Impact of bargaining power on supply chain profit allocation: a game-theoretic study

Sanjay Prasad (IBM India Private Limited, Bengaluru, India)
Ravi Shankar (Department of Management Studies, Indian Institute of Technology Delhi, New Delhi, India)
Sreejit Roy (IBM India Private Limited, Bengaluru, India)

Journal of Advances in Management Research

ISSN: 0972-7981

Publication date: 15 July 2019

Abstract

Purpose

The purpose of this paper is to study the impact of bargaining powers of firms in supply chain coordination. It studies selected aspects of bargaining powers, namely, impatience, breakdown probability and outside options, and uses a bargaining-theoretic approach to analyze surplus allocation in a coordinated supply chain.

Design/methodology/approach

This paper proposes one-supplier one-buyer infinite horizon supply chain coordination game, where suppliers and buyers negotiate for the allocation of supply chain surplus arising out of supply chain coordination. Various aspects of the bargaining power of the negotiating parties are modeled and the paper studies impact of power levels on the results of the bargaining game.

Findings

A significance of impatience on the bargaining process and the surplus split has been established. This paper also demonstrates a rather counter-intuitive aspect of bargaining that the impatience (as perceived by the other party) can improve the bargaining position and therefore share of profits.

Research limitations/implications

This paper has limited its analysis to three key components of bargaining power. Future works can study other aspects of bargaining power, namely information asymmetry, learning curve, inside options, etc. Further, the paper has considered an infinite horizon model – this assumption can be relaxed in future research.

Practical implications

Equations to derive optimal split of the surplus have been derived and can be leveraged to design an autonomous bargaining agent to discover equilibrium profit splits in a cloud or e-commerce setting. Further, insights from this paper can be leveraged by managers to understand their relative bargaining power and drive to obtain the best profit split.

Originality/value

This paper establishes that impatience (in terms of counter-offer probability) has a significant impact on the bargaining position and on the split of the surplus that the firm can get for themselves. It establishes the advantage of higher levels of impatience, provided the other party recognizes the impatience and factors it in their decision-making process.

Keywords

Citation

Prasad, S., Shankar, R. and Roy, S. (2019), "Impact of bargaining power on supply chain profit allocation: a game-theoretic study", Journal of Advances in Management Research, Vol. 16 No. 3, pp. 398-416. https://doi.org/10.1108/JAMR-10-2018-0096

Publisher

:

Emerald Publishing Limited

Copyright © 2019, Emerald Publishing Limited


1. Introduction

A supply chain consists of the number of organizations or enterprises linked to each other. Each element in the chain performs the function of increasing the utility of the commodity before passing it on to the next. Some elements/enterprises in the supply chain may not be as powerful as the others. Increasing globalization and specialization over the last two decades has led to a further increase in strategic sourcing, alliances and inter-firm collaborations.

Kreutter et al. (2012) discuss consolidation in the IT services outsourcing industry and predicts that the industry is moving toward value creation networks (VCNs), flexible networks organized toward a business problem. The success of these VCNs (IT services supply chains) will partly depend on the firm’s ability to effectively integrate technology and services from multiple vendors and develop sustainable partnerships and a strong ecosystem.

Most common models of supply chain power structures commonly discussed in the literature are Stackelberg and vertical integration (VI). In a Stackelberg supply chain, organizations operate in a non-co-operative environment and Stackelberg leader gives a take-it-or-leave-it option to other players. In a VI supply chain, decisions are taken as if the whole supply chain is one single organization. However, as Baron et al. (2016) point out, in many supply chains, the final decision is the outcome of a bargaining process between parties involved. Respective bargaining powers of firms decide the outcome and respective splits of the supply chain profit. Factors, which influence supplier’s bargaining power, are the importance of the raw material, substitute availability, alternate suppliers, supplier concentration and volumes ordered, whereas factors determining customer’s bargaining power are the number of buyers, integration of the customer base, purchasing power and availability of substitution. Nair et al. (2011) study the influence of bargaining power and inter-firm relationship on investments in strategic assets. Ramsay (2004) makes a case for the study of negotiation in the supply chain management field and concludes that the journey from uninformed adversarial buyer–supplier attitudes toward enlightened co-operative relations in supply chains may be not as real as it is assumed in popular supply chain literature.

While supply chain coordination problem has been well examined by the researchers, there is an insufficient amount of work on how supply chain surplus needs to be distributed in consideration of differing levels of bargaining power of supply chain partners. This paper studies the distribution of supply chain surplus, arising due to coordination, among supply chain partners. It takes a bargaining-theoretic approach to profit allocation problem, and studies the impact of individual bargaining powers of negotiation parties. It incorporates selected aspects of bargaining powers, namely impatience, breakdown probability and external options, and develops a model to derive equations for optimal profit allocation.

Next section presents the literature review and establishes the research problem. Section 3 introduces the bargaining model and presents salient modeling aspects. Section 4 formulates the base model and analyzes it. Section 5 extends the bargaining model and incorporates the additional aspect of power, namely outside options. Section 6 further extends and generalizes the model developed in Sections 4 and 5. Section 7 concludes and presents managerial implications.

