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Non Cooperative Games Five Lectures
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Transcript of Non Cooperative Games Five Lectures
CSE593 Game Theory for Computer Science
Dr Ganesh Neelakanta Iyer
http://ganeshniyer.com
• Introduction to the course
• Grade details
• Non-cooperative Games
– Normal form
– Extensive form
– Applications in CS
– Coordination games
What is Game Theory About?
• Analysis of situations where conflict of interests are present
Goal is to prescribe how conflicts can be resolved
2
2
Game of Chicken
driver who steers away looses
What should drivers do?
©All Rights Reserved, Ganesh Neelakanta Iyer August 2012
4
Game Theory:
Applications
• Economics: Oligopoly markets, Mergers
and acquisitions pricing, auctions
• Political Science: fair division, public
choice, political economy
• Biology: modeling competition between
tumor and normal cells, Foraging bees
• Sports coaching staffs: run vs pass or
pitch fast balls vs sliders
• Computer Science: Distributed systems,
Computer Networks, AI, e-Commerce
Why Game Theory (GT) for CS?
• Integral part of – AI, e-commerce, networking
• Internet calls for analysis and design of systems that span multiple entities, each with its own information and interests – GT is is by far the most developed theory of such
interactions
• Technology push – The mathematics and scientific mind-set of game
theory are similar to those that characterize many computer scientists
Computer Science and Game Theory. Y. Shoham. Communications of the ACM, 51(5), August, 2008
Course Objectives • Introduce the concepts of Game Theory
• Emphasize on Modern Computer Science Applications
– Computer Networks, Cloud Computing
– Social Media, Internet Marketing Strategies
– Security Mechanisms
Rough Syllabus Basic concepts, definitions, utilities, classification of games, classic examples, typical application scenarios. Non-cooperative games: Extensive form games, dominant strategy equilibrium, Nash equilibrium and related concepts, Game theoretic modeling, analysis, and mitigation of security risks Repeated games: monitoring, discounting, Cloud provider’s reliability. Bayesian Games: Games of In-complete Information, Signaling Games, Applications, Congestion games: Multi-cast routing, Potential Games, Energy minimization in mobile cloud computing. Bargaining Theory: Nash Bargaining Solution, Resource allocation in Cloud computing, Multimedia resource management Coalitional Game theory: Core, Shapely Value, Revenue maximization in Mobile Cloud networks, cooperation between multiple networks. Mechanism design: Social Choice Theory, Incentive compatible mechanisms, profit maximization, cost sharing, pricing and investment decisions in Internet, multi-cast pricing, Mechanism Design and Computer Security, Multi-cast cost sharing Auction Theory: Social media marketing (Google Ad Words, Facebook Ads), Cloud brokers, online auctions (eBay), sponsored search. Network Games: Routing, flow control, congestion control, revenue sharing between Internet service providers, fairness, charging schemes and rate control Evolutionary Games: ESS, Congestion control, Application deployment in Cloud Computing Coordination games – Sustaining marketer-consumer cooperation. Miscellaneous topics: Combinatorial Games, NIM Games, Future directions and remarks
PRE-REQUISITE: Basic mathematics, Basic Computer Science
Outline of the Course
• Non co-operative Games – Two player Games – Repeated Games – Congestion Games – Baysean Games
• Cooperative Games – Auctions – Bargaining – Coalitions
• Advanced Topics – Social Choice and Voting Theory – Evolutionary Games – Mechanism Design – Network Games – NIM Games
http://messengyr.deviantart.com/art/GAME-THEORY-149275884
Grading Structure
• Exams: – 2 Mid-terms (15%+15%) – Final Exam (40%)
• Assignment 1: (10%) – Given a specific paper, you need to write a critique on
it
• Assignment 2: (20 %) – Option 1: Given a CS topic, detailed survey on how
game theory is applied and write a technical report – Option 2: Write a program to model certain game
theory topic suggested by lecturer
Grading Absolute Grading
Grade Marks
A 95-100
A- 85-94
B 75-84
B- 65-74
C 55-64
C- 45-54
D 40-44
Fail
0-39
Key Takeaways
• Understand the importance of Game Theory in modern Computer Science
• Understand to use Game Theory to model various research problems
• Model various real-life situations even outside of CS domain
– E.g. Airline pricing systems, politics
How the course will be?
• Some Mathematics
– Minimal equation solving
– No calculus
– No Algebra
– Some discrete mathematics
– Some probability
• Some real-life applications modeling
• If you wish to code, yes coding
Intended audience
• This course is NOT for
– Those who came to know more on Computer Games
– Those who expect a lot of coding and programming
• This course is for
– Those who wish to understand how many real-world things work
– Those who are doing/wish to do research in Computer Science (M.S., PhD etc)
References • Slides and additional notes/materials will be given wherever required
• No single text book could be sufficient
• Game Theory in Wireless and Communication Networks: Theory, Models, and Applications, Zhu Han, Dusit Niyato, Walid Saad, Tamer Baar, Are Hjørungnes, Cambridge Publications, 2011
What is Game Theory?
• Study of how people interact and make decisions
• “…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…”
• The study of strategic interactions among economic (rational) agents and the outcomes with respect to the preferences (or utilities) of those agents
TCP Back off Game TCP Congestion Control - AIMD
Algorithm AIMD
Additive Increase Multiplicative Decrease
Increment Congestion Window by one packet per RTT Linear increase
Divide Congestion Window by two whenever a timeout occurs Multiplicative decrease
Source Destination
…
60
20
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0
KB
T ime (seconds)
70
30 40 50
10
10.0
TCP Backoff Game
• Should you send your packets using – Correctly-implemented TCP ( which has a “backoff”
mechanism) or
– using a defective implementation (which doesn’t)?
• This problem is an example of what we call a two-player game: – Both use a correct implementation: both get 1 ms delay
– One correct, one defective: 4 ms for correct, 0 ms for defective
– Both defective: both get a 3 ms delay.
Self Interested Agents
• What does it mean to say that an agent is self-interested?
– not that they want to harm others or only care about themselves
– only that the agent has its own description of states of the world that it likes, and acts based on this description
Self Interested Agents
• Each such agent has a utility function
– quantifies degree of preference across alternatives
– explains the impact of uncertainty
– Decision-theoretic rationality: act to maximize expected utility
20
What is a game?
Players: who are the decision makers? • People? Governments? Companies? • Somebody employed by a Company?...
