Broadcast in Ad hoc Wireless Networks with Selfish Nodes: A Bayesian Incentive Compatibility Approach

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1 Broadcast in Ad hoc Wireless Networks with Selfish Nodes: A Bayesian Incentive Compatibility Approach N. Rama Suri Research Student, Electronic Enterprises Laboratory, Dept. of Computer Science and Automation, Indian Institute of Science, Bangalore, India. nrsuri@csa.iisc.ernet.in Y. Narahari Research Supervisor, Electronic Enterprises Laboratory, Dept. of Computer Science and Automation, Indian Institute of Science, Bangalore, India. hari@csa.iisc.ernet.in Abstract We consider the incentive compatible broadcast ICB) problem in ad hoc wireless networks with selfish nodes. In this paper, we design Bayesian incentive compatible Broadcast BIC-B) protocol to address this problem. VCG mechanism based payment schemes have been popularly used in the literature to design several dominant strategy incentive compatible DSIC) protocols for ad hoc wireless networks. There are a few important limitations in using a DSIC broadcast protocol for the ICB problem since it requires the network to be bi-connected and it is not budget balanced [5]. Our proposed BIC-B protocol overcomes these difficulties. We prove the optimality of our proposed incentive compatible scheme for the BIC-B protocol. We finally compare and contrast the BIC-B protocol and a DSIC broadcast protocol. Keywords Ad hoc wireless networks, broadcast, selfish nodes, rationality, VCG mechanisms, dagva mechanism, budget balance, incentive compatibility. I. INTRODUCTION An ad hoc wireless network is a collection of wireless nodes forming a temporary network without the aid of any centralized administration or standard support services. These networks attempt to form multi-hop networks without preconfigured network topologies. There is peer-to-peer interaction among the nodes. Each node is a host as well as a router since ad hoc wireless networks are distributed in nature [6], [7]. With the availability of multiple routes to the same node, they can perform route selection, based on various metrics such as robustness and energy cost. A. The Incentive Compatible Broadcast ICB) Problem In this paper, we study broadcast in the ad hoc wireless networks. Broadcast is useful in route discovery, paging a particular host, or sending an alarm signal. So, it is interesting to study broadcast in which to consider incentives because the individual wireless nodes could belong to independent and self-interested users in many typical applications of ad hoc wireless networks. We call nodes selfish if they belong to independent users and their objective is to maximize their individual goals. Most of the existing protocols for broadcast assume that the nodes follow the prescribed protocol without any deviation in the sense that they forward the transit packets. As the nodes in ad hoc wireless networks are powered by batteries which provide a limited amount of energy and forwarding the transit packets consumes battery power, selfish nodes may not be ready to forward the transit packets. For this reason, the existing broadcast protocols may not work in the presence of selfish nodes. So, there is a need to compensate the transit costs of the selfish nodes to stimulate them to cooperate with each other in forwarding the packets. But, in the real world applications, we do not know the incurred costs of the nodes since the incurred cost of a node depends only on its consumed resources like CPU cycles, bandwidth, etc. Designing a payment scheme to reimburse the transit costs of the nodes in such settings is very challenging and interesting problem. We call this problem as incentive compatible broadcast ICB) problem. B. Motivation In this paper, we design a Bayesian incentive compatible broadcast BIC-B) protocol to solve the ICB problem in ad hoc wireless networks with selfish nodes. We now give the motivation for our work. VCG mechanisms are also called as dominant strategy incentive compatible DSIC) mechanisms [3]. DSIC schemes for unicast, multicast, and broadcast protocols [1], [2], [8], [11] have been proposed in the literature. We call a protocol DSIC protocol, if the protocol uses an incentive scheme that is DSIC. Very recently, there are studies [4] on the design of DSIC protocols for ad hoc wireless networks. However, there are two major difficulties in using DSIC broadcast protocols. D1) The underlying graph of ad hoc wireless network must be bi-connected for the DSIC broadcast protocols to work. D2) DSIC broadcast protocols are not budget balanced [5]. The difficulty D1) implies that DSIC broadcast protocols work only for those ad hoc wireless networks for which the underlying graph of that network is bi-connected. This means

