978-1-4244-8385-3/11/$26.00 ©2011 IEEE MSIE 2011 1029

Research on the Cooperation Between Supply Chain Enterprises Based on Game Theory

Mu Weizhe Management School

Harbin University of commerce Harbin, China

muweizhe@126.com

Abstract—In 21 century, the market competition tends to compete between supply chain and supply chain. The competitive power of the supply chain comes from the cooperation of supply chain company. This paper made research on the cooperative relationship between supply chain enterprises, analyzed the non-cooperative and cooperative game strategy between supply chain enterprises, compared the cooperation efficiency in both cases. The efficiency of most supply chain is low under non-cooperative strategy, and cooperative game can make the supply chain get greater return in the theory. The conclusion is that long-term cooperation in the supply chain can lead to greater benefits for the supply chain enterprises. Therefore, cooperative game strategy can improve the overall efficiency of the supply chain, and also allows for maximum individual interests. The results make all parties to obtain higher returns, achieve win-win situation.

Index Terms—game theory, supply chain, cooperative game, prisoner's dilemma

I. INTRODUCTION The relationship between the supply chain enterprises is a

mixture of competition and cooperation. Fundamentally speaking, the supply chain enterprises whether to cooperate is the game issue of the supply chain. We know that a supply chain enterprise's benefit depends not only on its own behavior, but also on the transaction with which the behavior of the other supply chain. The game mode is divided into two categories: cooperation and non-cooperation. The cooperation is not only for their own interests but focus more on common interests, and reaches a binding agreement. Either party makes its own strategy choices; do not know what strategy the other party will choose, but each party will predict the strategy what the other party will choose. To achieve long-term interests, the rational parties will establish the target of maximize the common interests.

The cooperative game theory in supply chain management has been studied. Christy and Gorut [1] set up a game theory frame model, to describe the relationships between buyers and supplier in the supply chain. Cachong and Lariviere[2] demonstrated the problem whether there is no cost signal based on the promises that manufacturers imposed supplier. In the game analysis and supply chain partner selection, Lin xudong and Zhu shunquan[3] established a pay game matrix, according to the Nash equilibrium theory, to determine the scope of the choice, then choose the best partners. In the supply affect the

game analysis of trust, Zhou yong, Zheng pi e and Zhang hao[4] point out that the trust was the key enterprise relationship in the supply chain. The trust was divided into participants trust and control trust; they constitute the basis of the supply chain. Chen yuhua and Zhu shunquan[5] proved that when the players with long-term profits for the target, cooperation was the optimal strategy. This illustrated the possibility and necessity to establish long-term cooperation relations in the supply chain. Chen yanru, Jiang yangsheng and Pu yun[6] in a class of supply chain join game model, illustrated the supply chain was a new type of relationship mode between manufacturers, and supply chain management was essentially a management about the relationship management.

II. THE ANALYSIS OF THE GAME STRATEGY BETWEEN SUPPLY CHAIN ENTERPRISES

A. The Analysis of the Non-cooperative Game Strategy Between Supply Chain Enterprises

1) The equilibrium analysis of the non-cooperative Game Strategy

Assume that a supply chain contains only one supplier and one manufacturer. The supplier provides raw materials to the manufacturer; the manufacturer obtains products by processing raw materials. The market price of product is decided by the quantity of products the manufacturers to sell. Assume that the raw material which the supplier provides is the only raw material which the manufacturer production product needs. The depreciation is zero.

And assume that to produce one unit of product requires one unit of raw materials. That is the input-output ratio is 1:1, and divides the bilateral benefit equally. The two enterprises carry on two stage complete information dynamic game, the step is:

a) The manufacturer purchases raw material to the supplier, the supplier determines their optimal bidding first.

b)The manufacturer determine their own optimal order quantity after considers the market demand curve, the quotes of supplier and the production costs of these products.

c) The supplier provided raw material to the supplier on time.

