Game Theoretic Approaches to Analyzing Wireless Networks : Research Issues

1@ 2007, D. Saha, Indian Institute of Management Calcutta (IIM-C), India

WNMC’2007, Kolkata, IndiaMarch 10, 2007

Game Theoretic Approaches to Analyzing Wireless

Networks: Research Issues

Debashis Saha, Debashis Saha, PhDPhD

Professor, MIS & CS Group,

Indian Institute of Management (IIM), Calcutta, India

Joka, D. H. Road, Calcutta 700 104, India

[email protected]



Outline• Overview of Game Theory

• Players, actions, payoffs– Representations of games

• Wireless Networks (WiNets)• A case for applying game theory?

– Literature review

• Game Theoretic Models of WiNets• Four sample problems

– Analyses

• Concluding Remarks• Research Issues



Overview of Game Theory



What is a game?• A game is a structured or semi-structured activity,

usually undertaken for enjoyment and sometimes also used as educational tools. – Key components of games are goals, rules, challenge, and

interactivity. • [Ref: http://en.wikipedia.org/wiki/Game]

• A game is an interactive decision problem– “A game is a form of art in which participants, termed

players, make decisions in order to manage resources through game tokens in the pursuit of a goal.” (Greg Costikyan)



Is this a game?



Is this a game?



Is this a game?

• When is it a game?



What is Game Theory ?

• A Branch of Applied Mathematics

• It describes and studies interactive decision problems– Studies strategic interactions among rational

players, where players choose different actions in order to maximize their returns.



Game + Theory• Various types of games exist (e.g. card, board, sport,

war, etc.)• Game Theory deals with games having the following

properties:– Two or more players– Choice of action involves a strategy– One or more outcomes– Outcome depends on the chosen strategies: i.e., strategic

interaction• Rules out:

– Games of pure chance– Games without strategic interaction



Five Elements of a Game

1. Set of Players

2. Set of Actions

3. Set of Strategies

4. Set of Outcomes

5. Payoff or Utility

The basic notions of game theory include:• players (decision makers)• choices (feasible actions)• payoffs (benefits, prizes, rewards, etc)• preferences over payoffs (objectives)

Game theory is concerned with determining when one choice is better than another choice for a particular

player (strategy).



Modeling Wireless Network as Game

Wireless Network GameNodes

Power Levels

Algorithms

Players

Actions

Utility Functions+Learning

Structure of game is taken from the algorithm and the environment

void update_power(void){

/*Adjusting power level*/int k;

}

[Laboratoire de Radiocommunications et de Traitement du Signal]



Actions

Ai – Set of available actions for player i

ai – A particular action chosen by i, ai Ai

A – Action Space, Cartesian product of all Ai

A=A1 A2· · · An

a – Action tuple – a point in the Action Space

A-i – Another action space A formed from

A-i =A1 A2· · · Ai-1 Ai+1 · · · An

a-i – A point from the space A-i

A = Ai A-i

A1= A-2

A2 = A-1

a

a1 = a-2

a2 = a-1

Example Two Player Action Space

A1 = A2 = [0 )

A=A1 A2

b

b1 = b-2

b2 = b-1



Utility Function

Maps action space to set of real numbers.

:iu A R

Also known as Objective Function or Payoff Functions

# Quantifying actions brings the problem into the domain of conventional mathematics

# After quantification, all sorts of valuable mathematical operations can be introduced.



Utility Functions [contd.]

# Note that the quantification operation is not unique as long as relationships are preserved.

# Many people map relationships to [0,1].

Example

Jack prefers Apples to Oranges

JackApples Oranges Jack Jacku Apples u Oranges

a) uJack(Apples) = 1, uJack(Oranges) = 0

b) uJack(Apples) = -1, uJack(Oranges) = -7.5



Example Game• Matching Numbers

– Allen and Brian (A & B)

– Each can choose to put out one finger or two fingers. • If they match, Allen gives Brian a dollar. If they’re

different, Brian gives Allen a dollar.

• Components– Player Set: N = {A,B}

– Action Sets: AA = AB = {1,2},

A = {{1,1},{1,2},{2,1},{2,2}}

– Utility Functions: 1

1A B

AA B

a au

a a

1

1A B

BA B

a au

a a



How to solve a game ?

• We would like to solve a game…

• Solving a game consists of – trying to predict the strategy of each player,

• considering the information the game offers and

• assuming that the players are rational.

• Several ways to solve a game. – Simplest way relies on strict dominance.



Iterated Dominance/Strict Dominance

• Strategy B strictly dominates A: choosing B always gives a better outcome than choosing A, no matter what the other player(s) do.

• Iteratively eliminate dominated strategies.

• Solution is always unique.



