An application of cooperative game among container terminals of one port

European Journal of Operational Research 203 (2010) 393–403

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastic and Statistics

An application of cooperative game among container terminals of one port Naima Saeed a,*, Odd I. Larsen b,1 a b

Logistics at Molde University College, Norway Faculty of Economics, Informatics and Social Sciences, Molde University College, Post Box 2110, 6402 Molde, Norway

a r t i c l e

i n f o

Article history: Received 6 November 2008 Accepted 24 July 2009 Available online 30 July 2009 Keywords: Game theory Two-stage game Bertrand model Competition and cooperation Characteristic function Core

a b s t r a c t In this paper a two-stage game that involves three container terminals located in Karachi Port in Pakistan is discussed. In the first stage, the three terminals have to decide on whether to act as a singleton or to enter into a coalition with one or both of the other terminals. The decision at this stage should presumably be based on the predicted outcome for the second stage. The second stage is here modelled as a Bertrand game with one outside competitor, the coalition and the terminal in Karachi Port (if any) that has not joined the coalition. Furthermore, three partial and one grand coalition among the three terminals at Karachi Port are investigated. The concepts of ‘‘characteristic function” and ‘‘core” are used to analyse the stability of these coalitions and this revealed that one combination does not satisfy the superadditivity property of the characteristic function and can therefore be ruled out. The resulting payoffs (profits) of these coalitions are analysed on the basis of ‘‘core”. The best payoff for all players is in the case of a ‘‘grand coalition”. However, the real winner is the outsider (the terminal at the second port) which earns a better payoff without joining the coalition, and hence will play the role of the ‘‘orthogonal free-rider”. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction To foster competition one option available to public authorities is to introduce privatisation in sectors where firms are owned by the government or to allow private firms to compete with existing publicly owned firms on equal terms. The same is true for the port sector in which privatisation is increasingly emerging, especially in the ownership and operation of container terminals. The objective of privatisation is mainly to increase efficiency and performance. Two ports in Pakistan – Karachi and Qasim – have recently experienced a process of privatisation in the ownership and operation of their container terminals. Karachi Port has three terminals: 1. Karachi International Container Terminal (KICT); 2. Pakistan International Container Terminal (PICT); 3. A conventional terminal with priority berthing for geared vessels (KPT).2

* Corresponding author. Tel.: +47 71214268; fax: +47 71214100. E-mail addresses: [email protected], [email protected] (N. Saeed), [email protected] (O.I. Larsen). 1 Tel.: +47 71214219; fax: +47 71214100. 2 This public terminal does not have any specific name like KICT or PICT. In the official records of Karachi Port Authority, it is simply written ‘‘Geared Vessels”, probably because only geared vessels call at this terminal. For simplicity it is mentioned as KPT in this paper. Moreover, as this terminal is owned by the port authority, both the port and KPT are referred to as KPT. 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.07.019

The first two terminals have recently been privatised. The third one is still owned and operated by the port authority. Port Qasim has mainly operated as a landlord port and in 1996 a new private container terminal was opened. This terminal is known as Qasim International Container Terminal (QICT). Strong competition now exists between the four terminals of the two ports. This competition has resulted in QICT taking several customers away from Karachi Port. In order be stronger in their competition with QICT, the three terminals at Karachi Port could form a coalition. The formation of a coalition could yield three main advantages for the concerned parties: 1. Market power (higher prices): with the formation of a coalition, terminals will gain market power and therefore will be able to charge higher prices from their customers. 2. Better utilisation of combined capacity: if three terminals can serve as a single unit, combined capacity will increase compared to the capacity of individual terminals. This increased capacity will reduce the average waiting time of container vessels especially when the individual terminals are approaching capacity. This is an effect that follows from elementary queuing theory. If we have three servers with independent demands the total waiting time will be higher than in a situation with three servers that serve the total demand. 3. Usage of storage facilities: the private terminals at Karachi Port which do not have storage facilities will be able to utilise the sheds of the public terminal. This will result in an increase in efficiency and reduction in total cost.

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To analyse the cooperation, economists have used a fairly new game-theoretic approach in which cooperation is the outcome of a non-cooperative strategic behaviour of the players involved in the negotiations (see Bloch, 1996; Carraro and Siniscalco, 1993; Yi, 1997; Olieman and Hendrix, 2006). The game is therefore a two-stage game: in the first stage, agents decide non-cooperatively whether or not to sign the agreement (join the coalition); in the second stage, those players which join the coalition act cooperatively, whereas non-signatories play a non-cooperative Nash game against the group of signatories (Carraro and Marchiori, 2002). The objective of this paper is to analyse the different combinations of coalitions among the three terminals at Karachi Port on the basis of this type of two-stage game. In the first stage, the three terminals at Karachi Port have to decide whether to act singly or to join the coalition. Furthermore, different combinations of possible coalitions (for instance: between KPT and KICT, KPT and PICT, PICT and KICT or all three terminals as one unit) are analysed. In all coalitions, signatories decide cooperatively on what prices to set and how to use their combined capacity in order to maximise the coalition surplus, while the coalition and singletons compete with one another in a non-cooperative way; this is the second stage. Furthermore, we will check the stability of these suggested coalitions with the help of concepts of ‘‘characteristic function” and ‘‘core” of the cooperative game. Since the first stage decision (when players have to decide whether to join the coalition or not) depends on the predicted outcome for the second stage, we will study the problem by backward induction. The rest of the paper is organised as follows: in Section 2 we present a detailed description of the case study and the information and data available on the terminals. In Section 3 we present a number of research works related to the application of game theory to the port sector, and the difference between our research and previous research. In Section 4 we present a model for a Bertrand game for the second stage and its parameters. Section 5 presents the basic concepts of cooperative game theory. Section 6 constitutes a numerical example and is followed by a conclusion and policy implications in Section 7. 2. Background information about the case study Some detail about the four container terminals are given as follows: 1. Karachi International Container Terminal (KICT): this terminal at Karachi Port has been in operation since 1998. The giant shipping line, American President Line (APL) invested in KICT on a build-operate-transfer (BOT) basis. BOT is the classic case of concessions in which the public sector does not lose ownership of the port infrastructure, and new facilities built by private firms are transferred to the public sector after a specified period of time. Now the terminal has been bought and is managed by Hutchison, Hong Kong. 2. Pakistan International Container Terminal (PICT): this terminal at Karachi Port was privatised in August 2002. It was also developed on a BOT basis, specifically build, operate and transfer after 21 years. It is the only container terminal in Pakistan sponsored and owned by Pakistanis.3 3. Karachi Port Trust (KPT, public terminal): this is a conventional terminal with priority berthing for geared vessels, mainly feeders. Unlike KICT and PICT, this terminal does not have modern equipment like gantry cranes to handle containers. Regular container service started at this terminal in 1973. 3

See http://www.pictcntrtrack.com/.

Fig. 1. Location of terminals at two ports.

