Prisoner’s dilemma game model for e-commerce

Applied Mathematics and Computation 292 (2017) 128–144

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Prisoner’s dilemma game model for e-commerce Jalal Eddine Bahbouhi, Najem Moussa∗ LAROSERI, Department of Computer Science, University of Chouaib Doukkali, EL Jadida, Morocco

a r t i c l e

i n f o

Keywords: e-commerce Trust Small world Prisoner’s dilemma

a b s t r a c t This study investigates how the user’s exchanges affect people’s trust in electronic commerce and empower them to interact with foreign customers. Online exchanges often occur between strangers who cannot rely on past behavior or the prospect of future interactions to establish mutual trust. Game theorists have formalized this problem as a prisoner’s dilemma and predict mutual noncooperation. In this paper, we introduce a social network based model to promote the cooperation in the prisoner’s dilemma game. For this study, we implemented an agent-based simulation framework, which models different types of behaviors in online auctions. Agents who represent buyers or sellers are located in the nodes of a small world graph. Each link weight between the agent and its neighbor symbolizes how much it trusts this neighbor. A link with trust inferior to some cut-link threshold will be removed from the graph. Our results show that the evolved structure of the graph induces considerable variation in the level of cooperation and the profit in the e-commerce system. This shows that the outcome of our co-evolutionary game, in terms of cooperative behavior, strongly depends on topology but also on the update rule used in the trust between users. Co-evolution of game dynamics in interconnected networks is also studied, where two networks are interconnected, and players have interactions not only with others in the same network, but also with players in the other network. We will demonstrate that the interdependence between networks promotes the cooperation in both networks. But, the degree of promotion changes as a function of interdependence degree. © 2016 Elsevier Inc. All rights reserved.

1. Introduction The appearance of the Internet helped to unify and approach the world, and becomes an important business platform for trading goods and services between consumers. More specifically, the expansion in information and communication technologies produces a great advancement to e-commerce. This emergence has greatly improved the perspective of commercial behavior of traditional business. Today, consumers have very easy ways to interact with each other through Internet,via social networks and peer-to-peer networks. So, e-commerce is growing at an impressive rate in the world, and daily life becomes ideal with online shopping, providing people the possibility to purchase commodities and to get the services they want. These developments are attracting more and more people to come online and interact with each other, which makes e-commerce grow at an exponential rate. However, this development in the e-commerce did not prevent the existence of some problems in the system; resulting from the anonymity and insecurity such as deception, collusion, shill bidding, cheating in online trading, or even fraud ∗

Corresponding author. E-mail addresses: [email protected] (J.E. Bahbouhi), [email protected], [email protected] (N. Moussa).

http://dx.doi.org/10.1016/j.amc.2016.07.018 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.

