A truthful auction mechanism for channel allocation

(1)

O R I G I N A L A R T I C L E

A truthful auction mechanism for channel allocation

in multi-radio, multi-channel non-cooperative wireless networks

Zuying Wei^•Tianrong Zhang ^•Fan Wu^• Xiaofeng Gao^•Guihai Chen^•Ping Yi

Received: 8 November 2012 / Accepted: 8 April 2013 / Published online: 25 August 2013

Springer-Verlag London 2013

Abstract Due to users’ fast-growing demands, wireless spectrum is becoming a more and more scarce resource.

However, the state of spectrum usage shows that while large chunks of spectrum are left idle at many places, many emerging wireless applications cannot get enough spectra to provide their services. In contrast to existing truthful mechanisms for channel redistribution, which achieve strategy-proofness at the price of lowered system performance, we propose SHIELD, which not only guarantees strategy-proofness in the process of channel redistribution, but also achieves high system performance. Our evaluation results show that SHIELD outperforms the existing mechanisms, in terms of spectrum utilization and user satisfaction ratio. Here, channel utilization represents the average number of radios allocated to channels, and buyer satisfaction ratio shows the percentage of buyers who get at least one channel in the auction.

Keywords Wireless network Channel allocation Mechanism design Game theory

1 Introduction

As the fast development of the communication technolo- gies, the spectrum is becoming a more and more scarce resource. It is expected that global mobile data traffic will increase 26-fold between 2010 and 2015 [6]. To adapt the fast growth of data traffic over wireless links, next gen- eration wireless applications need more spectrum to carry their services. However, traditional spectrum management makes new wireless network applications face the plight of increasingly scarce spectrum resources. Currently, almost every country has a specific department for regu- lating spectrum usage, e.g., Federal Communications Commission (FCC) [12] in the US and Radio Adminis- tration Bureau (RAB) in China [31]. FCC and RAB divide available wireless spectrum into a number of bands and grant the right of using each band within a specified geographical area to a particular business organization or wireless application. Traditional static spectrum allocation has been unable to meet the growing demand for wireless broadband services [13]. On one hand, frequency bands for wireless communications have almost been fully allocated [11]. On the other hand, already allocated spectrum is not fully utilized. For example, measurement results show that in downtown Berkeley, the utilization of spectrum up to 3 GHz is only about 32 %, while for the spectrum of 3–5 GHz, the utilization is less than 1 % [49].

Therefore, to improve the spectrum utilization, we need to find a more efficient mechanism to redistribute the idle spectrum to the wireless applications that need the spectrum resource.

Z. Wei T. Zhang F. Wu (&) X. Gao G. Chen

Shanghai Key Laboratory of Scalable Computing and Systems, Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, China

e-mail: fwu@cs.sjtu.edu.cn Z. Wei

e-mail: zu_ying_hi@sjtu.edu.cn T. Zhang

e-mail: ztr1211@sjtu.edu.cn X. Gao

e-mail: gao-xf@cs.sjtu.edu.cn G. Chen

e-mail: gchen@cs.sjtu.edu.cn P. Yi

School of Information Security Engineering, Shanghai Jiao Tong University, Shanghai, China

e-mail: yiping@sjtu.edu.cn DOI 10.1007/s00779-013-0708-5

(2)

A usual way to implement spectrum redistribution is to use auction, by which the spectrum owner (seller) gets profit through leasing idle spectrum to the wireless applications (buyers) who need the spectrum. In the literature, there are a number of auction mechanisms proposed for dynamic spectrum/channel redistribution, e.g., [43,55–57].

These auction mechanisms target at guaranteeing strategy- proofness of the spectrum auction. Intuitively, an auction mechanism is strategy-proof, if it is the best strategy for each buyer to truthfully report her valuation of the good as the bid, no matter what the others do, and nobody’s individual-rationality is hurt. Two commonly used metrics for evaluating the efficiency of a spectrum auction mechanism are spectrum utilization and buyer satisfaction ratio. Here, spectrum utilization captures the average number of buyers (or radios if the buyers have multiple radios) correspond to each channel, and buyer satisfaction ratio represents the percentage of buyers who get at least one channel in the auction. Although most of existing channel auction mechanisms achieve strategy-proofness, they provide low guarantee for the allocation efficiency in terms of spectrum utilization and buyer satisfaction ratio.

In this paper, we propose SHIELD, which is a strategy- proof and highly efficient channel auction mechanism for multi-radio wireless networks. SHIELD not only guarantees strategy-proofness, but also achieves high performance compared with existing mechanisms. SHIELD divides the buyers into non-conflicting groups, in which every pair of buyers is well separated and can do the transmission on the same channel simultaneously, and gives larger groups higher precedence to be allocated a channel. We also do some improvements for SHIELD and name the mechanism as Fair-SHIELD. Fair-SHIELD achieves fairness in repeated auctions. In this paper, we make the following key contributions.

• First, we model the problem of channel redistribution as a sealed-bid auction and propose a simple but efficient channel auction mechanism, namely SHIELD.

• Second, we prove that SHIELD is a strategy-proof channel auction mechanism.

• Third, we do some improvements for SHIELD and design Fair-SHIELD. Fair-SHIELD can achieve fairness in repeated auctions.

• Forth, we do extensive simulations to compare the performance of SHIELD with existing representative channel auction mechanisms, such as SMALL and VERITAS. Evaluation results verify that SHIELD guarantees strategy-proofness and show that SHIELD outperforms existing representative channel auction mechanisms in terms of spectrum utilization and user satisfaction ratio. We also do simulations to show the performance of Fair-SHIELD, and the evaluation

results show that Fair-SHIELD achieves fairness in repeated auctions.

We organize the rest of this paper as follows. In Sect.2, we present the game model for the problem of channel redistribution and review some important solution concepts from game theory. In Sect. 3, we give the detailed description of SHIELD. In Sect.4, we show the design of Fair-SHIELD, which can achieve fairness in repeated auctions. In Sect.5, we illustrate simulation results of our auction mechanism. In Sect.6, we discuss related works. In Sect. 7, we conclude the paper and point out potential directions for future work.

2 Preliminaries

In this section, we show our game-theoretic model and review some closely related solution concepts from game theory.

2.1 Game-theoretic model

We model the problem of channel redistribution as a sealed-bid auction, in which there are a spectrum seller and a number of buyers, as shown in Fig. 1. The seller holds m idle wireless channels, denoted by C¼ fc1; c2; . . .; cmg:

The seller wants to lease her idle channels to buyers to get some profit. A channel can be leased to multiple buyers, who are not conflicting with each other according to an adequate signal-to-interference- and-noise ratio (SINR).

