Purchasing and Supply Chain Management

International Journal of Production Economics 200 (2018) 50–67

Contents lists available at ScienceDirect

International Journal of Production Economics

journal homepage: www.elsevier.com/locate/ijpe

Ambiguity aversion in buyer-seller relationships: A contingent-claims and social network explanation

Yongling Gao a, Tarik Driouchi b,*, David J. Bennett c,d

a Central University of Finance and Economics, Beijing, China b King's College London, University of London, London, UK c Aston University, Birmingham, UK d Chalmers University of Technology, Gothenburg, Sweden

A R T I C L E I N F O

Keywords: Buyer-seller relationships Real options Supply networks Social networks Multiple-priors Ambiguity

* Corresponding author. King's College London, U E-mail addresses: [email protected] (Y. Gao), tari

https://doi.org/10.1016/j.ijpe.2018.02.004 Received 14 December 2016; Received in revised form 31 Available online 9 February 2018 0925-5273/© 2018 Elsevier B.V. All rights reserved.

A B S T R A C T

Negotiations between buyers and sellers (or suppliers) of goods and services have become increasingly important due to the growing trend towards international purchasing, outsourcing and global supply networks together with the high uncertainty associated with them. This paper examines the effect of ambiguity aversion on price ne- gotiations using multiple-priors-based real options with non-extreme outcomes. We study price negotiation be- tween a buyer and seller in a dual contingent-claims setting (call option holding buyer vs. put option holding seller) to derive optimal agreement conditions under ambiguity with and without social network effects. We find that while higher ambiguity aversion raises the threshold for commitment for the seller, it has equivocal effects on the buyer's negotiation prospects in the absence of network control. Conversely when network position and relative bargaining power are accounted for, we find the buyer's implicit price (or negotiation threshold) de- creases (or increases) unequivocally with increasing aversion to ambiguity. Extending extant real options research on price negotiation to the case of ambiguity, this set of results provides new insights into the role of ambiguity aversion and network structures in buyer-seller relationships, including how they influence the range of nego- tiation agreement between buyers and sellers. The results also help assist managers in formulating robust buying/ selling strategies for bargaining under uncertainty. By knowing their network positions and gathering background information or inferring the other party's ambiguity tolerance beforehand, buyers and sellers can anticipate where the negotiation is heading in terms of price negotiation range and mutual agreement possibilities.

1. Introduction

The relationships between buyers and sellers of goods and services have come under increasing scrutiny in the literature since the results and consequences of negotiations between them can be critical to the competitiveness and integrity of firms operating within international networks. Examples of relevant issues that have been investigated include trust (Schoenherr et al., 2015; Hemmert et al., 2016), transaction costs (Schneider et al., 2013; Abd Rahman et al., 2009), ethics and social responsibility (Goebel et al., 2012; Govindan et al., 2016). In this paper we examine the behavioural issue of ambiguity, which is a concern involving both sides during negotiations between buyers and sellers. As a type of uncertainty beyond probabilistic risk, ambiguity characterises commitment and transactional situations where future outcomes are not known with certainty or high confidence (Ellsberg, 1961; Ghosh and Ray,

niversity of London, SE19NH, U [email protected] (T. Driouch

January 2018; Accepted 7 February

1997). When faced with ambiguity buyers and sellers are unsure about their future prospects and are doubtful about the probabilities of future events and their subsequent realisations, displaying ambiguity aversion and pessimism (Hazen et al., 2012; Abdellaoui et al., 2015). This is more so in negotiation cases where commitment is irreversible and trans- actional arrangements are fraught with uncertainty on both sides. The ambiguity aversion bias of each party can distort pricing dynamics resulting in suboptimal relationships between buyers and sellers. Network positions and relative bargaining power are also key to these linkages. This paper studies the effect of ambiguity on price negotiations between buyers and sellers, with and without network control, using real options theory (Trigeorgis, 1996; Driouchi and Bennett, 2012; Char- alambides and Koussis, 2018) and social network principles (Braun and Gautschi, 2006). In our research we use the term “seller” because it re- lates to commercial transactions where price is one of the main criteria

K. i), [email protected] (D.J. Bennett).

2018

2

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

used in negotiation. However, in the literature the terms “supplier” and “seller” are often used interchangeably within the same context of buying products and services (Oosterhuis et al., 2013; Esmaeilia and Zeep- hongsekul, 2010).

Several recent papers have been devoted to the study of the real op- tion value of flexible decision making in buyer-seller relationships, optimal contracting and price negotiation (Li and Kouvelis, 1999; Kam- rad and Siddique, 2004; Fotopoulos et al., 2008; Moon et al., 2011). Focusing on buyer-seller interaction and negotiation, Yao et al. (2010) and Jiang et al. (2008, 2010) show how each party's real options deter- mine contractual outsourcing arrangements under risk whereas Moon et al. (2011) examine the impact of risk-neutral optionality on negotia- tion performance. Moon et al. (2011) in particular present a bilateral negotiation model under risk-neutrality with optimal selling (buying) rules. They propose the idea of an implicit zone of possible agreement (IZOPA) and obtain negotiation agreement probabilities using real op- tions (i.e. contingent-claims). They find that the negotiation range and probability of agreement between buyers and sellers are narrower in the presence of optionality than in its absence. What is missing from this growing literature, however, is an explicit recognition of the role of in- dividual behaviour or miscalibration and network position in negotiation decisions and, especially, how ambiguity affects option-based price negotiation and its investment outcomes. Given that negotiation exer- cises are often influenced by ambiguity, behavioural factors and social network effects, it is important to account for negotiators' beliefs, rela- tional characteristics, psychology and uncertainty preferences (e.g. pessimism) in the decision making process.1

Our paper addresses this gap in research by investigating how ne- gotiations between a buyer and seller are affected by their ambiguity and social network position (our ‘Extensions and additional results’ in Section 4 examines the case of multiple sellers). A search of the literature reveals that this is the first paper to integrate real options, ambiguity and social networks principles in bilateral negotiation and buyer-seller interaction. We contribute to extant literature on real options in buyer-seller re- lationships (e.g. Moon et al., 2011; Zheng and Negenborn, 2015) by providing novel decision-making and production economics insights into how ambiguity aversion and social network effects alter the relationships among uncertainty, real options and price negotiation outcomes. We also add to buyer-seller literature concerned with behaviour, ambiguity and information asymmetry (e.g. Esmaeilia and Zeephongsekul, 2010; Hazen et al., 2012; Schoenherr et al., 2015; Hemmert et al., 2016) by developing new theoretical propositions for empirical research. We analyse the ef- fects of negotiators' ambiguity aversion on their real options prospects, with and without network control, using a multiple-priors expected utility (MEU) with non-extreme outcomes (hereafter called NMEU) in continuous-time. Adjusting for uncertainty aversion in probabilistic appraisal, this utility specification is related to the maxmin expected utility (MEU) covered in recent ambiguity-based real options research, such as Nishimura and Ozaki (2007), Trojanowska and Kort (2010), and Moreno (2014). The MEU satisfies the dynamic consistency constraint and can reflect both the present value and option value effects domi- nating the timing of commitment but its assumption of complete pessi- mism might be considered too extreme in a number of cases. Our NMEU framework is also indirectly linked to the α-maxmin expected utility (α-MEU) used to study infrastructure projects (Gao and Driouchi, 2013), corporate investments (Schr€oder, 2011) and supplier contracting (Gao, 2017). The α-MEU utility is useful in examining the impact of ambiguity attitudes on decision outcomes from the present value perspective but partly suffers from dynamic inconsistency and can result in timing thresholds for either extremely pessimistic or optimistic agents ðα ¼

1 The research findings of Ghosh (1994) and Zwick and Lee (1999) go along these lines and suggest that to enhance the descriptive power of negotiation models, risk preferences, information incompleteness and tolerance for ambi- guity need to be included in the analysis.

51

0 or 1Þ. This means that both the MEU and α-MEU models are concerned with extreme attitudes towards ambiguity but ignore situations, such as bargaining and price negotiation, where unsure decision makers might still have some confidence in their probability judgments (i.e. realization of their risk-based estimates) while caring about the worst case scenario (i.e. uncertainty aversion).

Motivated by the above decision making issues, we rely on the NMEU heuristic to solve the optimal commitment and flexible timing problem for any level of ambiguity aversion while satisfying dynamic consistency. Our NMEU utility evaluates and combines the worst case in negotiators' minds with the standard probabilistic case. Separating risk from uncer- tainty (Ellsberg, 1961; Abdellaoui et al., 2015; Agliardi et al., 2015), we present conditions for negotiation agreement under ambiguity and incorporate negotiators' aversion to uncertainty and network position in the real options analysis to show how they affect investment outcomes and optimal agreement.2 We deliberately do not investigate or discuss the role of risk aversion in the real options dynamics since these effects have been well documented in the literature (see e.g. Henderson and Hobson, 2002; Hugonnier and Morellec, 2007).

We extend uncertainty-neutral findings from recent studies, in particular those of Nagarajan and Bassok (2008), Moon et al. (2011) and Zheng and Negenborn (2015), and the Nash bargaining model consid- ering social network effects by Braun and Gautschi (2006) to the case of ambiguity aversion. We contribute to extant literature on buyer-seller interaction (e.g. Bichescu and Fry, 2009; Birkeland and Tungodden, 2014) by examining the link between ambiguity aversion and mutual agreement while considering negotiators' flexibility and discretion regarding optimal investment choice and contract timing in the negoti- ation exercise. We find that in the absence of network effects, ambiguity and ambiguity aversion do not necessarily have symmetric effects on negotiation outcomes under the NMEU. This impact is reversed in the presence of network control. Thus, we add realism and generality to the analysis by explicitly allowing for miscalibration in the uncertain nego- tiation and accounting for the structural positions of buyers and sellers in the supply chain network, and show why standard risk-neutral or normative contingent-claims assessment might be incomplete for the appraisal of commitment situations where true uncertainty, cognition and vagueness determine outcomes and heterogeneous behaviour.

In addition, many real options studies on incomplete information tend to assume that the partial information about the uncertainty vari- ables is generally symmetric (see also Grenadier and Wang, 2005; Nishihara and Shibata, 2008; Shibata and Nishihara, 2011; Feng et al., 2014; Grenadier et al., 2016). We consider this information to remain private in our setting and design incentives and signalling mechanisms for the buyer (seller) to elicit the true level of ambiguity aversion of his (or her) counterpart in the presence of information asymmetry. This is documented later in Section 4.3.

The paper is organized as follows. Section 2 presents the real options negotiation problem. Section 2.1 introduces notation, assumptions and our multiple-priors utility specification. Section 2.2 produces the policies for option exercise under ambiguity aversion. Section 2.3 identifies the negotiation's implicit zone of achievable agreement (IZOAA) or negoti- ation range, studying its optimal conditions, and the effects of ambiguity aversion and probabilistic ambiguity on the threshold for negotiation. In Section 3, we derive the Nash bargaining solution under ambiguity aversion by considering real options, structural autonomy and network positions in the price negotiation in the context of a negatively connected

In contrast to extant research on buyer-seller relationships, we do not use “risk” and “uncertainty” interchangeably in this paper. By ambiguity we refer to uncertainty, beyond probabilistic or measurable risk, as defined by Ellsberg (1961) and as discussed in Asano and Shibata (2011) and Nishimura and Ozaki (2007). Our paper is the first real option study to consider such dimension of uncertainty, and aversion towards it, in B2B buyer-seller interaction and price negotiation.

Table 1 Variable definitions.

Notation Definition

i i ¼ 1 and 2 denote the seller and buyer. X Negotiation price that connects the buyer and seller. Xi, Xnci X1 ðX2Þ are the seller's (buyer's) implicit reservation prices. Xnci is the

implicit reservation price with network control for each party. Si S1 and S2 are the seller's costs and buyer's revenues. μi, σi μi and σi are the growth rate and volatility of Si. σi > 0. κi κi is the probabilistic ambiguity surrounding the drift term of Si, κi � 0.

When κi ¼ 0, the corresponding geometric Brownian motion of SiðtÞ is denoted by ~SiðtÞ.

ρi ρ1 and ρ2 reflect the degrees of ambiguity aversion regarding seller's expected costs S1 and buyer's expected revenues S2, respectively. ρi 2 ½0; 1�.

WiðtÞ The NMEU of SiðtÞ under ambiguity. λi λi is the NMEU-based ambiguity multiplier. It connects Si at time t and the

subjective expected value WiðtÞ, where i ¼ 1; 2. r r is the discount rate. ε The correlation coefficient between dB1ðtÞθ1 and dB2ðtÞθ2 . ξlj and γlj ξlj ðγljÞ is l's relative negotiation power with respect to j in a negatively

(positively) connected network. We use γ to denote each seller's relative negotiation power vis-�a-vis the buyer in a positively connected network in Section 4.2.

4 In practical terms, this implies that the exact rates of return on buyers' and

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

network. Section 4 extends our analysis to account for outside options (e.g. exiting or opting out from the negotiation), multilateral negotiation between one buyer (an assembler) and multiple sellers, and mechanism design and incentives under asymmetric information. The final section concludes with a summary of results and implications. Proofs and addi- tional results are found in the Appendix and Supplementary Material.

