Game Theory for Wireless Communications -...

IEEE International Symposium on Personal, Indoor andMobile Radio Communications

2008

Game Theory forWireless Communications

Merouane Debbah (*) and Samson Lasaulce (**)(*) Alcatel-Lucent Chair on Flexible Radio

(**) LSS (CNRS–Supelec–Paris 11)[email protected], [email protected]

15 September 2008

Outline

PART I

1. Introduction2. Historical Development of Game Theory3. Static Games4. Dynamic Games5. Optimization Results for Game Theory6. Wireless Communication Metrics for Game Theory

PART II

1. Shannon Rate Efficient Power Allocation Games2. Energy Efficient Power Control Games3. Handover Games4. Medium Access Control Games5. Concluding Remarks6. References and Reading

1

Tutorial: Game Theory for Wireless Communications

PART I

2

Presentation

1. Introduction

3

4G Mobile Internet

Uplink: Up to 7 Mbps/TerminalDownlink: Up to 15 Mbps/Terminal

4

The spectral efficiency barrier: energy and bandwidth

C = W log(1 + SNR)

SNR = CEb

N0

limC→0

Eb

N0=

2C − 1

C= ln(2)

The two degrees of freedom (Energy and Bandwidth) are the barrier limit ofcommunication.

There is minimum energy to transmit a natural unit of information over a noisy channel.

5

Is there a limit to spectral efficiency?

”As the number of users in the network increases, interference becomes the

bottleneck”...the resources must be shared...

6

Mobile Flexible Networks: from b/s/Hz to b/s/Hz/m2

Mobile Flexible Networks do not consider wireless resources as a ”cake to be shared”among users but take benefit of the high number of interacting devices to increase thespectral efficiency frontier.

More devices represent more opportunities to schedule information which enhances theglobal throughput.

Interference is considered as an opportunity rather than a drawback by exploitingintelligently the degrees of freedom of wireless communications:

• Space (MIMO Network)• Frequency/time (Cognitive Network)• User (Opportunistic Network).

7

MIMO Network

”We build to many walls and not enough bridges”, Isaac Newton

Combat interference through frequency/space reuse or power control.

Exploit interference through coordination and cooperation.

8

MIMO Network

In theory, the only limiting factor for increasing the spectral efficiency is the number ofbase stations.

The technology, adequate for dense networks, relies on sophisticated tools of dirty paper

coding and cooperative Beam-forming.

9

Cognitive Networks

Cognitive networks coordinate the transmission over various bands offrequencies/technologies by exploiting the vacant bands/technologies in idle periods.

It requires base stations able to work in a large range of bandwidth for sensing thedifferent signals in the network and reconfigure smartly.

Example: They will sense the different technologies, the energy consumption of the

terminals and reconfigure (changing from a GSM to UMTS base station if UMTS terminals

are present) to adapt to the standards or services to be delivered at a given time.

10

Opportunistic Networks

11

Opportunistic Networks

As the number of users increase, the spectral efficiency of opportunistic network

increases, as the probability of having a user with a good channel increases with the

number of users.

12

Flexible Networks: a Shannon historic perspective

THREE LANDMARK PAPERS FROM Bell-Labs

1948 ”A Mathematical Theory of Communication”, C. Shannon, Bell System TechnicalJournal, Vol. 27 (July and October 1948), pp. 379-423 and 623-656.

1949 ”Communication Theory of Secrecy Systems”, C. Shannon, Bell System TechnicalJournal, Vol. 28 (1949), pp. 656-715.

1950 ”Programming a Computer for Playing Chess”, C. Shannon, Philosophical Magazine,Series 7, Vol. 41 (No. 314, March 1950), pp. 256-275

13


14


15


16


17


18


Claude Shannon and his mouse Theseus, in one of the first experience on artificial

intelligence in 1950. Shannon’s mouse appears to have been the first learning device of

this level. Like the learning mouse, flexible radio intends to build learning networks, just on

a much larger, and more complex scale.

19

The big dilemma

One of the most challenging problems in the development of this technology is to managecomplexity. The key is to develop:

• The right abstractions to reason about the spatial and temporal dynamics of complexsystems

• Understand how information can be processed, stored, and transferred in the systemwith bounded delay.

We refer to this as the ”theoretical foundations of Mobile Flexible Networks”.

20

Interdisciplinary Research

The research is highly interdisciplinary and is a blend of

• Statistical inference methods to build devices which would carry plausible reasoning(Maximum Entropy methods,..).E. T. Jaynes, ”Probability Theory: The Logic of Science”, Cambridge, 2003.

• Random matrix theory techniques to reduce the dimensionality of the problem i.e findthe parameters of interest in a network rather than optimizing through simulations with1 billion parameters.V.A. Marchenko and L.A. Pastur, ”Distribution of Eigenvalues for Some Sets of RandomMatrices”, Math USSR Sb, 1967, vol.1, pp. 457–483.

• Game theoretic techniques (based on rational players) to promote decentralized/adaptiveresource allocation schemes.J. von Neumann and O. Morgenstern, ”Theory of Games and Economic Behavior”,Princeton, NJ: Princeton University Press, 1944.

21

Interdisciplinary Research

The research is highly interdisciplinary and is a blend of

• Control theory with the use of feedback mechanisms in the realm of cybernetics.N. Wiener, ”Cybernetics”, John Wiley, 1948.

• Physics to study how information can be processed, stored, and transferred in thenetwork.C. H. Bennett and R. Landauer, ”Fundamental Physical Limits of Computation”,Scientific American 253:1 48-56, July 1985.

• Network Information theory to understand the fundamental limits achievable withintelligent devices.R. G. Gallager, ”Basic Limits on Protocol Information in Data Communication Networks”,IEEE Transactions on Information Theory, 22, 385-399, 1976

22

New Mathematical Tools

• Statistical Physics• Convex Optimization• Random Matrix theory

23

Intelligent Game Networks

From a game perspective, where and how to split the intelligence...which is the purpose ofthe course.

24

Presentation

2. Historical Development of Game Theory

25

Game Theory

26

Game Theory

Theme of the Academy Award-winning film ”A Beautiful Mind” inspired by the Nobel Prize

(Economics) winning mathematician John Nash.

27

Some notions on Game theory

What is Game Theory?

Basically, Game Theory is the mathematics of strategy.

Game Theory aims to help us to understand situations in which decision makers interact.

The primary theory is the Minimax Theorem which basically says that if all the players of agame play the best, most rational strategy, the resulting outcome of the game ispredictable.

28

Limitations of Game theory

Is it useful?

Players are considered rational and should determine what is the best

Game Theory aims to help us to understand situation in which decision makers interact.

The primary theory is the Minimax Theorem which basically says that if all the players of agame play the best, most rational strategy, the resulting outcome of the game ispredictable.

29

Applications of Game Theory

• Game theory and Economics (Auman, Schelling (Nobel Prize in 2005)).• Game Theory and War (Cold war, war on terrorism).• Game theory and Philosophy (morality from self-interest).• Game theory and Social Science (Explanation for the democratic peace).• Game theory and biology (Evolutionnary Stable Strategy introduced by J. M. Smith in

1982) .• Game theory and computer science (modelling interactive computations).

30

Game Theory and Networking

Previous applications of game theoretical tools to all areas of networking.

• Security issues modeled as zero-sum games (Kodialam et al., 2003).• Non-zero sum games in the context of access to a shared wireless channel (Altman et

al., 2004, MacKenzie et al., 2003).• Evolutionary games (Bonneau et al., 2005, Menasche et al., 2005).• Nash Bargaining (Fang et al., 2005, Han et al., 2005).• Potential games (Sandholm et al., 2001, Shakkottai et al., 2006).• Match making games applied to output queueing in switches (Chuang et al., 1999).• S-modular games in the context of power control in wireless networks (Altman et al.,

2003).• Non-cooperative repeated games and dynamic games (Cao et al., 2002, Felegyhazi et

al., 2005).

First formulation of the communication process as a game:

N. M. Blachman, ”Communication as a Game”, in Proc. IRE WESCON conf., pages

61-66, August 1961.

31

Landmark papers

Antoine Augustin Cournot, ”Researches into the Mathematical Principles of the Theory ofWealth”, 1838

J. Von Neumann, ”Zur Theorie der Gesellschaftsspiele”, Mathematische Annalen 100,295-320, 1928

J. von Neuman and O. Morgenstern, ”Theory of Games and Economic Behavior”,Princeton, NJ: Princeton University Press, 1944.

J. F. Nash, ”Equilibrium points in n-points games” Proc. of the National Academy ofScience, Vol. 36, no. 1, pp. 48-49, January 1950.

J. F. Nash, ”The Bargaining problem”, Econometrica, vol. 18, no. 2, pp. 155-162, April1950.

J. F. Nash, ”Two-person cooperative games”, Econometrica, vol. 21, no. 1, pp. 128-140,January 1953.

32

The Birth of Game Theory

John von Neumann, 1903-1957

J. von Neumann made important contributions in quantum mechanics, functional analysis,informatics, economics as well as many other fields of mathematics and physics.

33

The Birth of Game Theory

Oskar Morgenstern, 1902-1977

Oskar Morgenstern worked on the application of mathematics to economy and laid with J.