4. Qasim International Container Terminal (QICT): the fourth player is in the second Port, Port Muhammad Bin Qasim. It was built on a build-own-operate (BOO) basis. In the case of BOO, parts of the seaport are transferred to the private operators for development. Initially, Maersk invested in QICT, which has now been bought by Dubai port investors. This terminal was incorporated in 1996 and is located approximately 45 km from Karachi (see map, Fig. 1). The figures4 related to operation, in terms of the number of ships and TEUs (20 foot equivalent units) handled, are given for each terminal in Tables 1 and 2. Tables 1 and 2 show that, since 2003, QICT has been the biggest terminal in terms of throughput of ships and containers. Although KPT lacks modern equipment it handles a considerable number of ships and containers. The reason is that feeder and geared vessels tend to prefer this terminal due to its low handling charges. Moreover, only KPT has sheds to store goods de-stuffed from containers. The average market share (in terms of TEUs handled) for all terminals, calculated for the years 2001–2006, and the handling charges (obtained with the help of a questionnaire5) are given in Table 3 and 4: The average market share for PICT over the whole period is quite low, but has increased in recent years. This might be due to the fact that this terminal was privatised quite recently. 3. Related literature Research related to the application of game theory to ports is very limited. Anderson et al. (2008) developed a game-theoretic best response framework for understanding how competing ports will respond to development at a focus port, and whether the focus port will be able to capture or defend market share by building additional capacity. They applied this model to the investment and competition currently occurring between the ports of Busan and Shanghai. Unlike the analyses on which port expansion plans are typically based, the authors explicitly account for the incen4 Data about the number of TEUs handled and the number of ships that called at all three terminals at Karachi Port was obtained from the Port Authority. Data for QICT was collected from the official website of the terminal, www.qict.net. Moreover, data for capacity at each terminal was collected from the official website of each terminal; see http://www.qict.net, http://www.kpt.gov.pk/, http://www.kictl.com/, http:// www.pictcntrtrack.com/. 5 Questionnaires filled out by shipping agents working for foreign principals in Karachi, Pakistan; for more information see the author’s previous paper (Saeed, 2008).

N. Saeed, O.I. Larsen / European Journal of Operational Research 203 (2010) 393–403 Table 1 Number of ships that called at the terminals. No. of ships

2001– 2002

2002– 2003

2003– 2004

2004– 2005

2005– 2006

PICT KICT KPT QICT

208 351 340 131

155 345 369 239

122 336 335 386

229 346 330 449

270 406 325 520

Table 2 Total containers handled at the terminals. Total TEUs

2001– 2002

2002– 2003

2003– 2004

2004– 2005

2005– 2006

PICT KICT KPT QICT

112,805 335,645 267,442 172,000

80,570 337,916 311,471 289,000

90,600 393,818 340,335 419,000

206,764 416,518 288,654 519,000

300,994 559,194 283,962 585,000

Table 3 Market share and handling charges.

Market share (%) Handling chargesa (in US $)

QICT

KICT

PICT

KPT

31 76

32 94

13 69

23 54

a A World Bank Report (see http://www.wds.worldbank.org/external/default/ WDSContentServer/WDSP/IB/2006/08/14/000160016_20060814091138/Rendered/ PDF/36523.pdf) mentions charges of $113 per container for Karachi Port and $105 for QICT. These may be more appropriate in the sense that the questionnaire responses represent a small proportion of the total shipping agents.

Table 4 General parameters of demand. Level of demand

Log sum parameter

Price parameter

1550,000

0.010

0.050

tives and opportunities for fellow competitors to respond to investments (or the threat thereof) or to defend or appropriate market share through a game-theoretic response framework. However, the authors did not apply a two-stage Bertrand competition model. They instead, in order to develop a game-based analysis of Busan and Shanghai port development policies, abstracted from the pricing game, focusing instead on strategy in the development game given the observed or projected prices. Kaselimi and Reeven (2008) apply the Hotelling location model to inter-port competition. They use this model to develop a framework for linking the strategic interdependence between ports’ and potential terminal providers’ investment desirability. Moreover, considering the importance of the role of the users’ cost in selection of a port, the authors explicitly include it in the model in order to examine how changes in the users’ cost affect potential operators’ decisions. However, the users’ cost consists here of port dues charged by the port authority and the service fee of the service providers. Zhang (2008) considers both the quantity of competition and price of competition between ports and examines the interaction between hinterland access conditions and port competition. Competition between ports is treated as competition between alternate intermodal transportation chains, while the hinterland access conditions are represented by both the corridor facilities and the inland roads. When ports compete in quantities, an increase in corridor capacity will increase the port’s own output, reduce the

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rival port’s output and increase the port’s own profit. On the other hand, an increase in inland road capacity may or may not increase the port’s own output and profit, owing to various offsetting effects. Finally, inland road pricing may or may not increase the port’s own output and profit. De Borger et al. (2008) analyse the interaction between the pricing behaviour of the ports and optimal investment policies in port and hinterland capacity. They use the framework of a two-stage game in capacities and prices. The main focus here is on a governance structure where capacity decisions are public, but pricing decisions are private. The game is analysed by backwards induction. The authors obtained the following results. First, profit-maximizing ports internalise hinterland congestion in so far as it affects their customers. Second, investment in port capacity reduces prices and congestion at both ports, but increases hinterland congestion in the region where the port investment is made. Investment in a port’s hinterland is likely to lead to more port congestion and higher prices for port use, and to less congestion and a lower price at the competing port. Third, the induced increase in hinterland congestion strongly reduces the direct benefits of extra port activities. Finally, imposing congestion tolls on the hinterland road network raises both port and hinterland capacity investments. Our research is different from the previously mentioned research done in the same field in the following aspects. First, in none of the research is the multinomial logit model used to analyse the game outcome by solving a numerical example with the help of real data. Second, in all previous research the authors discussed inter-port competition, but we focus on container terminals and inter- as well as intra-port competition. Third, to our knowledge, no one has analysed the possibilities of coalitions between terminals as presented in this paper. Fourth, we solve the Bertrand game in terms of prices but we also analyse the outcome from the perspective of users’ benefits when coalitions are formed. We also take into account the effects on capacity utilisation but not on capacity expansion as dealt with in previous research. Finally, in previous research on ports only non-cooperative games were analysed. In this paper we also try to include both cooperative and non-cooperative aspects.

4. The model 4.1. The demand for container terminal services In our model we treat the competition between terminals as a Bertrand game and we also use the outcome of Bertrand games to investigate the payoff (profit) to different coalitions. Bertrand game is a natural choice in this setting. We are dealing with a service industry where competitors offer similar, but not quite homogeneous services from the perspective of individual customers. In order to detail the structure of the Bertrand game, the demand function faced by each port must be made explicit. In the following we will use the term ‘terminal users’ for the agents who pay the cost of container freight and handling and make the choice of which terminal to use. Different ports and terminals can rarely be considered as perfect substitutes from a user perspective. In addition to the terminals charges for handling and storing containers, the user will have additional costs which we may call other user costs (OUC). Among the components of OUC we have: – Inland transport (i.e. rail and trucks) cost for containers to and from terminals within Pakistan. – Freight rates charged by container lines, in particular any surcharges related to port and terminal efficiency.

