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Journal of Software Engineering and Applications, 20 12, 5, 30-35 doi:10.4236/jsea.2012.512b007 Published Online December 2012 (http://www.SciRP.org/journal/jsea) Copyright © 2012 SciRes. JSEA Reliable Multi-path Routing in Selfish Networks with Hidden Information and Actions Gang Peng, Mingrui Zou, Sammy Chan* Departmentof Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China. Email: *eeschan@cityu.edu.hk Received 2012 ABSTRACT In this paper, we propose a novel game-theoretical solution to the multi-path routing problem in wireless ad hoc net- works co mprising selfis h node s with hidd en infor mation and a ctions. B y incorpora ting a suitable traffic allo catio n poli- cy, the proposed mechanism results in Nash equilibria where each node honestly reveals its true cost, and forwarding subgame perfect equilibrium in which each node does provide forwarding service with its declared service reliability. Based on the generalised second price auction, this mechanism effectively alleviates the over-payment of the well-kno wn VCG mechanism. The effectiveness of this mechanism will be shown through simulati ons. Keywords: Wireless ad hoc network; non-cooperative networks; hidden information; hidden action; mechanism design; GSP auction 1. Introduction Wireless ad hoc networks consist of a set of autonomous nodes distributed in a certain area, and do not have a central coordinator to support communication among the nodes. Wireless devices rely on each other to forward packets to the destinatio n. Initiall y, protocol design often assumed that nodes would always follow the protocol. These kinds of design worked well because such wireless devices were usually owned by a single entity and were supposed to cooperate with each other. With great ad- vancement of mobile communication and computing, more and more small and convenient mobile devices, such as smart phones and tablet PCs, come into everyday life. Wireless ad hoc networks can possibly be formed by these devices owned by different individuals. Due to li- mitations of memory, energy supply and computing re- sources, such devices may behave selfishly and do not follow the protocol. This results in non-cooperative net- works in which each node relies on other nodes to for- ward its packets and yet is not willing to spend its own resources to forward packets for other nodes. To ensure normal operation in non-cooperative wire- less networks, it requires selfish nodes to participate in the routing protocol to establish available paths and for- ward packets if it is along a chosen path to a destination [18]. Since selfish nodes are mainly interested in max- imizing their own payoffs, one common approach to deal with them is to design an appropriate payment mechan- ism to reward cooperation. That is, if a cooperating node receives a payment more than its expended cost in for- warding a packet, then it is likely to follow the routing protocol and forward packets for other nodes. However, an important issue of this approach is to ensure that each node hone stly repo rts its cost expenditure in forwarding a packet, otherwise, traffic sources will be asked to make unrealistic payments. In many literatures [2,13,14], me- chanis ms were designed for soliciting the truthful cost declaration from selfish nodesso that a certain optimal routing structure could be bui lt to connect a so urce node and a destination node. The problem can be modeled as a hidden information game. In addition, another important issue is to ensure that intermediate nodes indeed forward data packets when they are asked to [17]. Unfortunately, as shown in [18], no dominant strategy solution exists in which every node always forwards others’ packets. When packet loss occurs dur ing for wardi ng, i t is dif ficult for other nodes to distinguish whether a failure is due to natural hazard, or due to intentionally dropping by a node. Even if the protocol deploys monitoring mechanism to allow the senders or the receivers to ascertain the loca- tion of the failure, they may still be unable to attribute the causes of failure to either the deliberate action of the intermediate node, or some external factors beyond the contr ol of the no de, suc h as ne twork co ngestion, channe l interference, or data corruption. This problem is referred to as the hidden action problem. The VCG payment mechanism has been applied in single-path routing [3,5,6,15,16], and multi-path routing *Corresponding author. Reliable Multi-path Routing in Selfish