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Module J
Game theory in Wireless Networks
Julien Freudiger,Márk Félegyházi,
Jean-Pierre Hubaux
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Upcoming networks vs. mechanisms
X X X X X
X X X X
X X X X X X X
X X X X X X X X
X X X X X X X
X X X X X X X
X X X X X X
X X X X
Naming and addressing
Discoura
ging
greedy o
p.
Small operators, community networks
Cellular operators in shared spectrum
Mesh networks
Hybrid ad hoc networks
Autonomous ad hoc networks
Security
associa
tions
Securin
g neighbor disc
overy
Secure
routin
g
Privac
y
Enforcing P
KT FW
ing
Enforcing fa
ir MAC
Vehicular networks
Sensor networks
RFID networks
Part I Part IIIPart II
Upcoming wireless networks
Security and cooperation mechanisms
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Brief introduction to Game Theory
• Discipline aiming at modeling situations in which actors have to make decisions which have mutual, possibly conflicting, consequences
• Classical applications: economics, but also politics and biology
• Example: should a company invest in a new plant, or enter a new market, considering that the competition may make similar moves?
• Most widespread kind of game: non-cooperative (meaning that the players do not attempt to find an agreement about their possible moves)
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Game 1: Guess what?
1. Choose an integer number in [0,100]
2. The winner is: gets closest to the 2/3 of the average
3. Take a piece of paper and write• your name (e.g., Julien FREUDIGER)• your number (e.g., 99)
4. Price: 4 extra points in Quiz I• if several winners, price is divided equally
5. We play the game again at the end of the course !
Remark : Assume truthful game– rational student => write best guess– no (rational) student discussed solution with friends
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Classification of games
Non-cooperative Cooperative
Static Dynamic (repeated)
Strategic-form Extensive-form
Perfect information Imperfect information
Complete information Incomplete information
Cooperative
Imperfect information
Incomplete information
Perfect information: each player can observe the action of each other player.
Complete information: each player knows the identity of other players and, for each of them, the payoff resulting of each strategy.
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Cooperation in self-organized wireless networks
S1
S2
D1D2
Usually, the devices are assumed to be cooperative. But what if they are not?
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Static (or “single-stage”) games
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Example 1: The Forwarder’s Dilemma
?
?
Blue Green
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E1: From a problem to a game
• users controlling the devices are rational = try to maximize their benefit
• game formulation: G = (P,S,U)– P: set of players– S: set of strategy functions– U: set of payoff functions
• strategic-form representation
• Reward for packet reaching the destination: 1• Cost of packet forwarding: c (0 < c << 1)
(1-c, 1-c) (-c, 1)
(1, -c) (0, 0)
BlueGreen
Forward
Drop
Forward Drop
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Solving the Forwarder’s Dilemma (1/2)
' '( , ) ( , ), ,i i i i i i i i i iu s s u s s s S s S
iu Ui is S
Strict dominance: strictly best strategy, for any strategy of the other player(s)
where: payoff function of player i
strategies of all players except player i
In Example 1, strategy Drop strictly dominates strategy Forward
(1-c, 1-c) (-c, 1)
(1, -c) (0, 0)
BlueGreen
Forward
Drop
Forward Drop
Strategy strictly dominates ifis'is
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Solving the Forwarder’s Dilemma (2/2)
Solution by iterative strict dominance:
(1-c, 1-c) (-c, 1)
(1, -c) (0, 0)
BlueGreen
Forward
Drop
Forward Drop
Drop strictly dominates ForwardDilemma
Forward would result in a better outcomeBUT }
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Repeated Iterative Strict Dominance
(3, 9) (0, 6) (0, 4) (1, 2)
(1, 3) (3, 6) (6, 1) (6, 3)
(4, 2) (4, 1) (2, 2) (8, 3)
(3, 7) (4, 5) (2, 6) (4, 7)
BlueGreen
A
B
X Y V W
C
D
Strict dominance: strictly best strategy, for any strategy of the other player(s)
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Example 2: The Joint Packet Forwarding Game
?Blue GreenSource Dest
?
No strictly dominated strategies !
• Reward for packet reaching the destination: 1• Cost of packet forwarding: c (0 < c << 1)
(1-c, 1-c) (-c, 0)
(0, 0) (0, 0)
BlueGreen
Forward
Drop
Forward Drop
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E2: Weak dominance
?Blue GreenSource Dest
?
'( , ) ( , ),i i i i i i i iu s s u s s s S
Weak dominance: strictly better strategy for at least one opponent strategy
with strict inequality for at least one s-i
Iterative weak dominance (1-c, 1-c) (-c, 0)
(0, 0) (0, 0)
Blue
Green
Forward
Drop
Forward Drop
BUT
The result of the iterative weak dominance is not unique in general !