2. The negotiation problem under ambiguity

2.1. Problem description and assumptions

We consider a (bilateral) price negotiation setting in which the de- cision to buy or sell goods within global supply networks has a long-term impact, incurs sunk costs and is at least partly irreversible. Due to un- certainty, there is a noticeable option value in delaying commitment and keeping options open (Trigeorgis, 1996; Roemer, 2004; Driouchi et al., 2010). Our model is based upon the IZOPA3 under risk-neutrality of Moon et al. (2011) and Jiang et al. (2008). We extend these authors' findings to the case of ambiguity using a multiple-priors expected utility with non-extreme outcomes (e.g. Chateauneuf et al., 2007). This ambiguity-based utility specification is equivalent to a weighted average between a risk-neutral utility and the minimal outcome of a multiple-priors utility (i.e. worst case scenario) (see e.g. Chateauneuf et al., 2007; Fonseca and Rustem, 2012). This subjective utility should be more reflective of cognitive or behavioural biases affecting buyer-seller assessments than those of rational or uncertainty-neutral counterparts. In the bilateral price negotiation, the seller (called she) is uncertain about the costs of producing a certain good to be sold at a price X to a buyer (called he) who is uncertain about the future revenues generated by investing in X. A typical representation/illustration of this situation would be the case of two supply chain actors negotiating over the price and potential distribution of a specific good or service. Revenues and costs are difficult to predict and follow lognormal diffusions with ambiguous drifts but unambiguous volatilities. Despite their ambiguity, both agents have some confidence in their probability judgments. Although interacting, parties do not have private information about counterparties' uncertain quantities. The case of slotting allowances/fees or new product introductions in retailing closely matches this price negotiation model. The buyer (e.g. retailer or wholesaler) holds a call option to exchange X for S2 paying X in exchange for future revenues S2. The seller (e.g. manufacturer) holds a put option to exchange operating costs S1 for product/contract price X. The put (call) option is in the money when X > S1 ðS2 > XÞ. Buyers and sellers negotiate based on their individual attitudes towards ambiguity and their sentiment (ambiguity aversion or pessimism) regarding the future fluctuations of their sto- chastic variables (i.e. revenues for the buyer and costs for the seller, respectively). Section 3 adds a social network dimension to this problem (see Table 1). For the remainder of the paper, a seller's (buyer's) ambi- guity aversion will refer to situations where the seller (buyer) is pessi- mistic about their operating costs (revenues).

X, Xi and Xnci are the decision variables. Other variables in the table are exogenous and assumed to be constant. The following assumptions are adopted.

Assumption 1. The buyer's demand is assumed to be fixed and normalized to 1.

Assumption 2.1. In line with standard real options literature (Nem- bhard et al., 2005; Wu and Liou, 2011; De Waegenaere and Wielhouwer, 2011), costs and revenues follow two separate lognormal diffusions. Due to negotiators' lack of confidence in their probability estimates we consider parametric uncertainty in the drifts of the Brownian motions. This type of vagueness or probabilistic ambiguity determines the level of

3 Their notion of the IZOPA is based on the concepts of negotiation range and contract zone in economics (e.g. Fudenberg and Tirole, 1983).

52

ambiguity aversion of each negotiating party.

Assumption 2.2. The seller's costs ðS1Þ and buyer's revenues ðS2Þ follow ambiguous Brownian motions B1ðtÞ and B2ðtÞ, which are defined on a probability space ðΩ; F T; ℙÞ. ðF tÞ0�t�T is a standard filtration for B1ðtÞ and B2ðtÞ. Ambiguity in the seller's costs ðS1Þ and buyer's revenues ðS2Þ is modelled by the set of priors P i ¼ fQθii

��θi ¼ ðθiðtÞÞ 2 Θig. Qθii is derived from the reference probability measure Qi using the density generator θi (see definitions in Chen and Epstein, 2002; Nishimura and Ozaki, 2007; Riedel, 2009). 8θi 2 Θi are restricted to the non-stochastic range Ki ¼ ½ � κi; κi�, where κiðκi � 0Þ stands for the probabilistic ambi- guity surrounding the drift terms of the geometric Brownian motions used to model the seller's costs and buyer's revenues.4 For any θi 2 Θi the Ito processes of S1 and S2 to the general sets P 1 and P 2 yield under ambiguity:

dSiðtÞ ¼ ðμi � σiθiÞSiðtÞdt þ σiSiðtÞdBiðtÞθi ð8t � 0; 8θi 2 Θi; i ¼ 1; 2Þ (1)

where μi � σiθi is the expected growth rate of Si and σi its volatility. Parameters μi and σi are assumed to be constant. σi > 0. The drift term is affected by the ambiguity parameter θi, i ¼ 1; 2. We assume the corre- lation coefficient between dB1ðtÞθ1 and dB2ðtÞθ2 to be ε. Let ~SiðτÞ denote the geometric Brownian motion under the benchmark probability mea- sure Qi. The expectation of ~SiðtÞ under risk-neutrality reflects the case of non-extreme outcomes for the uncertain decision maker.

Assumption 3. In their appraisal of economic prospects, the buyer and seller account for both the risk-neutral reward and the minimal/pessi- mistic outcome of a multiple-priors utility (i.e. combination of risk- neutral and worst case scenarios). Our NMEU specification combines the worst case in negotiators' minds with the risk-neutral outcome, thus adjusting for uncertainty aversion in probabilistic appraisal.

Assumption 3.1. For the seller, we use ρ1 with 0 � ρ1 � 1 to denote her degree of ambiguity aversion or worst case appraisal regarding future operating costs. In line with extant real options and optimal stopping research, time horizon T is assumed to approach infinity. The NMEU value of S1ðtÞ with respect to Qθ11 can be expressed as 5:

sellers' commitment to a certain price or contractual arrangement are unknown to each party. 5 The derivation of the supremum of these costs can be easily obtained based

on Nishimura and Ozaki (2007).

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

W1ðtÞ ¼ ρ1 sup θ12½�κ1;κ1�

EQ θ1 1 � ∫ ∞ t S1ðτÞe�rðτ�tÞdτjF t

� þ ð1

� ρ1ÞE � ∫ ∞ t ~S1ðτÞe�rðτ�tÞdτjF t

� ¼ λ1S1ðtÞ (2)

where λ1 ¼ ρ1r�μ1�σ1κ1 þ 1�ρ1 r�μ1, κ1 � 0, λ1 is the NMEU-based ambiguity

multiplier which connects S1 at time t and the subjective value W1ðtÞ, λ1 2 R. r is the discount rate, μ1 þ κ1σ1 < r. E½∫

∞ t ~S1ðτÞe�rðτ�tÞdτjF t� cor-

responds to the risk-neutral expectation of S1ðτÞ. When ρ1 ¼ 1, the NMEU value coincides with the maxmin heuristic of Gilboa and Schmeidler (1989) or the case of pure pessimism or extreme ambiguity aversion.

Assumption 3.2. For the buyer, we use ρ2ðρ2 2 ½0; 1�Þ to denote his degree of ambiguity aversion about future revenues which reflects the weight attributed to the worst case for investment. Consequently under κ2-ignorance, the NMEU value of S2ðtÞ can be written as:

W2ðtÞ ¼ ρ2 inf θ22½�κ2;κ2�

EQ θ2 2 � ∫ ∞ t S2ðτÞe�rðτ�tÞdτjF t

� þ ð1 � ρ2ÞE

� ∫ ∞ t ~S2ðτÞe�rðτ�tÞdτjF t

� ¼ λ2S2ðtÞ (3)

where λ2 is the NMEU-based ambiguity multiplier of S2ðtÞ incorporating buyer's attitude towards ambiguity, λ2 ¼ ρ2r�μ2þσ2κ2 þ

1�ρ2 r�μ2. In the absence

of ambiguity (ρ2 ¼ 0 or κ2 ¼ 0), W2ðtÞ simplifies to a risk-adjusted per- petuity.

ρ1 and ρ2 consider the trade-off in negotiators' minds between the worst and risk-neutral scenarios and represent the degrees of ambiguity aversion of sellers and buyers towards price negotiation. Each parameter depends on individual ambiguity attitudes and is reflective of subjective beliefs about the accuracy of probability estimates.ρi should help deter- mine the direction of the negotiation process and its outcomes in terms of negotiation range and mutual agreement occurrence.

2.2. Ambiguity and buyer-seller real options

The seller's or manufacturer's problem is to determine the optimal selling conditions to maximize her opportunity value under NMEU am- biguity F1ðtÞ:

F1ðtÞ ¼ max t0�t

� ρ1

� e�rðt

0�tÞX � sup θ12Θ1

EQ θ1 1 � ∫ ∞ t0 S1ðτÞe�rðτ�tÞdτjF t

�� þ ð1 � ρ1Þ

� � e�rðt

0�tÞX � E � ∫ ∞ t0 ~S1ðτÞe�rðτ�tÞdτjF t

��� (4)

Then her put opportunity value F1ðtÞ can be expressed as:

F1ðtÞ ¼ max t0�t

fX � W1ðtÞ; Jðt0Þg (5)

where Jðt0Þ ¼ max t0�tþdt

( e�rðt

0�tÞX � " ρ1 sup

θ12Θ1 EQ

θ1 1 ½∫ ∞t0 S1ðτÞe�rðτ�tÞdτjF t� þ

ð1 � ρ1ÞE½∫ ∞ t0 ~S1ðτÞe�rðτ�tÞdτjF t�

#) .

The seller's put option value under the NMEU ambiguity specification can be written as (see the recursive structure/properties of the option value and derivation of the solution in Appendix A):

F1ðW1ðtÞÞ ¼ � A1λ

β1 1 S1ðtÞβ1 if S1ðtÞ > S*1

X � λ1S1ðtÞ if S1ðtÞ � S*1 (6)

where S*1 ¼ β1ðβ1�1Þλ1 X, A1 ¼ � ðλ1S*1Þ

1�β1

β1 , λ1 ¼ ρ1r�μ1�σ1κ1 þ

1�ρ1 r�μ1, β1 ¼

1 2 �

ζ1 σ21 �

53

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 �

ζ1 σ21

�2 þ 2r

σ21

s < 0, ζ1 ¼ ρ1ðμ1 þ σ1κ1Þ þ ð1 � ρ1Þμ1. A1, λ1 and β1 are

constants. Different from the option value derived by Moon et al. (2011) under

risk-neutrality, Eq. (6) accounts for the effect of ambiguity on decision making and shows how option exercising properties are affected by sellers' subjective beliefs λ1, probabilistic ambiguity κ1 and ambiguity aversion ρ1. Ignoring such behavioural effects can result in biased in- vestment triggers (e.g. premature commitment or late real option exer- cise) and suboptimal negotiation decisions if the negotiation process is characterized by vagueness and information incompleteness. S*1 is the critical trigger value of selling the good under ambiguity. The seller ex- ercises the put option only when costs S1ðtÞ � S*1.

The buying opportunity value under ambiguity F2ðtÞ is:

F2ðtÞ ¼ max t0�t

� ρ2

� inf θ22Θ2

EQ θ2 2 � ∫ ∞ t0 S2ðτÞe�rðτ�tÞdτjF t

� � e�rðt0�tÞX

� þ ð1 � ρ2Þ

� E � ∫ ∞ t0 ~S2ðτÞe�rðτ�tÞdτjF t

� � e�rðt0�tÞX

�� (7)

Using the same logic as above, the buyer's call option value under the NMEU specification can be written as:

F2ðW2ðtÞÞ ¼ � A2λ

β2 2 S2ðtÞβ2 if S2ðtÞ < S*2

λ2S2ðtÞ � X if S2ðtÞ � S*2 (8)

where S*2 ¼ β2ðβ2�1Þλ2 X, λ2 ¼ ρ2

r�μ2þσ2κ2 þ 1�ρ2 r�μ2, A2 ¼

ðλ2S*2Þ 1�β2

β2 , β2 ¼ 12 �

ζ2 σ22 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2 �

ζ2 σ22

�2 þ 2r

σ22

s > 1, ζ2 ¼ ρ2ðμ2 � σ2κ2Þ þ ð1 � ρ2Þμ2. S*2 is the critical

trigger value of buying the good under ambiguity. The buyer will exercise the call option only when revenues S2ðtÞ � S*2.

Otherwise, he will delay commitment until S2ðtÞ � S*2. Here again, ex- ercise conditions and option value are influenced by the buyer's subjec- tive beliefs λ2 and ambiguity parameters ρ2 and κ2. Ignoring the interaction effects of these variables on option value can result in erro- neous investment outcomes (e.g. premature exercise, late commitment or an impasse) if the negotiation is fraught with ambiguity. Option value and optimal exercise policies are affected by the buyer's ambiguity aversion and probabilistic ambiguity through individual and subjective factors β2 and λ2. Considering these cognitive factors in the uncertain price negotiation allows us to know how the worst case in the negotiator's mind affects her/his judgment about the timing of commitment and the likelihood of mutual agreement.

Using the above results (eqs. (6) and (8)), we next identify the implicit zone of achievable agreement (IZOAA) under ambiguity and its existence conditions, and study the effect of ambiguity aversion on the price negotiation range and the thresholds for mutual agreement. The IZOAA corresponds to the range of negotiation where buyers and sellers are likely to reach agreement and avoid impasse.

2.3. Ambiguity and the negotiation range

From eqs. (6) and (8), the seller (buyer) will agree to sell (buy) when S1ðtÞ � S*1 ¼ Xð1�1=β1Þλ1 (S2ðtÞ � S

* 2 ¼ Xð1�1=β2Þλ2 ). We refer to X1 and X2 as

implicit reservation prices for the negotiating seller and buyer. For given S1ðtÞ and S2ðtÞ:

X1 ¼ ð1 � 1=β1Þλ1S1ðtÞ � X (9)

X2 ¼ ð1 � 1=β2Þλ2S2ðtÞ � X (10) The optimal buying and selling strategies are to sell when X1 � X and

to buy when X2 � X. Thus, the region ½X1; X2� stands for the IZOAA or negotiation range under ambiguity for buyers and sellers. It nests the

Fig. 2. Effects of buyer's ambiguity on his implicit reservation price. Here (r,μ2, σ2,S2) ¼ (0.08, 0.04, 0.15, 25).