Networks with Hidden Information and Actions Copyright © 2012 SciRes. JSEA 31 [5] scenarios to deal with hidden information and hidden actions. However, VCG suffers from the inevitable over-payment problem [4,10]. Efforts to alleviate this problem have been made in single-path routing scenario [12]. A game method proposed in [5,16] stimulates co- operation from intermediate nodes to overcome the hid- den-action problem with hidden information. They mainly considered the single-path scenario. In [8,9], we assumed that all links and nodes are reliable, and have no hidden action. The generalised second price (GSP) auc- tion in multi-path routing was proposed to d eal wit h hi d- den information of selfish nodes. In this paper, we will extend the G SP a uction in mu lti-pat h routi ng to take i nto account more complex conditions such as hidden action of a node. We assume that the link failures are indepen- dent among different links, and aim to design protocols that can eliminate the hidden action without using addi- tional monitoring scheme. As discussed in [8,9], al- though GSP achieves lo wer over-pa yment than VCG, its existing form used in Internet advertising does not guar- antee each node to reveal its true cost. Therefore, we also need to design a mechanism which results in Nash equi- libria for all nodes to behave honestly. The remainder of this paper is organized as follows. Section 2 discusses the abstracted network representation and system model. Section 3 builds the mechanism and designs the algorithm that can efficiently deal with the hidden action and hidden information problem and en- sure reliable multi-path routing in the link layer. Section 4 evaluates their effectiveness through simulations. Sec- tion 5 concludes this paper. 2. System Model 2.1. Network Model A wireless network is formed by a finite number of nodes, denoted by V = {1, 2,...,n}. The existence of the directed edge (i, j) ∈E between node i and node j is de- pendent on the transmission power. We assume that each node i has a set Ti of discrete transmission power levels. For any i, j∈V, there is a minimum power level Tij at which node i could transmit packets to node j. If Tij≤ max(Ti), then we say node j is reachable from node i. In this way, the network could be represented by a weighted directed graph G = {V, E, W}, where W is the set of weights representing the cost on each edge. Each l ink (i, j) ∈E is assi gned an inher ent val ue η(i, j) ∈ [0, 1], denoting the reliability of the link (i, j). The reliability of a link means the probability of decoding the packet correctly when node i has sent a packet through the wireless channel to node j. In other words, node j will receive the data correctly with probability η(i, j) due to possible corruption by interference, after node i sends data to node j. We assume that η(i, j) is public informa- tion i n this paper. Each node i∈V is assigned an inherent value γi∈ [0, 1], indicating the service reliability that a data packet will be successfully forwarded by i if a path includ ing i i s chosen for packet forwarding. In other words, node j will receive the data correctly with probability γiif the link (i,j) is completely reliable, when node ishould forward data to node j. We assume that γi is private for each node i. Node i can choose to declare this private information correctly or incorr e c tly in or derto maximize its o wn utility. For convenience, we denote δ(i, j)= γi · η(i, j) as the value of the corresponding service from node i to node j. We consider implementing a routing protocol that will reliably route data from a source node S to a target node D through multiple paths. Such a routing protocol with selfi sh part icipati ng nodes co mpri ses of the ro uting sta ge and forwarding stage, which are modelled as routing subgame and forwarding subgame [18]. 2.2. Least Cost Path Construction The player set of this multi-path routing game are the intermediate nodes. We assume each intermediate node incurs a per-packet cost for forwarding traffic with the corresponding service reliabilityγi, and thi s cost a nd γiare private to itself. The reliable minimum cost multi-path routing pro blem is to find a set of least cost p aths (LCPs) connecting S and D such that the expected total cost spent by all nodes (including the retransmissions) is mi- nimized, for the g iven probabilities δ(i, j)= γi ·η(i, j). When the link layer reliability is implemented, it means that the retransmission is completed by the upstream node till a packet is successfully received by the