Analysis of profit allocation strategies for competing networks by applying cooperative game theory within an agent-based model

Analysis of profit allocation strategies for competing

networks by applying cooperative game theory

within an agent-based model

G. Bas

Jaffalaan 5, PO Box 5015 2600 GA Delft, The Netherlands

Email: g.bas@tudelft.nl

T.E. Van der Lei

Jaffalaan 5, PO Box 5015 2600 GA Delft, The Netherlands

Email: t.e.vanderlei@tudelft.nl

Abstract—In many industries there are possibilities to co-operate with supply chain partners in order to realise greater profits for the value chain as a whole. The responsibility for the allocation of profits realised by the cooperative network can be given to a third party, a network orchestrator. However, the allocation strategy that is applied for the network might determine what partners are attracted to the network and thereby what network profits are. Thus for a cooperative network to be successful in an environment of multiple competing networks it is essential that the right profit allocation strategy is applied for the partners in the network. In this paper we present an agent-based model of cooperative networks that compete for the best network partners. This model resembles different notions from cooperative game theory, but applies them to compare different allocation strategies in a competitive environment. We compare five different types of allocation strategies: allocation based on Gately point, Proportional allocation, Residual allocation, Even allocation, and Random allocation. Results suggest that Residual, Evenly, and Random allocation outperform the other two allocation strategies. The outcomes of this research should be studied into greater detail to increase the insights into the dynamics that lead to these results. Future work could also focus at expanding the model to explore factors other than the allocation strategy, such as transaction costs or penalties for leaving a network prematurely, that could be of relevance for the network profit.

I. INTRODUCTION

In future we may see the emergence of more and more cooperative networks of otherwise competing companies. One innovation that is expected to enable the trend towards the emergence of cooperative networks is the development of factories in containers [1]. The size of these factories makes that they have a much lower throughput than the traditionally deployed world-scale facilities, but their size also makes them highly mobile and easy to redeploy throughout the world providing competitive advantage through mobility and fast access to upcoming markets.

One way in which these factories can be deployed is in a cooperative network with supply chain partners, situated at a shared site. Such a local production network consists of the entire value chain of processing steps, and can process

raw materials directly into finished products without the need to ship intermediate products around the globe or the need for non-value-added operations carried out for transport. The management of these networks can be entrusted to a network orchestrator that acts as a third party trustee and is responsible for aligning operations, resolving issues, and allocation profits among the network partners. This mode of operations allows companies to be more agile in an increasingly uncertain business environment. Another benefit is that this mode of operations is expected to reduce the environmental impact of the operations due to reduced transport movements [2].

The profit of a cooperative network emerges from the quality of technology applied by the partners in that network, their operational and organisational competencies, and the extent to which they collaborate. Factories in containers can be redeployed easily in other networks, which makes that the networks have to compete with each other for the services of the partners that operate these factories. In this competitive interaction all networks attempt to attract the best performing factories in order to maximise the network profit. The networks use their profit allocation strategy to determine the share of the network profit that is offered to potential network partners, who then choose to join the network that offers them the highest individual profit. This bidirectional relation between the network profits and the network partners suggests that there might be a positive feedback loop that increases the profit of well performing networks and decreases the profit of less performing networks. A well performing network has a high network profit, which means that it has more profit to offer to potential network partners. This allows the network to attract better performing partners, thereby increasing the network profit.

Although the technology to realise local production networks of factories in containers has materialised, companies are reluctant to implement this new mode of operating. One of the reasons for this reluctance is that potential network partners have little experience with operating in a cooperative network and have limited insights in how the profits of the network should be allocated among the network partners. With

Fig. 1. Three Networks orchestrators (rectangles) with connected to them their network partners (circles), and one partner not in a network.

regard to the allocation strategy and the subsequent shares attributed to network partners there are two counteracting forces: 1) the share offered should be high enough to attract better performing partners to the network, and 2) the share offered should not be so high that it is more profitable for incumbent well-performing partners to leave for another network. To overcome the reluctance, that comes from unfamiliarity with operating in a cooperative network, it is important that insights are obtained into the profits each allocation strategy generates in an environment of competing cooperative networks. However, the need to balance the two counteracting forces makes it non-trivial to determine what allocation strategy results in the highest sustainable network profits.