2. Literature review

Marek and Konecka (2009) point out that “Customer is King” theory is not necessarily true, and customers must keep in mind that excessive bargaining may lead to the shrinking of supplier base, thereby robbing the customer of his powers. Further, Maloni and Benton (2000) advise that establishing a good, strong relationship between the supplier and customer where both are engaged in mutual co-operation will benefit the supply chain and the two parties as well. On the other hand, Cox (1999) points out that business is not about a passing value to customers, but it is about appropriating value for oneself as much as possible and sustainable. Therefore, while a company like Toyota, operating in a competitive environment, would consistently work on delighting customers, some other firms like Microsoft, Intel and Cisco in IT industry were in a position in late 1990s and first half of 2000, where it could get away with just satisfying their customers (Cox, 1999). Further, Kim et al. (1999) point out that the perception gap between supply chain partners impacts the partnership needed for a successful buyer–supplier relationship. They also study a Korean semiconductor manufacturer and its suppliers to establish that these perceptions are driven by the partner’s production capability and the product requirement.

2.1 Bargaining power

Power of a firm in the supply chain dictates how much value it must share with other supply chain entities. Further, Kim and Heungshik (2005) also state that each supply chain partner’s resource commitment to activities such as quality improvement and new product development may vary based on the balance of bargaining power.

A concept of bargaining power has its roots in labor economics, more specifically the analysis of the relationship between employer and employee (Autushka-Sikorski, 2014). However, the concept itself has not yet been clearly and uniquely defined. Weber (1947) defines power as “the probability that one actor within a social relationship would be in a position to carry out his own will despite resistance.” Pfeffer (1981) defines power as “a relationship among social actors in which one social actor, A, can get another social actor, B, to do something that B would not otherwise have done.” Fletcher (1961) sums up bargaining power as market-imposed conditions, benefits and constraints on the negotiating parties. Wolff (2016) studies the bargaining powers of buyers and sellers on the online diamond market and finds that the average bargaining power of buyers is less than that of sellers.

Based on the work done by Muthoo (2000, 2001), Blokhuis et al. (2008) identify several factors as the most important components of the bargaining power of negotiating parties, namely impatience, breakdown risk, outside options, commitment tactics, inside options, information asymmetry, reputation, learning curve and future negotiation opportunities. Galinsky and Magee (2006) define three types of power in negotiations, first being lack of dependence on other (or best alternative to a negotiated agreement), second being role or position in the organizational hierarchy and last being a psychological sense of power.

Based on the coverage in literature and analytical tractability, the following three factors are chosen for analysis in this paper:

  1. Risk of breakdown: while bargaining, there is a risk of the negotiations breaking down in a random manner for one reason or another. Reasons for such a breakdown may be due to a player getting fed up with protracted negotiation, a third-party intervention, additional information or any other random reason (Muthoo, 2001). Zwick et al. (1992) demonstrate that persons not willing to take the risk of negotiations to end (leaving them with nothing) are inclined to be satisfied with a smaller part of the payoff (risk aversion).

  2. Outside options: during a negotiation process, either player can quit negotiation taking up the best option available elsewhere. This best option is known as the outside option. Binmore et al. (1989) conclude that having a credible alternative strengthens the negotiation position with respect to the other party. Cunyat (1998) establishes the impact of an outside option on a bargaining game, and further argues that the changes due to the outside option depend on the nature of the outside option and whether it is available to one or both players.

  3. Impatience: McLeish and Oxoby (2007) define inter-temporal discounting as the measure of impatience. They define inter-temporal discounting as “the manner in which individuals trade-off future and present costs and benefits, with the value of future costs and benefits being lower than that of identical, but more temporally proximate, costs and benefits.” Rambaud and Torrecillas (2016) state that decreasing discount rates mean decreasing impatient. If one actor has a different inter-temporal discounting rate than the other, there is an impact on the bargaining power.

2.2 Bargaining for decision making

There has been a great deal of research in planning and coordination issues in the supply chain. The majority of this research has been a model-based approach on supply chain contracting (Cachon, 2003). This line of research focuses on devising coordination mechanisms that provide each player incentive to work in a way that the supply chain behaves like a vertically integrated supply chain resulting into supply chain surplus as opposed to an uncoordinated supply chain. Further, there has been a fair amount of work applying the game-theoretic approach to supply chain coordination (Esmaeili et al., 2009; Huang et al., 2011). However, these works do not adequately cover profit sharing within the supply chain. There have been some recent works leveraging bargaining models to address profit sharing problem within a coordinated chain (Ertogral and Wu, 2001; Hermel, 2013; Baron and Berman, 2014; Monroy et al., 2017). Pan and Choi (2016) build and study an agent-based two-phase (co-operative phase and competitive phase) negotiation model on price and delivery date in a fashion supply chain.

Negotiation or bargaining is another form of decision making with two or more parties, with potentially differing goals, need to make decision collaboratively. Therefore, bargaining parties make concessions to achieve a common ground for all parties involved (Kersten et al., 1991). Bargaining theory provides a powerful tool for the analysis of supply chain coordination problems, and to develop mechanisms to ensure that none of the parties deviate from the agreement even in the absence of a legally enforceable contract. There have been two main streams of research on bargaining: axiomatic (co-operative game) models, and strategic (non-co-operative game) models.