2 that there cannot exist any cut vertex in the corresponding underlying graph. As the number of nodes in the ad hoc wireless network increases, sustaining this condition is difficult. We say that an incentive scheme is budget balanced if the sum of incentives and transit costs of the nodes is equal to the sum of charges collected from the nodes. This implies that a protocol based on such incentive scheme is self-started by the nodes in the network. So, we can conclude from the difficulty D2) that an external entity has to inject some amount of money into the system to start any DSIC broadcast protocol. We note that there is a specific setting where DSIC schemes are budget balanced [5]. But, such a specific setting may not be relevant for the ad hoc wireless networks. These two difficulties limit the applicability of a DSIC broadcast protocol in ad hoc wireless networks with selfish nodes. So, we design a BIC-B protocol to solve the ICB problem that overcomes the above two difficulties. C. Contributions of the Paper To the best of our knowledge, this is the first time a Bayesian incentive compatible protocol for ad hoc wireless networks is being explored. In particular, we design a Bayesian incentive compatible broadcast protocol. The following are the contributions of this paper. 1) We propose a Bayesian incentive compatible broadcast BIC-B) protocol to solve the ICB problem in ad hoc wireless networks with the selfish nodes. 2) We show that our solution approach effectively tackles the two difficulties D1) and D2) mentioned in the previous section. 3) We prove the optimality of our proposed incentive compatible pricing scheme that is used in the BIC-B protocol. We also prove a few other properties of this protocol. 4) We present a discussion in which we compare and contrast the BIC-B protocol with a DSIC broadcast protocol. D. Outline of the Paper We organize this paper in the following way. We review the related work in Section 2. In Section 3, we explain the network model and the problem that we are addressing. We propose the BIC-B protocol in Section 4. We prove a few properties of the BIC-B protocol in Section 5. In Section 6, we present a discussion on a DSIC broadcast protocol and the BIC-B protocol. We conclude the paper in Section 7 by presenting a few directions to the future work. II. RELATED WORK Routing in the presence of selfish nodes has been one of the major issues in the design of ad hoc wireless networks. The recent works of Anderegg and Eidenbenz [1], Eidenbenz, Santi, and Resta [2], and Wang and Li [11] address the design of DSIC or strategy-proof) unicast protocols for ad hoc wireless networks with selfish nodes. We cannot adopt these solution approaches to the incentive compatible broadcast problem since each node is an intended receipt of the packets) in broadcast. Note that the intermediate nodes on the unicast path are not the intended receipts of the packets). In Wang and Li [10], the authors propose DSIC multicast protocols for ad hoc wireless networks with selfish nodes. Wang and Li [10] assumed that the nodes in the multicast set forward the packets for free among themselves. In this paper, we do not make any such assumptions. That is, each node incurs a cost for forwarding a packet. Suri [8] proposed an incentive mechanism, Immediate Predecessor Node Pricing Mechanism IPNPM), to stimulate cooperation among the selfish nodes for broadcast in ad hoc wireless networks. According to IPNPM, a node pays only to its immediate predecessor node in the broadcast tree that spans all the nodes. The reason is that all the nodes in the broadcast tree are intended recipients of the broadcast packets). In the literature, the dominant strategy incentive compatibility is popularly used to design incentive schemes for unicast, multicast, and broadcast protocols in ad hoc wireless networks with selfish nodes. All the models that we discussed in this section suffer from the difficulties similar to D1) and D2) mentioned above. In this paper, we propose the BIC-B protocol for the ICB problem such that it overcomes these difficulties. III. THE MODEL The ad hoc wireless network under consideration is represented by an undirected graph G N, E), where N {1, 2,..., n} is the set of n wireless nodes and E is the set of communication links between the nodes. There exists a communication link between two nodes i and j, if a node i is reachable from node j, and node j is also reachable from node i. Thus we have an undirected graph representation. We assume that each wireless node has an omnidirectional antenna and a single transmission of a node can be received by any node within its vicinity, i.e. all its neighbors in G. We note that all nodes in the graph G are connected. We do not impose any restrictions on the topology of the graph G. We assume that the nodes in the ad hoc wireless network are owned by intelligent and selfish individuals and their objective is to maximize their individual goals. For this reason, they may not participate in the network functions, like forwarding the packets, since such activity might consume the node s resources, say battery power, bandwidth, CPU cycles, etc. Let each node i incur a cost θ i for forwarding a packet. We also call θ i as type. For simplicity, we assume that θ i is independent of the neighbor from which the packet is received and the neighbor to which the packet is destined. We can represent θ i as the weight of node i in the graph G. This implies that we work with a node weighted graph in our model. Moreover the transit cost of a node depends only on its consumed resources and is independent of the other nodes. Thus transit costs of the nodes are statistically independent of one another. We assume that wireless nodes do not collude with one another to improve their utilities. This assumption is a standard one in all economic based approaches.