Cooperative game theory is concerned with solving games in which players can collaborate in a coalition and have to impute the payoffs (or costs) of the coalition among them. An example is a group of people that have gone to dinner and need to decide how to split the bill. The payoffs of possible network configurations are the starting point of a cooperative game and the outcomes are the coalition that materialises and the imputation of payoffs [3]. To determine the imputation of payoffs game theorists have developed solution concepts (e.g. the Kernel [4], the Nucleolus [5], and the Shapley Value [6]) that are premises for working together [3]. Solution concepts are typically based on believes of fairness or expected behaviour in a bargaining situation and can be thought of as guidelines for the allocation of profits. There have been comparative studies of different solution concepts, but they only compare the outcome of a game solved multiple times with different solution concepts [7]. No study has been performed that compares solution concepts when applied in the same game, as would be the case for a system of competing local production networks that apply different solution concepts.

The system of cooperative networks that compete for network partners (referred to as “our system”) resembles different notions from cooperative game theory. First, the network partners in our system can be regarded players in a cooperative game, like the networks in our system can be regarded coalitions in a cooperative game. Second, in

cooperative games the players have to decide with whom to collaborate in a coalition and in our system the partners have to decide with whom to collaborate in a cooperative network (i.e. which network to join). Third, where cooperative games are concerned with the imputation of payoffs among the players that form a coalition, our system is concerned with the allocation of network profits among the network partners that participate in a cooperative network. Fourth, like in cooperative games the payoff of a coalition is determined by the players that form that coalition, the profit of a network in our system is the resultant of the partners that participate in that network. Fifth, as cooperative games impute payoffs among the players in a coalition based on a solution concept, in our system network profits are allocation according to allocation strategies. However, some aspects of our system deviate from typical cooperative games. First, where the formed coalitions and the imputation of payoffs are the primary outcomes of a cooper-ative game, they are merely means to an end in our system. In our system we are primarily interested in the profitability of the allocation strategies applied by those networks. The imputation of the profits and which partners participate in which networks are only needed for these profits to materialise. Second, cooperative game theory typically applies one solution concept in a cooperative game, where we study the profitability of different allocation strategies in a system of competing cooperative networks. In order to model this competition among networks with different allocation strategies, our system has to represent multiple allocation strategies simultaneously. One way to look at this to consider each cooperative network as a separate cooperative game with a single solution concept. Our system, then, is a collection of cooperative games that are connected to each other through the players deciding which game to play (i.e. which network to join).

To determine the outcomes of a cooperative game, the game can be played with human players or it can be simulated. One way of simulating games is by means of an agent-based model that simulates the behaviour of the individual players. Agent-based models simulate systems of heterogeneous agents that make individual decisions and interact with each other [8]. This type of models are considered by some to be underpinned by game theory [9], which makes them well suited to simulate games. The specification of an agent-based model requires the definition of agents, their decision rules, and their interactions; the system behaviour then emerges from the (inter)actions of the agents [10]. For our system we would have to define the cooperative networks, the network partners, and the rules that specify how the networks are formed (i.e. how agents decide to join a particular network). Simulating such a system (i.e. letting the agents decide which networks to join) then leads to the emergence of the profits of the networks and subsequently the profitability of the different allocation strategies.

In this paper we present an agent-based model to asses the relative profitability of different allocation strategies, which are based on solution concepts from cooperative game theory. This model consists of cooperative networks that have different profit allocation strategies and compete with each other for network partners in order to maximise the network profit. The network partners operate factories that differ in their performance, which they want to deploy in the network

that allows them to maximise their profitability. We simulate the network formation negotiations between the networks and the potential network partners in order to determine what allocation strategy allows networks to attract the best performing factories and maximise the network performance. This paper is structured as follows: in section II we describe the agents, their behaviour, and their interactions, as they are modelled in the agent-based model. This is followed by section III in which we present the experiment performed to assess the performance of the allocation strategies. In the same section we also discuss the outcomes of the experiment. We conclude this paper with a discussion of the results, what their applicability and limitations are, and in what ways the model could be extended in future work.

II. MODEL DESCRIPTION

The model1 consists of network partners that collaborate in a cooperative network in order to produce a product that can be sold at a particular profit [12]. By collaborating with multiple partners the produced product can be marketed further downstream, which allows the network to a achieve a higher profit. The network profits are allocated to the partners that participate in the network on basis of the network’s allocation strategy. In order to attract the best performing partners and maximise the network profit, the networks offer profits to eligible network partners that they could earn if they would join the network. The proposed network partners consider the offers of different networks and join the network that offers them the highest profit.

