Download simultaneous - move games

SIMULTANEOUS - MOVE GAMES
A game is simultaneous-move if the players choose their actions
without any information on the other players’ actions.
This does NOT necessarily mean that the players are acting at
exactly the same time (even though that may very well be the case).
Example: the interaction between Beckham and Rüstü
is a simultaneous-move game
(even though they do not act at exactly the same time)
because Rüstü does NOT observe where Beckham kicked the ball
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A game (in strategic form or normal form) is
1. A set of players
N
2. For each player i in N, a set of his strategies:
Si
3. For each player i in N, his payoff function:
ui
Example: the Rock-Paper-Scissors game
N = {1 , 2}
S1 = S2 = {r , p , s }
u1(r , s) = u1(p , r) = u1(s , p) = 1
u1(r , r) = u1(p , p) = u1(s , s) = 0
u1(r , p) = u1(p , s) = u1(s , r) = -1
u2 is defined similarly
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Example 2
Guessing half of the average
(10 players)
1. Explain the rules of the game
2. Write it in normal form
3
To illustrate simple simultaneous-move games
we use a
Game Table
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Example: the Rock-Paper-Scissors game
draw a table:
Player 1 takes rows
Player 2 takes columns
PLAYER 2
R
P
S
R
PLAYER
1
P
S
Observe: each cell corresponds to a possible outcome
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Then place in each cell
the payoffs agents get from
that outcome
PLAYER 2
PLAYER
1
R
P
S
R
0, 0
-1, 1
1, -1
P
1, -1
0, 0
-1, 1
1, -1
0, 0
S
What about the halfof the average
game?
Row player
-1, 1
: -1
Column player: 1
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A tale of two prisoners:
Tchaikovsky
Confess
(Defect)
Deny
(Cooperate)
Confess
(Defect)
10 yr, 10 yr
1 yr, 25 yr
Deny
(Cooperate)
25 yr, 1 yr
3 yr, 3 yr
Conductor
ATTENTION: THIS IS NOT A GAME TABLE
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Prisoner’s Dilemma
Tchaikovsky
Confess
(Defect)
Deny
(Cooperate)
Confess
(Defect)
-10,
-10
-1,
-25
Deny
(Cooperate)
-25,
-1
-3,
-3
Conductor
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Students doing a project together
Student 2
Goof off
(Defect)
Work hard
(Cooperate)
Goof off
(Defect)
1,
1
3,
Work hard
(Cooperate)
0,
3
2,
0
Student 1
2
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Two firms (duopolists), choosing prices
Firm 2
Low
(Defect)
Low
(Defect)
High
(Cooperate)
600, 600
1200, -200
-200, 1200
1000, 1000
Firm 1
High
(Cooperate)
Other examples: the arms race, common property, etc.
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Battle of the Sexes
Wife
Football
Soap opera
Football
2,
1
0,
Soap opera
0,
0
1,
0
Husband
2
Other examples: two politicians determining position on an issue
two merging firms choosing between PC and MAC
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Matching Pennies
Veli
Head
Head
Tail
1,
-1
-1,
1
-1,
1
1,
-1
Ali
Tail
Example: leader and follower firms choosing product appearances
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The Stag hunt ( Rousseau, in Discourse on the Origin
and Foundations of Inequality Among Man )
Hunter 2
Stag
Hare
Stag
2,
2
0,
1
Hare
1,
0
1,
1
Hunter 1
Other example: arms race
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The Chicken game
DEAN
Swerve
(Chicken)
Straight
(Tough)
Swerve
(Chicken)
0, 0
–1, 1
Straight
(Tough)
1, –1
–2, –2
JAMES
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"The Gift of the Magi" (O. Henry)
is about a young couple who are short of money but desperately
want to buy each other Christmas gifts. Unbeknownst to Jim, Della
sells her most valuable possession, her beautiful hair, in order to
buy a platinum fob chain for Jim's watch; while unbeknownst to
Della, Jim sells his own most valuable possession, his watch, to
buy jeweled combs for Della's hair.
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The Gift of the Magi (O Henry)
Jim
Don’t buy
present
Don’t buy
present
Buy
present
0, 0
–1, 1
1, –1
–2, –2
Della
Buy
present
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Zero-sum game
A 2-player game is zero-sum if in each entry of the game table, the
agents' payoffs add up to zero.
Constant-sum game
A 2-player game is constant-sum if in each entry of the game table,
the agents' payoffs add up to the same constant.
Competitive game
A 2-player game is competitive if Player 1’s ranking of the strategy
profiles is the opposite of Player 2’s ranking.
Every zero-sum game is a constant-sum game and every constantsum game is competitive. However, every competitive game can be
made zero-sum by changing the payoff numbers.
In these games, the players have totally opposing interests.