Nash (1950) establishes a framework for the axiomatic bargaining solution. Osborne and Rubinstein (1990) summarize Nash’s approach as “[…] defines a ‘bargaining problem’ to be the set of utility pairs that can be derived from possible agreements, together with a pair of utilities which is designated to be the ‘disagreement point.’ A function that assigns a single outcome to every such problem is a ‘bargaining solution’ […].” Nash proposes that any bargaining solution should satisfy four basic axioms, and then derives the unique solution, also known as the “Nash Bargaining Solution,” which satisfies the stated axioms. Several authors have further revised and refined the Nash proposal by replacing, relaxing or adding to the axioms and the analysis (Kalai and Smorodinsky, 1975; Binmore, 1987). The axiomatic approach leaves out the actual process of negotiations while focusing on the expected outcome based on the defined solution properties based on the player’s attitude to risk.

Non-co-operative, sequential bargaining process is studied by modeling bargaining as a sequence of offers and counter offers (Stahl, 1972). Rubinstein (1982) first laid out a framework for non-co-operative bargaining models. He proposes an alternating-offer sequential bargaining procedure, where the players take turns in making offers and counter offers to one another until an agreement is reached. Gain to the player is discounted by time and therefor the player’s attitude to time drives a compromise. Binmore et al. (1986) model and study two strategic bargaining models, first with time preference and second with the exogenous risk of breakdown. They give a condition for unique perfect equilibrium for both strategic bargaining models and provide a guide for the application of Nash bargaining solution to economic models.

2.3 Research gaps and question

There have been some limited research works (Leng and Zhu, 2009) on how the supply chain surplus can be divided among the two parties. However, there is a need for more research in this area, as a perfect coordination mechanism is of no use, if the either party does not stick by it.

This paper studies the distribution of supply chain surplus, arising due to coordination, among the supply chain partners. It takes a bargaining-theoretic approach to profit allocation problem, and studies the impact of individual bargaining powers of negotiation parties. It incorporates selected aspects of bargaining powers, namely impatience, breakdown probability (more accurately, offer acceptance probability) and external options, and develops a model to derive equations for optimal profit allocation. A bargaining-theoretic approach is deployed to analyze the impact of bargaining power, especially impatience, outside options and breakdown probability on surplus allocation. Impatience and breakdown probability are also proxying for a firm’s financial stability, partnership options and the value that the firm puts on this partnership. For example, if a market-leader firm, say Firm A, is negotiating with another firm for a key technology partnership. The other firm may see it as a very strategic opportunity and will be impatient to close the deal. On the other hand, if another large firm, say Firm B, is also courting the same smaller firm for an exclusive partnership, Firm A may be more impatient to close an exclusive partnership and may be willing to settle with a lower pie of the surplus.

Accordingly, this paper considers the following research question:

RQ1.

What is the impact of bargaining power (measured in terms of breakdown probability, outside options, and inter-temporal discounting) on supply chain surplus split among negotiating parties?

3. Bargaining game

3.1 Model setting

This paper proposes to model the process of splitting coordination surplus as a bargaining game on the lines of the Rubenstein’s (1982) alternating-offer model. We consider a bargaining game where a buyer and a single sourcing supplier enter negotiations for a supply chain contract. We assume that both players are rational, self-interested and risk neutral (expected value maximizers), but they have different levels of bargaining power that is modeled in the base model by the probability of it accepting an offer. Like in the Rubenstein’s model, while multiple time periods are modeled for study, bargaining concludes immediately as soon as it starts since both parties are rational and can compute the equilibrium strategies of the other party. This paper concerns itself primarily with splitting of the system-wide surplus, additional profit generated due to co-operation and by implementing the system-optimal solution, under varying bargaining power levels.

3.2 Model novelty

This paper’s bargaining model, while based on Ertogral and Wu (2001), is significantly different from Ertogral and Wu (2001) and other related works (Binmore et al., 1986; Feng and Lu, 2013; Hermel, 2013; Baron and Berman, 2014) in the following ways:

  • Ertogral and Wu (2001) assume that both the parties have the same probability of offer acceptance. In real life, different parties have different sets of business priorities and different costs of capital. Therefore, equal probability of offer acceptance by both parties is a serious limitation of Ertogral and Wu (2001). We have modeled and analyzed different probabilities of offer acceptance for both the teams.

  • Ertogral and Wu (2001) have not considered the time discounting for the bargaining model. However, every firm has its own cost of capital and trade-off between current and future payoffs. Consequently, every party in the supply chain will have a different time discounting in real life. Therefore, our model not only considers time discounting, but it also allows for different time discounting for both the parties.

  • Hermel (2013) primarily focuses on outside options for the pivotal supply chain partner, and does not consider other factors such as breakdown probability and inter-temporal discounting. Our paper studies the equilibrium conditions under the simultaneous interaction of various factors driving the bargaining power of supply chain partners.

  • Baron and Berman (2014) assume bargaining powers as α and 1−α for the supply chain partners and solve for equilibrium conditions. However, they have not addressed how to obtain α. This paper has deep dived into the components of bargaining power and has defined the components which constitute the bargaining power. Consequently, the results of this paper will be easier to use in an industry setting.

  • Binmore et al. (1986) look at inter-temporal discounting and probability of breakdown separately. This paper studies an integrated model with inter-temporal discounting, outside options and breakdown probability.

In summary, this paper’s model focuses on the impact of bargaining power on supply chain surplus split and studies three dimensions of bargaining power (breakdown/counter-offer probability, outside option and inter-temporal discounting) simultaneously as compared to other models in the existing literature, which have considered only one of the above dimensions at a time.