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– Cost related to transport time, including the cost of container lease or rental. Container lease cost is included in this component because with the increase in transport time, the lease period will also increase which will result in increased costs. The difference between the first and third components is that, in the first case cost refers to what users pay for inland transportation of containers, while in the third case, cost refers to the costs which they have to bear due to time spend in the transportation of containers. Even if terminal charges are equal, differences in OUC may thus lead to different market shares for competing terminals. On the other hand, with differences in OUC, we may have persistent differences in terminal charges and market shares even in a competitive setting. In our model, we assume that OUC is composed of 2 components. One that is independent of the volume of containers handled by each terminal and one component that is an increasing function of the volume handled (and decreasing in rated capacity). The rationale for a variable component of OUC is two-fold: (1) The spatial aspect: Marginal customers will, on average, have longer transport distances and higher transport cost to the terminal than the average customer. (2) When the volume of containers approaches or exceeds the rated capacity we must expect different types of delays to increase. Some delays may affect the ship turn-round time and subsequently the freight rates due to congestion surcharges by shipping lines, while other types of delays may affect the dwell time of containers in port. A counteracting force may be that the level of service in terms of vessel calls will improve with the volume of containers handled. Within surface and air transport, this aspect is generally referred to as the Mohring effect (UNITE, 2003). Veldman and Rachman (2008) use the throughput share of the port to capture the Mohring effect. However, in this case with constant capacity we must expect that both (1) and (2) have a stronger negative impact on OUC than the positive ‘Mohring effect’. In general, the user cost function for terminal ‘i’, OUC ðiÞ, thus has the form:

OUCðiÞ ¼ C0i þ f



 Xi ; CAPi

ð1Þ

where C0i is the fixed component, X i is the volume handled by terminal ‘i’ and CAPi is the rated capacity of terminal ‘i’. f is an increasing function of the ratio. In our numerical implementation of the Bertrand model, the market share of each terminal is determined by an aggregate multinomial logit model6 and the demand (in TEUs) for all terminals combined is a function of the logsum from the logit model. We further assume that the capacity of each terminal is fixed as capacity expansion will involve major investment and take a long time for implementation if it is physically possible to expand at all. The use of a logit model presupposes that a ‘utility function’ can be assigned to each terminal. The utility functions in an aggregate logit model can be interpreted as a measure of the attractiveness of a terminal as perceived by the ‘average’ user. The utility functions used have the following simple form:

U i ¼ ai þ bfpi þ OUC i g;

ð2Þ

where U i is the ‘utility’ of terminal ‘i’ i = KICT, PICT, KPT and QICT, pi is charge per TEU in terminal ‘i’ including fee to Port Authority, i = KICT, PICT, KPT and QICT, OUC = other user cost at each terminal, b is the coefficient of price at terminals and ai is the alternative specific constant for terminal ‘i’ 6 For detail about logit model see Ben-Akiva and Lerman (1985) and Train (2003). Choice models have been used in Port and shipping too, e.g. Tiwari et al. (2003), Nir et al. (2003), Malchow and Kanafani (2004), and Magala (2004).

aPICT and aKPT ¼ 0; while aKICT and aQICT > 0: Alternative specific constant is included in the utility functions for KICT and QICT, to capture the attributes due to which these terminals capture high market share, compared to their competitors. As is apparent from Table 3, the average market share of these two terminals is high compared to PICT and KPT. As KPT is owned by the Port Authority, the port and KPT will be treated as an economic entity in the model and we do not distinguish between handling charges and fees. Thus what matters for KPT, is the combined revenue from fees paid by the private terminals and profits from their own terminal’s operation. The market share of terminal ‘i’ is given by the logit expression:

eUi Qi ¼ P U j je

i ¼ KICT; PICT; KPT and QICT:

ð3Þ

The logsum is defined by:

LS ¼ ln

X

! eU j :

ð4Þ

j

Total aggregate demand (in TEUs) for all the terminals is thus given by:

X ¼ AehLS ;

ð5Þ

where A and h are constants and 0 < h < 1, Individual demand for terminal ‘i’ is given by the equation:

X i ¼ X:Q i

i ¼ KICT; PICT; KPT and QICT:

ð6Þ

The demand faced by a terminal will thus depend on handling charges (including unit fee) and OUC for all terminals. The private terminals will keep the handling charge, but the revenue from fee is transferred to the Port Authority. Individual demand is elastic because change in price and other attributes of one terminal will shift the traffic between that terminal and other terminals. There will also be a slight effect on the total demand via the logsum. 4.2. Revenue/profit for terminal operators and port authorities The operating surplus of the private terminal ‘i’ is:

Pi ¼ ðpi  wi  ci Þ  X i ;

ð7Þ

where pi is the handling charge per TEU paid by the users, wi is the fee paid by private terminals per TEU handled and ci is the marginal cost per TEU. If the contract implies that unit fee is a percentage of the handling charge, the surplus is alternatively given by:

Pi ¼ ðpi  ð1  di Þ  ci Þ  X i ;

ð8Þ

where di is the fee and pi  (1  di ) is the share of the handling charges retained by the terminal. When we consider KPT and Karachi Port Authority as an economic entity, the profit for KPT is taken as:

P3 ¼ ðp3  c3 ÞX 3 þ w1 X 1 þ w2 X 2 ;

ð9Þ

where 1 = KICT, 2 = PICT and 3 = KPT. For any contract between the Port Authority and private terminal operator which will be viable in the long run we must have:

Pi ¼ ðpi  wi  ci Þ  X i P annual rent ðiÞ:

ð10Þ

That is, the operating surplus must be greater than the annual rent paid to the port authority. This constraint is set in the general terms; however it is not incorporated in the model to get the numerical solution presented in Section 6 because the constraint never becomes binding in our model. Insofar as wi (or di ) will influence the outcome of a game between competing terminals, i.e. the total revenue ðpi  X i Þ, a con-

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tract that specifies the magnitudes of di and annual rent constitutes an important strategic decision for a Port Authority that attempts to maximize total revenue. In a competitive situation with few players and an inhomogeneous product the outcome in terms of market shares and prices can often be treated as the result of a game where each player maximizes profit, but with due consideration of the expected reaction of its competitors. When the competitor’s actions are confined to setting the prices of their own product (service), the outcome can be modeled as Bertrand equilibrium (Pindyck and Rubinfeld, 2001). Whatever price, other terminals are charging terminal ‘i’s profit is maximized when the incremental profit from a very small increase in its own price is just zero. Thus, in order to find the best reply for player i, we differentiate his profit function with respect to pi and set the derivative equal to zero. Thus, the Bertrand Nash equilibrium is characterized by the first-order conditions:

@ Pi ¼ 0; @pi

i ¼ KICT; PICT; KPT and QICT:

ð11Þ

The profit function, say for terminal 1 is given by:

P1 ¼ ðp1  w1  c1 Þ  X 1 : hLS

Since X 1 ¼ Ae

ð12Þ

Q 1 (12) can also be written as:

P1 ¼ ðp1  w1  c1 Þ  AehLS Q 1 :

ð13Þ

By taking the derivative of Eq. (13) and setting it equal to zero we get the condition:

@ P1 @ðAehLS Q 1 Þ ¼ AehLS Q 1 þ ðp1  w1  c1 Þ ¼ 0: @p1 @p1

ð14Þ

In the appendix (Part A) we show that (14) can be rewritten as:

1 þ ½bðhQ 1 þ 1  Q 1 Þ  ðp1  w1  c1 Þ ¼ 0:

ð15Þ

This is the implicit reaction curve (pricing rule) for terminal 1 (i.e. KICT). The reaction function can not be given on a closed form in this model. The prices of the other terminals enters via Q 1 , see (2) and (3). Similarly, reaction curves for the other three terminals can be derived. Solving these reaction functions yield the Nash equilibrium in prices. 4.3. Cooperative game with external competitors In the case of cooperative game, three terminals within Karachi Port can establish different combinations of coalition. In this case, the profit function for each terminal will be different from Eq. (10). For instance, if all the terminals at Karachi Port decided to work under one decision unit, then the profit function of the coalition, for instance, for KICT will be as follows:

P1 ¼ ½X 1 ðp1  c1 Þ þ X 2 ðp2  c2 Þ þ X 3 ðp3  c3 Þ:

ð16Þ

This will give 3 conditions, one for each price. Again Bertrand Nash equilibrium is characterized by the firstorder conditions, thus by taking the derivative of Eq. (16) and setting it equal to zero we get the condition:

    hLS @ AehLS Q 2 @ P1 @ Ae Q 1 hLS ¼ ðp1  c1 Þ þ Ae Q 1 þ ðp2  c2 Þ @p1 @p1 @p1   @ AehLS Q 3 ðp3  c3 Þ ¼ 0: ð17Þ þ @p1 This condition can also be written as:

½bðhQ 1 þ 1  Q 1 Þðp1  c1 Þ þ 1 þ Q 2 ½bðh  1Þðp2  c2 Þ þ Q 3 ½bðh  1Þðp3  c3 Þ ¼ 0:

ð18Þ

397

In the appendix (Part B), complete derivation is shown. This is the reaction curve for KICT when all 3 terminals have formed a coalition within Karachi Port. Similarly, reaction curves for other two terminals can be derived. Moreover, in this case we have not considered the fee paid by private terminals to Karachi Port in the profit function because this is a matter of internal transfers within the coalition. Similarly in other combinations of coalition, fee of that terminal will not be included in the profit function which will become the partner with KPT. 5. Basic concepts of cooperative game theory Since the main objective of this paper is to analyse the possible combinations of cooperation (coalitions) among the terminals at Karachi Port, we will briefly describe the concept of the cooperative game and related criteria to check the stability of suggested coalitions. Cooperative game theory is basically concerned with those situations in which players can negotiate before the game is played about what to do in the game. Moreover, it is assumed that these negotiations can be concluded by the signing of a ‘‘binding agreement” (Binmore, 1992). There are two key points in a cooperative game: (i) the payoff for each coalition, and (ii) what payoff each player in the coalition should get (Neumann and Morgenstern, 1967). Moreover, cooperative coalitional games are divided into two categories: games with transferable utilities (e.g. money which can be transferred) and games with non-transferable utilities (Peleg and Sudholter, 2003). As the present case represents the situation where gain is in the form of profit which can be transferred between players, we will use the concept of ‘‘transferable utility”. However, in order to apply the concept of transferable utility, it is necessary to introduce the two important concepts of ‘‘characteristic function” and ‘‘core”, explained by Song and Panayides (2002), who applied the same concepts in shipping, as follows. 5.1. Characteristic function A coalitional game with transferable utilities on P (a set of players) is a function, known as characteristic function, which associates a value (worth) to every possible coalition. The characteristic function has the two following properties. The first property states that a coalition without any members has no worth

Uð/Þ ¼ 0:

ð19Þ

The second property is known as ‘‘superadditivity”

Uðp1 [ p2 Þ P Uðp1 Þ þ Uðp2 Þ:

ð20Þ

Eq. (20) states that the benefit which two players p1 and p2 can achieve by cooperating is at least equal or greater to that which they can achieve by working separately. 5.2. Core of an n-player game If all players in a game decide to work together then the question arises of how to divide the total profit. Different solution concepts are discussed in the literature (see Peleg and Sudholter, 2003; Friedman, 1977; Suijs, 2000; Curiel, 1997; Straffin, 1993). The most prominent among them are ‘‘core” and ‘‘Shapley value”. We will use the concept of core to analyse the situation in two ports. Core, proposed by Gillies (1953), is a generalisation of the ‘‘contract curve” of Edgeworth. It consists of those payoff vectors which are, in a specific sense, acceptable to all players (Friedman, 1977). The core rests upon the idea of an ‘‘imputation”. An imputation satisfies both ‘‘individual rationality” and ‘‘collective rationality”

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(Shubik, 1968). The formal definitions of these, as given by Song and Panayides (2002) are as follows. Let U be an n-person game in characteristic function form with players P ¼ ðp1 ; p2 ; . . . :pn Þ. An n-tuple ðx1 ; x2 ; . . . :xn Þ of real numbers is said to be an imputation if both the following conditions hold. 1. Individual rationality:

xi P Uðpi Þ i ¼ 1; 2 . . . n:

ð21Þ

The above equation indicates that each agent is only willing to participate in the coalition if it pays him at least as much as he can obtain on his own. 2. Collective rationality: n X

xi ¼ UðPÞ:

ð22Þ

i¼1

Eq. (22) states that the sum of payoffs of a group of n-players is equal to the value that is guaranteed by the characteristic function. Any payoff vector satisfying Eqs. (21) and (22) is named an imputation. The imputation can be thought of as a possible social arrangement that satisfies minimal conditions of rationality. Presumably, any ultimate arrangement will be drawn from the set of imputations. A drawback of the sets of imputations, however, may itself be quite large. Thus, ‘‘coalition rationality” can be argued to be a natural extension of the conditions of individual and collective rationality. Coalition rationality requires that the security level of every coalition defined by characteristic function be satisfied, formally:

X

xi P UðCÞ for all C in P;