Actions: What can the players do? • Enter a bid in an auction? • Decide whether to end a strike? • Decide when to sell a stock? • Decide how to vote? Strategies: Which action did I choose • actions which a player chooses to follow • I will sell the stock today, I will vote for NOTA
Payoffs: what motivates the players? • Do they care about some profit? • Do they care about other players?... Outcome: What is the result? • Determined by mutual choice of strategies
Defining Games: Two standard representations
• Normal Form (a.k.a. Matrix Form, Strategic Form) List what payoffs get as a function of their actions – It is as if players moved simultaneously
– But strategies encode many things...
• Extensive Form Includes timing of moves (later in course) – Players move sequentially, represented as a tree
• Chess: white player moves, then black player can see white’s move and react…
– Keeps track of what each player knows when he or she makes each decision
• Poker: bet sequentially – what can a given player see when they bet?
Defining Games: The Normal Form
• Finite, n-person normal form game: ⟨N, A, u⟩:
– Players: N = {1, … , n} is a finite set of n, indexed by i
– Action set for player i, Ai :
• a = (a1, … ,an) ∈ A = A1 X … X An is an action profile
– Utility function or Payoff function for player i: ui : A→ R
• u = (u11, …, un) , is a profile of utility functions
Normal Form Games The Standard Matrix Representation
• Writing a 2-player game as a matrix: – “row” player is player 1, “column” player is
player 2
– rows correspond to actions a1 ∈ A1, columns
correspond to actions a2 ∈ A2
– cells listing utility or payoff values for each
player: the row player first, then the column
• . QUESTION: Write TCP Backoff Game in matrix form
TCP Backoff Game in matrix form
Correct Defective
Correct
Defective Pla
ye
r 1
Player 2
-1,-1
-3,-3 0,-4
-4,0
• Should you send your packets using – Correctly-implemented TCP ( which has a “backoff” mechanism) or
using a defective implementation (which doesn’t)?
• This problem is an example of what we call a two-player game:
– Both use a correct implementation: both get 1 ms delay – One correct, one defective: 4 ms for correct, 0 ms for defective – Both defective: both get a 3 ms delay.
A Large Collective Action Game
• Players: N = {1, . . . , 10,000,000}
• Action set for player i Ai = {Revolt, Not}
• Utility function for player i: – ui(a) = 1 if #{j : aj = Revolt} ≥ 2,000,000
– ui(a) = −1 if #{j : aj = Revolt} < 2,000,000 and ai = Revolt
– ui(a) = 0 if #{j : aj = Revolt} < 2,000,000 and ai = Not • . Game
26
Prisoner’s Dilemma
• Two suspects arrested for a crime
• Prisoners decide whether to confess or not to confess
• If both confess, both sentenced to 3 months of jail • If both do not confess, then both will be sentenced
to 1 month of jail • If one confesses and the other does not, then the
confessor gets freed (0 months of jail) and the non-confessor sentenced to 9 months of jail
• What should each prisoner do?
27
Prisoner’s Dilemma: Revisited
• Two suspects arrested for a crime
• Prisoners decide whether to confess or not to confess
• If both confess, both sentenced to 3 months of jail
• If both do not confess, then both will be sentenced to 1 month of jail
• If one confesses and the other does not, then the confessor gets freed (0 months of jail) and the non-confessor sentenced to 9 months of jail
• What should each prisoner do?
Confess Not
Confess
Confess
Not
Confess Pri
so
ne
r 1
Prisoner 2
-3,-3
-1,-1 -9,0
0,-9
28
Prisoner’s Dilemma: Nash Equilibrium • Each player’s predicted strategy is the best response to the predicted strategies
of other players
• No incentive to deviate unilaterally
• Strategically stable or self-enforcing
Confess Not
Confess
Confess
Not
Confess Pri
so
ne
r 1
Prisoner 2
-3,-3
-1,-1 -9,0
0,-9
http://www.environmentalgraffiti.com/people/news-are-humans-selfish-concept-homo-economicus
PD in general form
• Prisoner’s dilemma is any game
with c > a > d > b
C D
C
D
Pla
ye
r 1
Player 2
a,a b, c
c, b d, d
Games of Pure Competition
• Players have exactly opposed interests – There must be precisely two players (otherwise they
can’t have exactly opposed interests)
• For all action profiles a ε A, u1(a) + u2(a) = c for some constant c – Special case: zero sum
• Thus, we only need to store a utility function for one player – in a sense, we only have to think about one player’s
interests
Let’s play a game
32
Rock-paper-scissors game
• A probability distribution over the pure strategies of the game
• Rock-paper-scissors game
• No pure strategy Nash equilibrium
• One mixed strategy Nash equilibrium – each player plays rock, paper and scissors each with 1/3 probability
Rock-paper-scissors game
Rock Paper Scissor
Rock 0,0 -1,1 1,-1
Paper 1,-1 0,0 -1,1
Scissor -1,1 1,-1 0,0
34
Definition: Normal form of a Game
• The normal-form (also called strategic-form) representation of an n-player game specifies the players' strategy spaces S1, …, Sn and their payoff functions u1…un. We denote this game by
G = {S1,…, Sn; u1,…, un} • Let (s1,…,sn) be a combination of strategies, one for each player. Then
ui(s1,…,sn) is the payoff to player i if for each j = 1,…,n, player j chooses strategy sj.
• The payoff a player depends not only on his own action but also on the actions of others! This inter-dependence is the essence of games!
QUESTION: Write the normal form representation of the game “Prisoner’s Dilemma.
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35
Question: Normal form representation of Prisoner’s dilemma
G = {S1,S2; u1,u2}
S1 = {Confess, Not Confess} = S2
u2(C,NC)= -9, u1(C,NC)= 0, …
Confess Not
Confess
Confess
Not
Confess Pri
so
ne
r 1
Prisoner 2
-3,-3
-1,-1 -9,0
0,-9
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36
Some Notations
http://ganeshniyer.com
niii
niii
iiniii
SXSXSXXSS
sssss
ssssssss
... ...
)...,,,...,(
),()...,,,,...,(
111
111
111
37
Definition: Strictly Dominated Strategy
In a normal-form game G = {S1,…, Sn; u1,…, un}, let si’ and si’’ ϵ Si. Strategy si’ is strictly dominated by strategy si” (or strategy si” strictly dominates strategy si’) if for each feasible combination of other player’s strategies, player i’s payoffs from playing si’ is strictly less than the payoff from playing si”. i.e.,
Rational players do not play strictly dominated strategies since they are always not optimal no matter what strategies others would choose.
