1
A Two-Tier Market for Decentralized Dynamic
Spectrum Access in Cognitive Radio Networks
Dan Xu, Xin Liu, and Zhu Han∗
Department of Computer Science, University of California, Davis, CA 95616
∗
Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77004
Abstract—Market mechanisms have been exploited as important
means for spectrum acquisition and access in cognitive radio networks. In this paper, we propose a two-tier market for decentralized
dynamic spectrum access. In the proposed Tier-1 market, spectrum
is traded from a primary user (PU) to secondary users (SUs) in a relatively large time scale to reduce signaling overhead. Then driven by
dynamic traffic demands, SUs set up the Tier-2 market to redistribute
channels among themselves in a small time scale. More specifically,
we use Nash bargain game to model the spectrum acquisition of SUs
in the Tier-1 market and derive the equilibrium prices. We then use
strategic bargain game to study the spectrum redistribution in the
Tier-2 market, where SUs can exchange channels with low overhead
through random matching, bilateral bargain, and the predetermined
market equilibrium prices. We disclose how various factors, such as
availability of channels and bargain partners, matching schemes, and
traffic dynamics, impact the market relationships. This work provides
better understanding on the spectrum market and valuable guidelines
to primary and secondary network operators.
I. Introduction
Dynamic spectrum access (DSA) enabled by cognitive
radio technology has the great potential to alleviate spectrum scarcity. A popular paradigm of DSA is where secondary users (SUs) opportunistically access under-utilized
primary user (PU) spectrum. In this paradigm, marketbased mechanisms have been widely studied because they
provide incentives for PUs.
Among those proposed market approaches, auction is believed to be effective for spectrum distribution. The current research works, e.g., in [1][2][3][4][5][6][16], target the
short-term and on-demand spectrum distribution in small
regions, with spectrum reuse considered, which is more flexible than the traditional FCC-style auction. Pricing mechanisms are also well studied, where a server determines the
prices for SUs to access, e.g., in [8][10][11][12][13][14][15].
However, the current approaches have limitations. Consider auction. Current works promote on-demand auctions
where a SU can request bandwidth according to its traffic
demands. However, the time overhead inherent in the auction procedure, including market setup time, bidding time,
and pricing clearing time, hampers the timely satisfaction
of SU traffic demands, which often change in a small time
scale, e.g., in multimedia communications. Moreover, auction loses its merits and becomes less efficient when there
are only a few bidders. Pricing mechanisms may incur low
signaling overhead since SUs are not involved in determining prices. However, a centralized server is needed to handle SUs’ admission control, to calculate the prices, and to
charge the SUs. Therefore they are difficult to be applied
in a scenario where SUs are in an ad hoc manner.
Observing the limitations, we propose a two-tier market
The work was in part supported by NSF through CAREER Award
#0448613 and Grant #0520126, and by Intel through a gift grant.
structure based on the decentralized bargain theory to enable distributed DSA. The two-tier structure is motivated
by the fact that spectrum trade from PUs to SUs often incurs relatively large singling overhead, since users usually
need the information of supplies and demands to determines prices. In our Tier-1 market, spectrum is traded
from a PU to SUs that targets a relatively large time scale
contracts to reduce the signaling overhead. The SUs that
leave the Tier-1 market with channels then enter the Tier-2
market, which is driven by the dynamic traffic demands of
SUs that vary in a small time scale. In the Tier-2 market,
spectrum is redistributed from the SUs with more channels (than that demanded by their current traffic) to those
with less channels. By random matching, bilateral bargain, and the predetermined market equilibriums prices,
the overhead is significantly reduced in the Tier-2 market
compared to auction markets. A SU only needs to match
with another SU and exchange the market type information. Both the Tier-1 and Tier-2 markets are in distributed
manners. We note that the Tier-1 market can be implemented by auction schemes. But bargain exhibits new spirits, including 1) decentralized structure, i.e., a PU and a
SU is matched randomly, and there is no centralized auctioneer needed; and 2) scalability, i.e., multiple PUs and
SUs can be involved, which can be referred to [19].
Our key contributions are summarized as follows.
1. We propose a two-tier market structure which targets
large time scale spectrum contracts from a PU to SUs in
Tier-1, and small time scale spectrum redistribution among
SUs to satisfy SUs’ dynamic traffic demands in Tier-2.
2. We use Nash bargain game to model the bilateral
bargain problem between a PU with multiple channels and
multiple SUs in the Tier-1 market. We further derive the
market equilibriums for different matching schemes. Our
problem is different from existing works on Nash bargain
markets [19] where a seller only has one piece of goods.
3. We propose the Tier-2 market with a distributed
and random matching structure. We use strategic bargain
game to model the transactions among SUs. We propose a
polynomial-time algorithm to derive the market equilibriums under two matching schemes.
4. We show how the market role of each type of users
and the market equilibrium are affected by different factors,
such as channel availability, matching scheme, and traffic
dynamics.
The rest of paper is organized as follows. In Section II,
we describe the system models. In Section III, we study the
Tier-1 market using Nash bargain game. In Section IV, we
2
study the Tier-2 market using strategic bargain game. Numerical evaluation is in Section V, followed by the related
work in Section VI and conclusions in Section VII.
II. Preliminaries and System Models
We consider one primary spectrum license holder with a
block of spectrum in a large geographical region, e.g., a TV
service provider who broadcasts on a set of TV channels
and serves a large area. A number of SUs are located in
the same area. The PU and SU models are as follows.
PU model: PU traffic on a channel follows the ON-OFF
model. We assume the idle time of PU channels is in a
relatively large time scale, e.g., in the order of hours.
SU model: We assume that each SU is selfish and seeks
to maximize its own utility. Each SU is equipped with a
cognitive radio and can use up to N PU channels. We consider SU traffic dynamics. Assume that the SU networks
are time-slotted, with a slot length of Ts . Traffic state
changes at the boundary of a time slot and remains constant during one time slot. We use j to denote the traffic
state of a SU, 0 ≤ j ≤ N , which represents the number
of channels needed to satisfy the traffic. Let ρjj 0 denote
0
the transition probability from state j to j . To simplify
the calculation, we assume each SU have the same traffic
transition probability matrix. We leave the heterogenous
traffic transition case in the future work. The time scale of
SU traffic dynamics is relatively small, e.g., in the order of
hundreds of milliseconds of multimedia communications.
