Multi-dimensional Analysis of Utilities in Pricing Game

Panehal Gajanani and Jain vipuf

Abstract-The utility for players or agents is the quantification of their happiness or satisfaction. It can be the

function of single parameter or many. In game theory literature, utility is the function of strategy they wish to play. In many cases, it is assumed that the utility for a player is the

function of single strategy. In this paper, we have considered

the multiple strategies of players to define their utilities. We have developed the model to analyze these utilities of players involved in pricing games (Supplier-buyer game). We have taken a numerical example to validate the model. Also we have

extended our analysis to the combination of pricing and production game (Supplier-supplier game) which serves as an input for supplier selection process.

I. INTRODUCTION

IN microeconomics theory, utility analysis for different feasible options plays an important role in decision

making. The utility for players is their happiness or satisfaction by choosing a best alternative amongst all other feasible alternatives [1]. The happiness can be expressed in terms of utilities which quantify players' happiness. In game theory settings, utilities are the payoff for each player by choosing a strategy out of set of feasible strategies [2] [3] [4]. The utility for a player is not only depend on its own

strategy but also the others strategies. For example,

Ui(Si,S_i)is the utility for ith player when its strategy is Si

[ (si' S -i) is the way to represent strategy profile of all i

players in game theory]. There are difference types of games those apply in the supply chain context. Mainly these are categories in to two types; production games and pricing

games. In production games, as shown in Fig. 1, the competition is between various suppliers for sharing the total

demand from the customers. Because of this competition, these games are also known as the horizontal competition games. In this game, the strategic behavior is simultaneous

which allows all the players in the game to act at the same instance of time. Here the utility is the function of demand for the product. One the other hand, in the pricing games

competition is for setting the price depending on the reveled

demand. As these two are related to each other, it is required to know what is the true demand and corresponding price.

Manuscript received March 25, 2011. (Write the date on which you submitted your paper for review.)

I Gajanan Panchal is Research scholar in Mechanical Engineering Department of Indian Institute of Technology Delhi-110016, INDIA (phone: +91-11-2659 1145; fax: +91-11-2658 2053; e-mail: gajananbpanchal@gmail.com).

2 Dr. Jain Vipul is Assistant Professor in Mechanical Engineering Department of Indian Institute of Technology Delhi-110016, INDIA (Phone: +91-11-1659 1145; Fax: +91-11-2658 2053; e-mail: vjain@mech.iitd.ac.in).

978-1-4577-0574-8/11/$26.00 ©2011 IEEE 359

The pricing games are the type of games which depict the most of the phenomenon of supply chain [5]. Fig. 2 shows the simple form of pricing game where there are two players; a supplier and a buyer. This falls in to the category

of sequential games. Here one of the players acts first and then the other. These games are also called as principal­agent games or the leader-follower games. Here the competition is between the two tiers of the supply chain. This competition is vertical competition and so the name, vertical competition games.

Supplier 1, rJ2

Fig I. Production game, depicting the competition among the set of

supplier over the share of the total demand q sE Q

Supplier

\V

Buyer

Fig 2. Pricing game, first supplier will make an announcement (a), and then buyer will put hislher willingness (w).

These players are having the utility corresponding to the strategy they select out of the feasible set of strategies. For example, suppliers who select the number of components to

supply will have the utility in terms of the revenue they get from selling those components.

Even though in supply chain network all players work together, it is found many times that players are rational in

nature and are intend to optimize their own objective/s [6] [7]. Because of this rationality, a game is induced in these

players. One is interested to know the equilibrium of this game. Specially, when there is more than one strategy in deciding utility for a player [8]. In pricing game phenomenon, finding out solution to the given problem is the equilibrium. Here supplier is the leader and buyer is the follower. The supplierls makes announcements by stating of the form of utility function. The buyer can do the best by

responding to the supplier announcement (as)' This

response is the best response function of the buyer b, BRb(as)'

II. LITERATURE REVIEW

The concept of multidimensional utility got some attention because of its applications economic analysis [9]

[10]. In [10], author has explained the importance of considering the multi-dimensions in the utility. The author has also mentioned that multi-dimensional utility is the lexicographical ordering of the alternatives. An alternative is a vector of the different parameters. As the utility functions are monotonic in nature, one has large utility for at least one of the available alternatives.

The utility, in the supply chain context, is the revenue from the sales or the provided service, WLOG. Some of the researchers have considered the utility based on the single

dimensions [9] [10] [11]. In [11], the authors have mentioned utilities in terms of the service level. They have

defined service level in terms of two parameters; mean lead

time (MLT) and the standard deviation of lead time (SOL T).

In some of the literature, utility refers to the broader qualitative term, Quality of services (QoS) which aims to capture the satisfaction of the user, participant or agent of the supply chain in general [12] [13].

