model the relationship between the manufacturer and the retailer as a sequential noufacturer and the retailer as a sequential noncooperative game with the manufacturer ancooperative game with the manufacturer as the leader and the retailer as the follower.s the leader and the retailer as the follower.

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3.Stackelberg equilibrium3.Stackelberg equilibrium We first solve for the reaction function in the We first solve for the reaction function in the

second stage of the game:second stage of the game: is a concave function of is a concave function of Setting the first derivative of with respect Setting the first derivative of with respect

to to be zero:to to be zero:

Then we have Eq(5): Then we have Eq(5):

r ar

a

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We can observe that:

So the manufacturer can use his co-op advertising policy and his national brand name investment to induce the retailer to increase or decrease local advertising expenditure at a level he expects.

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Next the optimal value of and are Next the optimal value of and are determined by maximizing the determined by maximizing the manufacturer’s profit subject to the constraint manufacturer’s profit subject to the constraint imposed by Eq(5).Hence,the manufacturer’s imposed by Eq(5).Hence,the manufacturer’s problem can be formulated as problem can be formulated as

q t

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Substituting

into the objective yields the following problem (9):

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Solving Eq(9),and substituting the Solving Eq(9),and substituting the outcome into Eq(5),we have outcome into Eq(5),we have the unique equilibrium point of the two-stage game:

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Proposition 1:Proposition 1:If If (1)the manufacturer offers positive advertising (1)the manufacturer offers positive advertising

allowance to the retailer ,otherwise he will allowance to the retailer ,otherwise he will offer nothing;offer nothing;

(2)(2)

1

r

m

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Three implications:Three implications: (1) (1) if retailer’s marginal profit is high,retailer

has strong incentive to spend money in local advertising to stimulate the sales, even though the manufacturer only shares a small fraction of local advertising expenditures or doesn’t help;

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(2)the higher (the lower) the retailer’s ((manufacturer’s) ) marginal profit,the lower the manufacturer’s advertising allowance for the retailer;

(3)the increase of such that will cause an increase in the sales and then

will give the retailer incentive to do local advertising without manufacturer’s financial help.

1

r

m

20


In this game,the manufacturer holds In this game,the manufacturer holds extreme power and has almost complete extreme power and has almost complete control over the behavior of the retailer.control over the behavior of the retailer.

The relationship is that of The relationship is that of

an employer and an employeean employer and an employee!!

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4.Nash equilibrium4.Nash equilibrium

Recent studies in marketing have demonstrated that in many industries retailers have increased their power relative to manufacturers over the past two decades.

Especially,for durable goods such as appliances and automobiles, the retailer has more influence on the consumer’s purchase decision.

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In this section,we relax the leader-follower In this section,we relax the leader-follower relationship and assume a symmetric relatirelationship and assume a symmetric relationship between the manufacturer and the ronship between the manufacturer and the retailer.etailer.

The manufacturer and the retailer simultanThe manufacturer and the retailer simultaneously and noncooperatively maximize theeously and noncooperatively maximize their profits with respect to any possible strateir profits with respect to any possible strategies set by the other member .gies set by the other member .

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Hence,the manufacturer’s optimal Hence,the manufacturer’s optimal problem is:problem is:

The retailer’s optimal problem is:The retailer’s optimal problem is:

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It is obvious that the manufacturer’s It is obvious that the manufacturer’s optimal fraction level, ,is zero,because of optimal fraction level, ,is zero,because of its negative coefficient in the objective.its negative coefficient in the objective.

A Nash equilibrium advertising scheme A Nash equilibrium advertising scheme can be obtained by simultaneously solving can be obtained by simultaneously solving the following conditions:the following conditions:

t

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We then obtain the unique Nash equilibrium We then obtain the unique Nash equilibrium advertising scheme as follows:advertising scheme as follows:

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Three implications:Three implications: (1)since the manufacturer’s allowance (1)since the manufacturer’s allowance

policies does not influence the sales policies does not influence the sales response volume function, independent response volume function, independent actions taken by both members actions taken by both members simultaneously make no impact of the simultaneously make no impact of the sharing policies on the determination of the sharing policies on the determination of the retailer’s local advertisingretailer’s local advertising level; level;

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(2)(2)

0,0****

mm

aq

(3)(3)

0,0****

rr

aq

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4.Nash equilibrium4.Nash equilibrium Comparisons among results between two diffComparisons among results between two diff

erent noncooperative game:erent noncooperative game:

Proposition 2:Proposition 2:

(a)The manufacturer always prefers the leade(a)The manufacturer always prefers the leader-follower structure rather than the simultaner-follower structure rather than the simultaneous move structure;ous move structure;

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4.Nash equilibrium4.Nash equilibrium (b) If (b) If

the retailer prefers the simultaneous move the retailer prefers the simultaneous move game structure,otherwise he prefers the game structure,otherwise he prefers the leader-follower game structure.leader-follower game structure.

