Lesson overview
Chapter 5 Simultaneous Move Games with Pure Strategies …
Lesson I.8 Simultaneous and Continuous Move Theory
Each Example Game Introduces some Game Theory
• Example 1: Continuous Variables
• Example 2: Risk in Nash Equilibrium
• Example 3: Rationality in Nash Equilibrium
• Example 4: Rationalizability
• Example 5: A Rationalizable Solution
• Review Problems
Lesson I.9 Simultaneous and Continuous Move Applications
BA 592 Lesson I.8 Simultaneous and Continuous Move Theory
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Example 1: Continuous Variables
Best-response analysis for finding all the Nash equilibria of
simultaneous-move games can be extended from games with a
finite number of available strategies to a continuum of strategies.
To calculate best responses in this type of game, we find, for each
possible value of one player’s strategy, the value of the other
player’s strategy that is best for it. The continuity of the sets of
strategies allows us to use algebraic formulas to show the best
responses as curves in a graph, with each player’s strategy on one
of the axes. The Nash equilibrium of the game is where the two
curves meet.
BA 592 Lesson I.8 Simultaneous and Continuous Move Theory
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Example 1: Continuous Variables
Walmart and Costco compete on price to sell some goods that are
perfect substitutes(like emergency food supplies in a weatherproof bucket), while other goods are imperfect substitutes (like
prepared pizzas). Suppose Walmart sets the price Px of their
good at the same time Costco sets the price Py of their good. And
suppose, given these prices, demands are
Qx = 44 -2 Px + Py
Qy = 44 -2 Py + Px
(The terms + Py and + Px indicate the goods are substitutes --better gross substitutes.) And suppose unit supply costs are $8.
Compute best-response curves for this Price Competition Game,
then intersect the curves to find the Nash equilibria. Can the two
firms each increase profit if they can commit to different
strategies?
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Example 1: Continuous Variables
Given prices Px and Py of the two goods, use demand and cost to
compute profits:
Px = (Px - 8) Qx = (Px - 8)(44 -2 Px + Py)
= - 8(44 + Py) + (60 + Py)Px – 2(Px)2
Py = (Py - 8) Qy = (Py - 8)(44 -2 Py + Px)
= - 8(44 + Px) + (60 + Px)Py – 2(Py)2
Since Px is a concave function of Px and Py is a concave function
of Py, profits Px(Px) and Py(Py) are maximized where their
derivatives are zero.
0 = Px’(Px) = (60 + Py) – 4Px
0 = Py’(Py) = (60 + Px) – 4Py
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Example 1: Continuous Variables
Best-response curves for the Price Competition Game are the
solutions to those first-order conditions:
0 = (60 + Py) – 4Px
0 = (60 + Px) – 4Py
meaning solve the first equation for best-response curve
Px = 15 + .25Py
and the second equation for best-response curve for
Py = 15 + .25Px
Substituting the first curve into the second yields
Py = 15 + .25(15 + .25Py) = 18.75 + 0.0625 Py
and the unique solution Py = 20. Hence, the first curve yields
Px = 15 + .25(20) = 20. The Nash equilibrium is thus
(Px,Py) = (20,20).
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Example 1: Continuous Variables
Can the two firms each increase profit if they can commit to
different strategies?
At (Px,Py) = (20,20),
Px = (Px – 8) Qx = (Px – 8)(44 – 2 Px + Py) = 12(24) = 288
Py = (Py – 8) Qy = (Py – 8)(44 – 2 Py + Px) = 12(24) = 288
But at non-equilibrium prices (Px,Py) = (P,P),
Px = (Px – 8) Qx = (P – 8)(44 – 2 P + P) = – 352 + 52P – P2
Py = (Py – 8) Qy = (P – 8)(44 – 2 P + P) = – 352 + 52P – P2
Since Px and Py are a concave function of P, profits Px(P) and
Py(P) are maximized where their derivatives are zero:
0 = Px’(P) = 52 – 2P and 0 = Py’(P) = 52 – 2P, so P = 26.
The two firms each increase profit if they can commit to increase
price from P = 20 to P = 26.
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Example 2: Risk in Nash Equilibrium
Risk neutral players’ evaluate uncertain outcomes by their
expected value. For example, if payoffs in a game are in dollars,
then a risk-neutral player would be indifferent between being
given $10 and flipping a coin for double or nothing. Later
lessons accommodate risk averse players (even risk loving
players), but for now consider just the simplest case of risk
neutral players.
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Example 2: Risk in Nash Equilibrium
What strategy should Walmart choose in the following game with
Costco?
Costco
Walmart
A
B
C
A
2,2
1,3
2,0
B
3,1
2,2
2,3
C
0,2
3,2
2,2
There is only one Nash Equilibrium: Walmart chooses A and
believes Costco chooses A. Is that that Nash equilibrium a
reasonable recommendation?