The area that the PU spectrum owner covers is divided
into a number of cells. In each cell, the PU spectrum owner
places an agent, named PUA, to handle spectrum transactions. We consider the market of a single cell.
In the Tier-1 market, a PUA generates contracts on a
number of idle channels every Tc time and sells them to
SUs. Tc is in the order of the idle time of PU channels and
is much larger than Ts . Then, due to dynamic SUs’ traffic
demands, the Tier-2 market is formed to exchange channels
among SUs in each SU time slot. The Tier-1 market is a
one-time market since a SU leaves when it obtains a set
of channels, while the Tier-2 market is a repeated market
since traffic varies from slot to slot.
We use Nash bargain game and strategic bargain game
to investigate the market equilibriums of the Tier-1 and
Tier-2 markets, respectively. Nash bargain game captures
the notion of social efficiency and fairness [19]. The solution is to maximize the product of all players’ gains over
the outcome of the disagreement. Nash bargain is simple,
unique, and avoids any specification of the bargain procedure. The models of the strategic bargain game embody
detailed bargain procedure, where a proposer starts the
game by proposing a price and the responder accepts or
rejects it. In the Tier-1 market, we apply Nash bargain
game because in this market, the bargain is a one-time
action between a seller and a buyer, and thus it is more
reasonable to implement a game driven by the players’ attitude toward the risk of breakdown, which is central in
the Nash bargain game. In the Tier-2 market, a SU can
bargain with different types of SUs in different time slots.
The attitude toward risk of breakdown is not as essential
as the Tier-1 market. In this case, it is more interesting to
explicitly model the strategic consideration of abandoning
the current partner and finding new partners. Moreover,
it is hard to specify the outcomes of disagreement in the
Tier-2 market that is needed in the Nash bargain game.
In bargain, a public value is assumed for a given piece
of goods, e.g., a channel in our case. Based on the public
value, a seller and a buyer negotiate a price. In practice,
a PU or a SU may have its private value of a channel at a
given time, e.g., due to channel diversity. In this paper, we
consider the transmission reward of a SU in a slot normalized over channel quality, denoted by R, as a public value
of a channel, which can also be measured by the PU. Note
in [5], the authors also assume the distribution of the private value of a SU is known publicly. In the Tier-1 market,
we use U as the public value of a piece of goods (a number of channels). The value U is regarded as the sum of
SUs’ transmission rewards over the transmission time Tc .
Therefore, in the Tier-1 market, for a piece of goods that
consists of n channels, we have U = nR TTsc .
III. Tier-1 market: The PU-SU bargain market
A. Models
In Tier-1, a piece of goods is defined as a fixed number of
channels, lasting Tc time. There are totally Ns SUs, each
desires one piece of goods. In practice, SUs may request
different numbers of channels. With some modifications,
our approach can still be applied. We leave it to our future
work.
The Tier-1 market has multiple negotiation time slots.
In a negotiation time slot, the PUA and a SU set up a
transaction through a control channel. The communication
between the PUA and SUs can be based on a contention
MAC protocol, or a contention-free protocol, e.g., the PUA
selects each SU sequentially. Note the control channel may
also be shared by other networks. We use θ to denote the
probability that a transaction is successfully set up in a
slot. We have 0 < θ ≤ 1.
A SU that obtains a piece of goods will leave the market. Given there are ns SUs in the market, the probability
that a SU is matched with the PUA to conduct the current
transaction is n1s . If the PUA and a SU cannot reach an
agreement in the current transaction, we consider two possible consequences: 1) Case I: The SU still enjoys the same
chance of bargaining with the PUA in the future as other
SUs that have not been engaged; 2) Case II: The PUA
refuses to bargain with the SU again. We refer to these
two cases as different matching schemes, which result in
different market equilibriums.
We use Nash bargain game to determine the market equilibrium for each transaction. Nash bargain solution (NBS)
maximizes the product of players’ utility gain over their
disagreement outcomes. Therefore, for each transaction,
we need to specify the outcome of disagreement. The PUA
and the SUs often have time preference on the transactions.
We use a constant 0 < γ < 1 to discount the future utilities
of both the PUA and the SUs.
Although we only consider one seller in the Tier-1 mar-
3
ket, the market is scalable to the case with multiple sellers
and buyers1 . To the best of our knowledge, the current
works on decentralized Nash bargain market only consider
where a seller has one piece of goods [19]. We consider a
seller with multiple pieces of goods, which is more suitable
for spectrum trading and more challenging to study.
B. Market equilibriums
We consider the market equilibrium for each transaction
the PUA and SUs conducted. A candidate for market equilibriums is a function f that either assigns a transaction a
price p or a disagreement D. There are three possibilities
for the disagreement D: 1) when both the PUA and the
SU choose disagreement, D = Dps ; 2) when the PUA disagrees, D = Dp ; and 3) when the SU disagrees, D = Ds .
The three cases have the same result, i.e., no goods is exchanged and both the parties receive a utility of 0 from the
current transaction. However, they come from different
situations that the PUA and the SU face. The function f
determines the outcome of the current transaction between
the PUA and the SU. Therefore, f relates to the current
market setting (nc , ns ), i.e., the number of pieces of goods
nc , and the number of SUs ns .
If f = D, the engaged SU does not leave the market
in Case I, and leaves in Case II. The difference is whether
the SU considers the future utility in determining f for
the current transaction or not. If the PUA pretends to
have only one piece of goods to sell in each transaction,
it does not consider the utility from future transactions.
We consider this case later. We first focus on the scenario
where the PUA discloses the true value of nc to SUs
In Case I, when the current market setting is (nc , ns ),
we use Vp (nc , ns ) and Vs (nc , ns ) to denote the expected
utility of the PUA and a SU, respectively. If the PUA
and the SU agree on a price p(nc , ns ), the market setting
evolves to (nc − 1, ns − 1) since the SU leaves with one
piece of goods. Since the number of slots needed to successfully set up the next transaction, denoted by mt , is a
geometrically distributed random variable P
with parameter
∞
θ, we can write Vp (nc , ns ) = p(nc , ns ) + mt =1 γ mt (1 −
θ)mt −1 θVp (nc − 1, ns − 1), where γ mt is the discount factor
when the delay is mt , and (1 − θ)mt −1 θ is its probability.
