Discrete Strategy Game-theoretic Topology Control in Wireless Sensor

Networks

Evangelos D. Spyrou

, Shusen Yang

and Dimitrios K. Mitrakos

School of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece

Institute of Information and System Science, School of Mathematics and Statistics,

Xi’an Jiaotong University, Xian City, China

Keywords:

Transmission Power, Transmission Reliability, Potential Game, Pareto Optimality, Packet Reception Ratio.

Abstract:

One of the most signiﬁcant problems in Wireless Sensor Network (WSN) deployment is the generation of

topologies that maximize transmission reliability and guarantee network connectivity while also maximising

the network’s lifetime. Transmission power settings have a large impact on the aforementioned factors. In-

creasing transmission power to provide coverage is the intuitive solution yet with it may come with lower

packet reception and shorter network lifetime. However, decreasing the transmission power may result in

the network being disconnected. To balance these trade-offs we propose a discrete strategy game-theoretic

solution, which we call TopGame that aims to maximize the reliability between nodes while using the most

appropriate level of transmission power that guarantees connectivity. In this paper, we provide the conditions

for the convergence of our algorithm to a pure Nash equilibrium as well as experimental results. Here we

show, using the Indriya WSN testbed, that TopGame is more energy-efﬁcient and approaches a similar packet

reception ratio with the current closest state of the art protocol ART.

1 INTRODUCTION

A signiﬁcant problem in Wireless Sensor Network

(WSN) topology management is to guarantee con-

nected network topologies that have a high transmis-

sion reliability. The simple approach would be to in-

crease the radio transmission power levels of uncon-

nected nodes. However, this is too simple and does

not account for the complexities of the wireless chan-

nel. An increase in transmission power might cause

an increase in interference, decreasing the number of

packets received (i.e. lowering Packet Reception Ra-

tio, or PRR). On the other hand, as we see in (Spy-

rou and Mitrakos, 2015b), if the distance between

the transmitter-receiver and interferer-receiver is dif-

ference by approximately a factor of 2, interference

does not cause packet loss. This indicates that a node

may select a high transmission power level, in order

to strengthen its signal, without suffering from packet

loss. There is a sweet spot in PRR related to trans-

mission power levels that can keep PRR to a high

level while not using a larger transmission power level

than necessary. The transmission power also affects

the energy consumption of the node, directly inﬂu-

encing the lifetime of the WSN (Antonopoulos et al.,

2009). In order to handle this trade-off we present a

discrete strategy distributed game-theoretic approach

that maximizes each node’s PRR while using the op-

timal transmission power from an optimisation prob-

lem; guaranteeing connectivity. We call our approach

TopGame.

Speciﬁcally, we focus on the trade-offs between

energy consumption, and PRR. We use game theory,

since it can appropriately describe the behavior of

selﬁsh nodes and ﬁnd an optimal solution in a dis-

tributed manner. Modeling systems with selﬁsh algo-

rithms have been shown to provide efﬁcient solutions

that improve network performance (Yeung and Kwok,

2006). We consider nodes to be individual players

that play selﬁshly in order to ﬁnd a best response for

their objectives. In this paper we present our model

and prove that this game is a potential game (Mon-

derer and Shapley, 1996). Potential games are games

where the incentive of players to change their strat-

egy can be expressed in a single global function, the

potential function. Potential games have been used

in wireless networks in a plethora of problems, in-

cluding power control (Heikkinen, 2006) (Spyrou and

Mitrakos, 2015a), cognitive radio (Neel et al., 2004),

gateway selection (Song et al., 2011) and channel al-

location (Chen et al., 2011). In our game-theoretic

formulation we prove that there is an equilibrium

Spyrou E., Yang S. and Mitrakos D.

Discrete Strategy Game-theoretic Topology Control inWireless Sensor Networks.

DOI: 10.5220/0006128700270038

In Proceedings of the 6th International Conference on Sensor Networks (SENSORNETS 2017), pages 27-38

ISBN: 421065/17

 2017 by SCITEPRESS – Science and Technology Publications, Lda. All r ights reserved

point. Next, we provide testbed results to show the

convergence of our proposed algorithm and we com-

pare it to the closest state of the art algorithm, Adap-

tive Robust Topology (ART)

, with respect to connec-

tivity, energy-efﬁciency and PRR. To our knowledge

this is the ﬁrst practical topology control game that

has been evaluated on a real testbed system, and show

the following:

• TopGame exhibits slightly lower network PRR

than ART, since it exploits a Transmission Relia-

bility metric to determine each node’s ﬁnal trans-

mission power.

• Using TopGame, the network’s relative energy

consumption is less by 5% than ART’s, due to

the fact that TopGame can use a per node trans-

mission power setting, which remains so after the

optimisation process. Also, connectivity is pre-

served.

• TopGame’s operation increases contention for ac-

cessing the wireless medium, since it keeps a

steady transmission power level and it includes

the bootstrapping period. This explains the

slightly less PRR of our approach.

• TopGame includes mathematical proofs to sup-

port the convergence of each node’s transmission

power in the form of the Nash equilibrium of a

potential game.

• We prove that the Price of Stability and Price

of Anarchy of TopGame is 1. This shows that

TopGame can ﬁnd the optimal equilibrium of the

game.