down- stream node. Consider a path P connecting the set of nodes {a1,a2,...,ah} wherea1is the so urce a ndah is the des - tination, respectively. The expected cost of sending a packet through P, ε(CP ), is given by ∑ − =+ + = 1 1),( 1 . ),( 1 )( 1 h taa tt P tt c aa C δ ε (1) Here, 1/δ(at,at+1) is the expected number of total trans- missions over link(at,at+1) including the initial trans- mission and all retransmissions, c (at,at+1) is the cost of sending a packet thro ugh l ink (at,at+1). For the sake of simplicity, we assume in this pa per that there is no collusion among the nodes. In the route dis- covery process, each node declares a cost for an outgoing link along with service reliability. Note that the declara- tion may not be honest. After obtaining all the link in- formation and constructing the network graph, the routing protocol orders the node-disjoint paths as the reliable LCP candidates according to the expected cost of each path. That is, after path ordering, ε(Ci) <ε(Cj) if i<j, where ε(Ci) is the expected cost of LCPi. Reliable Multi-path Routing in Selfish Networks with Hidden Information and Actions Copyright © 2012 SciRes. JSEA 32 2.3. System Objective Assume that the first mLCP candidates are selected for packet forwarding. A fraction of data traffic fiwill be forwarded through LCPi. The per-packet payment is cal- culated according to the routing decision, forwarding action and the bids placed by intermediate nodes. The bid of each intermediate node is kept confidential b y encr yp- tion and can only be exposed to the destination and source nodes. Once the route discovery process is fi- nished, nodes cannot change their bids before the trans- mission is completed or rerouting is triggered. Therefore, we model this auction as a simultaneous-move, one-shot strategic game. Following the definition in game theory [7], node i’s per-packet payoff or utility uiis given by , iii cpu −= (2) where ciis node i’s cost and piis the pa yment mad e by the source to node i. While it is obvious that e ach intermediate node i’s ob- jective is to maximize its utility by giving a proper bid, the goal of the entire system is to minimize the total transmission cost by allocating proper traffic among the m paths, subject to certain constraints {P(n)}, which represent a set of policies to be discussed in the follo wi ng section. 2.4. Policy Enhancement In the formulation presented above, the constraints rep- resent a set of policies {P(n)}. Such policies are an im- portant part of our mechanism design. The basic policies are as follows: P(1) : Since we consider multi-pa th r out in g i n thi s paper, naturally, the number of selected LCP candidates mis larger than one, but less than the total number of LCP candidates between the source and destination. P(2) : The fractions of data traffic carried by different selected LCP satisfy the conditions ∑= = m ii f 1 1 and f1>f2> ··· >fm> 0. The first condition follows directly from the fact that all packets need to be forwarded to the destination. Moreover, we emphasize that i∀ , fi>0 since any fi=0 will reduce m. Also, the fact that the fractions of traffic allocated to the mLCP candidates appear in des- cending ord er is compatible to the entire syste m’s goa l o f minimizing total cost, since LCP iwith larger i tends to introduce higher cost per packet. P(3): For any p<q, we require [ε(Cp+1) −ε(Cp)]fp> [ε(Cq+1) −ε(Cp)]fq> [ε(Cq+1) −ε(Cq)]fq.This policy governs how traffic are allocated among the m LCPs. The first inequality[ε(Cp+1) −ε(Cp)]fp> [ε(Cq+1) −ε(Cp)]fqsays that a player on a path with les s total cost tends to have a better utility than one on a path with larger total cost. The second inequality [ε(Cq+1) −ε(Cp)]fq> [ε(Cq+1) −ε(Cq)]fqsays that by overbidding to go from a path with less total cost to one with larger total cost, a player can- not increase its utility. As we will see later , this condition guarantees a truth-telling node’s utility if some other nodes are trying to game the routing protocol. 