QUESTION: What is the strictly dominated strategy and strictly dominant strategy for the game “Prisoner’s Dilemma?
iiiiiiii Ssssussu " ),,(),( "'
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Nash Equilibrium
NASH EQUILIBRIUM An important concept in game theory, a Nash equilibrium
occurs when each player is pursuing their best possible strategy in the full
knowledge of the other players’ strategies. A Nash equilibrium is reached when
nobody has any incentive to change their strategy. It is named after John Nash, a
mathematician and Nobel prize-winning economist.
John F. Nash, 1928 -
39
Definition: Nash Equilibrium
In the n-player normal-form game G = {S1,…, Sn; u1,…, un}, the strategies (s1*,…,sn*) are a Nash Equilibrium if:
QUESTION: What is the Nash Equilibrium strategy for the game of Prisoner’s Dilemma?
nissussu iii
Ss
iii
ii
..1),(),( ***
max "
In the n-player normal-form game G = {S1,…, Sn; u1,…, un}, the strategies (s1*,…,sn*) are a Nash Equilibrium if:
• What to do when it is not obvious what the equilibrium is?
• In some cases, we can eliminate dominated strategies.
• These are strategies that are inferior for every opponent action.
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Nash Equilibrium • “A strategy profile is a Nash Equilibrium if
and only if each player’s prescribed strategy is a best response to the strategies of others” – Equilibrium that is reached even if it is not the
best joint outcome
4 , 6 0 , 4 4 , 4
5 , 3 0 , 0 1 , 7
1 , 1 3 , 5 2 , 3
Player 2
L C R
Player 1
U
M
D
Strategy Profile:
{D,C} is the Nash
Equilibrium
**There is no
incentive for
either player to
deviate from this
strategy profile
Example
• A 3x3 example:
a b
b 80,26
57,42
35,12
73,25
Row
Column
a
c
c
66,32
32,54
28,27 63,31 54,29
41 http://ganeshniyer.com
Example
• A 3x3 example:
a b
b 80,26
57,42
35,12
73,25
Row
Column
a
c
c
66,32
32,54
28,27 63,31 54,29
c dominates a for the column player
42 http://ganeshniyer.com
Example
• A 3x3 example:
a b
b 80,26
57,42
35,12
73,25
Row
Column
a
c
c
66,32
32,54
28,27 63,31 54,29
b is then dominated by both a and c for the row player.
43 http://ganeshniyer.com
Example • A 3x3 example:
a b
b 80,26
57,42
35,12
73,25
Row
Column
a
c
c
66,32
32,54
28,27 63,31 54,29
Given this, b dominates c for the column player – the column player will always play b.
44 http://ganeshniyer.com
Example
• A 3x3 example:
a b
b 80,26
57,42
35,12
73,25
Row
Column
a
c
c
66,32
32,54
28,27 63,31 54,29
Since column is playing b, row will prefer c.
45 http://ganeshniyer.com
Example
a b
b 80,26
57,42
35,12
73,25
Row
Column
a
c
c
66,32
32,54
28,27 63,31 54,29
We verify that (c,b) is a Nash Equilibrium by observation: If row plays c, b is the best response for column. If column plays b, c is the best response by row.
46 http://ganeshniyer.com
Example #2
• You try this one:
a b
b 1,2
1,1
4,1
2,2
Row
Column
a
c
4,0
3,5
47 http://ganeshniyer.com
Summary of Nash Equlibrium
• Each player’s action maximizes his or her payoff given the actions of the others.
• Nobody has an incentive to deviate from their action if an equilibrium profile is played.
• Someone has an incentive to deviate from a profile of actions that do not form an equilibrium.
Implications
• In life, we react to other people’s choices in order to increase our utility or happiness – Ignoring a younger sibling who is irritating
– Accepting an invitation to go to a baseball game
– Proxy for a friend and take turns for the classes
• Once we react, the other person reacts to our reaction and life goes on – One stage games are rare in life
• Very rarely are we in a “NE” for any aspect of our lives – There is almost always a choice that can better our current
utility
Game with no pure NE
Left Right
Left 1/0 0/1
Right 0/1 1/0
Penalty Taker
Goalie
Penalty taking in football (soccer)
https://www.youtube.com/watch?v=RqGb1Gx0t9U#t=41
Games with multiple NE Compact Disc battle
• Battle for competing technical standards • Sony and Philips competing for a standard for CD
in late 1970s • Each wanted their own system
Std A Std B
Std A 5/4 1/0
Std B 0/1 4/5
Philips
Sony
In the end, the result was a mix of both
52
Battle of Sexes
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• At the separate workplaces, Ram and Sita must choose to attend either cricket or a movie in the evening.
• Both Ram and Sita know the following: Both would like to spend the evening together.
But Ram prefers the cricket
Sita prefers the movie
2 , 1 0 , 0
0 , 0 1 , 2 Ram
Sita
Movie
Cricket
Movie
Cricket
53
Mixed Strategy
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• A mixed strategy of a player is a probability distribution over player’s (pure) strategies. A mixed strategy for Ram is a probability distribution (p, 1-
p), where p is the probability of playing cricket, and 1-p is that probability of watching movie.
If p=1 then Ram actually plays cricket. If p=0 then Ram actually watches movie.
Battle of sexes Sita
Cricket Movie
Ram Cricket (p) 2 , 1 0 , 0
Movie (1-p) 0 , 0 1 , 2
54
Matching Pennies
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• Each of the two players has a penny. • Two players must simultaneously choose whether to
show the Head or the Tail. • Both players know the following rules: If two pennies match (both heads or both tails) then player
2 wins player 1’s penny. Otherwise, player 1 wins player 2’s penny.