3 Consider the task of broadcast in the current network settings. Assume that the source of the broadcast is node s. If not all remaining nodes are connected to node s, then a few nodes have to forward the broadcast packet to make it reach all the nodes in the network. As explained above, the nodes that are forwarding the packets) incur costs. We need a tree that spans all the nodes in the network and minimizes the cost to forward packets. We assume that the node s is the root of the tree. We call such tree as Source Rooted Broadcast Tree SRBT). In this paper, given a SRBT, we study the design of a Bayesian incentive compatible payment scheme for the ICB problem. A. Problem Statement Consider broadcast in an ad hoc wireless network represented by a node weighted undirected graph G N, E), where the weight of each node is the transit cost of that corresponding node. Node s N is the source of broadcast. Each node is required to announce its transit cost. Let θ θ 1,..., θ n ) be the announced cost profile of the nodes. In such settings, given a SRBT of the considered graph under θ, our goal is to design a Bayesian incentive compatible payment scheme for the ICB problem in ad hoc wireless networks with selfish nodes. We call the broadcast protocol with this payment scheme as Bayesian incentive compatible broadcast BIC-B) protocol. In what follows, we present the design of such a protocol that overcomes the two major difficulties D1) and D2). IV. BIC-B PROTOCOL BIC-B protocol is a broadcast protocol with a Bayesian incentive compatible scheme built into it. In this section, we study how to design such an incentive mechanism that implements the social choice function SCF) fθ) kθ), t 1 θ),..., t n θ)), θ Θ, where Θ is the set of all cost profiles, corresponding to the ICB problem. Here kθ), and t i θ), i N are interpreted in the following way. kθ) is the allocation rule that represents which nodes in the network have to forward the packet, given the profile θ of types and t i θ) is the payment received by the node i, given the profile θ of types. For any i N, if t i θ) > 0, then the interpretation is that i receives some positive amount and if t i θ) < 0, then the interpretation is that i pays some positive amount. Now assume that we are given the SRBT corresponding to the graph under consideration, when the cost profile is θ. A. Payment Scheme Based on SRBT In the SRBT, all the internal nodes forward the broadcast packet. We call such packet forwarding nodes as routers, and represent the set of routers by R. In the SCF f.) corresponding to the ICB problem, we define the allocation rule in the following way: θ Θ, i N, k i θ) 1, if i R 0, if i / R The valuation function, v i kθ), θ i ), of node i is its cost to forward a transit packet. From the allocation rule k.) of the SCF f.), we get, θ Θ, i N, v i kθ), θ i ) θ i, if i R 1) 0, if i / R 2) To compensate the incurred costs of the routers in the ad hoc wireless network, we need to determine payments to the nodes. We use the payment rule of the dagva mechanism or expected externality mechanism [5] to compute the payments to the nodes in our scheme. So, i N, θ Θ, we get [ ] t i θ) E θ i l i v lkθ), θ l ) - 1 j i E θ j [ l j v lkθ), θ l )] where n N. From 1), 2) we get, i N, θ Θ, ) 1 t i θ) E θ j θ l E θ i θ l n 1 j i l R, l j l R, l i [ 3) ] where E θ i l R, l i θ l is interpreted as the expected valuation to node i that would be generated by all the remaining nodes in its absence. We now give an interpretation [ of this ] payment rule. Assume that ξ i θ i ) E θ i l R, l i θ l, θ Θ, i N. Each node i distributes ξ i θ i ) equally among the remaining nodes. Figure 1 shows this interpretation for a graph with 3 nodes. Fig. 1. An Interpretation of the Payment Rule Let us represent a flow into a node with positive sign and a flow going out from a node with negative sign. Now the payment, t 1 θ 1, θ 2, θ 3 ), for node 1 can be written as t 1 θ 1, θ 2, θ 3 ) 1 2 ξ 2θ 2 ) ξ 3θ 3 ) ξ 1 θ 1 ) Similarly, we can write the payments to the remaining two nodes. This completes characterizing the payment rule in the SCF f.) for the ICB problem in ad hoc networks with selfish nodes.