ð23Þ

pi 2C

where C denotes all possible coalitions formed by a sub-set of the n players. For example, C may stand for the coalition ðp1 ; p2 Þ, or ðp1 ; p2 ; p3 Þ. The set of imputations satisfying the conditions of the coalition rationality constitute the core. The rationale behind condition (23) is as follows. Suppose that a coalition, C, forms and attempts to divide the value assigned to C by the characteristic function. Further, suppose that a sub-group of C, say C 1 , is offered a payoff less than what C 1 is worth according to the characteristic function. In this case C 1 would not accept the offer, since it can do better without the remaining members of C. Thus, for C 1 to remain in the coalition, it should receive at least as much as UðC 1 Þ. If this argument is extended to all possible conditions, then the condition of coalition rationality is required (Song and Panayides, 2002). 6. Numerical analysis By solving the equilibrium of the Bertrand game we can get the pricing rule set by the players, which will yield the Nash equilibrium. However, before this we need to make certain assumptions about the values for parameters a, b and marginal costs of terminals. 6.1. Assumptions about the parameters of the model 6.1.1. Assumed value for b b is the coefficient of price at ports or cost for customers (shipping lines). In other words, this is the coefficient of price of the choices faced by decision makers. No research has been done in which this value has been estimated, by discrete choice methodology, taking any port as a case study. However, there is some research, similar in many ways to the current case, in which its value has been estimated by researchers. For example, Polydoro-

poulou and Litinas (2007) developed the models which included a multinomial logit model with dependent variable the choice among different types of shipping lines and airlines. The result show that the parameter of travel cost of ships is negative and significant. An increase of 1 Euro in shipping cost will result in a utility reduction of 0.0516. The travel cost parameter of an aeroplane is 0.0648. Based on a unique data set of 760 air cargo transhipment routings to/from the Northeast Asian region in 2000, Ohashi et al. (2004) applied an aggregate form of multinomial logit model to identify the critical factors influencing the choice of air cargo transshipment route decisions. They selected five airports as the main competing transshipment points for the air cargo traffic in the NEA region: Beijing, Osaka, Shanghai, Seoul and Tokyo. The estimated value of the parameter for landing fee (the term reflects the total airport charges including landing/take off fee, aircraft parking, security charges, noise-related charges and cargo-handling charges) of airports is 0.039. Pereira et al. (2007) implemented a stated preference choice game which, asked airline passengers to choose between competing alternative airlines in the Portuguese air corridor between Madeira and Oporto. These alternative airlines differ, in a trade-off sense, and in several attributes, such as ticket cost, penalties for changing tickets, punctuality warranties, comfort, and so on. The estimated coefficient for cost (price for ticket) is 0.0259 and is statistically significant. In research conducted by Cantillo et al. (2006), this value is around 0.070 as a coefficient of cost, as attribute for three choices: taxi, bus and metro. Estimated values for mainline deviation costs (based on deviation from main navigation course for transshipment port) and feeder costs (costs of feeder transport stack-to-stack between feeder region and transshipment port), are 0.077 and 0.030, respectively, by Veldman and Rachman (2008). While the estimated value for a model with combined feeder and deviation costs is 0.032. One of the authors, Saeed (2009), estimated a logit model for container terminal selection by shipping companies on a dataset for the 4 terminals treated here and obtained a statistically significant parameter of 0.0624 for the container handling charge (in US$). Therefore, based on these values we assume that the value for price parameter, in our model, is 0.050. 6.1.2. Assumed value for a In general terms, Equation (2) can be written as by dividing utility into two additive parts. For instance, for two alternatives, A and B, the utilities can be written as follows:

U An ¼ V An þ eAn and U Bn ¼ V Bn þ eBn where n ¼ 1; . . . N: where N is the number of decision makers (or users); V An and V Bn are the systematic (or representative) components of the utility of A and B; and eAn and eBn are the random parts and are called the disturbance (or random components). a is the alternative–specific constant and reflects the mean of eBn  eAn ; that is, the difference between the utilities of alternatives A and B when ‘‘all else is equal”.

Table 5 Terminal-specific parameters.

Alt.spec. constant User cost constants Marginal cost Capacity Terminal fee

QICT

KICT

PICT

KPT

0.1 7 50 600,000 5% of pricea

0.5 5 55 525,000 6.03

0 5 55 400,000 12.54

0 40 27 300,000

a This is the 5% royalty of the load on/load off revenue on cargo exceeding 150,000 TEUs to Port Qasim. Information about the fee per TEU charged to KICT and PICT is obtained from a personal interview with authorities at KPT.

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The values of alternative specific constant for QICT and KICT, presented in Table 5, are arbitrary chosen. 6.1.3. Assumed value for cost The basis of all port tariffs should be short-run marginal cost, which measures the resources used up by supplying a unit of port service. However, strictly setting a price equal to the marginal cost is best only in a perfectly competitive free economy or in an efficient socialist economy (Bennathan and Walters, 1979). For the marginal costs of terminals, the average cost of PICT for 2005 is calculated. Figures are obtained from the annual report of the terminal.7 The terminal’s operating cost is Rs. 695,915,000 divided by total containers handled in 2005, i.e. 206,764 TEUs, which gives an average cost of US $57. On the basis of this figure, the marginal costs for the three private terminals are assumed. 6.1.4. Assumed value for h Demand for port calls, port transshipment and supplementary service is derived from demand for the goods involved and is thus a function of economic growth, industrial production and industrial trade (Meersman et al., 2003). Thus, a change in price and other attributes of one terminal will shift the traffic to that terminal from other terminals. It will not much affect the total demand, but will affect the market share of all four terminals. That is why the value for h is assumed to be quite low (0.01).

Table 6 Bertrand equilibrium when all terminals are independent.

Equilibrium price US $/TEU User cost US $/TEU Market share Profit (in millions of US $) a

With the available information, a model consisting of Eqs. (1)– (3), (12), (15), (18) (for each terminal) and (5) is solved using an equation solver. 6.2.1. Input parameters Tables 4 and 5 provide information about the input parameters used in the model. The values of the log sum parameter and price parameter are assumed on the basis of the literature described in Section 6.1. The values of user cost constants are also assumed; a high value is set for KPT because this is a conventional terminal and does not have modern equipment like gantry cranes to handle containers. Moreover, the user cost for QICT is set at US $7 because it suffers from a cost disadvantage of about US $2 per TEU as compared to KICT and PICT.8 The values for marginal costs for private terminals are explained in Section 6.1.3. However, the value for marginal cost for KPT is assumed and arbitrary chosen. Data about capacity is collected from the official website of each terminal.9 In the preceding section, Bertrand solutions for different combinations of coalition as well as the present situation are described. Since Karachi Port Authority has signed a contract with its private terminals, in which unit fee and annual rent are settled, negotiations about the coalitions could be settled within the contract, i.e. in the form of a binding agreement. 6.2.2. Independent terminals (present situation – case A) Tables 6 and 7 show the resulting Bertrand Nash equilibrium, which is characterised by the first-order conditions (see Eq. (11)) for the present situation. The market share and equilibrium prices for the four terminals are close to the actual figures presented in Table 3, and hence confirm the validity of the proposed model. 7

Available online at http://www.pictcntrtrack.com/docs/annualreport.pdf. 8 A World Bank Report (see http://www-wds.worldbank.org/external/default/ WDSContentServer/WDSP/IB/2006/08/14/000160016_20060814091138/Rendered/ PDF/36523.pdf. 9 See http://www.qict.net,http://www.kpt.gov.pk/, http://www.kictl.com/,http:// www.pictcntrtrack.com/.