-1 , 1 1 ,-1
1 ,-1 -1 , 1
Player 1
Player 2
Tail
Head
Tail
Head
Solving Matching Pennies
• Player 1’s expected payoffs If Player 1 chooses Head, -q+(1-q)=1-2q
If Player 1 chooses Tail, q-(1-q)=2q-1
Player 2
Head Tail
Player 1 Head -1 , 1 1 , -1
Tail 1 , -1 -1 , 1 q 1-q
1-2q
2q-1
Expected
payoffs
r
1-r
55 http://ganeshniyer.com
1 q
r
1
1/2
1/2
Solving Matching Pennies
• Player 1’s best response B1(q): For q<0.5, Head (r=1) For q>0.5, Tail (r=0) For q=0.5, indifferent (0r1)
Player 2
Head Tail
Player 1 Head -1 , 1 1 , -1
Tail 1 , -1 -1 , 1
q 1-q
1-2q
2q-1
Expected
payoffs
r
1-r
56 http://ganeshniyer.com
Solving Matching Pennies
• Player 2’s expected payoffs If Player 2 chooses Head, r-(1-r)=2r-1
If Player 2 chooses Tail, -r+(1-r)=1-2r
Player 2
Head Tail
Player 1 Head -1 , 1 1 , -1
Tail 1 , -1 -1 , 1
1-2q
2q-1
Expected
payoffs
r
1-r
q 1-q
Expected
payoffs 2r-1 1-2r
57 http://ganeshniyer.com
Solving Matching Pennies
• Player 2’s best response B2(r):
For r<0.5, Tail (q=0)
For r>0.5, Head (q=1)
For r=0.5, indifferent (0q1)
Player 2
Head Tail
Player 1 Head -1 , 1 1 , -1
Tail 1 , -1 -1 , 1 q 1-q
1-2q
2q-1
Expected
payoffs
r
1-r
Expected
payoffs 2r-1 1-2r
1 q
r
1
1/2
1/2
58 http://ganeshniyer.com
1 q
r
1
1/2
1/2
Solving Matching Pennies
• Player 1’s best response B1(q):
For q<0.5, Head (r=1)
For q>0.5, Tail (r=0)
For q=0.5, indifferent (0r1)
• Player 2’s best response B2(r):
For r<0.5, Tail (q=0)
For r>0.5, Head (q=1)
For r=0.5, indifferent (0q1)
Check r = 0.5 B1(0.5) q = 0.5 B2(0.5)
Player 2
Head Tail
Player 1 Head -1 , 1 1 , -1
Tail 1 , -1 -1 , 1
r
1-r
q 1-q
Mixed strategy
Nash equilibrium
59 http://ganeshniyer.com
Mixed Strategy
• Mixed Strategy:
A mixed strategy of a player is a probability distribution over player’s (pure) strategies.
Definition Let G be a n-player game with strategy sets S1,
S2 ,.., Sn. A mixed strategy i for player i is a probability
distribution on Si. If Si has a finite number of pure strategies,
i.e. } ... , ,{ 21 iiKiii sssS then a mixed strategy is a function
ii S: such that 1)(1
iK
jiji s . We write this mixed
strategy as ))( ..., ),( ),(( 21 iiKiiiii sss .
60 http://ganeshniyer.com
Mixed Strategy: Example
• Matching pennies
– Player 1 has two pure strategies: H and T ( 1(H)=0.5, 1(T)=0.5 ) is a Mixed strategy. That is, player 1 plays H and T with probabilities 0.5 and 0.5, respectively. ( 1(H)=0.3, 1(T)=0.7 ) is another Mixed strategy. That is, player 1 plays H and T with probabilities 0.3 and 0.7, respectively.
61 http://ganeshniyer.com
Find Mixed strategy NE • Row chooses “top” with probability p and bottom with probability 1-p.
• Column chooses “left” with probability q and “right” with probability 1-q.
Left
Right
Top
4, -4
1, -1
Bottom
2, -2
3, -3
• Players choose strategies to make the other indifferent.
– 4q+1(1-q)=2q+3(1-q)
– -4p-2(1-p)=-1p-3(1-p)
• The MS-NE is: p=.25, q=.5.
– The expected value of either Row strategy is 2.5 and of either Column strategy is –2.5
© 2003 Arthur Lupia
Mixed strategy NE
• A mixed strategy Nash Equilibrium does not rely on an player flipping coins, rolling, dice or otherwise choosing a strategy at random.
• Rather, we interpret player j’s mixed strategy as a statement of player i’s uncertainty about player j’s choice of a pure strategy.
• In games of pure conflict, where there is no pure strategy Nash equilibria, the mixed strategy equilibriums are chosen in a way to make the other player indifferent between all of their mixed strategies.
– To do otherwise is to give others the ability to benefit at your expense. Information provided to another player that makes them better off makes you worse off.
Rock, paper, scissor game
• Find the mixed strategy equilibrium
http://ganeshniyer.com 64
Equilibrium Concepts Move sequence:
static
dynamic
Information:
complete
incomplete
complete
incomplete
Appropriate Nash Equilibrium concept
Generic
Bayesian Subgame perfect
Perfect Bayesian, sequential
•What is the set of self enforcing best responses?
•The equilibrium concepts build upon those of simpler games.
• Each subsequent concept, while more complex, also allows more precise conclusions from increasingly complex situations
SELF-PRACTICE: Choosing Classes!
Suppose that you and a friend are choosing classes for the semester. You want to be in the same class. However, you prefer Microeconomics while your friend prefers Macroeconomics. You both have the same registration time and, therefore, must register simultaneously
Micro Macro
Micro 2 1 0 0
Macro 0 0 1 2 Pla
yer
A
Player B
What is the equilibrium to this game?
NOTE: This solution needs knowledge of convex optimization and concepts of Lagrange multipliers and KKT conditions. It is NOT IN SCOPE for EXAM
Note
• Solving “Choosing classes” requires knowledge of convex optimization and Lagrange multipliers. It is not in scope for this subject. However, if time permits, we will discuss it at the end of the semester
http://ganeshniyer.com 67
Micro Macro
Micro 2 1 0 0
Macro 0 0 1 2 Pla
yer
A
Player B
If Player B chooses Micro, then the best response for Player A is Micro
If Player B chooses Macro, then the best response for Player A is Macro
The Equilibrium for this game will involve mixed strategies!
SELF-PRACTICE: Choosing Classes!
