Download Game Theory intro

Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Game Theory intro
CPSC 532A Lecture 3
Game Theory intro
CPSC 532A Lecture 3, Slide 1
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Lecture Overview
1
Recap
2
Example Matrix Games
3
Pareto Optimality
4
Best Response and Nash Equilibrium
Game Theory intro
CPSC 532A Lecture 3, Slide 2
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Defining Games
Finite, n-person game: hN, A, ui:
N is a finite set of n players, indexed by i
A = A1 × . . . × An , where Ai is the action set for player i
(a1 , . . . , an ) ∈ A is an action profile, and so A is the space of
action profiles
u = hu1 , . . . , un i, a utility function for each player, where
ui : A 7→ R
Writing a 2-player game as a matrix:
row player is player 1, column player is player 2
rows are actions a ∈ A1 , columns are a0 ∈ A2
cells are outcomes, written as a tuple of utility values for each
player
Game Theory intro
CPSC 532A Lecture 3, Slide 3
implementation) and D (for using a Defective one). If both you and your colleague
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
adopt C then your average packet delay is 1ms (millisecond). If you both adopt D the
delay is 3ms, because of additional overhead at the network router. Finally, if one of
Games
in Matrix Form
you adopts D and the other adopts C then the D adopter will experience no delay at all,
but the C adopter will experience a delay of 4ms.
These consequences are shown in Figure 3.1. Your options are the two rows, and
the TCP
Backoff
written
as acell,
matrix
(“normal
your Here’s
colleague’s
options
are theGame
columns.
In each
the first
number represents
form”).
your payoff (or, minus your delay), and the second number represents your colleague’s
payoff.1
C
D
C
−1, −1
−4, 0
D
0, −4
−3, −3
Figure 3.1 The TCP user’s (aka the Prisoner’s) Dilemma.
It’s an
example
prisoner’s
dilemma.
Given
these
options of
what
should you
adopt, C or D? Does it depend on what you
think your colleague will do? Furthermore, from the perspective of the network operator, what kind of behavior can he expect from the two users? Will any two users behave
Game
Theory when
intro
CPSC 532A
Lecture
3, Slide 4
the same
presented with this scenario? Will the behavior change
if the
network
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Lecture Overview
1
Recap
2
Example Matrix Games
3
Pareto Optimality
4
Best Response and Nash Equilibrium
Game Theory intro
CPSC 532A Lecture 3, Slide 5
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
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 ∈ 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, it’s a one-player game
Game Theory intro
CPSC 532A Lecture 3, Slide 6
Example Matrix Games
Optimality
Best Response and Nash Equilibrium
theRecap
abbreviation
we must explicit state Pareto
whether
this matrix represents
a common-payoff
game or a zero-sum one.
Matching
A classical Pennies
example of a zero-sum game is the game of matching pennies. In this
game, each of the two players has a penny, and independently chooses to display either
heads or tails. The two players then compare their pennies. If they are the same then
player 1 pockets both, and otherwise player 2 pockets them. The payoff matrix is
shown
in Figure
One
player 3.5.
wants to match; the other wants to mismatch.
Heads
Tails
Heads
1
−1
Tails
−1
1
Figure 3.5
Matching Pennies game.
The popular children’s game of Rock, Paper, Scissors, also known as Rochambeau,
provides a three-strategy generalization of the matching-pennies game. The payoff
matrix of this zero-sum game is shown in Figure 3.6. In this game, each of the two
players can choose either Rock, Paper, or Scissors. If both players choose the same
Game Theory intro
CPSC 532A Lecture 3, Slide 7
Example Matrix Games
Optimality
Best Response and Nash Equilibrium
theRecap
abbreviation
we must explicit state Pareto
whether
this matrix represents
a common-payoff
game or a zero-sum one.
Matching
A classical Pennies
example of a zero-sum game is the game of matching pennies. In this
game, each of the two players has a penny, and independently chooses to display either
heads or tails. The two players then compare their pennies. If they are the same then
player 1 pockets both, and otherwise player 2 pockets them. The payoff matrix is
shown
in Figure
One
player 3.5.
wants to match; the other wants to mismatch.
Heads
Tails
Heads
1
−1
Tails
−1
1
Figure 3.5
Matching Pennies game.
Play this game with someone near you, repeating five times.
