A Game Theoretical Approach to Distributed Relay Selection in Randomized Cooperation

Abstract

In this paper, the problem of node management in the presence of randomized cooperation is tackled. First, game theory is exploited to model the problem of setting up a cluster of cooperative nodes in a wireless network as a multiplayer noncooperative game. In this game the set of players is made of all the nodes belonging to a potential relay cluster an d the set of actions for each player consists of two options only (characterized by different payoffs), namely transmitting a data packet or remaining silent. Then, a novel strategy for the management of node participation to a distributed cooperative link is derived. The proposed solution is fully distributed, is characterized by autonomous choices made by each potential relay and is of significant practical interest since it guarantees the participation of a proper number of nodes to a virtual antenna array (so avoiding an energy waste associated with an excessive number of cooperating nodes) without requiring any overhead for node management.

Full text

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST 20102611
A Game Theoretical Approach to Distributed
Relay Selection in Randomized Cooperation
Simone Sergi and Giorgio M. Vitetta, Senior Member, IEEE
Abstract—In this paper, the problem of node management in
the presence of randomized cooperation is tackled. First, game
theory is exploited to model the problem of setting up a cluster
of cooperative nodes in a wireless network as a multiplayer
noncooperative game. In this game the set of players is made of
all the nodes belonging to a potential relay cluster an d the set of
actions for each player consists of two options only (characterized
by different payoffs), namely transmitting a data packet or
remaining silent. Then, a novel strategy for the management
of node participation to a distributed cooperative link is derived.
The proposed solution is fully distributed, is characterized by
autonomous choices made by each potential relay and is of
significant practical interest since it guarantees the participation
of a proper number of nodes to a virtual antenna array (so
avoiding an energy waste associated with an excessive number
of cooperating nodes) without requiring any overhead for node
management.
Index Terms—Game theory, distributed randomized - orthogo-
nal space time coding, selfish nodes, ad hoc networks, cooperative
communication.
I. INTRODUCTION
I
instance, data transmission from a given source node to a
far destination node can involve other nodes acting as relays
to establish a reliable and energy efficient multihop link [1].
To enhance link performance in each hop, relays can be also
grouped to form clusters of cooperative transmitters; when
this occurs, the nodes of each cluster coordinate their data
transmissions according to a specific strategy, i.e. according
to a given distributed cooperative transmission technique.
An important example of this approach is offered by the so
called distributed space-time coding schemes, like distributed
orthogonal space-time coding (D-OSTC) [2]. Unluckily, the
implementation of distributed transmission methods is hin-
dered mainly by their significant complexity and by the large
overhead required for node management. This is due to the
need of identifying the nodes that can potentially join each
cluster and of assigning the available codewords to cluster
nodes in a proper fashion [2], [3].
Recently, a solution to the problem of codeword assignment
for D-OSTC has been proposed in [4], [5]. According to
this solution, known as distributed randomized – orthogonal
space-time coding (DR-OSTC), each node transmits a linear
N wireless ad hoc and sensor networks data communica-
tions usually require the cooperation of their nodes; for
Manuscript received September 16, 2009; revised December 15, 2009,
January 9, 2010, March 7, 2010, and April 30, 2010; accepted May 23, 2010.
The associate editor coordinating the review of this paper and approving it
for publication was O. Simeone.
The authors are with the Department of Information Engineering,
University of Modena e Reggio Emilia (e-mail: {simone.sergi, gior-
gio.vitetta}@unimore.it).
Digital Object Identifier 10.1109/TWC.2010.061810.091394
combination of multiple codewords, which are randomly se-
lected from a code matrix shared by all the network nodes.
In principle, the DR-OSTC scheme does not require an a
priori knowledge of the number of nodes contributing to
data transmission in each cluster; in practice, however, an
accurate code design and a proper number of nodes in each
distributed transmission are required to ensure a given outage
probability. These problems are analysed in detail in [6],
where the minimum number of nodes required to accomplish
a cooperative transmission and the code size needed to satisfy
given performance requirements are derived. However, as fas
as we know, what is still missing in the technical literature
about DR-OSTC is an efficient and low overhead strategy
for the selection of a proper set of nodes for cooperative
transmissions.