4. Base model

The base model considers a bargaining game where a buyer and a supplier enter negotiations for a supply chain contract. It is assumed that both players are rational, self-interested and risk neutral (expected value maximizers), but they have different levels of bargaining power that is modeled in the base model by the probability of it accepting an offer. Base bargaining model, therefore, considers only breakdown (counter-offer) probability aspect of the bargaining power discussed in Section 2.1.

Base bargaining model assumes that there are no chances of breakdown of bargaining. However, the time value of the money and individual impatience level will ensure that bargaining converges to an acceptable solution for all parties. Further, we also model the time value of the decision in the form of interest rate. However, interest rate is assumed to be same for both the parties in the base model and does not result in any power equation. The variable notation used in this paper is listed in Table I.

If ψb is higher than ψs, buyer is more eager as compared to the supplier to reach an outcome and will accept an offer sooner and vice versa. Therefore, ψb(ψs) is a proxy for the bargaining power of the buyer (supplier).

As in Ertogral and Wu (2001), the following additional notations are introduced:

  • Mb (Ms): maximum payoff the buyer (the supplier) receives in the subgame perfect equilibrium (SPE) of any subgame starting with her offer.

  • mb (ms): minimum payoff the buyer (the supplier) receives in the SPE of any subgame starting with her offer.

  • Si: expected payoff for the supplier in ith iteration, where i = 0, 1, …, and so on.

The base bargaining model is illustrated in Figure 1. All payoffs are from the supplier’s point of view. In Step 1, buyer makes an offer, S0, and supplier can either accept the offer with probability, ψs, or counter-offer with the probability (1−ψs). If the supplier accepts the offer, game stops. Otherwise, buyer can either accept the suppliers’ offer with probability, ψb, or counter-offer with the probability, (1−ψb), and the game continues. Figure 1 depicts the payoff for the supplier in each case. The following equation is for a scenario where supplier receives minimum payoff:

(1) S 0 = ψ s × ( π M b ) + ( 1 ψ s ) × ( ψ b × m s + ( 1 ψ b ) × S 1 ) .

S0 is the expected payoff of the supplier in the beginning and S1 is the expected payoff of the supplier in the first iteration. The first term in the RHS. of Equation (1) is expected split in case of offer acceptance by supplier, the second term denotes expected split in case of counter-offer by the supplier. Under equilibrium, time-discounted present value for expected payoffs in various periods will remain same. Hence, for equilibrium conditions, the following relation will hold true:

S 1 = ( 1 + δ ) × S 0 .

Further, under equilibrium conditions, buyer will offer the split as per supplier’s expected payoff in period 0. Hence, S0 will be equal to πMb. Therefore, Equation (1) can be further simplified as follows:

(2) S 0 = ψ s × ( π M b ) + ( 1 ψ s ) × ( ψ b × m s + ( 1 ψ b ) × ( 1 + δ ) × S 0 ) , or π M b = ψ s × ( π M b ) + ( 1 ψ s ) × ( ψ b × m s + ( 1 ψ b ) × ( 1 + δ ) × S 0 ) , or ( π M b ) ( 1 ψ s ) ( 1 ( 1 ψ b ) ( 1 + δ ) ) = ( 1 ψ s ) × ψ b × m s .

We introduce new variables k_b and k_s such that:

(3) k b = ( 1 ( 1 ψ b ) × ( 1 + δ ) ) k s = ( 1 ( 1 ψ s ) × ( 1 + δ ) ) .

The second term in the RHS of Equation (3) (say, for kb) is the probability of counter-offer by buyer multiplied by the time value of unit payoff in next period which can be interpreted as expected time value of the payoff in the next period when supplier makes an offer. Therefore, kb can be interpreted as the complement of expected time value of payoff in next period when supplier makes an offer. Further, Equation (2) can be rewritten as:

k b × M b + ψ b × m s = k b × π .

The above equation can be interpreted as follows. In the buyer-initiated bargaining game, the maximum possible payoff of buyer, Mb, is less than the overall supply chain surplus, π by a fraction, ψb/kb, of the minimum possible payoff of supplier, ms, in a supplier-initiated game. The fraction, ψb/kb, increases with the probability of offer acceptance by buyer, ψb, and the time value of money, δ.

Further, due to the symmetric nature of the relationship, the same exercise can be repeated for four scenarios corresponding to supplier initiated or buyer initiated, and minimum or maximum payoff for supplier, and we will get the following equations:

(4) k b × M b + ψ b × m s = k b × π ,
(5) k b × m b + ψ b × M s = k b × π ,
(6) ψ s × M b + k s × m s = k s × π ,
(7) ψ s × m b + ψ s × M s = k s × π .

Based on the above equations, it can be easily deduced that in equilibrium:

(8) X b = M b = m b X s = M s = m s ,
where Xb and Xs are the equilibrium payoffs for buyer and supplier, respectively. Further, we can solve the system of Equations (4)(7) to get the values for Xb and Xs in terms of k, p and π:
(9) X b = k s ( ψ b k b ) × π ψ b ψ s k b k s ,
(10) X s = k b ( ψ s k s ) × π ψ b ψ s k b k s .

