Pricing with risk sensitive competing container shipping lines: Will risk seeking do more good than harm?

Transportation Research Part B 133 (2020) 210–229

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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

Pricing with risk sensitive competing container shipping lines: Will risk seeking do more good than harm? ✩ Tsan-Ming Choi a, Sai-Ho Chung b, Xiaopo Zhuo c,∗ a

Business Division, Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong c Business School, Sun Yat-sen University, Guangzhou 510275, PR China b

a r t i c l e

i n f o

Article history: Received 9 April 2019 Revised 4 December 2019 Accepted 10 January 2020

Keywords: Container shipping lines Risk attitudes Demand volatility Price competition Mean-risk framework

a b s t r a c t Container-shipping-lines (CSLs) operate in a highly competitive market with high demand volatility. Unlike many other industries in which risk-averse attitude is dominating, it is well known that many CSLs exhibit risk-seeking behaviors. In this paper, by using the mean-risk formulation, we analytically study the effects of risk attitude and demand volatility on the service pricing game between two CSLs. We find that the equilibrium prices increase when CSLs can accept more risk, and being slightly risk-seeking can help maximize the expected profits of both CSLs. We discover the risk-attitude convergence effect, which theoretically reveals that the “optimal risk attitudes” for the CSLs move in the same direction. Therefore, our research suggests that both CSLs should hire slightly riskseeking managers in a competitive shipping market. In addition, we explore the impacts brought by demand volatility and find that the equilibrium prices increase (decrease) in the demand volatility when CSLs are risk-seeking (risk-averse). We uncover that when an individual CSL’s risk sensitivity level is small and its rival is risk-seeking, its own expected profit under the decentralized case could be larger than that in centralized case. Finally, for robustness checking of our research findings, we investigate the scenarios with multiplicative randomness of demand, asymmetric information on risk attitude and multiple (more than two) competing CSLs, and find that our main findings still hold. Counterintuitively, in the case with multiplicatively random demand, we analytically find that CSLs may not prefer a large market potential when they are risk-averse. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Motivation and background Being responsible for carrying 90% of the world’s merchandises, the shipping industry is the backbone of the global trade (UNCTAD, 2017). Among the seaborne trade, 52% of cargos by value are carried by container ships (Zheng et al., 2017).

✩ We sincerely thank the editors and anonymous reviewers for their critical and helpful comments on this paper. This paper was partially supported by a grant from the Research Grants Council of Hong Kong (General Research Fund) under the project code “PolyU 152131/17E”. Xiaopo Zhuo was funded by the National Natural Science Foundation of China (Grant numbers. 71431007 and 71901227). ∗ Corresponding author. E-mail address: [email protected] (X. Zhuo).

https://doi.org/10.1016/j.trb.2020.01.003 0191-2615/© 2020 Elsevier Ltd. All rights reserved.

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Fig. 1. Growth of demand in container shipping, 2006-2017 (Percentage) (UNCTAD, 2017).

Container shipping has experienced a booming growth over the past decades and Statista1 forecasts that global container market demand is projected to increase by about 4.7% between 2016 and 2019. With such a rapid increase, containershipping service operations have high demand for container shipping lines (CSLs). Container shipping is a capital-intensive industry, where freight rate competition is common (Wang et al., 2014). For instance, industrial news reports that competition among Intra-Asia routes will become more intense, where Intra-Asia is the world’s largest container-shipping trade. Philip Damas, head of the logistics practice of shipping consultancy Drewry, said that the price war between carriers in the container shipping market resulting in substantial reductions in freight rate2 . Generally, the shipping deployments, such as frequency and ship capacity setting, are similar among CSLs, which implies that the CSL competition is mainly on freight rate (Lee and Song, 2017). With a market environment of intense freight rate competition, pricing and risk management are critical in shipping service operations. Consultancy Mckinsey reports that pricing strategy is the most important factor to improve CSLs’ return on sales3 , and it is because vessel slots lose the opportunity to generate revenue if pricing inappropriately. Moreover, pricing strategies of individual CSLs will affect not only their own profits but also the market demands as well as profits of the other CSLs. For instances, price war on Asia-Europe routes leads to extremely low freight rates, which makes Maersk Line’s second quarter revenue in 2015 become 9.2 percent lower than that in the same period in 20144 . Therefore, it is of great significance for CSLs to price appropriately facing a highly competitive market. CSLs face high risk in shipping service operations, where both external and internal challenges exist. For example, external demand is uncontrollable by the CSLs and it is highly volatile and greatly influences CSLs’ profitability. The situation is even more complicated because the CSL market is very competitive. Fig. 1 exhibits the growth of demand in container shipping in the past decade, which is associated with a high volatility. Mckinsey5 says the demand of container shipping over the next 25 years is extremely uncertain and the earnings are more volatile. In addition, the freight rate is also highly volatile due to the fluctuation of bunker price and intense freight rate competition among CSLs. UNCTAD (2017) reports that the freight rate of Shanghai-Northern Europe markets slumped 45.8% in 2016. Shipping consultancy Drewry3 reports that freight rate in the major East-West trade routes are expected at least 30% decrease as the price war among CSLs persists. Internally, CSLs have their own operations constraints. For instance, they have their own operations costs. Therefore, these external and internal challenges make CSLs face high risk in operations. In the presence of a highly risky market situation, CSLs may take different risk attitudes, which can be classified into three types: Risk-averse, risk-neutral and risk-seeking. Traditionally, it is in fact true in many cases that shipping entrepreneurs facing risk tend to exhibit risk-averse behaviors. However, shipping entrepreneurs are famous for taking risks (Stopford, 2008). One example is that during the 1999 recession many new Panamax bulk carriers (PBCs) were ordered for

1 “Projected global container market demand growth between 2008 and 2019”, access on April 1, 2019, https://www.statista.com/statistics/253931/ global- container- market- demand- growth/. 2 “Drewry: Box Shipping Rates Fall as East-West Price War Rages On”, access on access on April 1, 2019, https://worldmaritimenews.com/archives/195010/ drewry- east- west- carrier- contract- rates- drop- amid- price- war/. 3 “The hidden opportunity in container shipping”, access on April 1, 2019, https://www.mckinsey.com/business-functions/strategy- and- corporate- finance/ our- insights/the- hidden- opportunity- in- container- shipping. 4 “Maersk Line warns competitors in the price war”, access on April 1, 2019, https://shippingwatch.com/carriers/Container/article7931266.ece. 5 “Uncertain outlook for container shipping”, access on April 1, 2019, https://www.americanshipper.com/Main/fullasd/Uncertain_outlook_for_container_ shipping_71758.aspx.

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delivery in 2002 and disastrously, their delivered spot earnings were only $5500 per day. However, just two years later in 2004, the average one-year T/C (time-charter) rate was $34,323 per day; and by 2007, it reached $51,0 0 0 per day. For the PBCs, taking order in 1999 is a significantly risk-seeking behavior. Existing research also highlights the fact that riskseeking behavior is common in the shipping industry (Lorange and Norman, 1973; Cullinane, 1991; Ishizaka et al., 2018). For example, estimation based on actual market data shows that shipping market participants exhibit risk-seeking behaviors (Ishizaka et al., 2018). With a highly unprofitable container-shipping environment over the past years, Mckinsey suggests that CSLs challenge the legacy and shake things up by bringing in new and controversial points of view, which somehow also reflects the attitude of risk-seeking. The classical approach for exploring supply chain risk is to employ an expected measure approach in the operational research literature (Chiu and Choi, 2016). However, risk should include both unfavorable outcomes and the associated level of uncertainty. The expected measure can only show the magnitude of “expected unfavorable outcomes” but not the level of uncertainty. In addition, the expected measure approach can only capture the risk-neutral attitude. Hence, the mean-risk approach is proposed to eliminate the theoretical flaws of the expected measure approach (Lau, 1980; Chiu and Choi, 2016). The mean-risk approach considers both the expected payoff (i.e. the “mean”) and the variance/standard-deviation of profit (i.e. the “risk”) in the analysis. It can capture both the level of uncertainty of unfavorable outcomes and the decision-maker’s different attitudes (i.e. risk attitudes) towards to unfavorable outcomes. In particular, the mean-risk theory is applied to analyze operations management problems in two perspectives: 1. As an analytical measure for risk aversion and being included in the optimization objective. 2. As a performance measure to capture the profit risk of the associated operations in transportation logistics (Choi et al., 2019). 1.2. Major findings Motivated by the fact that CSLs operate in a highly risky and competitive market, we build stylized game-theoretic model to analytically study how different risk attitudes of the CSLs and market competition affect the pricing decisions as well as the performance of the CSLs. We consider a basic model with two CSLs, indexed by CSL 1 and CSL 2, which compete with each other in freight rate in a specific shipping service route facing uncertain spot market demand6 . We adopt the mean-risk approach to characterize CSLs’ risk attitudes, where CSLs can be risk-seeking, risk-averse and risk-neutral. Our main findings are summarized as follows. In this paper, we first study the effects of risk attitude, demand volatility and competition on equilibrium prices. We find that CSL’s equilibrium price increases if it can accept more risk. CSL also determines a higher price when the rival can accept more risk. Since the rival determines a higher price when it can accept more risk, it leads to higher equilibrium prices under price competition. In addition, equilibrium prices increase in demand volatility when CSLs are both risk-seeking, while decrease in demand volatility when CSLs are both risk-averse. We provide the necessary and sufficient conditions to determine whether equilibrium prices are positively and negatively affected by the competition intensity. One interesting finding is that when CSL is sufficiently risk-seeking, its equilibrium price decreases in the competition intensity, while the equilibrium price increases in the competition intensity when it is sufficiently risk-averse. Second, effects of risk attitude, demand volatility and competition on performance (expected profit and standard deviation of profit) of CSLs are also studied. We find that CSL’s expected profit would be affected by its risk attitude. In particular, when CSL is risk-averse, its expected profit increases if it can accept more risk. Interestingly, there exists a theoretically optimal risk sensitivity coefficient to maximize CSL’s expected profit, which is slightly risk-seeking. We discover and characterize the risk-attitude convergence effect which indicates that a CSL’s optimal risk sensitivity coefficient is increasing in its rival’s in duopoly. Therefore, managers of liner shipping companies are encouraged to behave to be slightly risk-seeking in shipping service operations. This finding helps to explain why it is reported that in CSLs, risk seeking behaviors are present (Cullinane, 1991; Ishizaka et al., 2018). We explore the effects of demand volatility and find that CSLs get a larger standard deviation of profit with a higher demand volatility when they are risk-seeking. We consider a centralized case to examine how far the decentralized equilibrium results deviate from it. Interestingly, we find that for individual CSL, its expected profit with mean-risk objective function could be larger than that in the centralized case, which occurs when CSL’s risk sensitivity level is low and its rival is risk-seeking. In addition, we consider multiplicative randomness of demand and find that all our main findings in the basic model still hold. Interestingly, with multiplicative randomness of demand, we find that CSLs may not prefer a large market potential when they are risk-averse, which is contrary to that with additive randomness of demand. 1.3. Contribution statement and paper’s structure To the best of our knowledge, this paper is the first study that investigates competition between CSLs possessing different types of risk attitudes. We employ a mean-risk framework to model the competing CSLs’ different risk attitudes (risk-seeking, risk-neutral and risk-averse). This is the first time that the concept of mean-risk is introduced to measure 6 It is standard practice to consider two players in the operations management science literature (e.g., Shao et al., 2014; Guthrie et al., 2017), as the two-player case is fundamental when examining multi-player games and could uncover the impact of competition.