The paper is structured as follows: Section 2 pro-

vides the related work, Section 3 introduces topology

control in WSN, Section 4 describes game theory ba-

sics and potential games, Section 5 formally describes

TopGame, Section 6 shows the experimental results

obtained and Section 7 presents the conclusions.

2 RELATED WORK

The characteristics and behaviors of wireless links are

now more understood. There has been work measur-

ing the effects of varying power levels and showing

the irregularity of radio ranges and the lack of link

symmetry (Son et al., 2005) (Zhao and Govindan,

2003). The relationship between PRR and RSSI for

the Chipcon CC2420 radio was established in (Lin

et al., 2006). Subsequent work then looked at the

differences in behavior between indoor and outdoor

Note that by ART we mean the optimised ART.

networks, and ﬂuctuations in link quality over longer

durations of time (Hackmann et al., 2008).

Regarding of Topology Control (TC) speciﬁcally,

(Hackmann et al., 2008) contributes a comprehensive

review of this ﬁeld which we summarise. Given the

diversity of link behaviors inﬂuenced by their envi-

ronment, experimentation for much of the early TC

work was carried out using graph theory and simula-

tion studies for tractability reasons. Yet, this work

did not consider aspects like realistic radio ranges,

node distributions or node capability/capacities into

account, limiting their usefulness for real sensor net-

works (Li et al., 2005a; Li et al., 2005b) (Burkhart

et al., 2004) (Blough et al., 2007) (Gao et al., 2008).

For example, some have assumed that link costs are

proportional to link length, but in reality a more com-

plex relationship is evident (Son et al., 2005) (Gane-

san et al., 2002) (Zhao and Govindan, 2003). The

main competitors in the practical Topology control

area are PCBL (Son et al., 2005) and ART (Hack-

mann et al., 2008), which we introduce next.

PCBL was derived from link quality observations

showing that links with a very high PRR remain quite

stable. They then categorise links as blacklisted, mid-

dling or highly reliable. The power in the latter is

minimised to their lowest stable power setting while

the blacklisted are not used at all. The middling links

are those that lie between the two and are set to full

power. Given the expense of probing the network

to establish the link categories, this protocol cannot

work with dynamic routing protocols such as CTP

(Gnawali et al., 2009). CTP aims to ﬁnd the least

expensive routes through the network. To overcome

such link probing, link quality metrics have been

used to approximate PRR in ATPC (Lin et al., 2006).

Speciﬁcally there is a link between RSSI and PRR,

and LQI and PRR over a monotonically-increasing

curve. Further, linear correlations between transmis-

sion power levels and RSSI/LQI are observed at the

receiver but are different for each environment moni-

tored. Therefore, ATPC estimates the slope and uses

closed feedback to adjust the model to the current sit-

uation to achieve lower bound RSSI (PRR).

Hackmann et al., showed that RSSI and LQI can-

not always realistically estimate PRR in indoor envi-

ronments (Hackmann et al., 2008), nor can instanta-

neous probing represent the behaviors of a link over

time. They propose ART, which does not rely on es-

timates of link quality nor does it involve long boot-

strapping phases. Being more dynamic, ART adapts

link power to changes in the environment as well as

contention using a gradient. Also, where applications

expect acknowledgment messages, ART can piggy-

back these to reduce communication overhead. ART

SENSORNETS 2017 - 6th International Conference on Sensor Networks

selects the appropriate transmission power based on

the failures observed when the target PRR is 95% and

a contention gradient.

In (Hao et al., 2015), the authors proposed a

distributed topology control and channel allocation

game-theoretic algorithm. The main objective of

the work is the relief of interference and the energy

consumption balancing. They examined the connec-

tion between topology control and channel allocation.

They designed a game-theoretic model that takes into

account transmission power, energy consumption and

interference suffered by a node. They have proven the

existence of Nash Equilibrium and they developed an

algorithm that preserves connectivity by jointly set-

ting the transmission power and channel. Lastly, their

algorithm converges to Pareto optimality.

Tan et al. (Tan et al., 2015), suggested a topology

control scheme where every node tunes its transmis-

sion power adaptively, in order to use its harvested en-

ergy in an efﬁcient manner. The authors, proposed an

ordinal potential game model where high harvesting

nodes cooperate with the low harvesting nodes to en-

sure network connectivity. They proved the existence

of a Nash Equilibrium and they designed an algorithm

that achieves it.

Abbasi at al. (Abbasi and Fisal, 2015), investi-

gated the issue of topology control in wireless sen-

sor networks, in order to perform energy consumption

minimisation and energy balancing. Their approach

accomplished their objectives by adjusting transmis-

sion power on the nodes and preserving connectivity.

The authors utilised a game-theoretic scheme to ad-

dress energy welfare topology control. They showed

that their proposed game-theoretic solution is a poten-

tial game and it achieves a unique Nash equilibrium,

which is Pareto optimal as well.

Nahir et al. (Nahir et al., 2008), provided a game-

theoretical solution to the topology control problem,

by addressing three major issues: the price of estab-

lishing a link, path delay and path congestion prone-

ness. They established that bad performance due to

selﬁsh play in the considered games is signiﬁcant,

while all but one are guaranteed to have a Nash equi-

librium point. Furthermore, they showed that the

price of stability is typically 1; hence, often optimal

network performance can be accomplished by being

able to impose an initial conﬁguration on the nodes.

