Lesson overview
Chapter 4 Simultaneous Move Games with Pure Strategies I:
Discrete Strategies
Lesson I.5 Simultaneous Move Theory
Each Example Game Introduces some Game Theory
• Example 1: A Normal Form
• Example 2: Nash Equilibrium
• Example 3: Dominate Strategies
• Example 4: Iterated Dominance
• Example 5: Minimax
Lesson I.6 Simultaneous Move Problems
Lesson I.7 Simultaneous Move Applications
BA 592 Lesson I.5 Simultaneous Move Theory
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Example 1: A Normal Form
Game tables or normal forms condense the information in a game
tree or extensive form. Like the extensive form, the normal form
specifies strategies for every player and the outcomes of the
actions taken by all players. But unlike the extensive form, the
normal form does not model the order of the actions. Normal
forms are the simplest way to model games where actions are
simultaneous.
BA 592 Lesson I.5 Simultaneous Move Theory
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Example 1: A Normal Form
Walmart and Costco both sell emergency food supplies in a
weather-proof bucket that provides 275 delicious easy-to-prepare
meals, including potato soup and corn chowder. The unit cost to
both retailers is $75. The retailers compete on price: the lowprice retailer gets all the market and they split the market if they
have equal prices. Suppose they consider prices $75, $85, and
$95, and suppose market demands at those prices are 140, 100,
and 80.
Define the normal form for this Price Competition Game.
BA 592 Lesson I.5 Simultaneous Move Theory
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Example 1: A Normal Form
To begin, at Walmart price $95 and Costco price $85, Costco gets
the entire market demand of 100. Hence, Walmart makes $0 and
Costco makes $(85-75)x100 = $1,000.
Costco
Walmart
$75
$85
$95
$75
0,0
0,0
0,0
$85
0,0
500,500
0,1000
BA 592 Lesson I.5 Simultaneous Move Theory
$95
0,0
1000,0
800,800
4
Example 2: Nash Equilibrium
A Nash equilibrium are strategies for all players so that each
player’s strategy is a best response to the strategies of all other
players. Put another way, each player’s strategy is a best
response given his beliefs about the strategies played by the other
players, and those beliefs are, in fact, the actual strategies played
by the other players.
BA 592 Lesson I.5 Simultaneous Move Theory
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Example 2: Nash Equilibrium
Walmart and Costco consider two possibilities: Either continue
their price competition as described in Example 1, or offer the
following low-price guarantee: We guarantee lower prices than
any other store, and we do everything in our power to ensure that
you're not paying too much for your purchase. That's why we
offer our 110% Low Price Guarantee. If you find a lower
advertised price, simply let us know and we'll gladly meet that
price and beat it by an additional 10% of the difference!
To decide whether to offer that guarantee, reconsider the normal
form for the original Price Competition Game and find the Nash
equilibrium. Then, define the normal form for the Price
Competition Game modified by the 110% Low Price Guarantee,
and find the Nash equilibrium. Compare equilibria.
BA 592 Lesson I.5 Simultaneous Move Theory
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Example 2: Nash Equilibrium
The normal form for the original Price Competition Game has
two Nash equilibria: (Walmart price = $75, Costco price = $75)
and (Walmart price = $85, Costco price = $85).
Costco
Walmart
$75
$85
$95
$75
0,0
0,0
0,0
$85
0,0
500,500
0,1000
BA 592 Lesson I.5 Simultaneous Move Theory
$95
0,0
1000,0
800,800
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Example 2: Nash Equilibrium
The normal form for the Price Competition Game modified by
the 110% Low Price Guarantee has three Nash equilibria:
(Walmart price = $75, Costco price = $75) and (Walmart price =
$85, Costco price = $85) and (Walmart price = $95, Costco price
= $95).
Costco
Walmart
$75
$85
$95
$75
0,0
0,0
0,0
$85
0,0
500,500
500,500
$95
0,0
500,500
800,800
The new Nash equilibrium (Walmart price = $95, Costco price =
$95) is best for both firms and worst for consumers!
BA 592 Lesson I.5 Simultaneous Move Theory
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Example 3: Dominate Strategies
A dominate strategy for a player gives better payoffs for that
player compared with any other strategy, no matter what other
players choose for their strategies.
A weakly dominate strategy for a player gives at least as good
payoffs for that player compared with any other strategy, no
matter what other players choose for their strategies, and better
payoffs for at least one choice of strategies for the other players.
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Example 3: Dominate Strategies
The Prisoners’ dilemma described earlier is whether or not to
confess. If you don’t confess, you get 1 year if the other prisoner
does not confess and 15 years if he does. If you do confess, you
get 0 if the other prisoner does not confess and 5 years if he does.