Von Neumann the foundations of game theory.

34

Formal aspects of Game Theory

Number of players

• One player: Optimization, control theory.• Two players: Classical case and the best understood.• Many players: Rich and complicated due to coalitions.

Temporal structure

• Simultaneous: prisonner’s dilemma• Sequential: Chess.

35

Formal aspects of Game Theory

Random factors

• Deterministic: chess• Probabilistic: Stock market

Access to information

• Complete: chess• Incomplete: poker

36

Presentation

3. Static Games

37

Static games in Strategic Games

Strategic Games: A strategic game consists of:

• A set of players.• for each player, a set of actions.• for each player, preferences over the set of action profiles.

Example 1: the players may be firms, the actions prices and the preferences the firm’sprofit.

Example 2: the players may be candidates for political office, the actions campaignexpenditures and the preferences a reflection of the candidates’s probabilities of winning.

We consider for the moment only static games (the players have only one move as a

strategy).

38

Strategic Games: some notations

Players: Let P be the set of players. The subscript −i designates all the playersbelonging to P except i himself.

Remark: These players are often designated as being opponents of i. In a two player

games, player i has one opponent referred as j.

39


Strategy: Let S be the joint set of the strategy space of all players S = S1 ∗ ... ∗ SN . Si

corresponds to the pure strategy space of player i whereas the pure strategy of theopponents of player i is denoted by S−i = S/Si.

The set of chosen strategies constitutes a strategy profile s = s1, s2, ....

Remark: For each player, the strategy assigns zero probability to all moves except one (itclearly determines one move). The player could also use mixed strategies in the sensethat they choose different strategies with different probabilities.

40


Utility (or payoff function): Let ui(s) be the benefit of player i given the strategy profile s.In the two players case, we have U = u1(s), u2(s).

Remark: The payoff function represents a decision maker’s preferences in the sense thatif he prefers a ∈ S to b ∈ S then u(a) > u(b).

Remark: The decision’s maker’s preferences convey only ordinal information. They tell us

that he prefers the action a to the action b but not how much he prefers a to b.

41

Strategic Games: complete information

Definition: A game with complete information is a game in which each player has fullknowledge of all aspects of the game.

In particular, the players know:

• who the other players are.• what their possible strategies are• What payoff will result for each player for any combination of moves.

Remark: Complete information is different from perfect information. A game is of perfectinformation if all players know the moves previously made by all other players (applicationfor sequential games mainly). Complete information requires that every player know thestrategies and payoffs of the other players but not necessarily the actions!

42

Let us start: the prisoner’s dilemma

Context: Two suspects in a major crime are held in a separate cells.

• If they both stay quiet, each will be convicted of the minor offense and spend one yearin prison.

• If one and only one of them finks, he will be freed and used as a witness against theother who will spend four years in prison.

• If they both fink, each will spend three years in prison.

43

Modeling the game

Players: The two suspects

Actions: Each player’s set of actions is Quiet, Fink

Preferences: We need a function u1 such as:

u1(Fink, Quiet) > u1(Quiet, Quiet) > u1(Fink, Fink) > u1(Quite, Fink)

For example,

• u1(Fink, Quiet) = 3.

• u1(Quiet, Quiet) = 2.

• u1(Fink, Fink) = 1.

• u1(Quite, Fink) = 0.

44

Games and matrices

Suspect 2 Suspect 2Quiet Fink

Suspect 1 Quiet 2,2 0,3Suspect 1 Fink 3,0 1,1

• There are gains from cooperation (each player prefers that both players choose Quitethan they both choose Fink).

• However, each player has an incentive to ”free ride” (choose fink?) whatever the otherplayer does.

• In any case, for those who don’t understand game theory, YOU will always go to jail!

Question: How to solve it? In other words, how to predict the strategy of each player,

considering the information the game offers and assuming that the players are rational?

45

Methods to solve static games

• Iterated dominance.• Nash Equilibrium.• Mixed Strategies.

In all these techniques, we assume non-cooperative games. Cooperative games require

agreements between the decision makers and might be more difficult to realize (additional

signalization).

46

Illustration of the methods in basic examples

• Single hop Packet forwarding.• Joint packet forwarding.• Multiple access games.• Jamming games.

47

Single hop packet forwarding

Context: We consider two devices p1 and p2 wanting to send packets respectively to r1

and r2 in each time slot using the player as a forwarder.

• If player p1 forwards the packet of p2, it cost player p1 a fixed cost c (0 < c ≤ 1, c

represents the energy spent).• If he does so, he enables communication between p2 and r2, which gives p2 a reward

of 1.• What would be the pay-off?

Of course, each player is tempted to drop the packet to save resources. but the otherplayer would act the same way....

Question: Represent the Single hop packet dilemma forwarding through a matrix.

48


p2 p2

F Dp1 F (1-c,1-c) (-c,1)p1 D (1,-c) (0,0)

49

Joint packet forwarding

Context: A sender wants to send a packet to his receiver in each time slot. He needsboth devices p1 and p2 to forward for him.

• There is a forwarding cost c (0 < c ≤ 1) if a player forwards the packet to the sender.• If both players forward, then they each receive a reward of 1 (from the receiver or the

sender).• What would be the pay-off?

Question: Represent the joint packet dilemma forwarding through a matrix.

50


p2 p2

F Dp1 F (1-c,1-c) (-c,0)p1 D (0,0) (0,0)

51

Multiple Access Game

Context: Two players p1 and p2 want to send some packets to their receivers r1 and r2

using a shared medium.

• Players have a packet to send at each time slot and they can decide to transmit it or not.• Their transmission mutually interfere.• When p1 transmits, it incurs a cost of c.• The transmission is successful if p2 does not transmit. In this case, p1 gets a reward of

1 from the successful packet transmission.

Question: Represent the multiple access game through a matrix.

52


p2 p2

Q Tp1 Q (0,0) (0,1-c)p1 T (1-c,0) (-c,-c)

53

Jamming Game

Context: The objective of the malicious player p2 is to prevent player p1 from a successfultransmission by transmitting on the same channel in a given time slot. The objective of p1

is to succeed in spite of the presence of p2. We suppose that the wireless medium is splitinto two channel C1 and C2.

• The player receives a payoff of 1 if the attacker cannot jam.• The player receives a payoff of -1 if the attacker jams his packet.• We neglect the transmission cost as it applies to each payoff.

Question: Represent the multiple access game through a matrix.

54

Jamming Game

jammer jammerC1 C2

sender C1 (-1,1) (1,-1)sender C2 (1,-1) (-1,1)

55

Zero and non-zero sum games

The Single hop packet forwarding, the joint packet forwarding and the multiple accessgames are non-zero sum games as the players can mutually increase their payoff bycooperation.

Where is the conflict?

• Single hop packet forwarding: they have to provide the packet forwarding service foreach other.

• Joint packet forwarding: they have to establish the packet forwarding service.• Multiple Access Game: They have to share a common resource.

The Jamming game is a zero sum game since the reward of one player represents theloss of the other player.

56

Iterated Strict Dominance

Definition Strategy s′i of player i is said to be strictly dominated by his strategy si if

ui(s′i, s−i) < ui(si, s−i)∀s−i ∈ S−i

Exercise: Apply the technique of iterated strict dominance to the Single hop Packetforwarding and the packet forwarding game.

57

Iterated weak Dominance

A weaker condition is often needed.

Definition Strategy s′i of player i is said to be weakly dominated by his strategy si if

ui(s′i, s−i) ≤ ui(si, s−i)∀s−i ∈ S−i

with strict inequality for at least one s−i ∈ S−i.

The solution of iterated strict dominance is unique, which is not the case for the weakdominance case.

These techniques are useful to reduce the size of the strategy space.

Exercise: Apply the technique of iterated weakly dominance to the packet forwardinggame and to the multiple access game.

58

Best response framework

In the multiple access game,

• If player p1 transmits, then the best response of player p2 is to be quiet.• If player p2 is quiet, then p1 is better off transmitting a packet.

Definition Denote bi(s−i) the best response of player i to the opponent’s strategy vectors−i. The best response bi(s−i) of player i to the profile of strategies s−i is a strategy si

such that:

bi(s−i) = argmaxsi∈Siui(si, s−i)

One can see that if two strategies are mutual best responses to each other, then no playerwould have a reason to deviate from the given strategy profile.

This is the start of the Nash equilibrium

59

The Birth of the Nash Equilibrium

J. F. Nash, ”Non-Cooperative Games”, Doctoral Dissertation, Princeton University, 1950.

John Forbes Nash, Jr., 1928-

60

Nash Equilibrium

Definition The pure strategy profile s∗ constitutes a Nash equilibrium if, for each player i,

ui(s∗i , s

∗−i) ≥ ui(si, s

∗−i), ∀si ∈ Si

Expressed differently, a Nash equilibrium embodies a steady stable ”social norm”: ifeveryone else adheres to it, no individual wishes to deviate from it.

Remark: It can be shown that a solution derived by iterated strict dominance is a Nashequilibrium.

Exercise: What is the Nash equilibrium for the Prisonner’s dilemma?

Exercise: Show that the Jamming game:

• Cannot be solved by iterated strict dominance.• has no pure strategy Nash equilibrium.• Can we find an equilibrium of another nature in this case?