3. Reliable Multi-path Routing with GSP Auction 3.1. Payment Me ch ani sm Design A sel fish node has certain private information such as it s service cost and the service reliability. Thus, the actions taken by a selfish node are 1) declaring whether it can provid e suc h s ervices and 2 ) pr oviding the tr uthful tel ling of service cost and what kind of forwarding service. The main goal of this routing scheme (composed of routing subgame and forwarding subgame) is then to ensure that every selfish node fulfills its declared service cost and forwarding service. We design GSP mechanism to calculate the payment. The objective of such a mechanism is to stimulate the rational pla yers being honest without hurtin g their utility. In the link layer multi-path reliable routing game, GSP per-packet payment only considers the service cost and service reliability of other nodes on LCPk(excluding itse lf) and LCPk+1, and i s given by ),( )()( ] ),( )([)( 1 1 , kk i kkk k k kk i kkk GSP ki ji c CfC f f ji c CfC p k k δ εε δ εε +−= −− = + + (3) δ(ik,jk)is the link reliability asδ(ik,jk)=γik· η(ik,jk), and ε(Ck) is the expected cost of LCPkpath calculated by Equation (1). Although GSP achieves lower over-payment than VCG (which will be sho wn in Sectio n 4.2), GSP auction has an unpleasant flaw that generally it does not have any truth-telling equilibrium [1,11], where each node truth- fully reveals its private type. In our mechanism design, we eliminate this flaw by adding the polices discussed in Section 2.4 which control the tr a ffic alloc a tion among the selected LCP candidates. 3.2. Algorithms First, we design Algor ithm 1 to alloc a te tr a ffic among the selected paths between a source-destination (S−D) pair. We allocate traffic based on routing policy P(3) and make sure all selected mLCP paths can jointly carry the traff ic. The n, we desi gn the li nk layer multi-path re liable routing scheme given byAlgorithm 2. This scheme en- sures t he link layer re lia bility. Based on a similar approach in [8], it can be proved that, if Algor ithm 2 is used, t here are Nash equilibria for all player nodes to truthfully declare their costs, and for- warding packets according to the declared service relia- Reliable Multi-path Routing in Selfish Networks with Hidden Information and Actions Copyright © 2012 SciRes. JSEA 33 bilities by all player nodes is a subgame perfect equili- brium. Due to spac e limit, the p r oof is omitted in here . 4. Performance Evaluation To establish m LCPs according to our criteria, any exist- ing routing algorithm can be used. Here, we use the commonly used Dynamic Source Routing (DSR) proto- col as an example. The basic route discovery and route maintenance are handled by DSR as usual. The only modification is that instead of hop-count metric, the en- crypted bidding cost and offered service reliability of each node is included in the route request (RREQ) mes- sage. Accordingly, the route reply (RREP) message sent from destination to source includes the network graph constructed from the information gathered in RREQ message s. T hen the routing protocol determines f1,f2,...,fm by usin g Algorithm 1. In order to investigate the incentive aspect of our me- chanism, we have developed an event-driven simulator using C++ (Microsoft Visual Studio 2008, Ver. 9.0.21022.8 RTM) programming language. The general settings of our simulation experiments are as follows. There are 100 nodes and 250 links in the network. To generate a network topology, these 100 nodes are ran- domly connected by 250 links. The costs of links are uniforml y chose n from [1, 5]. The reliability of a link or a node is uniformly chosen from [0.001, 1]. For each intermediate node, the cost involved in forwarding a packet is dominated by the cost of the corresponding outgoing links. Each source node splits traffic among all but one of the available paths according to the specified rules. Without loss of generality, the packet generation rate at each source is assumed to be 1 packet per second, and the transmission time of a packet between two nodes is always 1 second. 4.1. Effects of Different Strategies In this scenario, we investigate the effects of different strategiesand we mainly look at two different aspects, namely, the credit balance and the utility. The credit bal- ance of each node is the payment received from other nodes minus the payment made to others. The utility of each node is the payment received from others minus the cost involved in forwarding packets. There are two ac- tions a node could choose to take during the auction, namely honest and cheating. By honest, it means that a node bids with its true cost and indeed provides its de- clared service reliability, while cheating means a node either understates or exaggerates its cost, and violate its declared service reliability. Therefore there are totally four combinations of different strategies depending on actions by a node itself and other s. While we expec t that when others behave honestly, the strategy of being hon- est always leads to a better utility than the strategy of cheating, we