To decide whether to confess, define the normal form for the
Prisoners’ Dilemma, and find any dominate strategies.
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Example 3: Dominate Strategies
Confess is a dominate strategy for each prisoner. In particular, it
is the only Nash equilibrium.
Prisoner 2
Don't C.
Prisoner 1
Confess
Don't C.
-1,-1
0,-15
BA 592 Lesson I.5 Simultaneous Move Theory
Confess
-15,0
-5,-5
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Example 3: Dominate Strategies
The Price Competition Game modified by the 110% Low Price
Guarantee has $95 as a weakly-dominate strategy for each player.
Although it is not the only Nash equilibrium, it is the
recommended solution.
Costco
Walmart
$75
$85
$95
$75
0,0
0,0
0,0
$85
0,0
500,500
500,500
BA 592 Lesson I.5 Simultaneous Move Theory
$95
0,0
500,500
800,800
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Example 4: Iterated Dominance
A dominated strategy for a player gives worse payoffs for that
player compared with some other strategy, no matter what other
players choose for their strategies. While dominate strategies are
the recommended choice to play games, dominated strategies
should never be chosen. Eliminating dominated strategies
reduces the game, and the new game may have further dominated
strategies, which can be eliminated, and so on.
A weakly dominated strategy for a player gives at least as bad
payoffs for that player compared with some other strategy, no
matter what other players choose for their strategies, and worse
payoffs for at least one choice of strategies for the other players.
Eliminating weakly-dominated strategies reduces the game, and
the new game may have further weakly-dominated strategies,
which can be eliminated, and so on.
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Example 4: Iterated Dominance
The original Price Competition Game has $75 as a weaklydominated strategy for each player. Eliminate that strategy to
solve the game.
Costco
Walmart
$75
$85
$95
$75
0,0
0,0
0,0
$85
0,0
500,500
0,1000
BA 592 Lesson I.5 Simultaneous Move Theory
$95
0,0
1000,0
800,800
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Example 4: Iterated Dominance
The reduced game has $85 as a dominate strategy for each player.
It is thus the recommended solution to the original Price
Competition Game after iterated elimination of weakly
dominated strategies.
Costco
Walmart
$85
$95
$85
500,500
0,1000
BA 592 Lesson I.5 Simultaneous Move Theory
$95
1000,0
800,800
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Example 5: Maximin
Zero-sum games with two players can be solved using special
analysis of the direct conflict of the players. Outcomes that are
good for one player are bad for the other player. Hence, each
player chooses his strategy by thinking: “Would this strategy be
the best choice for me, given that my opponent choose the
strategy that is worst for me?”
A maximin strategy for a player is the action that yields the
maximum payoff for that player given that the other player
choses their strategy that gives the first player the minimum
payoff.
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Example 5: Maximin
Dallas Cowboys and New York Giants consider possible
offensive (Cowboys) and defensive (Giants) plays as time runs
out in the Super Bowl. Coaches for both teams have computed
the probabilities percentages that the Cowboys win depending on
which plays are selected.
Defense
Run
Short Pass
Offense
Med. Pass
Long Pass
Run
2
6
6
10
Pass
5
5.6
5
3
BA 592 Lesson I.5 Simultaneous Move Theory
Blitz
13
10.5
1
0
17
Example 5: Maximin
Minimum (worst) outcomes for each offensive play are
computed, and the offense chooses the maximin strategy Short
Pass that maximizes those minimum payoffs.
Maximum (worst) outcomes for each defensive plays are
computed, and the defense chooses the minimax strategy Pass
that minimizes those maximum payoffs.
Defense
Run
Pass
Blitz
Run
2
5
13
Short Pass
6
5.6
10.5
Offense
Med. Pass
6
5
1
Long Pass
10
3
0
max = 10 max = 5.6 max = 13
BA 592 Lesson I.5 Simultaneous Move Theory
min = 2
min = 5.6
min = 1
min = 0
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Example 5: Maximin
Maximin and Minimax strategies are recommended in the
football game because they are a Nash equilibrium: Short Pass
for offense is a best response to Pass for defense, and Pass for
defense is a best response to Short Pass for offense. (Because the
game is zero sum, the game table need only list the payoffs for
the row player.)
Defense
Run
Pass
Blitz
Run
2
5
13
Short Pass
6
5.6
10.5
Offense
Med. Pass
6
5
1
Long Pass
10
3
0
max = 10 max = 5.6 max = 13
BA 592 Lesson I.5 Simultaneous Move Theory
min = 2
min = 5.6
min = 1
min = 0
19
BA 592
Game Theory
End of Lesson I.5
BA 592 Lesson I.5 Simultaneous Move Theory
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