A NOVEL SCHEME FOR DEVIATION DETECTION IN ASYNCHRONOUS DISTRIBUTED PRICING

International Journal of Network Security & Its Applications (IJNSA), Vol.3, No.4, July 2011
DOI : 10.5121/ijnsa.2011.3407 75
A NOVEL SCHEME FOR DEVIATION DETECTION IN
ASYNCHRONOUS DISTRIBUTED PRICING
S.S. Prasad1
, Rithika Baruah2
and Siddharth.A3
Department of Electronics & Communication Engineering, National Institute of
Technology, Jamshedpur, India
1
ssprasad@ieee.org
2
rithika_baruah@ieee.org3
sidaudhi@gmail.com
ABSTRACT
Modelling resource allocation problems in the form of non-cooperative pricing games takes into account
the difference between how much a given performance metric is valued and how much is paid for it. For the
convergence of the sum of all users’ payoff to a global maximum, the determination of the utility function is
essential. Although supermodularity conditions have been previously defined and determined to obtain
suitable utility functions, different utilities have significantly varying performance characteristics under
similar network parameters. In an ad-hoc framework, absence of a central authority leads to
uncontrollability of unfairness. Users could misbehave by broadcasting high price coefficients to force
other users to transmit at a lower power. This paper proposes an adaptation of the Asynchronous
Distributed Pricing Algorithm with a Deviation Detection Block that re-aligns the deviated system back
into the algorithm.
KEYWORDS
Game theory, asynchronous distributed pricing, distributed resource allocation, deviation detection.
1. INTRODUCTION
Wireless ad hoc networks are ‘impromptu’ networks comprising of autonomous users with
individually distinguishable Quality of Service (QoS) requirements. Applications of ad hoc
networks include ubiquitous commercial usage as well as emergency applications. A distributed
process is one that is not carried out centrally, but independently at every node. When resources
such as power, bandwidth, etc are allocated to a system of independent nodes having no central
unit controlling them (ad-hoc networks), the process is called Distributed Resource Allocation. In
order to arrive at a distributed process for resource allocation, challenges such as conflict and
non-cooperation among nodes must be overcome. Normally, nodes in this network do not
function on the principles of altruism and pursue individual goals. In the absence of a central
controller or algorithm with incentive mechanisms, performance degradation occurs in such
networks. One such scheme ensuring throughput fairness in the entire network is the
Asynchronous Distributed Pricing (ADP) suggested by Schmidt et al. in [1] rooted in the
principles of game theory.
Game theory, a branch of applied mathematics, deals primarily with decision making situations
with properly defined utilities. Game theory models individual, independent decision makers
whose actions potentially affect all other decisions in the game. Conflict and cooperation between
intelligent rational decision makers as defined in [2] is primarily replicated in game theoretical
calculations. Such situations are modeled on players which are assumed to be rational and
selfish, the precise definitions of which will be elaborated in the sections that follow. The

76
analytical tools in this theory are based on the assumption of strict adherence to a strategy based
on perceived goals. Game theory can be used to model existing systems as a particular type of
game in order to find optimum solutions to the models or better the performance of the models.
These models can also be used to study the properties of the system. Moreover, game theory can
also be used to implement new models to improve the system because in this case the objectives
of the game the fixed and the game is designed so as to attain the predefined goal set.
Primarily, game theory was invented with applications in economics although today, it finds
many applications including wireless engineering. Nowadays, game theoretic analysis has been
used in ad hoc networks in a wide range of applications. Examples include algorithms for power
control and waveform adaptation in the physical layer, medium access control as well as routing
in the network layer besides others [3, 4]. The pricing algorithms have been used for resource
allocation games ranging from bandwidth allocation, distributed beamforming, interference
pricing algorithms to power control games using virtual currency in [5-8]. This paper deals with
the problem of fair play among network entities chosen for a power control game aiming at
efficient resource allocation in ad hoc networks. Figure 1 depicts the evolutionary process of
game theory in wireless ad hoc networks.
Figure 1. Evolution of game theoretical applications for network optimization
The more the networks evolve, the more they tend to move towards decentralization without a
base station controlling the amount of power or the band of frequencies for operation of the nodes
of the network. E.g. wireless sensor networks, mobile ad hoc networks, and pervasive computing.
These networks are self-organizing multihop networks. Thus distributed decision making shall
potentially become the most important field of research in communications taking network
conditions as well as channel conditions into account as used in our thesis.
In C. H. Papadimitriou [16], the operation called price of anarchy was introduced which is the
difference of performance between selfish, local goal oriented networks and communal mode of
operation. Game theory can be used to assess this cost. In this thesis, we have used the case of
perfect information where all players know each others’ utilities and there is a certain goal that
each node tries to acquire, the goal of maximum utility. Future directions in this research include
the area of the type of games called, games of imperfect information which would rather focus on
1
2
3
4
Network
Optimization
Wireless Programming
Network Utility
Maximization
Distributed
Optimization
Game Theory