Furthermore, the authors express their concern re-

garding the computational tractability of their solu-

tion.

Komali et al. (Komali et al., 2008), analysed the

creation of energy efﬁcient topologies with two pro-

posed algorithms. Speciﬁcally, their game-theoretic

model speciﬁed that nodes have the incentive to pre-

serve connectivity with a sufﬁcient number of neigh-

bours and that the network will not partition. They

proved that their game is an exact potential game and

that a subset of the resulting topologies is energy ef-

ﬁcient. They addressed the major issue of fair power

allocation by providing the argument of efﬁcient allo-

cation vs fair allocation.

3 WSN AND TOPOLOGY

CONTROL

Wireless sensor networks are networks of small com-

putational devices ﬁtted with radio transceivers for

communication and sensors to capture data. Topol-

ogy control can be deﬁned by the construction of

a graph that represents the nodes and links in the

network that does not consist of any disjoint parts.

Good topology control mechanisms can be character-

ized by providing an energy efﬁcient network, offer-

ing high throughput and doing so with a low over-

head. Energy-efﬁciency equates to the use of the

minimum transmission power that guarantees con-

nectivity, where throughput can be maximised by re-

ducing interference and contention on the wireless

medium. However, minimum transmission power

does not guarantee a high reliability of transmission

resulting in high throughput. This is due to a weak

signal that may be signiﬁcantly inﬂuenced by a small

portion of interference.

For the most part, hitherto link asymmetry has

been ignored, and the use of different transmission

power levels when a node transmits to different neigh-

bours may cause undesired packet loss. In addition,

in a dense network, a node having a large number of

neighbours may not be able to cope with transmission

power changes when unicasting to different recipi-

ents in that neighbourhood while expecting to achieve

a high PRR as well. As observed by Ahmed et al.

(Ahmed et al., 2009), environmental effects and dif-

ferent node transmission powers are the major cause

of link asymmetry in WSNs.

4 GAME THEORY AND

POTENTIAL GAMES

Game theory studies mathematical models of conﬂict

and cooperation (Von Neumann et al., 2007), between

nodes in our work. Therefore, our meaning of the

term game corresponds to any form of social inter-

action between two or more nodes. The rationality

of a node is satisﬁed if it pursuits the satisfaction of

Discrete Strategy Game-theoretic Topology Control inWireless Sensor Networks

its preferences through the selection of appropriate

strategies. The preferences of a node need to satisfy

general rationality axioms, then its behavior can be

described by a utility function. Utility functions pro-

vide a quantitative description of the node’s prefer-

ences and the main objective is therefore the maxi-

mization of its utility function.

In this work, we focus on strategic non-

cooperative games, since we consider nodes to act

as selﬁsh players that want to preserve their interests.

The intuition behind this is that the nodes will reach

an optimal state, without having to pay a price to

maximize their payoffs. The Nash equilibrium is the

most important equilibrium in non-cooperative strate-

gic form games. It is deﬁned as the point where no

node will increase its utility by unilaterally changing

its strategy. It got its name from John F. Nash who

proposed it (Nash Jr, 1950).

In 2008, (Daskalakis et al., 2008) Daskalakis

proved that ﬁnding a Nash equilibrium is PPAD-

complete . Polynomial Parity Arguments on Directed

graphs (PPAD) is a class of total search problems

(Papadimitriou, 1994) for which solutions have been

proven to exist, however, ﬁnding a speciﬁc solution

is difﬁcult if not intractable. This development lead

researchers to concentrate a speciﬁc class of games

called ’Potential Games’, due to the important prop-

erties that pure Nash equilibria will always exist and

best response dynamics are guaranteed to converge.

This class of games consists of the exact and or-

dinal potential games. In this paper we utilise exact

potential games and refer the reader to (Monderer and

Shapley, 1996) for details on potential games. In or-

der to use exact potential games, it is essential to have

a potential function that has the same behavior as the

individual utility function, when a player unilaterally

deviates.

More formally:

A game GhN, A,ui, with N players, A strategy

proﬁles and u the payofffunction, is an exact potential

game if there exists a potential function

V : A → R (1)

subject to

∀i ∈ N,∀σ

−i

∈ A

−i

,∀σ

,σ

′

∈ A

(2)

where σ

is the strategy of player i, σ

′

is the deviation

of player i, σ

−i

is the set of strategies followed by

all the players except player i and A

−i

is the set of

strategy proﬁles of all players except i such as

V(σ

,σ

−i

) −V(σ

′

,σ

−i

) = u

(σ

,σ

−i

) − u

(σ

′

,σ

−i

)

(3)

5 TopGame

We developed the TopGame algorithm that aims to

guarantee connectivity, by locating the best response

of PRR and transmission power. The intuition behind

this research is that TopGame will force nodes to con-

verge to the best transmission power.

A WSN consists of a set of nodes N and each

node i ∈ N can switch its transmission power p

∈ P,

where k ∈ 3,7, 11,15,19,23,27, 31 and P is the set

of the available transmission power levels of our ex-

ample CC2420 transceiver. In this paper, we employ

4 transmission power levels, namely 11,15, 19,23, in

order to identify the PRR when transmission powers

that operate mostly on the gray area (Son et al., 2004)

are used. Let a vector P = (p

,...,p

|N|

) be an allo-

cation of the transmission power level of each sensor

node. The total number of possible power allocations

is 4

|N|

. The aim of this paper is to determine a power

allocation in a distributed way, which can achieve a

best response rade-off between network connectivity,

energy-efﬁciency and transmission reliability, using

game theory.