The popular children’s game of Rock, Paper, Scissors, also known as Rochambeau,
provides a three-strategy generalization of the matching-pennies game. The payoff
matrix of this zero-sum game is shown in Figure 3.6. In this game, each of the two
players can choose either Rock, Paper, or Scissors. If both players choose the same
Game Theory intro
CPSC 532A Lecture 3, Slide 7
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Rock-Paper-Scissors
3
Generalized matching pennies.
Competition and Coordination: Normal form games
Rock
Paper
Scissors
Rock
0
−1
1
Paper
1
0
−1
−1
1
0
Scissors
Figure 3.6
Rock, Paper, Scissors game.
...Believe it or not, there’s an annual international competition for
VG GL
this game!
Game Theory intro
VG
2, 1
0, 0
CPSC 532A Lecture 3, Slide 8
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Games of Cooperation
Players have exactly the same interests.
no conflict: all players want the same things
∀a ∈ A, ∀i, j, ui (a) = uj (a)
we often write such games with a single payoff per cell
why are such games “noncooperative”?
Game Theory intro
CPSC 532A Lecture 3, Slide 9
Recap that is maximally
Example Matrixbeneficial
Games
Pareto Optimality
Best Response and Nash Equilibrium
action
to all.
Because of their special nature, we often represent common value games with an
Coordination
abbreviated form ofGame
the matrix in which we list only one payoff in each of the cells.
As an example, imagine two drivers driving towards each other in a country without
traffic rules, and who must independently decide whether to drive on the left or on the
right. If the players choose the same side (left or right) they have some high utility, and
Which
side
of athe
should
you matrix
drive on?
otherwise
they
have
lowroad
utility.
The game
is shown in Figure 3.4.
Left
Right
Left
1
0
Right
0
1
Figure 3.4 Coordination game.
At the other end of the spectrum from pure coordination games lie zero-sum games,
which (bearing in mind the comment we made earlier about positive affine transformations) are more properly called constant-sum games. Unlike common-payoff games,
Game Theory intro
c
Shoham
and Leyton-Brown, 2006
CPSC 532A Lecture 3, Slide 10
Recap that is maximally
Example Matrixbeneficial
Games
Pareto Optimality
Best Response and Nash Equilibrium
action
to all.
Because of their special nature, we often represent common value games with an
Coordination
abbreviated form ofGame
the matrix in which we list only one payoff in each of the cells.
As an example, imagine two drivers driving towards each other in a country without
traffic rules, and who must independently decide whether to drive on the left or on the
right. If the players choose the same side (left or right) they have some high utility, and
Which
side
of athe
should
you matrix
drive on?
otherwise
they
have
lowroad
utility.
The game
is shown in Figure 3.4.
Left
Right
Left
1
0
Right
0
1
Figure 3.4 Coordination game.
Play this game with someone near you. Then find a new partner
Atand
the other
of the
spectrum
from in
pure
coordination games lie zero-sum games,
play end
again.
Play
five times
total.
which (bearing in mind the comment we made earlier about positive affine transformations) are more properly called constant-sum games. Unlike common-payoff games,
Game Theory intro
c
Shoham
and Leyton-Brown, 2006
CPSC 532A Lecture 3, Slide 10
0
Rock
Recap
−1
Example Matrix Games
Pareto Optimality
0
General Games: Paper
Battle of1 the Sexes
Scissors
1
−1
1
Best Response and Nash Equilibrium
−1
0
The most interesting games combine elements of cooperation and
competition.
Figure 3.6 Rock, Paper, Scissors game.
B
F
B
2, 1
0, 0
F
0, 0
1, 2
Figure 3.7
Battle of the Sexes game.
Strategies in normal-form games
We have so far defined the actions available to each player in a game, but not yet his
set of
strategies,
or his available choices. Certainly one kind of
strategy
is to3, select
Game
Theory
intro
CPSC
532A Lecture
Slide 11
0
Rock
Recap
−1
Example Matrix Games
Pareto Optimality
0
General Games: Paper
Battle of1 the Sexes
Scissors
1
−1
1
Best Response and Nash Equilibrium
−1
0
The most interesting games combine elements of cooperation and
competition.
Figure 3.6 Rock, Paper, Scissors game.
B
F
B
2, 1
0, 0
F
0, 0
1, 2
Figure 3.7
Battle of the Sexes game.
Play this game with someone near you. Then find a new partner
and play again. Play five times in total.