Buyers, such as WiFi access points, desperately need the channels to serve their customers. Suppose there are n buyers, denoted by N¼ f1; 2; . . .; ng. Each buyer has a private valuation of a channel, denoted by v¼ fv1; v₂; . . .; v_ng. Each buyer may equip with a single radio or multiple radios, so a buyer may bid for one or multiple channels. We assume that each channel is of the same value to each buyer. Therefore, we require that each buyer bids equally for each channel she requests. We also assume that the buyers do not cheat about the number of radios she has. Each buyer i2 N has a per-channel valuation v_i. The per-channel valuation can be the revenue gained by the buyer for serving her customers. The channel valuation v_i is private information to the buyer i. In the auction, each buyer submits her sealed per-channel bid b_i together with the number of radios r_ishe has to the seller/

auctioneer. The seller/auctioneer will decide auction result based on the buyers’ bids. Then, we can represent the buyers’ bid vector as b¼ ðb1; b2; . . .; bnÞ; and demand vector as r¼ ðr1; r₂; . . .; r_nÞ:

The seller/auctioneer uses a deterministic channel allocation algorithm to determine the channel allocation y¼

(3)

ðy1; y₂. . .; y_nÞ based on the bids. Here, yimeans that buyer i gets y_i channels in the auction. Then, the kth (k B y_i) radio of buyer i can work on the kth channel allocated to i. Each buyer i should pay for the channels she won in the auction with price p_i¼Py_i

k¼1p^k_i: Here, pik represents the charge to buyer i’s kth channel. The utility u_iof buyer i is defined as the difference between her valuation of allocated channels and the charge of using the channels:

ui¼X^yⁱ

k¼1

u^k_i ¼X^yⁱ

k¼1

ðvi p^k_iÞ ¼ vi yi pi:

Here, uik

represents the utility of buyer i gets on the kth channel. We assume that the buyers are rational and always want to maximize their own utilities. In contrast to an individual buyer’s objective, our auction mechanism aims to achieve high channel allocation efficiency, in terms of spectrum utilization and buyer satisfaction ratio.

Here, we use an example to show that traditional VCG auction model [7, 18, 36] cannot guarantee strategy- proofness for spectrum auction. VCG auction model sorts the bidders in non-increasing order and then allocates the channels to the bidders one by one using lowest indexed channel in each bidder’s available channel set. The charge to bidder i is the bid of the bidder who would get the channel if bidder i is absent. We model the interference among bidders using a conflict graph, which means that two bidders cannot use the same channel simultaneously if there is an edge between each other. Figure2 shows a simple example to illustrate that the VCG auction violates strategy-proofness. We assume that there are two channels waiting to be leased out. In Fig.2a, all the bidders bid truthfully and in Fig.2b, bidder E bids untruthfully.

Table1 shows the utilities of all the bidders when E bids truthfully and untruthfully. We can see that when E bids b_E= v_E= 4, he loses in the auction and get utility of 0.

When he bids b_E= 6 = v_E, he wins in the auction and gets the utility of 2. The bidder E can increase his utility by bidding untruthfully. We can see that the traditional VCG auction model cannot guarantee strategy-proofness.

In Sect.3.1, we will present our strategy-proof channel auction mechanism—SHIELD.

2.2 Solution concepts

We review two important solution concepts from game theory in this section.

Definition 1 (Dominant strategy [15,29]) A strategy s_iis player i’s dominant strategy, if for any s_i⁰= s_i and any strategy profile of the other players s_-i, her utility satisfies:

uðsi; s_iÞ uðs⁰_i; s_iÞ:

In our game model, each buyer is a player, and s_iis a buyer i’s bid.

A dominant strategy of a player is one that maximizes her utility regardless of what strategies the other players choose. Before defining strategy-proofness, we review the definition of incentive-compatibility and individual- Fig. 1 A simple and ordinal channel auction model, including a

seller (auctioneer) and multiple buyers

Fig. 2 A simple example which shows that traditional VCG auction model violates strategy-proofness. a When E bids truthfully, user A, B, C, D get the channel. b When E over bids, user A, B, D, E get the channel

Table 1 Utilities of all the bidders when E bids truthfully (Fig.2a) and untruthfully (Fig.2b)

Figure2a Figure2b

v_A= 10, b_A= 10, u_A= 6 v_A= 10, b_A= 10, u_A= 8 v_B= 7, b_B= 7, u_B= 3 v_B= 7, b_B= 7, u_B= 7 v_C= 2, b_C= 2, u_C= 2 v_C= 2, b_C= 2, u_C= 0 v_D= 5, b_D= 5, u_D= 1 v_D= 5, b_D= 5, u_D= 3 v_E= 4, b_E= 4, uE= 0 v_E= 4, bE= 6, uE= 2 Bold is for differences between the two cases

(4)

rationality. An auction mechanism is incentive-compatible if it is one’s dominant strategy for bidding real valuation.

Individual-rationality means that the buyers can always achieves at least as much utility from participating in the auction as staying outside.

Definition 2 (Strategy-proof mechanism [27, 34]) A mechanism is strategy-proof if it satisfies both incentive- compatibility and individual-rationality.

3 Design and analysis of SHIELD

In this section, we present detailed description of our channel allocation auction mechanism, namely SHIELD.

SHIELD follows the design methodology of SMALL [43].

However, with a novel winner selection method, SHIELD greatly improves spectrum utilization and buyer satisfaction ratio.

3.1 Design of SHIELD

We now present the design of SHIELD. SHIELD works in three steps: buyer grouping, winner selection and charge determination.

When a buyer i is equipped with r_i radio(s), we use r_i elementary buyer(s) to represent the buyer i (e.g., Fig.3).

We use N⁰ to denote the set of elementary buyers. There- fore, each elementary buyer requests only one channel.

Actually, a buyer who is equipped with one radio is an elementary buyer herself. Similar to [43, 56], SHIELD groups the buyers in a bid-independent way. SHIELD models the interference among buyers using a conflict graph. Each node in the graph is an elementary buyer. For each radio equipped by a buyer, we use an elementary buyer to represent it. Each edge in the graph represents that the two elementary buyers who interfere with each other.

Since the radios belonging to the same buyer have interference between each other, we connect the nodes/elementary buyers of the same buyer with each other to indicate the confliction. There are also conflicts across nodes/elementary buyers belonging to different buyers; we connect them with each other to represent the conflicts.

Then SHIELD divides all the elementary buyers into non- conflicting groups based on the conflict graph. We can use existing graph coloring algorithms (e.g., [42]) to figure out the grouping.

