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What is Game Theory? A Brief History of Game Theory Game Theory Basics with a suitable example An Interesting Analogy with Communication
Systems Non-Cooperative Game Theory in Wireless
Communications Research Coalitional Game Theory in Wireless Networks
Research Game Theory in Routing and Congestion
Control Game Theory in Network Security Scope of Further Research
Topics
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“…Game Theory is designed to address situations in which the outcome of a person’s decision depends not just on how they choose among several options, but also on the choices made by the people they are interacting with…”
“… Game theory is the study of the ways in which strategic interactions among economic (rational) agents produce outcomes with respect to the preferences (or utilities) of those agents ….”
Game Theory
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Game Theory: A Little History
• Cournot (1838), Bertrand (1883): Economics• J. von Neumann, O. Morgenstern (1944)
• “Theory of Games and Economic Behavior” • Existence of mixed strategy in 2-player game
• J. Nash (1950): Nash Equilibrium • (Nobel Prize in Economic Sciences 1994)
• Selten (1965): Subgame Perfect Equilibrium• Harsani (1967-68): Bayesian (Incomplete Information) Games
• The 80’s• Nuclear disarmament negotiations• Game Theory for Security (Burke)
• More recently:• Auction modeling, mechanism design• Routing, Congestion Control, Channel Access• Network Economics• Network Security• Biology
von Neumann 1903-1957
John F. Nash (1928)
O. Morgenstern 1902-1977
Game Theory in Communication Systems 5
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GAME = (P,A,U)
◦Players (P1; … ; PN): Finite number (N≥2) of decision makers.
◦Action sets (A1; … ;AN): player Pi has a nonempty set Ai of actions.
◦Payoff functions ui : A1x … xAN: R; i = 1;….;N
- materialize players’ preference, - take a possible action profile and assign to it a real number (von Neumann-Morgenstern).
Game Theory Basics
Game Theory in Communication Systems 6
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Cooperative and Non-Cooperative Symmetric and Asymmetric Zero-Sum and Non-Zero Sum Simultaneous and Sequential Static and Dynamic
Types of Games
Game Theory in Communication Systems 7
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What should Prisoner A do to minimize his maximum punishment when:
1. Prisoner B confesses? 2. Prisoner B stays quiet? What should Prisoner B do to minimize his
maximum punishment when:1. Prisoner A confesses? 2. Prisoner A stays quiet?
Strategizing!!: The Min-Max Algorithm
Game Theory in Communication Systems 9
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Routing, Congestion Control and Channel Access
Network Security
Application of Game Theory in Communication Systems
Game Theory in Communication Systems 11
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How do we apply an abstract Mathematical Tool like Game Theory in something as realistic like Communication Systems?
An Inevitable Question!!!
Game Theory in Communication Systems 12
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Communication Networks consists of several nodes which have to take decisions regarding several aspects like packet switching, packet forwarding, etc.
These nodes are considered as the players. Utility functions are often chosen to correspond to achieved connection rate or similar technical metrics.
An Interesting Analogy
Game Theory in Communication Systems 13
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Various studies have analyzed radio resource management problems in 802.11 WLAN networks.
In such random access studies, researchers have considered selfish nodes, who try to maximize their own utility (throughput) only, and control their channel access probabilities to maximize their utilities.
Medium Access Games for 802.11 WLAN
Game Theory in Communication Systems 15
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Power control refers to the process through which mobiles in CDMA cellular settings adjust their transmission powers so that they do not create unnecessary interference to other mobiles, trying, nevertheless, to achieve the required Quality of Service.
Power Control may be:1. Centralized2. Distributed
Power Control Games in CDMA systems
Game Theory in Communication Systems 16
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In such distributed settings, the mobiles can be considered to be selfish agents (players) who try to maximize their utilities (often modeled as corresponding throughputs).
Game theory is considered to be a powerful tool to study such scenarios.
Power Control Games in CDMA systems
Game Theory in Communication Systems 17
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Coalitional game theory is a branch of game theory that deals with cooperative behavior.
By cooperating, the players can strengthen their position in a given game as well as improve their utilities.
Coalitional game theory proves to be a powerful tool for modeling cooperative behavior in many wireless networking applications such as cognitive radio networks, wireless system, physical layer security, virtual MIMO.
