Game Theory Based Multi-attribute Negotiation
between MA and MSAs

Fang Yu, Toshiya Kaihara, Nobutada Fujii

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Fang Yu, Toshiya Kaihara, Nobutada Fujii. Game Theory Based Multi-attribute Negotiation between MA and MSAs. 19th Advances in Production Management Systems (APMS), Sep 2012, Rhodes, Greece. pp.535-543, 10.1007/978-3-642-40361-3_68 . hal-01470664

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# Game theory based multi-attribute negotiation between MA and MSAs

Fang Yu, Toshiya Kaihara, and Nobutada Fujii

Graduate School of System Information, Kobe University
1-1 Rokkodai,Nada, Kobe, Hyogo, 657-8501, Japan
yufang@kaede.cs.kobe-u.ac.jp, kaihara@kobe-u.ac.jp
nfujii@phoenix.kobe-u.ac.jp

**Abstract.** This paper focuses on the multi-attribute negotiation between Manufacture Agent (MA) and Material Supplier Agent (MSA) of supply chain network (SCN). A modified two-stage negotiation protocol is proposed based on the two-stage negotiation protocol proposed in the previous work. The negotiation between MA and MSAs, where the quantity of the order of MA depends on the demand of Consumer Agent (CA), are discussed to decide the final supplier and the final strategies. The strategies of the negotiation are the wholesale price of the product, the quantity of the order, and the lead time. The final solution is solved by finding the Stackelberg equilibrium of MA-Stackelberg game. Numerical case is provided to illustrate the proposed protocol.

**Keywords:** Multi-agent, supply chain, negotiation, game theory

## 1 Introduction

A Supply Chain (SC) can be defined as a system consists of suppliers, manufacturers, distributors, retailers, and customers, where materials flow downstream from suppliers to customers and information flows in both directions [1]. Game theory is a powerful tool for analyzing situations in which the decisions of multiple agents affect each agent's payoff [2]. It has become a primary methodology used in supply chain network (SCN) related problems.

Related to the topic, a lot of researches have been done. Hall et al. [3] modeled the manufacturer's capacity allocation problem of the SC with a manufacturer and several distributors. [4] considered a profit-maximizing retailer using Stackelberg game and Nash equilibrium. A game theoretic model of a three-stage supply chain consisting of one retailer, one manufacturer and one subcontractor was developed to study ordering, wholesale pricing and lead-time decisions in [5]. Sinha et al. [6] analyzed the coordination and competition issues in a two-stage SC distribution system where two vendors compete to sell differentiated products through a common retailer in the same market. Xia [7] studied the market competition and pricing strategies for suppliers in a SC with two competitive suppliers and multiple buyers. In the previous research, the single attribute negotiation between one Manufacture Agent (MA) and multiple Material Supplier Agents (MSA) has been discussed [8]. It was assumed that the quantity of the order of MA was fixed. However, in the real market, the quantity of MA is not fixed and it related to the demand of Consumer Agent (CA). Thus, this research extends to the multi-attribute negotiation between one MA and multi-MSA,
where the demand is based on the selling price of MA. In particular we focus on the situation of MA monopolies or oligopolies, as it is often the case in the trading business.

This paper is organized as follows: section 2 describes negotiation model and settings used in this research; section 3 describes the modified two-stage negotiation protocol; numerical example and analysis are given in section 4. In conclusion, the contributions and the directions of the future work are commented.

## 2 Model

All the used notations are shown as follows:

<table><tr><td>α<sup>Max</sup><sub>j</sub></td><td>maximum percentage of profit of MSA j'</td><td></td></tr><tr><td>β<sup>Max</sup></td><td>maximum percentage profit of MA</td><td></td></tr><tr><td>δ<sup>M</sup><sub>LT,w</sub></td><td>concession function of lead time of MA</td><td></td></tr><tr><td>δ<sup>M</sup><sub>Q,w</sub></td><td>concession function of quantity of MA</td><td></td></tr><tr><td>δ<sup>M</sup><sub>w,LT</sub></td><td>concession function of wholesale price of MA</td><td></td></tr><tr><td>δ<sup>S</sup><sub>w,Q</sub></td><td>concession function of wholesale price of SF<sub>j</sub></td><td></td></tr><tr><td>δ<sup>S</sup><sub>w,LT</sub></td><td>concession function of wholesale price of SF<sub>j</sub></td><td></td></tr><tr><td>γ<sub>j</sub></td><td>productivity of SF<sub>j</sub></td><td>π<sub>j</sub>[k] profit of MA at k</td></tr><tr><td>π<sup>S</sup><sub>j</sub>[k]</td><td>profit of SF<sub>j</sub> at k</td><td>π<sup>SM</sup><sub>j</sub>[k] profit of SF<sub>j</sub> takes strategy of MA at k</td></tr><tr><td>AC<sub>j</sub></td><td>combined ability of SF<sub>j</sub></td><td>PCA<sub>j</sub> maximum price of SF<sub>j</sub></td></tr><tr><td>cf</td><td>fixed cost per order of MA</td><td>PCI<sub>j</sub> minimum price of SF<sub>j</sub></td></tr><tr><td>cp<sub>j</sub></td><td>unit production cost of SF<sub>j</sub></td><td>PMA maximum price of MA</td></tr><tr><td>cs<sub>j</sub></td><td>set-up cost per order of SF<sub>j</sub></td><td>ps<sub>j</sub>[k] selling price of MA at k</td></tr><tr><td>cs<sub>j</sub></td><td>shortage cost of MA</td><td>Q<sup>S</sup><sub>j</sub>[k] quantity of SF<sub>j</sub> at k</td></tr><tr><td>D<sub>j</sub>[k]</td><td>demand of CA at k</td><td>Q<sup>M</sup><sub>j</sub>[k] quantity of MA at k</td></tr><tr><td>f<sub>Q</sub></td><td>function of quantity of SF<sub>j</sub></td><td>sv salvage value of unsold product of MA</td></tr><tr><td>h<sup>M</sup></td><td>holding cost of MA</td><td>TN deadline of the negotiation</td></tr><tr><td>h<sup>S</sup></td><td>holding cost of SF<sub>j</sub></td><td>TS time of each negotiation round</td></tr><tr><td>LT<sup>M</sup><sub>j</sub>[k]</td><td>lead time of MA at k</td><td>w<sup>M</sup><sub>j</sub>[k] wholesale price of MA at k</td></tr><tr><td>LT<sup>S</sup><sub>j</sub>[k]</td><td>lead time of SF<sub>j</sub> at k</td><td>w<sup>S</sup><sub>j</sub>[k] wholesale price of SF<sub>j</sub> at k</td></tr></table>

This research considers the multi-attribute negotiation between one MA and multiple MSAs, where MA should determine the quantity of his order based on the demand of CA. The negotiations between one MA and multiple MSAs, where the order requested by MA is too large for MSA to complete independently, have been discussed in the previous work [8]. It tried to find another way to resolve this problem which can maintain the integrity of the order. It assumed that the MSAs accepted only the orders which were able to fulfill independently, and tried to combine with the other MSAs as a coalition when the order was out of their abilities. Furthermore, it assumed that the quantity of the order of MA was fixed. However, in the real market, it must be related to the demand of CA. Thus, this research tries to extend the negotiation under fixed demand to the negotiation under the situation where the demand depends on the selling price of MA. Three main attributes considered in this research are the wholesale price of the product, the quantity of the order, and the lead time of the order. MA negotiates with the MSAs to reach agreements on the strategies of the three attributes. We assume that the demand of CA is in an additive form as (1),
where $a, b > 0$. It depends on the selling price of MA. $ps_j[k]$ is the selling price of MA of the negotiation between $SF_j$ at $k$.

$$D_j[k] = a - bps_j[k]. \qquad (1)$$

## 3 Negotiation Protocol

This research proposes a modified two-stage negotiation protocol based on the two-stage negotiation protocol proposed in [8]:

<figure><img src="image_2.png" /><figcaption>Fig. 1. Flowchart of modified two-stage negotiation protocol</figcaption></figure>

### Stage1: Negotiation among MSAs

- Step 1: MA forecasts the demand of CA, determines the initial price, quantity and lead time of the order which he wants to place, and then broadcasts the order to all the MSAs.