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

risk-neutral IZOPA found in prior studies. Setting implicit reservation prices can help initiate a profitable relationship between parties and capture their expectations of costs and revenues under ambiguity aver- sion. This aids in establishing whether the negotiation is successful or not. Given X1, a higher contract price X will generate higher benefits for the seller. For the buyer to make a profitable investment, X2 should exceed the contract price X. Proposition 1 summarizes how these implicit reservation prices are affected by changes in ambiguity aversion ρi and probabilistic ambiguity κi, where i ¼ 1; 2. Proposition 1. (The effect of ambiguity on implicit reservation prices)

An increase in the seller's ambiguity aversion ρ1 (probabilistic am- biguity κ1) increases her implicit reservation price X1 when κ1 > 0 ðρ1 > 0Þ. The effect of the buyer's ambiguity aversion ρ2 (probabilistic ambiguity κ2) on his implicit reservation price X2 is equivocal. If prob- abilistic ambiguity κi ¼ 0 (aversion ρi ¼ 0), changes in ρi ðκiÞ will not affect the implicit reservation price Xi, where i ¼ 1; 2. (See the proof in Appendix B).8>>>< >>>:

∂X1 ∂ρ1

> 0 if κ1 > 0

∂X1 ∂ρ1

¼ 0 if κ1 ¼ 0 ;

8>>< >>:

∂X1 ∂κ1

> 0 if ρ1 2 ð0; 1�

∂X1 ∂κ1

¼ 0 if ρ1 ¼ 0 :

∂X2 ∂ρ2

� < 0 and

∂X2ðκ2¼0Þ ∂ρ2

¼ 0; ∂X2∂κ2 � < 0 and

∂X2ðρ2¼0Þ ∂κ2

¼ 0. As shown in Fig. 1, Proposition 1 implies that in the presence of

ambiguity ðκ1 > 0Þ the seller will ask for a higher price (willingness-to- accept) for the good produced or to be delivered if she is more ambiguity averse about her future operating costs ð∂X1=∂ρ1 > 0Þ. This result is logical and in accord with risk aversion and maxmin dynamics. Similarly, a positive relationship exists between the seller's probabilistic ambiguity κ1 and her implicit reservation price X1 under increasing uncertainty aversion ðρ1 > 0Þ. This suggests that despite the presence of optionality, the IZOAA can become wider or narrower under ambiguity with changing aversion and that the risk-neutral IZOPA is likely to overstate mutual agreement prospects if the seller is increasingly pessimistic about her costs. While individual behaviour is key to negotiation outcomes, risk-neutral analysis ignores its effects on optionality and neglects the multiplicity of the IZOAA under ambiguity.

Proposition 1 states, on the other hand, that the buyer's ambiguity aversion ρ2 has an equivocal and non-monotonic effect on the implicit

Fig. 1. Effects of seller's ambiguity on her implicit reservation price. Here (r, μ1, σ1, S1) ¼ (0.08, 0.03, 0.15, 9).

54

reservation price X2 under probabilistic ambiguity κ2, as illustrated in Fig. 2. This can be explained by the specific properties of β2 in Eq. (8) and the potential interaction effects of r, μ2 and σ2 on optimal timing dy- namics. Such effects should be more pronounced under ambiguity. The resulting nonlinear association highlights the role of uncertainty (beyond just risk) on price negotiation outcomes and shows that the buyer's im- plicit price does not have to decrease with higher ambiguity aversion in the NMEU ambiguity specification. κ2 also has an equivocal effect on the buyer's implicit reservation price X2 under ambiguity aversion (i.e. when ρ2 is greater than zero). In other words, an optimistic buyer can poten- tially decrease his implicit reservation price and delay commitment in the presence of ambiguity later than a more pessimistic buyer. This implies that the IZOAA will not necessarily be narrower with higher ambiguity aversion from the buyer. This underlines the asymmetric effect of NMEU- based ambiguity on the negotiation's prospects. Under the standard MEU specification, buyer's willingness-to-pay (WTP) would be negatively related to ambiguity aversion as predicted by Hazen et al. (2012) in the context of remanufacturing. Changes in ambiguity aversion ρi (probabi- listic ambiguity κi) do not affect the implicit reservation prices Xi under risk-neutrality κi ¼ 0 ðρi ¼ 0Þ, where i ¼ 1; 2. The IZOPA is unique only in the absence of ambiguity.

We now turn to the effects of ambiguity aversion and ambiguity on the threshold for negotiation (i.e. joint options' exercise policy). Two parties involved in negotiation can reach agreement under ambiguity when the following condition is satisfied:

S2ðtÞ=S1ðtÞ � δKK (11)

where δKK ¼ ð1�1=β1Þλ1ð1�1=β2Þλ2, δKK denotes the ambiguity-based negotiation threshold of the ratio6 of the buyer's revenues ðS2ðtÞÞ to the seller's costs ðS1ðtÞÞ when call and put options are exercised. The threshold δKK rep- resents the minimum profit space for the negotiation to succeed and agreement to occur. Proposition 2 summarizes how this threshold is affected by changes in ambiguity aversion ρi and probabilistic ambiguity κi.

6 See Golan (2009) for an illustration of the negotiation threshold in employment contracts.

Fig. 4. Effects of buyer's ambiguity on the negotiation threshold δRK. Here ðr;μ1; σ1; μ2; σ2Þ ¼ ð0:08; 0:03; 0:15; 0:04; 0:15Þ.

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

Proposition 2. (The effect of ambiguity on the negotiation threshold) An increase in the seller's ambiguity aversion ρ1 (probabilistic am-

biguity κ1) increases the negotiation threshold δKK if κ1 > 0 ðρ1 > 0Þ. The effect of the buyer's ambiguity aversion ρ2 (probabilistic ambiguity κ2) on the negotiation threshold δKK is equivocal. When κi ¼ 0 ðρi ¼ 0Þ, changes in ρi ðκiÞ do not affect the negotiation threshold δKK, where i ¼ 1; 2. (See the proof in Appendix B).8>>>< >>>:

∂δKK ∂ρ1

> 0 if κ1 > 0

∂δKK ∂ρ1

¼ 0 if κ1 ¼ 0 ;

8>>< >>:

∂δKK ∂κ1

> 0 if ρ1 2 ð0; 1�

∂δKK ∂κ1

¼ 0 if ρ1 ¼ 0 :

∂δKK ∂ρ2

>¼ <

0 and ∂δKK ðκ2¼0Þ∂ρ2 ¼ 0; ∂δKK ∂κ2

>¼ <

0 and ∂δKK ðρ2¼0Þ∂κ2 ¼ 0.

Proposition 2 implies that under probabilistic ambiguity ðκ1 > 0Þ, the threshold for joint options' exercise will be higher the higher the seller's ambiguity aversion ð∂δKK=∂ρ1 > 0Þ. Fig. 3 illustrates this monotonic ef- fect. The positive association also holds between the seller's probabilistic ambiguity κ1 and the negotiation threshold δKK with higher uncertainty aversion. The negotiation process becomes more difficult if the seller is more pessimistic about her costs. On the other hand, and in line with the nonlinear effects highlighted in Proposition 1, higher ambiguity aversion from the buyer will not necessarily increase the negotiation threshold or likelihood of an impasse. This result differs from the one obtained by standard or uncertainty-neutral contingent-claims analysis and can be attributed to the asymmetric properties of the NMEU framework.

When κ1 ¼ 0 and κ2 ¼ 0, δKK reduces to the risk-neutral negotiation

threshold δRR ¼ ð1�1=β1RÞðr�μ2Þð1�1=β2RÞðr�μ1Þ, where β1R ¼ 1 2 �

μ1 σ21 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 �

μ1 σ21

�2 þ 2r

σ21

s < 0,

β2R ¼ 12 � μ2 σ22 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 �

μ2 σ22

�2 þ 2r

σ22

s > 1.

Let δKR denote the negotiation threshold with κ1 � 0 and κ2 ¼ 0 and δRK denote the negotiation threshold with κ1 ¼ 0 and κ2 � 0. The joint effects of ambiguity aversion ρi and ambiguity κi on these thresholds are illustrated in Figs. 3–4. Higher ambiguity aversion ρ1 (greater probabi- listic ambiguity κ1) from the seller results in a higher threshold δKR when κ1 > 0 ðρ1 > 0Þ as shown in Fig. 3. On the other hand, increasing aversion to uncertainty (greater probabilistic ambiguity) from the buyer does not induce a higher threshold δRK. Fig. 4 illustrates these equivocal effects.

Fig. 3. Effects of seller's ambiguity on the negotiation threshold δKR. Here ðr;μ1; σ1; μ2; σ2Þ ¼ ð0:08; 0:03; 0:15; 0:04; 0:15Þ.

55

The role played by r, μ and σ, and their interactions, in optimal timing dynamics are more important for call prospects than for puts under NMEU ambiguity. When κ1 ¼ 0 or ρ1 ¼ 0 (κ2 ¼ 0 or ρ2 ¼ 0), δKR ðδRKÞ is not affected by individual behaviour as shown in Figs. 3–4. Ignoring the interaction effects of ambiguity aversion and probabilistic ambiguity on real options dynamics in incomplete information settings, such as those of price negotiation, misses out the behavioural and subjective elements of buyer-seller interaction.

The relationship between the threshold under ambiguity δKK and its risk-neutral counterpart δRR can be expressed as follows:8>>>< >>>:

δKK � δRR if ð1 � 1=β1Þλ1 ð1 � 1=β2Þλ2

� ð1 � 1=β1RÞðr � μ2Þð1 � 1=β2RÞðr � μ1Þ

δKK < δRR if ð1 � 1=β1Þλ1 ð1 � 1=β2Þλ2

< ð1 � 1=β1RÞðr � μ2Þ ð1 � 1=β2RÞðr � μ1Þ

(12)

Eq. (12) highlights the role of subjective beliefs, in the form of the NMEU-based ambiguity multiplier, and ambiguity aversion in shaping mutual agreement. For agreement to be reached under ambiguity, the threshold will generally differ from δRR confirming that rational option pricing assumptions can lead to inflexible and suboptimal outcomes if individual behaviour, miscalibration and subjective beliefs are not accounted for in the uncertain negotiation. The above dynamics and propositions are based on the realistic setup that neither the buyer nor the seller is knowledgeable about other party's cost and revenue patterns or ambiguity parameters. We address this issue of asymmetric informa- tion in the presence of ambiguity aversion and its decision making im- plications in Section 4.3. Appendix C covers how the subjective probability of agreement is affected by ambiguity.

Overall the above results demonstrate that although optionality is a key element of negotiation when making purchases within supply net- works, the effects of ambiguity aversion and ambiguity on investment outcomes are equally important. Normative/rational predictions on the effects of uncertainty on implicit prices are indeed challenged when ambiguity, and aversion towards it, is part of the negotiation process. This is not surprising since psychology and the heterogeneity of indi- vidual attitudes towards uncertainty are known to significantly shape the direction of buyer-seller interaction (Ghosh, 1994; Moran and Ritov, 2002). Standard contingent-claims research on buyer-seller relationships and capital investment (e.g. Jiang et al., 2008; Driouchi et al., 2009;

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

Moon et al., 2011) neglects these so-called biases and frictional effects. We add realism to this literature by highlighting the moderating effects of ambiguity aversion and probabilistic ambiguity on real options dynamics and buyer-seller interaction. We find that under the NMEU ambiguity specification, uncertainty can have an asymmetric association with price negotiation outcomes. We also show that ambiguity aversion consistently influences negotiation performance when interacting parties, conscious of their options to delay commitment and negotiation agreement, are faced with ambiguity and information incompleteness.

While linear structures are useful for the analysis of bilateral nego- tiation and buyer-seller relationships, extant real options models omit to account for the relative positions and level of structural embeddedness of buyers and sellers in the supply chain network. As they influence both behaviour and strategy (Borgatti and Li, 2009; Kim et al., 2011), these factors should also play a role in negotiation dynamics. The next section adds a social network dimension to our negotiation problem under ambiguity.

3. The negotiation problem with network control and ambiguity

Building on the work of Braun and Gautschi (2006) on Nash bargai-

8>>>>< >>>>:

ξlj ¼ ln � νcj

lnðνclÞ þ ln � νcj if l and j negotiate in a negatively connected network

γlj ¼ ln � 1 � νcj

lnð1 � νclÞ þ ln

� 1 � νcj

if l and j negotiate in a positively connected network for l 6¼ j (15)

ning solutions in social networks, we account for the bargaining power of each party, based on their network position and relational features, in the uncertain negotiation. Consider an exogenous network, with the set of nodes ℕ ¼ f1; 2; ⋯; ng and m mutual ties in which the seller and buyer are embedded in. In this network, bargaining and exchange relations always coincide and negotiators have their own “network control”. In line with Braun and Gautschi (2006), we adopt the following assumption.

Assumption 4. The seller's (or buyer's) relative negotiation power re- sults from her (or his) network position in the network.7

The connectedness representation of the network is given by its N � N adjacency matrix А. The main diagonal elements of this matrix are equal to zero, i.e., all ¼ 0, l 2 ℕ. The relationship between members l and j is defined as follows: alj ¼ ajl ¼�

1 if there is a mutual tie between member l and j 0 otherwise

, where l 6¼ j and l; j 2 ℕ. The binary variable alj reflects whether l is connected with j. The normalized adjacency matrix А is denoted by the relational matrix R with main diagonal elements αll ¼ 0 for all l 2 ℕ. Its off-diagonal element αlj is derived as follows: αlj ¼ alj=

Pn k¼1akj for l; j; k 2 ℕ,

where αlj denotes l's level of “control” over j in the network and 0 � αlj � 1. The lth row of the matrix R reflects l's control over others. αlj ¼ 1ðαlj ¼ 0Þ means l has full (no) control over j.

To reflect how much power l has over other network members, the mean of l's control (i.e., l's network control level) over other parties in the

7 For tractability, we study relative bargaining power effects based on network position and using social network dynamics. It should be noted that horizontal competition and cooperation can also affect bargaining power (Sheu and Gao, 2014; Leider and Lovejoy, 2016). For example, when sellers make substitute (complementary) products, they might end up having a lower/higher bargaining power over the buyer. Buyers might also benefit from collective bargaining because of individual purchasing, sourcing or competition (Li, 2012; Heese, 2015). We thank an anonymous referee for this suggestion.

56

network is defined as:

cl ¼ 1 nl

Xn k¼1

αlk (13)

where nl denotes l's number of bargaining partners, cl is l's network control level. 0 < cl � 1.

We examine a network with negative connections in this section. The case with positive connections is covered in Section 4.2.8 The connection between the seller and buyer is negative (positive) if the buyer's exchange of resources with the seller precludes (promotes) transfers from (with) others (Yamaguchi, 1996). According to Binmore (1985) and Braun and Gautschi (2006), l's individual negotiation power (or market concentra- tion) ϖl can be defined as:

ϖl ¼ �

�1=lnðνclÞ if l faces a negatively connected relation �1=lnð1 � νclÞ if l faces a positively connected relation (14)

Then l's relative bargaining power vis-�a-vis j in the network can be calculated by ϖl

ϖlþϖj.

From Eq. (14), l's relative bargaining power can be expressed as:

where ν ¼ mþn1þmþn, ν reflects the network-specific weight based on existing nodes and mutual ties.

Equation (15) shows that equal network control levels result in

similar negotiation power in each network. Also note that ∂ξlj ∂cl

> 0 and ∂γlj ∂cl

< 0. This indicates that l's relative bargaining power vis-�a-vis j in-

creases (decreases) with l's network control in negatively (positively) connected networks if ν is unchanged .