In this paper a novel distributed strategy for managing
the node participation in a DR-OSTC based cooperative link
is derived resorting to a game theoretical modelling of the
considered problem [7]. In any scenario characterized by a
limited signal-to-noise ratio (SNR)1, the proposed strategy is
able to reach the needed diversity order on a cooperative link
and, at the same time, to avoid an energy waste deriving from
an excessive number of cooperative nodes. In addition, it is
characterized by the following relevant features: a) each node
is allowed to manage, in a completely autonomous fashion,
its contribution within a cluster of potential relays; b) no
transmission overhead is required for node management; c)
no prior information about the distribution of neighbouring
nodes or the channel statistics are needed for the distributed
management of network nodes. For the sake of completeness,
our analysis considers two different cases of node behavior:
in the first case, all the networks nodes are selfish, whereas in
the second one they are prone to cooperation.
The following part of this manuscript is organized as
follows. The network model is described in Section II. In
Section III some basic rules about node cooperation are given
and the participation dilemma of each node is formalized as
a noncooperative multiplayer game. Then, a learning strategy
for network nodes is described in Section IV and is exploited
in Section V to derive an optimal stochastic transmission
strategy. Performance results are illustrated in Section VI.
Finally, Section VII offers some conclusions.
II. NETWORK MODEL
To ease the derivation of the proposed strategy for the
management of node transmission and avoid diverting the
reader’s attention from the main contribution of this work, a
1Note that this situation is of relevant interest when dealing with energy
constrained distributed networks.
1536-1276/10$25.00 c ⃝ 2010 IEEE

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2612 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST 2010
two-hop link, i.e. a relay network, is analysed in the following.
In our scenario a source node is expected to communicate with
a destination through a set of 푁푇hierarchically equivalent and
rational potential relay nodes, each endowed with a single
antenna and operating in a decode and forward fashion. Each
node is also equipped with a battery having a finite stored
energy; therefore it needs to limit its power consumption for
data transmission so that its lifetime is extended as long as
possible. Each data packet sent by the source can be correctly
detected by a number of potential relays (correct detection
depends on both transmission power and channel state); then
each relay can decide if forwarding the packet toward the
destination or not. For energy saving it is advisable to let a
proper number of relay nodes contribute to packet forwarding.
As shown in the following, this goal can be achieved adjusting
the transmission probability 푃푡푥of the potential relay nodes,
so that a subset of nodes (selected in the set of 푁푇 available
nodes) actually plays, on the average, an active role in the
task of cooperative forwarding. The active nodes adopt the
DR-OSTC scheme proposed in [4] for data transmission;
this means that each node uses a single codeword randomly
selected from a common code matrix of proper size 퐿 for data
relaying2. It is well known that, in this scenario, the achievable
performance on a link in the presence of a limited SNR mainly
depends on the number 푀 of distinct codewords employed by
at least one node [6], [4]. For this reason, in the following the
performance enhancement deriving from the exploitation of
the same codeword from multiple transmitting nodes of the
same cluster is neglected. Moreover, to simplify our analysis,
the errors due to channel impairments (hence fading and noise)
are also neglected in the relays to destination multiple input
– single output (MISO) link, so that the only figure of merit
taken into account when assessing link quality is represented
by the achieved degree of diversity [6]. In our model the
following assumption are made: a) The packets transmitted by
the source contain a known preamble which can be exploited
by all the potential relays to achieve a rough synchronization
for their transmission [4]. This imperfect synchronization can
be deemed sufficient when considering delay tolerant ST codes
[8], [9]; however, a comprehensive discussion about this topic
is beyond the scopes of this paper (see [4] for a deeper analy-
sis). b) The whole set of potential relays is always of adequate
size, so that a reliable data transmission can be accomplished3
[6]. c) The actual outcome of any cooperative transmission
attempt towards the destination is sent to the potential relay
nodes through a single bit ACK/NAK feedback4. If a fixed and
known transmission power is assumed for each node, the same
feedback can be also exploited to estimate its mean channel
attenuation towards the destination itself.
2In the technical literature, more complex random rules, useful to achieve
a higher degree of diversity in the presence of few relay nodes, have been
proposed (e.g., see [5]); however, they are not taken into consideration here
for simplicity.
3In other words, it is assumed that the needed degree of diversity can
be certainly achieved by the whole set of nodes able to both decode the
transmission from the source and transmit towards the destination.
4In this work, for the sake of simplicity, the transmission of a single bit
feedback for each transmitted packet is considered. However, the learning
strategy proposed in Section IV can be easily generalized to the case of
cumulative ACK.