61

Mixed Strategies

Players can decide to play each the pure strategies with some probabilities.

Definition The mixed strategy σi(si) (also denoted σi) of player i is a probabilitydistribution over his pure strategies si ∈ Si.

Player’s i’s utility to profile σ is then given by:

ui(σ) =∑

si∈Si

σi(si)ui(si, σ−i)

62

Example with the multiple access game

Denote x the probability with which player p1 decides to transmit and y the probability forp2.

p1 and p2 stay quiet with probability 1− x and 1− y, respectively.

The payoff of player p1 is:

u1 = x(1− y)(1− c)− xyc = x(1− c− y)

The payoff of player p2 is:

u2 = y(1− x)(1− c)− xyc = y(1− c− x)

Each user wants to maximize his utility.

63

Example with the multiple access game

Analysis of the best response of p2 for each strategy of p1.

If x < 1− c, then u2 is maximized by setting y to the highest possible value i.e y = 1.

If x > 1− c, then u2 is maximized by setting y to the lowest possible value i.e y = 0.

However, if x = 1− c, any strategy of p2 is a best response. The game being symmetric,reversing the roles of the two players leads of course to the same result.

This means that (x = 1− c, y = 1− c) is a mixed-strategy Nash equilibrium for themultiple access Game.

Exercise: Show that for the jamming game a mixed strategy nash equilibrium dictateseach player to play a uniformly random distribution strategy (selecting one of the channelswith probability 0.5).

64

Importance of mixed strategies

J. Nash, ”Equilibrium points in n-person games”, Proceedings of the national academy ofSciences, vol. 36, pp. 48-49, 1950.

J. Nash, ”Non-cooperative games”, the annals of mathematics, vol. 54, no. 2, pp.286-295, 1951.

The importance of mixed strategies is mainly due to the following theorem.

Theorem. Every finite strategic-form game has a mixed strategy equilibrium

Remark: For mixed strategies, there is the underlying assumption that players areuncoordinated in their random choice.

65

The paper (11 pages)

66

Some useful theorems

Theorem: Consider a strategic-form game whose strategy spaces Si are nonemptycompact convex subsets of an Euclidean space. If the payoff functions ui are continuousin s and quasi-concave in si, there exists a pure-strategy Nash equilibrium.

This theorem is useful of games with an uncountable number of actions and the payoffsare continuous.

67

Some useful theorems

Theorem: Consider a strategic-form game whose strategy spaces Si are nonemptycompact subsets of a metric space. If the payoff functions ui are continous then thereexists a Nash equilibrium in mixed strategies.

The theorem is useful when the payoff functions are continuous but not necessarilyquasi-concave.

68

Equilibrium Selection

To solve a game, one has to investigate the existence of a Nash equilibria.

Three ways have been provided: iterated dominance, pure strategy Nash equilibria, mixedstrategy Nash equilibria.

The next steps are:

• To determine the uniqueness or not of the equilibrium.• When several equilibrium exists, one has to determine the most efficient.

Equilibrium selection intends to determine which one to choose by studying the efficiencyof the equilibrium.

69

Strategic Games: incomplete information

In games of incomplete information, the Nash equilibrium definition is refined to reflectuncertainties about player’s types, leading to the concept of Bayesian equilibrium.

70

Example of the Multiple Access Game

Both players in the multiple access game know that three Nash equilibria exist:

• Two pure strategy Nash equilibria (Q,T) and (T,Q).• One mixed strategy equilibrium.

What should be the final strategy?

71

Efficiency through Pareto optimality

A method to identify the desired equilibrium point in a game is to compare strategyprofiles using the concept of Pareto-optimality.

Definition. The strategy profile s is Pareto-superior to the strategy profile s′ if for anyplayer i ∈ N :

ui(si, s−i) ≥ ui(s′i, s

′−i)

with strict inequality for at least one player.

The strategy profile s∗ is pareto-optimal if there exists no other strategy that ispareto-superior to s∗.

Remark: One cannot increase the utility of player i without decreasing the utility of atleast one other player. In the Nash equilibrium, one can not UNILATERALLY change hisstrategy to increase his utility.

72


p2 p2

F Dp1 F (1-c,1-c) (-c,1)p1 D (1,-c) (0,0)

The Nash equilibrium (D,D) is not pareto-optimal. However, the strategy profilt (F,F) ispareto-optimal.

73


p2 p2

F Dp1 F (1-c,1-c) (-c,0)p1 D (0,0) (0,0)

(F,F) and (D,D) are Nash equilibria but only (F,F) is pareto optimal.

74


p2 p2

Q Tp1 Q (0,0) (0,1-c)p1 T (1-c,0) (-c,-c)

(T,Q) and (Q,T) are Nash equilibria and Pareto optimal.

The mixed strategy Nash equilibrium (x = 1− c and y = 1− c) results in the expectedpay-off (0,0) and is pareto inferior to the two pure strategy Nash equilibria.

It can be shown that there exists no mixed strategy profile that is Pareto-superior to allpure strategies

Why? Any mixed strategy of a player i is a linear combination of his pure strategies withweights that sum to one.

75

Jamming Game

jammer jammerC1 C2

sender C1 (-1,1) (1,-1)sender C2 (1,-1) (-1,1)

There exists no pure strategy Nash equilibrium. All pure strategies are pareto-optimal.

76

Can we cooperate without cooperating?

We only went through one type of equilibria in the case of non-cooperative games.

Nash equilibirum may be very efficient and cooperation is sometimes better off (Paretoequilibrium).

However, we don’t necessarily need cooperation. Coordination is enough (Correlatedequilibrium).

Example: (Aumann and Schelling, Nobel prize): People can coordinate rather well withoutcommunicating.

R. J. Aumann, ”Subjectivity and Correlation in Randomized Strategies”, Journal ofMathematical Economics, 1: 67-96, 1974.

• Consider the game where two people are asked to select a positive integer.• If they choose the same integer, both get an award, otherwise no award is given.• In such a setting, the majority tends to select the number 1. This number is distinctive,

since it is the smallest integer.

77

Correlated Games

Robert Ysral Aumann, 1930-

He received the Noble prize in economy in 2005 with Thomas Schelling for hiscontribution to the analysis of conflict and cooperation with Game theoretic tools.

78

Correlated Games

The correlated equilibria are defined in the context where there is an arbitrator who cansend (private or public) signals to the players.

These signals allow the players to coordinate their actions and perform jointrandomization over strategies.

The arbitrator can be a virtual entity (the players can agree on the first word they hear onthe radio) and generate signals that do not depend on the system.

A multi-strategy obtained using the signals is a set of strategies (one strategy for eachplayer which may depend on all the information available to the player including the signalit receives)

It is said to be a correlated equilibrium if no player has an incentive to deviate unilaterallyfrom its part of the multi-strategy.

A special type of ”deviation” can be of course to ignore the signals.

79

Evolutionary Games

M. Smith, ”Game Theory and the Evolution of Fighting”, Edinburgh University Press, 1972

M. Smith, ”Evolution and the Theory of Games”, Cambridge University Press, UK, 1987

T. L. Vincent and T. L. S. Vincent, ”Evolution and Control System design”, IEEE ControlSystems Magazine, 20(5): 20-35, October 2000

80

Optimality of the Nash Equilibirum

D. Braess, ”Uber ein paradoxen der werkehrsplannug”, Unternehmenforschung, 12,256-268, 1968.

In many cases, the Nash equilibrium solution is not optimal.

Adding more options to the players does not necessarily improve the performance(Braess paradox).

Incentive mechanisms (through punishment,..) can on the contrary push the playerstowards the desired solution.

C. Saraydar, N. B. Mandayam and D. J. Goodman, ”Efficient Power Control via Pricing inWireless Data Networks”, IEEE Trans. on Communications, 50(2): 291-303, February2002

81

The price of Anarchy

Centralized system: With a centralized system, one can have a global perspective withachieves a social optimum. This point is in many respect efficient but unfair.

Decentralized: In this case, players are in competition and one operating point is the Nashequilibrium. This point is in many respect not efficient but the players have no regret.

The Price of Anarchy (PoA), defined as the ratio of the cost function at equilibrium with thesocial optimum case, measures the price of not having a central coordination in thesystem

82

The Price of Anarchy

Suppose we have a system where cost functions C depend on the total demand x.

83

Social Optimum

The cost of participant is given by: C(x) =∑

a∈A ca(xa)xa

Definition: A social optimum is a feasible flow xso which minimizes C(.).

84

Wardrop equilibrium

J. G. Wardrop, ”Some theoretical aspects of road traffic research”, Proc. Inst. Civ. Eng.,Part 2, 1, 325-378, 1952

Definition: A wardrop equilibrium is a flow xNE such that nobody can switch to a path witha smaller travel time.

Characterization: (Smith (1979), Dafermos (1980)) The wardrop equilibrium ischaracterized by the variational inequality:

∑

a∈A

ca(xNEa )x

NEa ≤

∑

a∈A

ca(xNEa )xa

for all x. One can show that the wardrop equilibrium exists and is unique (Beckmann,Mcguire and Winsten, 1956)

85

Wardrop equilibrium

A. Haurie and P. Marcotte, ”On the Relationship between Nash-Cournot and WardropEquilibria,” Networks, vol. 15, pp. 295-308, 1985.