- Step 2: MSAs evaluate the order and check whether the order can be finished by themselves. If they can do it, themselves will be determined as the final coalition $SF_j$ and fed back to MA; if they cannot do it, then they can negotiate with the other MSAs to build a coalition $SF_j$ and feed back to MA. A cooperative game is used for coalition formation.

- Step 3: MA checks whether there exists any $SF_j$ can fulfill the order, if there exists, goes to the second stage, if there does not exist, then MA modifies the order and re-broadcasts the order.

### Stage2: Negotiation between MA and $SF_j$

- Step 4: MA starts to negotiate with $SF_j$. MA-Stackelberg game is introduced to find the final solution.

- Step 5: MA checks whether an agreement is reached, if the agreement is reached, negotiation ends, if the agreement is not reached, then MA checks whether he agrees with the strategies or not. If he agrees with it, he asks the MSA to modify his strategy, if he does not agree with it, then he adjusts his own strategy and gives a response.
– Step 6: MSA checks whether he agrees to modify his strategy or not. If he agrees to modify, he adjusts his own strategy, if he does not agree, then the negotiation ends and fails to find an agreement.

The flowchart of the modified two-stage negotiation is shown in Fig. 1. The processes in the left-hand side of the gray dash line are done by MA, and the processes in the right-hand side are done by MSA. The details of the negotiation protocol will be discussed in the following subsections.

## 3.1 Negotiation among MSAs

The negotiation among MSAs aims to find partners to establish a coalition when the order of MA is profitable but out of their abilities. It has been discussed in details in [8,9]. The final coalition SF<sub>j</sub> of MSA j will be determined after the negotiation.

## 3.2 Negotiation between MA and SF<sub>j</sub>

MA and SF<sub>j</sub> make their decisions sequentially in this part. They negotiate on the wholesale price, the quantity of the order, and the lead time. SF<sub>j</sub> wants to increase the wholesale price, quantity and lead time to improve his profit. However, MA wants to decrease the wholesale price to increase his profit. Therefore, the main point of this part is to find a balance between the profits of SF<sub>j</sub> and MA.

The negotiation between MA and SF<sub>j</sub> can be modeled as a Stackelberg game, where MA is indicated as a leader and SF<sub>j</sub> can be seen as a follower. In the formulation of a Stackelberg game, it is usually assumed that players' profit functions are common knowledge [10]. Thus, MA can use this common knowledge to construct SF<sub>j</sub>'s most profitable responses to all the possible decisions. Then, MA's profit function from corresponding decisions based on all the SF<sub>j</sub>'s responses can be explored. In the end, MA makes a decision which can maximize his profit. SF<sub>j</sub> then has a best response to matching the decision. We assume that both MA and SF<sub>j</sub> are rational, thus, they do not want to deviate from this set of decisions, which is called Stackelberg equilibrium in the game theory. The strategies of MA and SF<sub>j</sub> are determined in the following sections.

**Determination of the strategies of MA** From the analysis above, we can see that MA offers his strategies from low to high. The strategies of MA for SF<sub>j</sub> at round k are ($Q_j^M[k]$, $w_j^M[k]$, $LT_j^M[k]$), where:

$$w_j^M[k] = w_j^M[k-1] + \frac{PMA - w_j^M[k-1]}{(TN - kTS)/TS} + \delta_{w,LT}^M(LT_j^S[k-1]) \quad (2)$$

$$\delta_{w,LT}^M(LT_j^S[k-1]) = \begin{cases} \theta_z^{wLT}, & \text{if } LT_{ij}^S[k-1] \le x_1^{LT} \\ \theta_{z-1}^{wLT}, & \text{if } x_{z-1}^{LT} < LT_{ij}^S[k-1] < x_z^{LT} \\ 0, & \text{if } LT_{ij}^S[k-1] \ge x_z^{LT} \end{cases} \quad (3)$$

