3.2.1. Static Games of complete information: Dominant
Strategies and Nash Equilibrium in pure and mixed strategies
Strategic Behavior in Business and Econ
Outline
3.1. What is a Game ?
3.1.1. The elements of a Game
3.1.2 The Rules of the Game: Example
3.1.3. Examples of Game Situations
3.1.4 Types of Games
3.2. Solution Concepts
3.2.1. Static Games of complete information: Dominant
Strategies and Nash Equilibrium in pure and mixed strategies
3.2.2. Dynamic Games of complete information: Backward
Induction and Subgame perfection
Strategic Behavior in Business and Econ
Reminder
There are, basically, four different types
of games
S t at ic
Dyn amic
Co mplet e In f o r mat io n
In co mplet e In f o r mat io n
Player s have all r elevan t
N o t all player s have all t he
in f o r mat io n , an d mo ve
simulat en o usly
r elevan t in f o r mat io n , an d
mo ve simulat en o usly
Player s have all r elevan t
N o t all player s have all t he
in f o r mat io n , an d mo ve
r elevan t in f o r mat io n , an d
seq uen t ially
mo ve seq uen t ially
All games in a given category are represented and solved alike
Strategic Behavior in Business and Econ
Reminder
Solution Concepts
Co mplet e In f o r mat io n
In co mplet e In f o r mat io n
E q uilibr ium in Do min an t S t r at eg ies,
S t at ic
E limin at io n o f Do min at ed S t r at s.
Bayesian N ash E q uilibr ium
N ash E q uilibr ium
N ash E q uilibr ium,
Dyn amic
Backw ar d In duct io n ,
S ubg ame Per f ect E q uilibr ium
Bayesian Per f ect N ash E q uilibr ium,
S eq uen t ial E q ulibr ium
A solution of a game is called an Equilibrium of the game
Strategic Behavior in Business and Econ
Reminder
Static Games of Complete Information
All the players choose their strategies simultaneously. This does not
mean “at the same time” but “without knowing the choice of others”
Because of this simultaneity they can be represented by means of a
table (payoff matrix)
They are “one-shot games”, that is, they are played only once
All the players have all the information regarding who are the other
players, what are the own strategies and the strategies of the
others, what are the own payoffs and the payoffs of the others,
and what are the rules of the game
Strategic Behavior in Business and Econ
Solution concepts for this type of games
• Equilibrium in Dominant Strategies
When there is an “always winning” strategy
• Equilibrium by elimination of Dominated Strategies
When there are “worse than” strategies
• Nash Equilibrium
Works in any case
In pure strategies (players do not randomize)
In mixed strategies (players do randomize)
Strategic Behavior in Business and Econ
Reminder
An equilibrium of the game is a choice of strategies by all
the players that is stable, in the sense that
Given what the other players are doing, nobody
has any reason to change his or her own strategy
Strategic Behavior in Business and Econ
Example: Game with “always winning” strategy
(Dominant Strategy)
Prediction of Game Theory:
Both have a clear best
strategy
Philip Morris (player 2)
Not advertise
Advertise
Advertise no matter what
Not
advertise
50, 50
20, 60
60, 20
30, 30
Reynolds
(player 1)
Advertise
Strategic Behavior in Business and Econ
Example: Game with “worse than” strategies
(Dominated Strategies)
Prediction of Game Theory:
There is no “always winning”
strategy
Bar 2
Bar 1
$2
$4
$5
$2
$4
$5
10 , 10
12 , 14
15 , 14
14 , 12
20, 20
15 , 28
14 , 15
28 , 15
25, 25
(in thousands of dollars)
Strategic Behavior in Business and Econ
Prediction of Game Theory:
But there is a clearly bad
strategy: $2 is always worse
than $4
Bar 1
$2
$4
$5
Bar 2
$2
$4
$5
10 , 10
12 , 14
15 , 14
14 , 12
20, 20
15 , 28
14 , 15
28 , 15
25, 25
(in thousands of dollars)
Strategic Behavior in Business and Econ
Prediction of Game Theory:
If $2 is removed from the
game (it will never be used)
then the game is more clear
Bar 1
$2
$4
$5
Bar 2
$2
$4
$5
10 , 10
12 , 14
15 , 14
14 , 12
20, 20
15 , 28
14 , 15
28 , 15
25, 25
(in thousands of dollars)
Strategic Behavior in Business and Econ
Prediction of Game Theory:
Now $4 is clearly the best
strategy no matter what
my competitor does
Bar 1
$4
$5
Bar 2
$4
$5
20, 20
15 , 28
28 , 15
25, 25
(in thousands of dollars)
Strategic Behavior in Business and Econ
Prediction of Game Theory:
NOTICE: The “coincidence” of
red circles is (again) the stable
outcome
Bar 1
$4
$5
Bar 2
$4
$5
20, 20
15 , 28
28 , 15
25, 25
(in thousands of dollars)
Strategic Behavior in Business and Econ
Example: Game with no “always winning strategy
with no “worse than” strategies
(Nash Equilibrium)
Prediction of Game Theory:
There is no “always winning”
nor “worse than” strategies
Paul
Mary
Right
Left
Right
0, 0
-50 , -50
Left
-50 , -50
0,0
There are 2 equilibrium: (coincidence of red circles)
Both players driving on the right
Both players driving on the left
(but players do not randomize!)