Suppose that Player A has the following beliefs about Player B’s Strategy
Macro
Micro
r
l
Pr
Pr
Probabilities of choosing Micro or Macro
Player A’s best response will be his own set of probabilities to maximize expected utility
Macrop
Microp
b
t
Pr
Pr
)1()0()0()2(,
rlbrltpp
ppMaxbt
SELF-PRACTICE: Choosing Classes!
btbtrbltbt pppppppp 2112),,(
)1()0()0()2(,
rlbrltpp
ppMaxbt
Subject to
0
0
1
b
t
bt
p
p
pp Probabilities always have to sum to one
Both classes have a chance of being chosen
btbtrbltbt pppppppp 2112),,(
First Order Necessary Conditions
02 1 l
02 r
01 bt pp
01 tp
02 bp
02
01 0tp
0bp
0
0
b
t
p
p021
1
2
lr
rl
3
2
3
1 rl
Best Responses
3
2
3
1 rl
What this says is that if Player A believes that Player B will select Macro with a 2/3 probability, then Player A is willing to randomize between Micro and Macro
rbltpp
ppMaxbt
2,
Notice that if we 1/3 and 2/3 for the above probabilities, we get
bt
ppppMax
bt 3
2
3
2
,
If Player B is following a 1/3, 2/3 strategy, then any strategy yields the same expected utility for player B
3
2
3
1 rl pp
3
1
3
2 bt pp
0 1 rl pp 0 1 bt pp
1 0 rl pp 1 0 bt pp
It’s straightforward to show that there are three possible Nash Equilibrium for this game
Both always choose Micro
Both always choose Macro
Both Randomize between Micro and Macro
Note that the strategies are known with certainty, but the outcome is random!
Recap
• We learnt simultaneous Games
• Useful links: – http://www.eprisner.de/MAT109/Mixedb.html
– http://levine.sscnet.ucla.edu/Games/zerosum.htm
• What next?
– Sequential (dynamic) games of complete information
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Dynamic Games of Complete Information
Game Tree
Guess this place…
• Sequential moves are strategies where there is a strict order of play.
• Perfect information implies that players know everything that has happened prior to making a decision.
• Complex sequential move games are most easily represented in extensive form, using a game tree.
• Chess is a sequential-move game with perfect information.
Overview of Sequential Games
The E.T. “chocolate wars”
In the movie E.T. a trail of Reese's Pieces, one of Hershey's chocolate brands, is used to lure the little alien into the house. As a result of the publicity created by this scene, sales of Reese's Pieces tripled, allowing Hershey to catch up with rival Mars.
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Chocolate wars…the details
– Universal Studio's original plan was to use a trail of Mars’ M&Ms and charge Mars $1mm for the product placement.
– However, Mars turned down the offer, presumably because it thought $1mm was high.
– The producers of E.T. then turned to Hershey, who accepted the deal, which turned out to be very favorable to them (and unfavorable to Mars).
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Formal analysis of the chocolate wars
• Suppose:
– Publicity from M&M product placement increases Mars’ profits by $800 k, decreases Hershey’s by $100 k
– Publicity from Reases Pieces product placement increases Hershey’s profits by $1.2 m, decreases Mars’ by $500 k
– No product placement:
“business as usual”
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Extensive Form Games
• Also known as tree-form games
• Best to describe games with sequential actions
• Decision nodes indicate what player is to move (rules)
• Branches denote possible choices
• End nodes indicate each player’s payoff (by order of appearance)
• Games solved by backward induction (more on this later)
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Chocolate wars
Page 81
– Publicity from M&M product placement increases Mars’ profits by $800 k, decreases Hershey’s by $100 k
– Publicity from Reases Pieces product placement increases Hershey’s profits by $1.2 m, decreases Mars’ by $500 k
– No product placement: “business as usual”
[-500, 200]
[0, 0]
[-200, -100] buy
not buy
not buy
buy M
H
H
Chocolate wars [-500, 200]
[0, 0]
[-200, -100] buy
not buy
not buy
buy M
H
H
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Equilibrium strategies
– H chooses “buy”
– Anticipating H’s move, M chooses “buy”
Chocolate wars: summary
– Think about your competitor: Mars should think about Hershey, and vice versa
– Timing matters: Hershey had the last move. Outcome would be different if order of moves were different
– Key business insight: part of the benefit to Mars was to keep the opportunity from Hershey
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Entry Game Airline War on 1990s
• German domestic sector: Lufthansa is a monopolist
• Deregulation
– British Airways considered entering the market
– Deutsche BA
– Lufthansa threatened for a
price war
– Was this a credible threat?
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http://web.econ.ku.dk/cie/Workshops/Norio%20IV/pdf%20-%20papers/Hueschelrath-Predation.PDF
Entry Game • An incumbent monopolist faces the possibility of entry by a
challenger. • The challenger may choose to enter or stay out. • If the challenger enters, the incumbent can choose either to
accommodate or to fight. • The payoffs are common knowledge.
Challenger
In Out
Incumbent
A F 1, 2
2, 1 0, 0
The first number is the
payoff of the challenger.
The second number is the
payoff of the incumbent.
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Recap
Games
Non-cooperative Games
Static games of complete
infromation
Sequential games of complete information
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Sequential-Move Matching Pennies
• Each of the two players has a penny.
• Player 1 first chooses whether to show the Head or the Tail.
• After observing player 1’s choice, player 2 chooses to show Head or Tail
• Both players know the following rules: If two pennies match (both
heads or both tails) then player 2 wins player 1’s penny.
Otherwise, player 1 wins player 2’s penny.
Player 1
Player 2
H T
-1, 1 1, -1
H T
Player 2
H T
1, -1 -1, 1
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Dynamic (or Sequential-Move) Games of Complete Information
• A set of players
• Who moves when and what action choices are available?
• What do players know when they move?
• Players’ payoffs are determined by their choices.
• All these are common knowledge among the players.
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Definition: Extensive-Form Representation
• The extensive-form representation of a game specifies: the players in the game when each player has the move what each player can do at each of his or her
opportunities to move what each player knows at each of his or her
opportunities to move the payoff received by each player for each
combination of moves that could be chosen by the players
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Dynamic Games of Complete and Perfect Information
• Perfect information
All previous moves are observed before the next move is chosen.
A player knows Who has moved What before she makes a decision
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Game tree
• A game tree has a set of nodes and a set of edges such that
each edge connects two nodes (these two nodes are said to be adjacent)
for any pair of nodes, there is a unique path that connects these two nodes
x0
x1 x2
x3 x4 x5 x6
x7 x8
a node
an edge connecting
nodes x1 and x5
a path from
x0 to x4
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Game tree • A path is a sequence of distinct
nodes y1, y2, y3, ..., yn-1, yn such that yi and yi+1 are adjacent, for i=1, 2, ..., n-1. We say that this path is from y1 to yn.
