Module J Game theory in Wireless Networks Julien Freudiger, Mrk Flegyhzi, JeanPierre Hubaux

1
Module JGame theory in Wireless Networks
Julien Freudiger,Márk Félegyházi,Jean-Pierre
Hubaux
2
Upcoming networks vs. mechanisms
Security and cooperation mechanisms
Securing neighbor discovery
Upcoming wireless networks
Naming and addressing
Enforcing PKT FWing
Discouraginggreedy op.
Security associations
Enforcing fair MAC
Secure routing
Privacy
Small operators, community networks
Cellular operators in shared spectrum
Mesh networks
Hybrid ad hoc networks
Autonomous ad hoc networks
Vehicular networks
Sensor networks
RFID networks
Part I
Part III
Part II
3
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)

4
Game 1 Guess what?

• Remark Assume truthful game
• rational student gt write best guess
• no (rational) student discussed solution with
  friends

• Choose an integer number in 0,100
• The winner is gets closest to the 2/3 of the
  average
• Take a piece of paper and write
• your name (e.g., Julien FREUDIGER)
• your number (e.g., 99)
• Price 4 extra points in Quiz I
• if several winners, price is divided equally
• We play the game again at the end of the course !

5
Classification of games
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.
6
Cooperation in self-organized wireless networks
D2
D1
S2
S1
Usually, the devices are assumed to be
cooperative. But what if they are not?
7
Static (or single-stage) games
8
Example 1 The Forwarders Dilemma
?
Green
Blue
?
9
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 lt c ltlt 1)

Green
Forward
Drop
Blue
Forward
Drop
10
Solving the Forwarders Dilemma (1/2)
Strict dominance strictly best strategy, for any
strategy of the other player(s)
Strategy strictly dominates if
payoff function of player i
where
strategies of all players except player i
In Example 1, strategy Drop strictly dominates
strategy Forward
Green
Forward
Drop
Blue
Forward
Drop
11
Solving the Forwarders Dilemma (2/2)
Solution by iterative strict dominance
Green
Forward
Drop
Blue
Forward
Drop

Drop strictly dominates Forward
Dilemma
BUT
Forward would result in a better outcome
12
Repeated Iterative Strict Dominance
Strict dominance strictly best strategy, for any
strategy of the other player(s)
Green
X
Y
W
Blue
V
A
B
C
D
13
Example 2 The Joint Packet Forwarding Game
?
?
Source
Green
Dest
Blue
Green
Forward
Drop

• Reward for packet reaching the destination 1
• Cost of packet forwarding
• c (0 lt c ltlt 1)

Blue
Forward
Drop
No strictly dominated strategies !
14
E2 Weak dominance
Weak dominance strictly better strategy for at
least one opponent strategy
Strategy si is weakly dominates strategy si if
with strict inequality for at least one s-i
?
?
Source
Green
Dest
Blue
Green
Forward
Drop
Blue
Forward
Iterative weak dominance
Drop
BUT
The result of the iterative weak dominance is not
unique in general !
15
Repeated Iterative Weak Dominance
Weak dominance strictly better strategy for at
least one opponent strategy
Green
X
Y
W
Blue
V
A
B
C
D
16
Nash equilibrium (1/2)
Nash Equilibrium no player can increase its
utility by deviating unilaterally
Green
Forward
Blue
Drop
E1 The Forwarders Dilemma
Forward
Drop
Green
Forward
Drop
Blue
E2 The Joint Packet Forwarding game
Forward
Drop
17
Finding Nash Equilibria
Nash Equilibrium no player can increase its
utility by deviating unilaterally
Green
X
Y
W
Blue
V
A
B
C
D
18
Nash equilibrium (2/2)
Strategy profile s constitutes a Nash
equilibrium if, for each player i,
where
utility function of player i
strategy of player i
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
19
Example 3 The Multiple Access game
Time-division channel
Green
Quiet
Transmit
Blue
Reward for successfultransmission 1 Cost of
transmission c (0 lt c ltlt 1)
Quiet
Transmit
There is no strictly dominating strategy
There are two Nash equilibria
20
E3 Mixed strategy Nash equilibrium
p probability of transmit for Blue
q probability of transmit for Green

• objectives
• Blue choose p to maximize ublue
• Green choose q to maximize ugreen

is a Nash equilibrium
21
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

transmitter
two channels C1 and C2
jammer
Green
C1
C2
Blue
C1
C2
p probability of transmit on C1 for Blue
There is no pure-strategy Nash equilibrium
q probability of transmit on C1 for Green
is a Nash equilibrium
22
Theorem by Nash, 1950
Theorem Every finite strategic-form game has a
mixed-strategy Nash equilibrium.
23
Efficiency of Nash equilibria
Green
Forward
Drop
Blue
E2 The Joint Packet Forwarding game
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.
24
Efficiency
Pareto-optimality It is not possible to increase
the payoff of any player without decreasing the
payoff of another player.
Green
X
Y
W
Blue
V
A
B


C


D
25
How to study Nash equilibria ?

• Properties of Nash equilibria to investigate
• existence
• uniqueness
• efficiency (Pareto-optimality)
• emergence (dynamic games, agreements)

26
Dynamic games
27
Extensive-form games

• usually to model sequential decisions
• game represented by a tree
• Example 3 modified the Sequential Multiple
  Access gameBlue plays first, then Green plays.

