Overview
Overview
BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict
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Overview
Game Theory in Part III completes Game Theory in Part II for
those games where information is strategically revealed or
withheld.
In many games, a player may not know all the information that is
pertinent for the choice that he has to make at every point in the
game. His uncertainty may be over variables that are either
internal or external to the game.
BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict
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Overview
A player may be potentially uncertain about what moves the
other player is making at the same time he makes his own move;
we call that strategic uncertainty. All the simultaneous move
games in Part II were simple enough that that uncertainty was
resolved by eliminating dominated strategies.
Part III’s Lesson 5 considers games where strategic uncertainty
remains because uncertainty is not resolved by eliminating
dominated strategies, and because players’ interests conflict (as in
sports) so players conceal information about their own moves.
Lesson 6 considers games where players easily reveal
information about their own moves because players’ interests
align (as in setting a standard industry format).
BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict
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Lesson Overview
Lesson III.5 Strategic Uncertainty when Interests Conflict
Example 1: Unpredictable Actions
Example 2: Mixing with Perfect Conflict
Example 3: Mixing with Major Conflict
Example 4: Mixing with Minor Conflict
Summary
Review Questions
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Example 1: Unpredictable Actions
Example 1: Unpredictable Actions
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Example 1: Unpredictable Actions
Comment: Bob Gustavson, professor of health science and men's
soccer coach at John Brown University in Siloam Springs,
Arkansas, says “When you consider that a ball can be struck
anywhere from 60-80 miles per hour, there's not a whole lot of
time for the goalkeeper to react”. Gustavson says skillful goalies
use cues from the kicker. They look at where the kicker's plant
foot is pointing and the posture during the kick. Some even study
tapes of opponents. But most of all they take a guess — jump left
or right at the same time the kicker is committing himself to
kicking left or right.
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Example 1: Unpredictable Actions
Question: Consider a penalty kick in soccer. The goalie either
jumps left or right after the kicker has committed himself to
kicking left or right. The kicker’s payoffs are the probability of
him scoring, and the goalie’s payoffs are the probability of the
kicker not scoring. Those actions and payoffs define a normal
form for this Penalty Kick Game. Try to predict strategies or
recommend strategies.
Goalie
Kicker
Left
Right
Left
.1,.9
.4,.6
Right
.8,.2
.3,.7
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Example 1: Unpredictable Actions
Answer: To predict actions or
recommend actions, since
the game has simultaneous
Kicker
moves, first look for dominate
or dominated actions. There are none.
Goalie
Left
Right
Left
.1,.9
.4,.6
Right
.8,.2
.3,.7
Then look for a Nash equilibrium. There is none. If the Kicker
were known to kick Left, the Goalie guards Left. But if the
Goalie were known to guard Left, the Kicker kicks Right. But if
the Kicker were known to kick Right, the Goalie guards Right.
But if the Goalie were known to guard Right, the Kicker kicks
Left. So there is no Nash equilibrium.
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Example 1: Unpredictable Actions
Finally, look to see if any
action can be eliminated because
it is not rationalizable (that is, it
is not a best response to some
action by the other player.
Goalie
Kicker
Left
Right
Left
.1,.9
.4,.6
Right
.8,.2
.3,.7
But all actions are rationalizable.
On the one hand, it is rational to kick left if the Kicker believes
the Goalie jumps right. On the other hand, it is rational for the
Kicker to kick right if he believes the Goalie jumps left.
Likewise, it is rational for the Goalie to jump left if the Goalie
believes the Kicker kicks left, and it is rational for the Goalie to
jump right if the Goalie believes the Kicker kicks right.
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Example 1: Unpredictable Actions
Since there are no dominance
solutions and there are no
Left
Nash equilibria for this game
Kicker
Right
of simultaneous moves,
actions are unpredictable, and game theory has no
recommendation; either action is acceptable.
Goalie
Left
.1,.9
.4,.6
BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict
Right
.8,.2
.3,.7
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Example 2: Mixing with Perfect Conflict
Example 2: Mixing with Perfect
Conflict
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Example 2: Mixing with Perfect Conflict
Comment: Example 1’s choices for the goalie were jump left or
jump right. Call those actions because, in Example 2, strategies
are going to be more complicated; they will be probabilities for
taking specific actions --- say, jump left with probability 0.24 and
jump right with probability 0.76
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Example 2: Mixing with Perfect Conflict
Question: Consider the normal form below for the Penalty Kick
Game in soccer. Predict strategies or recommend strategies if this
game is repeated throughout the careers of the kicker and the
goalie.