Strategies in normal-form games
We have so far defined the actions available to each player in a game, but not yet his
set of
strategies,
or his available choices. Certainly one kind of
strategy
is to3, select
Game
Theory
intro
CPSC
532A Lecture
Slide 11
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Lecture Overview
1
Recap
2
Example Matrix Games
3
Pareto Optimality
4
Best Response and Nash Equilibrium
Game Theory intro
CPSC 532A Lecture 3, Slide 12
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Analyzing Games
We’ve defined some canonical games, and thought about how
to play them. Now let’s examine the games from the outside
From the point of view of an outside observer, can some
outcomes of a game be said to be better than others?
Game Theory intro
CPSC 532A Lecture 3, Slide 13
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Analyzing Games
We’ve defined some canonical games, and thought about how
to play them. Now let’s examine the games from the outside
From the point of view of an outside observer, can some
outcomes of a game be said to be better than others?
we have no way of saying that one agent’s interests are more
important than another’s
intuition: imagine trying to find the revenue-maximizing
outcome when you don’t know what currency has been used to
express each agent’s payoff
Are there situations where we can still prefer one outcome to
another?
Game Theory intro
CPSC 532A Lecture 3, Slide 13
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Pareto Optimality
Idea: sometimes, one outcome o is at least as good for every
agent as another outcome o0 , and there is some agent who
strictly prefers o to o0
in this case, it seems reasonable to say that o is better than o0
we say that o Pareto-dominates o0 .
Game Theory intro
CPSC 532A Lecture 3, Slide 14
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Pareto Optimality
Idea: sometimes, one outcome o is at least as good for every
agent as another outcome o0 , and there is some agent who
strictly prefers o to o0
in this case, it seems reasonable to say that o is better than o0
we say that o Pareto-dominates o0 .
An outcome o∗ is Pareto-optimal if there is no other outcome
that Pareto-dominates it.
Game Theory intro
CPSC 532A Lecture 3, Slide 14
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Pareto Optimality
Idea: sometimes, one outcome o is at least as good for every
agent as another outcome o0 , and there is some agent who
strictly prefers o to o0
in this case, it seems reasonable to say that o is better than o0
we say that o Pareto-dominates o0 .
An outcome o∗ is Pareto-optimal if there is no other outcome
that Pareto-dominates it.
can a game have more than one Pareto-optimal outcome?
Game Theory intro
CPSC 532A Lecture 3, Slide 14
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Pareto Optimality
Idea: sometimes, one outcome o is at least as good for every
agent as another outcome o0 , and there is some agent who
strictly prefers o to o0
in this case, it seems reasonable to say that o is better than o0
we say that o Pareto-dominates o0 .
An outcome o∗ is Pareto-optimal if there is no other outcome
that Pareto-dominates it.
can a game have more than one Pareto-optimal outcome?
does every game have at least one Pareto-optimal outcome?
Game Theory intro
CPSC 532A Lecture 3, Slide 14
equences
in Matrix
Figure
3.1. YourPareto
options
are the twoBest
rows,
andand Nash Equilibrium
Recap are shown
Example
Games
Optimality
Response
ue’s options are the columns. In each cell, the first number represents
or, Pareto
minus yourOptimal
delay), and the
second number
your
colleague’s
Outcomes
in represents
Example
Games
C
D
C
−1, −1
−4, 0
D
0, −4
−3, −3
Figure 3.1 The TCP user’s (aka the Prisoner’s) Dilemma.
e options what should you adopt, C or D? Does it depend on what you
lleague will do? Furthermore, from the perspective of the network operaof behavior can he expect from the two users? Will any two users behave
n presented with this scenario? Will the behavior change if the network
ws the users to communicate with each other before making a decision?
hanges to the delays would the users’ decisions still be the same? How
rs behave if they have the opportunity to face this same decision with the
part multiple times? Do answers to the above questions depend on how
Game Theory intro
CPSC 532A Lecture 3, Slide 15
As
an
example,
two
drivers
towards
each
other in a cou
equences
in Matrix
Figure
3.1. imagine
YourPareto
options
are thedriving
twoBest
rows,
andand
Recap are shown
Example
Games
Optimality
Response
Nash Equilibrium
traffic
rules,
and
who
must
independently
decide
whether
to
drive on the l
ue’s options are the columns. In each cell, the first number represents
right.