4 V. PROPERTIES OF THE BIC-B PROTOCOL We now prove a few properties of the BIC-B protocol. Lemma 1: For any i R and for any j / R, we have [ ] [ ] E θ i l R, l i θ l E θ j l R, l i θ l. Proof: The types of the nodes are statistically independent according to the current network model. Now it is easy to prove the lemma. Let the nodes in the set R be indexed by the set {1, 2,..., r}, where r R. Now for any j / R and i R, [ ] E θ j l R, l i θ l [ ]... l R, l i θ l fx 1 )...fx j 1 )fx j+1 )...fx n ) dx 1 )...dx j 1 )dx j+1 )...dx n ) [ ]... l R, l i θ l fx 1 )...fx r ) dx 1 )...dx r ) since the types are statistically independent) [ ]... l R, l i θ l fx 1 )...fx i 1 )fx i+1 )...fx r ) dx 1 )...dx i 1 [ )dx i+1 )...dx r ) since l R, l i l] θ does not include θ i ) [ ]... l R, l i θ l fx 1 )...fx i 1 )fx i+1 )...fx n ) dx 1 )...dx i 1 )dx i+1 )...dx n ) [since the types are statistically independent) ] E θ i l R, l i θ l Q.E.D. Lemma 2 For the BIC-B protocol, t i θ) 0, i / R, θ Θ That is, the nodes other than the routers will pay a positive amount of money for receiving the packets). Proof: If R is empty, then the source node can reach all the remaining nodes in the network within a single hop. In that case, all the payments will be 0. Thus, if R is empty, then we get t i θ) 0, i / R, θ Θ. Now we consider the case that, Let us assume that, j / R, [ Γ j E θ j l R, l j θ l E θ j [ l R θ ] l R > 0 4) ] since j / R) Since the types of nodes are statistically independent, the values of Γ j, j / R, are all the same. We represent this value with Γ. That is, Γ Γ j, j / R. 5) [ ] Now, let us assume that Υ i E θ i l R, l i θ l, i R. Then [ ] Υ i E θ i l R, l i θ l [ ] E θ j l R, l i θ l consequence of Lemma 3) < E θ j [ l R θ ] l since i R) Γ from equation 5)) So, we can conclude that Υ i < Γ, i R. 6) From the payment rule 3) of BIC-B protocol, we have i / R, 1 [ t i θ) j i, j N E θ j l R, l j l] [ θ ] - E θ i l R, l i θ l 1 [ j R E θ j l R, l j l] θ 1 [ + j i, j / R E θ j l R, l j l] [ θ ] - E θ i l R, l i θ l 1 j R Υ 1 j + j i, j / R Γ Γ since from equation 5)) 1 ) j R Υ j + N R 1 n 1 1 Γ 1 j R Υ 1 j - j R Γ 1 j R Υ j Γ) < 0, from 6)). This completes the proof the lemma. Q.E.D. Observation 1: In the proof of Lemma 4, we have i / R, 1 t i θ) j R Υ j Γ) where the right hand side expression is independent from i. Hence, using the BIC-B protocol, t i.), i / R are all the same. Call this value tθ). The immediate implication is that the payment made by each node i / R is the same. Lemma 3 The BIC-B protocol overcomes the two difficulties D1) and D2). Proof: We first explain how the BIC-B protocol overcomes the difficulty D1). Assume that we are using a DSIC broadcast protocol based on the VCG payment rule. Using this scheme to determine the payment to a node, we need to compute the marginal contribution of the node by removing that node and all the edges incident on it from the corresponding graph G. After the removal of the node and the corresponding edges, we need the remaining graph G to be connected. For this reason, we require the original graph G of the underlying ad hoc wireless network to be bi-connected in order to use a DSIC broadcast protocol. On the other hand, if we use the BIC-B protocol, the payment rule is such that it does not require any such condition on the original graph of the underlying network. The following example illustrates this idea. Let us consider the graph in Figure 4, which is not biconnected. We consider Θ 1, Θ 2, Θ 3, and Θ 4 are the type sets of the nodes 1, 2, 3, and 4 respectively in the graph.