Wardrop (1952) postulated that users in a network game select routes of minimal length.In the asymptotic regime, the non-cooperative game becomes a non-atomic one, in whichthe impact of a single player on the others is negligible.

In the networking game context, the related solution concept is often called Wardropequilibrium and is often much easier to compute than the original Nash equilibrium.

86

Price of Anarchy

The price of anarchy measures the impact of a lack of central coordination.

POA =C(NE)

C(SO)

Theorem: For affine costs,

POA ≤ 4

3

Selfishness drives the system close to optimality!

87

Proof

J. R. Correa, A. S. Shultz and N. E. Stier Moses, ”On the inefficiency of equilibria incongestion games”, in Proc. of the 11th Conf. on Int. Prog. and Comb. Opt. (IPCO’05),pages 167-181, 2005.

C(NE) =∑

a∈A

ca(xNEa )x

NEa ≤

∑

a∈A

ca(xNEa )xa

=∑

a∈A

ca(xa)xa +∑

a∈A

(ca(x

NEa )− ca(xa)

)xa

≤ C(x) +1

4

∑

a∈A

ca(xNEa )x

NEa

= C(x) +1

4C(NE)

88

General costs

J. R. Correa, A. S. Shultz and N. E. Stier Moses, ”A geometric approach to the price ofanarchy in nonatomic congestion games”, in Proc. of the 11th Conf. on Int. Prog. andComb. Opt. (IPCO’05), pages 167-181, 2005.

C(NE) ≤∑

a∈A

ca(xa)xa +∑

a∈A

(ca(x

NEa )− ca(xa)

)xa

≤ C(x) + β(C)C(NE)

Let

β(C) = max0<x<y

([ca(x)− ca(y)]x

ca(y)y

)

= max

(shared area

big rectangle

)

Theorem: If cost are drawn from a family of continuous costs C,

POA =C(NE)

C(SO)≤ (1− β(C))

−1

89

Bounds on the price of anarchy

Theorem. For polynomials of maximum degree p, the price of anarchy is bounded by:

degree 1 2 3 4 ... p

POA 4/3 1.626 1.896 2.151 ... Ω( pln(p))

90

Presentation

4. Dynamic Games

91

Dynamic Games

T. Basar and G. J. Olsder, Dynamic Non-Cooperative Game Theory, London/New York:Academic Press. Jan. 1995

In the static game, players move simultaneously without knowing what the other playersdo.

The case of sequential interaction, the framework falls in the realm of dynamic games.

These games are represented in an extensive form as opposed to the strategic form.

92

Complete information versus perfect information

A game with perfect information is game where the players have perfect knowledge of allthe previous moves in the game at any moment they have to make a new move.

It concerns only the previous moves. Hence, there is still a game.

93

Strategic versus extensive

In the strategic form, the game was represented by a matrix.

In the extensive form, the game is represented by a tree.

The root of the tree is the start of the game.

One level of the tree is a stage.

The sequences of moves define a path on the tree.

Each node has of course a unique history.

Each terminal node of the tree defines a potential end of the game called outcome and itis assigned a corresponding pay-off.

Finite-horizon games are considered (a game with a finite number of stages).

94

Classical example: Sequential Multiple Access Game

A basic technique in CSMA/CA protocol

The two devices p1 and p2 are not perfectly synchronized.

p1 decides to transmit or not.

p2 observes p1 before making his own move.

The strategy of p1 is to transmit (T) or to be quiet (Q).

95

Representation of the Sequential Multiple Access Game

p1 p2 payoffT T (-c,-c)T Q (1-c,0)Q T (0,1-c)Q Q (0,0)

How many pure Nash equilibria do we have?

96

Important Result

H. W. Kuhn, ”Extensive Games and the problem of information”, Contributions to theTheory of Games II, 1953.

Theorem. (Kuhn, 1953). Every finite extensive-form game of perfect information has apure strategy

97

Backward induction for Sequential Multiple Access Game

How do we solve the game?

If player p2 plays the strategy T then the best response of player p1 is to play Q.

however, T is not the best strategy of player p2 if player p1 chooses T

We can eliminate some possibilities by backward induction.

98

Backward induction for Sequential Multiple Access Game

Player p2 knows that he has the last move...p1 p2 payoffT T (-c,-c)T Q (1-c,0)Q T (0,1-c)Q Q (0,0)

Given all the best moves of p2, player p1 calculates his best moves as well.

It turns out that the backward induction solution is the historical move (T) then (Q).

99

Backward induction

The technique of backward induction is similar to the iterated strict dominance technique.

It helps reducing the strategy space but becomes very complex for longer extensive formgames.

The method provides a technique to identify Stackelberg equilibrium.

100

Stackelberg equilibrium

Definition. The strategy profile s is a Stackelberg equilibrium with player p1 as the leaderand player p2 as the follower if player p1 maximizes his payoff subject to the constraint thatplayer p2 chooses according to his best response function.

Application:

• If p1 chooses T then the best response of p2 is to play Q (payoff of 1-c).• If p1 chooses Q, then the best response of p2 is T (payoff of 0 for p1).

p1 will therefore choose T which is the Stackelberg equilibria.

101

Exercise

Discuss the extensive form for the other three cases.

102

Summary

Except for the case of repeated games (players interact several times), we have nowthe main tools to analyze different non-cooperative schemes in wireless networks.

We will deal with repeated games if time is left..

103

Presentation

5. Optimization Results for Game theory

104

Norm of a matrix

Definition. A norm T (H) on the space of N ×N matrices satisfies the followingproperties:

(1) T (H) ≥ 0 with equality if and only if H = 0 is the all zero matrix.

(2) For any two matrices H1 and H2,

T (H1 + H2) ≤ T (H1) + T (H2)

(3) For any scalar α and matrix H,

T (αH) =| α | T (H)

105

Hilbert-Schmidt norm

For an N ×N complex matrix W = (wij), the Hilbert-Schmidt norm (or Schur norm orEuclidean norm or Frobenius norm) of W is defined as:

|| W ||=√∑

ij

| wij |2

106

Affine function

An affine function consists of a linear transformation followed by a translation:

x → Ax + b

107

Compact

A subset X of a Euclidean space is compact if any sequence in X has a subsequencethat converge to a limit point in X.

The definition of compactness for more general topological spaces uses the notion of”cover”, which is a collection of open sets whose union includes the set X.

X is compact if any cover has a finite subcover.

A subset of Euclidean space Rn is called compact if it is closed and bounded.

108

Convex

A set X in a linear vector space is convex if, for any x and x′ belonging to X and anyλ ∈ [0, 1], λx + (1− λ)x′ belongs to X

A real-valued function f defined on an interval is called convex if for any two points x and yin its domain C and any t in [0,1], we have

f(tx + (1− t)y) ≤ tf(x) + (1− t)f(y)

A differentiable function of one variable is convex on an interval if and only if its derivativeis monotonically non-decreasing on that interval.

The convex hull for a set of points X in a real vector space V is the minimal convex set

containing X.

109

Quasiconvex

A function defined on a convex subset S of a real vector space is quasiconvex if wheneverx, y ∈ S and λ ∈ [0, 1] then

f(λx + (1− λy)) ≤ max (f(x), f(y))

A quasiconcave function is a function whose negative is quasiconvex.

110

Convex optimization

Stephen Boyd, Lieven Vandenberghe. ”Convex Optimization”. Cambridge UniversityPress. 2004

Given a real vector space X together with a convex, real-valued function f defined on aconvex subset L of X , the problem of convex optimization is to find the point x∗ in L

such that:f(x∗) ≤ f(x)

for all x ∈ X

Results: The following statements hold for convex optimization:

• the set of all (global) minima is convex.• if the function is strictly convex, then there exists at most one minimum.• if a local minimum exists, then it is a global minimum.

111

Karush-Kuhn-Tucker: sufficient conditions

Let f : Rn → R be the function to be minimized under the continuously differentiableconstraints gi : Rn → R, (i = 1, .., m) and hj : Rn → R, (j = 1, ..., l). If x∗ is a localminimum, then there exists constants µi(i = 1, ..., m) and λi(i = 1, ..., l) such as

f′(x∗)−

m∑

i=1

µig′i(x∗)−

l∑

j=1

λjh′j(x∗) = 0

withµi ≥ 0(i = 1, ..., m)

gi(x∗) ≥ 0, (i = 1, ..., m)

hj(x∗) = 0, (j = 1, ..., l)

µigi(x∗) = 0, (i = 1, ..., m)

112

Karush-Kuhn-Tucker: necessary conditions

Let f : Rn → R be convex and the constraints gi : Rn → R, (i = 1, .., m) andhj : Rn → R, (j = 1, ..., l) be respectively concave and affine functions. If there existsconstants µi(i = 1, ..., m) and λi(i = 1, ..., l) such as

f′(x∗)−

m∑

i=1

µig′i(x∗)−

l∑

j=1

λjh′j(x∗) = 0

withµi ≥ 0(i = 1, ..., m)

gi(x∗) ≥ 0, (i = 1, ..., m)

hj(x∗) = 0, (j = 1, ..., l)

µigi(x∗) = 0, (i = 1, ..., m)

then the point x∗ is a global minimum

113

Presentation

6. Wireless Communication metrics for Game theory

114

General Multiple Input Multiple Output Model

Model representing multiple-antennas, CDMA, OFDM, ad-hoc networks withcooperation,...

y = W P s + nReceived signal MIMO matrix Power emitted signal AWGN

N × 1 N ×K K ×K K × 1 ∼ N (0, σ2IN)

Let

W =

u U

, s =

[s1

x

]

The goal is to detect s.