Following Section 2, let i ¼ 1 and 2 denote our seller and buyer. The seller's relative negotiation power ξ12 is defined based on her relational features and structural position in the network. Fig. 5 illustrates examples of typical structures in which the seller and buyer might be embedded in (see Braun and Gautschi, 2006). For instance, the seller's network control level is 1 (1/3) in a 3-branch (stem) network indicating that she has complete (less) control over the buyer in this type of structure.

As shown by Braun and Gautschi (2006), the bargaining problem with network control between the seller and buyer can be written as: IðtÞ ¼ max X

ðλ2S2ðtÞ � XÞ1�ξ12 ðX � λ1S1ðtÞÞξ12 . Taking the first order of log IðtÞ with respect to X, the contract price is determined as follows:

X ¼ ð1 � ξ12Þλ1S1ðtÞ þ ξ12λ2S2ðtÞ (16) Equation (16) shows that the contract price with network control is

equal to the weighted sum of the subjective values of S1ðtÞ and S2ðtÞ in the non-extreme maxmin expected utility (NMEU) framework.

For a given negotiation power, the buyer's option value is a function of his network control level and ambiguity. The buyer's timing option can be rewritten as:

8 Similar to Braun and Gautschi (2006), we examine networks with either negative or positive relations. Negative (positive) connections are viewed as substitutable (complementary) (see e.g. Yamaguchi (2000) for mixed exchange networks with negative and positive connections).

Fig. 5. Illustration of network control levels and relative negotiation powers in negatively connected networks.

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

UðS1ðtÞ; S2ðtÞÞ ¼ ð1 � ξ12Þmax t0�t

e�rðt 0�tÞðλ2S2ðt0Þ � λ1S1ðt0ÞÞ; 0 (17)

� � For tractability, let z ¼ S2ðtÞ=S1ðtÞ. z is the ratio of buyer's revenues

S2ðtÞ to seller's costs S1ðtÞ when put and call options are exercised under ambiguity. The buyer's call option value with social network effects and the corresponding optimal time to purchase can be derived as (see the proof in Supplementary Appendix E):

UðS1ðtÞ; S2ðtÞÞ ¼ � ð1 � ξ12Þðλ2S2ðtÞ � λ1S1ðtÞÞ if S2ðtÞ=S1ðtÞ � z*

d2ðS2ðtÞÞbðS1ðtÞÞ1�b if S2ðtÞ=S1ðtÞ < z* (18)

where z* ¼ δ2b δ1ðb�1Þ ¼

λ1b λ2ðb�1Þ, δ1 ¼ 1=λ1, δ2 ¼ 1=λ2, λ1 ¼

ρ1 r�μ1�σ1κ1 þ

1�ρ1 r�μ1,

λ2 ¼ ρ2r�μ2þσ2κ2 þ 1�ρ2 r�μ2, δ1 and δ2 are the convenience yields of the seller and

buyer, d2 ¼ ð1�ξ12Þðz *Þ1�b

δ2b , b ¼ 12 � δ1�δ2σ22�2εσ1σ2þσ21 þ χ > 1, χ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2 � δ1�δ2σ22�2εσ1σ2þσ21

�2 þ 2δ1

σ22�2εσ1σ2þσ21

s .

The corresponding put option value with social network effects for the seller is:

RðS1ðtÞ; S2ðtÞÞ ¼ � ξ12ðλ2S2ðtÞ � λ1S1ðtÞÞ if S2ðtÞ=S1ðtÞ � z*1 d1ðS2ðtÞÞbðS1ðtÞÞ1�b if S2ðtÞ=S1ðtÞ < z*1

(19)

where z*1 ¼ z* ¼ δ2bδ1ðb�1Þ, d1 ¼ ξ12ðz*Þ1�b

δ2b .

The ratio of buyer revenues S2ðtÞ to seller costs S1ðtÞ, z*1 (that is equal to z*) denotes the profit space threshold with network control. Though affected by ambiguity aversion, this behavioural threshold does not seem to account for social network effects. When z is less than z*, the total option value is too low for cooperation or mutual agreement to occur. When z is larger than the threshold z*, it is worth cooperating.

Propositions 3a–b consider the joint effects of ambiguity aversion and relative bargaining power, in terms of relationship characteristics and network position, on implicit reserve prices Xnc1 ðtÞ and Xnc2 ðtÞ. These variables are more likely to be influenced by social network effects than the threshold.

Proposition 3a. (Implicit reservation prices with network control in negatively connected networks)

When the seller and buyer determine the negotiated share of coop- erative profits under ambiguity, their reservation prices with social network effects are:

Xnc1 ðtÞ¼ � b�1þ lnðνc2Þlnðνc1Þþlnðνc2Þ

� λ1S1ðtÞ b�1 and X

nc 2 ðtÞ¼

b� 1þ lnðνc2Þlnðνc1Þþlnðνc2Þ

� λ2S2ðtÞ

b .

Proposition 3a indicates that ambiguity aversion ρi and probabilistic ambiguity κi still affect the buyer's and seller's implicit reservation prices Xnci ðtÞ in the presence of social network effects. This is achieved through the option value parameter b and ambiguity multiplier λi. We also find that network control level ci and network scale m and n influence price negotiation outcomes under ambiguity. This is in accord with social network theory predictions and leads to Proposition 3b.

57

Proposition 3b. (Effect of ambiguity on implicit reservation prices in negatively connected networks)

The seller's implicit reservation considering network control Xnc1 is increasing in her ambiguity aversion ρ1 (probabilistic ambiguity κ1). The buyer's implicit reservation price considering network control Xnc2 is decreasing in his ambiguity aversion ρ2 (probabilistic ambiguity κ2). (See the proof in Supplementary Appendix F).

∂Xnc1 ðtÞ ∂ρ1

� 0; ∂X nc 1 ðtÞ ∂κ1

� 0; ∂X nc 2 ðtÞ ∂ρ2

� 0; ∂X nc 2 ðtÞ ∂κ2

� 0:

Recall: Xnc1 ðtÞ ¼ ðb � 1 þ ξ12Þ λ1S1ðtÞb�1 and Xnc2 ðtÞ ¼ ðb � 1 þ ξ12Þ λ2S2ðtÞ

b . Proposition 3b confirms that while ambiguity aversion ρi and prob-

abilistic ambiguity κi still affect the buyer's and seller's implicit reserva- tion prices Xnci ðtÞ in the presence of social network effects, network control level ci and network scale m and n moderate the effects of am- biguity (aversion) on implicit negotiation prices, making the relation- ships between them unequivocally monotonic for both buyers and sellers. Ambiguity aversion is, hence, negatively (or positively) related to buyer implicit prices for willingness-to-pay (or willingness-to-accept) outcomes when network positions are known (the WTP finding is in line with Hazen et al. (2012) and their Hypothesis 1). This is different from the asymmetric finding without network control of Proposition 2.

For illustration, let us assume that the seller and buyer are in the 3- Branch, Kite and Stem network structures introduced above. Their spe- cific positions are shown in Fig. 5. Figs. 6–7 highlight the effects of ambiguity aversion, probabilistic ambiguity and social network positions on price negotiation. Fig. 6 shows that the seller's implicit reservation price with network control Xnc1 ðtÞ increases as her ambiguity aversion ρ1 or probabilistic ambiguity κ1 rises. This positive relationship holds in all three network structures. This is consistent with our findings without social network effects (i.e. Proposition 1). We additionally observe that higher relative bargaining power for the seller is associated with even higher reservation prices in all three network structures.

On the other hand, a buyer with higher ambiguity aversion or higher probabilistic ambiguity would unequivocally decrease his implicit reservation price in the presence of bargaining power and network control as shown in Fig. 7. This is different from the equivocal and nonlinear effects observed in Section 2 for the buyer. This implies that familiarity with the social network structure and understanding of rela- tive bargaining powers provide information advantages to the buyer. The latter can use this information to decide his implicit reservation price unequivocally. The effect of ambiguity aversion becomes akin to that of risk aversion and maxmin MEU ambiguity when social network dynamics are accounted for. In other words, the asymmetric effect of NMEU am- biguity on the buyer's implicit price disappears in the presence of network control. This is explained by the profit sharing-based properties of eqs. (18) and (19), and by the dominant-negative effect of negotiation power on implicit prices. We indeed observe that higher relative bar- gaining power for the buyer is associated with lower reservation prices in all three network structures. This means that the narrowness of the

Fig. 6. Effects of seller's ambiguity on her implicit reservation price with network control. Here S1ðtÞ ¼ 9, r ¼ 0.08, μ1 ¼ 0.03, μ2 ¼ 0.04, ε ¼ 0:1, σ1 ¼ σ2 ¼ 0.15, ρ2 ¼ 0:5, κ2 ¼ 0:2. In Fig. 6.a, κ1 ¼ 0:2. In Fig. 6.b, ρ1 ¼ 0:5.

Fig. 7. Effects of buyer's ambiguity on his implicit reservation price with network control. Here S2ðtÞ ¼ 25, ρ1 ¼ 0:5, κ1 ¼ 0:2. In Fig. 7a, κ2 ¼ 0:2. In Fig. 7b, ρ2 ¼ 0:5. Other parameter values are the same as in Fig. 6.

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

IZOAA also depends on network structures. Adding new links or nodes to the network structure usually changes

the network control levels and the network-specific weights. This in- creases the difficulty of studying their comparative statics analytically. We observed that higher network control levels tend to increase relative negotiation powers and can create pricing advantages in negatively connected networks (see Figs. 5–7). Consider a supply chain network with n � 4 tiers (e.g. Waters, 2009). Suppose the tier number only affects the seller and buyer through their relative negotiation powers. If the seller and buyer are two of the most upstream entities in the network (see Fig. 8a), their network control levels stay unchanged at c1 ¼ 0:5 and c2 ¼ 0:75. Thus,

∂Xnc1 ∂ν < 0,

∂Xnc2 ∂ν < 0. In this most upstream case, the wholesaler's

relative bargaining power increases because of the addition of an inter- mediary (branch) in the network. Consequently, both the wholesaler (buyer) and the manufacturer (seller) decrease their implicit reservation prices. This is as if, due to a loss in relative bargaining power, the manufacturer is less ambiguity averse in this new structure.

Using the same logic as above, ∂Xnc1 ∂ν > 0,

∂Xnc2 ∂ν > 0 when the seller and

58

buyer are two of the most downstream entities (see Fig. 8b). Adding an intermediary in the network will lead to a higher (lower) relative bar- gaining power for the retailer (customer). Consequently, both the seller (manufacturing retailer) and buyer (customer) will increase their implicit reservation prices. This is as if the customer is relatively more ambiguity- seeking in this new structure. In the two cases, increasing network nodes and mutual ties strengthens the relative advantages of entities with higher network control levels despite the presence of ambiguity. Social network information might thus help to resolve some of the unknown uncertainty characterising the negotiation process.

4. Extensions and additional results

This section extends our previous modelling insights by considering outside options in the negotiation process (Section 4.1), sequential negotiation between one buyer and multiple sellers (Section 4.2), and mechanism and incentives design in the presence of asymmetric infor- mation (Section 4.3).

Fig. 8. Example of a narrow supply chain.

9 An alternative to this would be to calculate relative bargaining power by considering revoking commitment at a certain cost (see e.g. Muthoo, 1996; Nagarajan and Bassok, 2008) so that the assumption of zero disagreement values becomes less restrictive.

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

4.1. Ambiguity and the outside option

In Sections 2–3, we assumed that both the seller and buyer accept the price negotiation outcomes and commit to the contract if the optimal timing threshold is attained. However, the seller/buyer can also decide to exercise their outside options and exit the negotiation altogether. Here the outside option is viewed as the best alternative that a negotiator can go for if he or she withdraws unilaterally from the negotiation process (Binmore et al., 1986). The existence of outside options introduces new constraints to the problem as ambiguity also affects outside values (see e.g. Miao and Wang, 2011).

We adopt the outside option valuation model of Schr€oder (2011) and consider outside options for both sellers and buyers. Let the seller's (buyer's) outside option value V1ðV2Þ be random and follow a normal distribution N ðu1;bσ1ÞðN ðu2;bσ2ÞÞ. The set of likelihood distributions pi 2 ℛiðui; bσiÞ ¼ fN ðui; bσ iÞjui 2 ½ui � yi; ui þ yi�g is defined to capture ambi- guity in i's outside option value. For simplicity, we assume these to be independent of the seller's costs and buyer's revenues Si, where yi > 0, i ¼ 1; 2. We consider ambiguity in the mean of ui rather than the variance bσ i. The scope of the mean ui 2 ½ui � yi; ui þ yi� is defined based on κ-igno- rance in continuous-time and ε-contamination (e.g. Nishimura and Ozaki, 2007; Kopylov, 2016), where the ambiguity level yi reflects how confident i is in his/her probabilistic measure. Suppose the seller's (buyer's) ambiguity aversion ρ1ðρ2Þ is a trait that influences investment execution and outside option exercise. Then, the NMEU value of Vi can be written as: NMEUðViÞ ¼ ρi infpi2ℛiE

pi ½Vi� þ ð1 � ρiÞE½Vi� ¼ ui � ρiyi, where i ¼ 1; 2.

The NMEU version of outside option value Vi differs from the outside option value in Schr€oder (2011) by considering the mean ui , thus reflecting the influence of non-extreme prospects. NMEUðViÞ decreases with ambiguity aversion ρi and ambiguity level yi. When the ratio of buyer's revenues ðS2ðtÞÞ to seller's costs ðS1ðtÞÞ reaches z*, the seller and buyer maximize their utilities as follows:

bRðS1ðtÞ; S2ðtÞÞ ¼ maxfξ12ðλ2S2ðtÞ � λ1S1ðtÞÞ; NMEUðV1Þg (20) bUðS1ðtÞ; S2ðtÞÞ ¼ maxfð1 � ξ12Þðλ2S2ðtÞ � λ1S1ðtÞÞ; NMEUðV2Þg (21)

Eqs. (20) and (21) indicate that negotiation agreement is reached if ξ 12

� ξ12 � ξ12, where ξ12 ¼ NMEUðV1Þ

λ2S2ðtÞ�λ1S1ðtÞ, ξ12 ¼ 1 � NMEUðV2Þ

λ2S2ðtÞ�λ1S1ðtÞ. This

means the seller (buyer) will commit to the contract only if her (or his) negotiation power is not too low and the profit allocation policy is acceptable. Otherwise, the seller (or buyer) will opt out and exit the

59

negotiation. To ensure ξ12 > ξ12, the profit space λ2S2ðtÞ � λ1S1ðtÞ should be strictly greater than the sum of outside option values NMEUðV1Þ þ NMEUðV2Þ. Although relative bargaining power ξ12 does not directly affect optimal investment timing, it does determine whether both parties should proceed with the contract when considering their outside options.