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different risk attitudes in the liner shipping studies. The findings provide novel managerial insights to advance our understanding and knowledge towards the impacts brought by different risk attitudes to CSL operations. In particular, we discover the risk-attitude convergence effect. We find that when the CSLs hire slightly risk seeking operations managers, the expected profits of both CSLs will be maximized. This finding is in line with the very recent proposal in the literature on the probable benefits of hiring optimistic (and unrealistic) managers in a competitive market (Jiang and Liu, 2018). The organization of this paper is given as follows. We review the most related literature in Section 2. In Section 3, we construct the basic game-theoretic model between two competing CSLs. In Section 4, we derive the equilibrium prices, examine performance of CSLs, and conduct the equilibrium analysis. In Section 5, we explore the effects of risk attitude, demand volatility and competition on the CSLs’ equilibrium prices and performance. Extended models, which include the scenarios with multiplicative randomness of demand, asymmetric information on risk-attitude and multiple competing CSLs, are presented in Section 6. Concluding remark, managerial implications and future research are presented in Section 7. 2. Literature review This study is related on three primary streams of research: (1) revenue management in shipping liner operations; (2) different risk attitudes of decision makers; and (3) equilibrium pricing decisions of shipping liner companies under a competitive setting. We concisely review some related studies as follows. For studies on revenue management in shipping liner operations, a classic study is by Jansson and Shneerson (1985), who prove that economies of scale of trading density exist in the liner shipping industry. After that, Wang et al. (2015a) consider the situation when there exists more than one itinerary for containers shipment to the destination. By using the logit model, they model behaviors of customers and develop algorithms to determine the optimal itineraries and optimal freight rates. Wang et al. (2015b) investigate the seasonal shipping of liner containers and examine the profit optimization problem which considers multi-type of containers, number of containerships as well as the speed of containerships. Deng et al. (2015) analytically investigate the optimal pricing and product shipment scheduling problem in which a capacity-constrained supplier serves multiple clients. The authors derive an efficient algorithm to identify the optimal decision and uncover that customers having a larger waiting cost should be granted a higher priority in terms of shipping schedule. Goh and Chan (2016) conduct a study to reveal how carriers can find the minimum acceptable rates for back haul (BH) cargos. They highlight the relationship between variability of BH freight rates and other critical factors. Recently, Chen et al. (2017) focus on exploring the use of a public announcement called the “general rate increase (GRI)” and show that there is no evidence to prove that carriers in the liner shipping industry can affect spot rates by GRIs. Our paper relates to the above reviewed studies in which we also focus on revenue management and pricing for the liner shipping industry. However, different from all of them, we focus on the competitive scenario and we explore how risk-attitude of liner shipping companies affects the optimal pricing decisions as well as their performances at the equilibrium. A number of studies have concentrated on decision-makers’ risk attitudes in supply chains by employing the following three typical methods: value-at-risk (VaR, see the pioneering work Luciano et al. (2003) and Tapiero (2005)), conditional value-at-risk (CVaR, see Wu et al. (2006)) and mean-variance (MV, see Chiu and Choi (2016)). Choi and Chiu (2012) provide a comprehensive review on risk analysis employing VaR, CVaR and MV in stochastic supply chains. Agrawal and Seshadri (20 0 0a) consider a single-period inventory model in which a risk-averse retailer makes an order-quantity and a pricing decision under demand uncertainty. Agrawal and Seshadri (20 0 0b) consider a single-period model in which multiple risk-averse retailers purchase a single product from a common vendor and demonstrate that intermediaries in supply chains can reduce the financial risk faced by retailers using the mean-variance approach. Choi et al. (2008) analytically explore the supply chain coordination challenge when the agents can be risk-averse, risk-neutral or risk-seeking. The authors uncover that the larger the difference between the agents’ risk attitudes, the more difficult to achieve coordination by a supply contract. Choi et al. (2018) study the quick response supply chain with a stochastically risk sensitive retailer. They prove that ignoring the presence of different probable risk attitudes of the retailer (including the risk-seeking behavior) will lead to poor performance of the supply chain coordination contracts. In logistics supply chain management, most studies concentrate on decision-makers’ risk-averse behavior. For example, Liu and Wang (2015) consider a quality control game model in a logistics service supply chain, composed of a logistics service integrator and a functional logistics service provider. Exploring different combinations of risk attitude, the authors find that the integrator prefers a risk-seeking provider. Cui et al. (2016) employ the MV formulation in a two-tier logistics system to study a risk-averse retailer’s optimal decision of introducing its brand production, in which the effects of substitution factor, capital constraint and development cost are examined. Liu et al. (2016) incorporate risk aversion into a dual-channel supply chain to investigate supply chain agent’s pricing strategies and analyze the effects of risk aversion. Zheng et al. (2017) consider a case with one dominating shipping line which is risk-neutral and does not possess the capacity constraint, and the other shipping line is risk-averse with limited capacity, and study the effect of risk-averse attitude on shipping lines’ pricing strategies under competition. However, few studies consider decision-makers’ risk-seeking behavior. In this paper, we adopt the mean-risk approach because it is intuitive and analytically tractable to study CSLs’ different risk attitudes. It also fits our model well. For more details about the use of mean-risk analysis in supply chain and logistics management, refer to Chiu and Choi (2016). Employing the non-cooperative game theory, several studies explore the equilibrium pricing decisions of shipping liner companies under a competitive setting. For instance, Wang et al. (2014) conduct an analytical study to investigate the competition between two carriers in the liner container industry. By building and comparing the equilibrium solutions under

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different scenarios, they find that the Stackelberg equilibrium is a dominating one which also benefits the players most. Chen et al. (2016) consider the pricing problem in which carriers ship two kinds of shipments and need to decide the optimal pricing scheme. The authors study both competitive duopolies, and the single company (monopoly) models. They highlight that the market sensitivity to price and competition could significantly affect the optimal price. Peng et al. (2016) examine the vessel-cargo matching problem. The authors build a bidding game model and they interestingly reveal that the disadvantaged participant may gain more surplus in the bidding process. Most recently, Zheng et al. (2017) study the effect of risk-averse attitude on shipping line companies’ pricing strategies under competition. They model the case with one dominating shipping line company which is risk-neutral and does not possess the capacity constraint, and the other shipping line company is risk-averse with limited capacity. They establish the equilibrium and prove that under the uniformly distributed demand, the equilibrium prices of both companies will drop when the risk-averse company becomes more risk-averse. Similar to the above reviewed stream of studies on price competition in shipping liner companies, our paper also explores the competitive pricing issues in shipping liner operations. However, different from all of them, we consider different risk attitudes (including risk-averse, risk-neutral, and risk-seeking) in our analysis. Without assuming any specific form of demand distribution, we analytically derive all solutions in closed form and generate the corresponding managerial insights. Furthermore, we also highlight the role played by demand uncertainty. In modelling, we use the mean-risk formulation to model risk-averse, risk-neutral and risk-seeking attitudes, and also consider the multiplicative form of demand. For more studies on container transportation, please see the excellent recent review by Lee and Song (2017) 3. Basic model We consider two CSLs, indexed by CSL 1 and CSL 2, which compete with each other in freight rate for the spot market demand in a specific shipping service route. The long-term contracted price p0 is the shipping price per unit as a benchmark, which can be assumed to be constant (Zheng et al., 2017). Let i, j = 1, 2 and i = j. pi denotes the shipping price per unit for CSL i, which is a decision variable. The spot market demand faced by CSL i is denoted by Di . Di intuitively decreases in pi and increases in p0 , which is a common assumption in economics7 . To be specific, in the basic model, we employ the linear price dependent demand function to model the relationship between service demand and price under competition (Zhou and Lee, 2009; Zhang et al., 2010; Chen et al., 2016; Liu et al., 2016; Wang et al., 2017; Zheng et al., 2017; Wang et al., 2019)8 .