$$Q_j^M[k] = D_j[k] + \delta_{Q,w}^M(w_j^S[k-1]) \quad (4)$$

$$LT_j^M[k] = LT_j^M[k-1] + \delta_{LT,w}^M(w_j^S[k-1]) \quad (5)$$

$$ps_j[k] = w_j^M[k](1 + \beta^{Max}). \quad (6)$$
At each round of negotiation, MA should determine strategies of all the three attributes. MA determines the value of the first attribute according to his preferential choice and we assume that MA in this research firstly determines his price. MA not only considers the concession which related to the remnant negotiation time at round k (the second item of (2)), but also takes the strategies of SF<sub>j</sub> at k − 1 into account (the third item of (2)). He can increase certain amount of price if SF<sub>j</sub> can offer the product in shorter time. The quantity Q<sub>j</sub><sup>M</sup>[k] of MA depends on the demand of CA(the first item of (4)) and the increment according to the price of SF<sub>j</sub> (the second item of (4)). If SF<sub>j</sub> can offer lower price, MA will increase the total amount to buy. The lead time LT<sub>j</sub><sup>M</sup>[k] of MA related to the price of SF<sub>j</sub> (the second item of (5)). If SF<sub>j</sub> can offer lower price, MA will extend the lead time of his order. δ<sub>w,LT</sub><sup>M</sup> is a piece-wise function as (3), it means when LT<sub>j</sub><sup>S</sup>[k − 1] belongs to the range (x<sub>z − 1</sub><sup>LT</sup>, x<sub>z</sub><sup>LT</sup>), MA will give a concession of the price at the value of θ<sub>z − 1</sub><sup>wLT</sup>. All the threshold values x<sub>z − 1</sub><sup>LT</sup> of each segment and the related mapping values θ<sub>z − 1</sub><sup>wLT</sup> are determined by himself. δ<sub>Q,w</sub><sup>M</sup> and δ<sub>LT,w</sub><sup>M</sup> have the same form as (3).

**Determination of the strategies of SF<sub>j</sub>** Each SF<sub>j</sub> tries to determine his strategies to maximize his profit. The higher value of the three attributes, the better. Thus, he replies his strategies from high to low. However, it is better to reach an agreement than does not reach an agreement. Thus, SF<sub>j</sub> should also take the strategies of MA into account and then decide his strategies. The strategies of SF<sub>j</sub> are defined as follows:

$$
w_j^S[k] = w_j^S[k-1] - \frac{w_j^S[k-1] - PCI_j}{(TN - kTS)/TS} - \delta_{w,Q}^S(Q_j^M[k]) - \delta_{w,LT}^S(LT_j^M[k])(7)
$$

$$
Q_j^S[k] = f_Q(w_j^S[k]) \tag{8}
$$

$$
LT_j^S[k] = Q_j^S[k] / \gamma_j \qquad (9)
$$

$$
f_Q(w_j^S[k]) = \begin{cases} \theta_Q^{\text{max}}, & \text{if } w_j^S[k] \le \text{PCI}_j \\ \theta_Q[k], & \text{if } \text{PCI}_j < w_j^S[k] < \text{PCA}_j \\ \theta_Q^{\text{min}}, & \text{if } w_j^S[k] \ge \text{PCA}_j \end{cases} \tag{10}
$$

$$
\theta_Q[k] = AC_j - \frac{\left[ \frac{cs_j}{w_j^S[0]-cp_i} - AC_j \right] PCI_j}{PCA_j - PCI_j} + w_j^S[k] \frac{\left[ \frac{cs_j}{w_j^S[0]-cp_j} - AC_j \right]}{PCA_j - PCI_j}. \quad (11)
$$