It can be observed that the equilibrium payoffs for buyer(supplier) are a function of the probabilities of offer acceptance by itself, and that by the other player. Further, it is also driven by the time-discounting rate. Therefore, breakdown(counter-offer) probability of the players has an impact on the equilibrium strategy. Further, in the equilibrium conditions, a player will accept a proposal if there is no way the player can improve upon it in the future iterations. As this stands true for both players, an equilibrium strategy must be the one that constitutes Nash equilibrium in all iterations of the repeated game. We formulate the following two propositions for the base model:

P1.

Breakdown (counter-offer) probability of a player has a direct relationship with the player’s split of the surplus, ceteris paribus.

Proof. We can rewrite Equation (3) as:

k b = ( 1 ( 1 ψ b ) × ( 1 + δ ) ) = ψ b δ ψ b ¯ ,
where ψ b ¯ is equal to (1−ψb).

Using the above expansion of kb, we can rewrite Equation (9) as:

(11) X b = k s × δ × ψ b ¯ ψ b ( ψ s k s ) + k s × δ × ψ b ¯ × π = 1 ( ψ b / ψ b ¯ ) × ( ( ψ s k s ) / δ . k s ) + 1 × π .

By definition, if the buyer’s probability of offer acceptance, ψb, increases, ψ b ¯ decreases. Therefore, the denominator increases and Xb decreases. Hence, as the buyer (supplier)’s counter-offer probability increases (everything else remaining same), her split of the surplus goes up:∎

P2.

Breakdown (counter-offer) probability of a player has an inverse relationship with the other player’s split of the surplus, ceteris paribus.

Proof. To analyze the impact of supplier’s counter-offer probability on the buyer’s split of the surplus; we look at the fraction (ψsks/ks) more closely:

ψ s k s k s = δ ψ s ¯ ψ s δ ψ s ¯ = 1 ( ψ s / δ . ψ s ¯ ) 1 .

If ψs increases, (ψsks/ks) decreases. Therefore, the denominator in Equation (11) decreases and Xb increases. Hence, with an increase in the supplier (buyer)’s counter-offer probability (everything else remaining same), buyer (supplier)’s split of the surplus goes down.∎

4.1 Numerical analysis to study the impact of counter-offer probability on the split of surplus

We assume discounting rate, δ, to be 0.1 per period based on Schmidt (2013) and analyze supplier and buyer’s split under increasing supplier’s counter-offer probability, or, in other words, increasing probability of offer acceptance by supplier, ψs. As the bargaining problem is symmetric, all analysis results for supplier will hold true for buyer too.

Figure 2(a) and (b) illustrates the impact of increasing supplier’s counter-offer probability, (1−ψ_s). The supplier’s split is directly proportional to supplier counter-offer probability and buyer’s split is inversely proportional to the same as laid out in P1 and P2, respectively. Moreover, the relationship is mostly linear as expected from the proof laid out in the above two propositions. Further, it can be seen that if ψs = 1, i.e. supplier will accept any offer; its share of surplus is 0 and that of buyer is 1 as expected in the game theory.

It can be seen from Figure 3 that sum of supplier’s split and buyer’s split of surplus does not necessarily add to 1 (unless one of ψs and ψb is equal to 1 – interestingly there is no solution if both the probabilities are 1, meaning both will accept any offer). Sum of the splits is farther away from 100 percent, as the counter-offer probabilities are high, which means that parties are more patient indicating dragged negotiations. Difference between total surplus and the sum of supplier and buyer’s split of surplus is the opportunity for a supply chain intermediary to mediate the bargaining process and thereby increase profits for everyone involved. A supply chain intermediary can bring the expert knowledge of each party’s bargaining power, reduce information asymmetry and expedite the negotiation process. It will ensure that the sum of buyer and supplier splits of surplus comes as close to 1 as possible.

5. Model extension for outside options and negotiation breakdown

In this section, we extend the base model to account for outside options (and negotiation breakdown) for buyers and suppliers. The extended model considers both breakdown(counter-offer) probability and outside option aspects of the bargaining power discussed in Section 2.1.

We drop the consideration of time value of money, δ from the model (it will be reintroduced in Section 6). However, like Wu (2004) we assume that if the supplier/buyer does not accept the offer, there is an equal probability whether the negotiations will breakdown or it will continue to next round with a counter-offer. This also allows us to solve this model analytically and provide closed-form solutions. Ws and Wb are the external options/payoff for the supplier and buyer, respectively. We use Seq in the extended model to denote expected payoff for the supplier under equilibrium conditions. Figure 4 illustrates the bargaining model.

The equilibrium payoff for supplier is as follows:

(12) S e q = ψ s × ( π M b ) + ( 1 ψ s ) 2 W s + ( 1 ψ s ) 2 × ( ψ b × m s + ( 1 ψ b ) 2 × W s + ( 1 ψ b ) 2 × S e q ) , or ( π M b ) ( 1 ψ s ( 1 ψ s ) ( 1 ψ b ) 4 ) = ( 1 ψ s ) 2 × ψ b × m s + ( 1 ψ s ) ( 3 ψ b ) 4 W s , or ( π M b ) ( 3 + ψ b ) = 2 ψ b × m s + ( 3 ψ b ) × W S .

Equation (12) can be analyzed to see that the coefficient of supplier’s minimum payoff ((πMb) in case of buyer-initiated game, and ms in case of supplier-initiated game) contains a linear function of the probability of offer acceptance by buyer, ψb. Therefore, supplier’s minimum payoff varies directly with the counter-offer probability of buyer, (1−ψb).