Strategic Behavior in Business and Econ
Example: Game with no “always winning strategy
with no “worse than” strategies
(Nash Equilibrium)
Prediction of Game Theory:
There is no “always winning”
nor “worse than” strategies
Player 2
Rock
Player 1
Rock
Paper
Scissors
0, 0
+1, -1
-1, +1
Paper
-1, +1
0, 0
+1, -1
Strategic Behavior in Business and Econ
Scissors
+1, -1
-1, +1
0, 0
Example: Game with no “always winning strategy
with no “worse than” strategies
(Nash Equilibrium)
Prediction of Game Theory:
There is no “coincidence” of
red circles
Player 2
Rock
Player 1
Rock
Paper
Scissors
0, 0
+1, -1
-1, +1
Paper
-1, +1
0, 0
+1, -1
Players do randomize to play this game
Strategic Behavior in Business and Econ
Scissors
+1, -1
-1, +1
0, 0
Equilibrium in Dominant Strategies
A player has a Dominant Strategy if, regardless the
strategy chosen by the other players, that strategy is
always a best response (it has all red circles)
If every player has a Dominant Strategy, then
the predicted outcome of the game is the one that
corresponds to the players choosing that strategy
If some players have a Dominant Strategy and others
don't, the predicted outcome is that players with
dominant strategies will use them whereas players with
no dominant strategies will choose a best response to
them
Strategic Behavior in Business and Econ
Philip Morris (player 2)
Example:
Not advertise
Reynolds
(player 1)
Not
advertise
Advertise
Advertise
50, 50
20, 60
60, 20
30, 30
Philip Morris (player 2)
Example:
Not advertise
Reynolds
(player 1)
Not
advertise
Advertise
Advertise
50, 80
20, 60
60, 20
30, 30
Strategic Behavior in Business and Econ
Equilibrium by elimination of Dominated
Strategies
A player has a Dominated Strategy if, regardless the strategy chosen by the
other players, that strategy is always worse than some other strategy (it has NO
red circles)
If a player has a Dominated Strategy, the corresponding
row or column can be removed from the table
After the removal of one dominated strategy it might
happen that other strategies are also dominated.
The process of elimination of dominated strategies continues
until there are no more dominated strategies for any player
Strategic Behavior in Business and Econ
Example
Player 2
Player 1
V
W
X
Y
Z
A
4,-1
3,0
-3,1
-1,4
-2,0
B
-1,1
2,2
2,3
-1,0
2,5
C
2,1
-1,-1
0,4
4,-1
0,2
D
1,6
-3,0
-1,4
1,1
-1,4
E
0,0
1,4
-3,1
-2,3
-1,-1
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
V
W
X
Y
Z
A
4,-1
3,0
-3,1
-1,4
-2,0
B
-1,1
2,2
2,3
-1,0
2,5
C
2,1
-1,-1
0,4
4,-1
0,2
D
1,6
-3,0
-1,4
1,1
-1,4
E
0,0
1,4
-3,1
-2,3
-1,-1
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
Notice that (B,Z) is
the only outcome with
coincidence of red circles
V
W
X
Y
Z
A
4,-1
3,0
-3,1
-1,4
-2,0
B
-1,1
2,2
2,3
-1,0
2,5
C
2,1
-1,-1
0,4
4,-1
0,2
D
1,6
-3,0
-1,4
1,1
-1,4
E
0,0
1,4
-3,1
-2,3
-1,-1
There are no strategies with “all red circles”
That is, there are no Dominant Strategies
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
V
W
X
Y
Z
A
4,-1
3,0
-3,1
-1,4
-2,0
B
-1,1
2,2
2,3
-1,0
2,5
C
2,1
-1,-1
0,4
4,-1
0,2
D
1,6
-3,0
-1,4
1,1
-1,4
E
0,0
1,4
-3,1
-2,3
-1,-1
But there are strategies with NO red circles
That is, there are Dominated Strategies
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
V
W
X
Y
Z
A
4,-1
3,0
-3,1
-1,4
-2,0
B
-1,1
2,2
2,3
-1,0
2,5
C
2,1
-1,-1
0,4
4,-1
0,2
D
1,6
-3,0
-1,4
1,1
-1,4
E
0,0
1,4
-3,1
-2,3
-1,-1
But there are strategies with NO red circles
That is, there are Dominated Strategies
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
V
W
X
Y
Z
A
4,-1
3,0
-3,1
-1,4
-2,0
B
-1,1
2,2
2,3
-1,0
2,5
C
2,1
-1,-1
0,4
4,-1
0,2
D
1,6
-3,0
-1,4
1,1
-1,4
E
0,0
1,4
-3,1
-2,3
-1,-1
We can eliminate the dominated strategies !