Goalie
Kicker
Left
Right
Left
.1,.9
.4,.6
Right
.8,.2
.3,.7
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Example 2: Mixing with Perfect Conflict
Goalie
Answer: If the game were not
Left
Right
repeated, then since there are no
Left
.1,.9
.8,.2
dominance solutions and there
Kicker
Right
.4,.6
.3,.7
are no Nash equilibria (in pure
strategies) for this game of simultaneous moves, actions are
unpredictable, and game theory has no recommendation; either
action is acceptable. For example, the Kicker could kick Left and
the Goalie could jump Left.
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Example 2: Mixing with Perfect Conflict
But since the game is repeated,
actions need to become
unpredictable because
predictable actions can be
exploited.
Goalie
Kicker
Left
Right
Left
.1,.9
.4,.6
Right
.8,.2
.3,.7
For example, see how predicting actions helps the Goalie. If the
Kicker chooses Left predictably, the Goalie can choose Left and
keep the Kicker at payoff .1 and the Goalie at .9; and if the
Kicker chooses Right predictably, the Goalie can choose Right
and keep the Kicker at payoff .3 and the Goalie at .7.
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Example 2: Mixing with Perfect Conflict
The Nash equilibrium strategy for the Kicker is the mixed
strategy for which the Goalie would not benefit if he could
predict the Kicker’s mixed strategy. Suppose the Goalie predicts
p and (1-p) are the probabilities the Kicker chooses Left or Right.
The Goalie expects .9p + .6(1-p) from playing Left, and .2p +
.7(1-p) from Right. The Goalie does not benefit if those payoffs
equal, .9p + .6(1-p) = .2p + .7(1-p), or .6 + .3p = .7 - .5p, or
p = 1/8 = 0.125
Goalie
Kicker
Left
Right
Left
.1,.9
.4,.6
Right
.8,.2
.3,.7
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Example 2: Mixing with Perfect Conflict
The Nash equilibrium strategy for the Goalie is the mixed
strategy for which the Kicker would not benefit if he could
predict the Goalie’s mixed strategy. Suppose the Kicker predicts
q and (1-q) are the probabilities the Goalie chooses Left or Right.
The Kicker expects .1q + .8(1-q) from playing Left, and .4q +
.3(1-q) from Right. The Kicker does not benefit if those payoffs
equal, .1q + .8(1-q) = .4q + .3(1-q), or .8 - .7q = .3 + .1q, or
q = 5/8 = 0.625
Goalie
Kicker
Left
Right
Left
.1,.9
.4,.6
Right
.8,.2
.3,.7
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Example 2: Mixing with Perfect Conflict
Comment: Randomizing actions adds strategies (called mixed
strategies) that solve some games that have no dominance
solution or Nash Equilibrium (in pure strategies, where all
probability is on one particular action). For example, in the
Penalty Kick Game, there was no Nash equilibrium with pure
strategies, and there were multiple rationalizable pure strategies.
It turns out that most games have at least one Nash equilibrium in
mixed strategies.
In fact, the Penalty Kick Game has a unique Nash equilibrium in
mixed strategies. While any of the rationalizable strategies would
be reasonable if the game were played once, if instead the game
were repeated, then strategies in the unique Nash equilibrium are
the only way to play that guarantees the other player cannot gain
even if they used your history to correctly predict your strategy.
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Example 3: Mixing with Major Conflict
Example 3: Mixing with Major
Conflict
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Example 3: Mixing with Major Conflict
Comment: Employers are in conflict with (selfish, amoral)
workers, who want to steal or shirk (not work, or steal time).
However, the Work-Shirk Game is not one of total conflict (it is
not like the Penalty Kick Game) because monitoring workers
costs the employer but does not help the worker.
Because of the conflict, the other player exploiting your
systematic choice of strategy is to your disadvantage, and so there
is reason to follow mixed strategies in such games.