The utility of the leaving SU is U − p(nc , ns ). In the case
of f = D, the market setting remains as (nc , ns ). But
all the transactions are delayed by mt slots.
P∞ Therefore,
the disagreement leads to a utility pair of ( mt =1 γ mt (1 −
P∞
θ)mt −1 θVp (nc , ns ), mt =1 γ mt (1 − θ)mt −1 θVs (nc , ns )). Let
P∞
δ = mt =1 γ mt (1−θ)mt −1 θ thereafter. We define the market equilibrium as follows.
Definition 1: In Case I, for a market setting of
(nc , ns ), an outcome f ∗ is a market equilibrium if there
exist Vp (nc , ns ) and Vs (nc , ns ) such that
f = arg maxp∗ (nc ,ns ) (Vp (nc , ns ) − δVp (nc , ns ))×
(U − p∗ (nc , ns ) − δVs (nc , ns )),
(1)
where
1 It is reasonable to apply NBS, since there may be multiple PUA
competing in the market.
Vp (nc , ns ) = p∗ (nc , ns ) + δVp (nc − 1, ns − 1),
Vs (nc , ns ) =
(2)
µ
¶
U − p∗ (nc , ns )
1
+ 1−
δVs (nc −1, ns −1),
ns
ns
(3)

D , if δVp (nc , ns ) + δVs (nc , ns )


 ps
> δVp (nc − 1, ns − 1) + U,
f=
D
,
if
Vp (nc , ns ) < 0,

p


Ds , if p∗ (nc , ns ) > U,
(4)
and
Vp (nc , ns ) = δVp (nc , ns ), i.e.,Vp (nc , ns ) = 0, if f = D,
(5)
Vs (nc , ns ) = δVs (nc , ns ), i.e.,Vs (nc , ns ) = 0, if f = D.
(6)
Similarly, in Case II:
f = arg maxp∗ (nc ,ns )
(Vp (nc , ns ) − δVp (nc , ns − 1))
×(U − p∗ (nc , ns )),
(7)
Vp (nc , ns ) = p∗ (nc , ns ) + δVp (nc − 1, ns − 1),
Vp (nc , ns ) = δVp (nc , ns ), if f = D.
½
Dp , if Vp (nc , ns ) < 0,
and f =
Ds , if p∗ (nc , ns ) > U,
(8)
(9)
(10)
Remark: In (4), both parties choose to disagree if
they are potentially better off than to agree. The PUA
chooses Dp if its utility is negative. A SU chooses Ds if
p∗ (nc , ns ) > U since otherwise it gets a negative utility. In
Definition 1, the PUA may agree on a price p∗ (m, n) < 0
since the PUA can potentially obtain positive utility from
future transactions, which is different from the classical
model of bargaining on one piece of goods where a price p
is required in [0, U ]. In Case II, the disagreement results
in a utility pair of (δVp (nc , ns − 1), 0), since the engaged
SU must leave the market under disagreement. Therefore,
Dps is inherently included by Ds because the SU’s utility
cannot be better off.
Lemma 1: In Case I, f 6= Dps , ∀ (nc , ns ). Meanwhile,
we have the following equations for Case I and Case II
according to (1)(2)(3) and (7)(8), respectively.
In Case I:
U δ
f = p∗ (nc , ns ) = − Vp (nc −1, ns −1)+ans Vs (nc −1, ns −1),
2 2
(11)
U δ
Vp (nc , ns ) = + Vp (nc −1, ns −1)+ans Vs (nc −1, ns −1),
2 2
(12)
U
δ
Vs (nc , ns ) =
+
Vp (nc −1, ns −1)+bns Vs (nc −1, ns −1),
2ns 2ns
(13)
Vp (0, ns ) = Vp (nc , 0) = Vs (0, ns ) = Vs (nc , 0) = 0, ∀nc , ns .
(14)
2
(2ns −δ)(ns −1)δ
s −1)δ
where ans = − (n
and
b
=
.
ns
2(ns −δ)
2ns (ns −δ)
In Case II:
U δ
p∗ (nc , ns ) = + [Vp (nc , ns −1)−Vp (nc −1, ns −1)] (15)
2 2
4
U δ
δ
+ Vp (nc , ns −1)+ Vp (nc −1, ns −1), (16)
2 2
2
Vp (0, ns ) = Vp (nc , 0) = 0, ∀nc , ns .
(17)
Vp (nc , ns ) =
Proof: See technical report [20] due to page limits.
By Lemma 1, we rule out the action of Dps , ∀(nc , ns ).
We also derive the rule of calculating market equilibrium
prices p∗ (nc , ns ) by NBS. However, it remains unclear
whether the PUA chooses Dp or the SU chooses Ds under
a market setting. We further have the following results.
Proposition 1: In Case I, ∀(nc , ns ), when δ ≤ 0.82,
there exists a unique market equilibrium price p∗ (nc , ns ) ≤
U
2 determined by (11)(12)(13)(14), by which Vp (nc , ns ) ≥
0, Vs (nc , ns ) ≥ 0. In Case II, ∀(nc , ns ) and ∀δ, there exists a unique market equilibrium price p∗ (nc , ns ) ≤ U determined by (15)(16)(17), by which Vp (nc , ns ) ≥ 0.
Proof: Please refer to the technical report [20].
Note the proof of Proposition 1 is non-trivial for both
Case I and II, since the interactions among Vp (nc , ns ),
Vs (nc , ns ), and p∗ (nc , ns ) are complex. In addition, for
Case I, when 0.82 < δ < 1, we numerically studied the market equilibriums prices, and found that (11)(12)(13)(14)
always lead to a market equilibrium price.
The cases that PUA does not consider future utility: We
also study the case where the PUA pretends to have only
one piece of goods to sell in each transaction and does not
consider the future utility. Further, there are two cases
under disagreement. Case III: the SU stays in the market.
Case IV: the SU leaves. The results on the market equilibrium price are referred to [20]. In Section V, we compare
the market equilibrium prices in all four cases and disclose
how the market power of the PUA and SUs differs in each
case. Intuitively, refusing to bargain with the SU again
under disagreement enhances the power of the PUA and
therefore leads to a higher price in the current transaction.