Further, due to the symmetric nature of the relationship, the same exercise can be repeated for four scenarios corresponding to supplier initiated or buyer initiated, and minimum or maximum payoff for supplier, and as in the above section, we can deduce that:

(13) X b = M b = m b X s = M s = m s .

Therefore, the final equations are as follows:

(14) ( 3 + ψ b ) × X b + 2 × ψ b × X s + ( 3 ψ b ) × W s = ( 3 + ψ b ) × π ,
(15) ( 3 + ψ s ) × X s + 2 × p s × X b + ( 3 ψ s ) × W b = ( 3 + ψ s ) × π .

Solving the above equations, we get:

(16) X b = ( 3 + ψ s ) ( 3 ψ b ) ( 3 + ψ s ) ( 3 + ψ b ) 4 ψ s ψ b × ( π W s ) + 2 ψ b ( 3 ψ s ) ( 3 + ψ s ) ( 3 + ψ b ) 4 ψ s ψ b × W b ,
(17) X s = ( 3 + ψ b ) ( 3 ψ s ) ( 3 + ψ s ) ( 3 + ψ b ) 4 ψ s ψ b × ( π W b ) + 2 ψ s ( 3 ψ b ) ( 3 + ψ s ) ( 3 + ψ b ) 4 ψ s ψ b × W s .

Based on the above Equations (16) and (17), it can be observed that the equilibrium payoffs for buyer (supplier) are a function of the probabilities of offer acceptance by itself, and that by the other player. Further, it is also driven by the outside options; positively by its own outside and negatively by that of the other player. Further, we can propose the following propositions:

P3.

Buyer (supplier)’s split of surplus increases as its external option (payoff in case of breakdown) increases.

Proof. As the external option for buyer, Wb, increases, buyer’s split of surplus, Xb increases due to increase in the second term in the RHS of Equation (16):∎

P4.

Buyer (supplier)’s split of surplus increases as the external option (payoff in case of breakdown) of the other party decreases.

Proof. As the external option for supplier, Ws, increases, buyer’s split of surplus Xb decreases due to a decrease in the first term in the RHS of Equation (16).∎

5.1 Numerical analysis to study the impact of outside option on the split of surplus

We analyze supplier and buyer’s split under increasing supplier’s external option to a total surplus ratio (Ws/π).

Figure 5(a) and (b) illustrates the impact of increasing supplier’s external option. Supplier’s split is directly proportional to supplier’s external option and buyer’s split is inversely proportional to the same as laid out in P3 and P4, respectively. Further, as discussed in Section 4.1, the sum of supplier’s split and buyer’s split need not add up to 1, and this shortfall creates an opportunity for a supply chain intermediary to improve the profit of each party as compared to the disintermediated scenario of supply chain partners negotiating directly.

6. Model extension for differential inter-temporal discounting

In this section, we further extend and generalize the model developed in Sections 4 and 5 to incorporate differential inter-temporal discounting. This model now considers all three aspects of bargaining power, breakdown(counter-offer) probability, outside options and inter-temporal discounting, as discussed in Section 2.1. Further, in this section, we relax assumptions around the equal probability of breakdown and counter-offer.

As discussed in Section 2.1, power balance can be looked as a difference in impatience (inter-temporal discounting), breakdown probability and outside options. Previous sections modeled breakdown probability (0.5 ×(1 – probability of accepting an offer)) and outside options. This section further refines the model by bringing back in consideration discounting rate or time value of money, δ (dropped in Section 5). However, in this model, the discounting rate is different for both the parties. Therefore, we introduce two additional notations:

δ s : discounting rate for supplier ,
δ b : discounting rate for buyer .

The difference in discounting rate signifies the difference in impatience levels. Higher the discounting rate, more impatient is the concerned party. We use S0 in the extended model to denote expected payoff for the supplier in the beginning and S1 is the expected payoff of the supplier in the first iteration. For equilibrium conditions, the present value of expected payoffs in various periods will remain the same. Therefore, the following relation will hold true under equilibrium (from the supplier’s point of view):

S 1 = ( 1 + δ s ) × S 0 .

Further, this section drops the assumption made in Section 5 that if supplier/buyer does not accept the offer, there is an equal probability whether the negotiations will breakdown or it will continue to next round with a counter-offer. A new variable, λ, is introduced, which defines the conditional probability that a counter-offer will be made, given that the current offer has not been accepted.

It can be easily observed that Section 5 model is a special case (λ=0.5, δs=δr=0) of the generalized model in this section. Further, Section 4 model can also be obtained from this model by using λ=1 and δs=δr=δ.

Figure 6 illustrates the bargaining model (all payoffs are from the supplier’s point of view). It is very similar to the bargaining model in Figure 2 with the additional consideration of inter-temporal discounting rates.

The equilibrium payoff for supplier is as follows:

(18) S 0 = ψ s × ( π M b ) + ( 1 λ ) × ( 1 ψ s ) × W s + λ × ( 1 ψ s ) × ( ψ b × m s + ( 1 λ ) × ( 1 ψ b ) × W s + λ × ( 1 ψ b ) × S 1 ) .