Di = θi (a + ε ) − αi ( pi − p0 ) + γi ( p j − pi ),

(3.1)

where a is the base market potential, and ε is a random variable following a symmetric distribution with a variance σ 2 and a zero mean. For example, if ε follows the normal distribution, it represents the white noise. We assume that θ i ∈ (0, 1) represents the base market share of CSL i. α i measures the price sensitivity of CSL i, and γ i measures the competition intensity of the price for CSL i (McGuire and Staelin, 1983). Without loss of generality, we have α i > γ i .9 Let π i denote the profit of CSL i. With Eq. (3.1), we can find the profit functions for CSL i and list them in the following.

πi = ( pi − ci )Di = ( pi − ci )(θi (a + ε ) − αi ( pi − p0 ) + γi ( p j − pi )),

(3.2)

where ci is the shipping cost per unit for CSL i. In practice, the CSLs may have both fixed cost and marginal cost of shipping service. However, the fixed cost has straightforward effect on the equilibria and can be mathematically omitted. Hence, we merely consider the marginal shipping cost. Let E[π i ] and SD[π i ] denote CSL i’s expected profit and standard deviation of profit, respectively. We further analytically derive the expected profit and standard deviation (SD) of profit for each CSL as follows:

E [πi ] = ( pi − ci )(aθi − αi ( pi − p0 ) + γi ( p j − pi )),

(3.3)

SD[πi ] = ( pi − ci )σ θi .

(3.4)

Different from Zheng et al. (2017), we adopt the mean-risk approach to characterize CSLs’ risk attitude. We define the mean-risk objective function for each competing CSL below10 .

Ui ( pi ; p j ) = E [πi ] + λi SD[πi ] = ( pi − ci )



 θi (a + λi σ ) − αi ( pi − p0 ) + γi ( p j − pi ) ,

(3.5)

7 Note that in economics, when demand goes up with an increasing selling price, the situation is described by the presence of the Veblen effect (see, e.g., Leibenstein, 1950; Zheng et al., 2012). It mainly refers to some special situations when, e.g., conspicuous consumption exists in the market and there are strong social influences between different groups of customers. However, in the container shipping industry, there is no evidence showing that the Veblen effect commonly exists, and the majority of existing related studies all focus on the situation in which demand will decrease if the selling price increases. As such, in this paper, we follow the mainstream literature and also focus on this common case. 8 Note that this linear price dependent demand function is commonly seen in practice. Plus, it is consistent with the uniformly distributed valuation by customers. In this paper, we follow the extant literature and apply it directly. 9 It is intuitive that p0 can be combined mathematically with a. However, p0 and a have different effects on CSLs’ equilibrium. We keep p0 in the demand function to make the model more general and highlight its influence on CSLs’ spot price decisions. 10 In this paper, we employ the mean-SD objective function to reflect risk attitude. Other methods include VaR (Kouvelis and Li, 2018) and mean-variance (Chiu and Choi, 2016), etc. We adopt the mean-SD model because it can reflect different risk attitudes conveniently and analytically tractable.

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where λi is the risk sensitivity coefficient. Observe that λi measures the risk attitude of CSL i and reflects the influence of volatility to CSL i’s utility. If λi > 0, CSL i is risk-seeking and its utility increases with both the profit and the level of profit uncertainty. If λi < 0, CSL i is risk-averse and its utility increases with the profit but decreases with the level of profit uncertainty. If λi = 0, CSL i is risk-neutral and its utility increases with the profit and is independent of the level of profit uncertainty. A larger λi indicates that CSL i can accept more risk. If the absolute value of λi is larger, it means the corresponding level of risk sensitivity is higher. Since the demand uncertainty follows a symmetric distribution, the use of mean-SD approach is theoretically sound as minimizing SD is equivalent to minimizing the downside risk (e.g., Chiu et al., 2018; Wang et al., 2019). The event sequence is given as follows. The CSLs determine their optimal shipping price p∗i simultaneously to maximize their mean-risk objective function. The decision-makers of the CSLs are assumed to be rational and all the parameters are common knowledge. We extend the assumption of symmetric information and consider asymmetric information of risk attitude in the extended model. Before we identify the equilibrium prices of the risk sensitive CSL price competition game, we report the structural properties of the mean-risk objective functions Ui (pi ; pj ) in Lemma 1. Lemma 1. Under the basic model: (a) Ui (pi ; pj ) is concave functions of pi and pj , respectively. (b) The duopoly pricing game is supermodular and a unique Nash equilibrium exists. Lemma 1 (a) shows the concavity of the mean-risk objective functions of both CSLs. This feature is critical as it implies that we can find the best response functions by the first order conditions. The unique Nash equilibrium for the pricing game under the basic model exists (which is summarized in Lemma 1 (b)). After establishing Lemma 1, in Section 4, we derive in closed-form the equilibrium prices and conduct the respective analysis. 4. Equilibrium analysis Based on the concavity of the mean-risk objective functions of the competing CSLs, it is straightforward to find the Nash equilibrium and we have Proposition 1. For a notational purpose, let Ai = α i + γ i > 0, Bi = θ i (a + λi σ ) + α i p0 + Ai ci and B0 = 4A1 A2 − γ 1 γ 2 > 0. Proposition 1. (Equilibria) Under the basic model for each CSL i, (a) the equilibrium price is p∗i = expected profit is (2A j Bi +B j γi −B0 ci )θi σ B0

E[π ∗ ] i

=

(2A j Bi −B0 ci +B j γi )(Ai (2A j Bi +B j γi )+B0 (aθi + p0 αi −Bi )) B20

2A j Bi +γi B j ; B0

(b) the equilibrium

; (c) the equilibrium standard deviation of profit SD[πi∗ ] =

.

Proposition 1 (a) shows the closed-form expressions of the equilibrium prices under the pricing competition of CSLs. In this paper, we assume that pi > ci to make sure CSLs can make positive expected profits (or else they will not continue their business). We have Corollary 1 which highlights the role played by the base market share θ i and risk attitude λi of each CSL. Corollary 1. Under the basic model, we have (a) p∗i > p∗j if θi > θthreshold−1 = p∗ otherwise; j

(b) if the CSLs are symmetric with α 1 = α 2 , γ 1 = γ 2 , c1 = c2 and

2Ai B j −B j γi +Ai ci (−2A j +γ j )+ p0 αi (−2A j +γ j ) , and (2A j −γ j )(a+λi σ ) ∗ ∗ λ1 = λ2 , then pi > p j if θ i > 0.5, and p∗i

p∗i ≤

≤ p∗j

otherwise. Corollary 1 (a) shows that there is a threshold value θ threshold − 1 that characterizes CSLi’s market share into two forms. When CSLi’s market share is larger than the threshold value, CSLi prices higher than its rival does. It is because a large market share helps protect the demand even when the shipping price is high. This threshold value is associated with CSLi’s risk attitude. Interestingly, we find that the threshold value is decreasing in λi . This implies that when CSLi can accept more risk, it is easier to price higher than its rival does. In particular, if the CSLs are symmetric with α 1 = α 2 , γ 1 = γ 2 , c1 = c2 and λ1 = λ2 , Corollary 1 (b) gives a neat result on how market share of the CSL affects the equilibrium price in the CSL duopoly market. To be specific, the CSL that has a larger market share would tend to price higher than its rival does. We then conduct a performance analysis on CSLs. Proposition 1 (b) and (c) show the closed-form expressions of the equilibrium expected profits and equilibrium standard deviation of profits, respectively. Similarly, we have Corollary 2 which highlights the role played by the base market share θ i of each CSL. Corollary 2. Under the basic model, if the CSLs are symmetric with α 1 = α 2 = α , γ 1 = γ 2 = γ , c1 = c2 = cand λ1 = λ2 = λ, then (a) E[πi∗ ] > E[π j∗ ]if θ i > 0.5 and λ ∈ (λA ,λB ); or if θ i < 0.5 and λ ∈ ( − ∞, λA )∪(λB ,∞), and E[πi∗ ] ≤ E[π j∗ ] otherwise; (b) SD[πi∗ ] > SD[π j∗ ] if θ i > 0.5, and SD[πi∗ ] ≤ SD[π j∗ ] otherwise. Among them, λA =



and G3 = 4a(a+2( p0 − c )α )(α + γ ) + ( p0 − c ) 2

2

G2 −G3 G1

< 0, λB =

G2 +G3 G1

> 0, G1 =2(α + γ )σ ,G2 =(c − p0 )αγ ,

α2 γ 2.

Corollary 2 (a) implies that CSL i gets a larger equilibrium expected profit than CSL j when its market share is larger and the level of risk sensitivity is sufficiently low, or when its market share is smaller but the level of risk sensitivity is sufficiently high (see Fig. 2 for a graphical illustration). This interesting result highlights the role played by market share and level of risk sensitivity in affecting the equilibrium expected profits of the CSLs. CSL i prices higher when it has a

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Fig. 2. Analysis of equilibrium expected profit (E[πi∗ ] > E[π j∗ ]).