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[1–4]. These negative factors increase the hesitation of a trader before he chooses to participate in online auctions. Indeed, the absence of the physical presence of the participants, the lack of certainty about the reliability of services and the real existence of consumer products, generates a crisis of confidence among traders. Thus, the decision of who to trust and with whom to engage in a transaction becomes more difficult and falls on the shoulders of individuals when receiving goods without payment or receiving payment without sending goods. In such an environment, to ease consumers fears and help them in decision-making, various reputation management systems have been developed by auction sites to help users estimate the trustworthiness of unknown buyers and sellers. In general, the buyer or seller (or both) will receive a feedback score from their trading partner, at the end of each trade. A positive feedback score increases their reputation value by one point, whereas negative feedback will decrease their reputation value by one point. In theory, the trader with a higher reputation value simply remembers that he is responsible and trustworthy. Moreover, since positive feedback results from satisfactory deals,a high reputation value may also indicate the capability of providing a good service [5]. This can assist in deciding whether or not to engage in a transaction with this party. Reputation systems are particularly useful in cases where the opponent is unknown to the individual involved [6]. Another solution for trust problem has arisen with the emergence of Web 2.0 technologies, namely social commerce [7]. Social media has provided an enormous potential to transform e-commerce from a product oriented environment to a social and customer-centric environment [8].Social media refers to Internet-based applications built on Web 2.0, while Web 2.0 refers to a concept as well as a platform for harnessing collective intelligence [9]. Within this environment, customers have access to social knowledge and interactive shopping experience which puts the power of shared community in the customers hands, to support them in better understanding their on-line purchase needs, and in making more informed and perfect purchase decisions [10]. Meanwhile,online businesses are able to capture customers behaviors, which give them intuitions into their shopping experiences and expectations, and help them to develop successful business strategies [11]. Such ecommerce enhances customer participation [12], promotes customer relationships [13], and achieves greater economic value [14]. According to [15], trust is the probability upon which an individual expects another to perform actions on which the first individuals welfare depends. Trust has a key influence on the success of e-commerce. Online sellers need to construct a situation in which a consumer can be assured about any online transactions. Recent researches shows that trust positively influences a users intention to forge tighter and more effective links with business partners [16,17]. Indeed, trust reduces behavioral hesitation intended to buy or to sell in web sites and it gives power of control over the transaction to consumers. This power helps users to interact with others as they deliberate their intention to buy or to sell. That is, it is very likely that trust in online communities sustains users in their shopping behavior. Trust is an important aspect in e-commerce and when consumers have doubts about their deals,they try to reduce uncertainty by relying on trust through familiarity [16]. When people participate in transactions and ratings for other users, their level of familiarity is likely to increase. This brings more confidence to the next transaction. In addition, it has been confirmed that trust has a significant role in enhancing a persons intention to deal with on-line transactions [18,19]. When searching for new goods or services in an online environment,an individual needs to feel more confident and have less perceived risk [20]. Therefore, it is important to study the role of trust in e-commerce by developing new paradigms and models. In previous studies, some researchers have stated that there is a significant relationship between trust and online commerce behavior [21,22]. The neighborhood (familiarity) is the way used by people to minimize the feeling of uncertainty and build new relationships with others. Neighborhood provides people with information, often founded on prior interactions, experiences, and learning about the experiences of others. As such, neighborhood and trust complement each other. In addition, neighborhood reduces distance between people by establishing relations between unknown people, so makes for trust among them, and trust diminishes uncertainty by letting people hold relatively sure expectations about other people, so excluding unethical behavior [23]. Without neighborhood, trust cannot reach levels that can reflect favorable behavior. It provides a basis for deciding, and lets people create concrete ideas about future expectations, based on their previous interactions. Moreover, neighborhood presents an accumulation of very helpful and accurate impressions of someone who is not known. This is usually used to decide to build a new relationship, when the experience is favorable, or destroy it if it is not. The reason for this is that neighborhood measures the degree that prior experience has been understood. When experiences give favorable results, the neighborhood creates trust, but when the results are unfavorable, it ruins trust [24]. The aim of this paper is twofold. First, we wish to introduce an agent based model to simulate the effect of trust and uncertainty in the e-commerce systems. In such an environment,agents maintain the trust that they have in each other and communicate this with each other when necessary. When agents interact with each other, they ask their neighbors what trust they put in the opponent. This trust between agents and theirs neighbors can be reinforced or weakened, depending on the given information. Also agents may delete their relations with others, if the trust in them falls below a certain threshold, called cut-link. We shall show that this parameter plays an important role in the prisoners game model on a small world graph. Second, we wish to use this cut-link threshold as a measure of social trust in a community. Our study focuses on interconnecting two communities with different cut-link thresholds and examines how the strength of their interconnection can promote cooperation in both communities. Therefore, our study examine how the social trust in different communities can affect users use of e-commerce in interdependent networks. This paper extend existing approaches in the study of evolutionary prisoner’s dilemma game in a complex network. In the network, agents are supposed to make choices according to the trust, i.e agents will choose a strategy according to their