5 Now let us assume that the probability distributions, from which the players types are drawn, are uniform and let θ θ 1, θ 2, θ 3, θ 4 ) be the announced cost profile. Based on Fig. 2. Illustrative Example the type announced profile θ, we are given a SRBT. In this particular example, SRBT is same as the graph in Figure 4 since SRBT has to connect all the nodes in the network and there is only one way to connect all the nodes. Note that here N {1, 2, 3, 4}, R {2, 3}, n N 4. Now using the payment rule 3) of BIC-B protocol, the payment to node 2 is t 2 θ) 1 [ ] [ ] 3) j 2 E θ j l R, l j θ l E θ 2 l R, l 2 θ l Similarly, we can compute the payments to all the remaining nodes in the graph. An important observation is that it is not possible to compute the payments using the VCG payment rule of a DSIC broadcast protocol, since the graph is not biconnected. Now to show that the BIC-B protocol overcomes the difficulty D2), we need to show that [ i t iθ) 0, θ Θ. Let ] us assume that ξ i θ i ) E θ i l R, l i θ l, θ Θ, i N. Then, from 3), we get θ Θ ) 1 t i θ) ξ i θ i ) ξ j θ j ) n 1 j i n ) 1 n t i θ) ξ i θ i ) ξ j θ j ) n 1 i i1 i1 j i n ) 1 n ξ i θ i ) n 1)ξ j θ j ) n 1 i1 i1 n t i θ) 0 i1 That is, using the proposed scheme, the sum of the payments received and the payments made by the nodes in the network is zero. So, the payment scheme in section IV-A based on SRBT is budget balanced and hence the BIC-B protocol can be selfstarted. In this way, BIC-B protocol overcomes the difficulty D2). With this, we can conclude that the BIC-B protocol overcomes the two difficulties D1) and D2). Q.E.D. Lemma 4 The payment rule 3) based on SRBT is Bayesian incentive compatible and it is minimum among all Bayesian incentive compatible payment schemes based on SRBT. Proof: It is easy to see that the payment rule 3) based on SRBT is Bayesian incentive compatible since it is directly derived from the dagva mechanism, which is Bayesian incentive compatible. Now we need to show that the payment rule 3) based on SRBT is minimal among all payment schemes based on SRBT. We prove this by contradiction. Assume that there is another incentive compatible payment scheme ˆt ) that is minimal based on SRBT. That is, ˆt i θ) t i θ), θ Θ, i N Note that t i θ) and ˆt i θ) are the payments to node i using the payment schemes t ) and ˆt ) respectively, for all θ Θ. We build a contradiction by showing that ˆt ) is not incentive compatible for some node i under a cost profile θ θ i t i θ) + δ), where δ is a nonzero smallest possible positive quantity chosen such that t i θ) + δ) > θ i. Note that the notation θ i t i θ) + δ) indicates the cost profile θ obtained by replacing θ i with t i θ) + δ) in the profile θ. Assume that the node i is a leaf in the SRBT under θ. Then it implies that the node i continues to be a leaf in the new SRBT under θ, since SRBT minimizes the cost to forward the packets. Now, using the payment rule 3), we get ˆt i θ ) t i θ) 7) Now the utility of the node i under the cost profile θ by the payment scheme ˆt ) becomes ˆt i θ ) θ i ˆt i θ ) t i θ) + δ) < 0, due to the equation 7). That is, under the cost profile θ, when the node i reports its true cost, it gets a negative utility under the payment scheme ˆt ). Hence ˆt ) is not incentive compatible. So, from the above analysis, we can conclude that the incentive compatible payment scheme t ) based on SRBT is the minimum among all the Bayesian incentive compatible payment schemes based on SRBT. Q.E.D. Lemma 5 The payment scheme in the BIC-B protocol is interim individual rational. This lemma can be proved using the payment rule 3) and the fact that each node is an intended recipient of the broadcast packet. VI. ILLUSTRATIVE EXAMPLES Here we provide illustrative examples to explain the concepts that we presented in the previous sections. In the first example, we consider an ad hoc network for which the underlying graph is not bi-connected and in the second example, we consider