5.1 Connectivity Deﬁnition and

Measurement

In this paper we consider the small-world Model A

from (Ganesh and Xue, 2007), where there are N

nodes in the network and each one arbitrarily selects

m nearest neighbours to connect to. Essentially, we

utilise the variant of this small-world model, where

node locations are being modeled by a stochastic

point process. The number of neighbours consists

of nearest neighbours and shortcuts. A shortcut is an

edge between two nodes if either of the two nodes ex-

ist in the nearest neighbour set of the other. If a node

is connected by a nearest neighbour and a shortcut,

multiple edges are replaced by a single one. The pres-

ence of the shortcuts reduces the network diameter.

Furthermore, we have to note that m is the number of

neighbours a node has in terms of a spatial graph, and

(N − 1)p is the number of neighbours it has via short-

cuts. In order to ensure connectivity the quantities

m = (1 + δ)

2log(N) and Np = (1+ δ)

2log(N),

where δ > 0, are sufﬁcient. Hence connectivity is pre-

served with a smaller degree of (nearest neighbours

plus shortcuts). We select a degree of 6 for each each

node. It is well known that the node degree can be

reached by adjusting the transmission power; hence,

the transmission power level that satisﬁes connectiv-

ity satisﬁes the condition that more than 6 nodes exist

in the neighbourhood of each node.

SENSORNETS 2017 - 6th International Conference on Sensor Networks

5.2 Transmission Reliability (TR)

For a wireless link (i, j), the Packet Reception Ratio

PRR

i, j

is deﬁned as the ratio of the number of packets

received by node j over the number of packets sent by

node i. It can be expressed by approximation as

PRR

i, j

= (1− ξ

i, j

)

(4)

where l is the packet length in bits.

The Bit Error Rate (BER), which we denote as

i, j

, is given by the following formula (Fu et al., 2012)

i, j

1−

i, j

1+ γ

i, j

(5)

where γ

i, j

is the Signal-to-Interference-plus-Noise

Ratio (SINR) of the transmission from node i to node

j. γ

i, j

is given by

i, j

∑

t6=i,t6= j

t, j

+ N

(6)

where N

is the white noise and H

i, j

is the channel

gain of the wireless link (i, j) and H

t, j

is the channel

gain between the receiver and an interferer. Due to

the path loss, the larger the distance between nodes t

and j the smaller the H

t, j

. We focus on static WSNs,

hence, we assume that the channel is slow fading in

nature and the channel gain of every link remains con-

stant before the convergence of the TopGame algo-

rithm.

To measure the reliability of links around node i,

we deﬁne a new metric called Transmission Reliabil-

ity (TR

) as

, p

−i

) =

∑

j∈N

−i

),k∈N

−i

),k6=i

PRR

k, j



j∈N

−i

)

, p

−i

) − {i})



(7)

where p

is the power level of node i, p

−i

means the

power levels of all nodes except i, N

, p

−i

) is the

set of nodes such that ∀ j ∈ N

, p

−i

), PRR

i, j

> 0.

For instance, Figure 1 shows a sub-graph of

a WSN for a given transmission power allocation.

For each link (i, j) PRR

i, j

> 0. In this sub-graph

= 2, 15,7,11 and βTR

= (PRR

1,2

+ PRR

15,2

PRR

12,2

+ PRR

2,15

+ PRR

9,15

+ PRR

7,15

+ PRR

15,7

PRR

10,7

+ PRR

11,7

+ PRR

7,11

+ PRR

14,11

)/11.

In practice, everynode i can obtain TR

at run time

by every node j in N

calculating the

PRR

k, j

as the

average PRR

k, j

, k ∈ N

− {i} and periodically broad-

casting PRR

k, j

. Thereafter node i calculates TR

Figure 1: An example to explain Transmission Reliability

metric.

5.3 Utility and Potential Function

We deﬁne the utility function of each node i as,

) = TR

− c

(8)

where c

is the price assigned to each strategy played

by a node/player.

Our strategy domain consists of 4 strategies,

which are 11,15,19, 23, which correspond to the val-

ues in table 1. Notably, the 2 smallest and 2 largest

transmission power levels of the CC2420 radio have

been excluded. The main reason is to see TopGame

operate under medium to large SINR regime. The sec-

ond reason is to simplify TopGame.

Table 1: Transmission Power Levels and Values.

PA LEVEL dB mA

11 -10 11.2

15 -7 12.5

19 -5 13.9

23 -3 15.2

It is straightforward to see that the above utility

function has a minimum under the following condi-

tion of medium to high SINR values. We do the price

assignment in a similar way with (Candogan et al.,

2010). The prices assigned at every node has the value

1 except when it reaches its maximizer. Each node

then assigns the price given below:

= dif f(TR

) (9)

Hence, if we take the ﬁrst derivative to obtain the min-

imum, it follows that there is a local minimum. Since

we wish to maximize the function we simply take the

negative of (8).

) = c

− TR

(10)

Discrete Strategy Game-theoretic Topology Control inWireless Sensor Networks

Thereafter we wish to deﬁne the potential function

and prove that the game G is a potential game.