Fig. 3. Numerical analysis on the equilibrium expected profit.

larger market share (see Corollary 1 (b)), while a low level of risk sensitivity implies that the decline of demand due to an increasing price is slight. In contrast, CSL i prices lower when it has a smaller market share (see Corollary 1 (b)), which intuitively may hurt its expected profit margin. However, the increase of demand due to a decreasing price is significant when risk sensitivity level if high. Consequently, the increased demand makes CSL i obtain a larger expected profit than CSL j. When the assumption that the CSLs are highly symmetric is relaxed, we conduct a numerical analysis and show that the results in Corollary 2 can still exist. However, the threshold values in the conditions change. As shown in Fig. 3, we can still

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obtain that E[πi∗ ] > E[π j∗ ]if θ i > θ threshold − 2 and λ ∈ (λA ,λB ); or if θ i < θ threshold − 2 and λ ∈ ( − ∞, λA )∪(λB ,∞), in which

θ threshold − 2 , λA and λB can be calculated by the specific values of the given parameters. In addition, we explore the effects of demand volatility and have several observations. First, with a more volatile demand, the interval between λA and λB in

Corollary 2 becomes smaller. Second, with a more volatile demand, the effects of risk preference on CSLs’ expected profits are much more significant. Note that the asymmetric case is highly intractable and a numerical analysis can only provide evidence for the existence of such a case with a specific parameter setting. When the specific values of the parameters change, the corresponding conditions would also change. Regarding the equilibrium standard deviation of profits, Corollary 2 (b) gives a neat result on how market share of the CSL affects them. To be specific, CSL who has a larger market will price higher, which yields a higher standard deviation of profit. Similarly, this result can be qualitatively relaxed to a common situation. 5. Effects of risk attitude, demand volatility, and competition In Sections 3 and 4, we have constructed the expected profit, standard deviation of profit and equilibrium pricing decision of each CSL. Now, we conduct a deeper analysis to uncover how the risk attitude, demand volatility, and level of competition affect the equilibrium pricing decisions and performance of the CSLs. 5.1. Price Sensitivity Analysis From Proposition 1, the effects of risk attitude and demand volatility on equilibrium prices are summarized in Proposition 2. Proposition 2. (Price sensitivity) Equilibrium price of CSL i (a) increases in both its risk sensitivity coefficient and the rival’s; (b) increases (if λi >

−γi θ j λ j 2 A j θi

) or decreases (if λi <

λi < − σa ) in its market share.

−γi θ j λ j 2 A j θi

) in demand volatility; and (c) increases (if λi > − σa ) or decreases (if

Proposition 2 (a) exhibits that the equilibrium price of CSL i is increasing in both its risk sensitivity coefficient λi and the rival’s. It is because when CSL i can accept more risk, it tends to believe that demand may be larger and then sets a price higher to increase its profit. Interestingly, CSLi also prices higher when the rival can accept more risk. The reason is that, the rival prices higher when it can accept more risk, which pushes up CSLi’s equilibrium price in the duopoly market with price competition. Proposition 2 (b) implies that when both CSLs are risk-seeking (λi > 0, λj > 0), CSLi’s equilibrium price is increasing in the demand volatility. The intuition behind is clear. As demand becomes more volatile, CSLs price high to raise the expected profit. It is because they are risk-seeking and incline to believe that demand may be substantially larger. In contrast, when both CSLs are risk-averse (λi < 0, λj < 0), the CSL’s equilibrium price is decreasing in the demand volatility. As demand becomes more volatile, CSLs incline to believe that demand may be substantially smaller when they are risk-averse. Therefore, CSLs price low to compete for more demand, which leads to low equilibrium prices. Proposition 2 (c) characterizes CSL i’s optimal price sensitivity on its market share. Intuitively, we have CSL i’s optimal price increases in its market share if it is risk-seeking. A larger market share indicates a larger fluctuation of demand volume. CSL i prefers the high fluctuation of demand volume when it is risk-seeking, and then sets a high price. Similarly, CSL i does not like the high fluctuation of demand volume when it is sufficiently risk-averse, thus it sets a low price. Consequently, CSL i’s optimal price increases in its market share if it is risk-seeking, and decreases if it is sufficiently risk-averse. We also explore the effects of competition on equilibrium prices, and the results are summarized in Proposition 3 as follows. Proposition 3. (Price sensitivity) Assuming that α 1 = α 2 = α and p0 = 0, the equilibrium price of CSL i (a) decreases in the level of price sensitivity when both CSLs are not risk-averse; (b) is positively (if B0 ci − Bi (4Aj − γ j ) + 2α Bj > 0) or negatively (if B0 ci − Bi (4Aj − γ j ) + 2α Bj < 0) affected by the competition intensity γ i ; and is positively (if B0 cj − Bj (4Ai − γ i ) + 2α Bi > 0) or negatively (if B0 cj − Bj (4Ai − γ i ) + 2α Bi < 0) affected by the competition intensity γ j . In Proposition 3, note that α represents the degree of demand changes in response to the changes of price. Proposition 3 (a) states that CSLs’ equilibrium prices decrease in price sensitivity when they are both risk-seeking or riskneutral. We have discussed that when CSLs are risk-seeking, they incline to believe that demand may be substantially large. A large α implies that the demand will be significantly reduced due to a high price, and this is a situation that the riskseeking CSLs do not want to see. Therefore, the risk-seeking CSLs price low with an increasing α to increase the demand. Interestingly, we find that the CSLs’ equilibrium prices may increase in α (see Fig. 4 for a graphical illustration). For example, when CSLs are substantially risk-averse, they price high with an increasing α . A large α implies that the increase of price will significantly reduce the demand. However, CSLs incline to believe that demand may be substantially small when they are risk-averse. Therefore, the equilibrium prices of risk-averse CSLs increase in α , which protects their expected profit. Proposition 3 (b) and (c) state that the equilibrium prices would be positively or negatively affected by the competition intensity parameter γ 1 and γ 2 . To be specific, CSL i’s equilibrium price is increasing in γ i if B0 ci − Bi (4Aj − γ j ) + 2α Bj > 0, and is increasing in γ j if B0 cj − Bj (4Ai − γ i ) + 2α Bi > 0. Note that B0 ci − Bi (4Aj − γ j ) + 2α Bj is decreasing in λi ; we conclude

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Fig. 4. Sensitivity of equilibrium price on α .

that when CSL i is sufficiently risk-seeking, its equilibrium price decreases in γ i ; while the equilibrium price increases in γ i when it is sufficiently risk-averse. However, when CSL i is sufficiently risk-seeking, it may believe that demand may be substantially large, while a high competition intensity results in small demand. Consequently, CSL i prices low to increase the demand, which makes its equilibrium price decrease in its competitive intensity. In contrast, a high competition intensity results in small demand, which is the situation that CSL i inclines to believe when it is sufficiently risk-averse. Therefore, CSL i prices high to protect its profit, which makes its equilibrium price increase in its competitive intensity. Similarly, the term B0 cj − Bj (4Ai − γ i ) + 2α Bi is decreasing in λj . We can then conclude that when the rival CSL is sufficiently risk-seeking, CSL i’s equilibrium price decreases in γ j , whereas CSL i’s equilibrium price increases in γ j when the rival is sufficiently risk-averse. 5.2. Performance sensitivity analysis Effects of risk attitude on equilibrium expected profits are summarized in Proposition 4. Proposition 4. (Performance analysis) Under the basic model, we have (a) CSL i’s expected profit can increse (if p∗i − ci > 2 A j θ i λi σ

γi γ j

) or decrease (if p∗i − ci <

2 A j θ i λi σ

γi γ j

) in its risk sensitivity coefficient; (2) there exsits a risk sensitivity coefficient λ∗i max-

imizing CSL i’s expected profit, in which λ∗i = profit always increases λj .

γi γ j (2Ai A j ci −B0 ci +2A j p0 αi +B j γi +2aA j θi ) and 4A j (2Ai A j −γi γ j )θi σ

λ∗i increses in λj ; and (3) CSL i’s expected

Proposition 4 (a) exhibits that CSL i’s expected profit may be positively and negatively affected by its risk sensitivity coefficient λi . To be specific, we find that E[πi∗ ] always increases in λi when λi < 0 (i.e., the risk-averse case). Note that ( p∗i − ci )γi γ j − 2A j θi λi σ is decreasing in λi , which implies E[πi∗ ] decreases in λi when λi is sufficiently large (i.e., the riskseeking scenario). Therefore, when λ∗i =

γi γ j (2Ai A j ci −B0 ci +2A j p0 αi +B j γi +2aA j θi ) and given other market factors, CSL i’s expected 4A j (2Ai A j −γi γ j )θi σ

profit is maximized. The optimal risk sensitivity coefficient is λ∗i > 0, indicating a risk-seeking attitude (see Fig. 5 for a graphical illustration). We find that CSLs possessing the risk-seeking attitude can help yield the maximized expected profit, which seems counter-intuitive. Generally, managers of CSLs may tend to exhibit risk-averse behaviors when facing risk. However, our findings suggest that a slightly risk-seeking behavior helps CSLs to obtain the maximized expected profit in the duopoly market. Interestingly, we further find that CSL i’s optimal risk sensitivity coefficient is increasing in CSL j’s risk sensitivity coefficient. We refer this result as the “risk-attitude convergence effect”, which indicates that the optimal risk attitudes for CSLs in duopoly move in “the same direction”. In other words, if one CSL accepts more (less) risk, it is better off for the other one to accept more (less) risk in a competitive duopoly. Therefore, we advocate that both CSLs should hire slightly risk-seeking managers in a competitive shipping market. Observe that in the literature, manager’s optimism about demand is found to help increase firms’ performance in a competitive market under some conditions (Jiang and Liu, 2018), because entrepreneurs with optimistic bias believe that negative events are less likely to happen to themselves than to others. Similarly, we find that slightly risk-seeking behavior can increase CSLs’ performance. Our results are hence consistent “in spirit” with Jiang and Liu (2018) and we uncover that the optimal risk attitudes of the competitive CSLs move in the same direction. In contrast, CSL i’s expected profit is always increasing in the rival’s risk sensitivity coefficient λj (see Proposition 4 (c)). From Proposition 2 (a), we know that CSL i’s equilibrium price is increasing in the rival’s risk sensitivity coefficient. This

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Fig. 5. Sensitivity of CSL i’s expected profit on λi .