Iterated Dominance/Weak Dominance

• Strategy B weakly dominates A: There is at least one set of opponents' action for which B is superior, and all other sets of opponents' actions give A and B the same payoff.

• Solution depends on the sequence of eliminating weakly dominated strategies (due to multiple best responses).

• Might not result in a single solution profile.• Nevertheless, useful as to reduce the size of

the strategy space.



Prisoners' Dilemma (PD) Game

• Players: 2 Prisoners

• Actions: Prisoner 1: Confess, Deny Prisoner 2: Confess, Deny

• Strategies: Choose action simultaneously,

without knowing each other’s actions.

• Outcomes:Quantified in prison years

• Payoff:Fewer years == Better payoff

C

Deny? Confess?



Matrix Representation

• A matrix which shows the players, strategies, and payoffs.

• Presumed that players act simultaneously.

• PD example:P2 Confess

P2 Deny

P1 Confess 5, 5 0, 10

P1 Deny 10, 0 1, 1



Multi-access Protocols & PD Games

• MAC protocols (carrier sense)– Multiple nodes contend for the shared medium

• Can it be modeled as PD game?

• Say there are 2 stations– Like 2 prisoners

• Both sense channels

• If both transmits together, there is collision– Equal loss for both

• If one only transmits, it gets maximum payoff– Transmission is successful

• If both back off, channel remains idle



Matrix Representation

• A matrix for 2 nodes, their strategies and payoffs.

• Presumed that nodes act simultaneously.

Node2 back-off Node2 transmit

Node1 back-off 0, 0 0, 10

Node1 transmit 10, 0 -5, -5



Basic game theory models

• Normal form game– Also known as Strategic form game

• Do not capture sequencing– Simultaneous moves by all players

• Extensive form game– Game tree

• Captures “time”; players move in sequence– Information is available to latter players

• Repeated game

• Evolutionary game



Normal Form Games (Strategic Form Games)

, , iG N A u

In normal form, a game consists of three primary components (3-tuple)

N – Set of PlayersAi – Set of Actions Available to Player i

A – Action Space {ui} – Set of Individual Objective Functions

:iu A R

1 2 nA A A A



Representation of Games: Normal (Strategic) Form

• A matrix which shows the players, strategies, and payoffs.

• Presumed that players act simultaneously.

• PD example: P2 Confess

P2 Deny

P1 Confess 5, 5 0, 10

P1 Deny 10, 0 1, 1



Nash Equilibrium• To identify best responses, John Nash introduced the

concept of Nash Equilibrium.

• In a Nash equilibrium, none of the users can unilaterally change their strategy to increase their utility.

• Note: Any solution derived by iterated strict dominance is Nash equilibrium.



Normal form game: NE

1. A set of 2 or more players2. A set of actions for each player3. A set of utility functions that describe the players’

preferences over the action space

Components

a

b

A B

1,-1

-1,1

0,2

2,2

Player 1 Actions {a, b}Player 2 Actions {A, B}

States from which no playercan unilaterally deviate andimprove are Nash Equilibriums



Nash Equilibrium Solution Methods

• Direct Application of Definition

• Improvement Deviations

• Iterative Elimination of Dominated Strategies

• Best Response Analysis



Game Tree/Extensive Form• The games can be static or dynamic.

– In dynamic games the order of the moves/choices is important

• Here a game tree is a better representation than normal form

• Consider a simple game as this:– Player 1 chooses H or T– Player 2 chooses H or T (not knowing what Player 1

chooses).– If both choose the same Player 2 wins $1 from Player 1.– If they are different, Player 1 wins $1 from Player 2.

• We can draw this in extensive form as shown next– It can be shown in normal form too (like PD game)



Extensive Form



Extensive form games

1. A set of players.2. The actions available to each player at each decision

moment (state).3. A way of deciding who is the current decision maker.4. Outcomes on the sequence of actions.5. Preferences over all outcomes.

Components

Backward induction is a technique to solve a game tree of perfect information. It first considers the moves that are the last in the game, and determines the best move for the player in each case. Then, taking these as given future actions, it proceeds backwards in time, again determining the best move for the respective player, until the beginning of the game is reached.



Types of Games

• Symmetric and Asymmetric• Zero Sum and Non-Zero Sum• Simultaneous and Sequential• Perfect Information and Imperfect Information



Symmetric and Asymmetric

• Any game in which the identity of the player does not change the resulting game facing that player, is symmetric.

• E.g. prisoners’ dilemma, game of chicken, and battle of the sexes.

• General form:P2 L

P2 R

P1 U a, a b, c

P1 D c, b d, d



Zero Sum and Non-Zero Sum

• In zero sum game the total benefit to all players in the game, for every combination of strategies, always adds to zero.– i.e. A player benefits only at the expense of others.

• E.g. poker, chess, matching pennies.