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expectation of how much they trust other agents. Therefore, it is worthwhile to introduce agents’ trust into the prisoner’s dilemma game model and to study its effect on the emergence of cooperation in networks. Hence, this paper will study how the trust update rules in association with co-evolution game, changes the overall cooperative level. The remaining parts of our paper are structured as follows. We present related works in Section 2. Section 3 introduces the concepts of trust and e-commerce, and briefly discusses the consumer-provider dilemma, and then we introduce our agent-based simulation model for e-commerce. In Section 4, an evaluation of our model and a qualitative analysis of our empirical results are given for one network. Section 5 gives a simulation study of two interconnected networks. Finally, Section 6 concludes this paper. 2. Related works 2.1. Trust Trust and cooperation are an essential elements in human societies. They have received an important studies from several disciplines and different perspectives. Psychology [25,26], sociology [27], philosophy [28] and economy [29] are an illustration of disciplines that have studied trust and cooperation. However, the study of trust and cooperation in the discipline of computer science has acquired a great importance in the last few years. The enormous development of e-commerce have contributed to significantly increase the interest on the trust in this field. The computational models of trust have been built on two different sources of information: (i) the direct interactions and (ii) the information provided by other members of the society about experiences they had in the past [30–35]. These interactions help to recognize others types of social relations between their members [30]. A number of research on human societies have been used social networks for studying and analyzing trust [15,36]. These studies show that using the information obtained from the analysis of social network, it is possible to predict the behavior of individuals. There are many works on trust and reputation from the point of view of computer science [27,37,38]. Buskens et al. [27] in his paper addresses the way in which the level of trust in cooperative relations depends on network structures. Macy and Skvoretz [39] insert the genetic algorithm in a social network. He describe how trust could be more likely between people in a small network of frequent interaction. Baba [38] reveal the importance for successful networks of high levels of trust between network partners. One of the difficulties that has hindered previous research on trust has been a lack of clear differentiation among factors that contribute really to trust [40]. Kee and Knox [40] argued that to appropriately study trust there must be some meaningful incentives at stake and that the trustor must be cognizant of the risk involved. Trust is a belief or expectation that the word or promise by the merchant can be relied upon and the seller will not take advantage of the consumer’s vulnerability [41]. One important and fundamental concept related to the successful of every business and development of e-commerce is trust [42]. Quelch and Klein [43] noted, trust is a critical factor in stimulating purchases over the Internet.. Indeed, trust expresses feelings of security about risks or uncertainties in e-commerce environment [24,44,45]. A steady stream of literature has explored the critical role of trust in consumer adoption of the web [23,46–52]. Nevertheless, many studies have identified the building of trust as a fundamental and yet unresolved issue in the development of e-commerce [50,53,54]. Seligman has extended and clarified a number of Luhmanns ideas [24,55] and Earle and Cvetkovich [56] about the distinction between trust and confidence. Shapiro [54] refers to the need for controls in order to do not have opportunistic behaviors; however, the author also claims that such controls reduce trust in relationships. The Internet auction, as new form of online exchange in which most transactions occur among entities that have never met, have seen an increasing interest recently [57–62]. There are several other papers on online auction. Chen et al. [58] study procurement over price and transportation costs, but take the perspective of a third-party auctioneer rather than the buyer. Liu and Chen [63] study properties of slot auctions under incomplete information. Their setting restrict their attention to a model with a single slot and a binary type for bidder relevance (high or low). Lim and Tang [64] focus on the bidding behaviors in keyword auctions in a two firm environment, where the bidding prices can only take three discrete values. They have characterized the conditions under which both advertisers will submit high or low bids. Edelman et al. [65] introduce a concept they call locally envy-free equilibrium, which requires that each player cannot improve his payoff by exchanging bids with the player ranked one position above him. They also introduce the interesting concept of a Generalized English Auction and show that the unique perfect equilibrium of this game is the same as the locally envy-free outcome. 2.2. Prisoner’s dilemma In the original form of the Prisoner’s Dilemma Game, two agents decide whether to cooperate (C) or defect (D). They will get reward R when both of them choose to cooperate, and P if they both choose to defect, while the cooperator will receive S when he confronts a defector, who in turn will get T, where T > R > P > S. The dominant strategy in the classical PDG is that both agents choose to defect, despite it is better of both agents if they choose to cooperate. But contrary to the theoretical prediction, cooperation behavior is commonplace in society. To address this issue, in the past two decades, many extensions to the PDG have been proposed to resolve the dilemma and to explain the emergence of cooperation in the PDG [66–71]. One main approach is imposing a network structure on

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Table 1 Prisoner’s dilemma table.

C(Cooperate) D(Defection)

C(Cooperate)

D(Defection)

R/R T/S

S/T P/P

agents interactions. Nowak and May showed that the introduction of spatial structure via nearest neighbor interactions enabled the cooperators to form clusters on the square lattice and so protect themselves against the exploitation by defectors. Following this discovery, the impact of the spatial structure on the evolution of cooperation has been investigated in detail [69,70,72–76]. The PDG has been studied among a population which is structured by a network. The network structure provides a convenient way to describe the interconnection among agents in the system [77]. The relationship between network structure and evolutionary game theory has attracted much attention [66,67,78].Those evolutionary game theory models have demonstrated the ability of link reciprocity to promote the evolution of cooperation in group interactions [79]. Many Research on the PDG has used complex networks for studying the evolution od cooperation. Wu et al. [80] studied the cooperation on a NewmanWatts (NW) network where agents can voluntarily participate in the game. Also Fu et al. [81,82] studied the cooperation emergence on an NW network. Abramson and Kuperman [83] studied the PDG on a small-world network; they found that the changes in topologies of small-world, has effects on cooperation behaviors. Santos et al. [84–87], who found that a scale-free network can promote the emergence of cooperation. Unlike these workings, Vukov and Szab [88] used one-dimensional chain, for studying the maintenance of cooperation of the PDG. Now, it is widely accepted that the PDG on a complex network and heterogeneity tend to promote cooperation in the PDG. The network topology is proved to be a critical issue in cooperation emergence. In e-commerce the anonymity, insecurity, uncertainty and cheating, are factors that increase the hesitation of a trader in taking his strategy (cooperate or defect). This situation illustrates the problem of the Prisoners Dilemma where there are two players who cannot communicate with each other directly. Each player has two strategies, i.e. cooperation (C) and defection (D). Each strategy has a payoff, as shown in Table 1, where R denotes reward for cooperation, P punishment for mutual defection, T temptation to defect and S the suckers pay-off. In e-commerce each player must be the seller (A) or the buyer (B). In each game two players have to deal, one with a product that he try to sold to the buyer, and the other with the money that he try to buy the product from the seller. If both players choose to cooperate (i.e. A sends the product to B, and B sends money to A), both players receive a reward R = 1 (each one get what he want from the other). If the buyer chooses to cooperate and the seller chooses to defect, the latter gets the temptation payoff T = 2 (he will save his product plus he get the money), while the former is left with the suckers payoff S = 0 (he lose his money). And the same case if the seller chooses to cooperate and the buyer chooses to defect. If both players choose to defect (no one of them send anything to the other), both of them receive a punishment P = 0 (no one win anything). Throughout this paper, we will use R = 1, T = 2, S = 0 and P = 0 [89]. The strategies taken by the players define the outcome of the game. All the agents follow the mixed strategy for choosing their actions. A pure strategy is an action, or an action plan, which is chosen by each player with certainty. A mixed strategy is an assignment of a probability to each pure strategy. We assume that each player chooses a lottery on the set of pure strategies: he associates a probability to each strategy and left at random mechanism consisting of the composition Lottery care to choose a pure strategy. In this perspective, each player now aims to maximize expected gains by choosing the best lottery, i.e. the best mixed strategy. In the case of our game, player 1 has a probability x for choosing Cooperate (C) and a probability (1 − x ) for choosing Defect (D); for the player 2, these probabilities are respectively y and (1 − y ). The expected gain of player i (i = 1, 2) can be written as follows:

payo f f (i ) = xyUi (C, C ) + x(1 − y )Ui (C, D ) + (1 − x )yUi (D, C ) + (1 − x )(1 − y )Ui (D, D )

(1)

Where Ui represents the payoff of players i, i.e. Ui (X, Y) is the payoff of the player i for the strategy X of player 1 and strategy Y for player 2. 3. Model In the present paper, we study the evolutionary prisoner’s dilemma game on complex network. The WS small world network is used here, but the Barabsi–Albert Model give qualitatively the same results as WS. The model consists of a set of N agents connected through the graph approach proposed by Watts [90]. Each agent is located on some node of the weighted graph. In a small-world graph, there is a short path between any two nodes, and more exactly, the average shortest hop distance between two randomly chosen nodes is around six. The weight associated with any edge between two nodes a and b represents the strength of the connection between the two agents (how much agent a trust agent b : trust(a, b)). The graph is static throughout the simulation: no nodes are added or removed and the edge weights are variable (see Link adjustment). The network is a directed graph, we use two different edge weight values between every two agents (a, b): one associated with how much a trusts b and another associated with how much b trusts a. All weights used in this work are in the range [0, 1]. An example of the graph is depicted in Fig. 1.

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Fig. 1. Network Construct.

3.1. Trust evaluation Every agent i has a number m of links with other agents, which represent its neighborhood. Each link weight between the agent i and its neighbor k, symbolizes how much the agent i trusts the agent k. During the game, if i faces with an opponent j in its neighborhood, then it counts on link weight (trust) for deciding his strategy. When the agent i is confronted with a strange opponent j, that does not belonging to the agent’s neighborhood, it asks for advice about j from the best neighbor k. The choice for this specific neighbor is based on how much agent i trusts agent k. We represent the trust of agent i in agent k by trust(i, k). This measurement is integrated into the function eval (i, k, j ) that expresses the evaluation of neighbor k by agent i about opponent j, i.e.

eval (i, k, j ) = t rust (i, k ) ∗ t rust (k, j )

and t rust (i, j ) = max[eval (i, k, j )]

where k is the best neighbor chosen from neighborhood of agent i with the highest value of eval (i, k, j ). Asking is recursive, thus when agent k does not have j in his neighborhood, he then selects his best neighbor to ask for advice about j, and so on. After that, the trust value is propagated back to agent i, multiplied by the trust values in the information chain. For example, if the built chain goes from i via k1 to agent k2 to ... to agent kn who knows j then the propagated trust value received by i is:

t rust (i, j ) = t rust (i, k1 ) ∗ t rust (k1 , k2 ) ∗ ... ∗ t rust (kn−1 , kn ) ∗ t rust (kn , j ) 3.2. Execution of games The population of agents play a series of independent games that can take place between any two agents. At each time step, one couple of agents (i, j) is chosen randomly to perform a transaction between each other. The outcome of every game depends on the strategies of its players. The agent can choose cooperation (C) or defection (D). The action taken by an agent depends on the trust value propagated back through the chain established with its opponents for the current game. It can decide to use the received advice, and take a decision, to cooperate or not, according to the trust threshold (T). The result of the game is based on the estimation of the trust between the two agents:



if

t rust (i, j ) ≥ T

then i chooses Cooperate as strategy

if

t rust (i, j ) < T

then i chooses De f ect as strategy

(2)

If it is not possible to find an agent that knows the opponent j within (λ) steps (which we called here after the propagated chain limit), the chain will not be built and the agent i will use his default behavior (Cooperation). 3.3. Link adjustment When two agents (i, j) who have no relationship with each other, cooperatively play a game, a social link (wi j ) will be established between the two players. The weight of this newly created link is given by:



wi j =

t rust (i, j ) 1

if propagated chain limit <

otherwise

λ

(3)

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Table 2 Simulation parameters. Parameters

Values)

N: number of participants K: number max of neighbors

100 6 [0.0, 1.0] 6 0.5 [0.0, 0.1]

β

Steps limit when asking T Cut-Link threshold

Fig. 2. Changes in the Cooperation with time.

