Introduction to Game Theory: Normal Form Games
Economics 302 - Microeconomic Theory II: Strategic Behavior
Instructor: Songzi Du
compiled by Shih En Lu
(Chapters 6, 7, 9 and 11 in Watson (2013))
Simon Fraser University
January 22, 2016
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Why Games?
We want to model strategic behavior — conscious behavior arising
among a small number of competitors or players, in a situation where
all are aware of their (possibly conflicting) interests and
interdependence of their decisions.
This fits many economic situations where the “large number”
assumption of competitive markets fail: oligopoly, auctions,
bargaining, public goods, etc.
Also useful in other fields: biology, political science, computer science,
etc.
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What is a Game?
A game has players that each chooses from actions available to
him/her (given his/her information).
Example: Bob can either go to class or skip class, and the professor
can either give a pop quiz or not.
An outcome is a collection of actions taken by each player.
Example: (Go to class, Give quiz) is an outcome.
Each outcome generates a utility, or payoff, for each player. These
utilities are obtained from expected utility theory, so that a player’s
payoff from an uncertain situation is just the weighted average of her
payoffs from each outcome (where weights = probabilities).
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Simultaneous-Move Games of Complete Information
We first study simultaneous-move games of complete information.
Simultaneous-move means that each player picks her action without
knowing others’, and each player moves once.
Example: The game would not be simultaneous-move if the professor
decides whether to give a quiz after observing whether Bob is in class.
Complete information means that each player knows every player’s
payoff from each outcome.
Example: If lazy students’ payoffs differ from hard-working ones’, and
the professor does not know whether Bob is lazy, then the game
would not feature complete information.
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What is a Strategy?
In a game, each player plays a strategy, which in the case of
simultaneous-move game of complete information is a probability
distribution over her actions
A strategy specifies how the player will play his/her game.
Example: “Go to class” is one of Bob’s strategies. “Go to class with
probability 0.3 and Skip class with probability 0.7” is another.
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What is a Strategy?
In a game, each player plays a strategy, which in the case of
simultaneous-move game of complete information is a probability
distribution over her actions
A strategy specifies how the player will play his/her game.
Example: “Go to class” is one of Bob’s strategies. “Go to class with
probability 0.3 and Skip class with probability 0.7” is another.
A pure strategy is a strategy that puts all weight on an action
(probability 1).
A mixed strategy is just any strategy. “Mixed” is used to emphasize
that the strategy may not be pure.
A collection of each player’s strategy is called a strategy profile.
Example: (Go to class, 0.4 Give quiz + 0.6 No quiz) is a strategy
profile.
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The Normal Form
A convenient way to represent a two-player simultaneous move game
of complete information is through the normal form (also known as
strategic form, or game matrix).
Bob
Go to class
Skip class
Prof.
Give Quiz
0, 0
-5, -1
No Quiz
2, 6
5, 4
By convention, player 1 (Bob) picks the row, and player 2 (professor)
picks the column.
Each cell gives the payoffs of player 1 and player 2, in that order.
What do you expect the professor to do? What about Bob?
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Example: The Prisoner’s Dilemma (PD)
Not Guilty (Cooperate)
Guilty (Defect)
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Not Guilty (Cooperate)
-2, -2
-1, -5
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Guilty (Defect)
-5, -1
-3, -3
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Example: The Prisoner’s Dilemma (II)
What do you expect each prisoner to do?
What is the Pareto efficient outcome in the Prisoner’s Dilemma (PD)?
In a PD, defecting gives both players a higher payoff no matter what
the other one does, but (Defect, Defect) is worse for both than
(Cooperate, Cooperate).
Can you think of other situations that can be modeled as a PD?
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Dominance
For now, let’s only consider pure strategies.
A player’s strategy is strictly dominant if, for any combination of
actions by other players, it gives that player a strictly higher payoff
than all her other strategies. (“The best choice no matter what
others do”)
Examples: prisoner defecting, professor not giving a quiz.
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Dominance
For now, let’s only consider pure strategies.
A player’s strategy is strictly dominant if, for any combination of
actions by other players, it gives that player a strictly higher payoff
than all her other strategies. (“The best choice no matter what
others do”)
Examples: prisoner defecting, professor not giving a quiz.
A player’s strategy is strictly dominated if there exists another
strategy giving that player a strictly higher payoff for all combinations
of actions by other players. (“There’s something else that’s always
better”)
Examples: prisoner cooperating, professor giving a quiz.