Matching Pennies

P1/P2 Head Tail

Head 1, -1 -1, 1

Tail -1, 1 1, -1



Simultaneous and Sequential

• Simultaneous (a.k.a static games) games are games where both players move simultaneously.

• Sequential games (a.k.a dynamic games) are games where later players have some knowledge about earlier actions.



Information Perfection

• A game is one of perfect information if all players know the moves previously made by all other players. – E.g., chess

• Head/Tail game:– If Player 1 reveals his

choice before Player 2 chooses

• Here Player 2 wins always



Information Completeness• Often confused with Information Perfection• Complete information means that the players

know each element in the game definition:1. Who the other players are2. What their possible strategies are3. What payoff will result for each player for any

combination of moves



Wireless Networks &

Game Theory



Related Works• Game theory approaches have been used for

optimization and control of wireless networks [1]-[2]– as an alternative to traditional network optimization

[1] T. Alpcan, T. Basar, R. Srikant, E. Altman, ‘CDMA uplink power control as a noncooperative game’, in Proc. 40th IEEE Conf. Decision and Control, 197-202, 2001.

[2] C. Saraydar, N. B. Mandayam, and D.J. Goodman, ‘Efficient power control via pricing in wireless data networks’, IEEE Trans. Communication, vol. 50 (2), 291-303, 2002.



Related Works [contd.]

• In wireless communications literature, the tragedy of commons problem has typically been addressed by modeling power consumption (explicit energy) as a direct cost to the users [2]– A Nash equilibrium solution is obtained when users have an interest in

maximizing their own utility, defined as the ratio of rate to power• This approach, however, results in a sub-optimal resource allocation.

• This work has recently been extended [3] to capture the affect of receiver design where the authors show that receiver design can be used to induce a more efficient Nash equilibrium

[2] C. Saraydar, N. B. Mandayam, and D.J. Goodman, ‘Efficient power control via pricing in wireless data networks’, IEEE Trans. Communication, vol. 50 (2), 291-303, 2002.

[3] F. Meshkati, H. V. Poor, S. Schwartz, and N. Mandayam, “An energy efficient approach to power control and receiver design in wireless data networks,” IEEE Transactions on Communications, vol. 53, no. 11, pp. 1885–1894, November 2005.



M – the set of decision making radiosEi – the set of possible energy levels available to radio i ei – the energy level chosen by ie - the tuple of chosen energy levels of all radios in the networki – the set of signature waveforms available to radio i i – the chosen waveform of i - the tuple of chosen waveforms of all radios in the networkNi – noise power at node iij - the correlation between the signature waveform sequences of radios i and j. Note that ij necessarily equals ji.

Physical Layer Model



Radio Model

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G M A u

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eu a f c e

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i,j – path loss from i to j

vi – node of “interest” for node i



Some Wireless Network Games• Consider 2 rational players (i.e., nodes)

• Games correspond to the protocol stack

• The four games:1. Forwarder’s Dilemma

2. Joint Packet Forwarding

3. Multiple Access

4. Jamming



Forwards’ Dilemma

P2 Forward

P2 Drop

P1 Forward 1-c, 1-c -c, 1

P1 Drop 1, -c 0, 0

• Symmetric nonzero-sum



• Asymmetric, nonzero-sum

Joint Packet Forwarding

P2 Forward

P2 Drop


P1 Drop 0, 0 0, 0



Multiple Access• P1 and P2 are in the same transmission range,

and will interfere if they were to transmit at the same time.

• Should they transmit during a timeslot or stay quite?

• Symmetric, nonzero-sum

P2 Quite P2 Tx

P1 Quite 0, 0 0, 1-c

P1 Tx 1-c, 0 -c, -c



Jamming• Two channels

• Sender tries to send a packet on one of the channels

• Jammer tries to jam the sender

• Asymmetric, zero-sum

Jammer CH1

Jammer CH2

Sender CH1 -1, 1 1, -1

Sender CH2 1, -1 -1, 1



Game Theory Analyses



Steps in application of game theory• Develop a game theoretic model

– Solution of game’s Nash equilibrium yields information about the steady state and convergence of the network

• Does a steady state exist?– Uniqueness of Nash equilibrium

• Is it optimal?

• Do nodes converge to it?

• Is it stable?

• Does the steady state scale?



• Dropping strategy dominates Forwarding for both players.

• (D,D) is the solution!

Iterated Dominance/Strict Dominance

Forwards’ DilemmaP2

ForwardP2

Drop


P1 Drop 1, -c 0, 0


ForwardP2

Drop


P1 Drop 1, -c 0, 0


ForwardP2

Drop


P1 Drop 1, -c 0, 0




• Strict dominance can not be used to solve every game.