KICT

PICT

KPT

81.60 7.40 0.30 12.4

90.10 5.80 0.31 13.8

91.90 5.30 0.18 6.6

53.20 41.40 0.21 14.3a

Including fees paid by KICT and PICT.

Table 7 Total demand for two ports and combined profit for Karachi Port. Total demand in thousands of TEUs

1502

Combined profit of Karachi Port (terminals 2–4) in millions of US $ Logsum (LS)

34.7 3.14

Table 8 Bertrand equilibrium when KICT and KPT are cooperating.

Equilibrium price US $/TEU User cost US $/TEU Market share Profit in millions of US $ a

6.2. Bertrand results

QICT

QICT

KICT

PICT

KPT

79.60 7.70 0.35 13.5

91.60 5.50 0.32 17.6

92.20 5.30 0.19 7.1

64.00 40.20 0.14 11.2a

Excluding the fee paid by KICT and including the fee paid by PICT.

Moreover, from Table 6 it is clear that KPT has a 21% share of the total market, but its user cost is very high as compared to the private terminals. This means that users selecting this terminal suffer from a cost disadvantage not in monetary terms but in terms of a high waiting time. Furthermore, QICT is strongly competing with PICT and KICT. On the basis of these findings, one option for Karachi Port Authority is to establish a coalition with any one or both of its private terminals, to increase efficiency and market power. Hence, in the first stage players decide to join or not to join any one of the coalitions analysed in the preceding section. Next, we will discuss different combinations of coalitions and their stability, and also which one is preferable for the terminals at Karachi Port.

6.2.3. Cooperation between KPT and KICT at Karachi Port (case B) In this case, KPT has formed a coalition with one of the private terminal; that is, KICT at the same port. New Nash equilibrium results are presented in Tables 8, 9. Due to the benefits generated from the coalition (for instance, better utilisation of combined capacity, decrease in average waiting times, and so on) the reaction functions for the two terminals are shifted outward, i.e. the two terminals are able to charge higher prices as compared to the previous case. However, the reaction function of QICT is shifted downward. Thus, at the new equilibrium the prices for KICT and KPT are high while the price for QICT is low. As a result, QICT now captures a large market share and earns more profit as compared to case A. However, there is a slight effect on the price and market share of PICT.

6.2.4. Cooperation between KICT and PICT at Karachi Port (case C) In this case, the situation is the same as that discussed in the previous case. After formation of a coalition between KICT and PICT, the new equilibrium prices are higher for the two cooperating terminals and lower for the competing terminal (QICT, at the second port). However, there is a slight effect on the price and profit of

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Table 9 Total demand for the two ports and combined profit for Karachi Port.

Table 14 Bertrand equilibrium when all three terminals are cooperating.

Total demand in thousands of TEUs

1501

Combined profit of Karachi Port (terminals 2–4) in millions of US $ Logsum (LS)

35.9 3.22

Equilibrium price US $/TEU User cost US $/TEU Market share Profit in millions of US $ a

Table 10 Bertrand equilibrium when KICT and PICT are cooperating.

Equilibrium price US $/TEU User cost US $/TEU Market share Profit in millions of US $ a

QICT

KICT

PICT

KPT

80.40 7.90 0.37 14.7

96.70 5.30 0.28 14.9

103.40 5.00 0.12 6.6

53.80 42.10 0.23 14.0a

Including fees paid by KICT and PICT.

Table 11 Total demand for the two ports and combined profit for Karachi Port. Total demand in thousands of TEUs

1499

Combined profit of Karachi Port (terminals 2–4) in millions of US $ Logsum (LS)

35.5 3.32

Table 12 Bertrand equilibrium when PICT and KPT are cooperating.

Equilibrium price US $/TEU User cost US $/TEU Market share Profit in millions of US $ a

QICT

KICT

PICT

KPT

78.50 7.50 0.32 12.0

89.70 5.70 0.30 13.1

86.70 5.30 0.22 10.4

58.90 40.30 0.15 10.1a

QICT

KICT

PICT

KPT

83.00 8.50 0.42 18.1

102.00 5.30 0.28 20

102.00 5.10 0.17 12.2

74.00 40.10 0.12 8.6a

Excluding fee paid by PICT & KICT.

Table 15 Total demand for the two ports and combined profit for Karachi Port. Total demand in thousands of TEUs

1495

Combined profit of Karachi Port (terminals 2–4) in millions of US $ Logsum (LS)

41 3.60

shown in Table 17, is high compared to their joint profit when they were working independently. Hence this combination satisfies Eq. (20); that is, the superadditive property of the characteristic function. Moreover, the other two combinations (that is, when PICT and KICT have formed a coalition and when all three terminals have formed a coalition) also satisfy the superadditive property of the characteristic function. As is apparent from Table 17, the joint profits of the terminals after coalition are high as compared to when they were working independently. Specifically (all figures are in millions of US $:

28:8 > 28:1 ðfor the coalition of KPT and KICTÞ 21:5 > 20:4 ðfor the coalition of PICT and KICTÞ and 41:0 > 34:7 ðfor the coalition of KPT; PICT and KICTÞ However, for the case when KPT and PICT have formed a coalition, according to Table 17 their joint payoff is less after forming the coalition as compared to the payoff when they were working independently. That is:

Excluding fee paid by PICT and including fee paid by KICT.

Table 13 Total demand for the two ports and combined profit for Karachi Port. Total demand in thousands of TEUs

1503

Combined profit of Karachi Port (terminals 2–4) in millions of US $ Logsum (LS)

33.5 3.08

KPT, which is an outsider within the same port (see Tables 10 and 11). 6.2.5. Cooperation between KPT and PICT at Karachi Port (case D) In this situation, KPT have formed a coalition with PICT. This coalition will be discussed in detail later on while discussing the stability of all suggested coalitions. 6.2.6. Cooperation between all three terminals at Karachi Port (case E) In the last case, when all three terminals at Karachi Port have formed a coalition, at a new Nash equilibrium these terminals are able to charge high prices while their competitor, QICT, manages a low price and high profit. 6.2.7. Analysis of the stability of the coalitions In order to analyse the stability of the coalitions, individual profit as well as the joint profit of all four players in all cases are given in Tables 15–17. In order to check the stability of all coalitions, the concepts of characteristic function and core (as described in Section 5) are used. In case B, KPT and KICT have formed a coalition while PICT is working independently. The joint profit of these two terminals,