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Dominance
For now, let’s only consider pure strategies.
A player’s strategy is strictly dominant if, for any combination of
actions by other players, it gives that player a strictly higher payoff
than all her other strategies. (“The best choice no matter what
others do”)
Examples: prisoner defecting, professor not giving a quiz.
A player’s strategy is strictly dominated if there exists another
strategy giving that player a strictly higher payoff for all combinations
of actions by other players. (“There’s something else that’s always
better”)
Examples: prisoner cooperating, professor giving a quiz.
If a strategy is dominant, the player’s other strategies must be
dominated.
On the other hand, there may not be a dominant strategy, even
though some strategy is dominated.
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Dominance Solvability (I)
It makes sense to predict that a player will play her dominant strategy
if she has one, and will never play a dominated strategy.
This allows us to predict the outcome in the PD.
But in our first example, while we can predict what the professor
does, Bob doesn’t have dominant or dominated strategies.
Idea: take it a step further, and assume Bob can also deduce what
the professor does (plays his dominant strategy, which is not giving
the quiz).
Now we can predict what Bob will do (skip class).
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Dominance Solvability (II)
Iterated deletion of strict dominated strategies, or iterated strict
dominance (ISD): after deleting dominated strategies, look at
whether other strategies became dominated with respect to the
remaining strategies. If so, delete these newly dominated strategies,
and repeat the process until no strategy is dominated.
Example: “Going to class” is initially not dominated, but after “Give
quiz” is eliminated, it becomes dominated. Therefore, we eliminate
“Going to class” in the second iteration.
A game is dominance solvable if ISD leads to a unique predicted
outcome, i.e. only one strategy for each player survives.
Both our examples up to now are dominance solvable.
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Exercise
Find the set of strategies that survive ISD.
Top
Middle
Bottom
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Left
1, 7
5, 3
3, 0
Center
1, 1
6, 4
6, 5
Right
7, 0
5, 1
6, 0
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Remarks on Dominance Solvability
A solution through ISD is not as convincing as a solution in dominant
strategies: it assumes that players correctly anticipate what others
will not do (i.e., how the others will eliminate their actions).
However, ISD allows us to solve some games that don’t have a
dominant strategy for all players.
Moreover, the type of reasoning carried out in ISD seems feasible and
realistic, especially when the number of iterations is small. So it’s still
a pretty appealing concept.
Fact: the order in which ISD is carried out (which can vary since
there can be more than one dominated strategy at a time) does not
influence which strategies survive. (Brainteaser for math jocks: prove
this.)
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Traveler’s Dilemma
An airline has lost two suitcases belonging to Alice and Bob. Identical
antiques in both suitcases.
Each customer writes a claim value, simultaneously:
2 ≤ vA , vB ≤ n,
where n is the maximum possible value.
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Traveler’s Dilemma
An airline has lost two suitcases belonging to Alice and Bob. Identical
antiques in both suitcases.
Each customer writes a claim value, simultaneously:
2 ≤ vA , vB ≤ n,
where n is the maximum possible value.
If both claim the same value, each gets that amount.
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Traveler’s Dilemma
An airline has lost two suitcases belonging to Alice and Bob. Identical
antiques in both suitcases.
Each customer writes a claim value, simultaneously:
2 ≤ vA , vB ≤ n,
where n is the maximum possible value.
If both claim the same value, each gets that amount.
If Alice claims a smaller value (vA < vB ), Alice gets vA + 2, Bob gets
vA − 2.
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Traveler’s Dilemma
An airline has lost two suitcases belonging to Alice and Bob. Identical
antiques in both suitcases.
Each customer writes a claim value, simultaneously:
2 ≤ vA , vB ≤ n,
where n is the maximum possible value.
If both claim the same value, each gets that amount.
If Alice claims a smaller value (vA < vB ), Alice gets vA + 2, Bob gets
vA − 2.
If Bob claims a smaller value (vB < vA ), Alice gets vB − 2, Bob gets
vB + 2.
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Traveler’s Dilemma (II)
For simplicity, suppose n = 4. Write down the normal form.
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Traveler’s Dilemma (II)
For simplicity, suppose n = 4. Write down the normal form.
The strategy of claiming 4 is not strictly dominated by any other pure
strategy.