Proposition 1. The game G is a potential game. The

potential function is given by

V(p) =

∑

−

∑

, p

∈ A (11)

Proof. This comes as a result by taking the character-

isation of the potential games in (Monderer and Shap-

ley, 1996) where

∂V (p)

∂p

∂u

(p)

∂p

,i ∈ N.

V(p

, p

−i

)−V(p

′

, p

−i

) = u

, p

−i

)−u

′

, p

−i

∑

m∈N,m6=i

, p

−m

) − u

′

, p

−m

))

Since only one node can deviate

∑

m∈N,m6=i

, p

−m

)− u

′

, p

−m

)) = 0. Hence we

conclude that Γ is an exact potential game. This proof

comes as a result of the fact that given a strategy of

a node/player m, p

∈ N and an alternative strategy

′

∈ N and taking the assumption that the strategies

of all the other nodes remain the same, we have

, p

−i

) − u

, p

′

−i

) =

, p

−i

) −V

, p

′

−i

) (12)

where p

−i

is the transmission power strategy of all the

nodes excluding that of the node i. Hence, the game

is a potential game.

Remark 3.1: The potential function is signiﬁ-

cant since its maximisation, when a speciﬁc policy

is played, results in this policy being an equilibrium

of the designed game. In this work, the strategy set

is discrete; hence, in the case that the potential func-

tion satisﬁes particular types of concavity, such as the

Larger Midpoint Property (LMP) (Ui, 2008), the con-

verse is true as well. If a policy is an equilibrium,

it maximises the potential function. Thus, we may

consider the TopGame as the following optimisation

problem.

ˆp

= argmaxV

(i) (13)

As presented in (Altman et al., 2009), we consider

two n-dimensional vectors δ(1), δ(2). Deﬁnition 1:

(Marshall et al., 2010) A vector δ(2) majorises δ(1),

which we denote as δ(1) ≺ δ(2), if δ(2) is more ”un-

regular” in the following fashion:











∑

i=1

[i]

(1) ≤

∑

i=1

[i]

(2),k = 1,2,...,n − 1

∑

i=1

[i]

(1) =

∑

i=1

[i]

(2)

(14)

where δ

[i]

(m) is a permutation of δ

(m) satisfying the

condition δ

[1]

(m) ≥ δ

[2]

(m) ≥ ... ≥ δ

[n]

(m),m = 1,2

Equation (8) suggests that the largest element of

δ(2) is larger than the largest element of δ(1). Conse-

quently, the smallest element of δ(2) is smaller than

the smallest element of δ(1). Thereafter we proceed

in Schur convexity properties of majorisation.

Deﬁnition 2:: A function f : R

→ R is Schur con-

cave if δ(1) ≺ δ(2) suggests f(δ(1)) ≥ f(δ(2)). f is

Schur convexif the inequality suggests that f(δ(1)) ≤

f(δ(2)).

Deﬁnition 1 dictates that there is strong majori-

sation; however, at least one of the inequalities of

(8) is strict. Furthermore, Proposition C.2 of (Mar-

shall et al., 2010) dictates that a function f : R

→ R

that is symmetric and convex(concave), is also Schur-

convex (concave). Hence, we need to show that our

potential function is Schur-concave, in order to pro-

ceed with the majorisation properties.

Lemma 1. Function V is concave in N

Proof. It is obvious that the function is concave, since

if we take the second derivative test the ﬁrst term

will be set to 0 and the second term is a concave

term (raised to power) for medium to large SINR val-

ues. Note that for very high SINR values the second

derivative of (10) become positive and the function

becomes convex as we can deduct from (Meshkati

et al., 2006).

Proposition 2. If the function u(p) is concave then

the function V(p) is Schur concave.

Proof. The proof is given by using the following

corollary from (Marshall et al., 2010).

Corollary 5.0.1. Let φ(x) =

∑

i=1

g(x) where g is con-

cave (convex). Then φ is Schur-concave (convex)

Theorem 5.1. The Game G reaches the global opti-

mum via the potential function V(p) maximisation.

Proof. Recall that the potential V(p) Schur concave

and it satisﬁes the LMP. It follows that if p

∗

is a Nash

equilibrium strategy, then it maximises the potential

and is the global maximum. Assume that there is an-

other strategy proﬁle p

′∗

that maximises the poten-

tial and is the global maximum. This means by p

∗

majorises p

′∗

. Since V(p) is Schur concave it fol-

lows by deﬁnition that V(p

′∗

) ≥ V(p

∗

). Since, p∗

maximises the potential, this is only possible when

V(p

′∗

) = V(p

∗

). Hence, p

∗

is the global optimum.

This also comes as a result of the fact that we

have shown that there is a critical point in the function

SENSORNETS 2017 - 6th International Conference on Sensor Networks

V(p). It follows from (Jorswieck and Boche, 2006) -

Theorem 2.22 - that the critical point p

∗

is the global

optimum.

Notably, Schur concavity of V not only allows us

to capture the optimal policies, but it allows the com-

parison of the performance of two non-optimal strate-

gies, whenever one of the policies majorises the other.

Theorem 5.2. The price of stability of the game is 1

Proof. It follows from the previous theorem that

shows that the game reaches the global optimum.

