ECO 220 – Game Theory
Simultaneous Move Games
Objectives
• Be able to structure a game in normal form
• Be able to identify a Nash equilibrium
Agenda
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Definitions
Equilibrium Concepts
Dominance
Coordination Games
Applications
1
Definitions
• Simultaneous Move Games
– Games in which the players move within the
same information set without knowing what
the other players have done
– Occasionally this means moving at the same
time (Rock-Paper-Scissors) or may mean
moving at different times
– Games of imperfect information are played as
simultaneous move games
Definitions
• Discrete vs Continuous Strategies
– In games with discrete strategies there is a
finite number of distinct choices that the
players can make (enter or not enter)
– In continuous strategy games the actions that
the players can take actions along a
continuum (price of a product could be $1.00,
$1.01, $1.02 and so on
Definitions
• Pure Strategies vs Mixed Strategies
– In pure strategy games the player does one
action or the other with certainty
– In mixed strategy games the players randomly
choose their actions (such as in Rock-PaperScissors)
• We will focus initially on pure strategy
games with discrete choices
2
Definitions
• Normal (or strategic) form of the game is a
matrix representing the actions and
payoffs to the players
– The game tree that we looked at in the
sequential games is what is known as the
extensive form
A simple game
• We will use the game Rock-PaperScissors to illustrate these concepts
• Bart and Lisa have to decide how to spend
$50 that they got from their grandfather –
each knows what he or she wants to do
– If they do what Lisa wants, her payoff is +10
and Bart’s is 2
– If they do what Bart wants, his payoff is 8 and
Lisa’s is -3
A Simple Game
• Bart and Lisa cannot agree on a course of
action so they decide to play Rock-PaperScissors. Whoever wins RPS chooses
what to do with the $50
• The game in normal form is shown on the
next slide
3
A Simple Game
Lisa
Bart
Rock
Paper
Rock
(0,0)
(2,10)
(8,-3)
Paper
(8,-3)
(0,0)
(2,10)
Scissors
(2,10)
(8,-3)
(0,0)
Bart’s Payoff
Scissors
Lisa’s Payoff
Actions and Strategies
• In simultaneous move games actions are
the same as strategies
• In the sequential games players had to
enumerate all of their choices that they
could make once they knew what the other
had done
• In simultaneous move games they move
together so there is only one thing to do
Equilibrium Concepts
• The most frequently referred to equilibrium
concept is that of a Nash Equilibrium
– There are other special cases that we will look
at but they are all Nash equilibriums
• An outcome is a Nash equilibrium if no
player has any incentive to deviate from
his or her course of action given what the
other players are doing
4
Finding An Equilibrium
• There are several ways to find Nash
Equilibriums in pure strategies, when they
exist
– The most basic is a cell-by-cell analysis
– The best-response analysis in the textbook is
really a refinement of this
– Let’s see if there is a Nash Equilibrium in the
RPS game played by Lisa and Bart
A Simple Game
There is no Nash equilibrium in this game
Lisa
Bart
Rock
Paper
Rock
(0,0)
(2,10)
(8,-3)
Paper
(8,-3)
(0,0)
(2,10)
Scissors
(2,10)
(8,-3)
(0,0)
At (Rock, Rock) both players have
an incentive to choose differently.
Therefore, NOT NASH
Scissors
Bart has no
incentive to
deviate but Lisa
does.
NOT NASH
Lisa has no incentive to deviate
but Bart does.
NOT NASH
Finding Equilibrium
• In the cell-by-cell analysis we go through
each cell and see if one or more players
would be better off choosing a different
action
• Best Response Analysis is similar
– In each row, we find Lisa’s best response to
Bart’s choice of that row.
– We do the same with each column for Bart
5
Finding Equilibrium
Lisa
Bart
Rock
Paper
Rock
(0,0)
(2,10)
(8,-3)
Paper
(8,-3)
(0,0)
(2,10)
Scissors
(2,10)
(8,-3)
(0,0)
Bart’s best response to “Rock” played
by Lisa is to play paper
Scissors
Lisa’s best response to
“Paper” played by Bart
Best Response Analysis
• In the previous example, there were no
cells where that cell was the best
response for both players.