4.2. Sequential negotiation between one buyer and several suppliers

We next extend our bilateral negotiation problem to sequential multilateral negotiation situations involving one buyer and comple- mentary sellers. Several papers have examined cases of suppliers or a group of sellers supplying complementary components to a downstream firm (e.g. Nagarajan and Bassok, 2008; Nagarajan and So�si�c, 2008; Granot and Yin, 2008; He and Yin, 2015). Herein, we incorporate am- biguity and social network dynamics in the negotiation framework of Nagarajan and Bassok (2008) and relax the fixed channel profit assumption characterising their sequential negotiation. As before, we account for ambiguity in the sellers' costs and buyer's revenues and their respective network position features. This enables us to examine the ef- fect(s) of ambiguity aversion (and number of sellers) on profit allocation.

The buyer can be viewed as an assembler who buys one unit of complementary component from each seller and manufactures the final product. They are in a positively connected network in the sense that a deal between the buyer and one seller encourages the former to trade with other sellers. We consider a supply chain network with n nodes consisting of a buyer and n � 1 sellers, where n � 2. The buyer negotiates with the hth seller at stage h using Nash bargaining solutions, where h ¼ 1; ⋯; n � 1. From Eq. (15), γ ¼ γ12 denotes each seller's relative bar- gaining power in a positively connected network.9 The buyer's relative bargaining power is η ¼ 1 � γ.

The negotiation sequence is determined by the buyer. Each seller has her own subjective costs' expectations. The hth seller's NMEU-based

ambiguity multiplier is λh1 ¼ ρh1

r�μh1�σh1κh1 þ 1�ρ

h 1

r�μh1 > 0. We add the super-

script h to denote the hth seller. The total expected profit ΠðtÞ based on the NMEU specification can be

written as: ΠðtÞ ¼ λ2S2ðtÞ � Pn�1

h¼1λ h 1S

h 1ðtÞ, where ΠðtÞ > 0. To simplify

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

notation, we omit timing t from ΠðtÞ. Let Πh denote the total expected profit to be shared between the buyer and sellers h; h þ 1⋯n � 1, where Π1 ¼ Π. After n � 1 stages of negotiation, the buyer achieves his ex- pected profit πB ¼ Πn. Let Ψh denote the set of feasible alternatives when the buyer negotiates with the hth seller at stage h. We use πh to represent the level of profit that the hth seller achieves. In the first stage, the feasible set is defined by Ψ1 ¼ fðΠ2; π1Þ : Π2 þ π1 ¼ Π g. At stage h, Ψh ¼ fðΠhþ1; πhÞ : Πhþ1 þ πh ¼ Πh g. Then, the profit allocation rule between the buyer and the hth seller is determined through the gener- alized Nash bargaining solution: maxðΠhþ1;πhÞ2Ψh ðΠhþ1Þ1�γðπhÞγ. In line with Section 3, disagreement values of the buyer and sellers are assumed to be zero. The total surplus is split as follows: πh ¼ γΠh, Πhþ1 ¼ ð1 � γÞΠh. The profit distribution between the buyer and sellers is, thus, ob- tained via backward induction. The buyer's expected profit πB and the hth seller's expected profit πh can be written as:

πB ¼ ð1 � γÞn�1 λ2S2ðtÞ �

Xn�1 h¼1

λh1S h 1ðtÞ !

(22)

πh ¼ γð1 � γÞh�1 λ2S2ðtÞ �

Xn�1 h¼1

λh1S h 1ðtÞ !

(23)

Eqs. (22) and (23) illustrate the distribution of profits based on the supply chain member number, network position and ambiguity in sellers' costs and buyer's revenues. Compared with the optimal profit allocation in Nagarajan and Bassok (2008), our solutions consider the role of probabilistic ambiguity and ambiguity aversion in the negotiation. These solutions also add to recent real options literature on price negotiation (e.g. Moon et al., 2011; Zheng and Negenborn, 2015). It is intuitive to see that both the buyer's and sellers' profits decrease with revenues- and costs-related ambiguity aversion.

From Eqs. (22) and (23), the “procurement” price for the hth seller ~Xh and the total price paid by the buyer ~XB can be, respectively, expressed

as: ~Xh ¼ λh1Sh1ðtÞ þ γð1 � γÞh�1ΠðtÞ, ~XB ¼ λ2S2ðtÞ � ð1 � γÞn�1ΠðtÞ. The buyer will commit to higher prices if his ambiguity aversion is lower. The seller would ask for a higher price if her ambiguity aversion is higher.

Regarding the effects of multiple sellers, we follow Nagarajan and Bassok (2008) and discuss fixed versus adjustable negotiation sequences. In the fixed case, the buyer prefers fewer sellers as proved by Nagarajan and Bassok (2008) in their Theorem 4.1. We confirm this finding when considering network features. Note the buyer's expected profit πB is determined by ΠðtÞ and ð1 � γÞn�1. A smaller number of sellers increases the total expected profit ΠðtÞ and the value of ð1 � γÞn�1. For example, the term ð1 � γÞn�1 equals 0.25 in a Triangle network, while it amounts to 0.125 in a Full-4 network in Fig. 9. Consequently, the buyer benefits from a smaller number of sellers if the negotiation sequence is fixed. In the presence of a predefined negotiation sequence, the seller's expected profit decreases with her sequence h. For example, the first and second sellers'

Fig. 9. Examples of supply chains with a single buyer (assembler) and multiple suppliers (sellers).

60

expected profits are, respectively, 0:5ΠðtÞ and 0:25ΠðtÞ in the Triangular network.

If the negotiation sequence is adjustable, the buyer might encourage the sellers to pay for network positions in order to gain more profit share. Nagarajan and Bassok (2008) prove, in their Theorem 4.2, that when the sellers simultaneously compete for negotiation sequence and pay for their network positions10, at every Nash equilibrium, the expected profit of each seller equals πn�1 while the buyer's expected profit is ð1 � γð1 � γÞn�2ðn � 1ÞÞΠðtÞ. In such a setting, the buyer prefers to have more sellers (see Theorem 4.2 in Nagarajan and Bassok (2008)). We find that due to social network effects, the buyer does not necessarily benefit from a higher number of sellers. This is because a greater number of sellers decreases the total expected profit ΠðtÞ but might at the same time in- crease the 1 � γð1 � γÞn�2ðn � 1Þ term. For example, 1 � γð1 � γÞn�2ðn � 1Þ equals 0.5 in the Triangle network, while it is 0.625 in the Full-4 network in Fig. 9. As we study negotiation power from a social network perspective, the relationship between the number of sellers and the buyer's expected profits becomes equivocal if sellers compete and pay for negotiation position.

4.3. Asymmetric information and price negotiation under ambiguity

In the previous sections, we analysed investment timing and pricing decisions under ambiguity assuming that information was symmetric. However, information asymmetry is also known to influence buyer-seller interactions and their related transactional arrangements. There has been increasing interest surrounding issues of ambiguity aversion, asymmetric information and mechanism design in recent years (see Bodoh-Creed, 2012; Bose and Renou, 2014; Vierø, 2014; Wolitzky, 2016; Giraud and Thomas, 2017). We borrow from this literature to examine how asym- metric information affects our optimal timing and price negotiation outcomes under ambiguity. Our modelling builds on a rich and still growing stream of research on real options under incomplete information (e.g. Nishihara and Shibata, 2008; Shibata and Nishihara, 2011; Feng et al., 2014; Grenadier et al., 2016). We add to these studies by ac- counting for ambiguity and each party's private information about their own ambiguity aversion parameter ρi (and option value parameter b) in the negotiation.

Consider the buyer (principal) delegates the investment timing de- cision to the seller (agent) and determines the price contingent on the observable timing threshold. There are two types of sellers in the market in terms of their knowledge of their ambiguity aversion parameter ρi. We call the seller a high (in contrast to a low) type if her ambiguity aversion is ρ1H ðρ1LÞ with ρ1H < ρ1L. This means λ1H < λ1L. The probability of any seller belonging to the high type category is q1.

Let λ1w and bw denote w's NMEU-based ambiguity multiplier and option value parameter, where: bw ¼ 12 � δ1w�δ2σ22�2εσ1σ2þσ21 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2 � δ1w�δ2σ22�2εσ1σ2þσ21

�2 þ 2δ1w

σ22�2εσ1σ2þσ21

s , δ1w ¼ 1=λ1w, λ1w ¼ ρ1wr�μ1�σ1κ1 þ

1�ρ1w r�μ1 ,

w ¼ H or L denotes the high or low type, δ2 ¼ 1=λ2, λ2 ¼ ρ2r�μ2þσ2κ2 þ 1�ρ2 r�μ2.

Thus, the high (low) type seller has her own private information about the NMEU-based ambiguity multiplier and option value parameter. In line with Section 3, let zH ðzLÞ represent the ratio of buyer's revenues S2ðtÞ to seller's costs S1ðtÞ when the high (or low) type seller undertakes the contract. Note ðzðtÞ=zwÞbw is akin to a discount function (Grenadier and Wang, 2005; Feng et al., 2014). Assume zðtÞ < zw indicates that the contract is not implemented immediately. Since λ1w and bw are the seller's private information, the buyer's objective is to maximize his option value by observing investment timing zw and buying the product or service at price Xw:

10 Suppliers' coalitions and their stability are also discussed in Nagarajan and Bassok (2008).

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

max z ;z ;X ;X

qH

zðtÞ z

�bH ðλ2zHS1ðtÞ � XHÞ þ ð1 � qHÞ

zðtÞ z

�bL ðλ2zLS1ðtÞ � XLÞ

H L H L H L

(24)

subject to:

zðtÞ zH

�bH ðXH � λ1HS1ðtÞÞ �

zðtÞ zL

�bH ðXL � λ1HS1ðtÞÞ (25)

zðtÞ zL

�bL ðXL � λ1LS1ðtÞÞ �

zðtÞ zH

�bL ðXH � λ1LS1ðtÞÞ (26)

zðtÞ zH

�bH ðXH � λ1HS1ðtÞÞ � 0 (27)

zðtÞ zL

�bL ðXL � λ1LS1ðtÞÞ � 0 (28)

The terms λ2zHS1ðtÞ and λ2zLS1ðtÞ are the buyer's expected revenues if the seller belongs to the high type and low type categories, respectively. Constraints (25) and (26) mean that the high type seller is encouraged to undertake the contract at timing zH and the low type seller is induced to undertake the contract at timing zL. Constraints (27) and (28) are the participation constraints.

Solving this principal–agent problem and assuming z*L > bLλ1L

ðbL�1Þλ2 (see

proofs in Appendix D), we find that when information about the seller's pessimism and ambiguity aversion is asymmetric, the buyer's optimal policy is as follows:

ðz*H; X*H; z*L; X*LÞ ¼

bHλ1H ðbH�1Þλ2;

" λ1H þ

z*H z*L

�bH ðλ1L � λ1HÞ

# S1ðtÞ; z*L;

λ1LS1ðtÞ ! , where z*L is a solution to ð1 � qHÞzbH�bLL ðbLλ1L þ ð1 � bLÞλ2zLÞ þ

qHbHzðtÞbH�bL ðλ1L � λ1HÞ ¼ 0. This shows there is a similar functional form between the timing

trigger of the high type z*H and the timing trigger under symmetric in- formation z* in Eq. (18). Extant real options research (e.g. Nishihara and Shibata, 2008; Feng et al., 2014) documents that investment will usually

Fig. 10. Timing thresholds and prices under ambiguity and asymmetric information. same as in Fig. 6.

61

be deferred if managers belong to the low type category. We confirm this, in our buyer-seller and price negotiation setting, in the presence of asymmetric information concerning option value parameters and the degree of ambiguity. This is further illustrated in Fig. 10.a where timing threshold z*H is smaller than z

* L. We find an optimal incentives policy

exists only if z*L > bLλ1L

ðbL�1Þλ2. This implies that under ambiguity, the buyer

can implement the incentive contract only if the low type seller's timing threshold is relatively high. When probabilistic ambiguity is nil, ambi- guity aversion does not affect the seller's costs and there is a unique timing threshold (implicit price) as shown in Fig. 10.a (Fig. 10b). Let C1H denote the high type seller's expected costs where C1H ¼ λ1HS1ðtÞ. Her expected profits are shown in the grey area of Fig. 10.b. These profits are determined by the costs difference between the high and low type sellers

ðλ1L � λ1HÞS1ðtÞ and the portion

z*H z*L

�bH < 1. This means the buyer covers

part of the costs difference to encourage the high type seller to tell the truth. The incentives portion is, hence, affected by the ambiguity aver- sion of the buyer and that of each type of seller.

The seller can also act as a principal to induce the buyer to report his true type (good vs. bad). The good buyer has lower ambiguity aversion about his revenues than the bad buyer (see Appendix D). We, once again, find the functional form of the timing trigger for the good buyer z*G to be similar to the symmetric z*. z*G is also smaller than z

* B. The good buyer's

expected profits are determined by revenues difference between the two

types of buyers and the portion

z*G z*B

�bG < 1, where bG is the good buyer's

option value parameter. In the two above principal-agent cases, the timing trigger for the high

(good) type agent increases with her (his) ambiguity aversion about costs (revenues) and the principal's ambiguity aversion about revenues (costs). The timing trigger for the low type (bad) agent and implicit price of the high type (good) agent are nonlinear functions of the ambiguity aversion of the principal and that of each type of agent. The implicit price X*L of the low type seller increases with her pessimism about costs ρ1L. On the other hand, the implicit price X*B of the bad buyer is a nonlinear function of his optimism about revenues ρ2B since ρ2B affects X

* B through the NMEU-

based ambiguity multiplier λ2B and timing threshold z*B. The principal offers zero profit to the low (bad) type agent and positive profits to the

zðtÞ ¼ 1:3, qH ¼ 0:5, ρ1H ¼ 0:2, ρ1L ¼ 0:9. The other parameter values are the

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

high (good) type agent. These positive profits are contingent on the ambiguity aversion of the principal and that of each type of agent.