77
Bayesian equilibrium than Nash equilibrium. Cheating can be detected with the help of this model
of ignorance about other nodes as used in L. A. DaSilva and V. Srivastava [17].
Another model accounting for uncertainties regarding other players’ strategies are called games
of imperfect monitoring. Each player observing the actions of every other player at the end of
each stage in a repeated game is challenged in this type of games. E.g. in ad hoc networks, nodes
can deny forwarding packets for others to conserve their limited resources and monitoring also
requires and thus, forwarding of packets is not necessarily feasible. Instead of monitoring other
players’ actions, the nodes observe a random public signal at the end of every stage of the game
which is correlated to the actions of all the other players in the game. Distribution function of the
public signal depends on the action profile chosen by the other players but does not
deterministically reveal other players’ actions. In future, these games can be elaborately
researched upon. Another research area which is still in its infancy is the area of privately
monitored games where each player has a distinct assessment of others’ actions in a repeated
game format.
Asynchronous Distributed Pricing scheme addresses this problem by designing autonomous
allocation methods in each transmitter, based on the exchange of virtual interference prices. User
demands for service and associated priorities depend on a certain QoS metric as stated earlier as
well. It is known as the utility of the network. The network objective is then the sum utility over
all users. This objective is flexible enough to accommodate a wide range of performance metrics
through an appropriate assignment of utility functions. Each interference price is associated with
a particular receiver and indicates the marginal decrease in utility due to a marginal increase in
interference. Given a set of interference prices from nearby receivers, a transmitter then selects
resources according to a best response which maximizes its utility minus the total interference
cost. Applications of game theory to networking typically assume non-cooperative users, and
thus, these scenarios fit naturally within game theoretic frameworks. The motivation in ADP to
use game theory is to model the preferences of each user so that best response updates in the
resulting game lead to convergence to an optimal (utility-maximizing) allocation. The network
objective of finding a global optimum solution for the maximum sum utility over all users
accommodates a wide range of QoS metrics. At Nash equilibrium of a game, both the power and
price player chose not to deviate. The convergence of ADP algorithm, thus, can be ascertained by
showing that the best response updates of the game converge.
Our paper considers efficient resource allocation in ad hoc networks and hence, we deal with
distributed algorithms. Shi et al. in [8] have shown that ADP converges within the least possible
number of iterations as compared to previous distributed algorithms. However, for the class of
supermodular games defined and explained in [9], best response updates converge even when the
algorithm for power or price update is arbitrarily asynchronous. ADP has been proved to contain
a global optimum for supermodular games in [10].
In a game, selfish nodes try to maximize their own utilities by not considering the benefaction of
other nodes in the network. However, another group of nodes in a network exists in which the
nodes are called malicious nodes. These nodes try to harm the operation of the network by not
playing the game according to its rules. In a centrally monitored game, unfairness can be easily
avoided. However, selfishness modeled in a non-cooperative game theoretic with a distributed
framework can lead to uncontrollability of unfairness without extrinsic incentive mechanisms.
Virtual pricing is one such mechanism. In order to ensure cooperation in ad hoc networks, the
concept of nuglets was first introduced in [11]. Subsequently, they found applications in the
Terminodes project [12]. The ‘confidant’ protocol detected misbehavior by tracking of the
neighborhood nodes by each node and reporting misbehaviors to central controller as explained in
[13]. Validity of these reports is evaluated according to trust records in the reputation manager.
The ‘core’ protocol designed by Michiardi and Molva in [14] also functioned in similar lines. In