So does pretending to only have one piece of goods.
IV. Tier-2 market–A traffic driven SU market
In the Tier-1 market, a SU obtains a number of channels
which remain effective for Tc time. If a SU buys n channels
and pays p to the PUA, its cost in using a channel for
pTs
one SU slot is Cp = nT
. A SU’s state in a time slot is
c
characterized by (i, j), where i is the number of channels
and j is the traffic demands of the SU, 0 ≤ i, j ≤ N . Recall
that R denotes the reward that a SU reaps by transmitting
on one channel per time slot. We can write the utility of a
SU in state (i, j) for one time slot.
λij = min(i, j)R − iCP ,
(18)
where min(i, j)R is the transmission reward and iCP is
the channel cost. Since R > Cp (In the Tier-1 market,
p∗ (nc , ns ) < U , which is equivalent to R > Cp ), by (18),
given j, a SU’s utility is maximized in a slot if i = j.
Therefore, in the beginning of a time slot, when i > j,
the SU has incentive to sell channels to other SUs. When
i < j, the SU has incentive to buy channels from other SUs
to satisfy its traffic demand. Then a SU-SU bargain market
is formed among the SUs. This is the Tier-2 market, driven
by the traffic dynamics of SUs.
A. Models
Recall that the SU networks are time slotted and a SU’s
traffic state transits in the beginning of a slot. Since in the
Tier-1 market, the prices for different SUs may be different, which results in different Cp , we let those SUs have
the same or similar Cp entering the same Tier-2 market.
Note in our simulations, we observe that the Tier-1 market
equilibrium prices are very close.
We classify SUs in the market according to their states,
i.e., (i, j). Therefore, there are (N +1)2 types of SUs in the
market, each of which has a unique market role in channel
exchange. A SU’s type varies in different slots. Note SUs
have different experiences of market types, since they may
enter the market with different initial states, bargain with
SUs with different states, or stay in the market for different
time durations. We consider SUs with i < j as buyers, and
those with i > j as sellers. The number of channels of SUs
with i = j does not change.
A SU randomly probes another SU for a potential trade
over a control channel. This is referred to as the matching
phase and takes a small duration in the beginning of a slot.
Without specifying a concrete underlying MAC protocol,
we simply assume that a SU is successfully matched with
another one with probability ϑ, which is the same for each
SU in each slot. A longer matching phase leads to a larger
ϑ and thereby higher probability of channel exchange, at
the cost of reduced transmission time. We do not address
the problem of finding the optimal time for the matching
phase in the paper.
Conditional on the event of being matched, the probability that a SU is matched with a type-(i, j) SU is equal
to the fraction of the population of the type-(i, j) SUs in
the market, denoted by πij . The set {πij } describes the
distribution of market states. We first assume {πij } are
stationary in each slot and present a theoretical framework
that determines a set of static prices. We further discuss
the evolution of the market states and how to determine
{πij } given an initial market state and traffic transition
probabilities.
We use the strategic bargain game to determine the outcome of each encounter between a matched pair. In the
strategic bargain game, there are two players: a proposer
whose action is to propose a price on a number of goods,
and a responder whose action is either to accept or to reject
the proposed price. A SU in a matched pair either plays as
a proposer or a responder, each with probability 0.5. To be
a proposer is often more beneficial in the market. Although
this paper assumes each SU is honest, in our technical report [20], we describe an encryption protocol based on Hash
function without a third party to enforce equal opportunities of being a proposer, when SUs are not honest. Under
agreement, a seller sells a number of channels permanently
to a buyer, who can resell them later.
We next characterize the number of channels exchanged,
denoted by χkl
ij , between a type-(i, j) and a type-(k, l) SU.
We have χkl
ij 6= 0 only when i < j, k > l or i > j, k < l
holds. We consider the following scheme that aggressively
encourages channel trade. Initially, we set χkl
ij as
5
χkl
ij

 min(j − i, k − l), if i < j, k > l;
min(i − j, l − k), if i > j, k < l;
=

0, otherwise.
(19)
By (19), at least one SU’s current traffic demand is satisfied
with equality. Moreover, the total welfare of the matched
pair in the current slot is maximized. However, the future
traffic states may undermine the incentives of exchanging
χkl
ij channels in the current slot. Our framework incorporates the future states in determining the number of channels exchanged among any two types of SUs. As specified
later, a type-(i, j) SU and a type-(k, l) SU may not reach
a market equilibrium price by a χkl
ij . In this case, we rekl
duce χij by 1 and let them be less aggressive in channel
exchange, until they reach a market equilibrium.
When a matched pair agrees on a price, the transaction
is successfully concluded, and the pair breaks down. Each
party continues to randomly probe a SU in the market
in the next slot. The pair is not restored even both the
parties are not newly matched 2 . If a pair does not reach
an agreement in a slot, each party also continues to probe a
SU in the next slot. We consider two different cases if both
SUs are unmatched in the next slot. 1) Case I: Both SUs
remain unmatched. 2) Case II: The pair is restored and the
SUs continue to bargain (Note a little more communication
overhead is incurred here than in Case I). The division
of the two cases are similar to the two different matching
schemes in the Tier-1 market. Intuitively, not restoring
the relationship under disagreement enhances the market
power of a proposer in the current transaction, which is
similar to the effect of PUA’s refusing to bargain with the
SU again under disagreement in the Tier-1 market.
B. Market Equilibriums
We define a type of SU’s strategy as a function that assigns to every possible match either a price or a response,
according to the type of SU’s role in the match. We consider stationary strategies. That is, a type of SUs use the
same rule of behavior in every encounter with the same
type of the bargain partner. Further, we define the market
equilibrium as follows.
Definition 2: A market equilibrium is a pair of stationary strategies that is subgame perfect equilibrium (SPE) of
the bilateral bargain game between a matched pair.