Fig. 6. Sensitivity of CSL i’s expected profit on σ .

helps protect CSL i’s expected profit and consequently results in a larger expected profit. We then explore the effects of demand volatility and market share on equilibrium expected profits are summarized in Proposition 5. Proposition 5. (Performance analysis) CSL i’s expected profit (a) increases (if p∗i − ci > ci <

( ( p∗ i

θ i λi σ I j γi Ii ) in the demand volatility; and (b) increases (if

θ i λi σ I j ∗ γi Ii ) or decreases (if pi −

( ( p∗i − ci )γi γ j − 2A j θi λi σ )(a + λi σ ) > 0) or decreases (if − ci )γi γ j − 2A j θi λi σ )(a + λi σ ) < 0) in its market share, in which Ii = γ j λi θ i + 2Ai θ j λj .

Regarding the effects of demand volatility, it would positively and negatively affect CSL i’s expected profit. To be specific, we conduct a numerical study and find that there exists a demand volatility level which maximizes the CSLs’ expected profit (see Fig. 6 for a graphical illustration). In addition, we observe that given a certain level of demand volatility, a small level of risk sensitivity helps CSLs to obtain a large profit. In most cases, CSLs prefer a small demand volatility.

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Fig. 7. Sensitivity of CSL i’s expected profit on α .

10

10

10

9

9

9

8

8

8

7

7

7

6

6

6

5 0.0

0.2

0.4

0.6

0.8

1.0

5 0.0

0.2

0.4

0.6

0.8

1.0

5 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 8. Sensitivity of CSL i’s expected profit on γ .

With respect to the market share, effects of market share on CSL i’s expected profit are affected by its risk sensitivity coefficient. Generally, CSL i’s expected profit increases in its market share in most cases, which are consistent with the observed practices in the shipping industry. However, we find that when CSL i is sufficiently risk-averse or risk-seeking, its expected profit decreases in its market share. A large market indicates a large fluctuation of demand volume, and CSL i prices substantially high (low) to address the situation when it is sufficiently risk-averse (risk-seeking). A too high or too low shipping price hurts the expected profit, and consequently, CSL i’s expected profit decreases in its market share. Similarly, we also conduct a numerical study to explore the effects of competition on expected profit. Fig. 7 shows that CSLs’ expected profit is decreasing in the price sensitivity parameter α no matter what risk attitudes they hold. The rate of expected value drop becomes flattened when α becomes large. When α is small, we observe that its value change has significant impacts on CSLs’ expected profit. Regarding CSLs’ competition intensity, we assume that γ 1 =γ 2 =γ and the numerical results are summarized in Fig. 8, which shows that CSL i’s expected profit would be positively and negatively affected by its competition intensity. When the rival is risk-averse, CSL i’s expected profit decreases in γ (see Fig. 8 (a)); while when the rival is risk-seeking, CSL i’s expected profit increases in γ (see Fig. 8 (c)). In addition, we observe that when both CSLs are risk-neutral (see Fig. 8 (b)), CSL i’s expected profit decreases in γ . In addition, no matter what risk attitude the CSLs hold, we observe that the rate of expected profit value change becomes flattened when γ becomes large. This implies that under mild competition, the effects of competition intensity on expected profit are more significant. After exploring the effects of risk attitude, demand volatility and competition on CSLs’ expected profits, we investigate their effects on the standard deviation of profits. From the definition of standard deviation of profit SD[πi∗ ] = ( p∗i − ci )σ θi , we can easily derive that risk attitude and competition intensity parameters influence the standard deviation of profits

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Fig. 9. Expected profit deviation from the centralized case.

solely via the equilibrium prices. For example, we have the result that SD[πi∗ ] is increasing in λi and λj according to Proposition 2 (a), because

∂ SD[π ∗ ] ∂ SD[π ∗ ] ∂ p∗ i i i ∂ λi = ∂ p∗i ∂ λi =

∂ p∗

σ θi ∂ λi . Therefore, the effects of risk attitude and competition intensity i

parameters on standard deviation of profits and their driving forces are clear. We thus focus on the effects of demand volatility and derive Proposition 6. Proposition 6. (Performance analysis) SD[π i ] can increase (if 2A θ λ σ +γ θ λ σ − j i iB i j j 0

p∗i − ci > −

2A j θi λi σ +γi θ j λ j σ B0

) or decrease ( p∗i − ci <

) in demand volatility.

Proposition 6 states that demand volatility would positively and negatively affect the standard deviation of profit at the equilibrium. To be specific, we find that SD[πi∗ ] is increasing in σ if λi > 0 and λj > 0. That is, CSLs get a larger standard deviation of profit with a higher demand volatility when they are both risk-seeking, which is intuitive because of the increasing nature of standard deviation of profit margin ( p∗i − ci ) toward σ (see Proposition 2 (b)). Interestingly, we find that CSLs may also obtain a larger standard deviation of profit with a higher demand volatility when they are riskaverse. When CSLs are both slightly risk-averse, the standard deviation of profit margin ( p∗i − ci ) is decreasing in σ (see Proposition 2 (b)). However, given specific ( p∗i − ci ), we know that SD[π i ] always increases in σ (SD[πi∗ ] = ( p∗i − ci )σ θi ). Therefore, when CSLs are both slightly risk-averse, their standard deviation of profits also increases in the demand volatility. 5.3. Centralized case In this section, we compare the basic model (i.e., a decentralized model when the CSLs separately decide their decision) with the centralized case (in which there is a central planner who decides the optimal prices for both CSLs) and examine how far the decentralized equilibrium deviates from it. Different from the basic model, CSLs determine their prices to maximize the system profit (U1 (p1 ; p2 ) + U2 (p2 ; p1 )) in the centralized case. We assume that both CSLs are risk-neutral in the centralized case and add a bar (−) to the optimal solutions (e.g., p¯ ∗i , E[π¯ i∗ ], SD[π¯ i∗ ]) to indicate their counterparts in the basic model. The equilibrium results are summarized in Table A1. We assume that the CSLs are risk-neutral in the centralized case, and hence the effects of risk attitude and demand volatility on the equilibrium deviation from the centralized case are clear. For example, the deviation of CSL i’s equilibrium price is increasing in its own risk sensitivity coefficient and the rival’s (e.g.,

∂ ( p∗1 − p¯ ∗1 ) ∂ p∗ = ∂ λ1 ). It is intuitive that the total ∂ λi i

expected profits of CSLs in the centralized case are larger than that in the basic model. Note that we have found that CSL i’s expected profit is maximized when it is slightly risk-seeking in the basic model. Therefore, one natural question arises: For an individual CSL, will a slightly risk-seeking CSL i in the basic model (decentralized) obtain a larger expected profit than it can achieve under the centralized case? We therefore conduct a numerical study and derive some observations (see Fig. 9). First, when both CSLs are risk-neutral in the basic model, their expected profit is always smaller than that in the centralized case. It is because the CSLs have

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a common objective in the centralized case. Second, we find that a specific CSL i’s expected profit under the basic model can be larger than that in the centralized case, which occurs when CSL i’s risk sensitivity level is small and its rival is riskseeking. In this situation, the rival CSL’s expected profit is smaller than that in the centralized case. The possible reasons are as follows. Note that CSL i’s equilibrium price is increasing in its risk sensitivity coefficient and its rival’s. It is intuitive that when CSL i is risk-averse or risk-neutral, its equilibrium price is smaller than that in the centralized case. However, when CSL i is slightly risk-seeking, its own equilibrium price may be larger than that in the centralized case, which helps achieve a larger expected profit. Third, CSL i’s deviation of expected profit from the centralized case is increasing in its rival’s risk sensitivity coefficient. Similarly, the risk sensitivity coefficient that maximizes CSL i’s expected profit in the basic model also maximizes the its expected profit deviation. 6. Extended models 6.1. Multiplicative randomness11 In the basic model, we consider an additive model for demand randomness. In existing studies, multiplicative randomness of demand is also widely considered (Liu et al., 2007). In this section, we consider multiplicative randomness of demand, which is called the multiplicative model. We focus on the differences of equilibrium prices between the multiplicative model and basic model (P.S.: additive demand model). With multiplicative randomness, the demand function is specified as follows:

Dˆ i = θi (aεM ) − αi ( pˆ i − p0 ) + γi ( pˆ j − pˆ i ),

(6.1)

σM2

where ε M is a random variable following a symmetric distribution with a variance and a mean of 1. We add a hat (^) to the parameters and optimal solutions (e.g., pˆ∗i , E[πˆ i∗ ], SD[πˆ i∗ ]) in the multiplicative model to make them different from their counterparts in the basic model. Therefore, CSL i’s profit function is listed in the following:

πˆ i = ( pˆ i − ci )Dˆ i = ( pˆ i − ci )(θi (aεM ) − αi ( pˆ i − p0 ) + γi ( pˆ j − pˆ i )).