In addition, if the two agents know each other and play again cooperatively, their link will be reinforced. However, when an agent defects, the link with its opponent in this game is weakened or broken:



wi j =

wi j + ( 1 − wi j )/2 wi j − wi j /2

if j played cooperatively

if j played de f ectly

(4)

However, if the opponent j continues to commit treason during the subsequent games, and the link weight becomes below the cut-link threshold, the agent i cuts the link with it. 4. Simulation study on one network Simulation is one of the relevant approaches and suitable methods used to investigate agent based systems. In the following section, we will present our numerical results on a single network using the parameter settings listed in Table 2. Initially, the agents are placed in the nodes of the graph and the link weights between neighboring agents are randomly chosen from the interval [0.6; 0.8]. 4.1. Evolution of cooperation rate over time In this section, we shall study the evolution of the level of cooperation over time in the e-commerce system. Fig. 2 shows how the cooperation rate changes with time, to reach a stable value after some transient time. The blue line represents the agents who take the strategy cooperate and the red line for defect agents. So, we distinguish three phases in the time evolution of the network; illustrating the dynamic nature of link adjustment in the model. The first phase is when the cooperation level decreases, the second phase is when the level increases, and the last one is when it takes a stable value. In the beginning of the experiment, every agent has a table of neighbors whom he trusts; so the value of confidence is high for every neighbor. That makes the most of the chains created from an agent to his opponent, have a good value of trust, leading to the success of almost all the operations and therefore to have a good cooperation rate (ࣃ 0.90). Then, there is a decrease in the level of cooperation. The unique cause of this effect is the diminution in the chain of trust, because the players are far from each other, and/or the neighbors trust becomes low due to defect agents or to agents added to the neighborhood with a weak trust. Hence, the agents begin to cut the link with low confidence neighbors when it exceeds the cutting link threshold, and therefore these people become so far from each other. Fig. 3a illustrates a snapshot from the network at this time stage where we see that cooperative agents form a sparse network with weak clustering. Fig. 4a confirms this analyze where the clustering coefficient decreases in this part; indicating that the population becomes more

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Fig. 3. Coevolution of strategy and structure leads to high levels of public cooperation. Networks depict snapshots in time at (a) 40 0 0, (b) 12,0 0 0 and (c) 30,0 0 0 iterations. Accordingly, cooperators are depicted by green and defectors are depicted by red. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Fig. 4. We quantify the structural properties of the graphs in Fig. 2 by these characteristic: path length L and clustering coefficient C. L measures the typical separation between two vertices in the network, and C measures the cliquishness of a typical neighborhood.

scattered. Each link removed from a clustered neighborhood has an effect on the clustering coefficient C and on the path length L. Hence, we see that C drops rapidly, and L increases sharply (Fig. 4). We note that the simulations presented in Figs. 2, 3, 4, 5 and 6 are obtained with cut-link equal to 0.7. At the end of the first phase, all players drop their neighbors with a low trust rate, and keep only the neighbors with a great trust rate. Therefore, at that time, the agents that might give advice on which strategy to take, become very rare; and therefore players will be forced to use theirs default strategy (cooperate). This will promote created chains to be more credible and the population will become denser; leading to raise again the level of cooperation between the agents (second phase). This can be seen clearly from Fig. 4 where we show that the clustering coefficient increases, and the path between agents decreases. Stable behavior occurs just after, thus leading to a stable phase to take place in the system (third phase). This last stage corresponds to the non-equilibrium steady state in statistical physics, where all its statistical properties are considered time-independent. Fig. 3c illustrates a snapshot from the network at this time stage where we see that cooperative agents form a dense network with important clustering. Fig. 5 corresponds to the different time stages of nodes degree distribution in network, we can see that these distributions have different behaviors in each part of time. Hence, we see that at short time, nodes degree distribution follows an exponential decreasing, where almost all nodes are still have so little neighbors. More the simulation time increases, nodes get more neighbors, consequently the fraction of nodes with an important neighbors increase. When the time become long and the network approach from the steady state, the network come to be more cluster and nodes gathered an important neighbors, therefore we can perceive that the fraction of nodes with big degree is very high. This results is produced by cooperation between players, witch emerge to new link between players and so bigger neighborhood. Also the players can

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Fig. 5. Degree distributions in network at different time stages.

improve their topological position, for example by cutting links to defectors. This is confirmed by previous research Zimmermann and EguÃluz [91], which studied the Prisoner’s Dilemma game on adaptive networks. This evolutionary game is a mechanisms which favor cooperation under adaptive population structures, Tanimoto J [92], Tanimoto [93]. In Fig. 6, we show the time evolution of the total payoff in the e-commerce system. According to Eq. (4), the payoff of a single transaction depends on the cooperation rate of the players. So, the payoff decreases strongly, then it undergoes a slow enough increase to reach stationary values. 4.2. Effect of propagated chain length The length of the propagated chain is defined as the maximum number of hops beyond which the advice received is no longer considered. It is obvious that when the number of nodes to pass between the source and the target increases, the trust between them will diminish. For example, a path where a source is connected directly to the target may be expected to be more trusted than a path where there is an intermediate node between the source and the target. In this paragraph, we will analysis the effect of propagated chain length (λ) on the level of cooperation.