If
the
players
choose
the
same
side
(left
or
right)
they
have
some hig
or, Pareto
minus yourOptimal
delay), and the
second number
your
colleague’s
Outcomes
in represents
Example
Games
otherwise they have a low utility. The game matrix is shown in Figure 3.4
C
D
C
−1, −1
−4, 0
D
0, −4
−3, −3
Left
Right
Left
1
0
Right
0
1
3.4 Coordination game.
Figure 3.1 The TCP user’s (aka the Prisoner’s)Figure
Dilemma.
game
At you
the other
of the
from pure
coordination
games lie zero
eero-sum
options
what should
adopt,end
C or
D? spectrum
Does it depend
on what
you
which
(bearing
in
mind
the
comment
we
made
earlier
about
positive affine
lleague will do? Furthermore, from the perspective of the network operaonstant-sum
tions)
are
more
properly
called
constant-sum
games.
Unlike
common-pa
of behavior can he expect from the two users? Will any two users behave
ames
n presented with this scenario? Will the behavior change if the network
c before making
Shoham
and Leyton-Brown,
ws the users to communicate with each other
a decision? 2006
hanges to the delays would the users’ decisions still be the same? How
rs behave if they have the opportunity to face this same decision with the
part multiple times? Do answers to the above questions depend on how
Game Theory intro
CPSC 532A Lecture 3, Slide 15
As
an
example,
two
drivers
towards
each
other in a cou
equences
in Matrix
Figure
3.1. imagine
YourPareto
options
are thedriving
twoBest
rows,
andand
Recap are shown
Example
Games
Optimality
Response
Nash Equilibrium
traffic
rules,
and
who
must
independently
decide
whether
to
drive on the l
ue’s options are the columns. In each cell, the first number represents
right.
If
the
players
choose
the
same
side
(left
or
right)
they
have
some hig
Rock
Paper
Scissors
or, Pareto
minus yourOptimal
delay), and the
second number
your
colleague’s
Outcomes
in represents
Example
Games
otherwise they have a low utility. The game matrix is shown in Figure 3.4
0
Rock
Paper
Scissors
C
−1
D
1
Left
Right
C
1
−1, −1
0
−4, 0
−1 Left
1
0
D
−1
0, −4
1
−3, −3
0 Right
0
1
Figure 3.6 Rock, Paper, Scissors game.
3.4 Coordination game.
Figure 3.1 The TCP user’s (aka the Prisoner’s)Figure
Dilemma.
B
F
game
At you
the other
of the
from pure
coordination
games lie zero
eero-sum
options
what should
adopt,end
C or
D? spectrum
Does it depend
on what
you
which
(bearing
in
mind
the
comment
we
made
earlier
about
positive affine
lleague will do? Furthermore, from the perspective of the network operaB are2,more
1 properly
0, 0
onstant-sum
tions)
called
constant-sum
games.
Unlike
common-pa
of behavior can he expect from the two users? Will any two users behave
ames
n presented with this scenario? Will the behavior change if the network
c before making
Shoham
and Leyton-Brown,
ws the users to communicate
each
a decision? 2006
F
0, 0with 1,
2 other
hanges to the delays would the users’ decisions still be the same? How
rs behave if they have the opportunity to face this same decision with the
Figure 3.7 Battle of the Sexes game.
part multiple times? Do answers to the above questions depend on how
Game Theory intro
CPSC 532A Lecture 3, Slide 15
As
an
example,
imagine
two
drivers
driving
towards
each
other
in a cou
equences
in Matrix
Figure
3.1.
YourPareto
options
are the
twoat
rows,
andand
Recap are shown
Example
Games
Optimality
Best
Response
Nash
Equilibrium
competition;
one player’s
gain
must
come
the
expense
of the
other play
traffic
rules,
and
who
must
independently
decide
whether
to
drive
on the l
ue’s options are the columns.
In each
cell, the first number
As in the case
of common-payoff
games, represents
we can use an abbreviated m
right.
If
the
players
choose
the
same
side
(left
or
right)
they
have
some
hig
Rock
Paper
Scissors
or, Pareto
minus yourOptimal
delay),
and the
second number
your
colleague’s
Outcomes
in represents
Example
Games
represent
zero-sum
games,
in
which we
write
only one payoff value in ea
otherwise they have a low utility. The game matrix is shown in Figure 3.4
value represents the payoff of player 1, and thus the negative of the payof
representation is unambiguous,
Rock Note,
0 though, that
−1 whereas the
1 full matrix
Left
Right
C
D must explicit state whether
the abbreviation
we
this matrix represents a com
game or a zero-sum one.