Strategy si is weakly dominates strategy s’i if:
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Repeated Iterative Weak Dominance
(3, 9) (0, 6) (0, 4) (1, 2)
(1, 3) (3, 6) (6, 1) (6, 3)
(4, 2) (4, 1) (2, 2) (8, 2)
(3, 7) (4, 5) (2, 6) (4, 7)
BlueGreen
A
B
X Y V W
C
D
Weak dominance: strictly better strategy for at least one opponent strategy
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Nash equilibrium (1/2)
Nash Equilibrium: no player can increase its utility by deviating unilaterally
(1-c, 1-c) (-c, 1)
(1, -c) (0, 0)
BlueGreen
Forward
Drop
Forward DropE1: The Forwarder’s Dilemma
E2: The Joint Packet Forwarding game (1-c, 1-c) (-c, 0)
(0, 0) (0, 0)
BlueGreen
Forward
Drop
Forward Drop
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Finding Nash Equilibria
(3, 9) (0, 6) (0, 4) (1, 2)
(1, 3) (3, 6) (6, 1) (6, 3)
(4, 2) (4, 1) (2, 2) (8, 2)
(3, 7) (4, 5) (2, 6) (4, 7)
BlueGreen
A
B
X Y V W
C
D
Nash Equilibrium: no player can increase its utility by deviating unilaterally
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Nash equilibrium (2/2)
* * *( , ) ( , ),i i i i i i i iu s s u s s s S
iu Ui is S
where: utility function of player i
strategy of player i
( ) arg max ( , )i i
i i i i is S
b s u s s
The best response of player i to the profile of strategies s-i is a strategy si such that:
Nash Equilibrium = Mutual best responses
Caution! Many games have more than one Nash equilibrium
Strategy profile s* constitutes a Nash equilibrium if, for each player i,
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Example 3: The Multiple Access game
Reward for successfultransmission: 1
Cost of transmission: c(0 < c << 1)
There is no strictly dominating strategy
(0, 0) (0, 1-c)
(1-c, 0) (-c, -c)
BlueGreen
Quiet
Transmit
Quiet Transmit
There are two Nash equilibria
Time-division channel
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E3: Mixed strategy Nash equilibrium
objectives– Blue: choose p to maximize ublue
– Green: choose q to maximize ugreen
(1 )(1 ) (1 )blueu p q c pqc p c q (1 )greenu q c p
1 , 1p c q c
p: probability of transmit for Blue
q: probability of transmit for Green
is a Nash equilibrium
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Example 4: The Jamming game
transmitter:• reward for successfultransmission: 1• loss for jammed transmission: -1
jammer:• reward for successfuljamming: 1• loss for missed jamming: -1
There is no pure-strategy Nash equilibrium
two channels: C1 and C2
(-1, 1) (1, -1)
(1, -1) (-1, 1)
BlueGreen
C1
C2
C1 C2
transmitter
jammer
1 1,
2 2p q is a Nash equilibrium
p: probability of transmit on C1 for Blue
q: probability of transmit on C1 for Green
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Theorem by Nash, 1950
Theorem: Every finite strategic-form game has a mixed-strategy Nash equilibrium.
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Efficiency of Nash equilibria
E2: The Joint Packet Forwarding game
(1-c, 1-c) (-c, 0)
(0, 0) (0, 0)
BlueGreen
Forward
Drop
Forward Drop
How to choose between several Nash equilibria ?
Pareto-optimality: A strategy profile is Pareto-optimal if it is not possible to increase the payoff of any player without decreasing the payoff of another player.
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Efficiency
(3, 9) (0, 6) (0, 4) (1, 2)
(1, 3) (3, 6) (6, 1) (6, 3)
(4, 2) (4, 1) (2, 2) (8, 2)
(3, 7) (4, 5) (2, 6) (4, 7)
BlueGreen
A
B
X Y V W
C
D* *
*
*
Pareto-optimality: It is not possible to increase the payoff of any player without decreasing the payoff of another player.
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How to study Nash equilibria ?
Properties of Nash equilibria to investigate:• existence• uniqueness• efficiency (Pareto-optimality)• emergence (dynamic games, agreements)
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Dynamic games
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Extensive-form games• usually to model sequential decisions• game represented by a tree• Example 3 modified: the Sequential Multiple Access game:
Blue plays first, then Green plays.