5. Conclusions

Contributing to behavioural operations and production management research on buyer-seller relationships (e.g. Esmaeilia and Zeephongsekul, 2010; Hazen et al., 2012; Hemmert et al., 2016), this paper examines the real options and social network dynamics of bilateral (and multilateral) price negotiation under ambiguity by relying on a multiple-priors utility with non-extreme outcomes. Adjusting for uncertainty aversion in probabilistic appraisal, this utility combines the worst case in negotiators' minds with the risk-neutral case and provides flexible commitment thresholds for investment under ambiguity aversion. Besides extending risk-neutral insights from recent contingent-claims research, our results underline the moderating effects of individual behaviour and mis- calibration on the process of price negotiation and its performance. We find that ambiguity aversion and ambiguity do not necessarily have symmetric effects on pricing outcomes. Specifically, an increase in seller's ambiguity aversion increases her implicit reservation price and negoti- ation threshold with and without network control. On the other hand, the buyer's ambiguity aversion affects his implicit reservation price and the threshold for negotiation (un)equivocally in the absence (presence) of social network effects. The seller's (buyer's) probabilistic ambiguity af- fects her (his) implicit reservation price and negotiation threshold in a similar direction as her (or his) ambiguity aversion. This is because ambiguity aversion and probabilistic ambiguity dominate the influence of the worst case heuristic on decision making in the same direction. We confirm that standard option analysis with a single prior can lead to restrictive pricing outcomes and might overstate mutual agreement prospects and the range for negotiation. We, additionally, show that knowledge of network positions and other social network effects still play an important role in negotiation performance in the presence of ambi- guity. We also explore the case of one buyer and multiple sellers, examine the role of outside options, and consider the effect of information asymmetry in the various dynamics.

In terms of operations and production economics implications, our proposed real options frameworks provide quantitative insights into how ambiguity aversion and social network effects influence the range of negotiation agreement between buyers and sellers, and help formalise - using real options theory - recent predictions by Hazen et al. (2012) on the role of ambiguity tolerance (and perceived quality) in the decision to purchase remanufactured products. We add to this literature by

62

examining willingness-to-pay (WTP) and willingness-to-accept (WTA) decisions jointly and highlight the effect of social networks on the rela- tionship between ambiguity aversion and price negotiation outcomes in the context of B2B situations.

Our results also help inform how probabilistic ambiguity and pessi- mism (or other attitudes towards uncertainty) generally affect negotia- tors' behaviour, real options payoffs and investment outcomes in buyer- seller relationships, social network structures and other practical deci- sion making situations. By knowing their network positions and gath- ering background information or inferring the other party's ambiguity tolerance beforehand via cheap talk, buyers and sellers can anticipate where the negotiation is heading in terms of price negotiation range and mutual agreement possibilities despite the presence of ambiguity. This is especially useful for international operations and price negotiation situ- ations that involve buyers and sellers from different countries. Knowing the cultural characteristics of a country, including its degree of uncer- tainty avoidance (e.g. Hofstede, 2001), can help international managers identify suppliers and customers who might be more uncertainty-seeking (averse) in the international network or else plan, in a contingent-manner and considering relative bargaining powers, for potentially lengthy and difficult negotiations. Extensions of this work could consider further game-theoretic interactions, quantity/quality dynamics and account for the effects of second moment ‘uncertainty’, learning, horizontal compe- tition and cooperation on price negotiation outcomes and mutual agreement. Validating our frameworks using experimental principles can also provide interesting evidence on the emotional and perhaps irrational traits of price negotiation and highlight extra factors which could influ- ence buyer-seller decision making in the presence of ambiguity and social network effects.

Acknowledgments

We are grateful to the Editor, Professor Bart MacCarthy and two anonymous referees for their constructive and insightful comments and suggestions. Thanks are also due to Lenos Trigeorgis, Richard Arnott and Alex Preda for helpful feedback and suggestions. Yongling Gao would like to acknowledge the support of the National Natural Science Foun- dation of China (71201177; 71774182), Humanities and Social Science Research Project of Ministry of Education of China (17YJC630027) and Beijing Higher Education Young Elite Teacher Project (YETP0967). An early version of this paper was presented at the 16th International Annual Conference on Real Options in London.

Appendix A. Derivation of the seller's put option value under the NMEU

The selling opportunity value F1ðt þ dtÞ under ambiguity can be expressed as:

F1ðt þ dtÞ ¼ max t0�tþdt

� ρ1Gðt0Þ þ ð1 � ρ1ÞE

� ~Gðt0ÞjF tþdt

�� (A.1)

where Gðt0Þ ¼ e�rðt0�t�dtÞX � sup θ012Θ1

EQ θ01 1 ½∫ ∞t0 S1ðτÞe�rðτ�t�dtÞdτjF tþdt�, ~Gðt0Þ ¼ e�rðt

0�t�dtÞX � ∫ ∞t0 ~S1ðτÞe�rðτ�t�dtÞdτ.

The multiple-priors expected utility with non-extreme outcomes of F1ðt þ dtÞ is:

NMEU � e�rdtF1ðt þ dtÞjF t

¼ e�rdt max

t0�tþdt

( ρ1e

�rðt0�t�dtÞX � ρ1 sup θ12Θ1

EQ θ1 1

" ρ1 sup

θ012Θ1 EQ

θ0 1 1 � ∫ ∞ t0 S1ðτÞe�rðτ�t�dtÞdτjF tþdt

� þ ð1 � ρ1ÞE

� ∫ ∞ t0 ~S1ðτÞe�rðτ�t�dtÞdτjF tþdt

� jF t #

þ ð1 � ρ1Þe�rðt 0�t�dtÞX � ð1 � ρ1ÞE

" ρ1 sup

θ012Θ1 EQ

θ0 1 1 � ∫ ∞ t0 S1ðτÞe�rðτ�t�dtÞdτjF tþdt

� þ ð1 � ρ1ÞE

� ∫ ∞ t0 ~S1ðτÞe�rðτ�t�dtÞdτjF tþdt

� jF t #)

¼ max t0�tþdt

� e�rðt

0�tÞX � � ρ21M11 þ ρ1ð1 � ρ1ÞM12 þ ρ1ð1 � ρ1ÞM13 þ ð1 � ρ1Þ2M14

� (A.2)

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

where M11 ¼ sup θ12Θ1

EQ θ1 1

"" sup θ012Θ1

EQ θ0 1

1 ∫ ∞t0 S1ðτÞe�rðτ�tÞdτjF tþdt # jF t # , M12 ¼ sup

θ12Θ1 EQ

θ1 1 ½E½∫ ∞t0 ~S1ðτÞe�rðτ�tÞdτjF tþdt�jF t�, M13 ¼ E

" sup θ012Θ1

EQ θ0 1

1 ½∫ ∞t0

S1ðτÞe�rðτ�tÞdτjF tþdt�jF t # , M14 ¼ E½E½∫ ∞t0 ~S1ðτÞe�rðτ�tÞdτjF tþdt�jF t�.

Considering the time-consistency (or rectangularity) of the set of priors, we have M11 ¼ sup θ12Θ1

EQ θ1 1 ½∫ ∞t0 S1ðτÞe�rðτ�tÞdτjF t�. Since E½∫

∞ t0

~S1ðτÞe�rðτ�tÞdτjF tþdt� is a singleton set that is not affected by the density generators, M12 is equal to E½∫ ∞t0 ~S1ðτÞe�rðτ�tÞdτjF t�. Recall that sup θ012Θ1

EQ θ0 1

1 ½∫ ∞t0

S1ðτÞe�rðτ�tÞdτjF tþdt� denotes the expectation with respect to Q�κ11 conditional on F tþdt. This expectation stays unchanged at earlier times t. Then M13 ¼ sup θ12Θ1

EQ θ1 1 ½∫ ∞t0 S1ðτÞe�rðτ�tÞdτjF t�. In line with the law of iterated expectations and recursive utility under risk, we obtain M14 ¼ E½∫

∞ t0 ~S1ðτÞe�rðτ�tÞdτjF t� .

Then equation (A.2) becomes equivalent to:

NMEU � e�rdtF1ðt þ dtÞjF t

¼ max

t0�tþdt

8< :e�rðt0�tÞX �

2 4 ρ1 supθ12Θ1EQ

θ1 1 � ∫ ∞ t0 S1ðτÞe�rðτ�tÞdτjF t

� þð1 � ρ1ÞE

� ∫ ∞ t0 ~S1ðτÞe�rðτ�tÞdτjF t

� 3 5 9= ; ¼ Jðt0Þ (A.3)

Eq. (A.3) implies that once the decision maker commits to wait at time t, he (she) does not change his (her) plan when time elapses. Recall the NMEU-based ambiguity multiplier λ1 is a constant in the infinite time horizon. According to Eq. (1), dW1ðtÞ ¼ ðμ1 � σ1θ1ÞW1ðtÞdt þ

σ1W1ðtÞdB1ðtÞθ1 . Since the seller's put opportunity value F1ðtÞ depends on W1ðtÞ, we write F1ðW1ðtÞÞ for F1ðtÞ. According to Equations (A.3) and (5), the seller's put opportunity value can be defined as:

F1ðW1ðtÞÞ ¼ max t0�t

� X � W1ðtÞ; NMEU

� e�rdtF1ðt þ dtÞjF t

� ¼ max

t0�t fX � W1ðtÞ; ð1 � rdtÞ½F1ðW1ðtÞÞ þ NMEUðdF1ðW1ðtÞÞjF tÞ�g

¼ max t0�t

fX � W1ðtÞ; F1ðW1ðtÞÞ þ NMEUðdF1ðW1ðtÞÞjF tÞ � rF1ðW1ðtÞÞdtg (A.4)

where we approximate e�rdt using ð1 � rdtÞ and rely on F1ðW1ðtÞÞ þ NMEUðdF1ðW1ðtÞÞjF tÞ to estimate NMEUðF1ðt þ dtÞjF tÞ (see Nishimura and Ozaki, 2007; Trojanowska and Kort, 2010).

The NMEU satisfies dynamic consistency as the seller's option value is defined recursively in Eq. (A.4). In the waiting region, we have NMEUðdF1ðW1ðtÞÞjF tÞ ¼ rF1ðW1ðtÞÞdt. NMEUðdF1ðW1ðtÞÞjF tÞ can be expressed as follows:

NMEUðdF1ðW1ðtÞÞjF tÞ ¼ NMEU � F01 � ðμ1 � σ1θ1ÞW1ðtÞdt þ σ1W1ðtÞdB1ðtÞθ1

� þ 1 2 σ21ðW1ðtÞÞ2F001 dtjF t

� ¼ ζ1F01W1ðtÞdt þ

1 2 σ21ðW1ðtÞÞ2F001 dt (A.5)

where F01 ¼ ∂F1ðW1ðtÞÞ∂W1ðtÞ , F 00 1 ¼ ∂

2F1ðW1ðtÞÞ ∂W1ðtÞ2

, ζ1 ¼ ρ1ðμ1 þ σ1κ1Þ þ ð1 � ρ1Þμ1. This results in the following second-order ordinary differential equation:

1 2 σ21ðW1ðtÞÞ2F001 þ ζ1F01W1ðtÞ � rF1ðW1ðtÞÞ ¼ 0 (A.6)

The seller's trigger is subject to the value-matching, smooth-pasting, and boundary conditions: F1ðW*1Þ ¼ X � W*1, F01ðW*1Þ ¼ � 1, lim W1ðtÞ→∞

F1ðW1Þ ¼ 0. Thus, we obtain the seller's costs threshold and option value as expressed in Eq. (6).

Appendix B. Proofs of Propositions 1–2

Recall Xi ¼ 1 � 1

βi

� λiSiðtÞ, i ¼ 1; 2. Examining the effect of ambiguity aversion ρi on the implicit reservation price Xi :

∂Xi ∂ρi

¼ ∂Xi ∂βi

∂βi ∂ρi

þ ∂Xi ∂λi

∂λi ∂ρi

(B.1)

where i ¼ 1; 2, Xi ¼ 1 � 1

βi

� λiSiðtÞ, β1 ¼ 12 �

ζ1 σ21 � χ1 < 0, β2 ¼ 12 �

ζ2 σ22 þ χ2 > 1, χi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 �

ζi σ2i

�2 þ 2r

σ2i

s , ζ1 ¼ ρ1ðμ1 þ σ1κ1Þ þ ð1 � ρ1Þμ1, ζ2 ¼ ρ2ðμ2 �

σ2κ2Þ þ ð1 � ρ2Þμ2, ∂Xi∂βi > 0, ∂Xi ∂λi

> 0, ∂λ1∂ρ1 ¼ 1

r�μ1�σ1κ1 � 1

r�μ1 � 0, ∂λ2 ∂ρ2

¼ 1r�μ2þσ2κ2 � 1

r�μ2 � 0. The derivatives of β1 with respect to ρ1 and β2 with respect to ρ2 are:

∂βi ∂ρi

¼

8>>>< >>>:

β1κ1 σ1χ1

� 0 i ¼ 1

β2κ2 σ2χ2

� 0 i ¼ 2 (B.2)

Considering λ1 � 1r�μ1�σ1κ1, we obtain the derivative of the seller's implicit reservation price ðX1Þ with respect to ρ1:

63

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

∂X1 ∂ρ

¼ S1ðtÞ�β ð1 � β1Þ 1

r � μ � σ1κ1 � 1 r � μ �

λ1κ1 σ1χ

� S1ðtÞ�β ð1 � β1Þ 1

r � μ � σ1κ1 � 1 r � μ �

κ1 σ1χ ðr � μ � σ1κ1Þ

1 1

� 1 1

� 1

� 1

� 1 1

� 1 1

� ¼ S1ðtÞκ1

� ð1 � β1Þσ21χ1 � ðr � μ1Þ

� ð � β1Þσ1χ1ðr � μ1 � σ1κ1Þðr � μ1Þ

(B.3)

