3. Simultaneous move games
with pure strategies
In this section we shall learn
How to figure out the outcomes to expect in
simultaneous move game, by
Looking for clearly best strategies, ones that are always
played
Eliminating from consideration strategies that would never be
played.
Looking for strategies that allow your opponent to the least
harm to you.
How to reduce complicated games to something
much more tractable.
How to figure out what is likely to happen in games
where two or more put comes seem equally likely
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Games People Play.
Simultaneous move games
The prisoners dilemma.
A pair of criminals are suspected of carrying out a kidnapping, and
have been apprehended by the police. However, there is
insufficient evidence to convict them. The police can easily
convict them on a lesser charge which carries a 3yr sentence.
They are separated and individually offered the following
options. If one confesses the person who confesses gets 1yr
and the other 25yrs. If both confess they both get 10yrs.
If you are one of the kidnappers what
do you do?
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Simultaneous move games
The prisoners
dilemma.
Criminal #1
Confess
Deny
Confess
10,10
1, 25
Deny
25,1
3,3
Criminal
#2
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Payoff Matrices
Often referred to as the
game in normal form.
Payoff matrices are
particularly useful in
analyzing simultaneous
games.
As an example the game
rock-paper-scissors has a
payoff matrix of the form.
Player 2
Player
1
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Rock
Paper
Scissors
Rock
0,0
-1,1
1,-1
Paper
1,-1
0,0
-1,1
Scissors
-1,1
1,-1
0,0
Nash Equilibria
Thinking about simultaneous games
When you and your opponent have to play
simultaneously what do you do?
You reason as follows:
I wish to make the best reply I can to any choice my
opponent makes, and I know she is thinking the same
way.
I know she knows the way I’m thinking.
Hence we both expect each other to select mutual best
replies.
This defines a Nash equilibrium.
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Nash Equilibria
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Nash out of equilibrium
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Dominant Strategies
A dominant strategy is your best reply to
whatever the other players do.
If you have a dominant strategy, play it!
Unfortunately, you often don’t have a dominant
strategy.
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Simultaneous move games
Dominant strategies
Criminal #2
Confess
Deny
Confess
10,10
1, 25
Deny
25,1
3,3
Criminal #1
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Dominant Strategies
Example: Cigarette Advertising on TV
All US tobacco companies advertise heavily on TV.
1964
Surgeon General issues official warning that cigarette
smoking may be hazardous.
Cigarette companies’ reaction: fear of potential liability
lawsuits.
1970
Companies strike agreement: carry the warning label
and cease TV advertising in exchange for immunity
from federal lawsuits.
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Strategic Interactions
Players:
Strategies:
Payoffs:
Reynolds and Philip Morris
(Advertise, Do Not Advertise)
Companies’ Profits
Each firm earns $50 million from its customers
Advertising costs a firm $20 million
Advertising captures $30 million from competitor
How to represent this game?
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Dominant Strategies
Normal (Strategic) Form
PLAYERS
No Ad
Reynolds
Ad
Philip Morris
No Ad
Ad
50 , 50
20 , 60
60 , 20
STRATEGIES
30 , 30
PAYOFFS
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Dominant Strategies
Philip Morris
No Ad
Ad
No Ad
50 , 50
20 , 60
Ad
60 , 20
30 , 30
Reynolds
Best reply for Reynolds:
If Philip Morris advertises: advertise
If Philip Morris does not advertise: advertise
Regardless of what you think Philip Morris will do:
Advertise!
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Successive Elimination of
Dominated Strategies
If a strategy is dominated, eliminate it.
The size and complexity of the game is
reduced.
Eliminate any dominated strategies from
the reduced game.
Continue doing so successively.
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Successive Elimination of
Dominated Strategies
Example: Tourists & Natives
Two bars (bar 1, bar 2) compete
Can charge price of $2, $4, or $5
6000 tourists pick a bar randomly
4000 natives select the lowest price bar
Example 1: Both charge $2
each gets 5,000 customers
Example 2: Bar 1 charges $4
Bar 2 charges $5
Bar 1 gets 3000+4000=7,000 customers
Bar 2 gets 3000 customers
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Successive Elimination of
Dominated Strategies
Example: Tourists & Natives
$2
Bar 1 $4
$5
Bar 2
$2
$4
$5
10 , 10 14 , 12 14 , 15
12 , 14 20 , 20 28 , 15
15 , 14 15 , 28 25 , 25
in thousands of dollars
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Successive Elimination of
Dominated Strategies
Does any player have a dominant
strategy?
Does any player have a dominated
strategy?
Eliminate the dominated strategies
Reduce the normal-form game
Iterate the above procedure
What is the equilibrium?