20:5 < 20:9: Hence, this situation does not satisfy the superadditive property of the characteristic function. The formation of this coalition is not possible because at least one player can get a better payoff by working alone. Moreover, although cases B and E satisfy the superadditive property, the individual payoff of KPT seems to be less in the coalition than when it was working alone, as is apparent from Table 16. This apparently violates Eq. (21), that is, individual rationality. However, this is due to the fact that we have disregarded the fee paid by the private terminal which has formed the coalition with KPT. Thus, the low payoff in cases B and E, mentioned in Table 16, for KPT is due to ignorance of the fee paid by its partner terminal. For instance, if we assume that in case B profit will be distributed so that KPT will get at least the same payoff as it gets when working alone, even in that case the payoffs for KICT (US $14.5 million) and KPT (US $14.3 million) will be greater and equal, respectively to the payoff of case A. Hence, this case fulfils the condition of individual rationality. However, in case E, regardless of how profit will be distributed among the partners, the deficit in payoff for KPT due to the exclusion of the fee can be easily covered by the surplus earned after coalition. The set of imputations satisfying the conditions of coalition rationality or Eq. (23) constitute the core. In this case, the core does not exist because case D yields lower profits for at least one member of the coalition. Furthermore, solutions out of the core are unstable. For instance, if we compare the payoff of KICT in cases B and C, the

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N. Saeed, O.I. Larsen / European Journal of Operational Research 203 (2010) 393–403 Table 16 Profit of terminals for different combinations of coalitions within Karachi Port (in millions of US $).

a b c d

Profit (in millions of US $)

KICT

PICT

KPT

Combined profit (Karachi Port)

Case Case Case Case Case

13.8 17.6 14.9 13.1 20

6.6 7.1 6.6 10.4 12.2

14.3 11.2a 14.0b 10.1c 8.6d

34.7 35.9 35.5 33.5 41.0

A: All terminals are independent B: (KPT + KICT) and PICT is independent C: (PICT + KICT) and KPT is independent D: (KPT + PICT) and KICT is independent E: All terminals have formed a coalition

Excluding fee paid by KICT and including fee paid by PICT. Including fees paid by KICT and PICT. Excluding fee paid by PICT and including fee paid by KICT. Excluding fee paid by PICT and KICT.

Table 17 Joint profits of terminals that have formed a coalition (in millions of US $).

KPT and KICT PICT and KICT KPT and PICT KPT, PICT and KICT

Joint profit after coalition

Joint profit Before coalition

28.8 21.5 20.5 41.0

28.1 20.4 20.9 34.7

profit in case C is slightly better than in case B (as already mentioned, in case B it will get US $14.5 million, when profit is distributed in a way to overcome the deficit of KPT). Thus, KICT will prefer to form a coalition with PICT instead of KPT. On the other hand, if we look at the situation from the point of view of PICT, it gets a better payoff in case B when it is independent and the coalition is formed between KICT and KPT, as compared to case C when it forms a coalition with KICT. Thus, these two coalitions are not stable. However, the situation is feasible for all three terminals in the last case; that is, the grand coalition. In this situation no players can get a better payoff by joining any other coalition or by following a ‘‘go-it-alone” policy. Thus, in the first stage players should select the grand coalition. An important interpretation of the results obtained from different combinations of coalitions is from the viewpoint of the types of ownership of terminals. As mentioned in Section 2, KICT and PICT at Karachi Port and QICT at Port Qasim are private terminals. These two ports act as landlord ports and charge a fee to the private terminals per TEU handled. However, PICT pays a fee twice that of KICT. The reason for this is not clear, but it probably reflects the fact that the PICT contract is more recent. This difference in fee charged by port authority also influences the game outcome. For instance in case D, when KPT and PICT are partners, the joint profit of the two terminals is low but the individual profit of PICT is high as compared to case A. This is due to the fact that after the coalition, PICT does not pay a fee to KPT. This might be one reason for the low joint profit for the two terminals, because in this case KPT has formed a coalition with the private terminal, which is a great source of revenue for it in terms of paying twice the fee as compared to KICT. QICT is in the second port and remains outside of all the stable coalitions. However, in spite of this, it earned a better profit in all coalitions, as compared to case A when all are working independently (see Tables 6, 8, 10 and 14). This is due to the fact that it managed a low price, as compared to the private terminals at Karachi Port, in every coalition. Thus, this outsider plays the role of an ‘‘orthogonal free-rider” as opposed to a ‘‘non-orthogonal freerider”.10

10

In the case of orthogonal free-riding, free-riders benefit from the cooperative abatement of the coalition, but have no incentive to damage it. In the case of nonorthogonal free-riding, free-riders can damage the coalition (Carraro and Marchiori, 2002).

6.2.8. Users’ surplus In addition to profit, which is the payoff for terminal operators and port authorities, the situation can be analysed from the perspective of users. To estimate the users’ benefits, we apply ‘‘the rule of the half”. In this method a change in users’ surplus is estimated as changes in the generalised costs multiplied by the average demand before and after the formation of a coalition, as shown in Eq. (24):

US ¼

1 ðGC i  GC 0 Þ  ðX i þ X 0 Þ 2

ð24Þ

where US = users’ surplus; GC0 = generalized costs before coalition i. i = B, C, D, E.  GCi = generalised costs after formation of a coalition i. i = B, C, D, E.  X0 = demand before formation of a coalition i. i = B, C, D, E.  Xi = demand after formation of a coalition i. i = B, C, D, E. In our analyses, however, there are changes in the generalised costs both for public and private terminal users. Thus we have applied an alternative method of estimating the user’s surplus, based on the well known property of the logit model. Changes in the logsum of the logit model are as de la Barra (1989) amongst others point out conceptually equivalent to the traditional user’s surplus indicator shown in (24). Thus, (24) can be written as follows in Eq. (25):

US ¼

1 1 ðX i þ X 0 Þ  ðLSi  LS0 Þ 2 b

ð25Þ

where LS0 = logsum of the logit model before formation of a coalition i. i = B, C, D, E.  LS1 = logsum after formation of a coalition i. i = B, C, D, E.  b = the model parameter for user’s cost. The calculated users’ surplus in all four coalitions is presented in Table 18. In three coalitions, i.e. in cases B, C and E, the users’ surplus is negative. Thus the formation of any kind of coalition among terminals is not beneficial for users. In case D, users are better off with positive surplus, because in this case new Nash equilibrium prices for all three private terminals are low as compared to other cases (see Tables 12 and 13).