Assume that the demand curve which the manufacturer faces is

(mP b aQ a= + ＜0，b＞0) （1）

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In the formula:

mP —The market price of the product when the number of manufacturer's sales is Q

Q —Representatives of both the number of products, but also for the required amount of raw materials

For ease of distinction, assume that the number of products is cQ , the number of raw materials is yQ .

Assume that mC is the unit processing cost of product of

the manufacturer, and sP is the quoted price of the supplier. sP is the function of Q which is the number of raw materials required for the manufacturers. mR is the manufacturer's profit. Manufacturer's profit is not only the function of market prices and processing costs, but also the function of the quoted price of the supplier.

Because the supplier makes the first decision, the supplier's price is a real variable, is not a guess variable. Manufacturer determines the order quantity after he gets the quoted price of the supplier in the first phase. sC is the unit cost of the raw

materials from the supplier. sR is the supplier’s profit. sR depends not only on his own quoted price and production costs, but also on the manufacturer's order quantity. Then

( )m m s mR P P C Q= − − ∗ （2）

( )*s s sR P C Q= − （3）

Based on non-cooperative strategy, when the supplier considers the price, he considers that the manufacturer will maximize profits as a principle. The formula that manufacturers to maximize profits is

/ 0m dQdR =

The marginal revenue is 0:

2 0s mb aQ P C+ − − =

( ) / 2s mQ P C b a= + − （4）

So, the supplier will pull (4) into (3), then

( ) ( )* / 2s s s s mR P C P C b a= − + − （5）

To get the derivative:

( )2 / 2 0s s mP C b C a− − + =

Then

( ) / 2s s mP C b C= + − （6）

The quotes of supplier is sP , the manufacturer's order quantity is Q , pull (6) into (4), then

( ) / 2 / 2s m mQ C b C C b a= + − + −⎡ ⎤⎣ ⎦

Then

( ) / 4s mQ C C b a= + − （7）

At this point, the supplier's price and order quantity have been determined. The two sides will obtain the expected benefits according to their own calculated way. The manufacturer's expected benefits is:

( ) / 2 *m s m mR b aQ C b C C Q= + − + − −⎡ ⎤⎣ ⎦

Then

( )2 /16m s mR C C b a= − + − （8）

Similarly, the supplier's expected benefits is:

( )2 / 8s s mR C C b a= − + − （9）

Although their game strategy is non-cooperative, their actual benefits should be calculated according to the supply chain. Then,

( )*a m m sR P C C Q= − − （10）

Pull (7) into (10), then the common interest is:

( )23 /16a s mR C C b a= − + − （11）

To average, then

23( ) / 32m s s mR R C C b a= = − + − （12）

Then the game is over. The two sides both choose non-cooperation strategy, and we know this is the Nash equilibrium after the average. But in this Nash equilibrium, we do not make the supply chain to achieve Pareto optimal. That is not the average interests that based on the maximize profits as a starting point (10). The results suggest that competitive equilibrium do not make the supply chain to achieve Pareto optimal. In addition, it is the supplier to make pricing decisions first in the model. As a “first mover advantage”, the suppliers to improve their income by increasing supply prices, and the manufacturers have to reduce the number of orders to decline the overall revenue.

2) The instability of cooperation In above analysis, we simply assume that the benefits of

both sides to cooperate equally with the result that it is higher than the unilateral interests, when the strategy is non-cooperation. To the supplier, it is lower than the unilateral interests, when the strategy is non-cooperation. After the reunification of the supply chain, the increased revenue is all obtained by the manufacturer. It can be said that is the leverage punishment by the distribution of benefits. To punish the suppliers still choose non-cooperative strategies under “first mover advantage”.

