Readings
Readings
Baye “Mixed Strategies” (see the index)
Dixit Chapter 7
BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict
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Overview
Overview
BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict
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Overview
Part III 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 445 Lesson C.1 Strategic Uncertainty when Interests Conflict
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Overview
On the one hand, 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 1 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 2 considers
games where players easily reveal information about their
own moves because players’ interests align (as in setting a
standard industry format).
BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict
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Overview
On the other hand, a player may be potentially uncertain
about external circumstances, such as whether a job
candidate is smart and industrious or whether a product is
reliable.
Part III’s Lessons 3 and 4 considers games where players’
interests are aligned enough that they want to reveal
information but they conflict enough that it is costly to
credibly reveal information. For example, a job candidate
cannot simply state “I am smart and industrious”, but he
must signal that by mastering hard classes in college.
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Overview
Unpredictable Actions in games of simultaneous moves occur when there is not a
unique dominance solution and not a unique Nash equilibrium. Any one-shot
unpredictable action is acceptable.
Mixing given Perfect Conflict in repeated games is better than a systematic
choice of current actions, which makes future actions predictable and exploitable
by opponents (whose interests conflict).
Mixing given Major Conflict in repeated games is still better than a systematic
choice of current actions, which makes future actions predictable and exploitable
by opponents (whose interests conflict).
Mixing given Minor Conflict in games with unpredictable actions is worse than a
systematic choice that makes your actions predictable to your allies (whose
interests align).
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Example 1: Unpredictable Actions
Example 1: Unpredictable Actions
BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict
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Example 1: Unpredictable Actions
Overview
Unpredictable Actions in games of simultaneous moves
occur when there is not a unique dominance solution and
not a unique Nash equilibrium. Any one-shot unpredictable
action is acceptable.
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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 at the same time that the
kicker either kicks 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
moves, seek a solution in
three steps:
Goalie
Kicker
Left
Right
Left
.1,.9
.4,.6
Right
.8,.2
.3,.7
1) Eliminate dominated actions. That does not help here
since there are no dominated actions.
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Example 1: Unpredictable Actions
Goalie
2) Eliminate actions that are
Left
Right
not rationalizeable. That
Left
.1,.9
.8,.2
does not help here since
Kicker
Right
.4,.6
.3,.7
each action is rationalizeable
(each action is a best response to some action of the other
player).
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
Goalie
3) Look for a Nash equilibrium
Left
Right
in pure strategies (that is an
Left
.1,.9
.8,.2
action for each player in
Kicker
Right
.4,.6
.3,.7
which each player’s action is
a best response to the known action by the other player).
That does not help here since there is no Nash equilibrium.
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
Goalie
Since there are no dominance
Left
Right
solutions and there are no
Left
.1,.9
.8,.2
Nash equilibria for this game Kicker
Right
.4,.6
.3,.7
of simultaneous moves,
actions are unpredictable, and game theory has no
recommendation; either action is acceptable.
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Example 2: Mixing given Perfect Conflict
Example 2: Mixing given Perfect
Conflict
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Example 2: Mixing given Perfect Conflict
Overview
Mixing given Perfect Conflict in repeated games is better
than a systematic choice of current actions, which makes
future actions predictable and exploitable by opponents
(whose interests conflict).
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Example 2: Mixing given 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 given Perfect Conflict
Question: Consider the normal form below for the Penalty
Kick Game in soccer, where the goalie either jumps left or
right at the same time that the kicker either kicks left or
right. 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 given Perfect Conflict
Goalie
Answer: If the game were not
Left
Right
repeated, then since there
Left
.1,.9
.8,.2
are no dominance solutions
Kicker
Right
.4,.6
.3,.7
and there 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 given Perfect Conflict
But since the game is repeated,
actions need to become
unpredictable because
Kicker
predictable actions can be
exploited.
Goalie
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 given 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,
Goalie
or
p = 1/8 = 0.125
Kicker
Left
Right
Left
.1,.9
.4,.6
Right
.8,.2
.3,.7
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Example 2: Mixing given 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,
Goalie
or
q = 5/8 = 0.625
Kicker
Left
Right
Left
.1,.9
.4,.6
Right
.8,.2
.3,.7
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Example 2: Mixing given 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 given Major Conflict
Example 3: Mixing given Major
Conflict
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Example 3: Mixing given Major Conflict
Overview
Mixing given Major Conflict in repeated games is still better
than a systematic choice of current actions, which makes
future actions predictable and exploitable by opponents
(whose interests conflict).
BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict
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Example 3: Mixing given 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 given Major Conflict
Question: Consider the Work-Shirk Game for an employee
and an employer. Suppose that at the same time the
employee chooses to either work or shirk (not work) and
the employer chooses to either monitor the employee or
not monitor the employee. Suppose if the employee
chooses to work, he loses $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 given Major Conflict
Answer: First, complete the normal form below for the
Work-Shirk Game. For example, if the employee chooses
to work and the employer chooses to monitor, then the
employee loses $100 of happiness from the effort of
working but is paid $150, and the employer gains $400
from his employee but pays $80 for monitoring and pays
$150 to his employee.
Employer
Employee
Work
Shirk
Monitor Not Mon.
50,170
50,250
0,-80 150,-150
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Example 3: Mixing given Major Conflict
To predict actions or
recommend actions, since
the game has simultaneous
moves and is repeated, seek
a solution in four steps:
Employer
Employee
Work
Shirk
Monitor Not Mon.
50,170
50,250
0,-80 150,-150
1) Eliminate dominated actions. That does not help here
since there are no dominated actions.
2) Eliminate actions that are not rationalizeable. That
does not help here since each action is rationalizeable
(each action is a best response to some action of the
other player).