The total profit a network n obtains from its operations (Πn) is determined by three factors: the profit obtained per

partner in the network (Πp), the number of partners in the

network (|Pn|), and the performance factor of each partner

in the network (ηp). The profit obtained per partner in the

network is a global factor, fixed for all partners, and can be considered an aggregated representation of the market price for the products that the partners use in their operations and those they produce. A profit per partner of 10,000 implies that a single partner with a performance factor of 1 adds 10,000 value to the products it produces and therefore obtains 10,000 every time period. We assume that there is sufficient supply of raw materials and sufficient demand for finished products, so the networks only compete with each other over the network partners. The number of partners in the network (limited by a predefined maximum) specifies the number of value-adding processes performed by the network. This is assumed to be directly related to the profit the network obtains. The performance factor of a partner specifies the efficiency of a partner, implying that a better performing partner has lower costs and thus has higher profits. This factor represents the quality of technology deployed by the partner, its operational and organisational competencies, and how well it collaborates with others. The performance factor is defined to be 1 or higher. By multiplying the performance factors of the part-ners, the network performs better than the sum of its parts, representing the synergy achieved through the collaboration of the partners. This is in line with the superadditivity and

1_{The model is implemented in Netlogo [11]. The code can be requested} from the authors.

monotonicity assumptions that are common in cooperative game theory [13]. The profit function of a network n consisting of partners Pn is given as:

Πn=  Y p∈Pn ηp  |Pn|Πp (1)

The sequence of actions in this model is as follows:

1) The network orchestrators send offers to (some of) the eligible partners, specifying the profit they could obtain if they join their network (section II-A). 2) The partners that have received offers consider those

offers and decide whether to accept any of them or stay in their current network. This action consists of a sequence of sub-actions, which are performed by the partners. The best performing possible partner performs the full sequence of sub-actions (2a to 2d) until all possible network partners have decided to join or stay in their networks (section II-B).

a) The partner signs a contract with the network that offers the highest profit.

b) The partner deletes its remaining offers of other networks.

c) The network orchestrator of the network that the partner joined selects the best partners in its network by excluding the worst per-forming partner if the maximum number of partners is exceeded.

d) The network orchestrator deletes its remain-ing offers to other eligible partners.

3) The network orchestrators reallocate shares over the incumbent partners (section II-C).

4) The network orchestrators normalise the share allo-cated to the partners in its network (section II-D). 5) The networks operate, receive their profits, which are

distributed over the network partners by the network orchestrator (section II-E).

A. Sending offers to eligible network partners

The reconfiguration of networks starts with the network orchestrators that send offers to partners that are eligible to join their network. This implies that networks initiate the reconfiguration and that network partners cannot decide to leave a network without directly joining another network. Partners that are eligible to a particular network are those partners that do not participate in that specific network and that have a performance factor that is higher than the worst performing partner in the specific network.2 _{Networks that}

have a vacant position (i.e. the number of network partners is lower than the maximum number allowed) send offers to partners not already in a network every time period. Every couple of time periods (determined by the network’s chance to reconfigure) the networks with a vacant position also send offers to a certain percentage (determined by the rationality of the orchestrator) of eligible partners already in another networks. Networks that have no vacant position only send offers every couple of time period to a certain percentage

2_{If the network has a vacant position any partner is eligible for joining the} network.

of all eligible partners. An offer specifies the expected profit for the proposed partner if it decides to join the network. For this the orchestrator calculates the profit of the network including the proposed partner.3 The profit that the proposed partner can expect to make is a certain share of the network profit. The size of this share is determined by the allocation strategy used by the orchestrator. To calculate this share the network orchestrator has insight in the performance factors of all partners (including the proposed partner). Every network orchestrator uses one of these five allocation strategies:

• The first allocation strategy is based on the Gately point [14], which distributes the profit of a network so that the propensity to disrupt (by leaving the network) is minimised. The individual profit of the network partners is proportional to their marginal value to the network. To determine the share that a proposed partner o should be entitled to, first the marginal value of that partner to the network is determined. This is done by subtracting the network profit without the proposed partner (Πn−o) from the network profit

including the proposed partner (Πn). After this it is

determined how the marginal value of the proposed partner relates to the marginal value of all network partners Pn combined; this is the percentage of the

sum of marginal values all partners that is attributable to the proposed partner. This percentage is the share of the network profit that is allocated to the proposed partner under the Gately strategy (so,ga), and is

cal-culated through: so,ga= Πn− Πn−o P p∈PnΠn− Πn−p (2) • When the second allocation strategy is applied the profit is allocated Proportional to the performance fac-tor of the partner. For this the orchestrafac-tor determines what the performance factor of the proposed partner is compared to that of all partners in the network. The share that is allocated to the proposed partner under the Proportional strategy (so,pr) is calculated through:

so,pr =

ηo

P p ∈ Pnηp

(3) • The Residual strategy implies that the proposed part-ner gets allocated the profit that remains when all incumbent partners in the network have been allocated their profit. The profit allocated to the incumbent partners is fixed and the network profit is partially determined by the value that the proposed partner to the network profit, so the residual profit is to a large extent related to the marginal value of the proposed partner. The more value the proposed partners adds to the network profit, the higher the residual profit. The residual profit is calculated by subtracting the profit allocated to all other partners Pn−o(xp) in the network

from the network profit (Πn). To determine the share

3_{If the network has no vacant position and including the proposed partner} to the network would lead to exceeding the maximum number of partners, the orchestrator determines the profit for a network of the best performing partners. Since a proposed partners has a performance factor that is higher than the works performing incumbent partner it is included in this calculation and the worst performing incumbent partner is excluded from the network.

that then is allocated to the proposed partner (so,re),

the residual profit is divided by the network profit. This can be calculated through:

so,re=

Πn−P_p∈Pn−oxp

Πn

(4) • For the fourth strategy the profit is Evenly distributed over all partners that participate in the network. The share that is allocated to the proposed partner (so,ev)

is calculated through: so,ev=

1 |Pn|

(5) • The final strategy is Random, in which case the proposed partner gets allocated a Random share. This share is determined by drawing from a normal distri-bution with a mean value that is equal to the Evenly share and a standard deviation that is 25% of the Evenly share.

B. Network participation decision

After the offers are made by all network orchestrators, the partners that have received offers have to decide which network to join. Since network orchestrators do not get to select partners (besides selecting eligible partners to offer to) and will be joined automatically by the partner that accepts their offer, the order in which partners get to make their decision has a large effect on the outcome of the reconfiguration. We assume that network orchestrators want the best performing partner available in their network, and therefore the best performing partner has the most choice and gets to decide first, followed by the second best partner, and so on. In order to decide which network to join a partner considers all offers4 _received

and determines which of these allows it to obtain the highest profit. The partner selects the most profitable offer and signs a contract with the network that has made that offer. After this the partner discards all other offers that it has not accepted.

If a partner decides to join a network the orchestrator has to determine whether the number of partners in the network is not going to exceed the maximum allowed number of partners. If this is the case, the worst performing partner is removed from the network through terminating its contract.5 The remaining offers made to other eligible partners are withdrawn, effectively allowing the network to add only one partner per time period. In future work this assumption could be relaxed in order to be a more realistic representation of the system.

C. Reallocating shares

Adding a new partner to the network might, in combina-tion with some allocacombina-tion strategies, require the reallocacombina-tion

4_{The partner also considers (if applicable) the contract it has with its current} network. If we refer to offers we mean both the actual offers and the existing contract.

5_{The contracts that network partners have with a network thus have no} duration. The network orchestrator can determine unilaterally to terminate the contract immediately. On the other hand, a network partner also can determine unilaterally to terminate the contract immediately if it can sign a more profitable contract elsewhere.

of shares among the incumbent partners.6 _{To reallocate the}

shares among the incumbent partners, the network orchestrator performs the same calculations as for making the offers, but in this case it does this for all partners in the network. For each of the partners in its network the network orchestrator changes the share that is specified in the contract with that partner. D. Normalising shares

In the case of Random allocation of profits it is very likely that the sum of the share of all partners deviates from 100%. To prevent this from happening the shares are normalised. For this the orchestrator determines the sum of the share of all partners and divides the individual share of each partner (so) by that

sum. This results in a total share of 100% while preserving how the shares of the partners relate to each other. The normalised share of a partner o in network n (so,nor) is calculated through:

so,nor= so P p∈Pnsp (6) E. Operations

After the normalisation of the shares the reconfiguration of the networks is completed and the networks can operate. The focus of this model is on the allocation of profits and their ef-fect on the network profit, and therefore the operational aspects (products, information, etc.) are not considered. This implies that we assume that operations are performed optimally. The only aspect of operations that is considered is the profit that is realised as a result of them.