For the supplier, the cooperative supply chains will no returns through high pricing strategy to improve his income. Certainly there will no incentive to participate in co-operation. Because their products (raw materials) may still be sold at a price with sP even if it is not in the supply chain. In fact, the

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model ignored the two sides need to pay the appropriate co-management costs when the two sides cooperation. This will lead that the supplier’s gains not as many as uncooperative. (Here we assume that cooperation’s profit is greater than the sum of the cost to both sides.) Therefore, the profit distribution after the cooperation, for suppliers to be redistributed between the two sides, that is to be allocated in accordance with (8) and (9). The distribution of profits will take into account both efficiency and equity factors. The two sides will bargain to distribute profits. The potential costs of bargaining may be greater than the increased revenue under cooperation. So that the cooperation in fact can not be reached. Therefore, the cooperation is likely to be unstable.

B. The Analysis of the Cooperative Game Strategy Between Supply Chain Enterprises In reality, the supply chain management is more likely to

develop long-term relationship, fewer but the number of partners required, while the number of co-operation is very much. In the theory of modern supply chain management, the supply chain which is constitute of this small work groups under the long-term close co-operation, will be composed of more efficient production. It will continue to study the process of cooperation in the supply chain. Both of the process of cooperation reached between the parties and the distribution of benefits will be researched, making all parties in the supply chain to benefit from cooperation, to achieve Pareto improvement of game results.

If the Prisoner's Dilemma Game is infinitely repeated, some strategies will be better than non-cooperative strategies have been taken in each round. First, assume that the manufacturer adopt a “tit for tat” strategy. As a flip of the players hold “tit for tat” strategy, it will "punish" the upper hand that choose non-cooperative in any round opponent through the non-cooperation. But the punishment is measured, thus there will be a possibility to get a common return from a cooperative relationship.

If the prisoner's dilemma game is in the first round, then begin the second round by the same conditions. After the first game, the supplier believes that cooperation can get more revenue and assumes the other will also take the cooperation strategy. To get the derivative of (10), we can get the ideal order quantity:

( ) / 2m sQ C C b a= + − （13）

Pull (13) into (5), then get the derivative. We can get this price when the manufacturer gets this order quantity:

s sP C= （14）

At this point you can easily reach the interests of the supplier is 0, the interests of manufacturer equal to the supply chain, the manufacturer also adopt a cooperative strategy, so ( )*a m m m sR R P C C Q= = − − , pull (1) into it:

( )2 / 4a m sR C C b a= − + − （15）

Then equal distribution of interests:

( )2 / 8m s s mR R C C b a= = − + − （16）

At this time the game is over, both sides choose cooperation strategies. After the split, we know this is the Nash equilibrium, the supply chain reach the Pareto optimal in this Nash equilibrium. It is clear that the change is carried out under repeated games in long-term cooperation. There will not such a change, if the two sides in the last game stop cooperating. But the last act does not directly affect the judgments, the benefits of both sides should be the weighted average of the two game proceeds, and the weighted average (assuming the same weight) will enable the two sides recognized the benefits under cooperation strategies is higher then non-cooperative strategies. It makes the game to continue for the next round and be more inclined to cooperative strategy.

III. THE COMPARATIVE ANALYSIS OF THE EFFICIENCY BETWEEN NON-COOPERATION STRATEGY AND COOPERATIVE

STRATEGY

A. The Efficiency Analysis of Supply Chain Under Non-cooperative Strategy Non-cooperative game theory emphasizes that individuals

maximize the benefits; the result may be efficient, inefficient or even ineffective. Supply chain management requires the supply chain enterprises play to their strengths to get specialization advantages. Specialization create a huge potential for improve the efficiency of the supply chain, but it also implies a certain degree of information asymmetry, market monopoly and externalities. Therefore, the pursuit of the supply chain enterprises to self-interest often leads to inefficiencies in the whole supply chain or even the disintegration.

For example, in the condition of asymmetric information and externalities, the "bullwhip effect" of the supply chain shows that the inefficiencies of the supply chain were caused by individual rational decision. When the all supply chain enterprises make production or supply decisions only according to the demand information from its adjacent downstream enterprises, the demand information will amplify step by step along the supply chain from the consumer. When supply and demand information reach the original supplier, the actual needs of consumers has generated great deviation. To meet this demand, upstream supplier owns higher inventory than the lower supplier. Therefore, the overall revenue of supply chain will decline.