78
this protocol reputation values were assigned to other nodes at each node in their own separate
reputation tables which were maintained by the nodes according to their own observations. Thus,
misbehavior would lead to isolation and non-cooperation from other nodes in the game.
A set of coefficients of virtual price for the network which is an entity that can be regarded as a
payment for a desired payoff which varies with the interference that a terminal creates in the
network is mostly used in payment-based schemes. Users generating higher interference pay more
by transmitting at a lower power to give other users a fair throughput share. Some players may,
however, misbehave by broadcasting higher price coefficients so as to force other players to
transmit at a lower power. One type of games classified as the Prisoner’s Dilemma game, can be
used to provide incentives to the players to play fairly in these distributed games [15]. This study
has considered repeated games and fair broadcasting of the prices is ensured by incorporation of a
particular methodology that this study has introduced in the ADP algorithm. This paper also aims
to prove this analytically and hence improve the payoff of the players in a wireless game by
avoiding cheating even in a distributed framework.
2. PROBLEM FORMULATION
There are two entities in the ADP algorithm that are used for optimization – transmission power ܲ
and interference price ߨ. Each receiver announces a single interference price (ߨ௜), which is the
marginal cost of his own utility (‫ݑ‬௜) per unit interference. This is mathematically given as:
ߨ௜ = −
݀
݀‫ܫ‬௜
ሺ‫ݑ‬௜ሻ
Where ‫ܫ‬௜is the total interference power at ith
receiver.
While the price (ߨ௜) acts as a factor of the negative externalities, the new payoff function can be
determined by:
ෑሺܲ௜, ߨ௝ሻ = ‫ݑ‬௜ − ෍ ߨ௝ℎ௝௜
ଶ
௝ஷ௜
ܲ௜
By maximizing the resultant payoff ∏ሺܲ௜, ߨ௝ሻeach node maximizes its own utility and minimizes
the interference caused to other nodes in the system.
For the ADP algorithm, if the coefficients of relative risk aversion of each user’s utility satisfy
1 <= CRk(SINR) <= 2
for all feasible SINRs, then the resulting game is supermodular. Moreover, this case has a unique
global optimum and so it follows that the ADP algorithm will globally converge to the optimum
power allocation.
Hence, chosen utility functions must be supermodular to guarantee convergence of the algorithm.
One such example of a utility function that makes the resulting game supermodular is -1/x, with
CRk(x) value of 2.
The complete ADP algorithm is shown below in Figure 2.

79
Figure 2. Flowchart of the ADP algorithm
In the ADP algorithm, the users iteratively adapt their power allocations and announce new
interference prices. When a user adapts its power, it maximizes the resulting payoff assuming that
the power allocations and interference prices of the other users are fixed. The power update is
therefore the best response of an agent in the resulting game.
ADP being a distributed algorithm, each node in the network acts autonomously and updates its
utility asynchronously according to the network statistics. The network objective in ADP is
basically finding a global optimum solution for the maximum sum utility over all users,
accommodating a wide range of QoS metrics.
The information exchange (overhead) required for the algorithm is very limited. For every node,
the transmitter needs to know the interference prices from other receivers, its own SINR, and the
cross-channel gains to neighboring receivers. It does not need to know the other channel gains in
the network and other users’ utility functions. Also, to compute the interference price, the receiver
must know the direct channel gain and the interference-plus-noise power (or equivalently, the
SINR and the transmitted power).