1)(

m

r

m

r

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(a)The manufacturer’s brand name invest(a)The manufacturer’s brand name investment is higher at Nash than at Stackelbement is higher at Nash than at Stackelberg ;rg ;

(c) The manufacturer’s advertising allowa(c) The manufacturer’s advertising allowance for retailer is zero.nce for retailer is zero.

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(b) If (b) If

the retailer’s local advertising expendituthe retailer’s local advertising expenditure is higher at Nash than at Stackelberg, re is higher at Nash than at Stackelberg, otherwise it is lower at Nash than at Staotherwise it is lower at Nash than at Stackelberg.ckelberg.

1)( 1

m

r

m

r

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5. 5. An efficiency co-op advertising model

In this section we will retain the assumption of the symmetric relationship between the manufacturer and the retailer.

We will discuss the efficiency of manufacturer and retailer transactions in vertical co-op advertising agreements.

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We consider Pareto efficient advertising schemes in our co-op advertising arrangements.

A scheme is called Pareto efficient if one cannot find any other scheme (a,t,q) such that neither the manufacturer’s nor the retailer’s profit is less at (a,t,q) but at least one of the manufacturer’s and retailer’s profits is higher at (a,t,q) than at .

),,( 000 qta

),,( 000 qta

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Since and are quasi-concave, the set of Pareto efficient schemes consists of those points where the manufacturer’s and the retailer’s iso-profit surfaces are tangent to each other, i.e., for some >= 0

m r

35


This leads to to the following propositionroposition

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This theorem tells us that all Pareto efficient schemes are associated with a single local advertising expenditure and a single manufacturer’s brand name investment and with the fraction t of the manufacturer’s share of the local advertising expenditures between 0 and 1.

The locus of tangency lies on a vertical line segment at in (a,t,q) space.

*a *

q

),(**

qa

37



An advertising scheme is Pareto efficient if and only if it is an optimal solution of the joint system profit maximization problem.

This theorem tells us that, among all possible advertising schemes, the system profit (i.e., the sum of the manufacturer’s and the retailer’s profits) is maximized for Every Pareto efficient scheme, but not for any other schemes.

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Proposition 6:Proposition 6: (a)The system profit is higher at any Pareto

efficient scheme than at both noncooperative equilibriums;

(c) The local advertising expenditure is higher at any Pareto efficient scheme than at both noncooperative equilibriums;

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(b)If

then the manufacturer’s brand name investment is higher at any Pareto efficient scheme than at both noncooperative equilibriums,

otherwise the manufacturer’s brand name investment at any Pareto efficient scheme is higher than at Stackelberg equilibrium and is lower than at Nash equilibrium.

)(1r

m

m

r

40


Proposition 6 leads to the possibility that both the manufacturer and the retailer can gain more profits compared with Stackelberg equilibriums.

But it should be noted that not all Pareto efficient schemes are feasible to both the manufacturer and the retailer. Neither the manufacturer nor the retailer would be willing to accept less profits at full cooperation than with noncooperation.

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An advertising scheme An advertising scheme

is called feasible Pareto efficient ifis called feasible Pareto efficient if

Yqta ),,(**

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the feasible Pareto efficient set of advertising schemes.

Since only schemes satisfying (24) and Since only schemes satisfying (24) and (25) are acceptable for both the (25) are acceptable for both the manufacturer and the retailer when they manufacturer and the retailer when they do coordinate, we then calldo coordinate, we then call

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Referring to Referring to Proposition 2, Proposition 2, we know that:we know that:

(1)(1)

(2)If (2)If

then otherwise then otherwise

1)(

m

r

m

r

***mm

***rr ***

rr

44


Therefore Therefore

For the purpose of simplicity,we assume For the purpose of simplicity,we assume thatthat

1)(

m

r

m

r

45


Hence relationships in Eq(24) and (25) Hence relationships in Eq(24) and (25) can be rewritten ascan be rewritten as

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Let Let

Here we assume Here we assume 02 k

47


Then Then

and Z can be simplified asand Z can be simplified as

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5. 5. An efficiency co-op advertising model It can be shown that Therefore, for any given t which satisfies

we have

01 minmax tt

minmax ttt

This simply implies that there exist Pareto efficient advertising schemes such that both the manufacturer and the retailer are better off at full coordination than at noncooperative equilibrium.