Nevertheless, the strategy 4 is strictly dominated by the mixed
strategy consisting of claiming 3 with probability 1/2 and claiming
2 with probability 1/2.
Therefore, we can eliminate the (strictly dominated) strategy of
claiming 4.
After the strategy of claiming 4 is eliminated, strategy of claiming 3 is
strictly dominated by the strategy of claiming 2.
Therefore, a single strategy survives ISD: claiming 2 (for each player).
Exercise: write down the normal form for the traveler’s dilemma’s
with n = 5 and n = 6. Solve by ISD. Extra challenge: n = 100.
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Mixed Strategies and ISD
Some pure strategies may only be strictly dominated by mixed
strategies.
A player’s strategy is strictly dominated if there exists another
strategy, pure or mixed, giving that player a strictly higher payoff for
all combinations of strategies by other players. (“There’s something
else that’s always better”)
A pure strategy that is strictly dominated by a mixed strategy need to
be (iteratively) eliminated in ISD.
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Many Games Are Not Dominance Solvable
It’s easy to come up with a game where ISD doesn’t help at all.
Rowena
Pond
Timmie’s
Colin
Pond
1, 1
0, 0
Timmie’s
0, 0
1, 1
(This is a coordination game.)
Are we completely stuck?
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Nash Equilibrium (I)
If Rowena and Colin haven’t set a meeting place, are meeting each
other for the first time and are new at SFU, then it’d be hard to
predict their behavior.
But if they have communicated, have done this before or if there’s a
social norm, it seems likely that they will coordinate successfully.
Idea: We expect situations where everybody is playing her best
strategy, given (that he/she correctly anticipates) others’ strategies.
This way, nobody is making a mistake or has an incentive to switch.
Note: Just like for ISD, we are assuming that each player has correct
beliefs about what others are doing. But it’s a stronger assumption
here, because logic alone doesn’t allow us (or the players) to predict
behaviour.
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Nash Equilibrium (II)
A player i’s (pure or mixed) strategy σi is a best response to other
players’ strategies if, taking as fixed these other players’ strategies, σi
gives player i her highest possible payoff.
A Nash equilibrium (NE) is a strategy profile (σ1 , . . . , σn ) where
every player’s strategy is a best response to other players’ strategies.
Again, this means that nobody has a reason to switch (no incentive
to deviate), giving what everyone else is doing.
An NE in pure strategies or a pure-strategy NE is an NE where
every player’s strategy is pure.
Examples of pure-strategy NE: (Pond, Pond) and (Timmie’s,
Timmie’s) in the last example.
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Finding Pure-Strategy Nash Equilibria
Let’s find all the pure-strategy NEs in the following game:
Top
Middle
Bottom
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Left
0, 1
1, 1
2, 3
Center
0, 0
4, 3
6, 2
Right
5, 2
4, 0
3, 3
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Dumb
-10, -10
-10, -10
-10, -10
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Exercise
Find all the pure-strategy NEs in the following game:
Top
Middle
Bottom
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Left
3, 7
0, 0
4, 0
Center
9, 6
4, 0
1, 5
Right
1, 6
2, 0
0, 1
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Some Games Have No Pure-Strategy Nash Equilibria
Rock
Paper
Scissors
Rock
0, 0
1, -1
-1, 1
Paper
-1, 1
0, 0
1, -1
Scissors
1, -1
-1, 1
0, 0
But there might still be a mixed-strategy NE.
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Some Games Have No Pure-Strategy NE
We saw that Rock-Paper-Scissors has no pure-strategy NE. Here’s
another example:
Goalie
Left
Right
Kicker
Left
1, -1
-1, 1
Right
-1, 1
1, -1
What do soccer players actually do?
These are games where it’s important to keep the other(s) guessing.
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Optimality of Mixed Strategies
Goalie
Left
Right
Kicker
Left
1, -1
-1, 1
Right
-1, 1
1, -1
What would the goalie do if the kicker is more likely to play Left?
Right?
When will the goalie play both Left and Right with positive
probability?
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Optimality of Mixed Strategies
Goalie
Left
Right
Kicker
Left
1, -1
-1, 1
Right
-1, 1
1, -1
What would the goalie do if the kicker is more likely to play Left?
Right?
When will the goalie play both Left and Right with positive
probability?
If a player mixes between multiple pure strategies in a NE, then all of
these strategies (the ones played with positive probability) must be
best responses.