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Fig. 6. Variations in payoffs with time.

Fig. 7. The evolution of the level of cooperation for different values of λ with cut-link = 0.7.

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Fig. 8. Small World interpolation between a regular ring lattice and a random network by using a Random rewiring procedure, without altering the number of vertices or edges in the graph.

Fig. 9. The effect of small world topology on the cooperation level. We change p (beta) of small world from 0 to 1 and we note its effects. Each curve correspond to one value of beta.

Fig. 7a shows the cooperation level when λ = 1, which means that the source will search for advice about the target in their direct neighbors. In other words, the agents who could give advice on which strategy to take must be their neighbors; otherwise they will use theirs default strategy (cooperate). Hence, players will save in their neighborhood table just the neighbors that they trust. So, in this situation the level of error in evaluating the trust will be very low and consequently the cooperation level will be important. Fig. 7b depicts the level of cooperation when λ = 2. In this case the source will ask advice from their direct neighbors and/or the neighbors of their direct neighbors. These neighbors play the role of intermediates; and the decision of both players should depend on the information transmitted by the intermediate nodes. Thus, clearly the trust between the player and his opponent will be less than if they are directly connected, and consequently the level of error in evaluating trust may increase. As a result, a little diminishing in the time evolution of the cooperation level will appear. In Fig. 7c, we show the time evolution of the cooperation level when λ = 6. So the length of the propagated chain is now larger than the previous cases, and the trust becomes less important as we increase λ. Consequently, the cooperation level will decrease more. Moreover, we find that above this value (λ = 6), there will not be an effect on the level of cooperation.

4.3. Effect of network topology We perform another experiment where we focus on the effect of the topology of the network on the cooperation rate of the population. It is well known that when one increases the parameter β , different topologies can be formed in the watts model. Indeed, small world topology (β ࣃ 0.1) lies between the regular network and the random networks (Fig. 8). In this

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Fig. 10. Changes in the Cooperation with time, for different cut-link values.

experience, we fix the value of the cut-link threshold and we vary the special parameter β of the small-world network, from 0 to 1. For each value, we record the change in the cooperation rate (Fig. 9). We can see from Fig. 9 that the effect of varying the topology of the network is noticeable only in the first and the second phases of the curves of the cooperation rate. However, the third part does not sustain a great change. In the first part of Fig. 8, the curve for the β = 0 value, is higher than those for β = 0. Indeed, the regular lattice at β = 0 is highly clustered, where L grows linearly with the number of population [90]. That means that the clustering of the network enhances familiarity and increases trust between agents; and consequently, it increases the cooperation rate between people. In the second part of Fig. 8, we see that there is a decrease in the level of cooperation as we increase β , caused by loss of some connections between agents. Furthermore, the increase of β makes a drop in clustering and path length. So the people become further away from each other, and their trust and cooperation diminish more with the increase of β . Another important point that we can see from the figure is the fact that all curves are merged at the limit of simulation times. At short times, the level of cooperation is highly dependent on the network structure; but at long times, it converges to a stationary value. 4.4. Effect of cut-link In this subsection, we shall study the effect of varying the cut-link thresholds on the time evolution of the level of cooperation. Fig. 10 illustrates this effect, where we see clearly that important changes occur when varying cut-link thresholds at all time stages of the evolution. Furthermore, it is found that these variations are more important if the cut-link threshold is great. More interestingly, we will give more attention to the variation of the cooperation rate at steady states when we vary cut-link threshold. To do this, we keep the same parameters, but we change the cut-link threshold from 0 to 1, and we compute the mean cooperation and the mean of the profit at steady state, as a function of the cut-Link values (Fig. 11). Our diagrams in by Fig. 11 can be analyzed by distinguishing different regions. For small values of cut-link threshold (0 ≤ cut − link ≤ 0.2), the level of cooperation and the profit does not change significantly because each agent retains the majority of its neighbors, and has a wide choice for the installation of new relationships. For 0.2 ≤ cut − link ≤ 0.5 we see the level of cooperation and the profit decrease with increasing the cut-link threshold. When cut − link > 0.5, the level

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Fig. 11. Levels of Cooperation, Defection and Payoff as a function of cut-link at steady states.