80
Asynchronous Distributed Pricing is a distributed algorithm and hence, each node acts
autonomously and each node updates its utility asynchronously according to the network
statistics. The network objective in this problem is to find a global optimum solution for the
maximum sum utility over all users.
Every receiver declares its own interference price in the network which indicates the marginal
decrease in utility due to a marginal increase in interference. A transmitter selects power
according to a best response, which maximizes its utility minus the cost of interference incurred.
Users iterate between price and power updates until the algorithm has converged. The selfish
nature of nodes leads to false reporting of prices, resulting in unfair resource distribution wherein
nodes broadcast higher price coefficients so as to force other players to transmit at a lower power.
3. SYSTEM FRAMEWORK AND MATHEMATICAL PERSPECTIVE
A game is a set consisting of the following entities: a finite number of players or decision makers
(N), a set of possible actions of the players (Ai), and the set of utilities or consequences of the
actions (Ui).
The decision makers ensure the best possible utility calculated according to their preferences
while most of the games are designed so that the sum utility of all the players is a global
maximum. The preferences of the players are usually shown with the help of utility functions.
One of the most important solution concepts of a game which the game usually converges to is
the Nash equilibrium. Nash equilibrium is the set of actions from which no player deviates or
tends to deviate unilaterally because none of the players can benefit by changing strategies when
other players’ strategies are the same. But Nash equilibrium may or may not be the best
cumulative payoff. A system can be designed so as to converge the set of strategies to the best
cumulative payoff as well as the Nash equilibrium in a non-cooperative framework so that the
solution set reached to by the game is the best possible one as used in the game we have used in
this thesis.
An action vector a* = a1*,a2*,a3*,………..an* is a Nash equilibrium if
ui(ai*,a-i*) ≥ ui(ai,a-i*) for all ai and ui
We consider the following network framework for the problem. The network consists of N
“users” where each “user” is a single transmitter/receiver pair and each transmitter uses same
bandwidth. Each receiver is interested in the signal from its associated transmitter only. The
disturbance to the message signals comes from all other transmitters which constitute the
interference. In addition to interference, all the receivers experience equal amount of background
noise. The conditions of the wireless channel are reflected in the channel gains between each
transmitter and each receiver.
We have considered a system of 3 transmitter-receiver pairs with channel matrices hij where i is
the receiver and j is the transmitter. The system model is shown in Figure 3.
NjjNjj UANG ∈∈= }{,}{,

81
. Figure 3. System model and channel gains
We assume perfect channel estimation for all the nodes in our model. The channel is modeled as a
lossy channel for SINR calculations and SIR calculations. Each user needs to know adjacent
channel gains and interference prices only. This decreases the overall information exchange
overhead. User’s QoS are preferences given by a utility function ui (Ri (P)) where ui (·) is
increasing, twice differentiable and sufficiently concave function of Ri. Sufficient concavity is
defined in [18] which is modeled in supermodular games relies on the constraint that the utility
must neither be non-concave nor too concave.
Common QoS metrics used to perceive a user’s performance in a network are the received SIR
and SINR. ADP algorithm utilizes a virtual currency scheme to localize the optimization problem
faced in maximizing the sum utility of the network, by allowing the nodes to autonomously solve
the power optimization problem constrained by the strategy of the game and maximum of the
principal entity which is the transmitting power in a power game.
The resulting payoff function for power optimization suggested by the algorithm is,
∏ሺܲ௜, ߨ௝ሻ = ‫ݑ‬௜ − ∑ ߨ௝ℎ௝௜
ଶ
௝ஷ௜ ܲ௜
where ∏ is the net payoff, ui is the utility function, ∏j is the price announced by the jth
receiver, hji
is the cross channel gain between the ith
transmitter and the jth
receiver and Pi is the power
transmitted by the ith
transmitter.
The interference price is a virtual quantity which is the marginal cost of a user’s own utility per
unit interference as is given by:
ߨ௜ = െ
ௗ
ௗூ೔
ሺ‫ݑ‬௜ሻ
(1)
(2)

82
The choice of utility function is essential for the convergence of the algorithm. In [19], J. Yuan
and W. Yu have shown that the ADP power game becomes supermodular if the coefficient of
relative risk aversion factor lies between 1 and 2, resulting in a global optimum solution for the
sum utility.
We chose two main utility functions, which satisfy the necessary condition of supermodularity,
log(x) with CRk(x) = 1 and -1/x with CRk(x) = 2. However, in [20] it has been shown that rate
utilities also converge subject to a few constraints on the ADP algorithm. We consider a
diagonally dominant channel and the convergence thus obtained is a local optimum which may be
multiple depending upon the test cases considered. Also, we compared the results of the utilities
for both SIR and SINR service metrics.
3.1. Case I: u(x)=log(x)
As the optimization problem is solved locally, for individual nodes, the variable parameter is Pi
ෑሺܲ௜, ߨ௝ሻ = log ሺܽ௜ܲ௜ሻ െ ෍ ߨ௝ℎ௝௜
ଶ
௝ஷ௜
ܲ௜
∏ሺܲ௜, ߨ௝ሻ = log ሺܽ௜ܲ௜ሻ െ ܾ௜ܲ௜
where ܽ௜ =
௛௜௜మ
∑ ௛௜௝మ௉௝ାఙమ
ೕಯ೔
ܽ݊݀ ܾ௜ = ∑ ߨ௝ℎ௝௜
ଶ
௝ஷ௜
Maximizing (3), the condition for power update in ADP algorithm is,
ࡼ࢏ =
૚
࢈ܑ
It is notable that the interference price broadcasted by the user is equal to the interference power
seen by it. In case of SINR based utility functions, the price is given by the inverse of the sum of
the interference power and the noise variance at the receiver ሺߪଶ
ሻ.
࣊࢏ =
‫ە‬
‫۔‬
‫ۓ‬
૚
ࡵ૚
݂‫ݎ݋‬ ܵ‫ܴܫ‬ ܾܽ‫݀݁ݏ‬
૚
ࡵ૚ + ࣌૛
݂‫ݎ݋‬ ܵ‫ܴܰܫ‬ ܾܽ‫݀݁ݏ‬
3.1. Case I: u(x)=-1/x
Re-writing (1) in terms of the local variable Pi
∏ሺܲ௜, ߨ௝ሻ = െ
ଵ
௔೔௉೔
െ ܾ௜ܲ௜
(3)
(4)