Since ð1 � β1Þσ21χ1 > 2r and β1 < 0, ∂X1∂ρ1 > 0 if κ1 > 0 and ∂X1 ∂ρ1

¼ 0 if κ1 ¼ 0. Using the same logic as above, the derivative of X2 with respect to ρ2 is:

∂X2 ∂ρ2

¼ λ2κ2 β2σ2χ2

S2ðtÞ|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} �0

þ

1 r � μ2 þ σ2κ2

� 1 r � μ2

� 1 � 1

β2

� S2ðtÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

�0

> ¼ <

0 (B.4)

Eq. (B.4) also shows that when κ2 ¼ 0, the derivative of X2ðκ2 ¼ 0Þ with respect to ρ2 is equal to zero. Examining the effect of probabilistic ambiguity κi on the implicit reservation price Xi:

∂Xi ∂κi

¼ ∂Xi ∂βi

∂βi ∂κi|fflfflffl{zfflfflffl}

option�value effect

þ ∂Xi ∂λi

∂λi ∂κi|fflfflffl{zfflfflffl}

present�value effect

(B.5)

The derivatives of β1 with respect to κ1 and β2 with respect to κ2 are:

∂βi ∂κi

¼

8>>>< >>>:

β1ρ1 σ1χ1

� 0 i ¼ 1

ρ2β2 σ2χ2

� 0 i ¼ 2 (B.6)

The derivative of the ambiguity multiplier λi with respect to κi can be expressed as:

∂λi ∂κi

¼

8>>>< >>>:

ρ1σ1 ðr � μ1 � σ1κ1Þ2

� 0 i ¼ 1

�ρ2σ2 ðr � μ2 þ σ2κ2Þ2

� 0 i ¼ 2 (B.7)

Substituting ∂β1∂κ1 and ∂λ1 ∂κ1

in ∂X1∂κ1, we obtain the partial derivative of X1 with respect to probabilistic ambiguity κ1:

∂X1 ∂κ1

¼ ρ1S1ðtÞð � β1Þ

( ð1 � β1Þ

σ1 ðr � μ1 � σ1κ1Þ2

� λ1 σ1χ1

) � ρ1S1ðtÞ

ð � β1Þðr � μ1 � σ1κ1Þ2σ1χ1 � ð1 � β1Þσ21χ1 � ðr � μ1 � σ1κ1Þ

� (B.8)

Considering ð1 � β1Þσ21χ1 > 2r and β1 < 0, we prove that ∂X1∂κ1 > 0 if ρ1 2 ð0; 1� and ∂X1 ∂κ1

¼ 0 if ρ1 ¼ 0. From Eqs. (B.6) and (B.7), we get:

∂X2 ∂κ2

¼ ρ2S2ðtÞ β2σ2

( λ2 χ2

� ðβ2 � 1Þσ 2 2

ðr � μ2 þ σ2κ2Þ2 )

> ¼ <

0 (B.9)

Eq. (B.9) shows that the effect of the buyer's probabilistic ambiguity κ2 on his implicit reservation price X2 is equivocal. When ρ2 ¼ 0, the derivative of X2 ðρ2 ¼ 0Þ with respect to κ2 is equal to zero. This proves Proposition 1.

Next, we examine the effect of ambiguity aversion on the negotiation threshold.

As δKK ¼ X1S2ðtÞX2S1ðtÞ, we have

8>>>< >>>:

∂δKK ∂ρ1

> 0 if κ1 > 0

∂δKK ∂ρ1

¼ 0 if κ1 ¼ 0 and

8>>< >>:

∂δKK ∂κ1

> 0 if ρ1 2 ð0; 1�

∂δKK ∂κ1

¼ 0 if ρ1 ¼ 0 .

The effects of ρ2 and κ2 on δKK are equivocal:

∂δKK ∂ρ2

¼ �X1S2ðtÞX22 S1ðtÞ ∂X2 ∂ρ2

> ¼ <

0 and ∂δKK∂κ2 ¼ � X1S2ðtÞ X22 S1ðtÞ

∂X2 ∂κ2

> ¼ <

0. Note that ∂δKKðκ2¼0Þ∂ρ2 ¼ 0 and ∂δKKðρ2¼0Þ

∂κ2 ¼ 0. This proves Proposition 2.

Appendix C. The subjective probability of negotiation agreement under ambiguity

Since the seller is concerned about her costs and the buyer cares about his revenues, the negotiation agreement probability should be determined by their expectations of these quantities. Extending the single prior analysis of Moon et al. (2011) to the case of uncertainty and NMEU ambiguity, we examine how changes in ambiguity aversion and probabilistic ambiguity affect negotiators' subjective likelihood of agreement PKK. As defined in Eq. (1), S1ðtÞ and S2ðtÞ follow lognormal diffusions with ambiguous drifts and the two-dimension probability density function under our subjective probability measures is (assuming a possible correlation between costs and revenues):

64

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

exp � 1 2ð1�ϑ2Þ

� log S1�μ1N

σ1N

�2 � 2ϑ ðlog S1�μ1N Þðlog S2�μ2N Þ

σ1N σ2N þ � log S2�μ2N

σ2N

�2

gðS1; S2Þ ¼

� � �� 2πσ1Nσ2N

ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ϑ2

p S1S2

(C.1)

where μiN is the ambiguity-adjusted drift rate and σiN is the standard deviation of log Si under the NMEU.μ1N ¼ ζ1 � 12σ21

� t, ζ1 ¼ ρ1ðμ1 þ σ1κ1Þ þ ð1 �

ρ1Þμ1, μ2N ¼ ζ2 � 12σ22

� t, ζ2 ¼ ρ2ðμ2 � σ2κ2Þ þ ð1 � ρ2Þμ2, σiN ¼ σi

ffiffi t

p , for i ¼ 1; 2, 8t � 0, 8θi 2 Θi. We write S1 and S2 for S1ðtÞ and S2ðtÞ. The

subscript N implies log Si follows a normal distribution. ϑ is the correlation between S1 and S2. Ambiguity appears in Eq. (C1) both through the numerator and denominator.

We identify the process followed by S1S2 (e.g., Dixit and Pindyck, 1994):

dðS1S2Þ ¼ ðμ1 � σ1θ1 þ μ2 � σ2θ2 þ εσ1σ2ÞS1S2dt þ � σ1dB1ðtÞθ1 þ σ2dB2ðtÞθ2

� S1S2 (C.2)

where ε is defined by E½dB1ðtÞθ1 dB2ðtÞθ2 � ¼ εdt. Since the NMEU value and standard deviation of Si under ambiguity are given by NMEUðSiÞ ¼ Sið0ÞexpðζitÞ, stdðSiÞ ¼

Sið0Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expð2ζitÞ½expðσ2i tÞ � 1�

q , we can write:

NMEUðS1S2Þ ¼ S1ð0ÞS2ð0Þexp½ðζ1 þ ζ2 þ εσ1σ2Þt� (C.3) In line with Moon et al. (2011) but considering ambiguity in seller's costs and buyer's revenues, the correlation coefficient ϑ between S1 and S2 can be

expressed as:

ϑ ¼ NMEUðS1S2Þ � ½NMEUðS1Þ�½NMEUðS2Þ� stdðS1ÞstdðS2Þ

¼ expðεσ1σ2tÞ � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½expðσ21tÞ � 1�

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½expðσ22tÞ � 1�

p (C.4) Thus, the subjective probability of negotiation agreement under ambiguity PKK is obtained as follows:

PKK ¼ PKKð X � X1 and X � X2; t Þ ¼ PKK � S1ðtÞ � S*1 and S2ðtÞ � S*2

¼ ∫ Y

* 1

�∞∫ þ∞ Y* 2

exp � � 1

2ð1�ϑ2Þ ��

Y1�μ1N σ1N

�2 � 2ϑ ðY1�μ1N ÞðY2�μ2N Þ

σ1N σ2N þ � Y2�μ2N σ2N

�2�� 2πσ1Nσ2N

ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � ϑ2

p dY1dY2 (C.5)

where Y1 ¼ ln

S1ðtÞ S1ð0Þ

� , Y2 ¼ ln

S2ðtÞ S2ð0Þ

� , Y*1 ¼ ln

S*1

S1ð0Þ

� ¼ ln

X

ð1�1=β1Þλ1S1ð0Þ

� and Y*2 ¼ ln

S*2

S2ð0Þ

� ¼ ln

X

ð1�1=β2Þλ2S2ð0Þ

� . S*1 ¼ Xð1�1=β1Þλ1 , S

* 2 ¼ Xð1�1=β2Þλ2.

The above (Eq. (C.5)) indicates that the subjective probability of negotiation success is not unique when considering parties' real options, ambiguity aversion and ambiguity, it once again depends on the value of Si at time 0, the ambiguity multiplier or subjective beliefs λi, the discount rate r and the parameters of the Geometric Brownian motion(s) followed by Si as defined in Eq. (1). When probabilistic ambiguity ðκiÞ is greater than zero, there is a negative relationship between ambiguity aversion ðρiÞ and the subjective negotiation agreement probability, where i ¼ 1; 2. In the case of ambiguity averse negotiators ðρi 2 ð0; 1�Þ, the subjective probability of negotiation success decreases with increasing ambiguity ðκi; i ¼ 1; 2Þ, reflecting a con- servative approach towards negotiation.

When revenues and costs functions are independent, the subjective probability of negotiation agreement simplifies to the following closed-form solution under ambiguity PKKI (confirming the interaction effects of uncertainty aversion and ambiguity on negotiation performance):

PKKI ¼ PK � S1ðtÞ � S*1

PK � S2ðtÞ � S*2

¼ Φ

Y*1 �

� ζ1 � 12σ21

t

σ1 ffiffi t

p �

1 � Φ Y*2 �

� ζ2 � 12σ22

t

σ2 ffiffi t

p ��

(C.6)

where PKðS2ðtÞ � S*2Þ ¼ ∫ þ∞ Y*2

e �12ð½Y2�ðζ2�12σ22Þt�=σ2 ffitp Þ2

ð2πσ22tÞ 1=2 dY2, PKðS1ðtÞ � S*1Þ ¼ ∫

Y*1 �∞

e �12ð½Y1�ðζ1�12σ21Þt�=σ1 ffitp Þ2

ð2πσ21tÞ 1=2 dY1

Appendix D. Asymmetric information and price negotiation

We formulate the Lagrangian by considering the incentive compatibility constraint of the high type agent and the participant constraint of the low type agent as the follows (Grenadier and Wang, 2005; Shibata, 2009):

K1 ¼ qH zðtÞ zH

�bH ðλ2zHS1ðtÞ � XHÞ þ ð1 � qHÞ

zðtÞ zL

�bL ðλ2zLS1ðtÞ � XLÞ þ e1

" zðtÞ zH

�bH ðXH � λ1HS1ðtÞÞ �

zðtÞ zL

�bH ðXL � λ1HS1ðtÞÞ

# þ e2

" zðtÞ zL

�bL ðXL

� λ1LS1ðtÞÞ #

The first order condition of K1 with respect to XH and XL indicates that e1 ¼ qH and e2 ¼ 1 � qH þ qH

zL zðtÞ

�bL�bH . Recall zL > zðtÞ and bL > bH based on

∂b ∂ρ1

� 0 in Supplementary Appendix F. Then we know that e2 > 1. From the Kuhn-Tucker condition, constraints (25) and (28) are binding. We obtain the

65

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

solution X*H ¼ " λ1H þ

z*H z*L

�bH ðλ1L � λ1HÞ

# S1ðtÞ and X*L ¼ λ1LS1ðtÞ.

This implies X*H > λ1HS1ðtÞ and constraint (27) does not bind. The first order conditions of K1 with respect to zH and zL mean z*H ¼ bHλ1HðbH�1Þλ2 and z * L is

the solution to ð1 � qHÞzbH�bLL ðbL � 1Þλ2 �

bLλ1L ðbL�1Þλ2 � zL

� þ qHbHzðtÞbH�bL ðλ1L � λ1HÞ ¼ 0. Note λ1L > λ1H and bL > 1. To ensure the existence of a solution,

we assume that z*L > bLλ1L

ðbL�1Þλ2. To satisfy constraint (26), XH should be smaller than λ1LS1ðtÞ. Recall ∂z* ∂ρ1

� 0 from Supplementary Appendix F. As ρ1H < ρ1L,

we know that z*H < z * L and ðλ1L � λ1HÞ

1 �

z*H z*L

�bH! � 0. Then constraint (26) is satisfied.

Consider the seller (principal) delegates the investment timing decision to the buyer (agent). There are two types of buyers. The probability of any buyer belonging to the good type category is q2. We call the buyer good (bad) if his ambiguity aversion about revenues is ρ2G ðρ2BÞ with ρ2G < ρ2B. This indicates that λ2G > λ2B. Let λ2s and bs denote the NMEU-based ambiguity multiplier and option value parameter of type s agent, where bs ¼ 12 �

δ1�δ2s σ22�2εσ1σ2þσ21

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2 � δ1�δ2sσ22�2εσ1σ2þσ21

�2 þ 2δ1

σ22�2εσ1σ2þσ21

s , δ2s ¼ 1=λ2s, λ2s ¼ ρ2sr�μ2þσ2κ2 þ

1�ρ2s r�μ2 , s ¼ G or B. Assume that zðtÞ < zs. As bs and λ2s are the buyer's

private information, the seller asks for price Xs based on the observable contract timing zs as follows:

max zG;zB;XG;XB

q2

zðtÞ zG

�bG ðXG � λ1S1ðtÞÞ þ ð1 � q2Þ

zðtÞ zB

�bB ðXB � λ1S1ðtÞÞ;

subject to:

zðtÞ zG

�bG ðλ2GzGS1ðtÞ � XGÞ �

zðtÞ zB

�bG ðλ2GzBS1ðtÞ � XBÞ

zðtÞ zB

�bB ðλ2BzBS1ðtÞ � XBÞ �

zðtÞ zG

�bB ðλ2BzGS1ðtÞ � XGÞ

zðtÞ zG

�bG ðzGλ2GS1ðtÞ � XGÞ � 0

zðtÞ zB

�bB ðzBλ2BS1ðtÞ � XBÞ � 0

Using the same logic as above and when z*B > bBλ1

ðbB�1Þλ2B, the incentive policy under asymmetric information concerning the buyer's type can be defined

by ðz*G;X*G;z*B;X*BÞ ¼

bGλ1 ðbG�1Þλ2G;

" λ2Gz*G �

z*G z*B

�bG ðλ2G � λ2BÞz*B

# S1ðtÞ;z*B;λ2Bz*BS1ðtÞ

! , where z*B is the solution to ð1 � q2Þλ2BðbB � 1ÞzbG�bB�1B

� bBλ1

ðbB�1Þλ2B � zB � þ

ðbG � 1Þq2zðtÞbG�bB ðλ2G � λ2BÞ ¼ 0. From Supplementary Appendix F, we have ∂z*∂ρ1 � 0 and ∂z* ∂ρ2

� 0. Then we know that ∂z * H

∂ρ1H � 0 and ∂z

* H

∂ρ2 � 0, ∂z

* G

∂ρ1 � 0 and

∂z*G ∂ρ2G

� 0.