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Example 3: Mixing with Major Conflict
Question: Consider the Work-Shirk Game for an employee and
an employer. Suppose if the employee chooses to work, he
looses $100 of happiness from the effort of working, but he
yields $400 to his employer. Suppose the employer can monitor
the employee at a cost of $80. Finally, if the employee chooses to
not work and the employer chooses to monitor, then the
employee is not paid, but in every other case (“work” or “not
monitor”), then the employee is paid $150.
Predict strategies or recommend strategies if this game is
repeated daily.
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Example 3: Mixing with Major Conflict
Answer: First, complete the normal form below for the WorkShirk Game. For example, if the employee chooses to work and
the employer chooses to monitor, then the employee looses $100
of happiness from the effort of working but is paid $150, and the
employer gain $400 from his employer but pays $80 for
monitoring and pays $150 to his employee.
Employer
Employee
Work
Shirk
Monitor
Trust
50,170
50,250
0,-80 150,-150
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Example 3: Mixing with Major Conflict
To predict actions or
recommend actions, since
the game has simultaneous
Employee
moves, first look for dominate
or dominated actions. There are none.
Employer
Work
Shirk
Monitor
Trust
50,170
50,250
0,-80 150,-150
Then look for a Nash equilibrium in pure strategies. There is
none. If the Employee were known to Work, the Employer
Trusts. But if the Employer were known to Trust, the Employee
Shirks. But if the Employee were known to Shirk, the Employer
Monitors. But if the Employer were known to Monitor, the
Employee Works. So there is no Nash equilibrium.
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Example 3: Mixing with Major Conflict
Since the game is repeated,
actions need to become
unpredictable because
predictable actions can be
exploited.
Employer
Employee
Work
Shirk
Monitor
Trust
50,170
50,250
0,-80 150,-150
The Nash equilibrium strategy for the Employee is the mixed
strategy for which the Employer would not benefit if he could
predict the Employee’s mixed strategy. Suppose the Employer
predicts p and (1-p) are the probabilities the Employee chooses
Work or Shirk. The Employer expects 170p - 80(1-p) from
playing Monitor, and 250p - 150(1-p) from Trust. The Employer
does not benefit if those payoffs equal,
170p - 80(1-p) = 250p - 150(1-p), or -80 + 250p = -150 + 400p,
or p = 70/150 = 0.467
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Example 3: Mixing with Major Conflict
Employer
Employee
Work
Shirk
Monitor
Trust
50,170
50,250
0,-80 150,-150
The Nash equilibrium strategy for the Employer is the mixed
strategy for which the Employee would not benefit if he could
predict the Employer’s mixed strategy. Suppose the Employee
predicts q and (1-q) are the probabilities the Employer chooses
Monitor or Trust. The Employee expects 50q + 50(1-q) from
playing Work, and 0q + 150(1-q) from Shirk. The Employee does
not benefit if those payoffs equal,
50q + 50(1-q) = 0q + 150(1-q), or 50 = 150 – 150q,
or q = 100/150 = 0.667
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Example 4: Mixing with Minor Conflict
Example 4: Mixing with Minor
Conflict
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Example 4: Mixing with Minor Conflict
Comment 1: Blu-ray Disc is designed to supersede the standard
DVD format. The disc has the same physical dimensions as
standard DVDs and CDs. The name Blu-ray Disc derives from
the blue-violet laser used to read the disc. Blu-ray Disc was
developed by the Blu-ray Disc Association, a group representing
makers of consumer electronics, computer hardware, and motion
pictures.
During the format war over high-definition optical discs, Blu-ray
competed with the HD DVD format. Toshiba, the main company
supporting HD DVD, conceded in February 2008, and the format
war ended. In late 2009, Toshiba released its own Blu-ray Disc
player.
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Example 4: Mixing with Minor Conflict
Comment 2: The format war over high-definition optical discs
has The Blu-ray Disc Association in some conflict with Toshiba
since each group has gained expertise and lower costs in
producing a particular format and, so, each would gain if their
format were universally adopted. However, the Format War
game is not one of total conflict (it is not like the Penalty Kick
Game) or even of major conflict (like the Work-Shirk Game)
because both players loose most if neither format is universally
adopted.
Because conflict is less important than cooperation, the other
player exploiting your systematic choice of strategy is to your
advantage because you both want a universal format. So there is
less reason to follow mixed strategies in such games.