0)( tm 0)( tr

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We are interested in finding an advertising scheme in Z which will be agreeable to both the manufacturer and the retailer.

According to Proposition 6Proposition 6, for any Pareto scheme

where is a positive constant.

),,(**

qta

),,(),,(****

qtaqta rm

**

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This property implies that the more the manufacturer’s share of the system profit gain, the less the retailer’s share of the system profit gain, and vice versa.

So the manufacturer and the retailer will agree to change the local advertising expenditures to and the brand name investments to . However, they will negotiate over the manufacturer’s share of the local advertising expenditures .t

*a*

q

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6.Bargaining results6.Bargaining results

Assume that the manufacturer and the retailer agree to change local advertising expenditures to and brand name investments to from and , respectively, and engage in bargaining for the determination of reimbursement percentage to divide the system profit gain.

*a

*q *a *q

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A fraction closer to is preferred by the retailer, and a fraction closer to is preferred by the manufacturer.

maxtmint

To determine the division of the system profit gain,we must give some further assumptions.

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Since there is an environment uncertainty in sales volume,both members are assumed to be uncertain about the system profit gain, .

For each Pareto efficient advertising schemes, the uncertainty is represented in terms of a probability distribution for . We assume that both members agree on the probability distributions of interest.

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Suppose both the manufacturer and the retailer have preferences for the amount of shares of the system profit gain,which preferences are represented by each system member’s von Neumann–Morgenstern cardinal utility function for .),( rm

The manufacturer’s and the retailer’s The manufacturer’s and the retailer’s utility functions are denoted by and utility functions are denoted by and , respectively. , respectively.

),( rmmu ),( rmru

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We assume the utility functions are We assume the utility functions are additive, that is to say additive, that is to say it can be written in the form

where is the conditional utility function of member i (i=m, r) for (j =m, r).

iju

j

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6.Bargaining results6.Bargaining results It has been also shown that,for additive

individual utility functions, the system utility function, , is also additive under the linear aggregation rule. The form of us is as follows:

where is the vector of aggregation weights and .

),( rmmu

),( rmmu

0),( rm 1 rm

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In order to incorporate the manufacturer’s In order to incorporate the manufacturer’s and the retailer’s risk attitude into our anaand the retailer’s risk attitude into our analysis, we define the lysis, we define the Pratt–ArrowPratt–Arrow risk aver risk aversion function as follows:sion function as follows:

is the risk aversion function of membis the risk aversion function of member i (i = m, r) to the share of the er i (i = m, r) to the share of the jjth membth member (j =m,r).er (j =m,r).

ijr

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Here we present the Here we present the Nash(1950) bargaining Nash(1950) bargaining modelmodel determining the bargaining determining the bargaining reimbursement fraction over the line segment reimbursement fraction over the line segment of Pareto efficient solutions described byof Pareto efficient solutions described by

The bargaining outcome is obtained by The bargaining outcome is obtained by maximizing the product individual marginal maximizing the product individual marginal utilities over utilities over Pareto efficient locus.

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6.Bargaining results

To demonstrate this approach, we consider two degenerated exponential utility functions for the manufacturer and the retailer as follows:

where and are positive constant.mbrb

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Eqs. (37) and (38) imply that both the mEqs. (37) and (38) imply that both the manufacturer and the retailer have constananufacturer and the retailer have constant risk aversion functions with t risk aversion functions with and and rrr br )( mmm br )(

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Since the product ofSince the product of

and can be rewritten and can be rewritten as the form in terms of : as the form in terms of :

)( mmu )( rru m

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Taking the first derivative of with Taking the first derivative of with respect to and setting it to be zero:respect to and setting it to be zero:m

rmuu

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Now we consider several special cases.Now we consider several special cases. First, assume that both the manufacturer First, assume that both the manufacturer

and the retailer have the same degree of and the retailer have the same degree of risk aversion measures, i.e. risk aversion measures, i.e.