Thereafter, we will proceed with the derivation

of the Price of Anarchy (PoA) (Nisan et al., 2007),

in order to further check the optimality of the game.

Firstly, though, we start with the following result.

Deﬁnition 5.1. (Pareto efﬁcient) (Myerson, 1991)

A strategy proﬁle (p

OPT

, p

OPT

−i

), is considered to

be strongly Pareto efﬁcient if and only if there

exists no other strategy proﬁle (p

, p

−i

) such that

, p

−i

) ≥ u

OPT

, p

OPT

−i

),∀i ∈ N and u

, p

−i

) >

OPT

, p

OPT

−i

) for at least one node m. On the other

hand, a strategy proﬁle (p

OPT

, p

OPT

−i

) is weakly Pareto

efﬁcient if and only if there exists no strategy proﬁle

, p

−i

) such that u

, p

−i

) > u

OPT

, p

OPT

−i

),∀i ∈

N. We use the term Pareto efﬁcient for both weak and

strong cases.

Deﬁnition 5.2. A pure strategy NE is a Pareto efﬁ-

cient pure strategy NE if it is Pareto efﬁcient.

Theorem 5.3. A maximizer of V, which coincides

with the optimal solution of (11), is a Pareto efﬁcient

pure strategy NE.

Proof. We have shown previously that the game G

reaches the maximum which is a pure strategy NE.

Hence (p

OPT

, p

OPT

−i

) constitutes an optimal solution

of (11). There is no other strategy that maximises

the potential. That is that there is no strategy proﬁle

,...p

) ∈ P

i∈N

, such that

,...p

) = V(p

,...p

) > u

OPT

, p

OPT

−i

)

= V(p

OPT

, p

OPT

−i

),∀i ∈ N (15)

Thus, considering Deﬁnition 5.2, (p

OPT

, p

OPT

−i

) is

Pareto efﬁcient. Moreover, let us assume the ∀i ∈ N,

is an alternative strategy of node/player i, where

6= p

OPT

. Then, we obtain

, p

OPT

−i

) ≥ u

OPT

, p

OPT

−i

) (16)

We see that there is no node that can unilaterally

change its transmission power/ strategy, in order to

increase its utility. Furthermore, the strategy pro-

ﬁle (p

OPT

,..., p

OPT

) is also a pure strategy NE. To

summarise, (p

OPT

,..., p

OPT

) is a Pareto efﬁcient pure

strategy NE.

Since, the game G may have more than one pure

strategy NEs, we will check the optimality of the NE

to show the relationship between the local optimal

NE and the Pareto efﬁcient NE. Even though we have

shown that the Game G goes to the global optimum,

we will strengthen this proof even further, by evaluat-

ing the ratio between the highest utility and the worst-

case NE, namely the PoA.

Theorem 5.4. PoA = 1, i.e. a pure strategy proﬁle of

G is Pareto efﬁcient.

Proof. We assume that p

OPT

is a Pareto efﬁcient NE.

Also, assume that p

∗

is an arbitrary pure strategy NE

∗

= (p

∗

, p

∗

−i

). Then for any arbitrary node/player i,

we have

∗

) = V(p

∗

) = c

∗

− TR

∗

(17)

Note that u

∗

) ≥ u

OPT

, p

∗

−i

) according to the

deﬁnition of a game. Therefore, we have

∗

) = V(p

∗

) = c

∗

− TR

∗

≥ u

∗

) = V(p

OPT

) = c

OPT

− TR

OPT

(18)

Furthermore , since we have assumed that p

OPT

a Pareto-optimal pure strategy NE, ∀i ∈ N

V(p

OPT

) ≥ V(p

∗

) (19)

Combining (18) and (19) we have V(p

OPT

) ≥

V(p

∗

),∀i ∈ N. Hence, PoA = 1.

5.4 Algorithm Design

The TopGame algorithm is a cross-layer approach

that encapsulates information taken from the routing,

MAC and physical layers. In particular, the PRR and

neighbour information are obtained from the routing

layer, the transmission power used is acquired from

the radio and the MAC is responsible for triggering

the game to determine the topology in the case of

nodes failing or newly added to the network.

Initially, all nodes start communicating at their

maximum transmission power p

max

= 23. Node i col-

lects the neighbour information, such as current trans-

mission power levels used by its neighbours and their

respective PRR. This occurs simultaneously, since the

number of neighbours is determined via periodic bea-

cons being broadcast and the PRR obtained by unicas-

ting to random neighbours using a gossip-based pro-

tocol (Dammer and Hinrichsen, 2003). The nodes are

also synchronised using beacons with a ﬁreﬂy-based

(Breza and McCann, 2008) algorithm.

Node i iterates through its 4 available transmission

power levels, it computes TR for each power level and

Discrete Strategy Game-theoretic Topology Control inWireless Sensor Networks

it ﬁnally maximises its utility function u

. Note that

for practical reasons the pricing of each node’s utility

function is set to 1. The global optimum is accom-

plished as we can see in Theorem 5.1. Pseudocode of

TopGame is presented in Algorithm 1.

Algorithm 1: TopGame at node i.