SF<sub>j</sub> firstly decides his price according to the strategies of MA. (7) means SF<sub>j</sub> will give a discount if MA can increase the quantity of his order (the third item of (7)) or extend the lead time (the last item of (7)). The quantity (8) and lead time(9) of SF<sub>j</sub> strongly depends on his productivity. Thus, all the three attributes constraint each other. δ<sup>S</sup><sub>w,Q</sub>(Q<sup>M</sup><sub>j</sub>[k]) and δ<sup>S</sup><sub>w,LT</sub>(LT<sup>M</sup><sub>j</sub>[k]) are also piecewise functions as (3) but has opposite tendency (threshold and related mapping values). θ<sub>Q</sub>[k] is used to determine the minimum quantity to ensure the order will be profitable (for each order, minimum setup cost is needed).

**Determination of the final equilibrium** MA has his own preferences for
wholesale price, quantity and lead time and he is looking for the offer that best
satisfies these preferences. Both MA and SF<sub>j</sub> want to maximize their profits by choosing their preferred strategies. The profits of MA at k, where he agrees with the strategy of SF<sub>j</sub>, is defined as:

$$
\begin{equation}
\begin{split}
\pi_j^M [k] ={}& ps_j[k]D_j[k] + \operatorname{sgn}(Q_j^M[k] - D_j[k])(Q_j^M[k] - D_j[k])sv - \operatorname{sgn}(D_j[k] \\
& - Q_j^M[k])(D_j[k] - Q_j^M[k])cst - w_j^M[k]Q_j^M[k] - \frac{cfD_j[k]}{Q_j^M[k]} - \frac{hQ_j^M[k]}{2}
\end{split}
\tag{12}
\end{equation}
$$

where sgn(x) equals to 1, if x > 0, otherwise it equals to 0.

We can see that in this model MA is the leader and he has more decision power. Thus, the objective of the Stackelberg game is to find the equilibrium which can maximize the profit of MA. It can be transformed into finding the agreements on the strategies of the three attributes and so to maximize the profit of MA. However, the strategies must be accepted by SF<sub>j</sub>. Thus, the equilibrium of the negotiation between MA and SF<sub>j</sub> can be determined as the strategies (Q<sub>j</sub><sup>M</sup>[k], w<sub>j</sub><sup>M</sup>[k], LT<sub>j</sub><sup>M</sup>[k]) of MA at k. Moreover, these final strategies must meet the following conditions:

$$
\max \{ \pi_j^M[k] \} \tag{13}
$$

$$
\textbf{s.t.} \quad \pi_j^{SM}[k] \ge \pi_j^S[k], \text{ if } k < TN-1 \quad (14)
$$

$$
\pi_j^{SM}[k] > 0, \text{ if } k=TN-1 \qquad (15)
$$

$$
Q_j^M[k] \leq AC_j[k] \tag{16}
$$

where (14)-(15) are used for SF<sub>j</sub> to evaluate the acceptability of the strategies,
and these mean that the order must be profitable for SF<sub>j</sub>. (16) means the order
must be in ability of SF<sub>j</sub>. π<sub>j</sub><sup>SM</sup>[k] is the profit of SF<sub>j</sub> adopts the strategies of MA
at k (as (2)-(5)), π<sub>j</sub><sup>S</sup>[k] is the profit of SF<sub>j</sub> adopts his own strategies at k (as (7)-
(9)). The agreement can be reached only if the profit of taking the strategies of
MA is greater than the one of taking his own strategies. Finally, MA decides the
final supplier which can maximize his profit based on the equilibrium acquired
from (13)-(16).

The characteristics of the proposed protocol are:

- The agreement can be reached as long as (14)-(16) are satisfied, no matter MA cannot reach an agreement with SF<sub>j</sub> on the wholesale price or not.

- The agreement may not be reached even MA has reached an agreement with SF<sub>j</sub> on the wholesale price.

- The attributes may not be monotone changing.

4 Numerical case and analysis

A numerical case is provided to illustrate the processes of the multi-attribute
negotiation and how does the negotiation find the final equilibrium. Assume
there is one MA and 5 MSA in SCN and all the used parameters are defined as
Table 1.