Two different scenarios for the behavior of relay nodes
are considered in the following. In the first scenario (dubbed
scenario #1) all the nodes are selfish, whereas in the second
one (denoted scenario #2) they are prone to cooperation. If
we take into consideration the general case of an ad hoc
network consisting of peer nodes, the first scenario refers to
the situation in which each user owns its terminal and aims
only at carrying out its own data communications; in this case,
in the eyes of every potential relay, any cooperation effort is
perceived as a waste of personal resources so that, generally
speaking, it has to be properly stimulated. On the contrary, the
second scenario is well suited to describe, for example, sensor
networks, where the nodes are under the control of the same
central authority. In fact, in the last situation the only goals
of each node are the efficiency and the effectiveness of the
network it belongs to. It is important to note that in scenario
#1, proper policies for cooperation enforcement are typically
needed [10], [11]; in this case, all the solutions proposed in
the technical literature rely on the simple rationale that a node
acquires some credits (usually represented by virtual money
or a reputation level) when it cooperates with other nodes and,
in turn, can exploit these credits to obtain cooperation. On the
contrary, a simpler approach can be adopted for scenario #2.
In fact, in this case any concept of credit or reputation is no
more necessary, since, if a node is able to assess the benefits
brought to the network by its actions, it will make its choices
without expecting a personal reward.
III. GAME DESCRIPTION
A. Rules and description of the game
In the derivation of our strategy the following assumptions
are made:
1) The nodes belonging to the same potential relay cluster
do not exchange information, but are able to listen to a
common signal, originating from the destination node,
and carrying information about the status of the last
transmission attempt (ACK/NAK feedback).
2) When the nodes of a cluster of potential relays receive a
data packet, each of them autonomouslydecides whether
to forward it or not, provided that it is able to properly
decode the conveyed information.
3) The time axis is divided in slots to ease the modelliza-
tion of node actions. The slot length 푇푢is equal to the
duration of a data packet transmitted by network nodes.
Note that, generally speaking, the duration of the slot
period can appreciably influence system performance in
the presence of a time varying wireless channel.
Our main goal is achieving a degree of diversity equal to 푅
at the destination node, so that data decoding will be carried
out correctly without involving an excessive number of active
relays (and a consequent energy waste). To reach our goal,
we need to devise a node management strategy such that, if a
packet is received by a potential relay cluster consisting of 푁푇
nodes, an adequate number 푁(푡) of them decides to contribute
to data relaying in the 푡-th slot exploiting a DR-OSTC
scheme. Moreover, we are interested in devising a distributed
and noncooperative strategy, so that any explicit information
sharing among the 푁푇nodes is avoided. To achieve this target

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SERGI and VITETTA: A GAME THEORETICAL APPROACH TO DISTRIBUTED RELAY SELECTION IN RANDOMIZED COOPERATION2613
TABLE I
REPRESENTATION OF THE PARTICIPATION GAME FOR THE 푛-TH NODE IN
SCENARIO #1
푁−푛≤ 푅 − 2푁−푛≥ 푅 − 1
Pr{푀 ≥ 푅}푎푛
+ Pr{푀 < 푅}푐푛
0
푇푋푐푛
푁푂-푇푋0
we model the participation dilemma (i.e., joining or not the
set of nodes that accomplish a cooperative transmission) as a
multiplayer game; in such a game the set of players consists
of the 푁푇 nodes belonging to the potential relay cluster and
the action set of each player is made of two distinct options,
namely transmitting or remain silent. The payoffs associated
with different node actions depend on the node behavior.
Considering at first scenario #1, which refers to a network
populated by selfish nodes, it is reasonable to assume that
the benefit acquired by the 푛-th node is related to its active
contribution within a successful cluster transmission. In our
game model, such a benefit is also inversely proportional to
the number of cooperating relays; in addition, a cost related
to the energy spent for packet transmission (so depending on
the currently experiencedchannel condition) is charged to each
node irrespectively of the transmission success (further details
about the payoff definition are provided in Paragraph III-B).
For these reasons the following roles are established for the
payoffs:
∙ If the 푛-th node decides to take part to a cooperative
transmission of a data packet towards the destination
(푇푋 action) and the cluster it belongs to carries out this
task correctly, the associated payoff for this node is equal
to 푎푛.