(6.2)

CSL i’s expected profit and standard deviation of profit are given as:

E [πˆ i ] = ( pˆ i − ci )(aθi − αi ( pˆ i − p0 ) + γi ( pˆ j − pˆ i )),

(6.3)

SD[πˆ i ] = ( pˆ i − ci )aσM θi .

(6.4)

Similarly, it is straightforward to find the Nash equilibrium and we have Table A2 in Appendix A. When aσ M = σ , we have Bˆi = Bi (see Table A2), resulting in p∗i = pˆ∗i , E[πi∗ ] = E[πˆ i∗ ] and SD[πi∗ ] = SD[πˆ i∗ ]. In particular, when a = 1, with the same demand volatility (σ M = σ ), the equilibrium in the multiplicative model is the same as that in the basic model. We examine our main findings in the basic model and find that all of them still hold in the multiplicative model under similar conditions, which implies that our findings are robust. Despite the similarities, we also identify that there exists an important and interesting difference between the basic model and multiplicative model: The effects of market potential. We explore the effects of market potential on CSLs price decisions and have Proposition 7. Proposition 7. (a) With additive randomness of demand, CSL i’s equilibrium price increases in the market potential. (b) With multiplicative randomness of demand, CSL i’s equilibrium price can increase (if 2Aj θ i (1 + λi σ M ) + γ i θ j (1 + λj σ M ) > 0) or decreae (if 2Aj θ i (1 + λi σ M ) + γ i θ j (1 + λj σ M ) < 0) in the market potential. Proposition 7 implies that CSLs always price higher with a larger market potential in the basic model with additive randomness of demand. However, CSLs may price lower with a larger market potential in the multiplicative model. For example, when CSLs are both risk-averse (λi < σ−1 , λ j < σ−1 ), a larger market potential leads to lower equilibrium prices. For M M given market prices, CSLs’ standard deviation of profits are affected by the market potential in the multiplicative model (SD[πˆ i∗ ] = ( pˆ i − ci )aσM θi ). This indicates that the impacts of market potential on the mean-risk objective function are associated with CSLs’ risk attitude (Ui (pi ; pj ) = E[π i ] + λi SD[π i ]). However, CSLs’ standard deviation of profits are independent of the market potential in the basic model for a given market price (SD[π i ] = (pi − ci )σ θ i ). Therefore, when CSLs are both risk-averse and the market potential is large, though a high price may help achieve large expected profits, they have to price low to reduce the risk from standard deviation of profits. Similarly, it is easy to find that when CSLs are both risk-averse (λi < σ−1 , λ j < σ−1 ), a larger market potential leads to M M lower expected profits, which is also different from the ones under the additive randomness of demand case. Interestingly, CSLs may not prefer a large market potential when they are risk-averse and face the multiplicative randomness of demand. The reason is that: A large market potential drives CSLs to determine a low price to reduce the loss of standard deviation of profits, and a low price consequently leads to small expected profits. 11 Note that the additive randomness and multiplicative randomness cases respectively represent different models with different economic meanings (see Agrawal and Seshadri, 20 0 0a). As such, we have this extension to show the robustness of findings with respect to different form of demand randomness as well as uncover the corresponding insights.

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6.2. Asymmetric information on risk attitude In shipping practices, risk attitude for CSLs may not be directly observed. Therefore, in this section, we consider the case when the risk attitude is private information for each CSL. To explore the effects of asymmetric risk-attitude information, we assume that risk sensitivity coefficient λi is a random variable following a uniform distribution in the region [ui − τ i ,ui + τ i ], where ui is the average risk sensitivity coefficient and τ i is the deviation (Liu et al., 2016). Hence, the distribution funciton of λi is f (λi ) = 21τ . Because CSL i has incomplete information about CSL j’s risk attitude, the objective of CSL i is to determine i

the optimal price to maximize its expected objective value, i.e.,





max E Ui ( pi ; p j ) =



pi

ui +τi ui −τi

(E[π i ] + λi SD[π i ]) f (λ j )dλ j .

We add an underline (_) to the parameters and optimal solutions (e.g., p∗i , E[π ∗ ], SD[π ∗ ]) to make them different from i i their counterparts in the basic model. It is straightforward to find the Nash equilibrium (see the Appendix C), which is summarized in Table A3. From the equilibria, we can find that for CSL i, its optimal price and profits are affected by CSL j’s average risk sensitivity coefficient. Therefore, we can find that the results for CSL i in the basic model will still hold if we replace CSL j’s risk sensitivity coefficient by its average risk sensitivity coefficient. For example, results in Proposition 2 can be illustrated as CSL i’s optimal price increases in its risk sensitivity coefficient λi and CSL j’s average risk sensitivity coefficient. In particular, if CSL i’s average risk sensitivity coefficient is 0, i.e., risk-neutral, some results may be more direct and interesting. For example, from Corollary 1, we have pi > pj if θ i > θ threshold , and pi ≤ pj otherwise, in which θ threshold decreases in λi . In other words, when CSL i is risk-seeking, it may price higher even with a smaller market share. Increases of both market share and risk sensitivity coefficient have a positive effect on CSL i’s optimal price. In addition, CSL i’s optimal price increases in demand volatility when it is risk-seeking, while decreases when it is risk-averse (Proposition 2 (b)). 6.3. Multiple competing CSLs In practice, there may be multiple competing CSLs in the shipping market. Therefore, we consider a case with N competing CSLs in this section. Based on Eq. (3.1), the demand function is specified as follows (see, e.g., Fu and Zhang, 2010; Cai et al., 2012; Yang et al., 2015).

Di = θi (a + ε ) − αi ( pi − p0 ) + γi

N 

( p j − pi ), ∀i = 1, 2, ..., N and j = i.

(6.5)

j=1

We also assume that the CSLs are symmetric with price sensitivity (α 1 = α 2 =  = α N = α ), competition intensity (γ 1 = γ 2 =  = γ N = γ ) and marginal shipping cost (c1 = c2 =  = cN = c). Based on the concavity of the mean-risk objective functions, it is straightforward to find the equilibrium prices and we have Proposition 8. ( 2A+γ )Bi +γ

N 

(B j −Bi )

Proposition 8. With N competing CSLs, the equilibrium price for CSL i is pi = (2A+γ )(2A−(N−1 )γ ) , in which A = α + γ and Bi = θ i (a + λi σ ) + α i p0 + Aci . j=1

We conduct a price sensitivity and find that the analytical outcomes under the duopoly game can suggest similar results for a general multi-CSL setting. For example, we find the result that “p∗i is increasing in λi and λj ” still holds (j = 1, 2...N and j = i in the multiple-CSL setting). Other analytical outcomes suggest similar results in the multi-CSL setting with N  the corresponding conditions revised. For example, we have p∗i increases in σ if (2A + γ )θi λi + γ (θ j λ j − θi λi ) > 0, while j=1

decreases in σ if (2A + γ )θi λi + γ

N  j=1

(θ j λ j − θi λi ) < 0. Because the spot market price is decision variable, once the equilib-

rium price can be extended to the general multi-CSL setting, the analysis of expected profit and standard deviation of profit is similar and intuitive. Though a specific result may change especially the corresponding conditions, the findings under duopoly game can be easily extended. 6.4. Capacity-constrained CSLs12 In practice, the CSLs may be capacity-constrained when they compete in spot market (Zheng et al., 2017). Based on the basic model, we consider a symmetric case (α 1 = α 2 = α , γ 1 = γ 2 = γ and c1 = c2 = c) in which CSL 1 is capacity-constrained with the maximum capacity Q while CSL 2 is capacity-unconstrained. Each CSL i determines an optimal pi to maximize its 12

We sincerely thank an anonymous reviewer’s excellent comments that inspire us to include this important extension.

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objective value. Owing to the presence of capacity constraint, the optimization problem for CSL 1 is given below.

max U ( p1 ; p2 ) p1

s.t. E[D1 ] ≤ Q, p1 ≥ 0, p2 ≥ 0. Since CSL 2 is capacity-uncontrained, its objective function is the same as that in the basic model. We assume L(p1 ,p2 ,ϕ ) = U(p1 ; p2 ) + ϕ (Q − E[D1 ]) and apply the Kuhn-Tucker conditions as follows.