Figure3 shows a toy example. In Fig.3, buyer A is equipped with two radios. Node A¹and A²represent the two elementary buyers of buyer A. Similarly, B¹and B²represent the two elementary buyers of buyer B. C¹ and C² represent the two elementary buyers of buyer C. There are seven elementary buyers. There are many possible grouping

results, for example g1= {A¹, C²}, g2= {A², B¹, D} and g₃= {B², C¹}.

Without loss of generality, we assume that the elementary buyers have been divided into x non-conflicting groups by a given graph coloring algorithm:

G:fg1; g2; . . .; gxg:

Next, we discuss the very important step—winner selection. SHIELD sorts the buyer groups according to group size in non-increasing order as follows:

G⁰:jg⁰₁j jg⁰₂j jg⁰_xj:

In case of a tie, each tied group has an equal probability of being ordered prior to the others. SHIELD chooses the first m (or x, if x \ m) groups as winning groups. Furthermore, SHIELD sets the elementary buyers except the one with the smallest bid in each winning group as winners. In case of a tie, each tied elementary buyer has an equal probability of being selected as a winner. Algorithm 1 shows the pseudocode for the winner selection process.

We note that to achieve strategy-proofness, SHIELD sacrifices a buyer in each winning group. No matter how Fig. 3 A simple conflict graph, in which the node represents an elementary buyer and the link between two elementary buyers indicates the confliction

(5)

large the set of buyers is, the number of sacrificed buyers is bounded by m, which is the number of channels for leasing.

Finally, we determine the charges to the winners. The winners in each group are charged equally, and the charge is the smallest bid in that group. For each winner l2 g⁰_j; the payment for her is minfbqjq 2 g⁰_jg: Then the charge pito a buyer i2 N is the sum of charges to her winning elementary buyers. We note that since each elementary buyer will not be charged more than her bid. The seller’s income is the payments for all the buyers who get the channels:

Income¼Xⁿ

i¼1

pi:

To illustrative clearly, we use Fig.3 as an example. We can see that r_A= r_B= r_C= 2 and r_D= 1. We assume that b_A= 2, b_B= 5, b_C= 9, b_D= 1 and the number of channels m = 2. What is more, we assume the buyers bid truthfully to illustrate the procedure of channel allocation.

We use the three steps to allocate the channels to the four buyers.

– The same to above illustration, g₁= {A¹, C²}, g₂= {A², B¹, D} and g₃= {B²,C¹}.

– Here, G⁰: |g₂| C |g₃| C |g₁|. There are 2 channels now, and two groups (here, g₂ and g₃) are the winning groups. We assume group g₂gets channel 1 and g₃gets channel 2. The elementary buyer with the lowest bid in each group will lose the auction. Here, bD\bA²\bB¹ð1\2\5Þ in group g₂ and b_B2\b_C1ð5\9Þ; so the elementary buyer D and B² will lose in the auction and the elementary buyers A², B¹and C¹are the winners.

– The payment for each winner is the lowest bid in his buyer group. Here, the payment for elementary buyers A² and B¹ is elementary buyer D’s bid which is 1.

Similarly, the payment for elementary buyer C¹ is elementary buyer B²’s bid which is 5. We can see that p_A¹ = 0 and p_A² = 1, so p_A= 1 ? 0 = 1. Similarly, p_B= 1, p_C= 5 and p_D= 0.

In this example, buyer A, B and C gets one channel, respectively. We can easily get the utility of each elementary buyer: u_A= 2 - 1 = 1, u_B= 5 - 1, u_C= 9 - 5 = 4 and u_D= 0. The seller’s income is the payments for all the buyers who get the channels:

Income¼Xⁿ

i¼1

pi¼ 1 þ 5 þ 0 ¼ 6:

3.2 Analysis

In this section, we prove that SHIELD is strategy-proof, which means that reporting one’s truthful per-channel valuation as a bid is the best strategy of each buyer.

Lemma 1 SHIELD satisfies incentive-compatibility.

Proof SHIELD use a bid-independent grouping method to group the elementary buyers. The bid of buyer i will not affect the winning group selection method. Next, we will show that no matter how a buyer bids, she cannot increase her utility by bidding untruthfully. In other words, a buyer cannot increase her utility by misreporting.

If buyer i bids truthfully (i.e., b_i= v_i) and gets y_i channel. Then her utility is

ui¼X^yⁱ

k¼1

u^k_i ¼X^yⁱ

k¼1

ðvi p^k_iÞ:

Let t be the kth one of the elementary buyers generated from buyer i. Suppose t2 g⁰_j: If g⁰jis not a winning group, then t cannot be a winner no matter how buyer i bids. We then analyze the case, in which g⁰_jis a winning group. We prove that buyer cannot increase her utility get from elementary buyer t by bidding untruthfully. We distinguish two cases:

– If bi¼ vi¼ minfbsjs 2 gjg when bidding truthfully. In this case, the elementary buyer t would lose in auction or win with a charge equal to her valuation. So u_i^k= 0.

Let us see the utility get from the elementary buyer t if buyer i bids untruthfully. We further distinguish two cases:

If the bid b⁰_i\ bi, t will also lose in the auction and result in the utility u_i^k= 0.

If the bid b⁰_i[ bi, the utility on t is 0 when t still loses in the auction. If t wins in the auction, her utility will be u⁰_i^k= v_i- p⁰_i^k. That means the new lowest bid of the group minfbsjs 2 gjnftggbi¼ vi:

bi¼ vi: u^0k_i¼ vi p^0ki

¼ vi minfbsjs 2 gjnftgg

vi bi

¼ 0:

The utility on t will be non-positive. We can get that, if b_i¼ vi¼ minfbsjs 2 gjg; no matter how i bids, she cannot improve her utility got on t and her utility on t will be no more than 0.

– If bi¼ vi[ minfbsjs 2 gjnftgg; buyer t would win in the auction and get the utility u^k_i ¼ bi p^k_i ¼ bi minfbsjs 2 gjnftgg if i reports her bid truthfully. If i bids untruthfully, the utility of the buyer will be also u_i^kor will become 0.

– Assume i bids b⁰_i[ minfbsjs 2 gjg; t will win in the auction. Her utility will not change and is still u^0ki¼ vi minfbsjs 2 gjnftgg ¼ u^k_i because the lowest bid of

(6)

the group has not changed. But if she loses the auction her utility will be 0.

We can see that for an elementary buyer t, she cannot improve her utility no matter how i bids , which can be indicated as: u_i^kC u⁰_i^k. As we supposed above, the elementary buyer t is the kth one generated from buyer i. For buyer i, her utility is

u_i¼X^yⁱ

k¼1

u^k_iX^yⁱ

k¼1

u^0k_i¼ u⁰_i:

The above analysis shows that bidding truthfully is the buyers’ dominant strategy when participating in the auction. From the definition of incentive-compatibility, we can

draw the conclusion of Lemma 1. h

We next show that SHIELD satisfies individual- rationality.