∙ If the 푛-th node decides to take part to a cooperative
transmission of a data packet towards the destination
(푇푋 action), but the cluster it belongs to does not carry
out this task correctly, its payoff is equal to 푐푛 (this
denotes a waste of resources).
∙ If the 푛-th node decides not to join the cooperative
transmission of a data packet towards the destination
(푁푂 푇푋 action) its payoff is equal to 0 regardless of
the actual transmission outcome.
The derivation of the optimal transmission strategy for the 푛-th
node requires analysing the node point of view on its partici-
pation dilemma and, in particular, its subjective vision of the
multiplayer game described above. This game can be usefully
represented in normal form, so that the earnable payoffs can
be easily identified on the basis of the actions selected by the
푛-th node and its opponents: such a representation is provided
in Table I. In this table, the (random) parameter 푁−푛(푡)
represents the number of nodes, different from the 푛-th one,
that decide to take part to a cooperative transmission of a
given data packet in the 푡-th time slot, whereas 푀 denotes the
rank of the matrix generated stacking the codewords randomly
selected by the involved nodes (hence, the degree of diversity
achieved by the transmission). Unluckily, the game we are
analysing is characterized by a set of incomplete information
about 푁−푛(푡); this prevents us from computing the expected
payoff for the two different choices the 푛-th node can make.
In particular, it is worth nothing that in scenario #1 the payoff
expected by the 푛-th node for a packet transmission is given
by
휌(푁−푛(푡)) = Pr{푀(푡) ≥ 푅∣푁−푛(푡)}푎푛
+ Pr{푀(푡) < 푅∣푁−푛(푡)}푐푛.
In Section IV it is shown that the opponent strategy can
be estimated by the 푛-th node, so that the above mentioned
problem about the parameter 푁−푛(푡) can be solved modelling
it as a game of imperfect information (see [16, Sect. 3.7]). This
means that the 푛-th node is required to optimize its response
to the probability mass function Pr{푁−푛(푡)} as well as its
expected payoff (1). Actually, this is equivalent to playing a
strategy that optimally handles the case in which 푁−푛(푡) takes
a certain value on the basis of the probability of this event.
To ease the modelling, the same strategy can be obtained
considering the single payoff metric
(1)
훼(푁−푛(푡)) = Pr{푁−푛(푡)}휌(푁−푛(푡)),
which represents a weighted version of the expected payoff
(1). Then, the goal of the 푛-th node is to behave according
to a strategy (hence, in our case, to choose its transmission
probability 푃푡푥(푛,푡) in the 푡-th time slot), such that its mean
expected payoff
∑
is maximized. However, the dependency on the number of
nodes involved in a cooperative transmission makes the deriva-
tion of an equilibrium point for the game a non trivial task. To
tackle this problem, it is possible to consider the participation
dilemma as a repeated game. In fact, the game is continuously
repeated during the life of a given wireless link since the 푛-th
node has to take a decision about cooperating or remaining
silent in each time slot. If all the payoffs can be deemed
constant over a few turns of the game itself5, the game can
be deemed stationary [12] and this allows the 푛-th node
to acquire information about the behavior of its opponents
from their past moves. Note also that the payoffs shown in
Table I depend on the probability of the events {푀(푡) < 푅}
and {푀(푡) ≥ 푅}, i.e. on the probability that the available
degree of diversity is smaller than that needed for a correct
transmission or not, respectively. It is easy to understand
that these probabilities depend both on the overall number
푁(푡) of nodes involved in a cooperative transmission and on
the randomization rule adopted for the codeword assignment
[5]. For the randomization rule here considered, closed form
expressions to evaluate these probability are provided in
Appendix A.
When considering scenario #2, the rationale described so far
still holds. However, when the network is populated by nodes
prone to cooperation, it is reasonable to assume that each node
earns a constant reward for a correct packet transmission of the
cluster it belongs to, independently of its actual cooperation.
(2)
퐸(푇푋) =
푁−푛
훼(푁−푛(푡))
(3)
5The validity of this hypothesis depends on the rate of change of the
communication channel characterizing a given wireless link. In the following
we assume that the channel rate of change is so small to guarantee the
effectiveness of the learning strategy proposed in Section IV. The effects
of channel variations are assessed in Section VI via computer simulations.