• If P1 drops, P2 is indifferent

Joint Packet ForwardingP2

ForwardP2

Drop


P1 Drop 0, 0 0, 0




• For P2, Dropping strategy is weakly dominated by Forwarding.

• Solution is the strategy profile (F,F)


ForwardP2

Drop


P1 Drop 0, 0 0, 0


ForwardP2

Drop


P1 Drop 0, 0 0, 0


ForwardP2

Drop


P1 Drop 0, 0 0, 0



Nash Equilibrium

• Majority of games can not be solved by iterated dominance techniques.

• No strategy is dominated in this game.

Multiple AccessP2 Quite P2 Tx

P1 Quite 0, 0 0, 1-c

P1 Tx 1-c, 0 -c, -c



Nash Equilibrium/Best Response

• (T,Q) and (Q,T) are mutual best responses to each other.

• No player has a reason to deviate from the given strategy profile.

Multiple AccessP2 Quite P2 Tx

P1 Quite 0, 0 0, 1-c

P1 Tx 1-c, 0 -c, -c



Concluding Remarks



Research Issues in Network Layer

• Node participation versus Energy saving (battery life)– Switching interfaces to a sleep state affects network

operations• Network partition• Network congestion

– Individual benefits• Increased lifetime of nodes (inversely proportional)• Increase in throughput by participating (directly proportional)

– Individual losses• Loss of information for an ongoing session• Overhead involved in discovering location of other nodes on

waking up• Extra flow of route queries due to frequent topology changes



Transport layer

• Analyze congestion control algorithms for selfish nodes– Objective: Determine the optimal congestion

window additive increase and multiplicative decrease parameters

– Current efforts restricted to traditional TCP congestion control algorithms for wired networks

• Sensor networks– Incorporate the characteristics of the wireless

medium in the congestion control game



Other Open Problems• Coverage games

– Sensors collaborate to determine which sensors will remain at full power to provide necessary coverage

• Games of correlated equilibria– Beacons provide common basis for decision,

• symmetry breaking

• Games of population transition– Game based on movement into and out of defined

population• Basis for distributed detection is the reach-back through mobile

access points



Summary

• Covered some of the basics of game theory.

• Examined some simple examples that capture wireless networking problems and analyzed them to predict the behavior of players.

• Focused on non-cooperative games with complete information to address sensor network issues.



Discussion• Humans are not fully rational.

• Devices however can be modeled as rational decision makers.

• Modeling utility functions and costs correctly is difficult.

• Completeness of information, observable actions, and perfect recall may not be guaranteed in wireless networks.



References [Game Theory]• K. Dutta, Strategies and Games, Theory and Practice, MIT

Press, Cambridge Massachusetts, 1999.• D. Monderer and L. Shapley, “Potential games”, Journal of

Games and Economic Behavior, vol. 14, no. 0044, Pages: 124-143, 1996.

• “Game Theory”, http://en.wikipedia.org/wiki/Game_theory, Wikipedia – The Free Encyclopedia, 2006.

• “Game Theory”, http://plato.stanford.edu/entries/game-theory, Stanford Encyclopedia of Philosophy

• Rahul Garg, “An Introduction to Game Theory”, http://www.cse.iitd.ernet.in/~rahul/cs905, 2003.

• John Duff, “Introduction to Game Theory”, http://www.pitt.edu/~jduffy/econ1200/Lectures.htm, 2003.



References• A. Akella et al., “Selfish behavior and stability of Internet: A game theoretic

analysis of TCP,” Proceedings of ACM SIGCOMM Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications, August 2002, pp. 117-130.

• Robert Axelrod, “The Evolution of Cooperation,” Basic Books, Reprint edition, New York, 1984.

• L. Buttyan and J. P. Hubaux, “Nuglets: A virtual currency to stimulate cooperation in self organized mobile ad-hoc networks,” Swiss Federal Institute of Technology, Lausanne, Switzerland, Report no. DSC /2001/001, January 2001.

• S. Buchegger and J.Y. Le Boudec, “Performance analysis of the CONFIDANT protocol: cooperation of nodes – fairness in dynamic ad-hoc networks,” Proceedings of ACM MobiHoc, June 2002.

• Competitive routing in multiuser communication networks, A. Orda, R. Rom and N. Shimkin, IEEE/ACM Transactions on Networking, 1 (5) 1993

• How bad is selfish routing?, T. Roughgarden and E. Tardos, Journal of the ACM, 49 (2) 2002

• Selfish routing with atomic players, T. Roughgarden, ACM/SIAM Symp. on Discrete Algorithms (SODA) 2005



Thank You !!!



Any Doubt ?

• Please feel free Please feel free to write to me:to write to me:

[email protected]@iimcal.ac.in

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Game Theoretic Approaches to Analyzing Wireless Networks : Research Issues