Lemma 2 SHIELD satisfies individual-rationality.

Proof For an elementary buyer t, she can get 0 or higher utility through participating in the auction truthfully. So the utility of the buyer who is equipped with more than one radios can get 0 or higher utility too. That is to say:

truthfully participating in the auction is not worse than staying outside, which can be indicated as follows:

u^k_i 0ð1 k yiÞ;

ui¼X^yⁱ

k¼1

u^k_i 0:

Then the allocation mechanism satisfies individual-

rationality. h

Since SHIELD satisfies both incentive-compatibility and individual-rationality, we can draw the following conclusion from the definition of strategy-proofness.

Theorem 1 SHIELD is a strategy-proof channel auction mechanism.

4 Fairness in repeated auctions using fair-SHIELD We have shown the design of SHIELD used in single- round auction in Sect.3.1. We now consider the scenario, in which the spectrum can be allocated to buyers repeatedly in multiple-round auctions.

SHIELD sacrifices the buyer with the lowest bid in each winning group to achieve strategy-proofness in a single- round auction. However, in repeated auctions, the buyer with the lowest bid would never get the channel using the same method as used in the single-round auction. Hence, fairness [17] is a very important issue in repeated auctions.

Here, fairness means that each buyer can get the channel at

least once within certain amount of time in repeated auctions. Unfortunately, SHIELD cannot guarantee fairness in repeated auctions, and it results in starvation. Therefore, SHIELD cannot be directly used in repeated auctions. To achieve fairness, we make some improvements to SHIELD, and we use Fair-SHIELD, to indicate the mechanism used in repeated auctions.

We now show the design of Fair-SHIELD. In repeated auctions, the total utility Uiof each buyer i [ N is the sum of the utilities in all the rounds. We assume that we do the auction for a times, and the utility of a buyer i [ N can be calculated as follows,

Ui¼X^a

k¼1

u^k_i:

We use the similar method of grouping in Fair-SHIELD as in SHIELD here. First, we sort the groups in non- decreasing order according to the group size.

G⁰:jg⁰₁j jg⁰₂j jg⁰_xj:

In repeated auctions, we allocate the channels to the groups according to the order in G⁰ iteratively. In each round auction, m groups would be selected as winning groups. In the first round, the winning groups are g⁰₁; g⁰₂; . . .g⁰_m; In the second round, the winning groups are g⁰_mþ1; g⁰_mþ2; . . .; g⁰_2m: In the kth round, the set of winning groups is

fg⁰ððk1Þmþj1Þ mod xþ1jj 2 f1; 2; . . .; mgg:

Using this method, all the groups can be selected as winning groups in k round auctions.

We use the same method as in SHIELD to select winners. In addition, to achieve fairness in repeated auctions, we introduce the concept of virtual bid, which is the product of the bid and a random number, to give any buyer in a winning group some possibility to get the channel. The random number is distributed in (0, 1], and we use uðiÞ to indicate the random number of buyer i’s bid. Here, uðiÞ is regenerated in each round of auction. Then, the virtual bid b⁰_iof a player can be defined as follows:

b⁰_i¼ uðiÞ bi

Using the virtual bid, when the auction repeats for multiple-round auctions, the buyers can get the channel at least once in large enough numbers of auction rounds. In other words, Fair-SHIELD can achieve fairness in repeated auctions.

Finally, we determine the charge of the winners in each winning group. We charge the winners with different price in each winning group. Here, we set the charge to the winners in each winning group to be quotient of the lowest virtual bid divided by the random number of the winner’s virtual bid. For instance, assume the virtual bid b⁰_min¼

(7)

uðsÞ bsis the lowest virtual bid and winner i’s virtual bid is b⁰_i¼ uðiÞ bi in winning group g, then the charge to the winner i in winning group g is as follows:

p_i¼b⁰_min

uðiÞ¼uðsÞ bs

uðiÞ

¼uðsÞ uðiÞ bs:

Intuitively, since the charges to the winners are independent of the winners’ bids in each winning group, Fair-SHIELD can also achieve strategy-proofness. The proof is similar to that in Sect.3.2. So we do not show the details of the proof again.

Theorem 2 Fair-SHIELD is a strategy-proof channel auction mechanism in repeated auctions.

5 Evaluation results

In this section, we show the evaluation results. Since the channel allocation mechanism SMALL [43] and VERITAS [55] are similar to our mechanism SHIELD (both can work when each buyer is equipped with a single or three radios), we compare the performance of SHIELD with SMALL and VERITAS.

5.1 Metrics

We use the following three metrics to evaluate the performance of the channel auction mechanisms.

• Utility Utility is defined as the difference between a buyer’s channel valuation and charge. As we mentioned in Sect.3.2, a buyer may bid truthfully or untruthfully.

The utility reflect the impacts of buyers’ behaviors including bidding truthfully and untruthfully. We use this metric in our evaluations to verify that no buyer can increase her utility by misreporting.

• Spectrum utilization Average number of radios allocated to each channel.

• Buyer satisfaction ratio Buyer satisfaction ratio is the percentage of the buyers who get at least one channel in the auction. Buyer satisfaction ratio and spectrum utilization reflect the performance of a channel auction mechanism.

5.2 Evaluation setup

We now show the settings of the evaluation: We use a greedy graph coloring algorithm [41] to implement SHIELD. We assume that there are 6, 12, or 24 idle channels available and evaluate the cases in which every

buyer has a single radio or 3 radios. We vary the number of buyers from 20 to 400. The sized terrain area can be 1,000 9 1,000, 1,500 9 1,500 or 2,000 9 2,000 m. The interference range of each node is set to 425 m. We assume that buyers’ valuation per channel is randomly distributed in (0, 1].

5.3 Evaluation results

In our first set of evaluations, we show that SHIELD ensures that no buyer can increase her utility by misreporting the per-channel valuation. We set the number of buyers to 200. We randomly choose a buyer to show the results of honest reporting and misreporting. Since the utilities of the buyers when bidding truthfully and untruthfully is the same in most of the cases, to illustrate clearly, we just show the cases in which these two utilities are different. The simulation is repeated more than 1,000 times. Figure4 shows the results of buyer 55. Evaluation results of other nodes are similar to that of buyer 55 when bidding truthfully and untruthfully. When the two utilities are different, buyer 55 can always get a much higher utility when bidding truthfully. In Fig.4, we can also get that when buyer 55 bids truthfully, the utility is always non- negative, while bidding untruthfully can lead to negative utility. Therefore, a buyer cannot increase his utility by misreporting.