In extensive-form games, the notion of SPE rules out
the potential “incredible threat” and thereby enhances the
desirability of the strategies compared to the Nash equilibrium. The readers can refer to [19] for the formal definition
and more applications in the bargain games. For the sake
of efficiency in channels exchange, we restrict our attention
to the SPE strategies that rule out the action of rejection. Specifically, similar to Theorem 3.4 of the bargaining
game of alternative offer in Chapter 3 of [19], we derive a
SPE strategy where the optimal price that a type-(i, j) SU
2 This model is reasonable since after a successful transaction, two
SUs can less potentially satisfy each other in the next slot
proposes to a type-(k, l) SU, denoted by Pijkl , is accepted
immediately; meanwhile, the optimal price that the type(k, l) ask for, denoted by Akl
ij , is accepted by the type-(i, j)
SU. Both of them represent the amount of money that a
type (i, j)-SU gives to a type-(k, l) SU. Note Pijkl = −Aij
kl .
The SPEs are different in the two matching schemes we
consider.
B.1 Case I
Since Ts ¿ Tc , we assume an infinite time horizon in
Tier-2. Then a type of SU’s utility is independent of the
time slots with stationary strategies. We consider the utility of a SU as the sum of the utility in the current slot
and future slots. We discount the future utility by a factor
α, which can also be regarded as the probability that a SU
stays in the market in the next slot. A SU obtains revenues
from two aspects: transmission and bargain. We can write
the utility of a type-(i, j) SU when it is unmatched in a
slot as

à N N
!
N
XX
X
ρjj 0 ϑ
πkl ϕkl0 + (1 − ϑ)φij 0  ,
φij = λij +α 
ij
j 0 =0
k=0 l=0
(20)
where λij is the utility from the current slot by (18). In
the next slot, the type-(i, j) SU’s traffic transits to state
0
0
j with probability ρjj 0 . Then, as a type-(i, j ) SU, it will
be matched with a type-(k, l) SU with probability ϑπkl , or
remain unmatched with probability 1 − ϑ. In (20), we use
0
ϕkl
to denote the expected utility of a type-(i, j ) SU when
ij 0
it is matched with a type-(k, l) SU. We can write ϕkl
ij as
ϕkl
ij = −
Pijkl + Akl
ij
+ φkl
ij ,
2
(21)
P kl +Akl
where ij 2 ij is the expected payment from the type-(i, j)
0
SU to the type-(k, l) SU, φkl
ij = φi0 j and i is the new channel state of the type-(i, j) SU after it bargains with the
0
0
kl
type-(k, l) SU, i.e., i = i + χkl
ij if i < j, or i = i − χij if
i > j.
We next characterize the SPE price. When a type-(i, j)
SU is a proposer and offers a price Pijkl to a type-(k, l)
SU, the type-(k, l) SU only accepts Pijkl if the resulted utility is no less than that of rejection, by which in this case
the type-(k, l) SU does not continue to bargain with the
type-(i, j) SU even when both are unmatched in the next
slot. Therefore, the resulted utility is simply φkl . Then by
backward induction, the optimal price that the type-(i, j)
proposes should satisfy
φkl = Pijkl + φij
kl .
(22)
When a type-(i, j) SU is a responder, we have
kl
φij = −Akl
ij + φij .
(23)
To constitute a SPE, meanwhile, a type-(i, j) and a type(k, l) SU should be willing to propose Pijkl and Akl
ij , respecshould
respectively
satisfy
tively. That is, Pijkl and Akl
ij
ij
kl
kl
φkl
ij − Pij ≥ φij and φkl + Aij ≥ φkl .
(24)
6
Combining (22) and (23), both of the inequalities in (24)
are equivalent to
ij
kl
kl
Akl
ij ≥ Pij or φij − φij + φkl − φkl ≥ 0.
(25)
Combining (20)(21)(22)(23), we have a linear equation
set that can be reduced to calculate φij , where there are
(N + 1)2 equations and (N + 1)2 variables, i.e., φij , 0 ≤
i, j ≤ N . Note χkl
ij is set according to (19) initially. If
there is no solution which makes (25) hold for ∀(i, j), (k, l),
we reduce χkl
ij of the pair which does not satisfy (25) by
one and recalculate the prices until (25) is satisfied. When
kl
χkl
ij is reduced to 0, (25) must be satisfied since both Pij
ij
kl
and Akl
ij are equal to 0. When φij − φij + φkl − φkl <
0, it implies that the current channel exchange does not
favor the future utility due to traffic transition. This is the
intuition that we reduce χkl
ij to reach market equilibrium.
The market equilibriums have a nice property that the sum
of the welfare of a matched pair must be better off.
B.2 Case II
In this case, if the responder rejects the proposed price,
the bargain relationship either breaks down or continues in
the next slot. We first consider the event of breakdown.
Since the probability of a SU being newly matched in the
next slot is ϑ, the probability of breakdown under disagreement is ϑb = 1 − (1 − ϑ)2 . Conditional on the event of
breakdown, a type-(i, j) SU is either newly matched, w.p.,
(1−ϑ)ϑ
ϑ
. Therefore, we can write
ϑb , or unmatched, w.p.,
ϑb
the utility incurred by breakdown as
ωij =
N
N N
N
ϑ X XX
(1 − ϑ)ϑ X
ρjj 0 πkl ϕkl
ρjj 0 φij 0 ,
0 +
ij
ϑb 0
ϑb
0
j =0 k=0 l=0
j =0
(26)
where the representation of φij 0 and ϕkl
0 follows (20) and
ij
(21) respectively.
A type-(i, j) SU matched with a type-(k, l) SU proposes
a price Pijkl . As in Case I, the optimal Pijkl makes the type(k, l) SU choose to accept if the resulted utility is no less
than that of the rejection. Different from Case I, the utility
of the rejection is potentially from two aspects. One is
resulted from breakdown, i.e., ωij . The other is from the
transaction continued in the next slot, where the traffic
states of both the type-(i, j) SU and type-(k, l) SU transit
0
0
to j and l , respectively. Hence we have


N
N
0
X
X
ij
ij
φkl +Pijkl = λkl +α ϑb ωkl + (1 − ϑb )
ρll0
ρjj 0 ϕkl0  .
how to reduce the linear equation set in the technical report. To guarantee the type-(i, j) SU and type-(k, l) SU
kl
kl
would like to propose Pijkl and Akl
ij respectively, Aij ≥ Pij
kl
should also be satisfied. As in Case I, we set χij according
to (19) initially. Different from Case I, when χkl
ij = 0 for a
type-(i, j) and (k, l) pair, Pijkl and Akl
may
not
be equal to
ij
kl
0 by (27)(28). Then Akl
≥
P
may
not
be
satisfied
even
ij
ij
0
kl
when χkl
ij = 0. Therefore, in Case II, we reduce χij 0 > 0 by
0
0
kl
1, j , l = 0, . . . , N , if Akl
ij ≥ Pij is not satisfied given the
0
0
0
kl
kl
kl
current χkl
ij . When χij 0 = 0, j , l = 0, . . . , N , Aij ≥ Pij is
satisfied since both are 0.