115

Shannon Capacity

Mutual information M between input and output:

M(s; (y, W)) = M(s; W) + M(s; y | W)

= M(s; y | W)

= H(y | W)−H(y | s, W)

= H(y | W)−H(n)

The differential entropy of a complex Gaussian vector x with covariance R is given by

log2 det(πeR).

116

Shannon Capacity

In the case of Gaussian independent entries, since

E(yyH) = σ

2IN + WPWH

E(nnH) = σ

2IN

The mutual information per dimension is:

CN =1

N(H(y | W)−H(n))

=1

N

(log2det(πe(σ

2IN + WPWH))− log2det(πeσ

2IN))

=1

N

(log2det(IN +

1

σ2WPWH

)

)

117

MMSE Receiver

Model example :

y = WP12s + n

= up121 s1 + UP

12−1x + n

= us1 + n′

E(n′n′H) = (UP−1UH

+ σ2I) = QΛQH

Whitening filter:

y = Λ−12QHy = Λ−1

2QHup121 s1 + Λ−1

2QHn′

= gp121 s1 + b

b is a white Gaussian noise.

118

MMSE Receiver

y = Λ−12QHup

121 s1 + b

Define g = Λ−12QHu

The output SINR is maximized with:

gHy = gHgp121 s1 + gHb

As a consequence, the receiver is:

gHΛ−12QH

= uH(QΛ−1QH

)= uH

(UP−1U

H+ σ

2IN

)−1

Remark: The usual MMSE receiver is the unbiased one:

uH(WWH

+ σ2IN

)−1

=1

1 + uH (UUH + σ2IN)−1 u

uH(UUH

+ σ2IN

)−1

119

MMSE Receiver

After MMSE filtering, we obtain:

gHy = gHgp121 s1 + gHb

with g = Λ−12QHu

Signal to Interference plus Noise Ratio (SINR):

βN =(gHg)2p1E(| s1 |2)

gHg= gHg = p1u

H(UP−1U

H+ σ

2IN

)−1

u

.

120


PART II

121


1. Power Allocation Games

122

Shannon Rate-Efficient Power Allocation Games (1)

System features

• Multiple access channel (MAC): one base station (BS), K mobile stations (MS)• Multi-antenna mobile and base stations (MIMO)• Fast fading links• Global CSIR + global CDIT• Receiver: successive interference cancellation (SIC)• Existence of a coordination signal

For K = 2, S ∈ 1, 2. If S = 1 (resp. S = 2) user 1 (resp. 2) is decoded last.Notation: p = Pr[S = 2] = p, Pr[S = 1] = 1− p = p

The BS is assumed to follow the coordination signal

123


Game (general) features

• Players: mobile stations k ∈ 1, ..., K• Strategy of a player: his precoding matrix/matrices• Utility: individual Shannon transmission rate• Type of game:

Power allocation game(a) Static game(b) Non-cooperative game (selfish users)(c) With complete information(d) With rational users

Questions

I Can we predict what the players are going to do? I Uniqueness?I Is there an equilibrium? I Efficiency?

...by the way why is it a game?

124


Mathematical description of the game [belmega08a]

• Interaction/signal model: y(s)

(τ) =

2∑

k=1

Hk(τ)x(s)k (τ) + z

(s)(τ)

• Channel matrices: hk(i, j) i.i.d (complex Gaussian RVs)

• Strategies: Qk = (Q(1)k , Q(2)

k ) = (α(1)k PkI, α

(2)k PkI) [lasaulce07] with Q(s)

k =

E[x

(s)k x

(s),Hk

]

• Strategy sets: α(1)k ∈

[0, 1

pk

](pTr(Q(1)

k ) + pTr(Q(2)k ) ≤ ntPk → pα

(1)k + pα

(2)k ≤

1). Remark: space-time power constraint versus spatial power constraint. Game 6=optimization problem.

• Utility: uk(Q(1)1 , Q(2)

1 Q(1)2 , Q(2)

2 ) = pR(1)k (Q(1)

1 , Q(1)2 ) + (1− p)R

(2)k (Q(2)

1 , Q(2)2 ) where

R(1)1 (Q(1)

1 , Q(1)2 ) = E log |I + ρH1Q

(1)1 HH

1 |, R(1)2 (Q(1)

1 , Q(1)2 ) = E log |I +

ρH1Q(1)1 HH

1 + ρH2Q(1)2 HH

2 | − E log |I + ρH1Q(1)1 HH

1 |, etc.

Is there a pure Nash equilibrium in this game?

125


Existence of a pure Nash equilibrium

I Theorem 1 [rosen65]. Assumptions: (1) the strategy sets are compact and convex; (2)each player’s utility is continuous in his strategy and those of the others; (3) each player’sutility is concave in his strategy. Conclusion: there exists at least one pure Nashequilibrium.

Question for the audience: why not considering mixed Nash equilibria?

Is there only one pure Nash equilibrium?

126


Uniqueness of the Nash equilibrium

I Theorem 2 [rosen65]. Assumptions: (1)–(3); (4) the diagonal strict concavity conditionis met i.e. D > 0 with

D = (α′′1−α

′1)

[∂u1

∂α1

(α′1, α

′2)−

∂u1

∂α1

(α′′1, α

′′2)

]+(α

′′2−α

′2)

[∂u2

∂α2

(α′1, α

′2)−

∂u2

∂α2

(α′′1, α

′′2)

]> 0

for all (α′1,α′′1) ∈ A21 and (α′2,α′′2) ∈ A2

2 such that either α′1 6= α′′1 or α′2 6= α′′2 .Conclusion: there is a unique pure NE.

In our context proving Theorem 2 amounts to proving the following lemma.

I Lemma 1 [belmega08b]. Assumptions: let A′, A′′, B′ and B′′ be Hermitian andnon-negative matrices such that either A′ 6= A′′ or B′ 6= B′′. Conclusion: Tr(M) > 0

where

M = (A′′−A′)[(I + A′

)−1 − (I + A′′

)−1

]+(B′′−B′

)[(I + B′

+ A′)−1 − (I + B′′

+ A′′)−1

].

127


Determination of the Nash equilibrium

• Optimization problem (say for user 1): under the power constraint compute

arg maxQ

(1)1 ,Q

(2)1

u1(Q(1)1 , Q(2)

1 Q(1)2 , Q(2)

2 ) =

arg maxQ

(1)1 ,Q

(2)1

pE log |I + ρH1Q

(1)1 HH

1 |+ pE log |I + ρH1Q(2)1 HH

1 + ρH2Q(2)2 HH

2 |

− pE log |I + ρH1Q(2)1 HH

1 |

• Problem 1: When implementable, usual numerical optimization techniques aregenerally not very attractive because they rely on computationally-intensive Monte-Carlo simulations (see e.g. [vu05])

• Problem 2: physical interpretations are not easy to make

Solution: find an approximation. Good news: when the system sizes increase quantitieslike 1

nrlog |I + ρH1Q

(1)1 HH

1 | converge a.s. for usual channel models.

128


Determination of the Nash equilibrium [belmega08b]

I Large system approximation of the utilities [sylverstein95, tulino05]Assumptions: nt −→ ∞, nr −→ ∞, and lim

nt→∞,nr→∞nt

nr

= c < ∞Result:

R(1)1 = nt log2(1 + ρ1α1γ1) + nr log2 (cγ1)− ntγ1 (cγ1 − 1) log2 e

R(1)2 = nt log2(1 + ρ1α1γ1) + nt log2

[1 + 2ρ2

1−(1−p)α2p γ2

]+ nr log2 (2cγ2)

−4ntγ2 (cγ2 − 1) log2 e− R(1)1

with γ1 =nr

nt

1

1 + ρα1P1

1+ρα1P1γ1

γ2 =nr

nt

1

1 + ρ

α1P11+2ρα1P1γ2

+[1−(1−p)α2]P2

p+2ρ[1−(1−p)α2]P2γ2

.

I Power allocation at the NE(a) Kuhn-Tucker optimality conditions(b) Fixed-point method + iterative algorithm

...what about the approximated game? See [dumont07]

129


Sum-rate efficiency of the Nash equilibrium [belmega08b]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 117.6

17.65

17.7

17.75

17.8

17.85

p

Ach

ieva

ble

sum

−ra

te fo

r th

e te

mpo

ral p

ower

allo

catio

n ga

me

P1=1 P

2=10 n

r=n

t=4 ρ=5dB

RsumC [centralized sum−capacity]

RsumNE (p) [achievable sum−rate at the NE]

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

P1=P P

2=10P n

r=n

t=4 ρ=5dB

P

Ach

ieva

ble

sum

−ra

tes

Unfair SIC (p=0), time paCentralizedFair SIC (p=1/2), time paSUD

130


Sum-rate efficiency of the Nash equilibrium: observations

• Sum-rate vs p: ∀p ∈ [0, 1], the gap between the sum-rate of a decentralized MIMOMAC with selfish users and the sum-capacity of the equivalent virtual MIMO network(EVMN) is very small.