Paper
1classical example
0
−1 Left game
1 is the0game of matching pen
C A−1,
−1
−4, 0 of a zero-sum
game, each of the two players has a penny, and independently chooses to d
then compare their pennies. If they are th
Scissors heads
−1 or tails. The
1 two players
0 Right
0
1
Dplayer
0, −4
−3,both,
−3 and otherwise
1 pockets
player 2 pockets them. The pay
shown in Figure 3.5.
Figure 3.6 Rock, Paper, Scissors game.
3.4 Coordination game.
Figure 3.1 The TCP user’s (aka the Prisoner’s)Figure
Dilemma.
Heads Tails
B
F
game
At you
the other
of the
from pure
coordination
games lie zero
eero-sum
options
what should
adopt,end
C or
D? spectrum
Does it depend
on what
you
which
(bearing
in
mind
the
comment
we
made
earlier
about
positive affine
lleague will do? Furthermore, from the perspective of the network operaHeads
1 games.
−1 Unlike common-pa
B are2,more
1 properly
0, 0
onstant-sum
tions)
called
constant-sum
of behavior can he expect from the two users? Will any two users behave
ames
n presented with this scenario? Will the behavior change if the network
c before
Shoham
and Leyton-Brown,
Tailsmaking
−1a decision?
1 2006
ws the users to communicate
each
F
0, 0with 1,
2 other
hanges to the delays would the users’ decisions still be the same? How
rs behave if they have the opportunity to faceFigure
this same
decision with the
3.5 Matching Pennies game.
Figure 3.7 Battle of the Sexes game.
part multiple times? Do answers to the above questions depend on how
Game Theory intro
CPSC 532A Lecture 3, Slide 15
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Lecture Overview
1
Recap
2
Example Matrix Games
3
Pareto Optimality
4
Best Response and Nash Equilibrium
Game Theory intro
CPSC 532A Lecture 3, Slide 16
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Best Response
If you knew what everyone else was going to do, it would be
easy to pick your own action
Game Theory intro
CPSC 532A Lecture 3, Slide 17
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Best Response
If you knew what everyone else was going to do, it would be
easy to pick your own action
Let a−i = ha1 , . . . , ai−1 , ai+1 , . . . , an i.
now a = (a−i , ai )
Best response: a∗i ∈ BR(a−i ) iff
∀ai ∈ Ai , ui (a∗i , a−i ) ≥ ui (ai , a−i )
Game Theory intro
CPSC 532A Lecture 3, Slide 17
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Nash Equilibrium
Now let’s return to the setting where no agent knows
anything about what the others will do
What can we say about which actions will occur?
Game Theory intro
CPSC 532A Lecture 3, Slide 18
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
Nash Equilibrium
Now let’s return to the setting where no agent knows
anything about what the others will do
What can we say about which actions will occur?
Idea: look for stable action profiles.
a = ha1 , . . . , an i is a (“pure strategy”) Nash equilibrium iff
∀i, ai ∈ BR(a−i ).
Game Theory intro
CPSC 532A Lecture 3, Slide 18
ue’s options are the columns. In each cell, the first number represents
Recap
Example Matrix Games
Pareto Optimality
Best Response and Nash Equilibrium
or, minus your delay), and the second number represents your colleague’s
Nash Equilibria of Example Games
C
D
C
−1, −1
−4, 0
D
0, −4
−3, −3
Figure 3.1 The TCP user’s (aka the Prisoner’s) Dilemma.
e options what should you adopt, C or D? Does it depend on what you
lleague will do? Furthermore, from the perspective of the network operaof behavior can he expect from the two users? Will any two users behave
n presented with this scenario? Will the behavior change if the network
ws the users to communicate with each other before making a decision?
hanges to the delays would the users’ decisions still be the same? How
rs behave if they have the opportunity to face this same decision with the
part multiple times? Do answers to the above questions depend on how
gents are and how they view each other’s rationality?
intro to many of these questions. It tells us that any rational
CPSC 532A Lecture 3, Slide 19
ry Game
givesTheory
answers
ue’s options are the columns. In each cell, the first number represents
Recap
Example
Matrix
Paretothe
Optimality
Bestor
Response
Nash
Equilibrium
right.and
If the
theGames
players number
choose
same side
(left
right)and
they
have
some hig
or, minus your delay),
second
represents
your
colleague’s
otherwise they have a low utility. The game matrix is shown in Figure 3.4
Nash Equilibria of Example Games
C
D
C
−1, −1
−4, 0
D
0, −4
−3, −3
Left
Right
Left
1
0
Right
0
1
3.4 Coordination game.
Figure 3.1 The TCP user’s (aka the Prisoner’s)Figure
Dilemma.
game
At you
the other
of the
from pure
coordination
games lie zero
eero-sum
options
what should
adopt,end
C or
D? spectrum
Does it depend
on what
you
which
(bearing
in
mind
the
comment
we
made
earlier
about
positive affine
lleague will do? Furthermore, from the perspective of the network operaonstant-sum
tions)
are
more
properly
called
constant-sum
games.