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Successive Elimination of
Dominated Strategies
Bar 2
$2
$4
$5
$2 10 , 10 14 , 12 14 , 15
Bar 1 $4 12 , 14 20 , 20 28 , 15
$5 15 , 14 15 , 28 25 , 25
Bar 1
$4
$4
$5
20 , 20 28 , 15
$5
15 , 28 25 , 25
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What do we do when looking for
dominant and dominated strategies
does not yield a Nash equilibrium?
Here’s the tourists and natives again but now the method of eliminating
dominated strategies does not find a Nash equilibrium. Strategies of playing
$2 can still be eliminate by the successive elimination of dominated
strategies, but this is all.
$2
Bar 1 $4
$5
Bar 2
$2
$4
$5
10 , 10 14 , 12 14 , 15
12 , 14 20 , 20 21 , 18
15 , 14 15 , 28 25 , 25
in thousands of dollars
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What do we do when looking for dominant
and dominated strategies does not yield a
Nash equilibrium?
Bar 2
$2
$4
$5
$2 10 , 10 14 , 12 14 , 15
Bar 1 $4 12 , 14 20 , 20 21 , 18
$5 15 , 14 15 , 24 25 , 25
Apply another solution method.
Simplify as much as possible.
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Maximin
Tourists & Natives again
Bar 2
Bar 1
$4
$4
20 , 20
$5
21 , 18
$5
15 , 24
25 , 25
The idea here is to minimize the damage the other player can do
to you.
Maximin – Choose the row or column that gives the highest
minimum payoff , Bar 1 -$4, Bar 2 - $4.
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Minimax Regret
Tourists & Natives for a third time
The regret matrix represents the difference between the a
given strategy and the payoff of the best strategy.
Bar 2
$4
$4
0,0
$5
-4,-2
$5
-5,-1
0,0
Bar 1
Minimax regret requires you minimize your maximum loss.
Conclusion – Bar 1 chooses $4 and bar 2 chooses $4.
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Simultaneous move games
Dominated strategies
and dominance
solvable games.
Deny is a dominated
strategy for both
players.
Confess is a
dominant strategy for
both players.
Criminal #2
Criminal
#1
Games People Play.
Confess
Deny
Confess
10,10
1, 25
Deny
25,1
3,3
Simultaneous move games
A real example.
Happy-Families produce gizmos
using widgets
purchased
from the All American Rubber Company. Currently Happy-Families and All
American have an exclusive contract. All American is the sole supplier to
Happy-Families, who are All American’s sole customer. Happy-Families
can choose to qualify other suppliers of widgets, which will give them a
competitive supply base and also place them in a stronger bargaining
position with All American. But All American can get themselves qualified
by Lexmark, a move that gives them a stronger bargaining position with
Happy-Families. The two moves are offsetting but qualification is
expensive for both sides.
This is a classic prisoners dilemma!
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Simultaneous move games
The minimax method: Finding a Nash equilibrium.
In a football game an offence and defense square off, the offence’s gain is the
defense’s loss so this is a zero-sum game
Defense
Offense
Run
Pass
Blitz
min
Run
2
5
13
2
Short pass
6
5.6
10.5
5.6
Medium pass
6
4.5
1
1
Long pass
10
3
-2
-2
max
10
5.6
13
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Simultaneous move games
The minimax method
Note first that medium pass is a weakly dominated, so we may eliminate it.
But no more strategies can be eliminated by this method so the game is not
dominance solvable.
Now check for the minimax and maximin strategies.
For defense the minimax is pass. For offense the maximin is short pass.
{short pass, pass} is Nash.
Defense
Offense
Run
Pass
Blitz
min
Run
2
5
13
2
Short pass
6
5.6
10.5
5.6
Long pass
10
3
-2
-2
max
10
5.6
13
Games People Play.
Simultaneous move games
Cell-by-cell inspection
Nash equilibria can be found simply by cell-by-cell inspection.
For each strategy of your opponents look down the column or
across the row for your best reply.
Then check to see if your opponent would wish to change strategy in
response to your choice. If not you have found a Nash equilibrium. If
she changes her strategy, keep looking.
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Simultaneous move games
Continuous strategies.
Often you can vary your choices continuously, instead of
choosing A or B you can choose anything in between.
In these circumstances you need to compute best
response functions to find the Nash equilibria.
The point where the best response functions cross is the
Nash.
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Simultaneous move games
Continuous strategies
Example: The Rubber Duckie
sellers, Bert and Ernie.
Burt and Ernie each sell rubber
duckies in the same town. They
each have to choose their prices
as best replies to each other. Bert
finds that if Ernie charges a slightly
higher price for his rubber duckies
it is a best reply for him to raise his
price by a little less than Ernie’s
price increment. Ernie follows
exactly the same strategy.