To simplify the problem, we can modify (27)(28) to let
kl
Pijkl = 0 and Akl
ij = 0 if χij = 0, i.e., the bargain game
between a type-(i, j) SU and a type-(k, l) SU ends immekl
kl
diately if χkl
ij = 0. In this case, Aij ≥ Pij is satisfied when
kl
χij = 0. Then following Case I, we only need to reduce
χkl
ij > 0 by 1. Algorithm 1 highlights the procedure of
computing the market equilibriums for Case I and Case II.
Algorithm 1 Computing the Tier-2’s market equilibriums.
1: Set χkl
ij according to (19), and loop continue = F ALSE.
2: Compute the stationary market state {πij }.
kl and Akl by (20)(21)(22)(23) for Case I; Compute
3: Compute Pij
ij
kl and Akl by (20)(21)(26)(27)(28) for Case II.
Pij
ij
4: for ∀ (i, j) and (k, l) do
kl
if Akl
5:
ij − Pij < 0 then
kl
6:
Set χij =χkl
ij − 1.
7:
Set loop continue = T RU E.
8:
end if
9: end for
10: if loop continue = T RU E then
11:
Goto 2.
12: else
kl and Akl .
13:
Output the market equilibrium prices Pij
ij
14: end if
In Algorithm 1, the calculation of market states {πij }
can be referred to the subsection of market evolution in
the technical report [20]. We give a simple sketch here.
Given an initial channel and traffic state, and traffic transition probabilities, the expected market states in slot t,
i.e., {πij (t)} only depends on {πi,j (t − 1)}, and can be calP TTsc
πij (t). Note
culated from {πij (0)}. We have πij = TTsc t=0
πij (t) is an expected value rather than an instance3 . A SU
cannot know the market states in slot t. By πij , we are
able to determine a set of fixed prices that SUs can follow,
which results in a low market overhead compared to the
market with price dynamics.
l0 =0
j 0 =0
Proposition 2: Algorithm 1 computes market equilib(27)
rium
prices for both Case I and Case II in polynomial time.
Similarly, we can write the equation on Akl
ij as
Proof: Please refer to the technical report [20].


Remark: In our simulations, we observe that for both
N
N
X
X
0
Case
I and II, Algorithm 1 is efficient in calculating the
kl
kl
kl
φij −Aij = λij +α ϑb ωij + (1 − ϑb )
ρjj 0
ρll0 ϕij 0  .
market
equilibrium prices. We observe the linear equation
j 0 =0
l0 =0
sets output a unique equilibrium prices solution under dif(28)
Combining (20)(21)(26)(27)(28), we have a linear equation
3 We observe by simulations that π (t) is usually very close to π
ij
ij
.
We
present
more
details
on
set to compute Pijkl and Akl
when
t becomes large.
ij
7
0
10
We have A10
01 − A01 =
0
α(1−α)(1−ϑ)2 R
Cp +
2
(1−α)][1−α+ αϑ
2 (π10 +π01 )]
αϑπ
1−α+ 2 01
> 0.
10
The fact A10
01 > A01 means that in Case I a proposer has
stronger market power than that in Case II. The intuition is
if the responder chooses to reject, the bargain relationship
of a pair breaks down in the next slot, no matter whether
they are newly matched.
The impact of population: By (29), we observe that
the larger π10 , the smaller A10
01 . That is, a larger fraction
of a type of SUs in the market will undermine the market
power of that type of SU. Being a majority in the market
implies that its bargain partner has more potential choices.
In the market, the ability to alternative partners enhances
a user’s market role.
The impact of traffic dynamics: We study how traffic dynamics affect the market equilibrium prices. We consider Case I and set ρ00 = ρ11 = ρ. For simplicity, we
consider π00 = π01 = π10 = π11 = 14 . We have 4
µ
¶
R − 2Cp
R
1
−
.
(31)
A01
=
10
2 1 − (a − 14 aϑ)(2ρ − 1)
1−a
We can see that a larger ρ leads to a larger A01
10 , which
indicates that keeping the same traffic state of a SU in the
next slot enhances
its market
power. In (31), when ρ <
i
h
R(1−α)
1
1
01
2 + 2α(1− ϑ ) 1 − R−2Cp , A10 < 0. In this case, the traffic
4
of the type-(0, 1) SU has a large probability of transiting
4 If ρ
00 = ρ11 = ρ, we can show for arbitrary distribution of each
type of SUs and arbitrary value of ρ, there always exists a set of
market equilibrium prices with χ10
01 = 1.
0.9
0.9
U=1, Nc=5
0.6
0.5
0.4
Case I
Case II
Case III
Case IV
0.3
0.2
0.1
U=1, Ns=15
0.8
0.7
Tier-1 market equilibrium prices
0.8
Tier-1 market equilibrium prices
ferent market settings with few reductions in χkl
ij . Because
the market equilibrium prices are predetermined, compared
to auction markets, the overhead is significantly smaller. A
SU only needs to randomly match another one to form a
bargain pair. The only information exchanged between the
pair is each SU’s market type. The payments can be cleared
when a SU leaves the market. The buyer enabled by the
cognitive radio immediately switches to the new channels.
Besides the property of having a low overhead, the Tier-2
market is in a distributed manner, which is more flexible
and scalable than a market with a centralized server.
We next consider the special case N = 1 to obtain insights on how the market equilibrium prices are affected by
various factors. When N = 1, there are four type of SUs
in the market, i.e., (0, 0), (0, 1), (1, 0), and (1, 1).
The impact of matching schemes: We first compare
the market equilibrium prices of the two matching schemes,
i.e., Case I and Case II. For simplicity, we consider the
constant traffic states, i.e., ρ01 = ρ10 = 0. We calculate
A10
01 , i.e., the price proposed by the type-(1 0) SU to the
type-(0 1) SU.