Appendix E. Supplementary material

Supplementary data related to this article can be found at https://doi.org/10.1016/j.ijpe.2018.02.004.

References

Abdellaoui, M., Klibanoff, P., Placido, L., 2015. Experiments on compound risk in relation to simple risk and ambiguity. Manag. Sci. 61 (6), 1306–1322.

Abd Rahman, A., Bennett, D., Sohal, A., 2009. Transaction attributes and buyer-supplier relationships in AMT acquisition and implementation: the case of Malaysia. Int. J. Prod. Res. 47 (9), 2257–2278.

Agliardi, E., Agliardi, R., Spanjers, W., 2015. Convertible debt: financing decisions and voluntary conversion under ambiguity. Int. Rev. Finance 15 (4), 599–611.

Asano, T., Shibata, A., 2011. Optimal pricing and quality choice of a monopolist under Knightian uncertainty. Int. J. Ind. Organ. 29 (6), 746–754.

Bichescu, B.C., Fry, M.J., 2009. Vendor-managed inventory and the effect of channel power. OR Spectr. 31 (1), 195–208.

Binmore, K.G., 1985. Bargaining and coalitions. In: Roth, A.E. (Ed.), Game-Theoretic Models of Bargaining. Cambridge University Press, Cambridge, pp. 269–304.

Binmore, K., Rubinstein, A., Wolinsky, A., 1986. The Nash bargaining solution in economic modelling. Rand J. Econ. 17 (2), 176–188.

Birkeland, S., Tungodden, B., 2014. Fairness motivation in bargaining: a matter of principle. Theory Decis. 77 (1), 125–151.

Bodoh-Creed, A.L., 2012. Ambiguous beliefs and mechanism design. Game. Econ. Behav. 75 (2), 518–537.

Borgatti, S.P., Li, X., 2009. On social network analysis in a supply chain context. J. Supply Chain Manag. 45 (2), 5–21.

66

Bose, S., Renou, L., 2014. Mechanism design with ambiguous communication devices. Econometrica 82 (5), 1853–1872.

Braun, N., Gautschi, T., 2006. A Nash bargaining model for simple exchange networks. Soc. Netw. 28 (1), 1–23.

Charalambides, M., Koussis, N., 2018. A stochastic model with interacting managerial operating options and debt rescheduling. Eur. J. Oper. Res. 267 (1), 236–249.

Chateauneuf, A., Eichberger, J., Grant, S., 2007. Choice under uncertainty with the best and worst case in mind: neo-additive capacities. J. Econ. Theory 137 (1), 538–567.

Chen, Z.J., Epstein, L., 2002. Ambiguity, risk and asset returns in continuous time. Econometrica 70 (4), 1403–1443.

De Waegenaere, A., Wielhouwer, J.L., 2011. Dynamic tax depreciation strategies. OR Spect. 33 (2), 419–444.

Dixit, A.K., Pindyck, R.S., 1994. Investment under Uncertainty. Princeton University Press, Princeton.

Driouchi, T., Bennett, D., 2012. Real options in management and organizational strategy: a review of decision-making and performance implications. Int. J. Manag. Rev. 14 (1), 39–62.

Driouchi, T., Bennett, D., Simpson, G., 2010. A path-dependent contingent-claims approach to capacity investments. Eur. J. Oper. Res. 201 (1), 319–323.

Driouchi, T., Pennesi, P., Kapletia, D., Cuthbert, R., 2009. Real options in service investments: scenarios from defence. Int. J. Serv. Sci. 2 (1), 1–28.

Ellsberg, D., 1961. Risk, ambiguity, and the savage axioms. Q. J. Econ. 75 (4), 643–669. Esmaeilia, M., Zeephongsekul, P., 2010. Seller–buyer models of supply chain

management with an asymmetric information structure. Int. J. Prod. Econ. 123 (1), 146–154.

Y. Gao et al. International Journal of Production Economics 200 (2018) 50–67

Feng, B., Yao, T., Jiang, B., Talluri, S., 2014. How to motivate vendor's RFID adoption beyond mandate? A retailer's perspective. Int. J. Prod. Res. 52 (7), 2173–2193.

Fonseca, R.J., Rustem, B., 2012. Robust hedging strategies. Comput. Oper. Res. 39 (11), 2528–2536.

Fotopoulos, S.B., Hu, X., Munson, C.L., 2008. Flexible supply contracts under price uncertainty. Eur. J. Oper. Res. 191 (1), 253–263.

Fudenberg, D., Tirole, J., 1983. Sequential bargaining with incomplete information. Rev. Econ. Stud. 50 (2), 221–247.

Gao, Y., Driouchi, T., 2013. Incorporating Knightian uncertainty into real options analysis: using multiple-priors in the case of rail transit investment. Transport. Res. Part B 55, 23–40.

Gao, Y., 2017. The impacts of the supplier's optimism levels on pricing decisions under Knightian uncertainty. Syst. Eng. Theory Pract. 37 (9), 2297–2305.

Ghosh, D., 1994. Tolerance for ambiguity, risk preference, and negotiator effectiveness. Decis. Sci. 25 (2), 263–280.

Ghosh, D., Ray, M.R., 1997. Risk, ambiguity and decision choice: some additional evidence. Decis. Sci. 28 (1), 81–104.

Gilboa, I., Schmeidler, D., 1989. Maxmin expected utility with a nonunique prior. J. Math. Econ. 18 (2), 141–153.

Giraud, R., Thomas, L., 2017. Ambiguity, optimism, and pessimism in adverse selection models. J. Econ. Theory 171, 64–100.

Goebel, P., Reuter, C., Pibernik, R., Sichtmann, C., 2012. The influence of ethical culture on supplier selection in the context of sustainable sourcing. Int. J. Prod. Econ. 140 (1), 7–17.

Golan, L., 2009. Wage signaling: a dynamic model of intrafirm bargaining and asymmetric learning. Int. Econ. Rev. 50 (3), 831–854.

Govindan, K., Shankar, M., Kannan, D., 2016. Supplier selection based on corporate social responsibility practices. Int. J. Prod. Econ. 200, 353–379. https://doi.org/10.1016/ j.ijpe.2016.09.003.

Granot, D., Yin, S., 2008. Competition and cooperation in decentralized push and pull assembly systems. Manag. Sci. 54 (4), 733–747.

Grenadier, S.R., Wang, N., 2005. Investment timing, agency, and information. J. Financ. Econ. 75 (3), 493–533.

Grenadier, S.R., Malenko, A., Malenko, N., 2016. Timing decisions in organizations: communication and authority in a dynamic environment. Am. Econ. Rev. 106 (9), 2552–2581.

Hazen, B.T., Overstreet, R.E., Jones-Farmer, A., Field, H.S., 2012. The role of ambiguity tolerance in consumer perception of remanufactured products. Int. J. Prod. Econ. 135 (2), 781–790.

Hemmert, M., Kim, D., Kim, J., Cho, B., 2016. Building the supplier's trust: role of institutional forces and buyer firm practices. Int. J. Prod. Econ. 180, 25–37.

Henderson, V., Hobson, D.G., 2002. Real options with constant relative risk aversion. J. Econ. Dyn. Control 27, 329–355.

Heese, H.S., 2015. Single versus multiple sourcing and the evolution of bargaining positions. Omega 54, 125–133.

He, Y., Yin, S., 2015. Joint selling of complementary components under brand and retail competition. Manuf. Serv. Oper. Manag. 17 (4), 470–479.

Hofstede, G., 2001. Culture's Consequences: Comparing Values, Behaviours, Institutions, and Organisations across Nations. Sage Publications.

Hugonnier, J., Morellec, E., 2007. Corporate control and real investment in incomplete markets. J. Econ. Dyn. Control 31 (5), 1781–1800.

Jiang, B., Yao, T., Feng, B., 2008. Valuate outsourcing contracts from vendors' perspective: a real options approach. Decis. Sci. 39 (3), 383–405.

Jiang, B., Talluri, S., Yao, T., Moon, Y., 2010. Breaking the winner's curse in outsourcing. Decis. Sci. 41 (3), 573–594.

Kamrad, B., Siddique, A., 2004. Supply contracts, profit sharing, switching, and reaction options. Manag. Sci. 50 (1), 64–82.

Kim, Y., Choi, T.Y., Yan, T., Dooley, K., 2011. Structural investigation of supply networks: a social network analysis approach. J. Oper. Manag. 29 (3), 194–211.

Kopylov, I., 2016. Subjective probability, confidence, and Bayesian updating. Econ. Theory 62 (4), 635–658.

Leider, S., Lovejoy, W.S., 2016. Bargaining in supply chains. Manag. Sci. 62 (10), 3039–3058.

Li, X., 2012. Group buying, buyer heterogeneity, and sellers' bargaining power. Decis. Sci. 43 (5), 761–783.

67

Li, C.L., Kouvelis, P., 1999. Flexible and risk-sharing supply contracts under price uncertainty. Manag. Sci. 45 (10), 1378–1398.

Miao, J.J., Wang, N., 2011. Risk, uncertainty, and option exercise. J. Econ. Dyn. Control 35 (4), 442–461.

Moon, Y.M., Yao, T., Park, S.S., 2011. Price negotiation under uncertainty. Int. J. Prod. Econ. 134 (2), 413–423.

Moran, S., Ritov, I., 2002. Initial perceptions in negotiations: evaluation and response to ‘logrolling’ offers. J. Behav. Decis. Mak. 15 (2), 101–124.

Moreno, O.M., 2014. Consumption of Durable Goods under Ambiguity. Banco de Mexico working papers (N� 2014-02).

Muthoo, A., 1996. A bargaining model based on the commitment tactic. J. Econ. Theory 69 (2), 134–152.

Nagarajan, M., Bassok, Y., 2008. A bargaining framework in supply chains: the assembly problem. Manag. Sci. 54 (8), 1482–1496.

Nagarajan, M., So�si�c, G., 2008. Game-theoretic analysis of cooperation among supply chain agents: review and extensions. Eur. J. Oper. Res. 187 (3), 719–745.

Nembhard, D.A., Nembhard, H.B., Qin, R., 2005. A real options model for workforce cross-training. Eng. Econ. 50 (2), 95–116.

Nishihara, M., Shibata, T., 2008. The agency problem between the owner and the manager in real investment: the bonus–audit relationship. Oper. Res. Lett. 36 (3), 291–296.

Nishimura, K., Ozaki, H., 2007. Irreversible investment and Knightian uncertainty. J. Econ. Theory 136 (1), 668–694.

Oosterhuis, M., Molleman, E., van der Vaart, T., 2013. Differences in buyers' and suppliers' perceptions of supply chain attributes. Int. J. Prod. Econ. 142, 158–171.

Riedel, F., 2009. Optimal stopping with multiple priors. Econometrica 77 (3), 857–908. Roemer, E., 2004. Managing asymmetric resource dependence and environmental risk in

relationships by real options. Manag. Rev. 15 (1), 89–106. Schneider, C., Bremen, P., Sch€onsleben, P., Alard, R., 2013. Transaction cost economics in

global sourcing: assessing regional differences and implications for performance. Int. J. Prod. Econ. 141 (1), 243–254.

Schoenherr, T., Narayanan, S., Narasimhan, R., 2015. Trust formation in outsourcing relationships: a social exchange theoretic perspective. Int. J. Prod. Econ. 169, 401–412.

Schr€oder, D., 2011. Investment under ambiguity with the best and worst in mind. Math. Financ. Econ. 4 (2), 107–133.

Sheu, J.-B., Gao, X.-Q., 2014. Alliance or no alliance—bargaining power in competing reverse supply chains. Eur. J. Oper. Res. 233 (2), 313–325.

Shibata, T., Nishihara, M., 2011. Interactions between investment timing and management effort under asymmetric information: costs and benefits of privatized firms. Eur. J. Oper. Res. 215 (3), 688–696.

Shibata, T., 2009. Investment timing, asymmetric information, and audit structure: a real options framework. J. Econ. Dyn. Control 33 (4), 903–921.

Trigeorgis, L., 1996. Evaluating leases with complex operating options. Eur. J. Oper. Res. 91 (2), 315–329.

Trojanowska, M., Kort, P.M., 2010. The worst case for real options. J. Optim. Theory Appl. 146 (3), 709–734.

Vierø, M.-L., 2014. Bait contracts. Oxf. Econ. Pap. 66, 702–720. Waters, D., 2009. Supply chain Management: an Introduction to Logistics. Palgrave

Macmillan. Wolitzky, A., 2016. Mechanism design with Maxmin agents: theory and an application to

bilateral trade. Theor. Econ. 11 (3), 971–1004. Wu, L.-C., Liou, F.-M., 2011. A quantitative model for ERP investment decision:

considering revenue and costs under uncertainty. Int. J. Prod. Res. 49 (22), 6713–6728.

Yamaguchi, K., 1996. Power in networks of substitutable and complementary exchange relations: a rational-choice model and an analysis of power centralization. Am. Socio. Rev. 61 (2), 308–332.

Yamaguchi, K., 2000. Power in mixed exchange networks: a rational choice model. Soc. Netw. 22 (2), 93–121.

Yao, T., Jiang, B., Young, S., Talluri, S., 2010. Outsourcing timing, contract selection and negotiation. Int. J. Prod. Res. 48 (2), 305–326.

Zheng, S., Negenborn, R.R., 2015. Price negotiation between supplier and buyer under uncertainty with fixed demand and elastic demand. Int. J. Prod. Econ. 167, 35–44.

Zwick, R., Lee, C.L., 1999. Bargaining and search: an experimental study. Group Decis. Negot. 8 (6), 463–487.