There are several applications of the utility based analysis of the problem in the wireless networks. In particular, utility in the wireless network is the ability to transfer the data from one or many primary users to one or more secondary users [14] [15] [16]. In [16], authors have considered the utility as a function of two dimensions; flow rate of the data and the reliability of the data. Here the authors have analyzed the utilities in terms of the weighted

average of the different parameters in the consideration. These weights are being calculated based on the questionnaire or the expert opinion.

In health care applications [17], authors have defined the

clinical utility as the usefulness of an intervention in the

clinical practice which also means the ethical doctrine of achieving the greatest good of the greatest number. Here the authors have developed the multi-dimensional utility as the judgmental analysis of four factors in the defining clinical utility; appropriateness, accessibility, practicability and

acceptability. After scanning through plethora of the literature, we have

found out that there is need for analysis of the multi­

dimensional utility as most of the decisions are based on

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more than one parameter. Also our focus is on the pricing games and the associated multi-dimensional utility should have properties like differentiability. This is because of the

utility for a buyer or a follower is the best response function to the utility function of the supplier or the leader. These best response functions are find out with the first order differential conditions of optimality.

III. LEARNING FROM THE LITERATURE

As pointed earlier, there is need for the utility analysis which considers more than one parameter for decision making. The specialty of utility based analysis is in its possibility of coming up with the social outcome. In literature, some of the research work is based on the considering the utility as weighted average of multiple parameters. Likewise, we have found some of the important observations in the literature as follows;

Important observations

• It is required to consider more than one parameter which can be related by bijection function (Bijection functions are the one-to-one and one-on-one functions) Doing so one can take social decisions.

• There is need to model the pricing game which applies in to most of the supply chain cases. For example, the game induced in between the supplier and buyer is pricing game

• The analysis based on the weighted average of the various parameters will end up in fmding subjecting estimates of the weights which will not lead to social outcome.

• The decision is being made sequential way, so the follower needs to have some idea of the former or leader in order to response effectively. We need to consider two different weighing systems for supplier and buyer separately.

This way it is very easy to extend the utility analysis for designing the mechanism for specific strategy. This process is known as the mechanism design. For example, if we want truth from each player as an equilibrium strategy for every player of the supply chain. This approach is the reverse of the game theory approach. In some literature it is referred as the reverse engineering of game theory [2] [3].

Also we have learned from the literature that we can take an advantage of this analysis for the supplier selection process. The outcome we get is the equilibrium tuple consisting of the parameter in consideration. For example,

(q*,p*) is the equilibrium tuple with quantity and

corresponding price are the two parameters. This will guide us to select the supplier in case of competition among the set of suppliers. This situation will consist of pricing and production competition games depicting the horizontal and vertical competition.

IV. MODEL FOR MULTI-DIMENSIONAL UTILITIES

Traditionally the utilities, in general, are considered to be the single dimensional. The utilities with only one strategy is somehow not depicting the real decision making practice.

Let, x E X be a one particular strategy and Y E Y is another

pair of strategy to decide the utility for player i EN. We are

interested in getting the pair x, y and utility function

maps this pair to a real number; u: X x Y � lR. Here the

cross product is bijection of the two vectors. Let the pair be

denoted by ai = xi' Yi and utility for player i will

be ui ai . For any player in a game, generalized

optimization problem formulation is as follow,

Subject to,

max Z = � u( a * i) for 'if i I

In supply chain, there are more parameters or dimensions

to decide up on the utility for a player. We have proposed the model for the utility with more than one parameter; we call it as multi-dimensional utilities function. In particular, pricing games consist of two players, namely supplier and buyer). For any two types of strategies, utility function for

supplier and buyer can be defmed as Us qs'ps

and ub qb' Pb ' respectively. Both the players in the

supply chain are rational so tend to maximize their utilities. This decentralized optimization problem leads to the unstable/unsocial outcome. As a supply chain manager, our job is to solve optimization problem for the social outcome. Without loss of generality, one can combine the utilities of

supplier and buyer to form the total utility; Ut qt, Pt us(qs'Ps) = (w-c)*q * P

Where, w>O is wholesale price from the supplier, c>O is the

cost to the supplier, m>O is the margin for the buyer and q=q, >0 is the total demand for the product. The total utility will now become,

ut(qt,Pt) = (w+m -c)*q * P

Let the above total utility be the objective function and with constrains on the quantity and the price we can formulate the optimization problem as follows,

'_We are considering simple pricing game with one supplier and one buyer

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Supplier

Buyer

Fig 3. Multi-dimensional utility analysis; first supplier will make an

announcement a � (q s ' P s)' and then buyer will put his/her

willingness w � (qb,Pb)

maxLU' =L(w.+m.-c)*q. for '<:liEN iI i I I I

Subject to,

max qb "?Cs

Where, Cs max

is the maximum capacity of the player.