Case I:
R
αϑπ10
R + Cp
A10
−
.
01 =
1−α
2 (1 − α)[1 − α + αϑ
2 (π10 + π01 )]
(29)
Case II:
0
αϑπ10 R + α(1 − α)(1 − ϑ)2 R
R − Cp
−
A10
. (30)
01 =
1−α
2(1 − α)][1 − α + αϑ
2 (π10 + π01 )]
0.7
0.6
0.5
0.4
0.3
Case I
Case II
Case III
Case IV
0.2
0.1
0.0
0.0
0
5
10
15
20
0
5
(a) Number of SUs Ns
10
15
20
(b) Number of Channels Nc
Fig. 1. Market equilibrium prices under different number of SUs and
number of pieces of channels.
TABLE I
Market equilibrium prices when Ns = 4.
Nc
1
2
3
4
5
P (Nc , 4)
0.5000
0.2097
0.0510
-0.0078
-0.0078
Vp (Nc , 4)
0.5000
0.6597
0.6168
0.4254
0.4254
∗
to state 0. Then by the bargain it is more likely to become
a type-(1, 0). The type-(0, 1) SU even proposes to pay a
type-(1, 0) SU. The observation indicates that fast traffic
dynamics may undermine the market power of a SU. In
the current slot, a SU as a buyer, may become a seller in
the next slot due to traffic dynamics. Then the incentive
to buy channels in the current slot is smaller. This is the
reason that in some cases χkl
ij is needed to be reduced to
reach a market equilibrium.
V. Evaluation
In this section, we numerically study properties of the
Tier-1 and Tier-2 markets based on the theoretical results.
We first focus on the Tier-1 market. In the Tier-1 market, we compare the market equilibrium prices of following
cases: 1) Case I: A ∧ B, 2) Case II: A ∧ (∼ B), 3) Case III:
(∼ A) ∧ B, and 4) Case IV: (∼ A) ∧ (∼ B), where A represents that the PUA is honest in announcing the number of
channels it has, and B represents that the PUA does not
threaten to abandon the engaged SU under disagreement.
We consider the value of a piece of goods as U = 1 and the
discount factor δ = 0.9. In Fig. 1(a), we fix the number
of channels as Nc = 5 , and vary the total number of SUs
Ns from 1 to 20. In Fig. 1(b), we have Ns = 15, and vary
Nc from 1 to 20. We observe that p∗ (Nc , Ns ) is generally
an increasing function of Ns , and a decreasing function of
Nc , which follows the supply-and-demand market rule that
the prices are decreased/increased by the increase of supply/demand. We also observe that Case IV leads to the
highest market equilibrium prices for the PUA, and Case
I results in the lowest. The results indicate that threatening not to bargain with the SU in the future under disagreement enhances the market power of the PUA. So does
pretending to only have one channel in each transaction.
The market equilibrium prices are higher in Case II than
in Case III, which implies that threatening to abandon the
engaged SU favors the PUA more, compared to pretending to have only one piece of channels. Note pretending
to have only one piece of channels is a tactic of market
manipulation and could be detected by SUs cooperation.
In Table I, we consider Case I and let Ns = 4. Interest-
8
The SU's average utility in a slot
The average market equilibrium price
10
Case I
Case II
12
10
8
Matching probability=0.5
Discount factor=0.5
6
TABLE II
Ratio between the sum of number of channels exchanged of
market equilibriums and that of the initial setting by (19).
Discount factor=0.5
8
Matching probability=0.9
5
Matching probability=0.1
2
10
Case I
Case II
0
(a) The upper limit of the number of channels a SU uses N
5
10
(b) The upper limit of the number of channels a SU uses N
Fig. 2. The impacts of matching schemes, i.e., Case I and Case II in
the Tier-2 market.
Matching probability=0.1
Discount factor=0.5
12
Agreement policy
Disagreement policy
Conditional agreement policy
10
8
6
4
2
Matching probability=0.9
Discount factor=0.5
Agreement policy
Disagreement policy
Conditional agreement policy
14
The SU's average Utility in a slot
The SU's average Utility in a slot
14
0
12
10
8
6
4
2
0
0
5
10
(a) The upper limit of the number of channels a SU uses N
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
1
1
1
0.998
0.984
0.944
0.855
ϑ = 0.5
1
1
1
1
1
1
0.996
0.977
0.925
ϑ = 0.9
1
1
1
1
1
1
0.999
0.992
0.968
4
0
0
α
ϑ = 0.1
6
0
5
10
(b) The upper limit of the number of channels a SU uses N
Fig. 3. Performance of a SU with three polices.
ingly, we observe the prices are negative when Nc = 4 and
5. In this case, the PUA does not choose to disagree since
Vp (4, 4) and Vp (5, 4) are positive. Nevertheless, Vp (3, 4) is
even larger than Vp (4, 4), which discloses that in some cases
the PUA prefers not to sell all the channels it has.
We next study the market equilibriums of the Tier-2
market. We first study the difference of Case I and II.
First, we consider thePaverage proposed market equilibAkl
2
2
)
ij
rium price, i.e., Ā = (ij),(k,l)
, where Np = (N +N
2Np
4
is the total number of pairs which have channel exchange.
We set the discount factor α and the matching probability
ϑ as 0.5, respectively. Transmission utility R is 10 while
Cp is 5. We vary N from 1 to 8. The traffic transition
probability matrix, the initial channel and traffic state of
each SU are generated randomly (Note the initial state is
needed to calculate πij ) in each of the 200 simulations. In
Fig. 2(a), we observe that Case I always leads to a larger Ā
than II, which indicates that in general market settings, a
proposer’s market power is stronger in Case I than in Case
II. We also observe that A is an increasing, and almost linear function of N . The intuition is that as N increases, a
SU has more types of alternative partners, which increases
the SU’s market power.
In Fig. 2(b), we study a SU’s performance in two cases.
We consider Tc =3000Ts . The SU stays in the market for
the whole Tc time. We measure the SU’s average utility
per slot. We set ϑ = 0.1 and 0.9, respectively. The other
settings are the same as Fig. 2(a). We observe the two
cases result in similar SU performance. The result show
that Case I and Case II are in similar efficiency in channel
exchange (Note in our simulation, we observe χkl
ij s are almost the same in the two cases). The similar performance
is also because that the SU has the similar experience of
market types in the two cases. When SUs enter the market with different initial state, and stay in the market for
different time durations, they have different experience of
market types. Different sets of market equilibrium prices
result in different utilities for them. In Fig. 2(b), we also
observe that a larger ϑ leads to a higher performance, since
market is more efficient in channel exchange.