an ad hoc network for which the underlying graph is bi-connected. Before presenting the examples, we explain how the payments are computed using the VCG based broadcast protocol [8], [9]. Consider the graph in Figure 8. Assume for computational simplicity that the transit costs of all the nodes are the same and its value is c. Now the SRBT is given in Figure 3 corresponding to the graph considered. Let node 1 be the source of broadcast. The fundamental difference between this scheme and that of BIC-B protocol is in the payment rule only. Their allocation rules are the same. Here, the broadcast packets follow the least cost paths corresponding to the nodes as specified in the SRBT. For example, the least cost path from node 1 to node 8 is represented by LCP θ, 1, 8) 1, 3, 6, 8). Now consider the situation at node 6. When node 6 receives the packet along LCP θ, 1, 6), according to this pricing mechanism, node 6 pays an amount, call it, p 3 1,6 to node 3. Here

6 way, Fig. 3. SRBT of the graph in Figure 8 Θ 1 {3, 4}, Θ 2 {7, 8} Θ 3 {11, 12}, Θ 4 {5, 6} Θ 5 {1, 2}, Θ 6 {13, 14} Θ 7 {17, 18}, Θ 8 {15, 16} Now let us assume that the players belief probability functions are uniform and θ 3, 7, 11, 5, 1, 13, 17, 15) is the announced cost profile. node 3 is the immediate predecessor of node 6 in the SRBT. The value of p 3 1,6 is computed using the VCG pricing rule?? in the following way. p 3 1,6 cost of 3 to forward a packet) + cost of LCP θ, 1, 6) without node 3 in network) - cost of LCP θ, 1, 6) with node 3 in network) c + 3c - 2c 2c. Fig. 5. Now consider the following illustrative examples. in i) Example 1: Let us consider the graph in Figure 4, which is not bi-connected. We recall that the types of nodes are their incurred transit costs. We consider the type sets of nodes in the following way, Θ 1 {10, 11}, Θ 2 {15, 16} Θ 3 {12, 13}, Θ 4 {7, 8} Now let us assume that the players belief probability functions are uniform and θ 10, 15, 13, 8) is the announced cost profile. Fig. 4. Illustrative Example 1 In this example, assume that the given SRBT is also the same the original graph. Now, the allocation rule is kθ) 0, 1, 1, 0). We note that N {1, 2, 3, 4} and R {2, 3}. The valuation functions of nodes are, v 1 kθ)) 0, v 2 kθ)) 15 v 3 kθ)) 13, v 4 kθ)) 0 Now we compute the payments using the payment rule 3) of BIC-B protocol. t 1 θ) 9.3, t 2 θ) 11.3 t 3 θ) 7.3, t 4 θ) 9.3 An important observation is that it is not possible to compute the payments using the payment rule of VCG based Broadcast Protocol, as the graph is not bi-connected. Example 2: Let us consider the bi-connected graph in Figure 5i). We consider the type sets of nodes in the following i) ii) i) Illustrative Example 2, ii) SRBT rooted at node 1 for the graph For this example, the given SRBT is shown in Figure 5ii). Now the allocation rule is, kθ) 0, 1, 1, 1, 1, 1, 0, 0). We note that N {1, 2, 3, 4, 5, 6, 7, 8} and R {2, 3, 4, 5, 6}. The valuation functions of nodes are, v 1 kθ)) 0, v 2 kθ)) 7 v 3 kθ)) 11, v 4 kθ)) 5 v 5 kθ)) 1, v 6 kθ)) 13 v 7 kθ)) 0, v 8 kθ)) 0 Now we compute the payments using the payment rule 3) of BIC-B protocol. t 1 θ) 5.64, t 2 θ) 2.92 t 3 θ) 7.5, t 4 θ) 0.64 t 5 θ) 3.92, t 6 θ) 9.7 t 7 θ) 5.64, t 8 θ) 5.64 Now we compute the payments using the payment rule of VCG based broadcast protocol [8], [9] in the following way, t 1 θ) 0, t 2 θ) 13 t 3 θ) 36, t 4 θ) 24 t 5 θ) 6, t 6 θ) 29 t 7 θ) 0, t 8 θ) 0 We can conclude the following from the above two payment rules. Using the BIC-B protocol, the highest payment to a router is 9.7 and that router is 6. Using the VCG based broadcast protocol, the highest payment to a router is 36 and that router is 3. The average payment to a router using the BIC-B protocol and the VCG based broadcast protocol are 4.15 and 21.6 respectively. So, it is clear that the payments to the routers are less using the BIC-B protocol. We show the same thing using the simulation experiments in the following section.