In our second set of evaluations, we compare the performance of SHIELD with SMALL and VERITAS, in terms of spectrum utilization and buyer satisfaction ratio.

Our evaluation results show that SHIELD performs better than SMALL and VERITAS. When the buyers are extre- mely sparse, SHIELD performs a little bit worse than VERITAS. The reason for this is that SHIELD sacrifices a buyer in each winning group. As the number of buyers increases, SHIELD performs better than VERITAS.

SHIELD always outperforms SMALL regardless of the number of buyers and the size of terrain area.

Figure 5 shows spectrum utilizations of SHIELD, SMALL and VERITAS under the condition that there are 6, 12 and 24 idle channels available. In this evaluation, we set the terrain area to 2,000 9 2,000 m. We can see from Fig.5 that when the number of buyers is small, SHIELD achieves a little bit lower spectrum utilization than VE- RITAS. This is because VERITAS does not need to sac- rifice any buyer. When the number of buyers is more than a critical value (e.g., 120 in Fig. 5a), SHIELD outperforms VERITAS. This is because non-grouping-based algorithm used in VERITAS fails to fully consider the whole network topology. Figure 5b and c show that, when each is equipped with three radios and the number of buyers is more than 80, SHIELD performs much better than SMALL and

(8)

VERITAS. SHIELD outperforms better than SMALL in most of the cases.

Figure6 shows the spectrum utilizations of SHIELD, SMALL and VERITAS in different terrain areas with the same buyer density when each buyer is equipped with a single radio or three radios. We assume there are 80, 180 and 320 buyers that are randomly distributed in the terrain areas when the size of terrain area is 1,000 9 1,000, 1,500 9 1,500, and 2,000 9 2,000 m, respectively.

We can see from Fig.6that SHIELD always performs not worse than SMALL and VERITAS. Especially when the terrain area is relatively large (1,500 9 1,500, and 2,000 9 2,000 m), SHIELD performs much better than SMALL and VERITAS.

Figure7shows the buyer satisfaction ratios of SHIELD, SMALL and VERITAS. In Fig.7a, b and c, there are 6, 12 and 24 idle channels available, respectively. In each figure, we show the simulation results when each buyer requests only one radio or three radios in 2,000 9 2,000 m terrain area. We can see from Fig.7 that when the number of buyers is very small, SHIELD performs a little worse than VERITAS. The reason for this is that SHIELD sacrifices a buyer in each winning group. When the number of buyers is a little larger, SHIELD and SMALL outperforms VE- RITAS. When each buyer is equipped with a single radios,

Fig.7b and c shows that SHIELD and SMALL get nearly the same buyer satisfaction ratios when the number of buyers is less than 125 and 310, respectively. This is -0.2

0 0.2 0.4 0.6 0.8 1

0 10 20 30 40 50

Utility

Run

Honest reporting Misreporting

(a)

-0.5 0 0.5 1 1.5 2 2.5

0 10 20 30 40 50

Utility

Run

Honest reporting Misreporting

(b)

Fig. 4 Utilities of buyer 55 if she bids truthfully and untruthfully when each buyer is equipped with a single radio or three radios in a terrain area of 2,000 9 2,000 m. a Each buyer is equipped with a single radio. b Each buyer is equipped with three radios

0 2 4 6 8 10 12 14 16 18

0 50 100 150 200 250 300 350 400

Spectrum Utilization

Buyers SHIELD-1R

SMALL-1R VERITAS-1R

SHIELD-3R SMALL-3R VERITAS-3R

(a)

0 2 4 6 8 10 12 14 16 18

0 50 100 150 200 250 300 350 400

Buyers SHIELD-1R

SMALL-1R VERITAS-1R

(b)

0 2 4 6 8 10 12 14 16 18

0 50 100 150 200 250 300 350 400

Buyers SHIELD-1R

SMALL-1R VERITAS-1R

(c)

Fig. 5 Spectrum utilizations of SHIELD, SMALL and VERITAS when there are 6, 12 and 24 channels provided. Each buyer is equipped with one radio or three radios, in a terrain area of 2,000 9 2,000 m. a There are 6 channels available. b There are 12 channels available. c There are 24 channels available

(9)

because SHIELD and SMALL use the same method of winner selection in one winning group. When each buyer is equipped with three radios, Fig.7a and b show that SHIELD achieves higher buyer satisfaction ratios than SMALL and VERITAS in most of the cases. Figure7a and b also show that when the number of buyers is very large, SHIELD and SMALL achieve more closer buyer satisfaction ratio and they both outperform VERITAS.

Figure8shows the buyer satisfaction ratios of SHIELD, SMALL and VERITAS for auctioning 12 channels in the same density of buyers in different terrain areas, when a buyer is equipped with one or three radios. We assume there are 80, 180 and 320 when the terrain area is 1,000 9 1,000, 1,500 9 1,500 and 2,000 9 2,000 m, respectively. The buyers are randomly distributed in the terrain area. We can see from Fig.8that SHIELD always performs not worse than SMALL and VERITAS.

Especially when the terrain area is relatively large (1,500 9 1,500, and 2,000 9 2,000 m), SHIELD performs much better than SMALL and VERITAS.

0 0.2 0.4 0.6 0.8 1 1.2

0 50 100 150 200 250 300 350 400

Buyer Satisfaction Ratio

Buyers SHIELD-1R

SMALL-1R VERITAS-1R

(a)

0 0.2 0.4 0.6 0.8 1 1.2

0 50 100 150 200 250 300 350 400

Buyers SHIELD-1R

SMALL-1R VERITAS-1R

(b)

0 0.2 0.4 0.6 0.8 1 1.2

0 50 100 150 200 250 300 350 400

Buyers SHIELD-1R

SMALL-1R VERITAS-1R

(c)

Fig. 7 Buyer satisfaction ratios of SHIELD, SMALL and VERITAS when there are 6, 12 and 24 channels provided. Each buyer is equipped with one radio or three radios, in a terrain area of 2,000 9 2,000 m. a There are 6 channels available. b There are 12 channels available. c There are 24 channels available

0 2 4 6 8 10 12 14 16 18

2000×2000 1500×1500

1000×1000

Terrain Area (meter×meter) SHIELD-1R

SMALL-1R

VERITAS-1R

(a)

0 2 4 6 8 10 12 14 16 18

2000×2000 1500×1500

1000×1000

SMALL-3R

VERITAS-3R

(b)

Fig. 6 Spectrum utilizations of SHIELD, SMALL and VERITAS for auctioning 12 channels with the same density of buyers in terrain areas with different sizes, when a buyer is equipped with one radio or three radios. a Each buyer is equipped with a single radio. b Each buyer is equipped with three radios

(10)

In our third set of evaluations, we do some experiments to show that Fair-SHIELD achieves fairness in repeated auctions. Here, we use the buyer satisfaction ratio in repeated auctions to evaluate the performance of Fair-SHIELD.