• We can also use the sequence of edges induced by these nodes to denote the path.
• The length of a path is the
number of edges contained in the path.
• Example 1: x0, x2, x3, x7 is a path of length 3.
• Example 2: x4, x1, x0, x2, x6 is a path of length 4
x0
x1 x2
x3 x4 x5 x6
x7 x8
a path from
x0 to x4
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Game tree • There is a special node x0
called the root of the tree which is the beginning of the game
• The nodes adjacent to x0 are successors of x0. The successors of x0 are x1, x2
• For any two adjacent nodes, the node that is connected to the root by a longer path is a successor of the other node.
• Example 3: x7 is a successor of x3 because they are adjacent and the path from x7 to x0 is longer than the path from x3 to x0
x0
x1 x2
x3 x4 x5 x6
x7 x8
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Game tree
• If a node x is a successor of another node y then y is called a predecessor of x.
• In a game tree, any node other than the root has a unique predecessor.
• Any node that has no successor is called a terminal node which is a possible end of the game
• Example 4: x4, x5, x6, x7, x8 are terminal nodes
x0
x1 x2
x3 x4 x5 x6
x7 x8
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Game tree
• Any node other than a terminal node represents some player.
• For a node other than a terminal node, the edges that connect it with its successors represent the actions available to the player represented by the node.
Player 1
Player 2
H T
-1, 1 1, -1
H T
Player 2
H T
1, -1 -1, 1
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Game tree
• A path from the root to a terminal node represents a complete sequence of moves which determines the payoff at the terminal node
Player 1
Player 2
H T
-1, 1 1, -1
H T
Player 2
H T
1, -1 -1, 1
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Represent a Static Game as a Game Tree: Illustration
• Prisoners’ dilemma (another representation of the game. The first number is the payoff for player 1, and the second number is the payoff for player 2)
Prisoner 1
Prisoner 2
Prisoner 1
NC Confess
-3, -3 -9, 0
Not Confess (NC) Confess
NC Confess
0, -9 -1, -1
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RECAP: Strategy
• A strategy for a player is a complete plan of actions.
• It specifies a feasible action for the player in every contingency in which the player might be called on to act.
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Strategy and Payoff in game tree
• In a game tree, a strategy for a player is represented by a set of edges.
• A combination of strategies (sets of edges), one for each player, induce one path from the root to a terminal node, which determines the payoffs of all players.
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Sequential-move matching pennies
• Player 1’s strategies Head Tail
• Player 2’s strategies H if player 1 plays H, H if player 1 plays T H if player 1 plays H, T if player 1 plays T T if player 1 plays H, H if player 1 plays T T if player 1 plays H, T if player 1 plays T
Player 2’s strategies are denoted by HH, HT, TH and TT, respectively.
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Strategy and Payoff • A strategy for a player
is a complete plan of actions.
• It specifies a feasible action for the player in every contingency in which the player might be called on to act.
• It specifies what the player does at each of her nodes
Player 1
Player 2
H T
-1, 1 1, -1
H T
Player 2
H T
1, -1 -1, 1
a strategy for
player 1: H
a strategy for player 2: H if player 1 plays
H, T if player 1 plays T (written as HT)
Player 1’s payoff is -1 and player 2’s payoff is
1 if player 1 plays H and player 2 plays HT
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Sequential-Move Matching Pennies
• Their payoffs
• Normal-form representation
Player 2
HH HT TH TT
Player
1
H -1 , 1 -1 , 1 1 , -1 1 , -1
T 1 , -1 -1 , 1 1 , -1 -1 , 1
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Sequential Games
In many games of interest, some of the choices are made sequentially. That is, one player may know the opponents choice before she makes her decision.
Micro Macro
Micro 2 1 0 0
Macro 0 0 1 2 Pla
yer
A
Player B
Consider the game: Let Player A choose first.
We can use a decision tree to write out the extensive form of the game
Player A
Player B Player B
(2, 1) (0, 0) (0, 0) (1, 2)
The second stage (after the first decision is made) is known as the subgame.
Player A moves first in stage one.
We can use a decision tree to write out the extensive form of the game
Player A
Player B Player B
(2, 1) (0, 0) (0, 0) (1, 2)
Suppose that Player A chooses Macro.
Player B should choose Macro
Now, if Player A chooses Micro
Player B should choose Micro
Player A knows how player B will respond, and therefore will always choose Micro (and a utility level of 2) over Macro (and a utility level of 1)
Player A
Player B Player B
(2, 1) (0, 0) (0, 0) (1, 2)
In this game, player A has a first mover advantage
Note: Simultaneous Move Games
Player A
Player B Player B
(2, 1) (0, 0) (0, 0) (1, 2)
Suppose that we assume Player A moves first, but Player B can’t observe Player A’s choice?
We are back to the original mixed strategy equilibrium!
Terrorists
Terrorists
President
(1, -.5)
(-.5, -1) (-1, 1)
(0, 1)
In the Movie Air Force One, Terrorists hijack Air Force One and take the president hostage. Can we write this as a game?
In the third stage, the best response is to kill the hostages
Given the terrorist response, it is optimal for the president to negotiate in stage 2
Given Stage two, it is optimal for the terrorists to take hostages
Terrorists
Terrorists
President
(1, -.5)
(-.5, -1) (-1, 1)
(0, 1)
The equilibrium is always (Take Hostages/Negotiate). How could we change this outcome?
Suppose that a constitutional amendment is passed ruling out hostage negotiation (a commitment device)
Without the possibility of negotiation, the new equilibrium becomes (No Hostages)
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Solving sequential games
• To solve a sequential game we look for the ‘subgame perfect Nash equilibrium’
• For our purposes, this means we solve the game using ‘rollback’ – To use rollback, start at the end of each branch and work
backwards, eliminating all but the optimal choice for the relevant player
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Subgame
• Its game tree is a branch of the original game tree
• The information sets in the branch coincide with the information sets of the original game and cannot include nodes that are outside the branch.
• The payoff vectors are the same as in the original game.