Objection: Is not C better because it guarantees the Nash
equilibrium outcome of 2?
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Example 2: Risk in Nash Equilibrium
Costco
Objection: Is not C better
because it guarantees the
A
Walmart
B
Nash equilibrium outcome of
C
2?
Response: Not necessarily. The Nash equilibrium
recommendation of A is appropriate when you absolutely believe
the other player also chooses A. The guaranteed payoff from C is
only relevant if your belief is not absolute but, rather, there is a
probability 1-p-q<1 that Costco chooses A, a small probability
p>0 that Costco chooses B, and a small probability q>0 that
Costco chooses C.
A
2,2
1,3
2,0
B
3,1
2,2
2,3
C
0,2
3,2
2,2
Given those non-absolute beliefs, Walmart thus expects payoff
2(1-p-q) + 3p from A, which is larger (A is better) than the
guaranteed payoff of 2 from C when 2(-p-q)+3p > 0, or p-2q > 0.
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Example 2: Risk in Nash Equilibrium
What strategy should Walmart choose in the following game with
Costco?
Costco
Walmart
Up
Down
Left
9,10
10,10
Right
8,9.9
-1000,9.9
There is only one Nash Equilibrium: Walmart chooses Down and
believes Costco chooses Left. Is that that Nash equilibrium a
reasonable recommendation?
Objection: Is not Up better because it avoids the possibility of
loosing 1000?
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Example 2: Risk in Nash Equilibrium
Costco
Walmart
Up
Down
Left
9,10
10,10
Right
8,9.9
-1000,9.9
Objection: Is not Up better
because it avoids the
possibility of loosing 1000?
Response: Not necessarily. The Nash equilibrium
recommendation of Down is appropriate when you absolutely
believe the other player chooses Left. The safer payoff from Up
is only relevant if your belief is not absolute but, rather, there is a
probability 1-p<1 that Costco chooses Left and a small
probability p>0 that Costco chooses Right.
Given those non-absolute beliefs, Walmart thus expects payoff
10(1-p) - 1000p from Up, which is larger (Down is better) than
the safer expected payoff of 9(1-p) + 8p from Up when 10(1-p) 1000p > 9(1-p) + 8p, or p < 1/1019.
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Example 2: Risk in Nash Equilibrium
Costco
Walmart
Up
Down
Left
9,10
10,10
Right
8,9.9
-1000,9.9
Objection: Would you risk the
possibility of loosing $1000
for a gain of $1?
Response: Have you ever driven at a higher speed (risking more
than $1000 if you get in an accident) to save a few minutes and
so gain $1 of time, which you could spend at work or spend at
home looking for a lost pack or gum or a lost battery so you do
not have to spend $1 to replace it.
To dispel any doubt, the European Road Safety Observatory says
a higher speed increases the likelihood of an accident. Very
strong relationships have been established between speed and
accident risk: The general relationship holds for all speeds and all
roads, ….
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Example 3: Rationality in Nash Equilibrium
What strategy should Sam’s Club choose in the following game
with Costco?
Costco
Sam's
R1
R2
R3
C1
0,7
5,2
7,0
C2
2,5
3,3
2,5
C3
7,0
5,2
0,7
There is only one Nash Equilibrium: (R2,C2). Is that the only
reasonable recommendation?
BA 592 Lesson I.8 Simultaneous and Continuous Move Theory
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Example 3: Rationality in Nash Equilibrium
Costco
There is only one Nash
Equilibrium: (R2,C2). Is that
R1
Sam's
R2
the only reasonable
R3
recommendation?
Some claim it is. Sam’s plays R2 because Sam’s believes Costco
will play C2. And why does Sam’s believe this? Because Sam’s
believes Costco to be rational, Sam’s must believe that Costco
believes that Sam’s is choosing R2, because C2 would not be
Costco’s best choice if Costco believed Sam’s would choose
either R1 or R3. Thus, some claim, in any rational process of
formation of beliefs and responses, Sam’s would only choose a
strategy that is part of a Nash equilibrium, where beliefs are
correct.
But that argument stops after one round of thinking about beliefs.
C1
0,7
5,2
7,0
C2
2,5
3,3
2,5
C3
7,0
5,2
0,7
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Example 3: Rationality in Nash Equilibrium
Costco
Sam's
R1
R2
R3
C1
0,7
5,2
7,0
C2
2,5
3,3
2,5
C3
7,0
5,2
0,7
R2 is, in fact, not the only reasonable recommendation for Sam’s.
R1 is reasonable if Sam’s believes that Costco will choose C3,
which Costco will choose if Costco believes that Sam’s will
choose R3, which Sam’s will choose if Sam’s believes that
Costco will choose C1, which Costco will choose if Costco
believes that Sam’s will choose R1, and so on.