Cooperative/Coalitional Game Theory in Wireless Networks Research
Game Theory in Communication Systems 18
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It’s a non-cooperative game where the goal of each user is to maximize it’s own bandwidth by selecting its path.
First, the existence of the Nash Equilibrium(NE) is determined because at NE no user has the incentive to change its routing strategy.
Routing in Max-Min Fair Networks: A Game Theoretic Approach
Game Theory in Communication Systems 19
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It is investigated how the selfish behavior of the users may affect the performance of the network as a whole.
A concept of observed available bandwidth is introduced on each link which allows a user to find a path with maximum bandwidth under max-min fair congestion control.
A game-based algorithm is formulated to compute the Nash Equilibrium (NE).
It is seen that by following the natural game course the network converges to an NE.
Routing in Max-Min Fair Networks: A Game Theoretic ApproachContd…
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Routing games◦ users determine
network routes◦ multi-path routing
and traffic splitting is possible
◦ users’ data rates are given and must be routed
Routing Games vs Congestion Control Games
Congestion games users determine their
data rate network routes are
given (single path)
Game Theory in Communication Systems 21
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Who is attacking our communication Systems?
Hackers Terrorists, Criminal Groups
Hacktivists
Disgruntled InsidersForeign Governments
?Game Theory in Communication
Systems 23
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Example: Remote AttackSecurity
Why Game Theory for Security?
Traditional Security Solutions
Attack Defense
Game Theory also helps:
Trust
Incentives
Externalities
Machine Intelligence
Attacker strategy 1 strategy 2 …..
Defender: strategy 1 strategy 2 …..
A mathematical problem! Solution tool: Game
Theory Predict players’ strategies, Build defense mechanisms, Compute cost of security, Understand attacker’s behavior, etc…
E.g.: Rate of Port Scanning IDS Tuning
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Example: Forwarder’s dilemma
Key Concepts
Forwarding has an energy cost of c (c<< 1)Successfully delivered packet: reward of 1
If Green drops and Blue forwards: (1,-c)If Green forwards and Blue drops: (-c,1)
If both forward: (1-c,1-c) If both drop: (0,0)
Each player is trying to selfishly maximize it’s net gain.
What can we predict?Game Theory in Communication
Systems 25
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Example: Forwarder’s dilemma
Key Concepts
Game:Players: Green, BlueActions: Forward (F), Drop (D)Payoffs: (1-c,1-c), (0,0), (-c,1), (1,-c)
Matrix representation: Actions of Green
Actions of Blue
Reward of BlueReward of Green
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Nash equilibrium:“…a solution concept of a game involving two
or more players, in which no player has anything to gain by changing his own strategy unilaterally…”
Equilibrium Concept
John F. Nash (1928)
Game Theory in Communication Systems 27
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3 Communication Security Game Models
Intruder Game
p
1-p
AliceTrudy
BobX Y Z
AvailabilityAttack
IntelligentVirus
aNormal traffic
Virus b
Xn
Detection
If Xn > l => Alarm
Game Theory in Communication Systems 28
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M’ ¹ M
Intruder (Trudy)
What if it ispossible that:
M
Intruder Game
Scenario:
Network
Source (Alice)
User (Bob)
M
Encryption is not always practical ….Formulation: Game between Intruder and User
Game Theory in Communication Systems 29
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Intruder Game: Binary
Y
• Payoffs:
• Strategies (mixed i.e. randomized)• Trudy: (p0,p1), Bob: (q0,q1)
Alice
TrudyBob
Intercept
• One shot, simultaneous choice game
• Nash Equilibrium?
Z
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What if the receiver (Bob) can verify the message?(by paying a cost and using a side secure channel)
p
1-p
AliceTrudy
BobX Y Z
Pay: V
Game Theory in Communication Systems 31
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Intelligent Virus GameScenario
aNormal traffic
Virus b
Xn
Detection
If Xn > l => Alarm, ….Assume a known
Detection system: choose l to minimize cost of infection + clean up
Virus: choose b to maximize infection cost
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Intelligent Virus Game (IDS)
Smart virus designer picks very large b, so that the cost is always high ….Regardless of !l
0 10 20 30 40 50 60 70 80 90 1001
1.2
1.4
1.6
1.8
2
2.2
2.4
(/sec)
Virus G
ain
: Lin
ear
0=5
0=10
0=15
b
Scenario
aNormal traffic
Virus b
Xn
Detection
If Xn > l => Alarm, ….