Firstly, we discuss about the details of the multi-attribute negotiation be-
tween one MA and one SF<sub>j</sub>. We take the negotiation between MA and coalition
<table><caption>**Table 1.** Parameter settings and threshold values of the concession functions</caption><thead><tr><th colspan="2">MSA (J=5)</th><th colspan="2">MA</th></tr></thead><tbody><tr><td>γ<sub>j</sub> ~U(100,300)</td><td>cp<sub>j</sub> ~U(7,8)</td><td>α<sup>max</sup> = 0.5 h<sup>M</sup> = 3</td><td>cst<sup>M</sup> = 5</td></tr><tr><td>β<sup>min</sup><sub>j</sub> = 0.2</td><td>β<sup>max</sup><sub>j</sub> = 0.5</td><td>α<sup>min</sup> = 0.3 psIn<sup>M</sup><sub>i</sub> ~U(13,14)</td><td>cf<sup>M</sup> = 100</td></tr><tr><td>h<sup>S</sup><sub>j</sub> = 3</td><td>cs<sub>j</sub> ~U(200,300)</td><td>sv<sup>M</sup> = 2</td><td>a ~U(1000,2000)</td></tr><tr><td></td><td></td><td></td><td>b ~U(0,100)</td></tr></tbody></table>

{12} as an example. The fluctuations of three attributes are shown as Fig. 2. We can see that:

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** The fluctuation of the multi-attribute negotiation</figcaption></figure>

- At $k = 3$, the profit of {12} takes the strategies of MA is greater than takes his own strategies (see the area marked by ellipse in Fig. 2(d), where $\pi^{SM}[3] > \pi^{S}[3]$), that means (14) is satisfied. However, the quantity of MA is also greater than the ability of {12} (see the quantity at $k=3$ of Fig. 2(b), where $Q^{M}[3] > AC[3]$), that means (16) is not satisfied. Therefore, the agreement is not reached and the negotiation goes by.

- At $k = 18$, MA reaches an agreement with {12} on the wholesale price (see the area marked by ellipse in Fig. 2(a)) and (14) is satisfied. However, (16) is still not satisfied because $Q^{M}[18] > AC[18]$ from Fig. 2(b). Therefore, the agreement is not reached and the negotiation goes by.

- At $k = 20$, the wholesale price of MA keeps unchanging, but he makes a concession of his lead time (see the area marked by ellipse in Fig. 2(c)) and then (16) is satisfied (see the area marked by ellipse in Fig. 2(b), where $Q^{M}[20] < AC[20]$). Therefore, both constraints (14) and (16) are satisfied.

Then we can get final equilibrium between MA and {12} is the strategies of MA at $k = 20$ where the strategies are (10.704, 4489, 16). What we should pay attention to are: 1) The equilibrium not always exists; 2) The order MA may become out of ability of SF<sub>j</sub> even it was in ability at the first time.
Similarly, we can get all the equilibriums and then MA decides the final supplier which can maximize his profit. In this case, the final supplier for MA is {32} with the final strategies (10.703, 6735, 16) and the profit equals to 25501.295.

## 5 Conclusion

The multi-attribute negotiation between MA and MSA, where the quantity of the order of MA depends on the demand of CA, was discussed in this paper. A modified two-stage negotiation protocol was proposed. MA-Stackelberg game was introduced to decide the final strategies by finding the Stackelberg equilibrium. A different criterion was proposed for MSA to decide whether accept the order or not. The strategies of MA at *k* was defined as final equilibrium if the constraints are satisfied. In this research, we only provided a method to find the equilibrium of multi-attribute negotiation. However, it cannot ensure the equilibrium always exists. There still exists the situation where the negotiation cannot find the equilibrium. For future work, we will take the dynamic coalition formation into account to improve the success rate of finding the equilibrium. Moreover, the performance of the proposed protocol will be discussed, including what effects will be caused by changing the parameter, what's the effect of the ability of MSA on the formation of the coalition.

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