• Remark. The social optimum corresponds to a global optimization. Does it coincidewith the sum-capacity?

• Sum-rate vs transmit power (P1 = P2 = P ): same conclusion.• SIC vs SUD: the network sum-rate when the BS implements a SIC is much better

than that obtained when SUD is implemented, regardless of the distribution of thecoordination signal. This comparison makes especially sense for the point p = 1

2 (faircomparison).

131


2. Power Control Games

132

Energy-Efficient Power Control Games (1)

System features

• MAC, single-antenna BS and MSs• Slow and flat fading channels (for MC-CDMA see [meshkati06])• Global CSIR + local CSIT

Review of the non-cooperative game features [goodman00]

• Strategy of a player (mobile station): his transmit power level

• Utility: ∀i ∈ 1, ..., K, ui(p1, ..., pK) =Ti

pi

=Rif(SINRi)

pi

(bit/J)

• Type of game: power control game, (a)–(d) (see Game 1)

133


About the Nash equilibrium

• Signal model: Y =

K∑

i=1

hiXi + Z where Z ∼ CN (0, σ2).

• Single-user decoding is assumed at the BS:

SINRi =pi|hi|2∑

j 6=i pj|hj|2 + σ2

• Transmit powers at the NE (for the non-trivial case where pNEi < P max ):

pNEi =

σ2

|hi|2β∗

1− (K − 1)β∗where β∗f ′(β∗) = f(β∗)

• Problems(A) Existence of a non-trivial equilibrium. Solution: CDMA. The denominator of pNE

i

becomes |hi|2[1− (K−1)β∗

N

].

(B) The NE can be inefficient [goodman00]• Solution 1 to (B): pricing [saraydar02]• Solution 2 to (B): introduce hierarchy [hayel08]

134


General objective of the hierarchical approach

Split the intelligence between the BS and MSs in order to find a trade-off between theglobal network performance reached at the equilibrium and the amount of signalingneeded to make it work.

1. Introducing hierarchy by choosing a leader [hayel08]

• One BS which implements single-user decoding (SUD)• K mobile stations• One MS is chosen to be the game leader• K − 1 followers who know what the leader has played (cognitive radio, broadcasting

signal from the BS, etc)

135


Definition (Stackelberg equilibrium, 2-user case). A strategy profile (pSE1 , pSE

2 ) is called a(pure) Stackelberg equilibrium if pSE

1 maximizes the single-variable utility of the leader andpSE

2 ∈ BR2(p1). Mathematically:

pSE1 = arg max

p1u1(p1, p2(p1))

pSE2 = arg max

p2u2(p

SE1 , p2).

NB: the notation BR stands for best response.

1.a. Questions

• Existence, uniqueness of a Stackelberg equilibrium (SE)?• From a user point of view, is it better to be chosen to be a leader or a follower?• With respect to the non-cooperative game what is the gain brought by introducing

hierarchy?• Do all the players benefit from this?

136


1.b. Answers

Proposition 1 (existence, uniqueness). Sufficient conditions: f ′′(0)f ′(0) ≥ 2 (K−1)β∗

1−(K−2)β∗ and

φ(x) = x[1− (K−1)β∗

1−(K−2)β∗x]

f ′(x)− f(x) has a single maximum in ]0, γ∗[, where γ∗ isthe positive solution of the equation φ(x) = 0.

pSEi = σ2

|hi|2γ∗(1+β∗)

1−(K−1)γ∗β∗−(K−2)β∗

∀j 6= i, pSEj = σ2

|hj|2β∗(1+γ∗)

1−(K−1)γ∗β∗−(K−2)β∗.

Comments: handmade proof; f(x) = (1− e−x)M passed the test.

137


1.b. Answers (continued)

Proposition 2. Every user has always a better utility by being chosen as a follower than aleader.

Proposition 3. Both the leader and followers improve their utilities with respect to thenon-cooperative setting.

Comments

• All the players benefit from hierarchy 6= duopoly in economics [hamilton90]• All the users not only obtain a better energy-efficiency in the proposed Stackelberg

game but also transmit with a lower power than in the non-cooperative game

138


2. Introducing hierarchy by implementing SIC at the BS [hayel08]

• The BS implements successive interference cancellation• There exists a coordination signal indicating the decoding order to the MSs• Always with global CSIR and local CSIT

2.a. Questions

• Existence, uniqueness of an equilibrium (NE/SE)?• For a given user, is it better to live in the SUD-based game or in the SIC-based one?

139


2.b. Answers

Existence and uniqueness of a Nash equilibrium

Proposition 4. Notation: the user who is decoded with rank K − i + 1 is assigned index i.

The unique NE is given by: ∀i ∈ 1, ..., K, p(SIC)i =

σ2

|hi|2β∗(1 + β

∗)i−1 with

p(SIC)i ≤ pmax

i . The proof is based on following two theorems.

Theorem 3 (see e.g. [fundenberg91]). Assumptions: (1) the strategy sets are compactand convex; (2) each player’s utility is continuous in his strategy and those of the others;(3) each player’s utility is quasi-concave in his strategy. Conclusion: there exists at leastone pure Nash equilibrium.

140


Existence and uniqueness of a Nash equilibrium (cont’t)

Theorem 4 [yates95] Assumptions: the best response (BR) correspondenceBR(p) =

(BR1(p), ..., BRK(p)

)is a standard function (positive, monotonous and

scalable). Conclusion: the NE is unique.

• Positivity: BRi(p) > 0

• Monotonicity: p ≥ p′ ⇒ BR(p) ≥ BR(p′)

• Scalability ∀α > 1, αBR(p) > BR(αp)

141


1.b. Answers (continued)

Proposition 5 (SIC vs SUD). Let di be the decoding rank for user i ∈ 1, ..., K. Alsodefine the critical decoding rank dc as follows:

dc = max

1,

K −ln

[1

1−(K−1)β∗]

ln(1 + β∗)

.

If the decoding rank of a user is greater than or equal to dc then his utility is better whenthe receiver implements SIC instead of SUD, it is worse otherwise.

Comments

• In the 2-user case users 1 and 2 will always perform better in the noncooperative gamewith SIC, independently of the value of β∗

• But in general, only the users who have a decoding rank greater than or equal to dc

will have a better utility in the SIC-based game• If β∗ → 0 only users 1 and 2 will perform better whereas all users will improve their

utilities in the SIC-based game with random CDMA if K−1N β∗ → 1, which occurs, for

example, when the load is small

142


3. Network performance analysis [hayel08]

• Now, the BS becomes a player (say the super-leader) who wants to maximize the overallnetwork performance

• IssuesIn the Stackelberg formulation with SUD: how to choose the best leaderIn the hierarchical game with SIC: how to choose the best decoding order

The answers to these questions depend on your vision of the world.

3.a. Performance metrics

• Social welfare [arrow63]:

w =

K∑

i=1

ui =

K∑

i=1

Ti

pi

• Equivalent virtual MIMO network (EVMN) energy-efficiency [meshkati06]:

v =

∑Ki=1 Ti∑Ki=1 pi

143


3.b. Result examples

Proposition 6 (Best decoding order for w). Assume a non-cooperative game with SIC. Thebest decoding order in the sense of the social welfare is to decode the users in theincreasing order of their energy weighted by the coding rate Ri|hi|2.

Proposition 7 (Best decoding order for v). Assume a non-cooperative game with SIC. Thebest decoding order in the sense of the equivalent virtual MIMO network energy-efficiency

is to decode the users in the decreasing order of their signal-to-noise ratio (SNR) |hi|2σ2 .

Comments

• With W (w) as a super-leader: the rich get richer• With Vladimir (v) as a super-leader: the poor get richer (Vladimir is more social than

W!)• Network performance measure: difference with Shannon transmission rate based

utilities

144


3. Handover Games

145

Handover Games (1)

System Model [belmega08c]

(a) Fixed network infrastructure

• Network of S parallel multiple access channels• S ≥ 2 base stations with non-overlapping bands of frequency• The BSs are connected through perfect communication links• BS characteristics

Noise power levelBandwidthNumber or receive antennas

146

Handover Games (2)

System Model (continued)

(b) Mobile stations

• K ≥ 2 mobile stations equipped with a cognitive radio• Each MS can sense the quality of its links with the different BSs• Each MS can share its transmit power between the BSs• Users are selfish and free to choose their power allocation in order to maximize their

individual transmission rateHard handover: best BS selectionSoft handover: optimal sharing between the different BSs

147

(Hard) Handover Games (3)

Game description for a simple model

• Interaction model: ∀s ∈ 1, ..., S, ys =

Ks∑

k=1

ds,k + zs with E|ds,k|2 = P , E|zs|2 =

Ns

• User’s strategy: ∀k ∈ 1, ..., K, P k = (P1,k, ..., PS,k)

• Strategy set: B = e1, ..., eS (canonical basis of RS)

• User’s utility (SUD): uk(P 1, ..., P K) = log[1 + P

Ns+(Ks−1)P

]

Existence of a Nash equilibrium

• In general there is no pure Nash equilibrium• However,

by assuming a population of users (non-atomic games): there are even several pureNE (see congestion games [milchtaich96][roughgarden04])or by allowing mixed strategies, there is at least one mixed NE [nash50]

148


Nash/Wardrop equilibrium for a dense network [belmega08c]

• Fraction of users in BS s: xs = KsK with K >> 1

• Utility of user k if he connects to BS s:

u(s)k (x1, ..., xS) = log2

[1 +

P

Ns + (Kxs − 1)P

]

• At a Nash equilibrium, all the users have the same utility:

xNE,SUDs = max

0,

1

S+

1

S

S∑

j=1

Nj −Ns

KP

• Resulting network sum-rate

RSUDsum (x

NE,SUD) =

S∑s=1

KxNE,SUDs log2

[1 +

P

Ns + (KxNE,SUDs − 1)P

]

• Special cases: (1) N1 = ... = NS; (2) KP >> Ns.