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Zero-sum games:
it is sufficient to write one number in every box
the payoff of row player
(b) Payoffs in zero-sum notation
PLAYER 2
PLAYER
1
PLAYER 2
R
P
S
R
0, 0
-1, 1
1, -1
P
1, -1
0, 0
-1, 1
1, -1
0, 0
S
-1, 1
Row player receives
PLAYER
1
R
P
S
R
0
-1
1
P
1
0
-1
S
-1
1
0
-1
Column player receives - ( -1 ) = 1
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Game tables with three players:
TALIA chooses:
Contribute
Don't Contribute
NINA
EMILY
NINA
Contribute
Don't
Contribute
5, 5, 5
3, 6, 3
Don't
6, 3, 3
4, 4, 1
EMILY
Contribute
Don't
Contribute
3, 3, 6
1, 4, 4
Don't
4, 1, 4
2, 2, 2
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ANALYZING GAMES
Which strategies will the agents choose?
What will be the outcome?
USE: the assumption of rationality
20
In-class game
Guessing half of the average
(10 players)
Play the game
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Discussion
• dominance,
• iterated elimination of dominated strategies,
• culmination in a Nash equilibrium
• players getting close to the Nash equilibrium with more
experience
• if you expect the others not to play equilibrium strategies,
then your best response might be different
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What will the players do?
Tchaikovsky
Defect
Cooperate
Defect
-10,
-10
-1,
-25
Cooperate
-25,
-1
-3,
-3
Conductor
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Strict Domination
• Take an agent (call him Bond, James Bond)
and two of his strategies: U and D.
• The strategy U strictly dominates the strategy D (for Bond) if
1. for every possible strategy profile of the other agents, playing U
yields a higher (>) payoff than playing D.
That is, independent of what the others do, playing U always gives
a higher payoff
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Given a strategic form game
G = ( N , (S1,…,Sn) , (u1,…,un) )
for an Agent i, U strictly dominates D if
1. for every strategy profile
s-i = ( s1 ,…, si-1 , si+1 ,…, sn )
of the agents other than i,
ui ( U , s-i ) > ui ( D , s-i )
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What will the players do in this game?
Tchaikovsky
Defect
Cooperate
Defect
-10,
-10
-1,
-25
Cooperate
-25,
-1
-1,
-3
Conductor
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Weak Domination
• Take an agent (call him Bond, James Bond)
and two of his strategies: U and D.
• The strategy U weakly dominates the strategy D (for Bond) if
1. for every possible strategy profiles of the other agents,
playing U yields an at least as high (≥) payoff as playing D
and
2. for at least one strategy profile of the other agents,
U yields a higher (>) payoff than playing D.
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Given a strategic form game G
For agent i, U weakly dominates D if
1. for every strategy profile s-i of the other agents,
ui ( U , s-i ) ≥ ui ( D , s-i )
and
2. for at least one s-i* of the other agents,
ui ( U , s-i* ) > ui ( D , s-i* )
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What are the dominance relations?
Row Player’s payoffs
U
5
-1
3
M
2
-5
2
D
3
2
1
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NOTE:
If
U strictly dominates D
then U weakly dominates D
Definition:
If
there is some strategy U that strictly (weakly) dominates D
then D is called a strictly (weakly) dominated strategy
Important:
Rational players do NOT PLAY their
strictly dominated strategies
They can sometimes play their
weakly dominated strategies
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Cooperate is weakly dominated in this game?
Can they sustain (Cooperate,Cooperate) ?
Tchaikovsky
Defect
Cooperate
Defect
-10,
-10
-1,
-25
Cooperate
-25,
-1
-1,
-1
Conductor
31
QUESTION
• If M is strictly dominated by U and
M is weakly dominated by D
what do I call M ?
32
If a strategy of Bond strictly dominates all of his other strategies
it is called a strictly dominant strategy.
If Bond has a strictly dominant strategy, he always plays it
If a strategy of Bond weakly dominates all of his other strategies
it is called a weakly dominant strategy.
However, Bond does not always play a weakly dominant strategy.
Why? The answer lies in what Bond expects others to play.
33
QUESTIONS
1. If U strictly dominates M and
U weakly dominates D
what do I call U?
2. How many strictly dominant strategies can a player have?
How many weakly dominant strategies can a player have?
34
BEST SITUATION
Strictly Dominant Strategy for each agent
(it strictly dominates all of her other strategies)
or
Weakly Dominant Strategy for each agent
(it weakly dominates all of her other strategies)
THEN WE KNOW WHAT TO DO !!!
We have a Dominant Strategy Equilibrium
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Definition: a Dominant Strategy Equilibrium of the game
G = ( N , (S1,…,Sn) , (u1,…,un) )
is a strategy profile (s1,…,sn) such that for every player i in N, si
is a dominant strategy of player i.
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Both players have dominant strategies in the
Prisoner’s Dilemma.
Tchaikovsky
Confess
(Defect)
Deny
(Cooperate)
Confess
(Defect)
-10,
-10
-1,
-25
Deny
(Cooperate)
-25,
-1
-3,
-3
Conductor
Thus (Confess, Confess) is a Dominant Strategy Equilibrium
37
1.