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Finding Mixed-Strategy NE
General procedure to find a Mixed-Strategy NE:
1
2
For each player, conjecture a set of strategies that he/she mixes over.
Calculate the mixing probabilities so that no one has incentive to
deviate (Nash Equilibrium).
Unfortunately, it is generally hard to find all mixed-strategy NE.
But in 2x2x2 games (2 players and 2 actions for each player), it’s not
too bad.
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Kicker-Goalie Revisited
Goalie
Left
Right
Kicker
Left
1, -1
-1, 1
Right
-1, 1
1, -1
Suppose the goalie plays Left with probability p. To find the NE, we
need to find the value of p that makes the kicker indifferent between
Left and Right.
Similarly, we need to find the mix of the kicker’s actions (playing Left
with probability q) that makes the goalie indifferent between Left and
Right.
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(Politically Incorrect) Example: Battle of the Sexes
Find all NE in the following game:
Girl
Ballet
Hockey
Guy
Ballet
3, 1
0, 0
Hockey
0, 0
1, 4
We can also compare the expected payoffs of NE.
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Another example: mixing over some (but not all) strategies
Find all NE in the following game:
D
E
F
A
4, 0
0, 4
3, 0
B
0, 4
4, 0
3, 0
C
0, 3
0, 3
3, 3
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Nash’s Theorem
Theorem (Nash, 1950)
For a game with a finite number of players and where each player has a
finite number of actions, a Nash Equilibrium always exists.
John F. Nash (born 1928)
PhD in mathemaitcs, Princeton (1950)
won the Nobel Prize in Economics, 1994
movie A Beautiful Mind
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Relating Dominance Solvability and Nash Equilibrium
The support of a NE always lies among ISD strategies:
For NE (σ1 , σ2 , . . . , σn ), if σ1 places positive probability on a strategy
s1 , s1 must be player 1’s best response given σ2 , . . . , σn .
Therefore, s1 cannot be strictly dominated. (strictly dominated ⇒ not
a best response for anything)
Therefore, strategies in the support of σi are never deleted during the
ISD iterations.
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Relating Dominance Solvability and Nash Equilibrium
The support of a NE always lies among ISD strategies:
For NE (σ1 , σ2 , . . . , σn ), if σ1 places positive probability on a strategy
s1 , s1 must be player 1’s best response given σ2 , . . . , σn .
Therefore, s1 cannot be strictly dominated. (strictly dominated ⇒ not
a best response for anything)
Therefore, strategies in the support of σi are never deleted during the
ISD iterations.
NE offers sharper prediction than ISD.
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Bigger Games
Remember that NEs cannot involve strategies that are eliminated by
ISD (this is true of all NEs, not just pure-strategy ones).
So finding all mixed-strategy NE is also feasible in games that reduce
down to 2x2x2 through ISD.
In games that don’t reduce that far, sometimes payoffs are nice and
symmetric like in Rock-Paper-Scissors. But even then, it’s more work
than for 2x2x2 games.
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Symmetry
A symmetric game is a game where the payoffs for playing a
particular strategy depend only on the other strategies employed, not
on who is playing them.
If one can change the identities of the players without changing the
payoff to the strategies, then a game is symmetric.
Examples: prisoner’s dilemma, traveler’s dilemma, rock-paper-scissor,
etc.
X
Y
X a, a c, d
Y d, c b, b
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Symmetry
A symmetric game is a game where the payoffs for playing a
particular strategy depend only on the other strategies employed, not
on who is playing them.
If one can change the identities of the players without changing the
payoff to the strategies, then a game is symmetric.
Examples: prisoner’s dilemma, traveler’s dilemma, rock-paper-scissor,
etc.
X
Y
X a, a c, d
Y d, c b, b
A symmetric Nash equilibrium is a Nash equilibrium where every
player plays the same strategy.
A symmetric game always has a symmetric Nash equilibrium, though it
may also process an asymmetric Nash equilibrium.
For a symmetric Nash equilibrium: only needs to check that a single
player has no incentive to deviate.
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Symmetric mixed-strategy NE
Rock
Paper
Scissors
Rock
0, 0
1, -1
-1, 1
Paper
-1, 1
0, 0
1, -1
Scissors
1, -1
-1, 1
0, 0
Suppose that each player plays rock with probability r , paper with
probability p, and scissor with probability s. Find the mix-strategy
Nash equilibrium.