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Example 3: Mixing given Major Conflict
Employer
3) Look for a Nash equilibrium
Monitor Not Mon.
in pure strategies (that is an
Work
50,170
50,250
action for each player in
Employee
Shirk
0,-80 150,-150
which each player’s action is
a best response to the known action by the other player).
That does not help here since there is no Nash equilibrium.
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 in pure strategies.
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Example 3: Mixing given Major Conflict
Employer
4) Look for a Nash equilibrium in
mixed strategies (that is
Monitor Not Mon.
probabilities for each player in
Work
50,170
50,250
Employee
which the other player’s expected
Shirk
0,-80 150,-150
values are equal for both of his
actions; in that sense, the other player cannot exploit his knowledge of
the first player’s probabilities).
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 given Major Conflict
Employer
Employee
Work
Shirk
Monitor Not Mon.
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 given Minor Conflict
Example 4: Mixing given Minor
Conflict
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Example 4: Mixing given Minor Conflict
Overview
Mixing given Minor Conflict in games with unpredictable
actions is worse than a systematic choice that makes your
actions predictable to your allies (whose interests align).
BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict
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Example 4: Mixing given 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 Bluray 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 given 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.
BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict
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Example 4: Mixing given Minor Conflict
Question: Consider the Format Game for The Blu-ray Disc
Association and Toshiba. Suppose that at the same time
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.
BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict
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Example 4: Mixing given 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 given Minor Conflict
To predict actions or
recommend actions, since
the game has simultaneous
moves and is repeated, seek
a solution in four steps:
Toshiba
Blu-ray
Blu-ray
HD
Blu-ray
110,100
0,0
HD
0,0
100,110
1) Eliminate dominated actions. That does not help here
since there are no dominated actions.
2) Eliminate actions that are not rationalizeable. That
does not help here since each action is rationalizeable
(each action is a best response to some action of the
other player).
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Example 4: Mixing given Minor Conflict
Toshiba
3) Look for a Nash equilibrium
Blu-ray
HD
in pure strategies (that is an
Blu-ray 110,100
0,0
action for each player in
Blu-ray
HD
0,0
100,110
which each player’s action is
a best response to the known action by the other player).
There are two. On the one hand, both players choose Bluray; on the other hand, both players choose HD.
If this were an exam question, you would only be
responsible for identifying one solution to this problem, so
identifying either of those Nash equilibria in pure strategies
is sufficient for full credit.
4) However, for a complete analysis, note there is also a
Nash equilibrium in mixed strategies.
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Example 4: Mixing given Minor Conflict
Toshiba
The Nash equilibrium mixed
strategy for Blu-ray Association is
Blu-ray
HD
the mixed strategy for which
Blu-ray 110,100
0,0
Blu-ray
Toshiba would not benefit if they
HD
0,0
100,110
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(1-p) 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 given 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 Bluray Association predicts q and (1-q) are the probabilities
Toshiba chooses Blu-ray or HD. Blu-ray Association
expects 110q + 0(1-q) 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 given Minor Conflict
Toshiba
Comment: The expected
Blu-ray
HD
payoff of 52.4 for each player
Blu-ray 110,100
0,0
in the mixed strategy Nash
Blu-ray
HD
0,0
100,110
equilibrium 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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Example 4: Mixing given Minor Conflict
Comment: Here is an ethical application of the game theory
of mixed strategies given minor conflict. Consider the
tragedy of Kitty Genovese. …
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Example 4: Mixing given Minor Conflict
Kitty Genovese was a New York City woman who was
stabbed to death near her home in the Kew
Gardens neighborhood of the borough of Queens in New
York City, on March 13, 1964. Although neighbors heard
her scream when she was first attacked, no one called the
police. The attacker then returned and murdered her.
The circumstances of her murder and the lack of reaction
of neighbors prompted investigation into the social
psychological phenomenon that has become known as
the bystander effect or “Genovese syndrome” and
especially diffusion of responsibility. --- But game theory
offers an alternative interpretation.
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Example 4: Mixing given Minor Conflict
Suppose the cost of each neighbor phoning the police is
$1. It might seem that each neighbor valued saving Kitty at
less than $1 since none of them called. But that is not
necessarily so. Here is what might have happened:
The value each person places on the life of the victim could
have been $1,048,577 (that is, each person would pay up
to $1,048,577 of their own money if that would save Kitty).
Suppose there were 21 neighbors that heard Kitty, and
each believed with probability 1/2 that any one of the other
neighbors would call the police. That means that each
neighbor Ned believes that there is probability ½ that any
one neighbor would not call, and so there is probability
(1/2)20 that none of them would call.
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Example 4: Mixing given Minor Conflict
Should neighbor Ned call? It is a gamble. It costs $1 to
call. The payoff is that with probability (1/2)20 you are the
only one to call, and so you save a life, which profits you
$1,048,576 (which is $1,048,577 minus the dollar for the
call).
The expected value of that gamble is (1/2)20$1,048,576,
which equals $1. So, there is no net expected gain and no
expected loss from making that call.
Thus, although it was a less than 1 out of a million chance,
each neighbor might have not called, even though each
values Kitty at more than a million dollars.
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Example 4: Mixing given Minor Conflict
Likewise, we may value saving someone’s life at a million
dollars, yet refrain from playing the lottery for $1 with the
idea that, if you win, the winnings will save that live.
Kitty Genovese’s death could have been more like a onein-a-million freak accident, rather than the lack of concern
among neighbors. Her neighbors may have valued here
deeply but still behaved optimally, and should have no
regrets that no one of them actually called the police.
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Review Questions
Review Questions
 You should try to answer some of the review
questions (see the online syllabus) 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 445 Lesson C.1 Strategic Uncertainty when Interests Conflict
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BA 445
Managerial Economics
End of Lesson C.1
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