7 VII. SIMULATION EXPERIMENTS In this section, we show the efficacy of the proposed BIC-B protocol for the ICB problem. In our simulation experiments, we compare the performance of the BIC-B protocol against a VCG based broadcast protocol. A. Simulation Model VCG based broadcast protocol is based on the dominant strategy equilibrium of the underlying game and BIC-B protocol is based on the Bayesian Nash equilibrium of the underlying game. It is clear from [5] that every dominant strategy equilibrium is also a Bayesian Nash equilibrium and the other way is not true. For this reason, we first find the dominant strategy equilibrium of the underlying game formed by the nodes in the network and we compute the payments to the nodes using BIC-B protocol and VCG based broadcast protocol [8], [9]. We work with a randomly generated graph of an ad hoc wireless network with the number of nodes n 5, 10, 15, 20, 25, 30, 35, 40. According to our network model presented in Section IV, the graph is node weighted where these weights are the transit costs of the nodes chosen independently and uniformly from a range [1, 50]. After the nodes announce their types, we first compute the least cost paths to all the nodes from the source node and then construct an SRBT. Using the SRBT, we can decide the set of routers. This fixes the allocation rule. Then we compute payments to the nodes using the payment rule of the appropriate broadcast protocol. In all our simulation experiments, the results for the performance metrics are averages taken over 100 random instances. B. Analysis of Simulation Experiments We consider two performance metrics. The first metric is average payment to routers APR). This specifies the payment on an average to each router for forwarding the transit packets. The graph in Figure 6 shows the comparison of the BIC-B protocol and the VCG based broadcast protocol using APR. In the figure, the lower corresponds to the BIC-B protocol. It is clear from the figure that the BIC-B protocol performs better than the VCG based broadcast protocol. This metric conveys the following significant information. The system wide payment made by nodes to forward a broadcast packet is less using the BIC-B protocol. So, it reduces the demand on the nodes that are making the payments. The second performance metric, we follow, is the worst overpayment ratio WOR). We first define overpayment ratio as the ratio of payment made by a node to its least cost path value from the source node of broadcast. Now, we can define WOR as the maximum over the overpayment ratios of all the nodes in the network. That is, WOR max i t i COSTi,s) where COSTi, s) is the cost of the least cost path to node i from the source node s. We compare the WOR of BIC-B protocol and the VCG mechanism based broadcast protocol Fig. 6. Comparison of the Average Payment to Routers using VCG based Broadcast Protocol and BIC-B Protocol in Figure 7. We note that the lower curve corresponds to the BIC-B protocol in the figure. From the graph, it is easy to see that the worst overpayment ratio in the network is higher using the VCG based broadcast protocol than BIC-B protocol. WOR conveys the following significant information. When a node receives a packet from a router, then clearly the payment made by the receiver node to the router is higher than the value of its least cost path, since it has to give incentives to the router to make it reveal the true transit cost. If we take a ratio of the payment to the value of least cost path, from the Figure 7, this ratio is less than 2 times over all the nodes using the BIC-B protocol and it is higher than 6 times over all the nodes using the VCG based broadcast protocol. This says that nodes end up with very high payments than actually what their value of the corresponding least cost path. Fig. 7. Comparison of the Worst Overpayment Ratio of VCG based Broadcast Protocol and BIC-B Protocol VIII. BIC-B PROTOCOL VERSUS DSIC BROADCAST PROTOCOL In the previous sections, we gave an approach to design a BIC-B protocol for the ICB problem in ad hoc wireless networks, where the payment scheme is designed using a