Buyer satisfaction ratio of repeated auctions is the percentage of the all the buyers who get at least one channel in repeated auctions. We assume that there are 200 buyers, and 12 idle channels waiting to be leased out. We evaluate the cases in which each buyer is equipped with one or three radios. The size of the terrain area is 2,000 9 2,000 m.

Figure9shows the performance of Fair-SHIELD, when each buyer is equipped with one or three radios. We can see from Fig.9 that Fair-SHIELD achieves very high buyer satisfaction radio in repeated auctions. When each buyer is equipped with a single radio, Fair-SHIELD achieves more than 50 % buyer satisfaction ratio if the auction repeats only once. What is more, Fair-SHIELD achieves more than 90 % buyer satisfaction ratio if the auction repeats more than 2 times. When the auction repeats more than 24 times, Fair- SHIELD achieves nearly 100 % buyer satisfaction ratio.

The reason for this is that we use virtual bid to determine the winners in each winning group, and this method lets all the buyers have the possibility to get channels when the auction repeats for many enough times. When each buyer is equipped with three radios, the evaluation results are similar. In general, Fig. 9 shows that Fair-SHIELD achieves very high satisfaction ratio in repeated auctions.

6 Related works

In this section, we review the related works on channel allocation with cooperative participants and non-cooperative participants.

6.1 Existing works with cooperation participants

Generally, channel assignment schemes in cellular networks can be categorized as fixed channel assignment (FCA), dynamic channel assignment (DCA) and hybrid channel assignment (HCA), which is a combination of FCA and DCA. Many works have been done for wireless LANs(WLANs). For example, Mishra et al. [28] explored the use of channel hopping to improve the fairness and performance of overlapping 802.11 network deployments.

Many works about the channel allocation problems have also been done in wireless mesh networks. For instance, Kodialam and Nandagopal [23,24] considered the problem of optimal channel assignment, scheduling and routing using a linear programming technique. Rad et al. [30]

formulated joint channel allocation, interface assignment and MAC problem. On the optimal problem of the network throughput, Alicherry et al. [2], Raniwala et al. [32] and Kodialam et al. [24] took the routing into account with channel allocation.

0 0.2 0.4 0.6 0.8 1

0 5 10 15 20 25 30

Turn

SHIELD-1R SHIELD-3R

Fig. 9 Buyer satisfaction ratios of Fair-SHIELD, when each buyer is equipped with one or three radios

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2000×2000 1500×1500

1000×1000

SMALL-1R

VERITAS-1R

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3

2000×2000 1500×1500

1000×1000

SMALL-1R

VERITAS-1R

(b)

Fig. 8 Buyer satisfaction ratios of SHIELD, SMALL and VERITAS for auctioning 12 channels with the same buyers density in terrain areas of different sizes, when a buyer is equipped with one radio or three radios. a A buyer is equipped with a single radio. b A buyer is equipped with three radios

(11)

The spectrum allocation problem and improvement on this problem have been researched in many other wireless networks too. Kyasanur and Vidya [25] proposed a flow- based routing and channel assignment approach for a single interface in ad hoc networks. Vedantham et al. [35]

investigated the granularity of channel assignment deci- sions that gives the best trade-off in terms of performance and complexity in ad hoc networks. Ding et al. [9]studied distributed routing, relay selection and spectrum allocation in cognitive and cooperative ad hoc networks. Authors in [22,51] study the spectrum management problem in use of cognitive radio.

6.2 Existing works with non-cooperative participants The related works showed in above section requests the users to cooperate with each other, while an another cate- gory of works considers the case with non-cooperative participants. Related works in this section can be divided into two sub-categories including static auction and online auction.

6.2.1 Static auction

Felegyhazi et al. [14] studied Nash equilibria in static multi-radio multi-channel allocation game. After that Wu et al. [45] proposed a strongly dominant strategy equilib- rium to improve the performance of channel allocation, and the scheme achieves the optimal system throughput in single-hop wireless networks. Han et al. [21] presented a distributed algorithms for simultaneous channel allocation of individual links and packet-scheduling, in software- defined radio (SDR) wireless networks.

In recent years, Zhou et al. proposed TRUST [56] and VERITAS [55], which are based on spectrum auction and achieve truthfulness. TRUST is a general framework for truthful double spectrum auction, which takes both buyers and sellers into account and achieves good performance.

VERITAS focuses on the buyers and the circumstances, under which buyers request multiple channels. The most closely related work is SMALL [43], which also focuses on the buyers except that it lets the seller to set reserve price to protect her interest.

Gao et al. [16] studied the problem of competitive channel allocation among devices which use multiple radios in the multi-hop system. Xu et al. [46] designed an efficient channel allocation algorithm in different cases, such that the social efficiency can be approximately max- imized. Xu et al. also designed a polynomial-time approximation scheme to maximize the social efficiency.

Wang et al. [40] proposed a competitive spectrum sharing scheme based on the auction theory in cognitive radio networks. Mahmoud and Gupta [1] designed a polynomial-

time truthful spectrum auction that offers a performance guarantee on the expecter revenue for Bayesian setting.

Yang et al. [48] designed a truthful auction mechanism for the cooperative communication, named TASC. TASC is individually rational and budget-balanced, where wireless node can trade relay services. Wu et al. [44] studies the problem of adaptive-width channel allocation from a game- theoretic perspective and achieve higher system-wide throughput than that when system is in NE.

6.2.2 Online auction

Many works have been done for online auctions. Hajiag- hayi et al. [19] considered online auctions with a limited supply and presented value and time strategy-proof mechanisms with constant efficiency and revenue com- petitiveness. Hajiaghayi et al. [20] gave a characterization for the online allocation rules that are truthfully imple- mentable. Recently, Li [26] used a game-theoretic model to increase the rebate incentive mechanism. Wang et al.

[37] proposed TODA which is a truthful online double auction for spectrum allocation in wireless networks. Xu et al. [47] designed an efficient spectrum channel allocation and auction method for online wireless channel scheduling to decide whether to grant each user’s exclu- sive usage and how much will be charged. Deek et al. [8]

proposed a truthful online spectrum auction design called Topaz. Topaz can distribute spectrum efficiently while discouraging bidders from misreporting their bids or time report.