Game Theory in Communication Systems 33
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Intelligent Virus Game (IPS) Modified Scenario
aNormal traffic
Virus b
XnDetection
If Xn > l => Alarm
• Detector: buffer traffic and test threshold• Xn < l process• If Xn > l Flush & Alarm
• Game between Virus (b) and Detector (l)
Game Theory in Communication Systems 34
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Consider a tree with € links and n nodes. Let Ƭ be the set of spanning trees.
To get all the nodes connected in a cycle-free way, the Network Manager/Defender chooses a spanning tree TϵƬ of the network
The attacker simultaneously chooses a link eϵ€ to attack
The attacker wins if the attacked link belongs to the chosen spanning tree; the Defender wins elsewise
Tree-Link Game
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Game( modeled as a one-shot 2 player game)◦ Graph = (nodes V, links E, spanning trees T)
Defender: chooses T T
Attacker: chooses e E (+ “No Attack”)
◦ Rewards Defender: -1eT Attacker: 1eT - µe (µe cost of attacking e)
Model
Example:
Defender: 0Attacker: - µ2
Defender: -1Attacker: 1- µ1
– Defender : choose a distribution on T, to minimize the expected attack loss
--Attacker: Choose a distribution on E, to maximize the attack gain
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Let’s Play a Game!
Graph Most vulnerable links
Chance 1/2
Chance 4/7>1/2
a)
b)
c)
Assume: zero attack cost µe=0
1/2
1/2
1/7
1/7
1/7
1/7
1/71/7
1/7
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Critical Subset of Links
• Definition 1&2: For any nonempty subset E Ε
1. M(E) = min{| TE|, TТ} (minimum number of links E has in common with any spanning tree)
2. Vulnerability of E (E) = M(E)/|E| (minimum fraction of links E has in common with any spanning tree)
• Definition 3: A nonempty subset C Ε is said to be critical if (C) = maxE Ε((E))
(C has maximum vulnerability) vulnerability of graph ((G)) := vulnerability of critical subset
123 4
567
E={1,4,5}|T E|=2M(E) =1
Defender: choose trees that minimally cross critical subset
(E) = 1/3
(G)=1 (G)=1/2 (G)=4/7
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Critical Subset Attack Theorem
Theorem 1:There exists a Nash Equilibrium where
• Attacker attacks only the links of a critical set C, with equal probabilities
• Defender chooses only spanning trees that have a minimal intersection with C, and have equal likelihood of using each link of C, no larger than that of using any link not in C. [Such a choice is possible.]
There exists a polynomial algorithm to find C [Cunningham 1982]
Theorem generalizes to a large class of games.Game Theory in Communication
Systems 40
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Some implications
If ν ≤ 0: Attacker: “No Attack”Defender can invest to make µ highDeter attacker from attacking• Need to randomize choice of tree
Edge-Connectivity is not always the right metric!
ν= 3/4 ν= 2/3 ν= 3/5
2/3 > 3/5
Network in b) is more vulnerable than network in c)Additional link
Network Design
a) b) c)
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Availability Games◦ Critical set
Vulnerability ((G)): a metric more refined than edge-connectivity
Analyzing NE helps determine most vulnerable subset of links Importance in topology design Polynomial-time algorithm to compute critical set
◦ Generalization Set of resources for mission critical task
Most vulnerable subset of resources.
Conclusion
Intruder and Intelligent Virus Games:• Most aggressive attackers are not the most dangerous ones• Mechanisms to deter attackers from attacking
Game Theory helps for a better understanding
of the Security problem!
Game Theory in Communication Systems 42
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A certain number of issues◦ Costs model
Not based on solid ground
◦ Mixed strategy equilibrium How to interpret it?
◦ Nash equilibrium computation In general difficult to compute
This is an “young” research field!
Game Theory for Airport SecurityARMOR (LAX)
Airports create security systems and terrorists seek out breaches.
Placing checkpoint Allocate canine units
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• Repeated versions of the games– More realistic models– Applications: Attack Graphs
• Collaborative Security– Team of Attackers vs Team of Defenders– Trust and Security– Role of Information
• Security of Cloud Computing– Are you willing to give away your information?
• Policing the Internet– Who is responsible for security flaws?
Further Scope of Research
Game Theory in Communication Systems 44