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Understanding the Bystander Effect
Motivation:
Thirty-eight people witnessed the brutal murder of Catherine (“Kitty”)
Genovese over a period of half an hour in New York City in March
1964.
During this period, none of them significantly responded to her screams
for help; none even called the police.
A crime is observed by n people.
Each person would like the police to be informed (value v ), but prefers
that someone else make the phone call (cost c). Assume v > c > 0.
Action of a person: call the police, or don’t call the police.
Note that we may reinterpret (relabel) “calling the police” as “helping
the victim.”
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Understanding the Bystander Effect (II)
Pure strategy: 1 person calls the police, the rest do not call.
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Understanding the Bystander Effect (II)
Pure strategy: 1 person calls the police, the rest do not call.
Symmetric mixed strategy: each person calls police with probability p.
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Understanding the Bystander Effect (II)
Pure strategy: 1 person calls the police, the rest do not call.
Symmetric mixed strategy: each person calls police with probability p.
v 1 − (1 − p)n−1 = v − c
1
p = 1 − (c/v ) n−1 .
Probability p that an individual calls the police decreases with the
group size n.
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Understanding the Bystander Effect (III)
What is the probability the crime is reported in equilibrium?
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Understanding the Bystander Effect (III)
What is the probability the crime is reported in equilibrium?
The probability that no one calls the police is
n
1
(1 − p)n = (c/v ) n−1 = (c/v )1+ n−1 ,
1
since p = 1 − (c/v ) n−1 .
Alternatively,
c
(1 − p)
v
because v (1 − (1 − p)n−1 ) = v − c from the equilibrium condition.
In any case, the probability that the crime is unreported increases
with the number n of bystanders.
(1 − p)n = (1 − p)n−1 · (1 − p) =
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Understanding the Bystander Effect (III)
What is the probability the crime is reported in equilibrium?
The probability that no one calls the police is
n
1
(1 − p)n = (c/v ) n−1 = (c/v )1+ n−1 ,
1
since p = 1 − (c/v ) n−1 .
Alternatively,
c
(1 − p)
v
because v (1 − (1 − p)n−1 ) = v − c from the equilibrium condition.
In any case, the probability that the crime is unreported increases
with the number n of bystanders.
(1 − p)n = (1 − p)n−1 · (1 − p) =
Bystander effect: more bystanders lead to less probability of
reporting the crime/helping the victim.
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Social Psychology vs. Game Theory (Osborne, 2003)
Social psychology:
1
2
3
“diffusion of responsibility”: the larger the group, the lower the
psychological cost of not helping.
“audience inhibition”: the larger the group, the greater the
embarrassment suffered by a helper in case the event turns out to be
one in which help is inappropriate.
“social influence”: a person infers the appropriateness of helping from
others’ behavior, so that in a large group everyone else’s lack of
intervention leads any given person to think intervention is less likely to
be appropriate.
These explanations amount to increasing the cost c and decreasing
the benefit v (of intevening/reporting the crime) with n.
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Social Psychology vs. Game Theory (II)
We have instead assumed that the cost c and the benefit v are
constant, independent of group size n. We conclude that the
bystander effect is something more fundamental, an inevitable
consequence of the strategic interaction of the bystanders.
Our Nash equilibrium analysis rests on the premise that whether a
person intervenes depends on how likely he/she thinks others would
intervene. This perfectly reasonable premise leads to the disturbing
conclusion of the bystander effect.
Advantage: game-theoretic analysis is universal, the same analysis is
applicable to the bystander effect as well as to market competition,
voting, bargaining, etc.
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Conceptual Review Questions
1
2
3
4
5
6
7
8
9
Is pure strategy a mixed strategy? And the converse?
What’s the difference between a strategy profile and a Nash
equilibrium?
In a mixed-strategy Nash equilibrium, why solve for indifference? Why
not do it for a pure-strategy Nash equilibrium?
Does it ever make sense to compare different players’ payoffs?
When you’re doing ISD, do comparisons between player 2’s payoffs
help determine whether you should delete a row?
Why can a NE, whether or not it is in pure strategies, only involve
strategies that survive ISD?
If a game is dominance solvable, what can you say about its NE?
If you need to find the NE of a game that’s bigger than 2x2x2, what
could you do first to reduce your workload?
Is game theory only about complete-information and
simultaneous-move games?
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