For the operations all networks determine their network profit (as in equation 1) and allocate it to their partners on the basis of the share that is specified in their contracts. The partners remember this profit for the next time period in order to be able to judge if an offer from another network is better than their current contract. The network adds the profit to its capital. The capital of a network indicates its financial performance over the entire simulation.

III. EXPERIMENTS

A. Experimental setup

To determine which allocation strategy enables network orchestrators to attract (and hold onto) the best performing partners and maximize the network profit, we simulate a sys-tem of multiple networks with different allocation strategies. The allocation strategies are attributed randomly to networks, which leads (on average) to an equal distribution of the alloca-tion strategies over the networks. In every time period during a simulation run each network obtains a certain profit (depending on the network partners that collaborate in the network), which is being added to the capital of the network. The performance of networks is measured over a period of time because the performance of a network is quite volatile, due to partners joining and / or leaving the network, and we are interested in the general performance of each allocation strategy. In order to get proper insights into how the networks perform over an extended period of time we use the accumulated capital as the

6_{For example, adding a partner to a network of 3 partners that has an Evenly} allocation strategy requires the share of the incumbent partners to drop from 33% to 25%.

TABLE I. PARAMETER SETTINGS

Parameter Value Runtime 1,000 Number of networks 150 Number of partners 750 Maximum number of partners per network 5 Profit per partner 1,000 Performance factor distribution unif(1,2) Rationality 0.05 Chance to reconfigure 0.2

indicator of how well networks have been performing in that period. The performance of an allocation strategy over a period of time is measured through the mean capital accumulated by networks that apply that particular allocation strategy. In future work other indicators could be considered, such as the stability of the network.

Because well performing networks have a higher change of attracting the best performing partners, thereby becoming even better performing, the capital that a network accumulates is highly path dependent (see Arthur [15] for an extensive discussion). Initially, the partners are distributed randomly over the networks, which in combination with the path dependence makes that the performance of a network (and subsequently that of an allocation strategy) is to a certain extent affected by randomness. This implies that in a single run a part of the difference in the performance of allocation strategies is caused by that randomness. To determine the difference in the performance of allocation strategies that is actually caused by the allocation strategy, multiple runs of the simulation should be performed. The law of large numbers learns us that a large number of trials should bring the average outcome closer to the expected value. Thus by running multiple runs of the simulation the actual effect of the allocation strategy on the network profit is uncovered. In our case we have run the simulation 500 times, which should reduce the randomness to acceptable levels.

Table I presents the parameter settings of the simulation. The profit per partner specifies the value added (and there-fore profit obtained) by a partner with a factory that has a performance factor of 1. The performance factor distribution defines the distribution from which the performance factor of individual factory is drawn. The rationality of a network is the percentage of non-vacant eligible partners that a network sends its offers to. Each network gets assigned the same rationality. The chance to reconfigure defines the chance that a certain network with no vacant positions decides to reconfigure. Networks with vacant positions reconfigure every time period, but they only consider partners not already in another network. B. Outcomes

For every simulation run we indicate the profitability of the allocation strategies over the period of time by measuring the mean capital of networks using that strategy at the end of the simulation. Figure 2 visually presents the profitability of each allocation strategy at the end of the simulation runs performed. Table II presents the descriptive statistics of the profitability of the different allocation strategies. The mean outcome of the simulation runs is particular relevant in that regard, because

TABLE II. DESCRIPTIVE STATISTICS OF ACCUMULATED CAPITAL USING DIFFERENT ALLOCATION STRATEGIES

mean sd %rsd Gately 4.12E+07 1.36E+06 0.033 Proportional 4.04E+07 1.38E+06 0.034 Residual 4.22E+07 2.15E+06 0.051 Evenly 4.18E+07 1.52E+06 0.036 Random 4.20E+07 1.45E+06 0.035

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4.0e+07 4.5e+07 5.0e+07

Gately Proportional Residual Evenly Random Allocation strategy

Capital

Fig. 2. Accumulated capital using different allocation strategies

this indicator is less affected by the randomness than the individual simulation runs and therefore provides insights in the actual profitability of the different allocation strategies.