Green
BlueT Q
T Q T Q
(-c,-c) (1-c,0) (0,1-c) (0,0)
Reward for successfultransmission: 1
Cost of transmission: c(0 < c << 1)
Green
Time-division channel
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Strategies in dynamic games
• The strategy defines the moves for a player for every node in the game, even for those nodes that are not reached if the strategy is played.
Green
BlueT Q
T Q T Q
(-c,-c) (1-c,0) (0,1-c) (0,0)
Greenstrategies for Blue: T, Q
strategies for Green: TT, TQ, QT and QQ
If they have to decide independently: three Nash equilibria
(T,QT), (T,QQ) and (Q,TT)
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Extensive to Normal form
Blue vs. Green
TT TQ QT QQ
T (-c,-c) (-c,-c) (1-c,0) (1-c,0)
Q (0,1-c) (0,0) (0,1-c) (0,0)
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Backward induction
• Solve the game by reducing from the final stage• Eliminates Nash equilibria that are increadible threats
Green
BlueT Q
T Q T Q
(-c,-c) (1-c,0) (0,1-c) (0,0)
Greenincredible threat: (Q, TT)
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Subgame perfection• Extends the notion of Nash equilibrium
Green
BlueT Q
T Q T Q
(-c,-c) (1-c,0) (0,1-c) (0,0)
Green
Subgame perfect equilibria: (T, QT) and (T, QQ)
One-deviation property: A strategy si conforms to the one-deviation property if there does not exist any node of the tree, in which a player i can gain by deviating from si and apply it otherwise.
Subgame perfect equilibrium: A strategy profile s constitutes a subgame perfect equilibrium if the one-deviation property holds for every strategy si in s.
Finding subgame perfect equilibria using backward induction
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Repeated games
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Repeated games
• repeated interaction between the players (in stages)• move: decision in one interaction• strategy: defines how to choose the next move, given
the previous moves• history: the ordered set of moves in previous stages
– most prominent games are history-1 games (players consider only the previous stage)
• initial move: the first move with no history• finite-horizon vs. infinite-horizon games• stages denoted by t (or k)
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Payoffs: Objectives in the repeated game
• finite-horizon vs. infinite-horizon games• myopic vs. long-sighted repeated game
1i iu u t
0
T
i it
u u t
0
i it
u u t
myopic:
long-sighted finite:
long-sighted infinite:
payoff with discounting: 0
ti i
t
u u t
0 1 is the discounting factor
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Strategies in the repeated game
• usually, history-1 strategies, based on different inputs:
– others’ behavior:
– others’ and own behavior:
– payoff:
1i i im t s m t 1 ,i i i im t s m t m t
1i i im t s u t
Example strategies in the Forwarder’s Dilemma:
Blue (t) initial move
F D strategy name
Green (t+1) F F F AllC
F F D Tit-For-Tat (TFT)
D D D AllD
F D F Anti-TFT
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The Repeated Forwarder’s Dilemma
(1-c, 1-c) (-c, 1)
(1, -c) (0, 0)
BlueGreen
Forward
Drop
Forward Drop
?
?
Blue Green
stage payoff
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Analysis of the Repeated Forwarder’s Dilemma (1/3)
Blue strategy Green strategy
AllD AllD
AllD TFT
AllD AllC
AllC AllC
AllC TFT
TFT TFT
infinite game with discounting: 0
ti i
t
u u t
Blue utility Green utility
0 0
1 -c
1/(1-) -c/(1-)
(1-c)/(1-) (1-c)/(1-)
(1-c)/(1-) (1-c)/(1-)
(1-c)/(1-) (1-c)/(1-)
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Analysis of the Repeated Forwarder’s Dilemma (2/3)
Blue strategy Green strategy
AllD AllD
AllD TFT
AllD AllC
AllC AllC
AllC TFT
TFT TFT
• AllC receives a high payoff with itself and TFT, but• AllD exploits AllC• AllD performs poor with itself• TFT performs well with AllC and itself, and• TFT retaliates the defection of AllD
TFT is the best strategy if is high !
Blue utility Green utility
0 0
1 -c
1/(1-) -c/(1-)
(1-c)/(1-) (1-c)/(1-)
(1-c)/(1-) (1-c)/(1-)
(1-c)/(1-) (1-c)/(1-)
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Analysis of the Repeated Forwarder’s Dilemma (3/3)
Theorem: In the Repeated Forwarder’s Dilemma, if both players play AllD, it is a Nash equilibrium.
Theorem: In the Repeated Forwarder’s Dilemma, both players playing TFT is a Nash equilibrium as well.
Blue strategy Green strategy Blue utility Green utility
AllD AllD 0 0
TFT TFT (1-c)/(1-) (1-c)/(1-)
The Nash equilibrium sBlue = TFT and sGreen = TFT is Pareto-optimal (but sBlue = AllD and sGreen = AllD is not) !