Then solving (40),we have Then solving (40),we have

bbb rm

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Therefore,the best Pareto advertising reimbursement is

So if the manufacturer and the retailer have the same degree of risk aversion measures, the model suggests that the members should equally share the system profit gain.

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Second, assume that the manufacturer Second, assume that the manufacturer has a higher degree of risk aversion has a higher degree of risk aversion measures than the retailer and measures than the retailer and

Then solving (40), we have Then solving (40), we have

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Therefore, Therefore, the best Pareto advertising reimbursement is

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So when the manufacturer’s degree of risk So when the manufacturer’s degree of risk aversion is higher than the retailer’s,he aversion is higher than the retailer’s,he receives a lower share of the system profit receives a lower share of the system profit gain,which is consistent with the result in gain,which is consistent with the result in the case of negotiation with bargaining the case of negotiation with bargaining power.power.

A similar analysis can be accomplished for A similar analysis can be accomplished for the case where the retailer’s degree of risk the case where the retailer’s degree of risk aversion is higher than the manufacturer’s .aversion is higher than the manufacturer’s .

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7.Concluding remarks7.Concluding remarks

This paper attempts to investigate the efficiency of transaction for the system of manufacturer–retailer co-op advertising in the context of game theory.

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7.Concluding remarks7.Concluding remarks There are three possible avenues for

future research: First, the single manufacture–retailer

system assumption can be relaxed to a duopoly situation of manufacturers who sell their products through a common monopolistic retailer who sells multiple competing brands with varying degrees of substitutability.

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7.Concluding remarks7.Concluding remarks Second, in our analysis we employed

nonlinear sales response function to satisfy the saturation requirement. The use of a linear sales response function may yield different and interesting results in the analysis for vertical co-op advertising agreements.

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7.Concluding remarks7.Concluding remarks

Finally, in our study the manufacturer’s spending on local advertising is characterized only by its reimbursement policy.We can incorporate another factor, accrual rate,into our model to yield some interesting results.

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Li,S.X., Z.M.Huang, J.Zhu, P.Y.K.Chau. 2002. Cooperative advertising, game theory and manufacturer-retailer supply chains. Omega 30,347–357.

Keyword:

Co-op advertising; Supply chains; Leader–follower relationship; Pareto efficiencies; Bargaining model.

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1. Introduction2. Interactive two-stage co-op advertising model3. Higher order Stackelberg equilibrium4. Fully coordinated co-op advertising model5. Concluding remarks

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3.3.Higher order Stackelberg equilibrium Now let us consider both the manufacturer’s

and the retailer’s profit functions again:

Given a value of , Eq (1) and (2) can be used to describe the manufacturer’s and the retailer’s iso-profit curves and , respectively, in the space .

a

),( tqrm

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3.3.Higher order Stackelberg equilibrium Let , both the manufacturer’s and th

e retailer’s iso-profit curves and pass through ,which corresponds to the manufacturer’s optimal strategy at the two-stage game structure (see Figure).

),( ** tq

*aa m r

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3.3.Higher order Stackelberg equilibrium

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This can be implemented by solving the following optimization problem:

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Solving problem (12) yields the manufacturer’s optimal solution as follows:

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However, since we suppose the retailer is a However, since we suppose the retailer is a rational person, the manufacturer’s strategy rational person, the manufacturer’s strategy change will force him/her to reconsider its change will force him/her to reconsider its strategy because is not its optimal strategy because is not its optimal strategy anymore.The retailer’s new strategy strategy anymore.The retailer’s new strategy can be obtained by considering the following can be obtained by considering the following problem:problem:

*aa

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which results in the retailer’s optimal strategy as

At this point, a new equilibrium point is obtained between the manufacturer and the retailer. We refer as higher order Stackelberg equilibrium.

),,( qta

),,( qta

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So both the manufacturer and the retailer are better at than at .),,( qta ),,( *** qta

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Extension to Advertising-Pricing Extension to Advertising-Pricing CooperationCooperation

Alexandre Neyret, Jinxing Xie (2003):Alexandre Neyret, Jinxing Xie (2003):

Co-op Advertising and Pricing Models ICo-op Advertising and Pricing Models In a Manufacturer-Retailer Supply Chn a Manufacturer-Retailer Supply Chainain

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