SALUTE: A Strategy-Proof Auction Mechanism forFlexible Multichannel Allocation

Xuewen Dong, Xiaozhou Yang, Yongzhi Wang, Ahmed Salem, Yulong Shen and Jianfeng MaEmail: xddongxuewen@gmail.com, hlgyxz@sina.com, yzwang@xidian.deu.cn

a.salem@stu.xidian.edu.cn, ylshen@mail.xidian.edu.cn, jfma@mail.xidian.edu.cnXidian University, Xi’an, P.R.China

Abstract—Recently a few strategy-proof spectrum auctionmechanisms for multichannel have been proposed. However, suchdesigns have some limitations related to the channel demandsand valuations. In this paper, we propose SALUTE, which is aStrAtegy-proof and fLexible mUltichannel aucTion mEchanism.SALUTE supports flexible combinatorial bids, with continuousor noncontinuous demands and random non-decreasing valua-tion functions. Extensive evaluation results show that SALUTEachieves good spectrum redistribution efficiency and preventsbidders from manipulating the auction.

I. INTRODUCTION

Truthful spectrum auctions not only assign channels to a setof bidders but also prevent these bidders from manipulating theauction. However, the existing strategy-proof auction mecha-nisms have many limitations related to the channel demandsand the channel valuations. In [1] and [2], the proposedstrategy-proof auctions allow each bidder to bid for a singlechannel. The PRIDE proposed in [1] and the VERITAS in[3] support multichannel auction while the same per-channelvaluation for each bidder must be guaranteed. In the combi-natorial auction mechanism in [4], the channel demands mustbe continuous and the valuation functions must be concavenon-decreasing with the increasing number of channels. Noneof them supports noncontinuous demands and random non-decreasing valuation functions, which can be happened with ahigh probability in reality. For example, the bid set {(1, $4),(3, $10)} shows this bidder just wants a single channel or 3channels with convex non-decreasing valuations.

In this work, we propose SALUTE, a strategy-proof andmultichannel auction mechanism. To the best of our knowl-edge, it is the first time to study the problem of dynamicspectrum allocation, with combinatorial auction for flexible(continuous or noncontinuous) demands and random (convexor concave) nondecreasing valuation functions. Moreover,SALUTE takes spectrum spatial reusability into considerationand achieves good redistribution efficiency.

II. SYSTEM MODEL

We consider that there is a set C = {1, 2, ..., c} of homoge-nous and orthogonal idle channels, which can be leased to thebidders denoted as N = {1, 2, ..., n}. The maximal channeldemand for all the bidders is d, and the complete demand set isD = {1, 2, ..., d}. Each bidder i (i ∈ N ) has a channel demandset Di which satisfies Di ⊆ D, and |Di| ≤ |D| = d, so thevalues of demand in Di can be continuous or noncontinuous.

We denote the valuation of bidder i for k channels as vki , thevaluation set of bidder i for his/her channel demand set Di asVi = {vki |k ∈ Di}, and the valuation profile of all the biddersas−→V = {V1, V2, ..., Vn}.

We can also denote the bid profile for the bidders as−→B =

{B1, B2, ..., Bn}. The bid set of bidder i is Bi = {bki |k ∈ Di}.Then according to the bid set of a bidder, we can get theincreased bid vector βi = {β1

i , β2i , ..., β

di } for bidder i and

bki =∑k

j=1βji , k ∈ Di. (1)

The objective of SALUTE is to allocate the channels in Cto the bidders in set N according to the bid profile

−→B , while

guaranteeing that any bidder can get his/her maximum utilitywhen bid truthfully.

III. SALUTE

The design of SALUTE consists of three main components:virtual group formulation, winner selection and charging.

The steps of virtual group formulation are as follows:1) Grouping bidders by the conflict graph as G ={g1, g2, ..., gm}. 2) Constructing virtual groups by bidders’different channel demands in each group gl ⊆ G, gjl = {i|i ∈gl ∧ j ∈ Di}. 3) Calculating the virtual group bid (VGB) σj

l

of the virtual group gjl (i.e. if |gjl | = 0, σjl = 0; if |gjl | 6= 0,

σjl = |gjl | · min{β

ji |i ∈ gjl }). 4) Recalculating each VGB

%jl ; In a group of our model, there may exist NoncontinuousDemands Virtual Group Set(NDVGS), in which one (orseveral continuous) original VGB equals to 0 and the nextoriginal VGB is larger than 0. We reassign the average of allthe original VGBs to the recalculated VGB for each virtualgroup in the NDVGS.