149


Illustration of the NE sum-rate efficiency

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

2

3

4

5

6

7

8

9

10

11

x1

Sum

Rat

es

S=2 K=20 P=10 N1=1 N2=10

Rsum SICRsum centralizedRsum SUD

Rsum*,SICRsum*,C

Rsum*,SUD

10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

P

PO

A

K=100, N1=2, N

2=1, N

3=3, N

4=1,N

5=2

SUD,S=2SIC, S=2SUD,S=3SIC, S=2SUD,S=4SIC, S=2SUD,S=5SIC, S=2

with POA ,RC

sum(x∗,C)

Rsum(xNE)≥ 1

150


Introducing a pricing technique to improve the NE sum-rate efficiency

Modified game (spontaneous NE → stimulated NE)

• System designer standpoint: maximize the overall network performance maxx

Rsum(x)

• New utility (cost function) for user k if he connects to BS s: cs,k(x) = p(τs,k(x)) + βs

whereτs,k(x) =

nkus,k(x) the time user k is connected to BS s for sending nk bits

p(.) the price per unit of connection time (positive, strictly increasing);• The system designer chooses βs such that xNE = x∗

If βs > 0 the system owner will charge more the users that connect to BS s (tax orpunishment)If βs < 0 the system owner will charge less the users that connect to BS s (subsidiesor reward)

• For βs = −p(τs,k(x∗s)) + p(τs,k(x

NEs )) the new NE is the desired state

• We have that:1 ≤ POA ≤ POA

151


Simulation example: performance improvement brought by introducing pricing

0 10 20 30 40 500

5

10

15

20

25

30

K=100, N1=2, N

2=1, N

3=4

P

Ach

ieva

ble

rate

s

C,S=2SUD, S=2SIC, S=2M−SUD,S=2C, S=3SUD, S=3SIC, S=3M−SUD, S=3

152

(Soft) Handover Games (8)

About the soft handover case

• User’s strategy: ∀k ∈ 1, ..., K, P k = (P1,k, ..., PS,k)

• Strategy set: P k ∈ [0, P ]S

• Does the case of soft handover correspond to the case of mixed strategies?• ... by the way, is hard handover just a special case of soft handover?

Soft handover vs Hard handover

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

16

18

20S=3 K=10 N=[1 2 3]

P

Ach

ieva

ble

rate

s

SH,CSH,SUDSH,SICHH,CHH,SUDHH,SICHH,M−SUD

153


4. Medium Access Control Games

154

Medium access control games (1)

System model [altman06]

• Multiple access channel: one BS, K MSs• BS: it does not schedule the (uplink) transmissions• Notation: let N = 0, 1K be the set of all 2K subsets of 1, ..., K. On each time

slot, a random subset of mobiles z(t) ∈ N is active (discrete time system).• Example: for a subset of size(z(t)) = 3 MSs for which user 1 is not active and users 2

and 3 are active: z = (011). Number of active MSs at time t: w(z) = 2.• Probability of being active: let pz, z ∈ N be the probability that the subset z of mobiles

is active on a time slot• Access protocol: slotted ALOHA

All MSs are synchronizedCollision: two or more MSs simultaneously attempt to send a packet then alltransmitted packets are lost (erasure)When a collusion occurs each concerned user wait a random number of time slotsbefore retransmitting

• Each MS has always a packet to transmit (saturation assumption)• There is a power constraint. If we denote by qi the probability that MS i sends a packet

when active we have that: qi ≤ qmaxi

155


Non-cooperative game description

• Players: MSs.• User’s strategy: qi.• Set of strategy Qi = [0, qmax

i ].• User’s utility (individual throughput conditioned on being active):

Θacti (q1, ..., qK) = EZ [Pr[z = ei]]

= EZ

qi

∏

j∈1,...,size(z),j 6=i

(1− qj)

= qi

∑

z∈Npz

∏

j∈1,...,size(z),j 6=i

(1− qj)

︸︷︷︸T

Nash equilibrium

T > 0⇒ there is a unique NE: ∀i ∈ 1, ..., K, qi = qmaxi

156


Coordination mechanism [altman06]

• The set of users is partitioned into M groups G1, ...,GM . User i belongs to Gj iffi = j − 1 mod M .

• Coordination signal: at each time slot t, the BS broadcasts a signal to all MSs in the formof a uniformly distributed discrete random variable S ∈ 1, ..., M, with K = nM .The RV S is assumed to be independent of Z.

• Correlated strategy: xi = (pi, qi) where on time slot t, an active mobile i transmits apacket with probability pi if and only if i ∈ GS(t). Otherwise it transmits with probabilityqi. Define Xi be the set of feasible strategies for user i

157


Correlated equilibrium (CE)

• A multi-strategy x = (x1, ..., xK) is said to be a correlated equilibrium if

∀i ∈ 1, ..., K, ∀x′i ∈ Xi, Θ

acti (x) ≥ Θ

acti (x

′i, x−i)

• Finding a correlated equilibrium becomes rapidly an intractable problem as the numberof users grows → we consider the special case of a symmetric channel where ∀i ∈1, ..., K, qmax

i = qmax → symmetric CE• It can be shown that x∗ = (p, q) with

q =

∣∣∣∣∣∣max

0, Mqmax−1

M−1

min

1, Mqmax

M−1

correspond to CE (see [altman06]), and the corresponding p is obtained by saturatingthe power constraint p

M +(1− 1

M

)q = qmax

158


Simulation example: system throughput vs probability of being active

Power constraint qmax = 0.25, K = 6

The coordination mechanism allows one to obtain higher values in the non-cooperativecase.

159


5. Concluding Remarks

160

Concluding Remarks (1)

On what we have learned

• EquilibriumVariety: NE, SE, WE, CE,...Existence, uniqueness: Nash, Rosen, Debreu-Fan-Glicksberg, Yates, with hands,...Determination: how to find it (optimization problem, approximation, ...)Efficiency: how to measure it, how to improve it (hierarchy, pricing, coordinationsignal, ...)

• Interesting (sometimes surprising) results for distributed wireless networksPOALeader/followerSUD vs SIC in energy-efficient PC gamesetc.

161

Concluding Remarks (2)

On what we have assumed

• Complete information• CSIR, CSIT• Scalability• Rationality

162

Last Slide

THANK YOU!

163


6. References and Reading

164

References for Part II

[lasaulce07] S. Lasaulce, A. Suarez, M. Debbah and L. Cottatellucci, “Power allocationgame for fading MIMO multiple access channels with antenna correlation”, ACM Proc. ofthe Intl Conf. on Game Theory in Comm. Networks (Gamecomm), pp. 1–9, Oct. 2007.

[belmega08a] E. V. Belmega, S. Lasaulce and M. Debbah, “Power control in distributedmultiple access channels with coordination”, IEEE Proc. of the Intl. Workshop on WirelessNetworks: Communication, Cooperation and Competition (WNC3), pp. 1–8, April 2008.

[rosen65] J. Rosen, “Existence and uniqueness of equilibrium points for concave n-persongames”, Econometrica, Vol. 33, pp. 520–534, 1965.

[belmega08b] E. V. Belmega, S. Lasaulce and M. Debbah, “Power Allocation Games forMIMO Multiple Access Channels with Coordination”, IEEE Trans. on WirelessCommunications, submitted, 2008.

[meshkati06] F. Meshkati, M. Chiang, H. V. Poor and S. C. Schwartz, “A game-theoreticapproach to energy-efficient power control in multi-carrier CDMA systems”, IEEE Journalon Selected Areas in Communications, Vol. 24, No. 6, pp. 1115–1129, June 2006.

165


[goodman00] D. J. Goodman and N. B. Mandayam, “Power Control for Wireless Data”,IEEE Personal Communications, Vol. 7, No. 2, pp. 48–54, April 2000.

[saraydar02] C. U. Saraydar, N. B. Mandayam and D. J. Goodman, “Efficient power controlvia pricing in wireless data networks”, IEEE Trans. on Communications, Vol. 50, No. 2,pp .291–303, Feb. 2002.

[hayel08] Y. Hayel, S. Lasaulce, R. El Azouzi and M. Debbah, “Introducing Hierarchy inEnergy Efficient Power Control Games”, ACM Proc. of the of the Intl. Conf. on GameTheory in Communications Networks (GAMECOMM), pp. 1–9, Oct. 2008.