are also interested in comparing them with the other two strategies experimentally. Therefore, we generate the cheating scenario as follows. When we de- cide a node to be cheating, with equal probability the reported cost is either increased or decreased based on Algorithm 2 Link layer multi-path reliab le routing Suppose S wants to send data to D through mul- tiple paths. Input: we ighted graph G= {V, E, W} where |V|=n, service cost of nodes, ci,i∈[1,n], service reliab ility of nodes, γi,i ∈[1,n], link reliability, η(i, j), (i, j) ∈E; Routing subgame: 1) Initialization: Sinitiates with bro adcasting a query for forwarding packets with certain QoS. On re- ceiving the query, each node player ideclares its ser- vice cost , which may not be its tr ue cost,to provide a forwarding service δ(i, j) on link (i, j) for each one of its out-going neighbors j, and the corresponding service reliabilityγi. For all links (i, j), we define its weight ω(i, j) as /(γi· ηi, j); 2) Calculation: Scomputes all pathsto Dusing Di- jkstra’s algorithm under each intermediate node’s claimed cost with the corresponding service relia- bility, and selects the first mLCP candidates; 3) Traffic allocation: Suses Algorithm 1to allocate traffic among the selectedLCP candidates; 4) Forwarding subgame: When an intermediate node ireceives a packet from its upstream node, it will forward the packet to the next-hop node jusing service reliability γior using some other forwarding reliabilit y. 5) Compensation: Send s the tr ansmissi on, and only if Dreceives the data correctly, Spays each inter- mediate node iin one of the selected LCPs with ε(C k+1 ) − ε(C k )+ ω(i k ,j k ). Algorithm 1 Traffic alloca tion Input: selected mpaths, cost of LCPi: ε(C1),...,ε(Cm); Output: the fraction of data traffic f1, ..., fm; do { 1) randomly generate f1, ..., fm, subject to ∑= = m ii f 11 ,fi>0,i ∈[1,m], 2) sortfisuch thatf1>f2>··· >fm. 3)for i: 1 to m − 2 if[ε(Ci+1) − ε(Ci)]fi>[ε(Ci+2) − ε(Ci)]fi+1>[ε(Ci+2) − ε(Ci+1)]fi+1, ﬂag = 1; else {ﬂag = 0; break;} } while {ﬂag == 0} Reliable Multi-path Routing in Selfish Networks with Hidden Information and Actions Copyright © 2012 SciRes. JSEA 34 the true cost, with a percentage uniformly chosen from [5%, 25%]. Its forwarding reliability is either increased or decreased based on the true reliability, with a percen- tage uniformly chosen from [5%, 25%]. We arbitrarily pick a node, say node 23, as the objects under considera- tion. Results shown in Figure 1 and Figure 2 confirm that the strategy “honest when others honest” outper- forms “cheating when others honest”. Also, we observe that for most of the time, the strategies when node is honest produce the best results. 4.2. Over-payment Alleviation Last, we evaluate the over-payment alleviation through simulation. We randomly generate 200 topologies using the same approach and each node honestly follows the protocol under link layer model. We evaluate the over- payment ratio comparing GSP with VCG. We vary the number of paths being used for packet forwarding. The overpayment ratio between GSP and VCG is computed for each case. According to the cumulative distribution function shown in Figure 3, it is observed that GSP al- ways has less over-payment. The improvement of GSP over VCG monotonously increases with the number of for ward ing p aths . For ins tance, as shown in Figure 3 , Figure 1. Credit balance of node 23 with different strategie s. Figur e 2. Ut il it ies of node 23 with different strategies.. Figure 3. CDF of over-payment ratio. with the use of GSP, the largest reduction of overpay- ment in VCG is ab out 55 % in the sett in g of m = 3. And it is about 60% for m = 4. The best result is m = 6, the re- duction of overpayment in VC G is at le a st 30%. 5. Conclusion We have proposed a novel game-theoretical solutions to the multi-path routing problem in non-cooperative net- works in which nodes have hidden information and ac- tion. By incorporating suitable traffic allocation polices, the proposed mechanisms are highly compatible to ex- isting routing protocols, which results in Nash equilibria where each node honestly reveals its true cost, and for- wards packets according to its declared service reliability. Compared to the commonly used VCG payment mech- anism, our GSP-based mechanism results in lower over- payment. 6. Acknowledgment The authors are grateful to Mr. Xueyuan Su for his con- structive comments to improve the quality of this manu- script. REFERENCES [1] B. Ed elman, M. 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