The winner selection of SALUTE need to follow threeprinciples. Principle 1: The priority of gjl is higher than gj+1

l ,i.e., a channel can be allocated to gj+1

l after gjl . Principle 2:The virtual groups in an NDVGS must be selected togetheror not. Principle 3: For any two virtual groups from differentgroups, the virtual group with larger VGB has a higher priority.Note that, the first two principles are a prerequisites forthe third applying principle. Then we introduce our heuristicgreedy winner selection algorithm. Generally, in each iteration,we try to allocate one channel to a virtual group with thelargest recalculated VGB in the candidate, which includes allthe smallest indices of unselected virtual groups in each group.

Sometimes, we allocate several channels to an NDVGS in oneiteration. The algorithm will not terminate until all channelshave been allocated or the candidate is empty.

Algorithm 1 Charging Algorithm - Charging(gjl )

Input: The set G contains all virtual groups, the virtual groupgjl , and the winner set W .

Output: The charge of gjl is denoted as p(gjl ).1: If gjl is not in W , then p(gjl )← 0, return p(gjl );2: If gjl is in W , get it’s index ε in winner set W ;3: G′ ← G− {gjl , g

j+1l , ..., gdl

l };4: Reselect a winner set W ′ in the new virtual group set G′;5: Get the minimal VGB minvalue from the εth winner to

the last winner in W ′, p(gjl )← minvalue, return p(gjl );

Design of the charging method for SALUTE is to maintainthe strategy-proofness (truthful) by calculating the charges ofeach virtual group. The algorithm calculating the charge ofvirtual group gjl are illustrated in Algorithm 1. We calculatebidder i’s charge pji for the channel allocated to the virtualgroup gjl as the average charge of p(gjl ) to |gjl |. Then, wesum the total charge of the bidder i for ai (the number ofchannels allocated to bidder i) channels as pi =

∑ai

j=1 pji .

The utility ui for bidder i is vaii − pi.

IV. A SIMPLE EXAMPLE

Fig. 1: A conflict graph with 5 bidders and bid vectors

Fig. 2: Virtual groups of g1

TABLE I: Virtual groups and original or recalculated VGBs

gjl : (gjl , σjl )g11 : ({}, 0) g21 : ({A,C}, 12) g31 : ({A}, 3)g12 : ({B,D,E}, 9) g22 : ({E}, 5) g32 : ({B,E}, 6)

gjl : (gjl , %jl )g11 : ({A,C}, 6) g21 : ({A,C}, 6) g31 : ({A}, 3)g12 : ({B,D,E}, 9) g22 : ({E}, 5) g32 : ({B,E}, 6)

A simple example to used to illustrate the process ofSALUTE. We assume that c = 3 and d = 3 for all bidders.The conflict graph and bidders’ bid vectors are shown in Fig.1,and so are two groups g1 = {A,C} and g2 = {B,D,E}.Fig.2 shows the three virtual groups of g1. Table I illustratesall the virtual groups and their original or recalculated VGBs.The winner set is W = {g12 , g11 , g21} selected by the winnerselection algorithm. Then, the charge of any winner canbe calculated through the charging algorithm. For example,for the winner g12 in W , we can get W ′ = {g11 , g21 , g31},p(g12) = %31 = 3 and pB = pD = pE = 3/3 = 1.

V. RESULTS AND FUTURE WORK

Fig. 3: Satisfaction ratios of SALUTE, PRIDE and VERITAS

Fig. 4: Utilities obtained bid truthfully and untruthfully

Satisfaction ratio, the percentage of bidders that is allocatedwith at least one channel, is used to evaluate spectrum allo-cation efficiency. In the simulations, the bidders are randomlydistributed in a 2000 × 2000 m2 terrain area and 20 channelswill be allocated. Fig.3 shows the satisfaction ratio when wevary the number of bidders from 5 to 50 with continuous chan-nel demands. The satisfaction ratio of SALUTE is higher thanthe other two methods. Fig. 4 shows the utilities of a bidderrandomly selected when he/she bid truthfully or untruthfullywith noncontinuous channel demands, which demonstrates thatSALUTE is a truthful auction mechanism. We plan to furtherdiscuss combinatorial auction with flexible demands underuntrustworthy auctioneer scenario.

ACKNOWLEDGEMENT

This work is supported by National High TechnologyResearch and Development Program (863 Program) (No.2015AA016007, 2015AA017203), Shaanxi Science and Tech-nology Coordination and Innovation Project(2016TZC-G-6-3), National High Technology Research and DevelopmentProgram (863 Program) (Grant No. 2015AA017203), The KeyProgram of NSFC-Guangdong Union Foundation (U1135002).

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