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Subgame perfect equilibrium & credible threats
• Proper subgame = subtree (of the game tree) whose root is alone in its information set
• Subgame perfect equilibrium – Strategy profile that is in Nash equilibrium in
every proper subgame (including the root), whether or not that subgame is reached along the equilibrium path of play
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Backwards induction
• Start from the smallest subgames containing the terminal nodes of the game tree
• Determine the action that a rational player would choose at that action node – At action nodes immediately adjacent to terminal nodes, the player
should maximize the utility, This is because she no longer cares about strategic interactions. Regardless of how she moves, nobody else can affect the payoff of the game.
Replace the subgame with the payoffs corresponding to the terminal node that would be reached if that action were played
• Repeat until there are no action nodes left
Repeated Games
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Reputation
Reputation is intimately bound up with repetition.
For example:
1. Firms, both small and large, develop reputations for product quality and after sales service through dealings with successive customers.
2. Retail and Service chains and franchises develop reputations for consistency in their product offerings across different outlets.
3. Individuals also cultivate their reputations through their personal interactions within a community.
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Definition of a repeated game
These examples motivate why we study reputation by analyzing the solutions of repeated games.
When a game is played more than once by the same players in the same roles, it is called a repeated game.
We refer to the original game (that is repeated) as the kernel game.
The number of rounds count the repetitions of the kernel game.
A repeated game might last for a fixed number of rounds, or be repeated indefinitely (perhaps ending with a random event).
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Games repeated a finite number of times
We begin the discussion by focusing on games that have a finite number of rounds.
There are two cases to consider. The kernel game has:
1. a unique solution
2. multiple solutions.
For finitely repeated games this distinction turns out to be the key to discussing what we mean by a reputation.
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Two-Stage Repeated Game
• Two-stage prisoners’ dilemma Two players play the following simultaneous move game
twice The outcome of the first play is observed before the second
play begins The payoff for the entire game is simply the sum of the
payoffs from the two stages. That is, the discount factor is 1.
Player 2
L2 R2
Player 1 L1 1 , 1 5 , 0
R1 0 , 5 4 , 4
For ease of analysis, I represent the values here as
positive and numbers are representative
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Game Tree of the Two-stage Prisoners’ Dilemma
1
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
1+1 1+1
1+5 1+0
1+0 1+5
1+4 1+4
1 1 1 1
5+1 0+1
5+5 0+0
5+0 0+5
5+4 0+4
0+1 5+1
0+5 5+0
0+0 5+5
0+4 5+4
4+1 4+1
4+5 4+0
4+0 4+5
4+4 4+4
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Informal Game Tree of the Two-Stage Prisoners’ Dilemma
1
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
1
1 5
0
0
5 4
4
1 1 1 1 (1, 1) (5, 0) (0, 5) (4, 4)
1
1 5
0
0
5 4
4 1
1 5
0
0
5 4
4
1
1 5
0
0
5 4
4
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Infinitely repeated Prisoner’s Dilemma
• A game repeated infinitely
• Suppose the players play (C,C), (D,D), (C,C), (D,D),…. Forever
• We know the stage game payoffs 3,1,3,1,….
• Overall payoffs in a game with “x” repetitions can be represented as
3, 3 0, 9
9, 0 1, 1
cooperate defect
cooperate
defect
t
x
x
iuE1
)(
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Infinitely repeated Prisoner’s Dilemma
• In games of infinite repetitions there are two ways: Limit average reward: lim inft→∞(1/t)Σx=1..tE[ui
x] e.g. if payoffs are 3, 1, 3, 1, …, payoff is 2
Future-discounted reward:
• E.g. if stage payoffs are 3, 1, 3, 1, … and discount factor δ =.9, then payoff is 3 + 1*.9 + 3*.92+ 1*.93+ ... Delta takes into the account that “present” is more important than “future”. Definition of Nash Equilibrium though remains unchanged.
3, 3 0, 9
9, 0 1, 1
cooperate defect
cooperate
defect
1
1
3
2
21 )...x
i
t uuuu
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Infinitely repeated Prisoner’s Dilemma
• Tit-for-tat strategy: – Cooperate the first round,
– In every later round, do the same thing as the other player did in the previous round
• Trigger strategy: – Cooperate as long as everyone cooperates
– Once a player defects, defect forever
• What about one player playing tit-for-tat and the other playing trigger?
4, 4 0, 5
5, 0 1, 1
cooperate defect
cooperate
defect
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Few Examples in Computer Science
Example: Forwarder’s dilemma
One real application
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?
126 Source: Buttyan and Hubaux, “Security and Cooperation in Wireless Networks”
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Example: Forwarder’s dilemma
One real application
Game: Players: Green, Blue Actions: 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 Blue Reward of Green
127 Source: Buttyan and Hubaux, “Security and Cooperation in Wireless Networks”
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Qn 1
128 http://ganeshniyer.com
Qn 2 Identify the players (2 marks) Identify their strategies (3 marks) Determine the pay-off matrix (5 marks) Determine the Nash Equilibrium (5 marks)
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Game theoretic modeling, analysis, and mitigation of
security risks.
Who is attacking our communication Systems?
Hackers Terrorists, Criminal Groups
Hacktivists
Disgruntled Insiders Foreign Governments
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A lot of good effort!
131
Cryptography
Software Security
Intrusion Detection systems
Firewalls
Anti-Viruses
Risk Management Attack Graphs
Decision Theory Machine Learning
Information Theory Optimization
Hardware Security
• Some practical solutions
• Some theoretic basis
…
…
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Example: Remote Attack
Why Game Theory for Security?
Attack Defense
132
E.g.: Rate of Port Scanning IDS Tuning
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Security
Why Game Theory for Security?
Game Theory also helps:
Trust Incentives Externalities Machine Intelligence
133
…
Conferences (GameSec, GameNets) , Workshops, books, Tutorials,…
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…
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3 Communication Security Game Models
Intruder Game
p
1-p
Alice
Trudy
BobX Y Z
Availability Attack
134
Intelligent Virus
a Normal traffic
Virus b
Xn
Detection
If Xn > => Alarm
REF: Assane Gueye, Jean C. Walrand, Security in Networks: A Game-Theoretic Approach, Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-11, 2008 http://ganeshniyer.com
M’ M
Intruder (Trudy)
What if it is possible that:
M
Intruder Game
135
Scenario:
Network Source (Alice)
User (Bob)
M
Encryption is not always practical ….
Formulation: Game between Intruder and User
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136
Intruder Game: Binary
Y
• Payoffs:
• Strategies (mixed i.e. randomized)
• Trudy: (p0,p1), Bob: (q0,q1)
Alice
Trudy Bob
• One shot, simultaneous choice game
• Nash Equilibrium?