– There is no Nash equilibrium in this game.
Application 1
• Two players have to choose among four
actions each.
– Row player chooses R1, R2, R3, or R4
– Column player chooses C1, C2, C3, or C4
– The payoffs are shown in the following matrix
• Use cell by cell comparison as well as best
response analysis to find the equilibrium
6
Application 1
Column Player
Row Player
C1
C2
C3
C4
R1
(2, 4)
(2, 9)
(1, 7)
(0, 12)
R2
(5, 2)
(5, 3)
(2, 0)
(3, 1)
R3
(6, 0)
(7, 6)
(0, 2)
(1, 3)
R4
(1, 16)
(4, 8)
(4, 10)
(4, 6)
Dominance
• Another solution concept that can
sometimes isolate a unique equilibrium and
at others merely simplify the game for us
involves dominance
• A strategy (action) is dominant if it provides
a better outcome regardless of what the
other player does
• The Prisoners’ Dilemma illustrates this well
The Prisoners’ Dilemma
The Dominant Strategy Equilibrium
Rocko
Rat
Quiet
Rat
(-10,-10)
(0,-15)
Quiet
(-15,0)
(-2,-2)
Moose
Moose is always better off
ratting too so we know that
Quiet is strictly dominated and
he would never play it.
For Rocko, regardless of what
Moose does, he is better off
ratting. Therefore, keeping quiet
is strictly dominated and we
know he would never play it.
7
Dominance
• A dominant strategy equilibrium is always
a Nash equilibrium (but not all Nash
equilibriums are dominant strategy
equilibriums)
• Eliminating dominated strategies can
greatly simplify the game solving process
• The next game has a quick solution
Dominance
Column Player
Row Player
C1
C2
C3
C4
R1
(2, 4)
(2, 9)
(1, 7)
(0, 2)
R2
(5, 2)
(5, 3)
(2, 0)
(3, 1)
R3
(6, 0)
(7, 6)
(10, 2)
(7, 3)
R4
(1, 2)
(4, 8)
(4, 6)
(4, 6)
Dominance
• Much more frequently we won’t be so fortunate
to jump right to the solution in one go
– In many cases one player will have a dominant
strategy but the other will not
– It seems plausible that a player will recognize that
his/her opponent has dominated strategies and will
not play them
– This can allow us to apply iterative elimination of
dominated strategies
8
Iterated Elimination
Row does not
have a
dominant
strategy
Row Player
Column Player
C1
C2
C3
C4
R1
(12, 4)
(2, 9)
(1, 7)
(0, 2)
R2
(5, 2)
(5, 3)
(12, 0)
(3, 1)
R3
(6, 0)
(7, 6)
(10, 2)
(7, 3)
R4
(1, 2)
(4, 8)
(4, 6)
(14, 6)
Iterated Elimination
• If we look at the previous slide we can see
that Row does not have a dominant
strategy
• However, if we look at Column’s payoffs,
we see that column does have a dominant
strategy (so the others are dominated)
– If we eliminate the dominated strategies for
Column, we can find a dominant strategy for
Row
Iterated Elimination
Column Player
Row Player
C1
C2
C3
C4
R1
(12, 4)
(2, 9)
(1, 7)
(0, 2)
R2
(5, 2)
(5, 3)
(12, 0)
(3, 1)
R3
(6, 0)
(7, 6)
(10, 2)
(7, 3)
R4
(1, 2)
(4, 8)
(4, 6)
(14, 6)
9
Iterated Dominance – Application
• In the next game, there is a dominant
strategy equilibrium that requires a few
iterations to eliminate all the dominated
strategies
• Find the equilibrium in this game
– Does it matter whether you start by
eliminating Row or Column’s strategies first?
Iterated Elimination
Column Player
Row Player
C1
C2
C3
C4
R1
(12, 4)
(2, 9)
(1, 7)
(0, 2)
R2
(5, 2)
(5, 3)
(12, 0)
(3, 1)
R3
(6, 0)
(7, 6)
(10, 2)
(7, 3)
R4
(1, 2)
(4, 8)
(4, 6)
(14, 6)
Iterated Elimination
Column Player
C1
C2
C3
C4
R1
Row Player
R2
R3
R4
10