Here, the variables are P and q whose values are, as pointed out above, subject to the unstable solution. To

achieve the stable solution we can use the KKT conditions

for above problem.

max L(q,p,A,J1.) = LU. + LA (p -c)+ LJ1.b (C b -qb) i I S S S b

The interpretation of the Lagrange multipliers As and J1.b is

the discount in price and opportunity cost for the buyers.

V. NUMERICAL EXAMPLE

To demonstrate how the model will help us in decision making, we take simple example. Let, consider one supplier and one buyer case. Table 1 and Table 2 represent the data of quantity, price and corresponding utility for supplier and buyer respectively.

Utility for Quanity

supplied (qs) Selling Price

(ps) supplier

(Us)

10 30 250

20 25 400

30 20 450

40 15 400

50 10 250

Table l. Supplier data; quantity supplied, price and corresponding utility

Here, we are assuming fixed cost for supplier as 5 cost units. The utility for supplier is the revenue they get from selling units Qs. Here we are assuming range of quantity qs as one set that map in bijection function to corresponding price Ps. Clearly it can be seen that for supplier

pair qs'Ps == 30,20 will give maximum utility. And

because of rationality he will try to entice buyer for demand of 30 units for 20 cost units.

Quantity Willingness to Utility for ordered (Qb) pay for

-Qb (Pb) buyer (Ub)

5 25 100

10 22 170

25 18 325

40 12 280

60 10 300

Table 2. Buyer data; demand, willingness to pay and corresponding utility

Similar to supplier data, Table 2 shows the buyer data corresponding to the utility calculation with ordered quantity and willingness to pay for that. Here we assume fixed margin of 5 cost units for buyer which leads to profit for him/her. The utility in the table shows how much buyer has to pay to the supplier. So buyer wishes to buy only 5 units for 25 cost units. Assumption is that this decision is made considering the presented data. We can see the conflicting results of the utility of supplier and buyer if calculated separately. This misleads the concept of social outcome with the rational outcome. For this reason, the utility need to be calculated as the social objective of revenue maximization and not individual profit maximization. Table 3 shows the utility calculation with utility defined as suggested by our model. The calculation in

the Table 3 shows that the pair qs' Ps == 30,20 will

leads to revenue maximization with the social outcome. We can call this as the equilibrium pair or tuple.

Ordered Selling Willingness to Buyer Total quantity q price p payw margin m utility

5 30 25 5 150

10 30 22 8 330

15 25 18 7 405

20 25 18 7 540

25 20 18 2 425

30 20 12 8 690

35 15 12 3 455

40 15 12 3 520

45 10 10 0 225

50 10 10 0 250

55 10 10 0 275

60 10 10 0 300

Table 3. Utility calculation by the proposed model; considering various quantities, selling price of supplier, willingness to pay by buyer and variable margin of buyer

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The suggested equilibrium tuple and the solution from the supplier's point of view are same. This does not mean that the proposed model is dictatorial. The dictatorial function, in general, is the function which gives us the solution which coincides with at least one of the player of the game. Here in case undertaken, the dictatorial solution will coincides with either supplier's or buyer's solution. It is mere coincidental to fmd this situation.

We can extend this analysis to the supplier selection process when there are multiple suppliers competing for the total demand. Fig. 1 in the introduction shows one such scenario with n suppliers and one buyer. The decision maker will select the supplier which is close to the solution pair.

Here we are not calculating the Lagrange multiplier for the example but one can see the significance of the Lagrange multiplier. If supplier want to increase the capacity to satisfy the demand they have the opportunity cost. For example, in Table 2 we can see that the buyer is willing to order 60 units for 10 cost units which exceed supplier's capacity. Also for the buyer, price discount could be worked out if they order more than the 30 units.

VI. CONCLUSION AND FUTURE SCOPE

The decision maker has to decide best alternatives out of the bunch of feasible alternatives. The alternative with maximum utility is the best for any player involved in the process. In today's competitive world there is a demand for the multi-dimensional, lateral thinking in any kind of decision making. The multi-dimensional utility function will help in understanding the level of satisfaction, happiness etc. with respect to more than one parameter.

The proposed model will help decision maker in supply chain to decide upon the two parameters; quantity and price. This model is for the pricing game which is the sequential game between the supplier and buyer who are leader and follower respectively. The bijection function for defming the revenue for the supplier will take consideration of the two parameters involved in its utility. The Lagrange multipliers in the models give us an idea that the sensitivity analysis will leads to some interesting changes in the solution.