Time-division channel
Blue
Q
T
Reward for successfultransmission 1 Cost of
transmission c (0 lt c ltlt 1)
Green
Green
Q
Q
T
T
(0,0)
(-c,-c)
(1-c,0)
(0,1-c)
28
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.

Blue
strategies for Blue T, Q
Q
T
Green
Green
Q
Q
T
T
strategies for Green TT, TQ, QT and QQ
(-c,-c)
(1-c,0)
(0,1-c)
(0,0)
If they have to decide independently three Nash
equilibria
(T,QT), (T,QQ) and (Q,TT)
29
Extensive to Normal form
30
Backward induction

• Solve the game by reducing from the final stage
• Eliminates Nash equilibria that are increadible
  threats

Blue
Q
T
Green
Green
incredible threat (Q, TT)
Q
Q
T
T
(-c,-c)
(1-c,0)
(0,1-c)
(0,0)
31
Subgame perfection

• Extends the notion of Nash equilibrium

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.
Blue
Q
Finding subgame perfect equilibria using backward
induction
T
Green
Green
Q
Q
T
T
Subgame perfect equilibria (T, QT) and (T, QQ)
(-c,-c)
(1-c,0)
(0,1-c)
(0,0)
32
Repeated games
33
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)

34
Payoffs Objectives in the repeated game

• finite-horizon vs. infinite-horizon games
• myopic vs. long-sighted repeated game

myopic
long-sighted finite
long-sighted infinite
payoff with discounting
is the discounting factor
35
Strategies in the repeated game

• usually, history-1 strategies, based on different
  inputs
• others behavior
• others and own behavior
• payoff

Example strategies in the Forwarders Dilemma
36
The Repeated Forwarders Dilemma
?
Green
Blue
?
Green
Forward
Drop
Blue
Forward
Drop
stage payoff
37
Analysis of the Repeated Forwarders Dilemma
(1/3)
infinite game with discounting
38
Analysis of the Repeated Forwarders Dilemma
(2/3)

• 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 !
39
Analysis of the Repeated Forwarders Dilemma
(3/3)
Theorem In the Repeated Forwarders Dilemma, if
both players play AllD, it is a Nash equilibrium.
Theorem In the Repeated Forwarders Dilemma,
both players playing TFT is a Nash equilibrium as
well.
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 Prisoners Dilemma
  (Repeated Forwarders Dilemma) in pairs
• number of rounds is finite but unknown
• TFT was the winner
• second round TFT was the winner again

41
Mechanism Design
42
Game Theory Analysis vs. Mechanism Design
analysis
system
behavior
mechanism design
43
Algorithmic Mechanism Design
behavior (truthful)
system (protocols)
algorithmic mechanism design

• Questions
• distributed implementation
• complexity
• convergence speed

44
Example 5 Stable matching in cellular networks
(CSM)
B
A
C
45
E5 Cellular stable matching (CSM)

• N base stations and M mobiles (MgtN)
• 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 NM
• Stable marriage problem

46
E5 Cellular stable matching simplified (CSM),
(1/2)

• NM

B
a
c
A
C
b
47
E5 Cellular stable matching simplified (2/2)

• preference matrix

mobiles
a
b
BS
c
A
B
C
48
E5 Existence of CSM

• Theorem (Gale-Shapley, 1962) There always exists
  a stable matching
• Proof (constructive) The Gale-Shapley algorithm

49
E5 CSM The Gale-Shapley algorithm
mobiles
a
b
BS
c
A
B
C
user-centric
network-centric
a
A
3 -
- 1
A
a
1 -
- 3
B
b
- 1
b
3 -
B
1 -
- 3
C
c
3 -
- 1
C
c
1 -
- 3
50
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

51
E5 Back to CSM

• The Gale-Shapley algorithm generalized
• quota at each BS (q2)

mobiles
b
a
d
e
c
f
BS
A
B
C
A
a
b
B
c
C
d
e
f
52
Stable Matching

• Applications
• WiFi networks
• Load-balancing for processors
• Students to schools
• Job-hunting

53
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

mobiles
a
b
BS
c
A
B
C
A
a

• Accept only your favorite peer

b
B
C
c
54
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

55
E6 ASM simplified
a
b

• preference matrix

d
c
mobiles
mobiles
a
c
d
b
a
b
c
d
a
b
c
d
56
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

57
Who is malicious? Who is selfish?
Harm everyone viruses,
Big brother
Selective harm DoS,
Spammer
Cyber-gangster phishing attacks, trojan horses,
Greedy operator
Selfish mobile station
? Both security and game theory backgrounds are
useful in many cases !!
58
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

59
Game 2 Guess what again?

• Remarks about the game
• truthful
• each (rational) student writes his best guess
• no (rational) student should have discussed the
  solution with friends

• Choose an integer number in 0,100
• The winner is gets closest to the 2/3 of the
  average
• Take a piece of paper and write
• your name (e.g., Julien FREUDIGER)
• your number
• Price 2 extra points in Quiz I
• if several winners, price is divided equally