In the case of non-cooperation, the manufacturers and suppliers all make their efforts to maximize heir own interests. Plans to make the interests of manufacturers

It makes the scheme interests of the manufacturer is (8), the interests of the supplier is (9). But in the end the interest only is (12), this clearly is low efficient. The two sides are more likely to cause conflicts because of the profit is ( )2 / 32s mC C b a− + − .It is obviously that most of the supply

chain is low efficiency under non-cooperative strategy.

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B. The Efficiency Analysis of Supply Chain Under Cooperative Strategy The cooperation strategy emphasizes that collective

rationality, efficiency, impartiality and fairness. The concept of cooperation is closer to the concept of Pareto optimality. In game theory, to the supply chain which is composed of two or more members for the common purpose of action, members set the expense of their own utility function, and create a collective utility function which is a new, and used to determine their collective behavior.

It is found that, in the analysis of the two-stage complete information dynamic game between the supplier and manufacturer, the cooperation makes the interests of the supply chain actually increased ( )2 /16s mC C b a− + − . This shows that the two sides do not have to fight for the interests of ( )2 / 32s mC C b a− + − . Despite the effectiveness of the supplier only the same with their own best interests, he has omitted the transaction costs when the manufacturer will accept the high price of sP . So the next round he will still be tending to cooperation strategy. For the manufacturer, the higher profit will be more exciting. It’s a typical Nash equilibrium. Therefore, in terms relative to the non-cooperative game, cooperative game can make the supply chain for greater return in the theory. However, as each supply chain enterprise is independent economic entity, the final decision making is to maximize their individual expected profit as the goal. Therefore, we studied the cooperative game should not only adhere to individual rationality of the supply chain, but also the cooperative behavior model that meet the supply chain collective rationality.

The organization form of long-term relationships provides a way to solve the problem of "prisoner's dilemma" game theory. Although the cooperation in repeated games just based on individual rationality and common knowledge, the game results also enhance the collective efficiency. Because in the organization of long-term relationship, the long-term gains from mutual cooperation may remove the contradiction between short-term incentive and the behavior that not only consistent with individual rationality but also consistent with collective rationality. In the long-term relationship, supply chain organization should be committed to create the common faith to trust that others will cooperation. Cooperation is not always the dominant strategy, only everyone believes that others will cooperate, the cooperation is the individual rational choice. Therefore, in order to build a successful supply chain partnership between members, it means that the choice of supply chain enterprise is right in the early stage of cooperation, and the full range of communication and cooperation from the strategic level to the operating level after the supply chain formed. Meanwhile, both sides should work to enhance their

core competitiveness, making the parties strengthen rely on each other, and the cooperation relations will become increasingly closer.

IV. CONCLUSION We analyzed the game relations between the supply chain

enterprises, and analyze the non-cooperation and cooperation strategies between the supply chain enterprises by Prisoner theory. The results show that in the premise of meeting the individual interests, the competition between supply chain enterprises led to the low efficiency of the overall supply chain. The parties in the repeated game mechanism, to the cooperation strategy, will both to improve the overall efficiency of the supply chain, and also allows for maximum individual interests. The results make all parties to obtain higher returns, achieve win-win situation. The use of the repeated games theory can get Pareto optimal state. Through the comparison of the benefits to both sides, we have come to the conclusions that long-term cooperation in the supply chain can lead to greater benefits for the supply chain enterprises. But there is a problem that the supply chain enterprises will be lack of enthusiasm.

1) The non-cooperation strategy of the supply chain enterprises which is chosen in the state that the information is complete of the both sides, and it will be impact of supply chain enterprises to re-cooperation.

2) If the supply chain makes the two sides to reach agreement on long-term cooperation, the two sides will be tend to try cooperative strategy after weighing the interests. In principle, the cooperation strategy is always the most optimal efficiency. And in the long-term cooperation, the supply chain enterprises which adopt full cooperation strategy will always be more efficiency than the supply chain enterprises which has the history of non-cooperative strategy.

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