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
V
W
X
Y
Z
A
4,-1
3,0
-3,1
-1,4
-2,0
B
-1,1
2,2
2,3
-1,0
2,5
C
2,1
-1,-1
0,4
4,-1
0,2
We can eliminate the dominated strategies !
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
V
W
X
Y
Z
A
4,-1
3,0
-3,1
-1,4
-2,0
B
-1,1
2,2
2,3
-1,0
2,5
C
2,1
-1,-1
0,4
4,-1
0,2
Now there are strategies with NO red circles
for Player 2
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
V
W
X
Y
Z
A
4,-1
3,0
-3,1
-1,4
-2,0
B
-1,1
2,2
2,3
-1,0
2,5
C
2,1
-1,-1
0,4
4,-1
0,2
Now there are strategies with NO red circles
for Player 2
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
V
W
X
Y
Z
A
4,-1
3,0
-3,1
-1,4
-2,0
B
-1,1
2,2
2,3
-1,0
2,5
C
2,1
-1,-1
0,4
4,-1
0,2
We can eliminate, again, the Dominated Strategies
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
X
Y
Z
A
-3,1
-1,4
-2,0
B
2,3
-1,0
2,5
C
0,4
4,-1
0,2
We can eliminate, again, the Dominated Strategies
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
X
Y
Z
A
-3,1
-1,4
-2,0
B
2,3
-1,0
2,5
C
0,4
4,-1
0,2
Now we find new Dominated Strategies
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
X
Y
Z
A
-3,1
-1,4
-2,0
B
2,3
-1,0
2,5
C
0,4
4,-1
0,2
Now we find new Dominated Strategies
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
X
Y
Z
B
2,3
-1,0
2,5
C
0,4
4,-1
0,2
The process of elimination of Dominated Strategies
continues
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
X
Y
Z
B
2,3
-1,0
2,5
C
0,4
4,-1
0,2
The process of elimination of Dominated Strategies
continues
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
X
Z
B
2,3
2,5
C
0,4
0,2
The process of elimination of Dominated Strategies
continues
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 1
Player 2
X
Z
B
2,3
2,5
C
0,4
0,2
The process of elimination of Dominated Strategies
continues
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 2
B
X
Z
2,3
2,5
Player 1
The process of elimination of Dominated Strategies
continues
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 2
B
X
Z
2,3
2,5
Player 1
The process of elimination of Dominated Strategies
continues
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 2
Z
B
2,5
Player 1
The process of elimination of Dominated Strategies
continues
Strategic Behavior in Business and Econ
Example
Look for the best
replies
Player 2
Z
B
2,5
Player 1
The final (predicted) outcome of the game is
Player 1 chooses B and gets a payoff of 2
Player 2 choose Z and gets a payoff of 5
(Notice that this was the only outcome with “coincidence”
of red circles)
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The order of elimination of Dominated Strategies does
not affect the final outcome of the process
Sometimes the process of elimination continues until a
unique outcome survives, sometimes it stops earlier
If some outcome of the game has a “coincidence of red
circles”, then it will survive the process of elimination
Strategic Behavior in Business and Econ
Nash Equilibrium
after John F. Nash Jr. (1928-)
There are (many) games in which the
two previous solution concepts can
not be used.
That is, there (many) games with no
Dominant Strategies nor Dominated
Strategies.
The ideal would be to have a solution
concept that can be used in any game
and that always works. That is the
“Nash Equilibrium”
Strategic Behavior in Business and Econ
Nash Equilibrium
A Nash Equilibrium is a combination of strategies by the
players with the special feature that:
All players are playing a best reply to what the
other players are doing
Notice that, since all the players are playing a “best reply”,
nobody will want to change his choice of an strategy !!!