S0 is the expected payoff of the supplier in the beginning and S1 is the expected payoff of the supplier in the first iteration. The first term in the RHS of Equation (18) is expected split in case of offer acceptance by supplier, the second term denotes the expected value of outside option and the third term is expected split in case of counter-offer by the supplier. Further, under equilibrium conditions, buyer will offer the split per supplier’s expected payoff in period 0. Hence, equilibrium payoff for supplier in buyer-initiated negotiation is (πMb). Therefore, Equation (18) can be simplified as:

( π M b ) ( 1 ψ s ) ( 1 λ 2 ( 1 ψ b ) ( 1 + δ s ) ) = ( 1 ψ s ) ( λ × ψ b × m s + ( 1 λ ) ( 1 + λ ( 1 ψ b ) ) × W s ) ,
or:
(19) ( π M b ) ( 1 λ 2 ( 1 ψ b ) ( 1 + δ s ) ) = ( λ × ψ b × m s + ( 1 λ ) ( 1 + λ ( 1 ψ b ) ) × W s ) .

Equation (19) can be analyzed to see that the coefficient of supplier’s minimum payoff ((πMb) in case of buyer-initiated game, and ms in case of supplier-initiated game) contains a linear function of the probability of offer acceptance by buyer, ψb. Therefore, supplier’s minimum payoff varies directly with an impatience level of buyer, (1−ψb). Further, we introduce two new variables, gbs and gsb, such that:

(20) g b s = 1 λ 2 ( 1 ψ b ) ( 1 + δ s ) g s b = 1 λ 2 ( 1 ψ s ) ( 1 + δ b ) .

The second term in the RHS of Equation (20) (say, for gbs) is the probability of counter-offer by buyer multiplied by the time value of unit payoff for supplier in the next period, which can be interpreted as the expected time value of payoff for supplier in the next period when supplier makes an offer. Therefore, gbs can be interpreted as the complement of expected time value of payoff for supplier in next period when supplier makes an offer. Further, it can be seen that gbs (gsb) is an enhancement of kb(ks) introduced in the analysis of base model in Section 4 to account for differential time-discounting rates. Further, using Equations (8) and (20), Equation (19) can be simplified to:

(21) g b s × X b + λ × ψ b × X s + ( 1 λ ) × ( 1 + λ × ψ b ¯ ) × W s = g b s × π .

The above equation can be interpreted as follows. Equilibrium payoff of buyer, Xb, is less than the overall supply chain surplus, π, by a fraction, (λ×ψb/gbs), of equilibrium payoff of supplier, Xs, and another term which is the function of supplier’s outside option and its own counter-offer probability. The fraction, (λ×ψb/gbs), increases with the probability of offer acceptance by buyer, ψb, and supplier’s time value of money, δs.

As supplier-initiated negotiation is symmetrical, we can get the equation involving buyer’s split as follows:

(22) g s b × X s + λ × ψ s × X b + ( 1 λ ) × ( 1 + λ × ψ s ¯ ) × W b = g s b × π .

Solving Equations (21) and (22), we get:

(23) X b = g s b g b s g s b λ 2 ψ s ψ b × ( ( g b s λ × ψ b ) × π ( 1 λ ) × ( 1 + λ × ψ b ¯ ) × W s ) + λ × ( 1 λ ) × ψ b * ( 1 + λ × ψ s ¯ ) g b s g s b λ 2 ψ s ψ b × W b ,
(24) X s = g b s g b s g s b λ 2 ψ s ψ b × ( ( g s b λ × ψ s ) × π ( 1 λ ) × ( 1 + λ × ψ s ¯ ) × W b ) + λ × ( 1 λ ) × ψ s * ( 1 + λ × ψ b ¯ ) g b s g s b λ 2 ψ s ψ b × W s .

It can be quickly checked that Equations (9) and (10) (equilibrium conditions for the base model in Section 4) can be derived from Equations (23) and (24) by substituting λ=1 and δs=δr=δ. Further, Equations (16) and (17) (equilibrium conditions for model in Section 5) can be derived by substituting λ=0.5 and δs=δr=0.

Based on Equations (23) and (24), it can be observed that the equilibrium payoffs for buyer(supplier) are a function of the probabilities of offer acceptance by itself, and that by the other player. Further, it is also driven by the outside options: positively by its own outside and negatively by that of the other player. Further, we can propose the following propositions:

P5.

Buyer (supplier)’s split of surplus increases as its impatience, measured in time discounting rate, increases.

Proof. As the buyer’s discounting rate, δb, increases, only term in Equation (23) that changes is gsb, which decreases as per Equation (20). Examining first term in the RHS of Equation (23), (gsb/(gbsgsbλ2ψsψb)) can be rewritten as (1/(gbs−(λ2ψsψb/gsb))). With an increase in δb, the second term of denominator increases which makes denominator decrease due to negative sign and thus resulting in an overall increase.

Similarly, the second term in the RHS of Equation (23) can be quickly examined to see that an increase in δb decreases its denominator and thus causing it to increase. Since both terms increase due to increase in δb, it can be concluded that the buyer’s split of surplus, Xb increases with its impatience or discounting rate δb:∎

P6.

Buyer (supplier)’s split of surplus decreases as the impatience (measured in discounting rate) of the other party decreases.

Proof. As the supplier’s time-discounting rate, δs increases, its split of surplus, Xs increases due to P3. As Xb and Xs are coming out of the same surplus π, therefore, buyer’s split, Xb, decreases with an increase in δs (and consequently Xs).∎

The next section will further establish the above-mentioned propositions through numerical illustration.