The results in figure 2 and table II indicate that the most profitable allocation strategy is the Residual strategy. The second best strategy is the Random strategy; the profitability is 0.5% lower than that of the Residual strategy. The Random strategy is closely followed by the Evenly strategy, which has a profitability that is 0.9% lower than that of the Residual strategy. The fourth best performing strategy is the Gately strategy, with a profitability that is 2.4% lower than that of the best performing strategy. The worst performing strategy is the Proportional strategy, which has a profitability that is 4.3% lower than the profitability of the Residual strategy.

In order to test the statistical significance of the differences in profitability of the different allocation strategy we perform an analysis of variance (ANOVA) test. The ANOVA test is used to compare 2 or more groups on basis of their average, which makes this test suited for this situation. Table III presents the results of the ANOVA test. The p-value of 0.000 indicates that the zero hypothesis, stating that there is no difference between the mean capital accumulated by networks using different allocation strategies, is rejected. This means that there is a statistical significant difference between the allocation strategies. However, this does not imply that the differences between all allocation strategies are statistical significant.

To test whether the differences in profitability of any

TABLE III. ANOVAANALYSIS OF ALLOCATION STRATEGIES

Df Sum Sq Mean Sq F value Pr(>F) Variable 4 1.03E+15 2.58E+14 1.01E+02 0.000 Residuals 2495 6.39E+15 2.56E+12

TABLE IV. TUKEYHONESTSIGNIFICANTDIFFERENCES ANALYSIS

diff lwr upr p adj Gately-Proportional 7.72E+05 4.96E+05 1.05E+06 0.000 Evenly-Proportional 1.42E+06 1.15E+06 1.70E+06 0.000 Random-Proportional 1.55E+06 1.27E+06 1.83E+06 0.000 Residual-Proportional 1.77E+06 1.49E+06 2.04E+06 0.000 Evenly-Gately 6.50E+05 3.74E+05 9.27E+05 0.000 Random-Gately 7.77E+05 5.01E+05 1.05E+06 0.000 Residual-Gately 9.94E+05 7.18E+05 1.27E+06 0.000 Random-Evenly 1.27E+05 -1.49E+05 4.03E+05 0.720 Residual-Evenly 3.44E+05 6.77E+04 6.20E+05 0.006 Residual-Random 2.17E+05 -5.91E+04 4.93E+05 0.201

combination of allocation strategies is statistically significant we perform a Tukey Honest Significant Differences Analysis (Tukey HSD). A Tukey HSD analysis is used to find what means of the categories are significantly different from each other, making it suited for our purpose. Table IV presents the results of the Tukey HSD analysis. The results of this analysis show that for almost any pair of allocation strategies the means of these strategies is significantly different. The two exceptions are the pair of the Random and the Evenly strategy (a p-value of 0.720) and the pair of the Residual and the Random strategy (a p-value of 0.201). This implies that the mean values of these two pairs do not differ significantly and that we cannot draw any conclusions on the relative performance of these allocation strategies.

On basis of these outcomes we can conclude that the Residual, the Random, and the Evenly strategy have a signif-icant higher profitability than the Gately and the Proportional strategy. Also can we conclude that the Residual strategy is sig-nificantly more profitable than the Evenly strategy. However, we cannot draw any conclusions on how the Random strategy relates to the Residual and the Evenly strategy. Of the Gately and the Proportional strategy we have found that the Gately strategy performs significantly better than the Proportional strategy.

These outcomes are relatively unexpected since one would expect the strategies that allocate profits to network partners according to their relative performance (Gately and Propor-tional) to be the most profitable. The rationale for this may be that when profits are not allocated according to the relative performance of partners, better performing network partners are underpaid and less performing partners are overpaid. That would cause the well performing network partners to leave for networks that reward their performance better, leaving their old networks with only the less performing partners. Because the less performing partners are overpaid, the network does have sufficient profits left to outbids networks that allocate profits according to the relative performance of their partners. However, from the outcomes we conclude that Gately and Proportional are the worst performing strategies, which is opposed to what we expected.