Unlike
common-pa
of behavior can he expect from the two users? Will any two users behave
ames
n presented with this scenario? Will the behavior change if the network
c before making
Shoham
and Leyton-Brown,
ws the users to communicate with each other
a decision? 2006
hanges to the delays would the users’ decisions still be the same? How
rs behave if they have the opportunity to face this same decision with the
part multiple times? Do answers to the above questions depend on how
gents are and how they view each other’s rationality?
intro to many of these questions. It tells us that any rational
CPSC 532A Lecture 3, Slide 19
ry Game
givesTheory
answers
ue’s options are the columns. In each cell, the first number represents
Recap
Example
Matrix
Paretothe
Optimality
Bestor
Response
Nash
Equilibrium
right.
If the
theGames
players
choose
same side
(left
right)and
they
have
some hig
Rock
Papernumber
Scissors
or, minus your delay),
and
second
represents
your
colleague’s
otherwise they have a low utility. The game matrix is shown in Figure 3.4
Nash Equilibria of Example Games
0
Rock
Paper
Scissors
C
−1
D
1
Left
Right
C
1
−1, −1
0
−4, 0
−1 Left
1
0
D
−1
0, −4
1
−3, −3
0 Right
0
1
Figure 3.6 Rock, Paper, Scissors game.
3.4 Coordination game.
Figure 3.1 The TCP user’s (aka the Prisoner’s)Figure
Dilemma.
B
F
game
At you
the other
of the
from pure
coordination
games lie zero
eero-sum
options
what should
adopt,end
C or
D? spectrum
Does it depend
on what
you
which
(bearing
in
mind
the
comment
we
made
earlier
about
positive affine
lleague will do? Furthermore, from the perspective of the network operaB are2,more
1 properly
0, 0
onstant-sum
tions)
called
constant-sum
games.
Unlike
common-pa
of behavior can he expect from the two users? Will any two users behave
ames
n presented with this scenario? Will the behavior change if the network
c before making
Shoham
and Leyton-Brown,
ws the users to communicate
each
a decision? 2006
F
0, 0with 1,
2 other
hanges to the delays would the users’ decisions still be the same? How
rs behave if they have the opportunity to face this same decision with the
Figure 3.7 Battle of the Sexes game.
part multiple times? Do answers to the above questions depend on how
gents are and how they view each other’s rationality?
intro to many of these questions. It tells us that any rational
CPSC 532A Lecture 3, Slide 19
ry Game
givesTheory
answers
ue’s options are the columns.
In each
cell, the first number
represents
AsMatrix
in theGames
case
of common-payoff
games,Best
weResponse
can use an
abbreviated
m
Recap
Example
Paretothe
Optimality
Nash
Equilibrium
right.
If the
the second
players
choose
same side
(left
or right)and
they
have
some hig
Rock
Papernumber
Scissors
or, minus your delay),
and
represents
your
colleague’s
represent
zero-sum
games,
in which we
write
only one payoff value in ea
otherwise they have a low utility. The game matrix is shown in Figure 3.4
value represents
the payoff
of player 1, and thus the negative of the payof
Nash Equilibria
of Example
Games
Note,
though,
that
whereas
the
representation is unambiguous,
Rock
0
−1
1 full matrix
Left
Right
C
D must explicit state whether
the abbreviation
we
this matrix represents a com
game or a zero-sum one.
Paper
1classical example
0
−1 Left game
1 is the0game of matching pen
C A−1,
−1
−4, 0 of a zero-sum
game, each of the two players has a penny, and independently chooses to d
then compare their pennies. If they are th
Scissors heads
−1 or tails. The
1 two players
0 Right
0
1
Dplayer
0, −4
−3,both,
−3 and otherwise
1 pockets
player 2 pockets them. The pay
shown in Figure 3.5.