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Example 4: Mixing with Minor Conflict
Question: Consider the Format Game for The Blu-ray Disc
Association and Toshiba. Suppose each player either adopts the
Blu-ray format or the HD format. Suppose if both adopt the same
format, then both gain $100 million from customers that value the
convenience of having a universal format. Suppose if they both
adopt the Blu-ray format, then The Blu-ray Disc Association
gains an extra $10 million since their expertise with that format
gives them lower production costs. Finally, suppose if they both
adopt the HD format, then Toshiba gains an extra $10 million
since their expertise with that format gives them lower production
costs.
Predict strategies or recommend strategies if this game is
repeated yearly.
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Example 4: Mixing with Minor Conflict
Answer: First, complete the normal form below for the Format
Game. For example, if The Blu-ray Disc Association and
Toshiba both adopt HD, then both gain $100 million from
customers that value the convenience of having a universal
format, and Toshiba gains an extra $10 million since their
expertise with the HD format gives them lower production costs.
Toshiba
Blu-ray
Blu-ray
HD
Blu-ray
110,100
0,0
HD
0,0
100,110
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Example 4: Mixing with Minor Conflict
To predict actions or
recommend actions, since
the game has simultaneous
Blu-ray
moves, first look for dominate
or dominated actions. There are none.
Toshiba
Blu-ray
HD
Blu-ray
110,100
0,0
HD
0,0
100,110
Then look for a Nash equilibrium in pure strategies. There are
two. On the one hand, both players choose Blu-ray; on the other
than, both players choose HD.
There is also a Nash equilibrium in mixed strategies.
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Example 4: Mixing with Minor Conflict
Toshiba
The Nash equilibrium mixed
Blu-ray
HD
strategy for Blu-ray Association
0,0
is the mixed strategy for which Blu-ray Blu-ray 110,100
HD
0,0
100,110
Toshiba would not benefit if
they could predict Blu-ray Association’s mixed strategy. Suppose
Toshiba predicts p and (1-p) are the probabilities Blu-ray
Association chooses Blu-ray or HD. Toshiba expects 100p + 0(1p) from playing Blu-ray, and 0p + 110(1-p) from HD. Toshiba
does not benefit if those payoffs equal,
100p + 0(1-p) = 0p + 110(1-p), or 100p = 110 - 110p,
or p = 110/210 = 0.524
The expected payoff for Toshiba (whatever its strategy) is thus
100p + 0(1-p) = 0p + 110(1-p) = 52.4
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Example 4: Mixing with Minor Conflict
Toshiba
The Nash equilibrium mixed
Blu-ray
HD
strategy for Toshiba is the mixed
Blu-ray 110,100
0,0
strategy for which Blu-ray
Blu-ray
HD
0,0
100,110
Association would not benefit if
they could predict Toshiba’s mixed strategy. Suppose Blu-ray
Association predicts q and (1-q) are the probabilities Toshiba
chooses Blu-ray or HD. Blu-ray Association expects 110q + 0(1q) from playing Blu-ray, and 0q + 100(1-q) from HD. Blu-ray
Association does not benefit if those payoffs equal,
110q + 0(1-q) = 0q + 100(1-q), or 110q = 100 – 100q,
or q = 100/210 = 0.476
The expected payoff for Blu-ray Association (whatever its
strategy) is thus 110q + 0(1-q) = 0q + 100(1-q) = 52.4
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Example 4: Mixing with Minor Conflict
Toshiba
Comment: The expected payoff
Blu-ray
HD
of 52.4 for each player in the
0,0
mixed strategy Nash equilibrium Blu-ray Blu-ray 110,100
HD
0,0
100,110
is less than if both players had
agreed to one format or the other. That is a general lesson in
games with only minor conflict of interest. The players are better
off resolving the strategic uncertainty. The remaining lessons
take up the problem of revealing information.
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Summary
Summary
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Review Questions
Review Questions
 You should try to answer some of the following questions
before the next class.
 You will not turn in your answers, but students may request
to discuss their answers to begin the next class.
 Your upcoming cumulative Final Exam will contain some
similar questions, so you should eventually consider every
review question before taking your exams.
BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict
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BA 210
Introduction to Microeconomics
End of Lesson III.5
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