Is in this sense that a Nash Equilibrium is stable
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Strategic Behavior in Business and Econ
Nash Equilibrium
In practical terms
A Nash Equilibrium is where the best replies of
the players coincide
That is, a Nash Equilibrium is where the red circles
coincide (in the Table representation of the game)
Strategic Behavior in Business and Econ
Example: The Battle of the Sexes
Pat and Chris want to go out together after work
They work on different places and before going to work
they couldn't find any agreement on where to go
The options were go to the Opera of go to the Football
They both would like to go to a place together, but Pat
prefers the Opera whereas Chris likes the Football better
Thus, the situation is that after work (5 pm) each must
decide where to go without knowing the choice of the
other
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The environment of the game
Players:
Strategies:
Payoffs:
Pat and Chris
Opera or Football
In this case we must “define” the payoffs
in such a way that represent the game
described
(see the Table in the next slide)
The Rules of the Game
Timing of moves
Nature of conflict and interaction
Information conditions
Simultaneous
Coordination
Symmetric
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The game represented
Chris
Look for the best
replies
Opera
Opera
Football
3,1
0,0
-1 , -1
1,3
Pat
Football
Strategic Behavior in Business and Econ
The game represented
Chris
Look for the best
replies
Opera
Opera
Football
3,1
0,0
-1 , -1
1,3
Pat
Football
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The game represented
Chris
There are two Nash
Equlibria
Opera
Opera
Football
3,1
0,0
-1 , -1
1,3
Pat
Football
Strategic Behavior in Business and Econ
The game represented
Pat and Chris going
both to the Opera
is stable
Chris
Opera
Opera
Football
3,1
0,0
-1 , -1
1,3
Pat
Football
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The game represented
Pat and Chris going
both to the Football
is stable
Chris
Opera
Opera
Football
3,1
0,0
-1 , -1
1,3
Pat
Football
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The game represented
Any other outcome
is unstable
Chris
Opera
Opera
Football
3,1
0,0
-1 , -1
1,3
Pat
Football
Strategic Behavior in Business and Econ
The game represented
This game calls for
coordination
Chris
Opera
Opera
Football
3,1
0,0
-1 , -1
1,3
Pat
Football
Strategic Behavior in Business and Econ
Example: The Rock-Paper-Scissors Game
Player 2
Rock
Player 1
Rock
Paper
Scissors
0, 0
+1, -1
-1, +1
Paper
-1, +1
0, 0
+1, -1
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Scissors
+1, -1
-1, +1
0, 0
Example: The Rock-Paper-Scissors Game
Look for the best
replies
Player 2
Rock
Player 1
Rock
Paper
Scissors
0, 0
+1, -1
-1, +1
Paper
-1, +1
0, 0
+1, -1
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Scissors
+1, -1
-1, +1
0, 0
Example: The Rock-Paper-Scissors Game
There are not “always good”
nor “always bad” strategies.
Player 2
Rock
Player 1
Rock
Paper
Scissors
0, 0
+1, -1
-1, +1
Paper
-1, +1
0, 0
+1, -1
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Scissors
+1, -1
-1, +1
0, 0
Example: The Rock-Paper-Scissors Game
And there is no coincidence
of red circles. !!
There is no stable outcome !
Player 2
Rock
Player 1
Rock
Paper
Scissors
0, 0
+1, -1
-1, +1
Paper
-1, +1
0, 0
+1, -1
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Scissors
+1, -1
-1, +1
0, 0
Example: The Rock-Paper-Scissors Game
In this games, players choose
their strategy “at random”
This will be the equilibrium
Player 2
Rock
Player 1
Rock
Paper
Scissors
0, 0
+1, -1
-1, +1
Paper
-1, +1
0, 0
+1, -1
Strategic Behavior in Business and Econ
Scissors
+1, -1
-1, +1
0, 0
Nash Equilibria are stable outcomes of the game
The concept of Nash Equilibrium does not tell how
that outcome is reached
In case of more than one Nash Equilibria, we do not
know which one will be the one chosen by the players
(Advanced Game Theory has more solution concepts to
“select” among several Nash Equilibria)
An Equilibrium in Dominant Strategies is also a Nash
Equilibrium
A Nash Equilibrium survives the process of elimination
of Dominated Strategies
A Nash Equilibrium is a one-for-all solution
A Nash Equilibrium always exists, in any game
Strategic Behavior in Business and Econ
Summary
If you have a Dominant Strategy use it, and expect your
opponent to use it as well
If you have Dominated Strategies do not use any of them,
and expect you opponent not to use them as well
(eliminate them from the analysis of the game)
If there are neither Dominant Strategies nor Dominated
Strategies, look for Nash Equilibria and play accordingly.
Expect your opponent to play according to the Nash
Equilibrium as well
If there are multiple Nash Equilibria, further analysis is
required
Strategic Behavior in Business and Econ