6.1 Numerical analysis to study the impact of discounting rates on the split of surplus

We analyze supplier and buyer’s split under increasing supplier’s impatience, i.e. discounting rate, δs.

Figure 7(a) and (b) illustrates the impact of increasing supplier’s impatience in terms of inter-temporal discounting rate. The supplier’s split increases with supplier’s impatience level and buyer’s split decreases with supplier’s impatience level as laid out in P5 and P6, respectively.

7. Conclusion and future works

Most of the research literature on supply chain contracts has an implicit assumption that negotiating firms have the same level of bargaining power during the bargaining process. Frequently, supply chain is modeled as a Stackelberg game with one firm designated as the leader, and the other firm as profit-maximizing under the terms set by the leader. This paper studies the impact of bargaining power on supply chain surplus allocation. Three different aspects of bargaining power, namely inter-temporal discounting, outside options and breakdown risk, are modeled together. It formulates a bargaining game between the supplier and buyer and solves it using Rubenstein’s bargaining game.

This paper establishes that impatience (in terms of counter-offer probability) has a significant impact on the bargaining position and on the split of the surplus that the firm can get for themselves. It is common wisdom that a more patient firm can wait longer before accepting an offer and will thus get a better deal, whereas a more impatient firm accepts the counter-offer more readily and will get a worse deal. However, this paper establishes that a more impatient firm can improve its split of surplus, provided the other party recognizes the impatience and factors it in their decision-making process. Therefore, supply chain entities should work to devise appropriate messaging to convey their outside options, inter-temporal discounting and breakdown risk.

Solution to the bargaining problem also reveals an important strategic insight. Player’s impatience is advantageous to it, only when the other player recognizes it. Therefore, posturing and messaging in advance of the negotiations become very important.

This paper concludes by closed-form deriving equations for split of the surplus for the buyer and the supplier, respectively, in consideration of selected aspects of bargaining power. Work done in this paper is very significant in the current era of automation and artificial intelligence. Equations derived in this paper can be used to design an autonomous bargaining agent to discover equilibrium profit splits, and offer the same to the other party, thus reducing the negotiation efforts.

This paper has limited its analysis to three key components of bargaining power. Future works can study other aspects of bargaining power, namely information asymmetry, learning curve, inside options, etc. Further, the paper has considered an infinite horizon model – this assumption can be relaxed in future research.

Figures

A tree to illustrate bargaining game (all payoffs from supplier’s point of view)

Figure 1

A tree to illustrate bargaining game (all payoffs from supplier’s point of view)

Impact of counter-offer probability of supplier, (1−ψs), on supplier and buyer’s split of surplus

Figure 2

Impact of counter-offer probability of supplier, (1−ψs), on supplier and buyer’s split of surplus

Total split as function of counter-offer probability of supplier, (1−ψs)

Figure 3

Total split as function of counter-offer probability of supplier, (1−ψs)

A tree to illustrate bargaining game with outside options (all payoffs from supplier’s point of view)

Figure 4

A tree to illustrate bargaining game with outside options (all payoffs from supplier’s point of view)

Impact of (Ws/π) ratio on split of surplus

Figure 5

Impact of (Ws/π) ratio on split of surplus

A tree to illustrate bargaining game with external options, breakdown and inter-temporal discounting

Figure 6

A tree to illustrate bargaining game with external options, breakdown and inter-temporal discounting

Impact of supplier’s discounting rate, δs, on split of surplus (λ=0.5)

Figure 7

Impact of supplier’s discounting rate, δs, on split of surplus (λ=0.5)

Key variable used in the base model

Symbol Description
ψs Probability of supplier accepting an offer
ψb Probability of buyer accepting an offer
δ Interest rate per period
Π Total supply chain surplus generated due to co-operation

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Corresponding author

Dr Sanjay Prasad is the corresponding author and can be contacted at: skprasad@gmail.com

About the authors

Dr Sanjay Prasad is employed with IBM India as Chief Data Scientist. Sanjay has rich industry experience and has worked for large organizations like Coal India Ltd, Steel Authority of India Ltd, i2 Technologies, etc. He is actively involved in the academic circles and has delivered lectures in institutes like IISc Bangalore and RIMS Bangalore. He also received Emerald Literati Award for Outstanding Reviewer in May 2013. He has developed a strong interest and expertise in supply chain management, services engineering, data mining, machine learning, game-theoretic modeling and optimization modeling.

Professor Ravi Shankar is “Amar S. Gupta Chair Professor of Decision Science” and Group Chair of Operations and Supply Chain Management in the Department of Management Studies (DMS), Indian Institute of Technology (IIT) Delhi, India. He is also Program-Coordinator of MBA (Telecom Technology & Management) at Bharti School of Telecom Technology & Management, IIT Delhi. His areas of interest include decision science, business analytics and Big Data, operations and supply chain management, project management, total quality management & six sigma, sustainable freight transportation, strategic technology management, telecom system planning & design, knowledge management, etc.

Sreejit Roy is Vice President in IBM India. In this role, he is responsible to manage delivery across all industry verticals at IBM Client Innovation Centre, India. He has delivery accountability for both Global and India domestic clients, and across technology domains, i.e. ERP, custom applications, cloud, digital, mobile, cognitive and analytics. Sreejit received a PGDM from Indian Institute of Management Calcutta and BE (Mechanical) from the University of Jodhpur, Rajasthan.