[dumont07] J. Dumont, W. Hachem, S. Lasaulce, P. Loubaton and J. Najim, “On thecapacity achieving covariance matrix of Rician MIMO channels: an asymptotic approach”,IEEE Trans. on Inform. Theory, submitted, under revision, http://arxiv.org/abs/0710.4051.

[vu05 ]M. Vu and A. Paulraj, “Capacity optimization for Rician correlated MIMO wirelesschannels”, Proc. of Asilomar Conference, pp. 133–138, Asilomar, Nov 2005.

166


[sylverstein95] J. W. Sylverstein and Z. D. Bai, “On the empirical distribution ofeigenvalues of a class of large dimensional random matrices”, Journal of MultivariateAnalysis, Vol. 54, No. 2, pp. 175–192, 1995.

[tulino05] A. Tulino and S. Verdu, “Impact of antenna correlation on the capacity ofmulti-antenna channels”, IEEE Trans. on Inform. Theory, Vol. 51, No. 7, pp. 2491–2509,July 2005.

[hamilton90] J. Hamilton and S. Slutzky, “Endogenous timing in duopoly games:Stackelberg or Cournot equilibria”, Games and Economic Behavior, Vol. 2, pp. 29–46,1990.

[arrow63] K. J. Arrow “Social choice and individual values”, Yales University Press, 1963.

[fundenberg91] D. Fudenberg and J. Tirole, “Game Theory”, MIT Press, 1991.

167


[yates95] R. D. Yates, “A framework for uplink control in cellular radio systems”, IEEEJournal on Selec. Areas in Comm., Vol. 13, No. 7, pp. 1341–1347, Sep. 1995.

[belmega08c] E. V. Belmega, S. Lasaulce and M. Debbah, “Decentralized handovers incellular networks with cognitive terminals”, IEEE Proc. of the 3rd International Symposiumon Communications, Control and Signal Processing (ISCCSP), Malta, March 2008.

[milchtaich96] I. Milchtaich, “Congestion Models of Competition”, American Naturalist, Vol.147, No.5, pp. 760–783, 1996.

[roughgarden04] T. Roughgarden and E. Tardos, “Bounding the inefficiency of equilibria innonatomic congestion games”, Games and Economic Behavior, Vol. 47, pp. 389–403,2004.

[nash50] J. F. Nash, “Equilibrium points in n-points games”, Proc. of the NationalAcademy of Science, Vol. 36, no. 1, pp. 48–49, Jan. 1950.

[altman06] E. Altman, N. Bonneau and M. Debbah, “Correlated Equilibrium in AccessControl for Wireless Communications”, Networking, Coimbra, Portugal , May 2006.

168

Reading

A. MacKenzie, ”Game Theory for Wireless Engineers,” Morgan & Claypool, 2006

D. Fudenberg and J. Tirole, Game Theory, MIT Press, Cambridge, MA, 1991.

M. Felegyhazi and J.-P. Hubaux, ”Game Theory in Wireless Networks: A Tutorial”, inEPFL technical report, LCA-REPORT-2006-002, February, 2006

T. Basar and G. J. Olsder, ”Dynamic Non-Cooperative Game Theory”, New York,Academic Press, 1999.

E. Altman, T. Boulogne, R. El Azouzi, T. Jimenez and L. Wynter, ”A survey on networkinggames”, Computers and Operations Research, 2005.

169

Reading

M. Smith, ”Evolution and the Theory of Games”, Cambridge University Press, Cambridge,UK, 1987

G. Owen, ”Game Theory”, 3rd. ed. Academic Press, Burlington, MA, 2001

R. B. Myerson, ”Game Theory, Analysis of Conflict”, Harvard University Press,Cambridge, MA, 1991.

T. Basar and G. J. Olsder, ”Dynamic Noncooperative Game Theory”, New York, AcademyPress, 1999

M. J. Osborne and A. Rubenstein, ”A Course in Game Theory”, MIT Press, 1994

170

Reading

T. Basar, ”The gaussian test channel with an intelligent jammer,” IEEE Transactions onInformation Theory, 1983.

D. Monderer and L. S. Shapley: ”Potential Games”, Games and Economic Behavior, 14,pp. 124143 (1996)

H. Yaiche, R. R. Mazumdar and C. Rosenberg, ”A Game theoretic framework forbandwidth allocation and pricing in broadband networks,” IEEE/ACM Trans. Networks, vol.8, no. 5, pp. 667-678, Oct. 2000.

R. Menon, A. MacKenzie, R. Buehrer and J. Reed, ”Game Theory and InterferenceAvoidance in Decentralized Networks,” SDR Forum Technical Conference November,2002.

C. U. Saraydar, N. B. Mandayam and D. J. Goodman, ”Efficient power control via pricing inwireless data networks”, IEEE Trans. on Comm., vol. 50, no. 2, pp. 291-303, Feb. 2002.

171

Reading

R. La and V. Anantharam, ”Optimal Routing Control: Repeated Game Approach, IEEETans. on Automatic Control, 2002.

V. Srinivasan, P. Nuggehalli, C. Chiasserini, and R. Rao, ”Cooperation in Wireless Ad HocNetworks”, IEEE Infocom 2003.

D. P. Palomar, J. M. Cioffi and M. A. Lagunas, ”Uniform power allocation in MIMOchannels: a game theoretic approach,” in IEEE Transactions on Information Theory, vol.49, No.7, July 2003.

R. J. La and V. Anantharam, ”A game-theoretic look at the Gaussian multiaccesschannel”, Proceedings of the march 2003 DIMACS workshop on Network InformationTheory, vol. 66, 2004, pp. 87-106.

C. W. Sung and W. S. Wong, ”A noncooperative power control game for multirate CDMAdata networks”, IEEE transactions on Wireless Communications, vol.2, pp. 186-194,January 2003.

172

Reading

R. La and V. Anantharam, “A game-theoretic look at the Gaussian multiaccess channel,”in Proc. of the DIMACS Workshop on Network Information Theory, New Jersey, NY, USA,Mar. 2003.

J. Cai and U. Pooch, “Allocate fair payoff for cooperation in wireless ad hoc networks usingShapley value,” in Proc. Int. Parallel and Distributed Processing Symposium, Santa Fe,NM, USA, Apr. 2004, pp. 219227.

A. Kashyap, T. Basar and R. Srikant, ”Correlated Jamming on MIMO Gaussian FadingChannels,” IEEE Transactions on Information Theory, 50(9): 2119-2123, September 2004.

F. Meshkati, H. Poor, S. Schwarz and N. Mandayam, ”An Energy-Efficient Approach toPower Control and Receiver Design in Wireless Data Networks”, IEEE Trans. onCommunications, vol. 53, no. 11, pp. 1885-1894, Nov. 2005.

F. Meshkati, H. Poor, S. Schwarz and N. Mandayam, ”A non cooperative power controlgame in delay-contrained multiple access networks,” Proceedings of the IEEEInternational Symposium on Information Theory (ISIT), Adelaide, Australia, September,2005.

173

Reading

E. Altman, N. Bonneau, M. Debbah, and G. Caire, ”An evolutionary game perspective toALOHA with power control, ”Proceedings of the 19th International Teletraffic Congress,2005.

R. Etkin, A. Parekh and D. Tse, ”Spectrum Sharing for Unlicensed bands”, Proceedings ofthe Allerton Conference on Communication, Control and Computing, Monticello, IL, Sep.28-30, 2005

S. Shafiee and S. Ulukus, ”Correlated Jamming in Multiple Access Channels,” Conferenceon Information Sciences and Systems, Baltimore, MD, March 2005

L. Lai and H. E. Gamal, ”The water-Filling Game in Fading Multiple Access Channels,”submitted to IEEE Transactions on Information Theory, 2006

E. Altman, N. Bonneau and M. Debbah, ”Correlated Equilibrium in Access Control forWireless Communications,” in Networking 2006, Coimbra, Portugal, May, 2006.

174

Reading

L. Lai and H. El Gamal, ”On Cooperation in Energy Efficient Wireless Networks: The Roleof Altruistic Node,”, submitted to the IEEE Transactions on Wireless Communications,Aug. 2006.

G. Scutari, S. Barbarossa and D. P. Palomar, ”Potential Games: A framework for vectorpower control problems with couples constraints”, IEEE International Conference onAcoustics, Speech and Signal Processing, Toulouse, France, 2006.

M. Felegyhazi, J. B. Hubaux and L. Buttyan, ”Nash equilibria of packet forwardingstrategies in wireless ad-hoc networks,” IEEE transactions on Mobie Computing, vol.5,May 2006.

Z. Han and V. Poor, “Coalition games with cooperative transmission: a curse for the curseof boundary nodes in selfish packet-forwarding wireless networks,” in Proc. Int. Symp. onModeling and Opt. in Mobile Ad Hoc and Wireless Networks, Limasol, Cyprus, Apr. 2007

F. Meshkati, H. V. Poor and S. C. Schwartz, ”Energy-Efficient Resource Allocation inWireless Networks: An overview of game theoretic approaches,” IEEE Signal Process.Magazine, Vol. 24, pp. 58-68, May 2007.

175

Game Theory for Wireless Communications -...

Documents

Transcript of Game Theory for Wireless Communications -...