Require: A

= {p

, p

,..., p

max

}

Require: degree = 6, p

= p

max

1: for i = 1 to N

i,p

2: get p

−i

,PRR

i,k

∈ N

3: N

← N

4: compute TR

5: u

, p

−i

) = c

∗ p

− TR

6: end for

7: ˆp

= argmaxu

(m)

In the case of the addition or a failure of node,

nodes that detect a change in their neighbourtable ini-

tiate TopGame from the start, since their TR will be

affected by the topological change. This is due to the

fact that TopGame is a repeated game only on topo-

logical alterations.

5.4.1 Message Overhead

The message overhead per transmission power con-

sists of the sum of the broadcast messages for syn-

chronisation and the unicast messages transmitted to

each of the neighbours of every node. That is,

TopGame

= f

sync

∗ N + N ∗ f

link

∗ m∗ M (20)

where N is the number of nodes, m is the window

of the unicast transmission to obtain PRR and M is

the number of neighbours of every node for a given

transmission power.

Since f

sync

= Θ(1), f

link

= Θ(N) and M = Θ(1),

provided some constants c

we have: f

sync

≤

, f

link

≤ c

∗ N,M ≤ c

. Therefore, 20 can be ex-

pressed as follows:

TopGame

≤ f

sync

∗ N + N ∗ f

link

∗ m∗ M

⇒ O

TopGame

≤ c

∗ N + c

∗ N ∗ m∗ c

⇒ O

TopGame

≤ N ∗ (c

+ m∗ c

∗ c

)

⇒ O

TopGame

= O(N)

6 EXPERIMENTAL EVALUATION

AND RESULTS

In order to evaluate TopGame, in comparison to ART,

we performed 120 minute experiments using 50 nodes

selected at random on Indriya. The data rate of the

nodes was 250 kbps and each node transmits 4 pack-

ets per second. In addition each node calculates its

PRR over a window of 8 packets. Each node ﬁnishes

iterating through all the transmission power levels and

the utility function of each power level has been ob-

tained in order to proceed with the maximisation.

Our aim is to show that converging to lower trans-

mission powers, may provide similar reception per-

formance. In this section we will provide results

that show that TopGame approachesART (Hackmann

et al., 2008) in terms of PRR and is slightly better in

energy-efﬁciency while ensuring that the network is

connected.

6.1 Performance and Energy

Consumption

Initially, we obtained the average PRR and relative

energy consumption, in order to evaluate both algo-

rithms globally. The network average PRR is pro-

vided in Figure 2 (a). Speciﬁcally, we observe that

TopGame exhibits an average network PRR of 44.7%,

while ART 48.1%. Moreover, the standard deviation

of ART is higher than TopGame’s by 3%.

The difference in the PRR between the two

schemes is not quite signiﬁcant; However, we have

shown that a game theoretic algorithm, with a more

systematic approach, exhibits similar performance

with a state-of-the-art practical algorithm such as

ART. Further, the formation of less links does indi-

cate that TopGame uses its utility function that ﬁnds

the sweet spot per node. On the other hand, ART ﬂuc-

tuates between its two packet failure thresholds; thus,

forming more links. As we have seen in a previous

section, contention is related to the number of neigh-

bours (K) of each node. Table 2 presents the average

degree of the network and the number of links that are

formed with TopGame and ART.

Table 2: Average K and formed links.

Average K Number of Links

ART 7 360

TopGame 7 338

In order to examine whether the difference in the

PRR average of TopGame and ART is a result of

channel collision we performed 2 hour experiments

measuring the Clear Channel Assessment (CCA) fail-

ures. Figure 2 (c) presents the CCA failures ratio of

TopGame and ART. Brieﬂy, a CCA operation occurs

when the MAC layer receives a packet to transmit,

then it instructs the physical layer to check channel

availability (CCA) in two consecutive slots. If the

channel is found to be available in both slots, the node

proceeds with its transmission. Otherwise, the node

SENSORNETS 2017 - 6th International Conference on Sensor Networks

(a) Average PRR

(b) Average Relative Energy

Consumption

Figure 2: TopGame and ART average Relative Energy, Mean PRR and CCA failures Mean of 50 nodes.

attempts CCA again after a random back-off, which it

repeats a certain number of times and it calls a failure

of access to the upper layer. Hence, with TopGame

exhibiting nearly 10% more CCA failures than ART,

it is natural to assume that the difference in average

PRR comes from a higher interference and collisions

of TopGame of the bootstrapping period, since it ini-

tially forms a larger number of links that are included

in the graph. Speciﬁcally, TopGame exhibits 43%

failures, while ART’s percentage is 33.6%.

In our experiments we were unable to directly

monitor the energy consumed by the listening and

transmitting periods of each node. Thus, we decided

to use unicast communications as an indicator to cal-

culate the relative average energy. Making the as-

sumption that all nodes spend the same amount of en-

ergy in listening, to get a rough idea of relative energy

consumption, we added the number of unicast mes-

sages transmitted by ART and TopGame with their

respective transmission powers and multiplied them

with the corresponding mA radio energy consump-

tion. The relative energy consumption of the two al-

gorithms can be seen in Figure 2 (b). TopGame con-

sumes 5% less energy than ART, including the boot-

strapping period.