What are the domination relationships between players’ strategies?
2.
Which are the dominant strategies? Strict or weak?
3.
Which are the dominated strategies? Strict or weak?
4.
Any dominant strategy equilibrium?
COLUMN
ROW
Left
Center
Right
Up
1, 2
2, 5
1, 3
Level
0, 5
1, 3
1, 6
Down
1, 1
3, 1
5, 2
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What about the Battle of the Bismarck Sea?
JAPANESE NAVY
U.S.
AIR FORCES
North
South
North
2
2
South
1
3
Note that this is a zero-sum game
39
What about the Chicken game?
DEAN
Swerve
(Chicken)
Straight
(Tough)
Swerve
(Chicken)
0, 0
–1, 1
Straight
(Tough)
1, –1
–2, –2
JAMES
40
What do you do when there is no
dominant strategy equilibrium ?????
Successively Eliminating Strictly Dominated Strategies
(because players never play them)
The table gets smaller and smaller and smaller and smaller and smaller
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Successive elimination of strictly dominated strategies
PIERCE’S PIZZA PIES
DONNA’S
DEEP DISH
High
Medium
Low
High
60, 60
36, 70
36, 35
Medium
70, 36
50, 50
30, 35
Low
35, 36
35, 30
25, 25
42
CAUTION: When dominance is weak, the result depends on
the order of elimination.
COLUMN
L
ROW
C
R
T
1, 1
1, 1
0, 0
B
0, 0
1, 2
1, 2
HOW ABOUT GAMES LIKE THE CHICKEN GAME?
43
Nash Equilibrium (J. Nash, 1951)
A strategy profile where
an agent, given her opponents’ strategies,
can NOT INCREASE her payoff by switching to another strategy
That is, a strategy profile where
each agent’s strategy is a best response to the other agents’ strategies
Definition: a Nash Equilibrium of the game
G = ( N , (S1,…,Sn) , (u1,…,un) )
is a strategy profile (s1*,…,sn*) such that for every player i in N,
ui( si* , s-i* ) ≥ ui( si , s-i* )
for every si in Si.
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Interpretation:
Imagine a population of many decision-makers, who repeatedly
interact (assume the role of a game player)
In time, they will form beliefs about how opponents behave.
With experience, they will find the best actions for themselves.
A Nash equilibrium is designed to model a steady state of social
interaction (in a sense, a social norm) that emerges out of the
players gaining experience in playing repeatedly.
Example:
Driving (more specifically, the crossroads game where you either
stop or pass)
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The Prisoners’ Dilemma:
Student 2
Goof off
(Defect)
Work hard
(Cooperate)
Goof off
(Defect)
1,
1
3,
Work hard
(Cooperate)
0,
3
2,
0
Student 1
2
The incentive to free-ride eliminates the possibility of the
mutually desirable outcome.
46
Battle of the Sexes
Wife
Football
Soap opera
Football
2,
1
0,
Soap opera
0,
0
1,
0
Husband
2
Both outcomes are stable social norms.
47
Matching Pennies
Veli
Head
Head
Tail
1,
-1
-1,
1
-1,
1
1,
-1
Ali
Tail
48
The Tennis Game
HINGIS
SELES
DL
CC
DL
50
80
CC
90
20
49
The Stag hunt
Hunter 2
Stag
Hare
Stag
2,
2
0,
1
Hare
1,
0
1,
1
Hunter 1
One outcome better than the other, but coordinated
deviations are not allowed.
Will talk about focal equilibria.
50
IN-CLASS game:
• Two players
• Each proposes a division of 1000 YTL
• If proposals match, they get what is proposed
• Otherwise, they get nothing
Assume: each player’s objective is to maximize her monetary gain.
51
Consider the following coordination game:
COLUMN
BACH
STRAVINSKY
BACH
2,
2
0,
0
STRAVINSKY
0,
0
1,
1
ROW
Is one of the equilibria more likely?
It is called a focal equilibrium.
52
The Chicken Game
DEAN
Swerve
(Chicken)
Straight
(Tough)
Swerve
(Chicken)
0, 0
–1, 1
Straight
(Tough)
1, –1
–2, –2
JAMES
53
B
A
FIGURE 4.13 Lottery
0
1
2
0
0, 0
0, 5
0, 0
1
5, 0
0, 0
– 5, 0
2
0, 0
0, – 5
– 5, – 5
54
Copyright © 2000 by W.W. Norton & Company
COLUMN
ROW
EXERCISE 4.7
Left
Middle
Right
Top
3, 1
2, 3
10, 2
High
4, 5
3, 0
6, 4
Low
2, 2
5, 4
12, 3
Bottom
5, 6
4, 5
9, 7
55
Copyright © 2000 by W.W. Norton & Company
B
A
EXERCISE 4.9
1
2
3
1
10, 10
0, 0
0, 0
2
0, 0
15, 15
0, 0
3
0, 0
0, 0
15, 15
56
Copyright © 2000 by W.W. Norton & Company