40
Experiment: Tournament by Axelrod, 1984
• any strategy can be submitted (history-X)• strategies play the Repeated Prisoner’s Dilemma
(Repeated Forwarder’s Dilemma) in pairs• number of rounds is finite but unknown
• TFT was the winner• second round: TFT was the winner again
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Mechanism Design
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Game Theory: Analysis vs. Mechanism Design
analysis
mechanism design
behaviorsystem
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Algorithmic Mechanism Design
algorithmic mechanism design
behavior(truthful)
system(protocols)
Questions:• distributed implementation• complexity• convergence speed
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Example 5: Stable matching in cellular networks (CSM)
A
B
C
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E5: Cellular stable matching (CSM)
• N base stations and M mobiles (M>N)• Each BS and mobile has preference lists• Unstable: If BS A and B serve mobile a and b,
respectively, although a prefers B and B also prefers a.• Simplified version: N=M
• Stable marriage problem
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E5’: Cellular stable matching simplified (CSM’), (1/2)
• N=M
A
B
C
b
a
c
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E5’: Cellular stable matching simplified (2/2)
• preference matrix
1,3 2,2 3,1
3,1 1,3 2,2
2,2 3,1 1,3
BS
mobiles
A
B
a b c
C
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E5’: Existence of CSM
• Theorem (Gale-Shapley, 1962): There always exists a stable matching
• Proof (constructive): The Gale-Shapley algorithm
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E5’: CSM – The Gale-Shapley algorithm
A
B
C
a
b
c
A
B
C
a
b
c
network-centric user-centric
1 -
1 -
1 -
- 3
- 3
- 3
3 -
3 -
3 -
- 1
- 1
- 1
1,3 2,2 3,1
3,1 1,3 2,2
2,2 3,1 1,3
BS
mobiles
A
B
a b c
C
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E5’: Optimality of CSM
• Optimal: Each participant (in a group) is at least as well off as in another matching
• The Gale-Shapley algorithm is male (BS)-optimal
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E5: Back to CSM
• The Gale-Shapley algorithm generalized– quota at each BS (q=2)
1,3 2,2 4,3 6,1 3,2 5,1
6,1 1,3 5,1 3,3 4,1 2,2
2,2 3,1 6,2 1,2 5,3 4,3
BS
mobiles
A
B
a b c
C
d e f
A
B
C
abcdef
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Stable Matching
• Applications:– WiFi networks– Load-balancing for processors
– Students to schools– Job-hunting
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Truthful Stable Matching
• Theorem (Roth, 1982):– The Gale-Shapley algorithm is truthful for males
• Theorem (Gale-Sotomayor, 1985):– Women can cheat such that they get a better partner
1,3 2,2 3,1
3,1 1,3 2,2
2,2 3,1 1,3
BS
mobiles
A
B
a b c
C
– Accept only your favorite peer
A
B
C
a
b
c
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E6: Stable matching in ad hoc networks (ASM)
• M mobiles• Each mobile has preference lists• Unstable: If mobile a and c communicate with mobile
b and d, respectively, although a prefers c and c prefers a.
• Stable roommate problem
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E6’: ASM simplified
• preference matrix
- 1,3 2,2 3,1
3,1 - 1,3 2,2
2,2 3,1 - 1,3
1,3 2,2 3,1 -
mobilesmobiles
a
b
a b c
c
a b
c d
d
d
a b
c d
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Discussion on game theory
• Rationality• Utility function and cost• Pricing and mechanism design (to promote
desirable solutions)• Infinite-horizon games and discounting• Reputation• Cooperative games• Imperfect / incomplete information
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Who is malicious? Who is selfish?
Both security and game theory backgrounds are useful in many cases !! Both security and game theory backgrounds are useful in many cases !!
Harm everyone: viruses,…
Selective harm: DoS,… Spammer
Cyber-gangster:phishing attacks,trojan horses,…
Big brother
Greedy operator
Selfish mobile station
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Conclusion
• Game theory can help modeling greedy behavior in wireless networks
• Discipline still in its infancy• Alternative solutions
– Ignore the problem– Build protocols in tamper-resistant hardware
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Game 2: Guess what again?
1. Choose an integer number in [0,100]
2. The winner is: gets closest to the 2/3 of the average
3. Take a piece of paper and write• your name (e.g., Julien FREUDIGER)• your number
4. Price: 2 extra points in Quiz I• if several winners, price is divided equally
Remarks about the game:• truthful:
– each (rational) student writes his best guess– no (rational) student should have discussed the solution
with friends