83
where ܽ௜ =
௛௜௜మ
∑ ௛௜௝మ௉௝ାఙమ
ೕಯ೔
ܽ݊݀ ܾ௜ = ∑ ߨ௝ℎ௝௜
ଶ
௝ஷ௜
Maximizing (4),
=൐ ࡼ࢏ =
૚
ඥࢇ࢏࢈࢏
.
The parameter ai incorporates the noise variance factor in case of SINR based utility function.
The interference price of the user is found to be independent of its interference power and noise,
and is given by:
࣊࢏ =
૚
ࢎ࢏࢏
૛ ࡼ࢏
From Eqn.6. we have obtained an expression for the value of interference price of any receiver
for the -1/x utility function. The successful computation of this price requires the computing node
to know the transmission power and the direct channel gain of the ith
node.
With the help of this expression, this study modelled a deviation detection block shown in Figure
4, which can be executed by any node subject to certain conditions. This block is executed only
when the change in the reported price of a node exceeds a specific threshold value, which is
computed by comparing the change in price of other nodes. The block compares the value of the
price computed internally, from Eqn. 6, and the value of the price reported by the node under
investigation. Deviation can be ascertained when both the values do not match.
Figure 4. Flowchart for Deviation detection
Once cheating is detected, the algorithm is set back on track using the true value of interference
price computed by the cheating detection block. The threshold value of price change can be set as
the mean difference in all the other truthfully reported prices.
∆‫݌‬் =
∑ ∆ߨ௝
ே
௝ୀଵ,௝ஷ௜
ܰ
(6)
(5)

84
where ‫݌‬் is the threshold value of interference price, N is the number of truthfully reported prices
and is the price reported by the jth
node and the deviating node is the ith
node.
This paper has assumed that the number of deviating nodes is very less compared to the total
number of nodes in the system, as the threshold value of reported prices is more accurate when
the N is large. One important point to note is that the deviation detection block incorporated does
not change the course of ADP algorithm; rather it simply re-aligns it back to its original path.
Hence, the convergence of the adapted ADP algorithm still holds.
Achieving maximum performance out of such a system involves in controlling the nodes’
selfishness where they deviate from the algorithm to achieve higher individual performance but
degrading the performance of the system. The -1/x utility function presented an opportunity for
detecting such deviations. The theoretical block proposed for cheating detection makes use of
limited overhead and information exchange in order to detect any node’s deviation from the
algorithm and re-align the system to the actual ADP algorithm.
4. CONCLUSIONS
The game theoretical approach to QoS based distributed resource allocation acts as a preferable
alternative to the centralized scheme owing to its advantages of reduced overhead and
information exchange. Applications of these networks are found in ubiquitous civilian and
commercial usage, where nodes typically belong to different authorities and may not pursue a
common goal. Achieving maximum performance out of such a system involves in controlling the
nodes’ selfishness where they deviate from the algorithm to achieve higher individual
performance but degrading the performance of the system. The use of -1/x utility function in
asynchronous distributed pricing algorithm presented an opportunity for detecting such
deviations. The theoretical block proposed for cheating detection makes use of limited overhead
and information exchange in order to detect any node’s deviation from the algorithm and re-align
the system to the actual ADP algorithm. The limitations of this approach to fair power control in
ad-hoc networks is that the information exchange involved in the deviation detection block is
higher than that of the unchanged ADP algorithm. Also, its utility specific nature makes the use
of -1/x utility function mandatory.
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