Game-theoretic methods are also used in studying the media access problems in wireless networks, and there are also other works on strategy-proofness in wireless networks. There are many examples including wireless sensor networks [5,50] and ad hoc networks [3,4,10,33,38,39, 52–54].

7 Conclusion and future work

In this section, we draw our conclusion and discuss the future work. In this paper, we have proposed SHIELD, which is a strategy-proof and highly efficient channel auction mechanism for multi-radio wireless networks. We have proven its strategy-proofness and have implemented SHIELD. We have made some improvements for SHIELD and implemented Fair-SHIELD in repeated auctions. Our evaluation results have shown that SHIELD can achieve higher performance compared with existing channel auction mechanisms, and Fair-SHIELD can achieve fairness in repeated auctions. For future work, it will be interesting to investigate the problem of the collusion resistance in designing wireless channel auction mechanisms.

(12)

Acknowledgments This work was supported in part by the State Key Development Program for Basic Research of China (Grant No.

2014CB340303, 2012CB316201), in part by China NSF Grant 61170236, 61272443, 61133006, 61073152, in part by Shanghai Science and Technology fund 12PJ1404900 and 12ZR1414900, and in part by Key Project of Educational Science Research of Shanghai of China A1120. The opinions, findings, conclusions and recom- mendations expressed in this paper are those of the authors and do not necessarily reflect the views of the funding agencies or the government.

References

1. Al-Ayyoub M, Gupta H (2011) Truthful spectrum auctions with approximate revenue. In: Proceedings of 30th Annual IEEE Conference on Computer Communications (INFOCOM), Shanghai, China, Apr. pp. 2813–2821

2. Alicherry M, Bhatia R, Li L (2005) Joint channel assignment and routing for throughput optimization in multi-radio wireless mesh networks. In: Proceedings of The Eleventh International Con- ference on Mobile computing and networking (MobiCom), Cologne, Germany

3. Anderegg L, Eidenbenz S (2003) Ad hoc-VCG: a truthful and cost-efficient routing protocol for mobile ad hoc networks with selfish agents. In: Proceedings of the Ninth International Con- ference on mobile computing and networking (MobiCom), San Diego, CA

4. Ben Salem N, Buttyan L, Hubaux JP, Jakobsson M (2003) A charging and rewarding scheme for packet forwarding in multi- hop cellular networks. In: Proceedings of the Fourth ACM Symposium on mobile ad hoc networking and Computing (Mo- biHoc), Annapolis, MD

5. Chen J, Xu W, He S, Sun Y, Thulasiraman P, Shen X (2010) Utility-based asynchronous flow control algorithm for wireless sensor networks. IEEE J Sel Areas Commun 28(7):1116–1126 6. Cisco Visual Networking Index (2011) Global mobile data traffic

forecast update, 2010–2015

7. Clarke EH (1971) Multipart pricing of public good. Public Choice 11(1):17–33

8. Deek LB, Zhou X, Almeroth CK, Zheng H (2011) To preempt or not: tackling bid and time-based cheating in online spectrum auctions. In: Proceedings of 30th Annual IEEE Conference on Computer Communications (INFOCOM), Shanghai, China, pp. 2219–2227

9. Ding L, Melodia T, Batalama SN, Matyjas JD (2010) Distributed routing, relay selection, and spectrum allocation in cognitive and cooperative ad hoc networks. In: proceedings of the Seventh Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks(SECON 2010), Boston, Massachusetts,USA, June 21–25 pp. 484–492 10. Eidenbenz S, Resta G, Santi P (2005) Commit: a sender-centric

truthful and energy-efficient routing protocol for ad hoc networks with selfish nodes, In: Proceedings of the 19th International Parallel and Distributed Processing Symposium (IPDPS), Denver, CO 11. FCC Online Table of Frequency Allocations (2008)

12. Federal Communications Commission (FCC),http://www.fcc.gov/

13. Federal Communications Commission Spectrum Policy Task Force (2002) Report of the spectrum efficiency working group 14. Fe´legyha´zi M, Cˇ agalj M, Bidokhti SS, Hubaux J-P (2007) Non-

cooperative multi-radio channel allocation in wireless networks, In: Proceedings of 26th Annual IEEE Conference on Computer Communications (INFOCOM), Anchorage, AK

15. Fudenberg D and Tirole J (1991) Game theory. MIT Press, Cambridge

16. Gao L, Wang X (2008) A game approach for multi-channel allocation in multi-hop wireless networks,. In: Proceedings of the 9th ACM International Symposium on Mobile Ad Hoc Net- working and Computing (MobiHoc), Hong Kong, China, pp. 303–312

17. Gopinathan A, Li Z, Wu C Strategyproof (2011) auctions for balancing social welfare and fairness in secondary spectrum markets, in Proceedings of 30th Annual IEEE Conference on Computer Communications (INFOCOM), Shanghai, China, 3020–3028

18. Groves T (1973) Incentives in teams. Econometrica 41(4):

617–631

19. Hajiaghayi MT, Kleinberg R, Parkes DC (2004) Adaptive limited-supply online auctions. In: Proceedings of the ACM Sym- posium on Electronic Commerce (EC)

20. Hajiaghayi MT, Kleinberg RD, Mahdian M (2005) Online auctions with reusable goods. In: Proceedings of the ACM Sympo- sium on Electronic Commerce (EC)

21. Han B, Kumar VSA, Marathe MV, Parthasarathy S, Srinivasan A (2009) Distributed strategies for channel allocation and scheduling in software-defined radio networks. In: Proceedings of 28th Annual IEEE Conference on Computer Communications (IN- FOCOM). Rio de Janeiro, Brazil

22. Hou YT, Shi Y, Sherali HD (2007) Optimal spectrum sharing for multi-hop software defined radio networks. In: Proceedings of 26th Annual IEEE Conference on Computer Communications (INFOCOM), Anchorage, AK

23. Kodialam M, Nandagopal T (2005) Characterizing achievable rates in multi-hop wireless mesh networks with orthogonal channels. IEEE/ACM Transaction on networking 13(4):868–880 24. Kodialam M, Nandagopal T (2005) Characterizing the capacity region in multi-radio multi-channel wireless mesh networks, in Proceedings of The Eleventh International Conference on Mobile Computing and Networking (MobiCom), Cologne, Germany 25. Kyasanur P, Vaidya N (2005) A routing protocol for utilizing

multiple channels in multi-hop wireless networks with a single transceiver, In: Proceedings of the Second International Confer- ence on quality of service in heterogeneous wired/wireless networks (QShine), Orlando, FL