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Simultaneous move games
Continuous strategies
Example – The pizza sellers, Bert and Ernie.
Bert's Price
Nash equilibrium
Bert's best response
Ernie's best response
Ernie's Price
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Simultaneous move games
Three player games
The street garden game
Three gardeners, Tom, Dick, and Condoleezza
are planning a street garden. Each has to make a voluntary
contribution of their time to planting pretty flowers. The payoffs
they receive from this fine project are as follows.
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Simultaneous move games
Three player games
The street garden game
The payoffs to the three gardeners are
Tom doesn’t contribute
Tom contributes
Condoleezza
C
Dick
D
C
D
5,5,5
3,6,3
6,3,3
4,4,1
Condoleezza
Dick
Find the Nash equilibria.
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C
D
C
3,3,6
1,4,4
D
4,1,4
2,2,2
Simultaneous move games
Three player games
The street garden game
Equilibrium
Tom doesn’t contribute
Tom contributes
Condoleezza
C
Dick
D
C
D
5,5,5
3,6,3
6,3,3
4,4,1
Condoleezza
Dick
Find the Nash equilibria.
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C
D
C
3,3,6
1,4,4
D
4,1,4
2,2,2
Simultaneous move games
Changing the game
Changing it again!!
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Simultaneous move games
Multiple Equilibria and Focal Points
The chicken game.
Two cars drive straight at each other at great speed, if the cars
crash you are killed, the worst possible outcome. If your
opponent swerves aside and you continue in a straight line you
show him to be a chicken, you gain and he loses, but he loses
less than being killed! If you both swerve you miss each other
and there is no gain or loss on either side.
What do you do?
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Simultaneous move games
Multiple Equilibria and Focal Points
The chicken game.
There are two Nash equilibria
both associated with one player
choosing straight and the other
swerve.
Player #2
Straight
Swerve
Straight
-2,-2
1,-1
Swerve
-1,1
0,0
How is an equilibrium
selected?
Player
#1
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Simultaneous move games
Multiple Equilibria and Focal Points
In problems such as the chicken game we need a way
of selecting an equilibrium. Possibilities include
Rule making – the driver from the south should swerve.
Focal points – historically the driver from the south has always
swerved. So both drivers expect this.
Strategic moves – Driver #1 fixes his steering so he can only go
straight!!
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Simultaneous move games
Multiple Equilibria and Focal Points
In games such as the chicken game coordination
difficult. If player #1 plays swerve she still prefers the
other to swerve too! She may swerve but wants the
other to believe she will not. This makes equilibrium
selection particularly difficult
In other games where both individuals prefer both
equilibria to either of the alternatives coordination is
easier.
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Simultaneous move games
Multiple Equilibria and Focal Points
Battle of the sexes
A husband and wife decide to go out for the evening. The husband
proposes they go to watch wrestling, the wife proposes they go to see a
play. They agree that neither activity will be any fun alone, and that both
are better than staying home with the children.
Wife
Husband
Wrestling
Play
Wrestling
2,1
0,0
Play
0,0
1,2
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Simultaneous move games
Multiple Equilibria and Focal Points
Battle of the sexes
In this case there are lesser obstacles to coordination. The
players have no incentive to misrepresent their desire for
coordination. Agreeing is always better than disagreeing.
Wife
Wrestling
Play
Wrestling
2,1
0,0
Play
0,0
1,2
Husband
Games People Play.
Multiple Equilibria and Focal Points
an Happy-Families Example
Suppose that Happy-Families purchases a crucial component from a
particular supplier. It is noticed by a Happy-Families tech support guy
that a high percentage of the problems reported with the
final product are associated with the failure of this
component. But the component is being manufactured
to exactly the tolerances specified by Happy-Families.
If the supplier reengineers their component and
Happy-Families reengineer how it interfaces with the product, a
critical weakness in the final product can be eliminated. The product
will last longer and will achieve a higher level of performance. It can
thus be sold for a higher price. For both Happy-Families and the
supplier the reengineering is costly and only of value if both engage in
the process. This game has two equilibria as we see below.
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Multiple Equilibria and Focal Points
an Happy-Families Example
In the absence of any reengineering, Happy-Families can sell a million units a year
of the final product for $200 a unit. Suppose that Happy-Families and the critical
component manufacturer both reengineer, then a million units of the new superior
product can be sold for $230 a unit. The cost of reengineering are $10 million each
for Happy-Families and the supplier. If any gains are split evenly between HappyFamilies and the supplier this gives the payoff matrix
Supplier
HappyFamilies
No Change
Reengineer
No Change
0,0
0,-10
Reengineer
-10,0
5,5
Clearly there are two equilibria {NC,NC} and {R,R}.
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