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2614IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 8, AUGUST 2010
TABLE II
REPRESENTATION OF THE PARTICIPATION GAME FOR THE 푛-TH NODE IN
SCENARIO #2
푁−푛≤ 푅 − 2푁−푛= 푅 − 1
Pr{푀 = 푅}´ 푎푛
+Pr{푀 < 푅}´ 푐푛
0
푁−푛≥ 푅
Pr{푀 ≥ 푅}´ 푎푛
+Pr{푀 < 푅}´ 푐푛
Pr{푀 ≥ 푅}´푏푛
푇푋´ 푐푛
푁푂-푇푋0
In this circumstance, a node earns a positive payoff (given by
푏푛) also when it decides not to join a cooperative transmission
of a data packet towards the destination (NO TX action), but
the active cooperating nodes correctly carry out this task. The
representation of this modified game is provided in Table II.
Note that, in this case, the expected differential payoff
´ 휌(푁−푛(푡)) =Pr{푀(푡) ≥ 푅∣푁−푛(푡)}´ 푎푛
+Pr{푀(푡) < 푅∣푁−푛(푡)}´ 푐푛
−Pr{푀(푡) ≥ 푅∣푁−푛(푡)}´푏푛
has to be employed in place of 휌(푁−푛(푡)) (1) in the derivation
of the node participation strategy (and, consequently, in eqs.
2 and 3).
(4)
B. Payoff evaluation
1) Payoffs in scenario #1:
Evaluation of 푎푛 - In our model the payoff 푎푛 represents
the gain that the 푛-th cooperating node can earn through
the correct transmission of a data packet; it depends on two
relevant parameters, namely the estimate of the energy needed
for this transmission and the amount of credits originating
from this action [7]. In the following we assume that the
overall amount of credits earned by the 푛-th node thanks to
its transmissions until the end of the 푡-th slot over a specific
link is equal to
푃표푣(푛,푡) = 퐵
퐹푤(푛,푡)
푗∈퐶푇퐹푤(푗,푡),
∑
(5)
where 퐵 is the overall amount of credits made available by the
source node to reward the whole potential relay set, 퐹푤(푛,푡) is
the number of packets that the node has contributed to forward
until the end of the 푡-th slot and∑
which the 푛-th node belongs to, over the same time interval6.
This choice ensures that the source node can define a priori the
amount of credits 퐵 to be assigned to a relay stage to gain
its cooperation, independently of the fraction of nodes that
will really contribute to a packet transmission on the second
hop. Moreover, it will contribute to limit the overall number
of potential relays cooperating in a successful transmission,
so that the efficiency of the link itself is ensured.
푗∈퐶푇퐹푤(푗,푡) is the overall
number of packets sent by the whole potential relay set 퐶푇,
6An objective control of the effective number of transmissions carried out
by each node is not trivial since it is not easy for an external observer to
understand which nodes have actually participated to a specific transmission.
In this work, this issue is not discussed further; it is assumed, however,
that the nodes manage their transmission counter without cheating or that
an effective distributed control system is realizable. For example, a simple
policy cancelling any credit exchange if the sum of the potential relay requests
exceeds the credits assigned to them by the source discourages dishonest
behaviors in the case of rational - non malicious nodes.
For all the assumptions made above, if 퐸표푣(푛,푡) denotes
an estimate of the overall energy consumed by the 푛-th node
until the end of the 푡-th slot, the overall benefit acquired by
the node over the given time interval can be expressed as
푓 (푃표푣(푛,푡)) − 푔 (퐸표푣(푛,푡)),(6)
where 푓(⋅) and 푔(⋅) are monotonic increasing functions
having the specific purpose of making the two terms
appearing in the right-hand side of (6) homogeneous
and characterized by similar ranges (so that they play
comparable roles in the evaluation of the payoffs). If
one or more packet is successfully forwarded by the
푛-th node in the (푡 + 1)-th slot, 퐹푤(푖) and
will increase by 1 and by the number 퐹푤,푠푙표푡(푡 + 1)
of nodes participating to the cooperative transmission
respectively (an estimation technique for this parameter is
illustrated in Sec. VI), so that (see (5)) 푃표푣(푛,푡 + 1) =
퐵 (퐹푤(푛,푡) + 1)/[∑
퐸푠푙표푡(푛,푡 + 1), where 퐸푠푙표푡(푛,푡 + 1) represents an estimate
of the energy needed for the transmission in the (푡 + 1)-th
slot; such an estimate can be easily computed if the channel
state in the previous slot is known.