One possible explanation for the unexpected outcomes is that the partners in the Residual, Evenly, and Random (RER) are less often transferred to another network, because their performance factor is lower and therefore they are less often eligible to join another network. The transfer of a partner

TABLE V. DESCRIPTIVE STATISTICS OF NETWORK COMPOSITION

partners performance spread Gately 4.734 1.512 0.464 Proportional 4.740 1.504 0.449 Residual 4.625 1.560 0.419 Evenly 4.750 1.517 0.396 Random 4.744 1.520 0.409

causes a (albeit temporary) large decrease in performance, because it is missing one partner. This could explain the lower overall performance of the Gately and Proportional (GP) networks, even though they attract better performing partners. Table V presents the average number of partners and the average performance factor of these partners in networks with certain allocation strategies. This data shows that the performance of the GP networks is actually lower that of the RER networks and that the average number of partners in the network is not higher to the extent that it explains the difference in profitability.7 _{So partners in GP networks}

are not more eligible for transfer and the GP networks do not have a vacant position more often because of the transfer of a partner. Even more, the average number of partners for Residual networks is the lowest of all while its performance is the highest. Based on this we can conclude that the unexpected outcomes are not caused by the partners in RER networks being less often eligible to transfer.

Another explanation could be that GP networks spend too big a share on their best performing network partners, leading to insufficient remainders to attract other well performing partners. RER networks do not (or to a limited extent) allocate profits according to the performance of the network partners, which might lead to more balanced networks, i.e. networks where the spread8 _{of performance is relatively low. It might}

be that, despite their average performance factor being higher, unbalanced networks perform less than balanced network, e.g. an unbalanced network of a 1.1 and 1.9 partner has a perfor-mance of 2.09 (1.1 times 1.9) with an average perforperfor-mance of its partners of 1.5, while a balanced of network of two 1.45 partners has a performance of 2.10 with an average performance of its partners of 1.45. The data in table V shows that the spread of GP networks is higher than that of RER networks. This could to a certain extent explain the unexpected outcomes. Whether there are more explanations for these outcomes, and to what extent the spread affects the profitability of the allocation strategies, requires further research into the exact dynamics of the system.

IV. CONCLUSIONS AND DISCUSSION

In this paper we set out to determine which profit al-location strategy enable cooperative networks to maximise their profits sustainably. For that we developed an agent-based model consisting of networks that compete to attract the best performing partners. We defined five allocation strategies and compared the accumulated capital of the networks using these strategies. We found that the Residual, Evenly, and Random strategies are the best performing strategies and that the

7_{The difference in all values in table V are tested with the ANOVA test} and the Tukey HSD analysis, and are found to be significantly different from each other.

8_{The spread is defined as the highest performance factor of a partner minus} the lowest performance factor of a partner.

Gately and Proportional strategies are the worst performing. These outcomes are unexpected since we would expect that the strategies that allocate profits according to the partners’ performance (i.e. Gately and Proportional) would be the best performing. These results can partially be explained by the spread of the performance of network partners in Gately and Proportional networks being higher. Further research into to what extent the spread affects the network profits is needed.

This paper contributes to the study of emerging cooperative networks, which in practice is being hampered by the unfamil-iarity of operating in tightly interconnected networks of supply chain partners. This study has indicated that an agent-based model can be used to research the impact of factors that affect the profit of cooperative networks and to explore the intra-and inter-network dynamics of a system consisting of multiple competing networks. Obtaining more insights into the factors that affect network profits and into the dynamics that lead to these profit is very likely to reduce the reluctance to operate in a cooperative network.

This work can be continued into multiple directions: 1) extending the experiments performed on the current model, 2) extending the model to include more factors that might affect the network profit, or 3) extending the model in order to be a more truthful representation of reality.

The experiments performed on the current model could be extended by considering more allocation strategies than the five we researched in this study. In future works more advanced allocation strategies derived from cooperative game theory could be considered. The implementation of these allocation strategies requires more sophisticated algorithms. Solution concepts that could be considered are the Shapley Value, the Nucleolus, or the Kernel. A second extension to the experiments could be extending the performance indicators used to measure the performance of networks. In this study we used the network profit as the sole indicator of network performance. The discussion of the experiment’s outcome showed that other indicators, such as the average number of partners in a network or the spread of the partner’s performance in a network, could improve the insights obtained in the network’s dynamics.