This is due to the fact that the nodes do not ﬂuc-

tuate on a per packet basis and they are not targeting

a very high PRR value as dictated in the ART thresh-

olds; hence, TopGame is slightly more energy efﬁ-

cient. We present an example of two links from both

TopGame and ART, in order to show the difference in

the switching between transmission powers and the

convergence of TopGame. From the ﬁgures 3 and 4,

it is clear that ART switches its transmission power

according to packet drops; hence, the Tx ﬂuctuation

in the ﬁgure. On the other hand, TopGame collects

TR for each available transmission power and con-

verges to the transmission power maximizing the util-

ity function. Also, we are not aware of the energy cost

Figure 3: ART and TopGame Node 13 Tx levels.

of the continuous Tx switch. We assume it is negli-

gible. Note that TopGame is repeated only when a

neighbourhood change is detected.

Recall that ART’s intention is to reach the target

PRR of 95%, yet we observe that its reception qual-

ity is signiﬁcantly lower. TopGame also does not at-

tain this lofty ﬁgure. We believe that is the case for

our scheme because of the bimodal distribution of

802.15.4 link qualities (Srinivasan et al., 2007).

By looking at the Cumulative Density Functions

(CDF) of the two algorithms in ﬁgure 5, we observe

that TopGame has a slightly higher probability of

forming poorer quality links of PRR lower than 20%.

ART has a lower probability of forming medium to

high quality links. Furthermore, TopGame exhibits a

sightly higher probability of establishing links with

PRR over 80%. It would be strong to claim that

TopGame is better than ART; however, approaching

the numbers of ART is signiﬁcant, since it relies in

concrete theoretical basis.

Finally, we compared the RAM and ROM over-

head of TopGame with ART. Table 3 shows that

Discrete Strategy Game-theoretic Topology Control inWireless Sensor Networks

Figure 4: ART and TopGame Node 41 Tx levels.

TopGame consumes 2348 more ROM bytes than

ART, while it produces an overhead in RAM of 342

bytes.

Table 3: RAM and ROM (bytes).

RAM ROM

TopGame 3426 25228

ART 3084 22880

6.2 Connectivity

After we obtained the results we evaluated connectiv-

ity ofﬂine using method that determines whether the

resulting graph is connected. This was to evaluate the

connectivity model we discussed in a previous sec-

tion. We have shown that the average degree of each

node is greater than 6 nodes. We derived the avail-

able links from TopGame and ART’s data sets and we

created their respective adjacency matrices. There-

after, we used the matrices to ﬁnd a zero eigenvalue.

In the case that the corresponding eigenvector has 0s,

then a sum of non-zero number of rows/columns of

the adjacency is 0 (Horn and Johnson, 2005). Hence,

the degrees of these nodes are 0 and the graph is dis-

connected. Both TopGame and ART resulted in fully

connected graphs.

7 CONCLUSIONS

Compared with the state of the art protocol ART, we

showed that TopGame is a general solution provid-

ing efﬁcient robust Topology Control minimising the

costs of communications while ensuring connectivity.

First, we evaluated ART and TopGame on Indriya us-

ing 50 nodes to determine the average PRR and the

average relative TX power. We evaluated connectiv-

ity based on a method that checks the eigenvector of

Figure 5: CDF for ART and TopGame.

each algorithms adjacency matrix (links) and deter-

mined that the resulting graph is connected. The ex-

periments on Indriya showed that TopGame reduces

power consumption compared with ART without sig-

niﬁcantly degrading link quality. Macro-benchmarks

comparing TopGame to ART protocol indicated that

TopGame provides guaranteed connectivity and ex-

hibited slightly lower PRR than its competitor and

5% improvement on energy consumption. The corre-

sponding Probability Density Functions showed that

TopGame has a slightly higher probability of hav-

ing links of low quality (< 20%) than ART. More-

over, ART has a lower probability of creating links

of > 20 − 80% PRR. Finally, TopGame has a better

probability of creating high PRR links (> 80%) This

is a promising factor of the comparison between the

two algorithms in terms of performance, since the av-

erage network PRRs were not signiﬁcantly different.

In terms of energy consumption, we presented results

that show that TopGame converges to lower transmis-

sion power levels than ART making it more energy-

efﬁcient.

The main differences between ART and TopGame

are that ART establishes per-link power levels while

TopGame establishes power settings for a given

neighbourhood of nodes, and thus can be seen as non

link-based. A node running ART will have to switch

between transmission powers to transmit packet to

its neighbours. This has an impact on the transmis-

sion power selection in larger networks, since the

target PRR (95%) is not reached and nodes select

high transmission powers. TopGame’s power is set

to cover the neighbourhood and therefore has no such

switching overhead. ART obtains data and makes de-

cisions by indirectly considering link asymmetry in

that; hence ART selected higher transmission power

levels. In fact, their unoptimised version not using

the contention gradient verify these phenomena and

SENSORNETS 2017 - 6th International Conference on Sensor Networks

even the improved version also shows a decrease in

PRR. However, link asymmetry is taken into account

in TopGame; where bi-directional information helps

to ensure both the connectivity of all nodes and that

we will converge at a Nash Equilibrium. Finally, in

terms of implementation, ART is closely coupled to

CTP whereas, though TopGame is slightly more ex-

pensive in terms of speed and footprint, it is agnostic

to WSN Operating System or stack implementations

and is therefore more generally applicable. We aim to

interface our approach with CTP or other state of the

art routing protocol such as the Backpressure Collec-

tion Protocol (BCP) (Moeller et al., 2010).

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