The theoretical contribution of this paper is to develop the model for the combination of the two types of supply chain games: production and pricing games. This paper proposed the model for the supplier selection considering the rationality of each player in the game. The conventional method of supplier selection process is to select the supplier considering the centralized approach giving rise to counterfeiting risk. The decentralized approach of supplier selection will make sure that the social objective considering the rationality and selfishness of each participating player/supplier.

The numerical example considered in this paper explains the supplier selection process. This is the base for the much complicated process of supplier selection when the supply chain manager looks for more than just one or two parameters in the social and individual utility functions. For example, utility could be a function of quantitative

parameters like loss of revenue, risk of backing out from the supplier etc. and also some qualitative parameters like quality of the product/service provided. Developing supplier selection model considering the more than one parameters and with rational behavior of each participants will help the supply chain managers/practitioners in finding out the conflict free solution.

In future, one can add more parameters like quality, which is the fuzzy in nature, and develop the model. Also one can consider some of the important factors that affect the utility of the supplier or revenue of the supplier. We have limited ourselves to just develop the multi-dimensional utilities. One can compare the two approaches; the performance based analysis of the utilities and the analysis based on proposed model and come up with the guidelines about the applicability of these two.

REFERENCES

[1] J. R Green and J. Laffont, Incentives in Public DeciSion-Making. New York: North-Holland Publishing Company, 1980, ch. 2. Fundenberg, J. Tirole, Game Theory. USA: MIT Press, 2005, ch. 1.

[2] J. Laffont, D. Martimort, The Theory of Incentives- The Principal­Agent Model. USA: Princeton University Press, 2006, ch. 2.

[3] A. Mas-Colell, M. D. Whinston and J. R Green, Microeconomic Theory. New York: Oxford University Press, 2006, ch. 22.

[4] K. Kogen and C. S. Tapiero, Supply Chain Games: Operations Management and Risk Valuation. New York: Springer, 2007, ch. 2.

[5] M. P. Wellman, J. Estelle, S. Singh, Y. Vorobeychik, C. Kiekintveld and V. Soni, "Strategic Interactions in a Supply Chain Game", Computational Intelligence, vol 21, pp. 1-26, 2005.

[6] M. P. Wellman, W. E. Walsh, P. RWurman and J. K. Mackie-Mason, "Auction Protocol for Decentralized Scheduling", Game and Economic Behavior, vol. 35, pp. 271-303, 2001.

[7] G. Panchal and V. Jain, "A Review on Select Models for Supply Chain Formation", International Journal of Business Peiformance and Supply Chain Management, to be published.

[8] G. Panchal and V. Jain, "Models for supply chain formation", International Coriference on Latest Trends in Simulation Modeling andAnalysis, COSMA '09, NIT Calicut, Kerala, India, pp.71-80, December 2009.

[9] C. E. Ferguson, "The Theory of Multidimensional Utility Analysis in Relation to Multiple-Goal Business Behavior: A Synthesis", Southern EconomicJournal, vol. 32, pp. 169-175, 1965.

[10] Y. Narahari and N. K. Srivastava, "A Bayesian incentive compatible mechanism for decentralized supply chain formation", 9th IEEE International Conference on E-Commerce Technology, National Center of Sciences, Tokyo, Japan, pp.315-322, July 2007.

[11] A. S. Kalelkar and R E. Brook, "Use of Multidimensional Utility Functions in Hazardous Shipment Decisions", Accident Analysis Prevention, vol. 10, pp. 251-265, 1978.

[12] N. Paspallis, K. Kakousis and G. A. Papadopoulos, "A Multi­dimensional Model Enabling Automatic Reasoning for Contect-aware Pervasive Applications", In the Proceedings of the Workshop for Human Control of Ubiquitous Systems (HUCUBIS' 08) in corljunction with the 5'" Annual International Conference on Mobile and Ubiquitous System: Computing, Networking and Services (Mobiquitous; 08), Trinity College Dublin, Ireland, ACM Press, July 2008.

[13] K. J. Ray Liu and B. Wang, Cognitive Radio Networking and Security-A Game-Theoretic view. UK, Cambridge University Press, 2011, ch. 2.

[14] D. Niyato and E. Hossain, "A Game-Theoretic Approach to Competitive Spectrum Sharing in Cognitive Radio Networks", In the proceedings of WCNC 2007, Canada, pp. 16-20, 2007.

[15] J. Lee, M. Chiang and A. R Calderbank, " Price-Based Distributed Algorithms for Rate-Reliability Tradeoff in Network Utility Maximization", IEEE Journal on Selected Areas in Communications, vol. 24, pp. 962-975, 2006.

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[16] A. Smart, "A Multi-dimensional Model of Clinical Utility", International Journal for Quality in Health Care, vol. 18 number 5, pp377-382, 2006.