In this study the allocation strategy has been considered the only factor that affects the network profit. However, other factors could also have an impact on a network’s profit. One of these factors is the transaction costs involved with the reconfiguration of networks. Replacing one partner for another partner will certainly lead to a loss of production, resulting in temporarily lower profits. Transferring these transaction costs to the leaving partner might result in more stable networks that are more profitable over time. A second factor that could be of relevance for the network profit is the duration and other terms of contracts. In this study the contracts could be terminated unilaterally by any one of the two parties involved (network and partner) at no costs. This is caused by the assumption that the contracts bear no costs. Considering the duration of a contract and adding a penalty for terminating the contract before the expiration date of the contract might lead to other allocation strategies performing better.

Besides extensions of the model to research more factors that are of relevance to the network profit, the model could also

be extended to be a more realistic representation of reality and increase the reliability of the outcomes. One assumption that could be relaxed is that there is no difference in the types of partners that should be included in the network (i.e. partners performing different processes), nor are the operational aspects of the partners considered. We assume that there are sufficient partners available for the different processes, so that the competition for every process step is the same. This allows us to abstract from reality and assume that there are no different process steps and that only the number of network partners determines what product is being produced (represented by the profit that can be obtained). In reality it may be that the network partners for a particular process step are relatively scarce, increasing the competition for these partners. This could have a significant impact on the capital that the networks accumulate over time and the relative performance of the different allocation strategies. A second extension to improve the reality of the model could be to include strategic behaviour of network partners. In the current model the network partners are maximizing the profit of the current time period, while in reality it may occur that partners are willing to take a loss in order to realize future profits. For example, partners might be willing to lower their share of the profit in order to attract better performing partners, thereby increasing future (network and individual) profits. Including these considerations might alter the relative performance of the different allocation strategies.

ACKNOWLEDGMENT

The authors would like to thank S.W. Cunningham for his discussions on cooperative game theory.

REFERENCES

[1] F3_{Factory, “Welcome to the F}3 _{factory,” 2014. [Online]. Available:} http://f3factory.com/scripts/pages/en/home.php

[2] G. Bas, D. Van der Linden, B. Vannieuwenhuyse, M. Oudshoorn, and T. Van der Lei, “When big is no longer beautiful. exploring the competitiveness of factories in containers using an agent-based simulation model of a global supply chain,” forthcoming.

[3] M. J. Osborne and A. Rubinstein, A course in game theory. MIT press, 1994.

[4] M. Davis and M. Maschler, “The kernel of a cooperative game,” Naval Research Logistics Quarterly, vol. 12, no. 3, pp. 223–259, 1965. [5] D. Schmeidler, “The nucleolus of a characteristic function game,” SIAM

Journal on applied mathematics, vol. 17, no. 6, pp. 1163–1170, 1969. [6] L. S. Shapley, “A value for n-person games. Contribution to the Theory of Games,” in Annals of Mathematics Studies, H. Kuhn and A. Tucker, Eds., 1953, pp. 307–317.

[7] D. Gately, Investment Planning for the Electric Power Industry: An Inte-ger Programming Approach, ser. Research report. Dept of Economics. The University of Western Ontario. University of Western Ontario, 1970.

[8] C. R. Shalizi, “Methods and techniques of complex systems science: An overview,” in Complex systems science in biomedicine. Springer, 2006, pp. 33–114.

[9] Y. Shoham and K. Leyton-Brown, Multiagent systems: Algorithmic, game-theoretic, and logical foundations. Cambridge University Press, 2009.

[10] A. Borshchev and A. Filippov, “From system dynamics and discrete event to practical agent based modeling: reasons, techniques, tools,” in Proceedings of the 22nd international conference of the system dynamics society, no. 22, 2004.

[11] U. Wilensky, “Netlogo,” Center for Connected Learning and Computer-Based Modeling, Northwestern University. Evanston, IL, 1999. [Online]. Available: http://ccl.northwestern.edu/netlogo/

[12] S. Derikx and H. Koopstra, “Business model for local production networks,” Tech. Rep., 2014.

[13] G. Owen, Game theory. Academic Press, 1995.

[14] D. Gately, “Sharing the gains from regional cooperation: A game theo-retic application to planning investment in electric power,” International Economic Review, pp. 195–208, 1974.

[15] W. Arthur, Increasing Returns and Path Dependence in the Economy, ser. Ann Arbor Paperbacks. University of Michigan Press, 1994.