Since the overall benefit for the 푛-th node after a
correct transmission in the (푡 + 1)-th slot is given by
푓 (푃표푣(푛,푡 + 1)) − 푔 (퐸표푣(푛,푡 + 1)), the payoff 푎푛 can be
defined as the difference between the last quantity and (6),
i.e. as
∑
푖퐹푤(푖)
푘∈퐶푇퐹푤(푘,푡) + 퐹푤,푠푙표푡(푡 + 1)].
Moreover, we have that 퐸표푣(푛,푡 + 1)=퐸표푣(푛,푡) +
푎푛≜ [푓 (푃표푣(푛,푡 + 1)) − 푔 (퐸표푣(푛,푡 + 1))]
− [푓 (푃표푣(푛,푡)) − 푔(퐸표푣(푛,푡))],
since this expresses the additional benefit acquired by the 푛-
th relay node for the transmission of the new packet. This
definition deserves the following comments: a) it implies that
푎푛 > 0 (푎푛 < 0) corresponds to an actual reward for the
node (a damage for it); b) it does not take into account the
willingness of the 푛-th node to spend its residual resources for
data transmission (hence, to earn credits for its future needs).
To circumvent the last problem, (7) can be generalized as
(7)
ˆ 푎푛 ≜ [푤푡푥(푛) ⋅ 푓 (푃표푣(푛,푡 + 1)) − 푤푒푛(푛) ⋅ 푔 (퐸표푣(푛,푡 + 1))]
− [푤푡푥(푛) ⋅ 푓 (푃표푣(푛,푡)) − 푤푒푛(푛) ⋅ 푔 (퐸표푣(푛,푡))],
where the weight 푤푡푥(푛) (푤푒푛(푛)) measures the willingness
of the 푛-th node to cooperate (to save energy). Note that, since
the functions 푓(⋅) and 푔(⋅) are generic, the real influence of
these weights on the payoffs depends on their ratio, i.e. on the
parameter
퐾푛≜푤푡푥(푛)
푤푒푛(푛),
which can be interpreted as a risk affinity [7] for the 푛-th node.
A large risk affinity pushes the 푛-th node to cooperate with the
aim of gaining credits to acquire the neighbour’s support in
the near future; a small risk affinity, instead, can be interpreted
as an appreciable energy avidity, which pushes the node to
cooperate scarcely and only when its channel conditions are
favourable.
Evaluation of 푐푛 - When a set of relays takes part to a
transmission attempt without reaching the diversity order 푅
(8)
(9)

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SERGI and VITETTA: A GAME THEORETICAL APPROACH TO DISTRIBUTED RELAY SELECTION IN RANDOMIZED COOPERATION2615
needed to ensure a correct detection of the forwarded packet,
the transmission fails and the energy spent by each involved
node is wasted. The payoff 푐푛assigned to the 푛-th node for
an unsuccessful transmission can be evaluated resorting to the
approach described in the previous Paragraph for 푎푛; the only
difference is that, in the new situation, there is no reward
deriving from the node action. Therefore, the payoff 푐푛 can
be expressed as (see (7))
푐푛≜ 푔(퐸표푣(푛,푡)) − 푔(퐸표푣(푛,푡 + 1))
or, introducing the above mentioned weights, as
(10)
ˆ 푐푛≜ 푤푒푛(푛) ⋅ [푔 (퐸표푣(푛,푡)) − 푔(퐸표푣(푛,푡 + 1))].
Note that the payoff ˆ 푐푛(11) is insensitive to the risk affinity
factor 퐾푛(9).