Table 18 Users’ surplus for all the coalitions (in thousands of US $). Coalitions

Users’ surplus

Case Case Case Case

2402.4  5401.8 1803  13786.2

B: (KPT + KICT) and PICT is independent C: (PICT + KICT) and KPT is independent D: (KPT + PICT) and KICT is independent E: All terminals have a formed coalition

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7. Conclusions and policy implications In this paper an analysis of possible combinations of coalitions among container terminals at one port is presented. The game is divided into two-stages. In the first stage, three terminals at Karachi Port have to decide whether to act singly or to join the coalition. In the second stage, players who have decided to become part of the coalition play a non-cooperative game against non-members. Four possible combinations of coalitions are presented. Again, each game is solved numerically in terms of the Bertrand model. The results of these coalitions are analysed on the basis of two concepts of the cooperative game: characteristic function and core. Three out of four combinations yield a high joint profit for the members of the coalition and hence fulfil the condition of superadditivity property of the characteristic function. However, in case D, in which KPT and PICT join the coalition while KICT is independent, the joint profit for the members is low as compared to case A. Hence, formation of this coalition is not possible. This can be seen to be reasonable. In this situation we suggested KPT as the partner of PICT rather than KICT. However, PICT has a low handling capacity as compared to KICT, due to low storage and other equipment such as yard gantries and yard tractors (See Containerization International Yearbook, 2006; Containerization International Yearbook, 2007; Containerization International Yearbook, 2008). Hence, it proved to be weak partner for KPT, and it is more profitable for KPT to join its strong partner, KICT, in a coalition. The results of the possible coalitions can also be analysed on the basis of the concepts of individual rationality and collective rationality. It is found that, in cases B and E, individual rationality is apparently not satisfied because the payoff for one member, that is KPT, is low as compared to case A. This is due to the fact that we have not considered the fee paid by partner terminals to KPT in these cases. However, in case B, if profit is distributed among the partners so that it will cover the loss in payoff to KPT due to the exclusion of its partner fee, then individual rationality criteria will be fulfilled. This constitutes a proposal of one way of distributing profit among the partners in the coalition, so that it is distributed in such a way that the deficit in the payoff of KPT is covered. Case D yields low profit for at least one member of the coalition, hence coalition rationality is not satisfied and the core in this case does not exist. Button (1999) showed that the core of the shipping market does not always exist. However, no previous research has analysed the existence of the core in the port sector. Other important aspects of these findings are that in two coalitions, i.e. cases B and C, the formation of coalitions has shifted the reaction function of players (who have formed the coalition) outward, while the reaction function of their main competitor, QICT, is shifted downward. As a result, new equilibrium prices for terminals (members of coalitions) are high while the price for QICT is low in comparison with case A. However, case E, the formation of a coalition resulted in an outward shift of reaction function of all three terminals at Karachi Port but, unlike cases B and C, the reaction function of QICT also shifted outward. Hence, at the new equilibrium, the prices of all terminals at the two ports are high as compared to case A. However, the new equilibrium price for QICT is low as compared to the other three terminals. Thus, ultimately the result is same; that is, high prices for members of the coalition and low prices for the outsider. These results are consistent with the findings obtained by Zhang (2008), in which the author proved that under price competition an increase in the corridor capacity by region 1 will shift its port’s reaction function outward and port 2’s reaction function will shift downward. As a result, at the new equilibrium, the price of port 1 rises whilst the price of port 2 falls. According to the author the net impact on port 1’s profit is undetermined. Because an in-

crease in corridor capacity yields two offsetting effects on the port’s own profit, it reduces port 1’s profit owing to the price drop at port 2 and the ensuing demand shift away from port 1 – a negative strategic effect. On the other hand, capacity investment improves port 1’s profit via a reduction in its hinterland’s road delays – a positive direct effect. In our case the net effect on the profits of all players is determined. For instance, in coalitions A, B, C and E, despite high prices and a consequently low market share, the joint profit of the terminals which have formed the coalition is high as compared to case A. Similarly, the profit for the main competitor (QICT) in the second port is considerably higher with low prices. Thus, QICT, which is in a second port and not part of any coalition, still benefits from these cooperations among terminals at Karachi Port. In every coalition QICT’s payoff was higher as compared to case A. This is due to the fact that after the coalition every terminal at Karachi Port ended up with a high price while the price of QICT is low; thus it managed to capture a high market share and to earn better profit. Thus it behaves as an orthogonal free-rider in the cooperative game played among the competitors. Moreover, unlike for QICT, the prices at the new equilibrium for PICT and KPT when they are the outsider within the same port in cases B and C are high with a slight effect on profit. However, these two coalitions are not stable because the members of these two coalitions are better off either by joining any other coalition or by remaining independent. The only stable coalition for all players is the grand coalition, i.e. when all terminals in Karachi Port are members of the coalition. This coalition generates high benefits for the terminals, but the users are worst off in this case. Acknowledgement The authors are grateful to the three anonymous referees for productive comments that lead to improvements in this article. Appendix. Part A

P1 ¼ ðp1  w1  c1 Þ  AehLS Q 1

ð13Þ

By taking the log of last term of Eq. (13) we get:

lnðAehLS Q 1 Þ ¼ lnðAÞ þ hLS þ U 1  LS By taking the derivative of above equation with respect to P1 we get:

@ðAehLS Q 1 Þ ¼ AehLS Q 1 ½bðhQ 1 þ 1  Q 1  @p1

ð13aÞ

By substituting Eq. (13a) in Eq. (14) we get

1 þ ½bðhQ 1 þ 1  Q 1 Þ  ðp1  w1  c1 Þ ¼ 0

ð15Þ

Appendix. Part B

P1 ¼ ½X 1 ðp1  c1 Þ þ X 2 ðp2  c2 Þ þ X 3 ðp3  c3 Þ This will give 3 conditions, on for each price:

@ P @ðAehLS Q 1 Þ ¼ ðp1  c1 Þ þ AehLS Q 1 @p1 @p1 þ

@ðAehLS Q 2 Þ @ðAehLS Q 3 Þ ðp2  c2 Þ þ ðp3  c3 Þ ¼ 0 @p1 @p1

From Part A we have:

@ðAehLS Q 1 Þ ¼ AehLS Q 1 ½bðhQ 1 þ 1  Q 1 Þ @p1

ð16Þ

N. Saeed, O.I. Larsen / European Journal of Operational Research 203 (2010) 393–403

The third (and fourth) term is the cross derivatives. hLS

@ðAe Q 2 Þ ðp2  c2 Þ ¼ AehLS Q 2 ½bðhQ 1  Q 1 Þðp2  c2 Þ @p1 This should give us:

@ P1 ¼ AehLS Q 1 ½bðhQ 1 þ 1  Q 1 Þðp1  c1 Þ þ AehLS Q 1 @p1 þ AehLS Q 2 ½bðhQ 1  Q 1 Þðp2  c2 Þ þ AehLS Q 3 ½bðhQ 1  Q 1 Þðp3  c3 Þ ¼ 0 hLS

Now: Ae

ð17Þ

Q 1 Cancels out, leaving

½bðhQ 1 þ 1  Q 1 Þðp1  c1 Þ þ 1 þ Q 2 ½bðh  1Þðp2  c2 Þ þ Q 3 ½bðh  1Þðp3  c3 Þ ¼ 0

ð18Þ

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