of cooperation and the profit increases and returns back to the high values. Here, the cut-link threshold is high, which means that the neighborhood of any agent is built based on great values of trust; thereby encouraging further cooperation between agents. In the reality, when people have a higher level of trust in their neighborhood, then the agents will have a great confidence and less perceived risk in dealing with these people. This article deals with the study of the trust between people in order to predict their intentions to perform on-line transactions; thus this model is a reasonable starting point for developing a framework for trust in e-commerce. In addition, our results confirm some proof for the impact of revenue, education and developed country on trust and its related factors. On one hand, when people, as in New Zealand, those with higher incomes and higher education, are less worried about e-commerce technology, usually perceive less risk toward each other and with the foreigners; leading therefore to have a successful e-commerce [5]. This scenario can match what is happening in the model for small values of cut-link threshold (0 ≤ cut − link ≤ 0.2) of the diagram presented in Fig. 11. On the other hand, when people impose a high level of trust in e-commerce even if they perceive considerable risk, or have wide experience in the internet, their concern about security and privacy becomes more important, and thus it may reduce levels of trust. It seems that the higher the user’s Internet experience, the more they are likely to form a big neighborhood, which may help to improve their trust in e-commerce. This situation agrees with our numerical finding in region of Fig. 10 (0.6 ≤ cut − link ≤ 1.0) where a maximum cooperation and profit could be obtained in the model. When people suffer from economic, social, financial and physiological conditions, their partnership with a well-known partner or an unknown partner may reduce the perception of trustworthiness. So the country’s situation affects the social trust, and social trust affects the e-commerce in that country. Therefore a people with a low-trust environment are less likely to form committed transaction relationship. This is exactly what shows the region of Fig. 11 (0.2 ≤ cut − link ≤ 0.5) where the level of cooperation is decreasing rapidly. It is likely that the benefits of e-commerce are changing from one country to another. For example, in countries with high social trust, people are ready to enter into a transaction with foreign partners through the Internet. As a result, users can gain a lot from deals on the internet. In contrast, the profits from e-commerce in low-trust countries will not be as high as those in high-trust countries. In low-trust countries, users are not ready to enter into a transaction with foreign partners; so, they simply transact with existing partners. Consequently, even though many other users have adopted hightrust countries approaches, users sticking with their existing partners will not benefit from the e-commerce. Thus, in low trust countries, the neighborhood effects should not be considered a critical factor in people’decisions on the choice of the good strategy during their interactions with strangers [21]. Our model adopts social trust, and has been developed in order to study consumers’ behavior in e-commerce. Initially, the model emphasized the role of social relationships of individuals and the cooperation change through transferring information to other peers. Individuals are being rapidly cooperatives attracted to the information given by the others to help them to take the good choice. The empowerment earned by agents through the neighborhood makes them active users and encourages them to have social interactions with other agents. These social relationships drive value for both buyers and sellers. Furthermore, agents also intend to develop closer relationships with new agents, giving rise to better user relationship management. This model shows how the e-commerce is facilitated by these social interactions through the use of neighborhood. Indeed, agents feel closer to each other and encourage each other to have more participation. Our experiments tests significantly support the assertion that neighborhood will increase trust. In addition, this research indicates that neighborhood or familiarity is more likely to attract individuals, increases trust and drives consumers to purchase ecommerce goods. 5. Simulation study on two interdependent networks More recently, the interconnected networks have gained importance in many fields of scientific research including biology, physics and computer science [94–98]. Many published works have focused on the study of the cooperation between

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Fig. 12. Interdependent Networks - Blue arrows represent links inside S-Network while red arrows represent links inside W-Network. Black arrows represent links between S-network and W-network. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Fig. 13. Time evolution of cooperation in two interdependent networks for different values of p.

agents when all their interactions are within a single network. Here we consider two interdependent networks; each had its own set of agents, interacting not only between the agents of their own network but also with agents of the other network. Moreover, the interaction between agents within each network is similar to that we have studied in Section 3 of this paper. However, we distinguish the two networks by assigning different cut-link values to each network. Throughout this section, we shall fix the cut-link threshold in S-Network equal to 0.7 and in W-Network equal to 0.4. The objective of this part is two folds. The first is to try to model the e-commerce between different networks; a weak network (W-Network) in which people suffer from economic, social and financial conditions (0.2 ≤ cut − link ≤ 0.5) and a strong network (S-Network) where agents adopt high level of trust in their e-commerce (0.6 ≤ cut − link ≤ 1.0). The second is to determine how such influence could really help to promote maximum cooperation on both networks. Individuals on networks S and W can have interactions with their local opponents of the same network and with opponents in the other network, where network interdependence can influence the cooperation and the defection of individuals on both networks. Nevertheless, in the choice of strategy, depending on the interdependence between networks with a certain probability p, individuals of both networks can have advice from their local nearest neighbors, and also from their corresponding one on the other network. In other words, the interdependent networks are quantified according to a probability p which measures the strength of their interconnection. So, for p = 0, the network S has no connection with network W, and they cannot influence each other. In the opposite limit, p = 1, all the individuals on S and W are completely interconnected. For 0 < p < 1, the interdependence between S and W is subject to a random distribution (see Fig. 12). The number of links between the two networks is a random number whose mean value is equal to p ∗ n ∗ m, where n = 100 is the size of network S, and m = 100 is that of network W. At each time step, three couples of agents are chosen randomly to perform transactions between each others. Two couples are within each of the two networks while the third couple is between agents belonging to different networks.