Figure 3.6 Rock, Paper, Scissors game.
3.4 Coordination game.
Figure 3.1 The TCP user’s (aka the Prisoner’s)Figure
Dilemma.
Heads Tails
B
F
game
At you
the other
of the
from pure
coordination
games lie zero
eero-sum
options
what should
adopt,end
C or
D? spectrum
Does it depend
on what
you
which
(bearing
in
mind
the
comment
we
made
earlier
about
positive affine
lleague will do? Furthermore, from the perspective of the network operaHeads
1 games.
−1 Unlike common-pa
B are2,more
1 properly
0, 0
onstant-sum
tions)
called
constant-sum
of behavior can he expect from the two users? Will any two users behave
ames
n presented with this scenario? Will the behavior change if the network
c before
Shoham
and Leyton-Brown,
Tailsmaking
−1a decision?
1 2006
ws the users to communicate
each
F
0, 0with 1,
2 other
hanges to the delays would the users’ decisions still be the same? How
rs behave if they have the opportunity to faceFigure
this same
decision with the
3.5 Matching Pennies game.
Figure 3.7 Battle of the Sexes game.
part multiple times? Do answers to the above questions depend on how
gents are and how they view each other’s rationality?
The popular
of Rock,
Scissors,
as R
intro to many
CPSC 532A also
Lectureknown
3, Slide 19
ry Game
givesTheory
answers
of thesechildren’s
questions.game
It tells
us thatPaper,
any rational
ue’s options are the columns.
In each
cell, the first number
represents
AsMatrix
in theGames
case
of common-payoff
games,Best
weResponse
can use an
abbreviated
m
Recap
Example
Paretothe
Optimality
Nash
Equilibrium
right.
If the
the second
players
choose
same side
(left
or right)and
they
have
some hig
Rock
Papernumber
Scissors
or, minus your delay),
and
represents
your
colleague’s
represent
zero-sum
games,
in which we
write
only one payoff value in ea
otherwise they have a low utility. The game matrix is shown in Figure 3.4
value represents
the payoff
of player 1, and thus the negative of the payof
Nash Equilibria
of Example
Games
Note,
though,
that
whereas
the
representation is unambiguous,
Rock
0
−1
1 full matrix
Left
Right
C
D must explicit state whether
the abbreviation
we
this matrix represents a com
game or a zero-sum one.
Paper
1classical example
0
−1 Left game
1 is the0game of matching pen
C A−1,
−1
−4, 0 of a zero-sum
game, each of the two players has a penny, and independently chooses to d
then compare their pennies. If they are th
Scissors heads
−1 or tails. The
1 two players
0 Right
0
1
Dplayer
0, −4
−3,both,
−3 and otherwise
1 pockets
player 2 pockets them. The pay
shown in Figure 3.5.
Figure 3.6 Rock, Paper, Scissors game.
3.4 Coordination game.
Figure 3.1 The TCP user’s (aka the Prisoner’s)Figure
Dilemma.
Heads Tails
B
F
game
At you
the other
of the
from pure
coordination
games lie zero
eero-sum
options
what should
adopt,end
C or
D? spectrum
Does it depend
on what
you
which
(bearing
in
mind
the
comment
we
made
earlier
about
positive affine
lleague will do? Furthermore, from the perspective of the network operaHeads
1 games.
−1 Unlike common-pa
B are2,more
1 properly
0, 0
onstant-sum
tions)
called
constant-sum
of behavior can he expect from the two users? Will any two users behave
ames
n presented with this scenario? Will the behavior change if the network
c before
Shoham
and Leyton-Brown,
Tailsmaking
−1a decision?
1 2006
ws the users to communicate
each
F
0, 0with 1,
2 other
hanges to the delays would the users’ decisions still be the same? How
rs behave The
if they
have the
opportunity
to faceFigure
this same
decision
with
the
paradox
Prisoner’s
dilemma:
Nash
equilibrium
is game.
the only
Matching
Pennies
Figure
3.7 of
Battle
of the Sexes
game. the3.5
part multiple times? Do answers
to
the
above
questions
depend
on
how
non-Pareto-optimal outcome!
gents are and how they view each other’s rationality?
The popular
of Rock,
Scissors,
as R
intro to many
CPSC 532A also
Lectureknown
3, Slide 19
ry Game
givesTheory
answers
of thesechildren’s
questions.game
It tells
us thatPaper,
any rational