∂L ∂L ≤ 0, p1 ≥ 0, p1 = 0; ∂ p1 ∂ p1 ∂L ∂L ≤ 0, φ ≥ 0, φ = 0; ∂φ ∂φ Note that we only need to consider the situation in which p1 > 0 and ϕ > 0 hold because when ϕ = 0, all the equilibria and findings are the same as that in the basic model. By backward induction, the equilibrium prices are derived and presented in Proposition 9. Proposition 9. In the basic model with symmetric CSLs, when one CSL is capacity-constrained, the equilibrium prices are cαγ +cγ 2 −2Q (α +γ )+ p0 α (2α +3γ )+2aαθ1 +2aγ θ1 +aγ θ2 +γ θ2 λ2 σ −Q γ +c (α +γ )2 + p0 α (α +2γ )+aγ θ1 +a (α +γ )θ2 +(α +γ )θ2 λ2 σ , p2 = 2α 2 +4αγ +γ 2 2α 2 +4αγ +γ 2 (2α +3γ )α ( p0 −c )+2a(α +γ )θ1 +γ θ2 (a+λ2 σ ) θ 1 λ1 σ −Q (2α +γ )(2α +3γ ) + − α +γ . (α +γ )(2α 2 +4αγ +γ 2 ) 2α 2 +4αγ +γ 2

p1 =

and φ =

We examine the effects of risk attitude and demand volatility on the equilibrium prices and have several observations. First, interestingly, the equilibrium prices of both CSLs are independent of the capacity-constrained CSL’s risk sensitivity coefficient. This is because the equilibrium demand volume of the capacity-constrained CSL is D∗1 = Q, which is not affected by its risk attitude. The driving force is clear and important. The capacity-constrained CSL can not handle the cargos that exceed its capacity when it is risk-seeking and inclines to believe that demand would be large. Second, we show that the equilibrium prices of both CSLs increase in the capacity-unconstrained CSL’s risk sensitivity coefficient. This finding is consistent with that in the basic model. Third, the equilibrium prices of both CSLs increase (decrease) in the demand volatility when the capacity-unconstrained CSL is risk-seeking (risk-averse). Since the equilibrium prices are derived, the effects of risk attitude and demand volatility on the CSLs’ performance can also be explored. In addition, if both CSLs are capacity-constrained, for example, CSL i’s maximum capacity is Qi , the equilibrium findings and the associated managerial insights can also be derived in a similar way. 7. Conclusion 7.1. Concluding remarks and major findings Owing to the highly competitive market with high demand volatility, it is commonly known that container shipping lines (CSLs) face high risk in their shipping operations. Unlike many other industries in which risk averse attitude is dominating, it is well known that many CSLs are willing to take risk and exhibit risk seeking behaviors. In this paper, we develop a meanrisk framework to model the competing CSLs’ different risk attitudes, including risk-seeking, risk-averse and risk-neutral. We analytically investigate the effects of risk attitude, demand volatility and competition on service pricing decisions and performance of CSLs. We solve the game with CSLs possessing the mean-risk objective functions and obtain the equilibrium solutions, which are then compared to the centralized supply chain system with risk-neutral CSLs. The scenarios with multiplicative randomness of demand, asymmetric information on risk attitude, and multiple (more than two) competing CSLs are discussed to examine the robustness of our main findings. The following summarize the core insights revealed by this study. Impacts of CSL’s risk attitude on optimal pricing decisions: We examine the effects of risk attitude on equilibrium pricing decisions and find that CSL’s equilibrium price increases if it can accept more risk. It is because when CSL can accept more risk, it tends to believe that demand may be larger and then it prices higher to increase its profit. It is important to note that the CSL’s equilibrium price also increases if its rival can accept more risk. We then explore the effects of demand volatility and find that, the equilibrium prices increase in demand volatility when CSLs are risk-seeking, while decrease in demand volatility when CSLs are risk-averse. In addition, we explore the effects of competition and find that equilibrium prices increase in price sensitivity when CSLs are risk-seeking. Interestingly, equilibrium prices may also increase in price sensitivity when CSLs are risk-averse, which seems counter-intuitive. We also characterize the necessary and sufficient conditions on whether equilibrium prices will increase or decrease in the competition intensity. We find that when CSL is sufficiently risk-seeking, its equilibrium price decreases in the competition intensity; while when CSL is sufficiently risk-averse, its equilibrium price increases in the competition intensity. Impacts of CSL’s risk attitude on company performance: For the performance of CSLs, we find that CSL’s expected profit would be positively and negatively affected by its risk attitude, and when CSL is risk-averse, its expected profit always

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increases if it can accept more risk. Moreover, we find that the CSL’s expected profit always increases if its rival can accept more risk. Interestingly, there exists an optimal risk sensitivity coefficient to maximize CSL’s expected profit, which is found to reflect the slightly risk-seeking attitude. The risk-attitude convergence effect reveals that it is better off for one CSL to accept more (less) risk if its rival accepts more (less) risk. Therefore, a slightly risk-seeking behavior helps both CSLs to obtain larger expected profits in the competitive duopoly shipping market. This finding helps to explain why it is reported that in CSLs, risk seeking behaviors are present (Cullinane 1991; Ishizaka 2018). The expected profit would be positively and negatively affected by the demand volatility and competition level and hence, we conduct a numerical study and obtain some observations. One interesting observation is that when the rival is risk-averse, CSL’s expected profit decreases in the competition intensity; while when the rival is risk-seeking, its expected profit increases in the competition intensity. We further explore the effects of demand volatility on standard deviation of profit and find that CSLs get a larger standard deviation of profit with a higher demand volatility when they are risk-seeking. Decentralized versus centralized setting: We consider a centralized case to examine how far the decentralized equilibrium in the basic model deviates from it. Interestingly, we find that for an individual CSL, its expected profit with mean-risk objective function could be larger than that in the centralized case, which occurs when the CSL’s risk sensitivity level is low and its rival is risk-seeking. In addition, by considering the uncertain market price, we conclude that a higher level of market volatility drives CSLs to price higher when they are risk-seeking, while price lower when they are risk-averse. Multiplicative versus additive demand randomness models: In the basic model, we consider the additive demand randomness model. We then consider multiplicative randomness of demand in an extended analysis and find that the core main findings with respect to the impacts of competition intensity, demand uncertainty, etc. in the basic model with the additive demand still hold under the multiplicative demand model. Interestingly, with multiplicative randomness of demand, the effects of market potential on CSLs’ price decisions and performance may be contrary to that with additive randomness of demand. For example, we find that CSLs may not prefer a large market potential when they are risk-averse with the multiplicative randomness of demand. 7.2. Managerial Implications and Future Studies With the above findings, we have the following managerial implications: First, in a market with high market volatility, CSLs should price high when they are risk-seeking, while price low when they are risk-averse. Second, a bit counterintuitively, risk-seeking behaviors can do more good than harm. We suggest that both CSLs should hire slightly risk-seeking managers, which would help obtain larger expected profits in the competitive duopoly shipping market. Third, according to the risk-attitude convergence effect, we suggest that the CSL accept more (less) risk if its rival decides to accept more (less) risk in the competitive duopoly shipping market. Fourth, when the rival is risk-seeking, competing fiercely in price may be beneficial. In contrast, when the rival is risk-averse, it is better not to compete in price in the liner shipping market. Our model captures the two key characteristics in shipping industry: (1) risk-seeking behaviors and (2) both spot market price and long-term contracted price exist. We focus on pricing decisions under demand uncertainties. Note that the meanrisk analysis can be conducted with other sources of uncertainties. For example, one may assume the oil price uncertainty or the existence of a future spot market for cargo capacity with a volatile price. These are important factors that deserve future research. Moreover, it will also be interesting to study competition games with other decisions, such as the cargo capacity and service quality. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.trb.2020.01.003. Appendix A. Equilibria in different models Table A1–A3

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T.-M. Choi, S.-H. Chung and X. Zhuo / Transportation Research Part B 133 (2020) 210–229 Table A1 Equilibria in centralized case. Equilibria Notations: B¯ i = aθi + αi p0 + Ai ci − c j γ j ; B¯ 0 =4A1 A2 − (γ1 +γ2 )2 ; Equilirium prices: 2A2 B¯ 1 + (γ1 + γ2 )B¯ 2 ; p¯ ∗1 = B¯ 0 2A1 B¯ 2 + (γ1 + γ2 )B¯ 1 ∗ ; p¯ 2 = B¯ 0 Equilirium expected profits: (2A2 B¯ 1 − B¯ 0 c1 + B¯ 2 (γ1 + γ2 ))(A1 (B¯ 2 (γ1 − γ2 ) − 2A2 B¯ 1 ) + B¯ 1 γ1 (γ1 + γ2 ) + B¯ 0 (a(1 − θ ) + p0 α1 )) ; E[π¯ 1∗ ] = B¯ 20 ¯ ¯ ¯ ¯ ¯ ¯ ¯ (2A1 B2 − B0 c2 + B1 (γ1 + γ2 ))(A2 (B1 (γ2 − γ1 ) − 2A1 B2 ) + B2 γ2 (γ1 + γ2 ) + B0 (aθ + p0 α2 )) ; E[π¯ 2∗ ] = B¯ 2 0

Table A2 Equilibria in the case with multiplicative randomness of demand. Equilibria Notations: Bˆi = aθi (1 + λi σM ) + αi p0 + Ai ci ; Equilirium prices: 2A j Bˆi + γi Bˆ j pˆ∗i = ; B0 Equilirium expected profits: (2A j Bˆi − B0 ci + Bˆ j γi )(Ai (2A j Bˆi + Bˆ j γi ) + B0 (aθi + p0 αi − Bˆi )) E[πˆ i∗ ] = ; B20 Equilirium standard deviation of profits: (2A j Bˆi + Bˆ j γi − B0 ci )θi aσM SD[πˆ i∗ ] = ; B0

Table A3 Equilibria in the case with asymmetric information on risk attitudes. Equilibria Notations: B j = aθ j ( 1 + u j σ ) + α j p0 + A j c j ; Equilirium prices: 2 A j B i + γi B j ; p∗ = i B0 Equilirium expected profits: (2A j Bi − B0 ci + B j γi )(Ai (2A j Bi + B j γi ) + B0 (aθi + p0 αi − Bi )) E[π ∗i ] = ; B20 Equilirium standard deviation of profits: (2A j Bi + B j γi − B0 ci )θi σ SD[π ∗i ] = ; B0

Appendix B. All proofs and important deviations

Proof. of Proposition 1. Under price competition, the two CSLs determine the optimal price p1 and p2 to maximize the mean-risk objective functions, which lead to the following first-order derivatives:

∂ U1 ( p1 ; p2 ) = a + (c1 + p0 − 2 p1 ) α1 + (c1 − 2 p1 + p2 )γ1 − aθ + (1 − θ ) λ1 σ , and ∂ p1 ∂ U2 ( p1 ; p2 ) = p0 α2 + p1 γ2 + c2 (α2 + γ2 ) − 2 p2 (α2 + γ2 ) + aθ + θ λ2 σ . ∂ p2 Based on the first-order derivatives, we have

∂ 2U1 ( p1 ;p2 ) ∂ 2U2 ( p1 ;p2 ) < 0 and < 0. Therefore, we solve the equations ∂ ( p1 )2 ∂ ( p2 )2 2A2 B1 +γ1 B2 2A1 B2 +γ2 B1 ∗ ∗

∂ U1 ( p1 ;p2 ) ∂ U2 ( p1 ;p2 ) = 0 and = 0, and get the equilibrium prices p1 = ∂ p1 ∂ p2

Bi = θ i (a + λi σ ) + α i p0 + Ai ci , B0 = 4A1 A2 − γ 1 γ 2 and i = 1, 2.