8 Bayesian incentive compatible mechanism. Here we compare and contrast the DSIC broadcast protocol and the BIC-B protocol in the following way. Underlying Network Structure: DSIC broadcast protocols require the underlying graph of the ad hoc network to be bi-connected. If the number of nodes in the network increases, sustaining such a condition may be difficult. On the other hand, the BIC-B protocol works for any topology of the ad hoc wireless network. Strategic Complexity: In DSIC broadcast protocol, selection of the best strategy by a node does not depend on the other nodes since the equilibrium of the game derived by the DSIC mechanisms is a dominant strategy equilibrium. We note that, in a dominant strategy equilibrium, any node can choose its best strategy irrespective of the behavior of the other nodes. On the other hand, in the BIC-B protocol, selection of the best strategy by a node is dependent on the other nodes, in the sense that a node reveals its true type provided all other agents reveal their types truthfully. For this reason, every node is assumed to share a common prior about the distribution of nodes types. So, we can conclude that the strategic complexity of a node is less in DSIC broadcast protocol than that of the BIC-B protocol. i) Fig. 8. i) Illustrative Example, ii) SRBT rooted at node 1 Payments Made by the Nodes: Using DSIC broadcast protocol, the payments made by the nodes may not be the same in the following sense. If we consider two distinct nodes in the network that are only receiving the packets), their payments may not be the same. For example, let us consider the graph in Figure 8i) and the corresponding SRBT is shown in Figure 8ii). Here node 1 is the source of broadcast. The payments to the nodes 7 and 8 are 2 and 3 respectively, using DSIC broadcast protocol [8]. On the other hand, using the BIC-B protocol, the payments to the nodes 7 and 8 are 15/7 and 15/7 respectively. Note that Observation 1 also conveys this fact. ii) graph is bi-connected. In this scenario, we can use both these broadcast protocols in the network. Now assume that an edge is broken in the network and it is not biconnected any more. In this situation, we cannot use DSIC broadcast protocol. But, still we can use the BIC-B protocol in that network. IX. CONCLUSIONS AND FUTURE WORK In this paper, we addressed the incentive compatible broadcast ICB) problem in ad hoc wireless networks with selfish nodes. We proposed a Bayesian incentive compatible broadcast BIC-B) protocol to solve this problem. Our design is such that it overcomes the major limitations arising from the use of DSIC broadcast protocols. We then proved the optimality of the pricing scheme that is built in the BIC-B protocol. We also proved a few other crucial properties that the BIC-B protocol satisfies. We finally presented a discussion on using the BIC- B protocol and DSIC broadcast protocols for ad hoc wireless networks. In this paper, the payments computation is performed in a centralized way, so we would like to study the design of a distributed algorithm for the payments computation. It is also very interesting to pursue the design of Bayesian incentive compatible protocols for the incentive compatible unicast and multicast problems in ad hoc wireless networks. REFERENCES [1] L. Anderegg and S. Eidenbenz. Ad hoc-vcg: A truthful and costefficient routing protocol for mobile ad hoc networks with selfish agents. In Proceedings of the 9th Annual ACM International Conference on Mobile Computing and Networking MOBICOM), pages , [2] S. Eidenbenz, P. Santi, and G. Resta. 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In Proceedings of the IEEE INFOCOM 2006 Student s Workshop, Barcelona, [9] N.R. Suri, Y. Narahari, and D. Manjunath. An efficient pricing based protocol for broadcasting in wireless ad hoc networks. In Proceedings of the International Workshop on Software for Wireless Communications and Applications SOFTWIM), COMSWARE, New Delhi, [10] W. Wang and X.Y. Li. Low-cost truthful multicast in selfish and rational wireless ad hoc networks. In Proceeding of the 1st IEEE International Conference on Mobile Ad-hoc and Sensor Systems MASS), [11] W. Wang and X.Y. Li. Truthful low-cost unicast in selfish wireless networks. In Proceedings of the 4th International Workshop on Algorithms for Wireless, Mobile, Ad Hoc and Sensor Networks WMAN) of IPDPS, Fault Tolerance: The BIC-B protocol is more fault tolerant than DSIC broadcast protocol in the following sense. Consider we have a network whose underlying