Z
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Intruder game: NE
137
0 1
Trudy
Bob Always trust
0 1
1
0 1
1text
01
1
Payoff :
Trudy
Always flip
1
1 q0
q1
1
1 q0
q1
Always decide (1); the less costly bit
1
1 q0
q1
Always decide (1); the less costly bit
Cost
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Intelligent Virus Game
138
Scenario
a
Normal traffic
Virus b
Xn
Detection
If Xn > => Alarm, …. Assume a known
Detection system: choose to minimize cost of infection + clean up
Virus: choose b to maximize infection cost
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Intelligent Virus Game (IDS)
139
Smart virus designer picks very large b, so that the cost is always high …. Regardless of !
Scenario
a
Normal traffic
Virus b
Xn
Detection
If Xn > => Alarm, ….
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Intelligent Virus Game (IPS)
140
Modified Scenario
a
Normal traffic
Virus b
Xn Detection
If Xn > => Alarm
•Detector: buffer traffic and test threshold • Xn < process
• If Xn > Flush & Alarm •Game between Virus (b) and Detector ()
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Availability Attack Models!
Tree-Link Game: • Given the topology of a network, characterized by an undirected graph, we
consider the following game situation:
• A network manager is choosing (as communication infrastructure) a spanning tree of the graph, and
• An attacker is trying to disrupt the communication tree by attacking one link of the network.
• Attacking a link has a certain cost for the attacker who also has the option of not attacking.
141 REF: Assane Gueye, , Jean C. Walrand, and Venkat A, “A Network Topology Design Game: How to Choose Communication Links in an Adversarial Environment?” UC Berkeley
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NOTE: In the mathematical field of graph theory, a spanning tree T of a connected, undirected graph G is a tree that includes all of the vertices and some or all of the edges of G
Model • 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)
142
Example:
Defender: 0 Attacker: - µ2 Defender: -1 Attacker: 1- µ1
– Defender : a on T, to minimize
– Attacker: b on E, to maximize
– One shot game
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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
1/7 1/7
1/7
143
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Security and Game Theory
• Secret Communications – Where to hide the bits
• Identification of attackers – Audit mechanisms for provable risk management
• Network Security – Deceptive Routing in Relay Networks
• System Defense – Security assessment for IT infrastructure
• Applications Security – Electricity Distribution Networks
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Coordination Games
Coordination Games
• Consider the following problem:
– A supplier and a buyer need to decide whether to adopt a new purchasing system.
new old
old 0,0
0,0
5,5
20,20
Supplier
Buyer
new
No dominated strategies! 146 http://ganeshniyer.com
Coordination Games
new old
old 0,0
0,0
5,5
20,20
Supplier
Buyer
new
• This game has two Nash equilibria (new,new) and (old,old) •Real-life examples: Chrome vs Firefox, Mac vs Windows vs Linux, others?
• Each player wants to do what the other does
• which may be different than what they say they’ll do • How to choose a strategy? Nothing is dominated.
147 http://ganeshniyer.com
Solving Coordination Games
• Coordination games turn out to be an important real-life problem – Technology/policy/strategy adoption, delegation of
authority, synchronization
• Human agents tend to use “focal points” – Solutions that seem to make “natural sense”
• e.g. pick a number between 1 and 10
• Social norms/rules are also used – Driving on the right/left side of the road
• These strategies change the structure of the game
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Finding Nash Equilibria – Simultaneous Equations
• We can also express a game as a set of equations.
• Demand for corn is governed by the following equation: – Quantity q = 100000(10 – p)
• Government price supports say that price p must be at least 0.25 (and it can’t be more than 10)
• Three farmers can each choose to sell 0-600000 lbs of corn.
• What are the Nash equilibria?
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Setup
• Quantity (q) = q1 + q2 + q3
• Price(p) = a –bq (downward-sloping line)
• Farmer 1 is trying to decide a quantity to sell.
• Maximize profit = price * quantity
• Maximize: pq1 =(a –bq) * q1
• Profit Pr= (a – b(q1 + q2 + q3)) * q1 =
= aq1 –bq12 –bq1q2 –bq1q3
Differentiate: Pr’ = a – 2bq1 –bq2 – bq3
To maximize: set this equal to zero.
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Setup
• So solutions must satisfy – a – b(q2 + q3) – 2bq1 = 0
• So what if q1 = q2 = q3 (everyone ships the same amount?) – Since the game is symmetric, this should be a solution.
– a – 4bq1 = 0, a = 4bq1, q1 = a/4b.
– q = 3a/4b, p = a/4. Each farmer gets a2 / 16b.
– In this problem, a=10, b=1/100000.
– Price = $2.50, q1=250000, profit = 625,000
– q1=q2=q3=250000 is a solution.
– Price supports not used in this solution.
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Setup • What if farmers 2 & 3 send everything they have?
– q2 + q3 = 1,200,000
• If farmer 1 then shipped nothing, price would be: – 10 - 1,200,000/100,000 = -2.
• But prices can’t fall below $0.25, so they’d be capped there.
• Adding quantity would reduce the price, except for supports.
– So, farmer 1 should sell all his corn at $0.25, and earn $125,000.
• So everyone selling everything at the lowest price (q1 = q2 =q3 = 600,000) is also a Nash equilibrium. – These are the only pure strategy Nash equilibria.
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Price-matching Example
• Two sellers are offering the same book for sale.
• This book costs each seller $25.
• The lowest price gets all the customers; if they match, profits are split.
• What is the Nash Equilibrium strategy?
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Price-matching Example
• Suppose the monopoly price of the book is $30.
– (price that maximizes profit w/o competition)
• Each seller offers a rebate: if you find the book cheaper somewhere else, we’ll sell it to you with double the difference subtracted.
– E.g. $30 at store 1, $24 at store 2 – get it for $18 from store 1.
• Now what is each seller’s Nash strategy?
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Price-matching example
• Observation 1: sellers want to have the same price. – Each suffers from giving the rebate.
• Profit Pr = p1 – 2(p1 – p2) = -p1 –2p2
• Differentiate: Pr’ = -1. – There is no local maximum. So, to maximize
profits, maximize price.
• At that point, the rebate 2(p1 – p2) is 0, and p1 is as high as possible. – The 2 makes up for sharing the market.
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