B0

and p2 =

B0

, where Ai = α i + γ i ,

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227

Substituting p1 and p2 to Eq. (3.3) and (3.4) yields

E



π1∗

 π2∗   SD π1∗   SD π2∗ E





=

(2A2 B1 − B0 c1 + B2 γ1 )(2A1 A2 B1 − B1 B0 + B0 p0 α1 + A1 B2 γ1 + aB0 (1 − θ ) )

B20 (2A1 B2 − B0 c2 + B1 γ2 )(2A1 A2 B2 − B2 B0 + B0 p0 α2 + A2 B1 γ2 + aB0 θ ) = , B20 (2A2 B1 + B2 γ1 − B0 c1 )(1 − θ )σ = and B0 (2A1 B2 + B1 γ2 − B0 c2 )θ σ = , B0

,

Thus, CSLs’ equilibrium prices, expected profits and standard deviation of profits can be derived. Proofs of Corollaries 1 and 2. From Proposition 1, by simple arithmetic manipulation, we have B j γi −Ai (2B j +ci (−2A j +γ j ))+(2A j −γ j )( p0 αi +θi (a+λi σ ) ) . B0

Clearly, p∗i > p∗j if θi > θthreshold−1 =

p∗i − p∗j =

2Ai B j −B j γi +Ai ci (−2A j +γ j )+ p0 αi (−2A j +γ j ) . (2A j −γ j )(a+λi σ )

If the

CSLs are symmetric with α 1 = α 2 , γ 1 = γ 2 , c1 = c2 and λ1 = λ2 , we have pi > pj if θ i > 0.5. Therefore, the results in Corollary 1 can be obtained. By assuming that the CSLs are symmetric with α 1 = α 2 = α , γ 1 = γ 2 = γ , c1 = c2 = c and λ1 = λ2 = λ, we have E[πi∗ ] − E[π j∗ ] =

(2θi −1 )(a(a+2(−c+ p0 )α )(α +γ )+(c−p0 )αγ λσ −(α +γ )λ2 σ 2 ) (2α +γ )(2α +3γ )

and SD[πi∗ ] − SD[π j∗ ] =

(2θi −1 )σ (2a(α +γ )+( p0 −c )α (2α +3γ )+2(α +γ )λσ ) . (2α +γ )(2α +3γ )

With respect to the expected profit, we have f(λ) = a(a + 2( − c + p0 )α )(α + γ ) + (c − p0 )αγ λσ − (α + γ )λ2 σ 2 G −G G +G max f(λ) > 0. By solving f(λ) = 0, we have λA = 2G 3 < 0, λB = 2G 3 > 0, G1 =2(α + γ )σ ,G2 =(c − p0 )αγ ,



1

1

G3 = 4a(a+2( p0 − c )α )(α + γ ) + ( p0 − c ) α 2 γ 2 . With respect to the standard deviation of profit, SD[πi∗ ] > SD[π j∗ ] if θ i > 0.5, and SD[πi∗ ] ≤ SD[π j∗ ] otherwise. Hence, the results in Corollary 2 can be obtained. 2

2

2A j Bi +γi B j . By B0 2A j θi λi +γi θ j λ j B0 ∂σ =

we

and and have

Proof of Proposition 2. CSL i’s equilibrium price is p∗i =

deriving the first-order derivatives, we have

∂ λi =

and

∂ p∗i

2 A j θi σ B0

∂ p∗i

γi θ j σ

> 0, ∂ λ = B > 0. Similarly, we have 0 j Proposition 2 can be obtained.

∂ p∗i

∂ p∗

Proof of Proposition 3. Similar to the proof of Proposition 2, we have ∂ γi = i

∂ p∗i 2 A j ( a + λi σ ) . Hence, the results in B0 ∂ θi =

B0 ci −Bi (4A j −γ j )+2α B j

∂ p∗ X In addition, we also have ∂αi = 21 , where B0

B20

∂ p∗

and ∂ γi = j

B0 c j −B j (4Ai −γi )+2α Bi B20

.

X1 = −8aα 2 − 4c j α 2 γi − 16aαγ j − 2aγi γ j − 4ci αγi γ j − 8c j αγi γ j − 2ci γi 2 γ j − c2γi 2 γ j − 8aγ j 2 − 2ci γi γ j 2 − 4c j γi γ j 2 + 8aα 2 θi − 8aαγi θi − 4aγi 2 θi + 16aαγ j θi − 2aγi γ j θi + 8aγ j 2 θi − 8α 2 λi σ − 16αγ j λ j σ − 2γi γ j λi σ − 8γ j 2 λi σ + 8α 2 θi λi σ + 16αγ j θi λi σ + 2γi γ j θi λi σ + 8γ j 2 θi λi σ − 8αγi θi λ j σ − 4γi 2 θi λ j σ − 4γi γ j θi λ j σ . ∂ p∗

If λi ≥ 0 and λj ≥ 0, we have X1 < 0 and ∂αi < 0. Therefore, the results in Proposition 3 can be derived. Proof of Proposition 4. By checking the first-order derivatives, we have increases in λi if p∗i − ci >

2 A j θ i λi σ

2 A j θ i λi σ

∂ E[πi∗ ] θi σ ∂ λi = B 0

(( pi − ci )γi γ j − 2A j θi λi σ ). Hence, E[πi∗ ]

γi γ j , and decreases in λi if pi − ci < γi γ j . By solving ( pi − ci )γi γ j − 2A j θi λi σ = 0, we obtain γ γ (2A A c −B c +2A p α +B γ +2aA j θi ) γ 2γ θ θσ ∂λ∗ ∂ E[π ∗ ] λ∗i = i j i j 4i A (02Ai A −jγ 0γ )i θ σj i . It is easy to find that ∂ λi = 4A (2A Ai −jγ jγ )θ σ > 0. We also have ∂ λ i = Bj (2Ai ( p∗i − 0 j i j i j i j j i j i j i i j ∗ ∗ ∗ γi γ j θi σ ∂ E[π ] ∂ (2Ai ( pi −ci )−θi λi σ ) ∂ E[πi ] ci ) − θi λi σ ). If λi ≤ 0, we have ∂ λ i > 0. Since = > 0 , we have > 0 . Hence, the results in B0 ∂ λi ∂λj j ∗

∗

Proposition 4 can be obtained. Proof of Proposition 5. Similar to the proof of Proposition 4, we have

( p∗i −ci )γi (γ j λi θi +2Ai θ j λ j )−θi λi σ (γi λ j θ j +2A j θi λi ) ∂ E[πi∗ ] and B0 ∂σ =

∂ E[πi∗ ] ( a + λi σ ) (( p∗i − ci )γi γ j − 2A j θi λi σ ). Hence, the results in Proposition 5 can be obtained. B0 ∂ θi =

Proof of Proposition 6. By exploring the first-order derivatives, we have Proposition 6 can be immediately derived.

B (c −p∗ )−2A j θi λi σ −γi θ j λ j σ ∂ SD[πi∗ ] = 0 i i . The results in B0 ∂σ

Proof of Proposition 7. Based on Table A2, the results can be obtained by the approach similar to the proof of Proposition 2. Proof of Proposition 8. Similar to the Proof of Proposition 1.

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Appendix C. Equilibrium deviation in the case with asymmetric information on risk attitudes From the first-order conditions,

∂ E[U1 ] ∂ E[U2 ] ∂ p1 = 0 and ∂ p2 = 0, we have the optimal response function for CSLs:

p1 =

a + c1 α1 + p0 α1 + c1 γ1 + E [ p2 ]γ1 − aθ + λ1 σ − θ λ1 σ , 2(α1 + γ1 )

(A.1)

p2 =

c2 α2 + p0 α2 + c2 γ2 + E [ p1 ]γ2 + aθ + θ λ2 σ . 2(α2 + γ2 )

(A.2)

Since CSL 1 has incomplete information of CSL 2’s risk attitude, CSL 1 needs to examine the expected risk attitude for CSL 2, i.e.,



E [ p2 ] =

ui +τi

ui −τi

c2 α2 + p0 α2 + c2 γ2 + p1 γ2 + aθ + θ λ2 σ f (λ2 )dλ2 . 2(α2 + γ2 )

(A.3)

Similarly, CSL 2 finds the expected risk attitude for CSL 1.



E [ p1 ] =

ui +τi

ui −τi

a + c1 α1 + p0 α1 + c1 γ1 + p2 γ1 − aθ + λ1 σ − θ λ1 σ f (λ1 )dλ1 , 2(α1 + γ1 )

(A.4)

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