We next study the impact of α. A smaller α implies that
SUs care more about the current utility and are thereby
more aggressive in channel exchange to improve the current welfare. Recall that the sum of a pair’s utility in
the current slot is maximized, by χkl
ij set in (19). We
study how α impacts the number of channels exchanged
between
two different types of SUs. We consider the metric
P
¯kl
P
(i,j),(k,l) χij
P
r =
, where
χ¯kl is the sum of the
χ
(i,j),(k,l)
χkl
ij
(i,j),(k,l)
ij
number of channels exchanged between two types of SUs
under our proposed market equilibriums, while χkl
ij is set
as in (19). The ratio rχ reflects the aggressiveness in channel exchange. We consider α from 0.1 to 0.9, N = 5, and
ϑ = 0.1, 0.5, and 0.9. The other settings are the same as in
Fig. 2(a). We list the results of Case I in Table II. Note
Case II has almost the same results. We observe that in
Table II, the ratio rχ is very close to or equal to 1, which
indicates that SUs are aggressive in channel exchange to
satisfy their current traffic demands. It also implies that
Algorithm 1 is very efficient in computing the market equilibriums, since few reductions in χkl
ij are needed to reach
the market equilibrium. We observe a larger α leads to
a smaller rχ , i.e., less aggressiveness in channels exchange,
since the future traffic states may undermine the incentives
of exchanging channels in the current slot. We also observe
that a large ϑ promotes the aggressiveness. The reason is
that the market is more efficient for channel exchange with
a larger ϑ. When the number of channels a buyer obtains
in the current slot is excessive for the next slot, it is more
likely to be able to sell them out in a more efficient market.
A SU’s equilibrium strategy is pairwise optimal when its
partner adopts the strategy. We next study the temporal
properties of the market equilibrium strategies. A SU has
an infinite number of polices, including stationary polices
and random policies. We consider a market where all the
SUs adopt the market equilibrium strategy. We compare
three polices of a new SU entering the market. They are
1) Agreement policy: the SU always adopts the equilibrium
strategies when matched with different SUs. 2) Disagreement policy: the SU always rejects a price offer and its
proposed price is rejected by its partner, and 3) Conditional agreement policy: in this policy, we assume the SU
knows the distribution of its current type (i, j) in each slot
t. It chooses disagreement when πi,j (t) < πi,j , since he has
a stronger market power with a smaller population. Otherwise he chooses the equilibrium strategy. We vary N from
1 to 10. The other simulation settings follow Fig. 2(b). We
9
also let the SU stay in the market for the whole 3000 slots.
We consider the time average utility. In Fig. 3, we observe
that policy 1) always leads to the highest performance for
different N , policy 3) follows, and the disagreement policy
is the worst. When the matching probability ϑ is larger,
the performance gap between policy 1) and 3) is larger. We
note policy 3) is based on the assumption that a SU knows
the current market state. In practice, a SU does not know
the current market states, it has no basis to deviate from
the market equilibriums.
VI. Related work
Auction-based spectrum trade and access have been intensively studied toward various targets, which mainly include incentive compatibility [2], [4], [5], [6], [16], spectrum
reuse [1], [2], [3], [4], [5], auctioneer’s revenue maximization [2], [5], [6], [16], social welfare maximization [3][6],
and collusion resistance [3]. In [7], the authors propose
auction-based power allocation schemes for the SUs with a
interference temperature protection for the PU.
Pricing based DSA mechanisms are also well studied in
[8], [9], [10], [11], [12]. In the pricing based DSA, the
SUs are less active in determining the charged prices, compared to the auction-based markets and our bargain market
where SUs can bid or propose a price. In pricing based DSA
paradigm, a centralized spectrum server is often needed.
Two-tier market structures have been considered in [14],
[15], [16], where wireless service providers (WSP) buy spectrum from a spectrum broker, and sell them to the end
users. Our two-tier structure has different notions. We
consider a two-tier structure in different time scales that
reduces signaling overhead and enables short-term DSA.
Moreover, there is no a centralized WSP in our market,
where SUs buy channels directly from a PUA and redistribute them in a distributed way.
In [17], a local bargain approach is used to allocate channels among neighbor nodes in ad hoc networks to improve
spectrum utilization and fairness. They consider cooperative users. There is no price notions. In our paper, SUs
are selfish and have different market roles. They need to
follow a set of equilibrium prices to exchange channels.
Economic approaches have been extensively exploited to
study selfish behaviors in wireless communications and networks. We do not present a comprehensive survey in this
paper. Note Nash bargain solution has been applied to a
lot works on network resource optimization. In [18], the
strategic bargain game is applied to the packets exchange
market in a network coding based P2P system, which partially inspires our work in the Tier-2 market. Interested
readers can refer to this paper as well as Chapter 2, 3, 6,
and 7 in [19] for a better understanding of the decentralized
bargain market and related approaches.
VII. Conclusions and future works
In this paper, we propose a two-tier market structure
that enables distributed DSA for SUs with fast varying
traffic demands. We first consider the Tier-1 market where
spectrum is traded from a PUA to a set of SUs in a
large time scale. We use Nash bargain game to derive its
market equilibriums. We then propose the Tier-2 market where spectrum is redistributed among SUs to satisfy
the dynamic traffic demands in a small time scale. We
use strategic bargain game to determine the Tier-2 market equilibrium prices, following which the SUs exchange
channels through random matching with a low overhead.
We use both analytical and numerical approaches to study
the supply-and-demand market properties. Generally, the
threat of abandoning the bargain partner, the ability of
finding alternative partners, being a minority in the market, and keeping the current traffic demands enhance the
market power of a user.
To simplify the problems and focus on the essential ideas
of decentralized bargain, we simplify some models in this
paper. In our future work, we will further consider the
following cases. First, multiple PUs and SUs trade multiple pieces of goods in Tier-1. Second, each SU has heterogenous traffic transition in Tier-2. They will make the
proposed schemes more interesting and applicable.
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