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Fig. 14. Time evolution of big and average cluster in two interdependent networks for p = 0.5.

Fig. 15. Cluster size distributions in two interdependent networks at different time stages.

In Fig. 13, we plot the time evolution of the level of cooperation for different values of the internetwork probability p. So, for all values of p, we see that the level of cooperation at steady states in the network S is usually higher than in the network W. In addition, the figure clearly shows that the interdependence between networks induces an improved level of cooperation for both networks. Particularly, when each agent of the W-Network forges a link with the S-network ( p = 1), the cooperation level in the interconnected system will be much enhanced. One of the key advantages of interconnected networks is that, individuals can find theirs trading partners easily. This characteristic will benefit individuals in low-trust networks. That is, individuals in a network with a low level of trust often are willing to transact with new partners which have a high level of trust. As a result, individuals in low-trust networks should be more likely to adopt interconnected networks e-commerce. From our simulation results, we argue that

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interconnected networks could be a solution to the problem of trust, because they promote the cooperation between peoples with different communities and are a source for diffusion of trust in a low social trust networks. We have seen that the level of cooperation is enhanced in both networks when we interconnected the two networks. In order to investigate deeply the structure of the interconnected network, we plotted the distributions of clusters in the two networks. Here, we define a cluster of cooperators as the set of all present cooperators connected to each other. Thus, picking any two individuals belonging to the same cluster, it is possible to find a path between them only passing over cooperative individuals. This means that individuals within a cluster can form a single community sharing the same interest and projects. In Fig. 14 we show the time evolution of the average cluster size of the cooperators in both networks. So, comparing Fig. 14a and 14b, we see that over a long time, almost all cooperators joined each others to form one big cluster. We can also understand from theses curves that the gathering of cooperators nodes in one big cluster, helps spreading cooperation in the network. According to Fig. 14, we notice that the evolution cooperation on network S and W is guided predominantly by the volume of such big clusters. We notice also that, at steady state, the big cluster in S is larger than in network W. This confirms well the proceeding result which states that level of cooperation at steady states in the network S is usually higher than in the network W. By focusing on Fig. 15, which corresponds to the different time stages of clusters distribution in each network, we can see that these distributions have different behaviors in both networks. Hence, we see that over a short time, the cluster size distribution follows an exponential decreasing, where almost all cooperative individuals are isolated (cluster-size = 1). The more the simulation time increases, the larger the more large clusters appear in both networks. However, at the steady state (long simulation time) the distributions in both networks are normal with large mean values. In addition, we see that the mean value in network S is larger than that of the network S. This analysis further demonstrates that agents in the network W tend to evolve over time to the situation where they establish cooperative relationships with trusted agents (network S) to forge tighter and more effective links with business partners to improve the total payoff. 6. Conclusion In this paper, we have introduced an agent based system that represents the relationships and the interactions of consumers in an e-commerce environment. We provide numerical simulations to illustrate the ability of the model to represent the influence of the neighborhood and trust on the consumer’s behavior and its impact on the collaboration rate and profit in the population. Our general result is that a trusted neighborhood gives the opportunities for participation and collaboration between users. Furthermore, our computer simulations have shown that the temporal evolution of collaboration rate behaves differently depending on whether the simulation time is short or long. At short times, the collaboration rate varies according to the change of the network structure. More precisely, we have showed that the collaboration rate increases with increasing the clustering coefficient of the network. This supports well the assertion that the neighborhood increases trust. In the other hand, our system converges towards a steady state in the limit of long simulation times. Moreover, we have found that this steady state does not depend on the initial underlying network; but depends strongly on the cut-link threshold. Our agent based system incorporates also the notion of interconnected networks. We have shown that the level of cooperation can be developed using interdependency between two different networks. We considered a network where agents adopt high level of trust in their e-commerce transactions and another network where agents tolerate low confidence. We have shown that this interdependence can promote the cooperation in both networks. More precisely, the more ties between the two networks are important more the cooperation will be enhanced in the system. This finding confirms that inappropriate changes in one network can have advantage or disadvantage, and very much unexpected consequences in another network [21]. Finally, we pointed out that the advantage of our model is that it is quite simple and general enough to be applied to a wide variety of on-line trust situations. In addition, we hope that the model can be used as a framework to study numerically and/or mathematically complex social situation in the e-commerce. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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