2) Payoffs in scenario #2:
In scenario #2, since the nodes aim at contributing to the
effectiveness of the network, the concepts of credit exchange
and cooperation market is no longer needed. Therefore, it is
reasonable to assume that in the 푡-th slot each node will earn
the same reward Γ(푡) after a correct transmission of the cluster
of potential relays which it belongs to, irrespective of its actual
contribution. In the following it is assumed that Γ(푡) = 훾 for
a correct transmission and Γ(푡) = 0 for a failure. The value of
the parameter 훾 influences the node behavior and, in particular,
its willingness to contribute to packet transmissions (it plays
a similar role as the coefficient 퐾푛 (9)); in fact, generally
speaking, a node will get more favourable to transmitting
as 훾 increases. The cost charged to a node is related to the
energy spent for packet transmissions in the same way as in
the previous case. Therefore, the distinct payoffs earnable by
the 푛-th node of a cluster are 3 and are associated with the
following events (all referring to the (푡 + 1)-th slot):
1) Correct transmission of a packet in the presence of a
real contribution from the 푛-th node - Each element of
the cluster earns a reward equal to 훾 for the achievement
of this goal, but this payoff is decreased by a quantity
proportional to the consumed energy. Then, following
the rationale illustrated in Paragraph III-B1, the payoff
for the 푛-th node is defined as7
(11)
´ 푎푛≜ 푓 (훾)−(푔(퐸표푣(푛,푡 + 1)) − 푔 (퐸표푣(푛,푡))). (12)
2) Correct transmission of a packet in the absence of a real
contribution from the 푛-th node - In this situation the
푛-th node does not consume energy, so that its payoff
can be defined as
´푏푛≜ 푓 (훾).(13)
3) Transmission failure in the presence of real contribution
from the 푛-th node - In this case the network does not
earn any benefit, so that this event represents a resource
waste for 푛-th node. For this reason, the node payoff
can be expressed as
´ 푐푛≜ −(푔 (퐸표푣(푛,푡 + 1)) − 푔(퐸표푣(푛,푡))).
7Note that, generally speaking, the functions 푓(⋅) and 푔(⋅) are different
from the those appearing in (6)-(11) and referring to scenario #1. Morover,
even in this case, for the sake of simplicity and generality, the precise structure
of these functions is not defined.
(14)
IV. STRATEGY PLAYED BY THE OPPONENTS
As illustrated in the next Section, the derivation of the
optimal transmission strategy to be played by the 푛-th node
of a relay cluster in the 푡-th slot requires an estimate of the
probability Pr{푁−푛(푡)}. In the following we show how this
knowledge can be acquired through a proper learning strategy
that takes advantage of the repetitiveness of the game itself;
in our derivation it is assumed that the degree of diversity 푅
needed on a wireless link is perfectly known to the potential
relay nodes (the effects of an imprecise estimation of 푅 will
be analysed in Section VI).
To begin, we note that the probability Pr{푁(푡)} can be
evaluated as
Pr{푁(푡)} =
퐿
∑
푚=1
Pr{푁(푡)∣푀(푡)}Pr{푀(푡)},(15)
provided that Pr{푀(푡)} and Pr{푁(푡)∣푀(푡)} are available.
The probability Pr{푀(푡)} can be easily estimated through the
observation of the last transmission outcomes and adopting a
Poisson model8for 푀(푡), so that
Pr{푀(푡)} = 퐹(푀;휆푀(푡)),(16)
where
퐹(푥;휆푀(푡)) ≜휆푥
푀(푡)푒−휆푀(푡)
푥!
.(17)
In Appendix B it is shown that 휆푀(푡) can be computed as
휆푀(푡) = 푄−1
(
푅,
푘(푡)
푘(푡) + 1
)
,(18)
where 푄−1(푅,휆) is the inverse of the regularized upper
incomplete gamma function (see [19, Sect. 2.7]) and
푘(푡) ≜Pr{푀 ≥ 푅}
Pr{푀 < 푅}
(19)
represents the ratio between the probability of a correct packet
transmission and that of a transmission failure; note that this
parameter represents the only information available to the 푛-
th node about the behavior of its opponents (hence, about
their transmission probability 푃푡푥) and can be easily estimated
from the observation of the last 푇0transmission attempts made
by the relay cluster (generally speaking, in a time varying
scenario, the optimal value of 푇0 depends on the channel
coherence time).
The problem of estimating Pr{푁(푡)∣푀(푡)} is more com-
plicated and is solved resorting to the Bayes theorem [14];
this leads to the estimate
Pr{푀(푡)∣푁(푡)}˜푃푟{푁}
∑
where˜푃푟{푁} represents the prior knowledge we can acquire
about the number of transmitting nodes from what is expected
to be the regime in a normal functioning of our system;
note that, in this case, prior beliefs acquire an operational
significance (see [14, Chap. 5]). In our scenario the developed
Pr{푁(푡)∣푀(푡)} ≃
푙Pr{푀(푡)∣푁(푡) = 푙}˜푃푟{푁 = 푙},
(20)
8This choice is motivated by the fact that this model provides a statistical
description of the number of discrete and independent events with known
mean [13].