26. Li LI (2010) Reputation, trust, and rebates: How online auction markets can improve their feedback mechanisms. J Econ Manag Strategy 19(2):303–331

27. Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford Press, Oxford

28. Mishra A, Shrivastava V, Agrawa D, Banerjee S, Ganguly S (2006) Distributed channel management in uncoordinated wireless environments, In: Proceedings of The Twelfth International Conference on mobile computing and networking (MobiCom), Los Angeles

29. Osborne MJ, Rubenstein A (1994) A course in game theory. MIT Press, Cambridge

30. Rad AHM, Wong VW, (2007) Joint channel allocation, interface assignment and mac design for multi-channel wireless mesh networks. In Proceedings of 26th Annual IEEE Conference on Computer Communications (INFOCOM), Anchorage, AK, May pp. 1469–1477

31. Radio Administration Bureau (RAB),http://wgj.miit.gov.cn/

32. Raniwala A, Gopalan K, Chiueh TC (2004) Centralized channel assignment and routing algorithms for multi-channel wireless mesh networks. ACM SIGMOBILE Mobile Comput Commun Rev (MC2R) 8(2):50–65

33. Srinivasan V, Nuggehalli P, Chiasserini C-F, Rao R (2003) Cooperation in wireless ad hoc networks, In: Proceedings of 22nd Annual IEEE Conference on Computer Communications (IN- FOCOM), San Francisco, CA

(13)

34. Varian H (1995) Economic mechanism design for computerized agents, in USENIX Workshop on Electronic Commerce 35. Vedantham R, Kakumanu S, Lakshmanan S, Sivakumar R (2006)

Component based channel assignment in single radio, multi- channel ad hoc networks In: Proceedings of the Twelfth Inter- national Conference on Mobile Computing and Networking (MobiCom), Los Angeles

36. Vickrey W (1961) Counterspeculation, auctions and competitive sealed tenders. J Finan 16(1):8–37

37. Wang S, PingXu ST, XiaoHua Xu, Li X, Liu X (2010) Toda:

truthful online double auction for spectrum allocation in wireless networks. In: Proceedings of the First IEEE International Sym- posium on New Frontiers in Dynamic Spectrum Access Networks (DySPAN)

38. Wang W, Eidenbez S, Wang Y, Li X-Y (2006) Ours–optimal unicast routing systems in non-cooperative wireless networks, in Proceedings of The Twelfth International Conference on Mobile Computing and Networking (MobiCom) Los Angeles

39. Wang W, Li X-Y, Wang Y (2004) Truthful multicast in selfish wireless networks. In Proceedings of the Tenth International Conference on Mobile Computing and Networking (MobiCom), Philadelphia, PA

40. Wang X, Li Z, Xu P, Xu Y, Gao X, Chen H-H (2010) Spectrum sharing in cognitive radio networks - an auction-based approach.

IEEE Trans Syst, Man Cybernetics, Part B 40(3):587–596 41. Welsh DJA, Powell MB (1967) An upper bound for the chromatic

number of a graph and its application to timetabling problems.

Comput J 10(1):85–86

42. West DB (1996) Introduction to graph theory, 2nd edn. Prentice Hall, New Jersey

43. Wu F, Vaidya N Small (2011) A strategy-proof mechanism for radio spectrum allocation. In Proceedings of 30th Annual IEEE Conference on Computer Communications (INFOCOM), Shanghai, China

44. Wu F, Singh N, Vaidya NH, Chen G (2011) On adaptive-width channel allocation in non-cooperative, multi-radio wireless networks. In Proceedings of 30th Annual IEEE Conference on Computer Communications (INFOCOM). Shanghai, China, 2804–2812

45. Wu F, Zhong S, Qiao C (2008) Globally optimal channel assignment for non-cooperative wireless networks, In Proceed- ings of 27th Annual IEEE Conference on Computer Communi- cations (INFOCOM), Phoenix, AZ

46. Xu P, Li X-Y, Tang S, Zhao J (2011) Efficient and strategy proof spectrum allocations in multichannel wireless networks. IEEE Trans Comput 60(4):580–593

47. Xu P, Xu X, Tang S, Li X-Y (2011) Truthful online spectrum allocation and auction in multi-channel wireless networks. In Proceedings of 30th Annual IEEE Conference on Computer Communications (INFOCOM), Shanghai, China

48. Yang D, Fang X, Xue G (2011) Truthful auction for cooperative communications. In: Proceedings of The Eleventh ACM Sym- posium on Mobile Ad Hoc Networking and Computing (Mobi- Hoc). Paris, France: ACM

49. Yang J (2004) Spatial channel characterization for cognitive radios, Master’s thesis, University of California, Berkeley 50. Yu Q, Chen J, Fan Y, Shen X, Sun Y (2010) Multi-channel

assignment in wireless sensor networks: a game theoretic approach, in Proceedings of 29th Annual IEEE Conference on Computer Communications (INFOCOM). San Diego, CA,USA 51. Zheng H, Peng C (2005) Collaboration and fairness in opportu-

nistic spectrum access. In Proceedings of IEEE International Conference on Communications (ICC)

52. Zhong S, Li LE, Liu YG, Yang YR (2005) On designing incentive-compatible routing and forwarding protocols in wireless ad- hoc networks–an integrated approach using game theoretical and cryptographic techniques. In Proceedings of The Eleventh Inter- national Conference on Mobile Computing and Networking (MobiCom), Cologne, Germany

53. Zhong S, Chen J, Yang YR (2003) Sprite, a simple, cheat-proof, credit-based system for mobile ad-hoc networks, in Proceedings of 22nd Annual IEEE Conference on Computer Communications (INFOCOM), San Francisco, CA

54. Zhong S, Wu F (2007) On designing collusion-resistant routing schemes for non-cooperative wireless ad hoc networks. In Pro- ceedings of The Thirteenth International Conference on Mobile Computing and Networking (MobiCom), Montreal, Canada 55. Zhou X, Gandhi S, Suri S, Zheng H (2008) ebay in the sky:

strategy-proof wireless spectrum auctions. In Proceedings of The Fourteenth International Conference on Mobile Computing and Networking (MobiCom), San Francisco, CA

56. Zhou X, Zheng H (2009) Trust: a general framework for truthful double spectrum auctions In: Proceedings of 28th Annual IEEE Conference on Computer Communications (INFOCOM), Rio de Janeiro, Brazil

57. Zhou X, Zheng H (2010) Breaking bidder collusion in large-scale spectrum auctions, in Proceedings of The Eleventh ACM Sym- posium on Mobile Ad Hoc Networking and Computing (Mobi- Hoc), Chicago, IL