Likewise, R3 is reasonable if Sam’s believes that Costco will
choose C1, which Costco will choose if Costco believes that
Sam’s will choose R1, and so on.
BA 592 Lesson I.8 Simultaneous and Continuous Move Theory
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Example 4: Rationalizability
What strategy should Row choose in the following game with
Column?
Column
C1
C2
C3
C4
R1
0,7
2,5
7,0
0,1
R2
5,2
3,3
5,2
0,1
Row
R3
7,0
2,5
0,7
0,1
R4
0,0
0,-2
0,0
10,-1
R1 is reasonable if R believes C will choose C3, which C will
choose if C believes R will choose R3, which R will choose if R
believes C will choose C1, which C will choose if C believes R
will choose R1, and so on.
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Example 4: Rationalizability
Column
R2 is reasonable if R believes
R1
C will choose C2, which C
R2
Row
will choose if C believes R
R3
R4
will choose R2, and so on.
R3 is reasonable if R believes C will choose C1, which C will
choose if C believes R will choose R1, which R will choose if R
believes C will choose C3, which C will choose if C believes R
will choose R3, and so on.
C1
0,7
5,2
7,0
0,0
C2
2,5
3,3
2,5
0,-2
C3
7,0
5,2
0,7
0,0
C4
0,1
0,1
0,1
10,-1
R4 is never reasonable no matter what R believes C will choose.
R1, R2, and R3 are thus Rationalizeable. But R4 is not, even
though R4 is not (strictly or weakly) dominated by any other row.
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Example 5: A Rationalizability Test
Rationalizability is confirmed when you hit a cycle, and you can
find all in this way when there is only a finite number of
strategies. To cover continuous moves, rationalizability test is
the iterated elimination of strategies that are not the best response
to any strategies by the other players.
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Example 5: A Rationalizability Test
Gloucester Massachusetts has two fishing boats the go out every
evening and return the following morning to bring their night’s
catch to the market. All fish has to be sold and eaten the same
day. Fish are plentiful, so each boat decides how many fish to
bring to the market. But each knows that, if the total that is
brought to the market is too large, the glut of fish means low
prices and low profits.
Suppose the price of fish is P = 60 – (G+M) if George Clooney’s
boat brings back G barrels and Mark Wahlberg’s boat brings
back M barrels. Suppose unit costs of production are 30 for
George, and 36 for Mark.
Find all rationalizable strategies for this Quantity Competition
(Cournot) Game.
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Example 5: A Rationalizability Test
Given quantities G and M of the two boats, use price and cost to
compute profits:
PG = ((60-G-M)-30)G = (30 – M)G – G2
PM = ((60-G-M)-36)M = (24 – G)M – M2
Since PG is a concave function of G and PM is a concave
function of M, profits PG(G) and PM(M) are maximized where
their derivatives are zero.
0 = PG’(G) = (30 – M) – 2G, or G = 15 – M/2
0 = PM’(M) = (24 – G) – 2M, or M = 12 – G/2
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Example 5: A Rationalizability Test
Use the best response curves G = 15 – M/2 and M = 12 – G/2.
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Example 5: A Rationalizability Test
First round of eliminations, only G in the closed interval [0,15] is
a best response to some M, and only M in the closed interval
[0,12] is a best response to some G.
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Example 5: A Rationalizability Test
Second round of eliminations, only G in [9,15] is a best response
to some M in [0,12], and only M in [4.5,12] is a best response to
some G in [0,15].
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Example 5: A Rationalizability Test
Third round of eliminations, only G in [9,12.75] is a best
response to some M in [4.5,12], and only M in [4.5,7.5] is a best
response to some G in [9,15].
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Example 5: A Rationalizability Test
Fourth round of eliminations, only G in [11.25,12.75] is a best
response to some M in [4.5,7.5], and only M in [5.625,7.5] is a
best response to some G in [11.25,12.75].
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Example 5: A Rationalizability Test
Fifth round of eliminations, only C in [11.25,12.1875] is a best
response to some M in [5.625,7.5], and only M in [5.625,6.375]
is a best response to some C in [11.25,12.75].
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Example 5: A Rationalizability Test
And so on as the intervals converge to a single point, which is the
Nash equilibrium solution (G,M) = (12,6) to the best-response
curves G = 15 – M/2 and M = 12 – G/2.
Therefore, (G,M) = (12,6) are the unique rationalizeable
strategies.
BA 592 Lesson I.8 Simultaneous and Continuous Move Theory
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BA 592
Game Theory
End of Lesson I.8
BA 592 Lesson I.8 Simultaneous and Continuous Move Theory
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