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.


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


36

Some Notations


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 " ),,(),( "'


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.


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


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.


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.


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.


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.


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


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

https://www.youtube.com/watch?v=RqGb1Gx0t9U

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


• 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


• 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


• 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


1 q

r

1

1/2

1/2


• 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



• 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



• 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


1 q

r

1

1/2

1/2


• 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


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 .


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.


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


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!


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


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


http://www.eprisner.de/MAT109/Mixedb.html

http://levine.sscnet.ucla.edu/Games/zerosum.htm


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.

Page 77

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).

Page 78

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”

Page 79

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)


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

Page 82

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

Page 83

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?


http://web.econ.ku.dk/cie/Workshops/Norio%20IV/pdf%20-%20papers/Hueschelrath-Predation.PDF

http://web.econ.ku.dk/cie/Workshops/Norio IV/pdf - papers/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.


Recap

Games

Non-cooperative Games

Static games of complete

infromation

Sequential games of complete information


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


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.


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


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


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


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


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


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


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


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


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


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.


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.


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.


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


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


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

http://www.imdb.com/title/tt0118571/photogallery

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)

110

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

111

112

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.

113

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

114

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


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.


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).


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.


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


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


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


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

)(



• 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



• 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



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”


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”


Qn 1


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)


Game theoretic modeling, analysis, and mitigation of

security risks.

Who is attacking our communication Systems?

Hackers Terrorists, Criminal Groups

Hacktivists

Disgruntled Insiders Foreign Governments


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

…

…


Example: Remote Attack

Why Game Theory for Security?

Attack Defense

132

E.g.: Rate of Port Scanning IDS Tuning


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…


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


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


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


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


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, ….


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 ()


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


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


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


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



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.


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


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?


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.


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.


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.


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?


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?


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.


THANK YOU!

Documents

Non Cooperative Games Five Lectures