Game Theory
Market Design
University of Notre Dame
Market Design
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Game Theory
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Game Theory
A game is composed of
Players : Those agents who take actions
Actions or Strategies : The choices which players can select
Payoffs : The numerical value that players associate with different
outcomes of the game, which are allowed to depend on each player’s
action
Market Design
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Game Theory
2 / 38
Game Theory
A game is composed of
Players : Those agents who take actions
Actions or Strategies : The choices which players can select
Payoffs : The numerical value that players associate with different
outcomes of the game, which are allowed to depend on each player’s
action
Timing : A description of which players take actions when
Market Design
(ND)
Game Theory
2 / 38
Game Theory
A game is composed of
Players : Those agents who take actions
Actions or Strategies : The choices which players can select
Payoffs : The numerical value that players associate with different
outcomes of the game, which are allowed to depend on each player’s
action
Timing : A description of which players take actions when
Information : A description of what players know, and when they
know it
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Game Theory
2 / 38
Game Theory
We begin with simultaneous-move games of complete information, in
which all players make their decisions at the same time and know
everything about the game and about each other.
We then add timing, giving us dynamic games of complete
information, where players make decisions in sequence, and cannot
revisit their earlier choices (think tic-tac-toe).
Finally, we add incomplete or imperfect information to get Bayesian
games (think poker).
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Game Theory
3 / 38
Game Theory
A game is composed of
A set of players i = 1, 2, ..., N
A set of feasible actions or strategies Ai for each player i
A payoff function ui (a1 , a2 , ..., aN ) for each player i giving his payoff
given the choices of all the players
Market Design
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Game Theory
4 / 38
Game Theory
A game is composed of
A set of players i = 1, 2, ..., N
A set of feasible actions or strategies Ai for each player i
A payoff function ui (a1 , a2 , ..., aN ) for each player i giving his payoff
given the choices of all the players
So we can just write a game as {Ai , ui (a1 , ..., aN )}N
i=1 .
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Game Theory
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Game Theory
Two more pieces of notation:
A strategy profile is a list of actions for each player:
a = (a1 , a2 , ..., aN ),
so we write the players’ payoffs as ui (a) if a occurs.
Market Design
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Game Theory
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Game Theory
Two more pieces of notation:
A strategy profile is a list of actions for each player:
a = (a1 , a2 , ..., aN ),
so we write the players’ payoffs as ui (a) if a occurs.
The notation
a−i = (a1 , a2 , ..., ai−1 , ai+1 , ..., aN )
is a strategy profile with the i-th player removed. It lets us focus on
player i’s incentives by holding a−i fixed and thinking about what i
should do:
ui (a) = ui (ai , a−i )
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Game Theory
5 / 38
Prisoners’ Dilemma
There are two burglars, who have been captured in the process of
committing a crime. They have been very careful, and actually do not
even know each other’s real name. The district attorney tells them:
“If you both remain silent, I have enough evidence to send each of
you to jail for two years. However, if one of you confesses and the
other tells the truth, I will give the confessor a lighter sentence,
sending him to jail for only one year, while I prosecute the other
aggressively and send him to jail for five years. If both of you confess,
there won’t be a trial, and you both get three years.”
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Game Theory
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Prisoners’ Dilemma
Row
Silent
Confess
Column
Silent
-2,-2
-1,-5
Confess
-5,-1
-3,-3
What are the Pareto optimal outcomes of the game? What do you think
the burglars do, and why? What other economic situations have similar
incentives?
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Game Theory
7 / 38
Battle of the Sexes
Two people have decided to go on a date. The two options are a
Football game, and the Ballet. The male prefers football, while the
female prefers ballet. They discuss which option they will pick, but
both happen to forget which they decided on. Worse, they both
forgot their smart phones at work, and the two events are about to
begin. Both prefer to be together rather than apart.
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Game Theory
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Battle of the Sexes
Male
Football
Ballet
Female
Football
2,1
0,0
Ballet
0,0
1,2
What are the Pareto optimal outcomes for the couple? What do you think
they do, and why? What if we made the payoffs to Ballet (10, 20)? What
other economic situations have similar incentives?
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Game Theory
9 / 38
Matching Pennies
You are waiting for a plane with a friend. Both of you have plenty of
pocket change, so you propose the following game: You both secretly
pick Heads or Tails. If both coins are heads, you get both coins. If
both coins are tails, your friend gets both coins.
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Game Theory
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Matching Pennies
You
Heads
Tails
Friend
Heads
1,-1
-1,1
Tails
-1,1
1,-1
What are the Pareto optimal outcomes? What do you think they do, and
why? What happens if we made the payoff to (Tails, Tails) equal to
(10, −1)?
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Game Theory
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The Strategic Form
The matrix of players/actions/payoffs that we’ve been using to describe
games is very helpful, since it summarizes all of the relevant information
from a game theory perspective. We call it the strategic form.
Row Player
U
D
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Column Player
L
urow (U, L), ucolumn (U, L)
urow (D, L), ucolumn (D, L)
Game Theory
R
urow (U, R), ucolumn (U, R)
urow (D, R), ucolumn (D, R)
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The Strategic Form
The matrix of players/actions/payoffs that we’ve been using to describe
games is very helpful, since it summarizes all of the relevant information
from a game theory perspective. We call it the strategic form.
Row Player
U
D
Column Player
L
urow (U, L), ucolumn (U, L)
urow (D, L), ucolumn (D, L)
R
urow (U, R), ucolumn (U, R)
urow (D, R), ucolumn (D, R)
So you can think of game theory as a generalization of price-taking or
perfectly competitive models where consumers have preferences over
bundles of goods or firms have preferences over quantities produced, to a
setting where agents have preferences over how the other agents act.
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Game Theory
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Best-Responses
Definition
A particular strategy ai∗ is a best-response for player i to a−i if, for any
other strategy ai0 that player i could choose,
ui (ai∗ , a−i ) ≥ ui (ai0 , a−i )
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Game Theory
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Best-Responses
The first thing you should do when you see any game in a strategic form is
to underline the players’ best responses. Consider the game:
l
r
u 3, ∗ −2, ∗
m 2, ∗ −5, ∗
d 2, ∗ −2, ∗
Market Design
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Game Theory
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Best-Responses
The first thing you should do when you see any game in a strategic form is
to underline the players’ best responses. Consider the game:
l
r
u 3, ∗ −2, ∗
m 2, ∗ −5, ∗
d 2, ∗ −2, ∗
So u is a best-response to l, and u and d are both best-responses to r .
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Game Theory
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Strategy Dominance
Definition
A strategy ai ∗ dominates a strategy ai0 for player i if, for any a−i that
player i’s opponents might use,
ui (ai ∗, a−i ) ≥ ui (ai0 , a−i ).
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Game Theory
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Strategy Dominance
Going back to our example,
l
r
u 3, ∗ −2, ∗
m 2, ∗ −5, ∗
d 2, ∗ −2, ∗
So u dominates m and d, and d dominates m. So we’d be justified in
predicting that the row player use the strategy u.
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Game Theory
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Dominant Strategies
Definition
A strategy ai∗ is a dominant strategy for player i if, for any profile of
opponent strategies a−i and any other strategy ai0 that player i could
choose,
ui (ai∗ , a−i ) ≥ ui (ai0 , a−i ).
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Game Theory
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Prisoners’ Dilemma
Row
Market Design
(ND)
Silent
Confess
Column
Silent
-2,-2
-1,-5
Game Theory
Confess
-5,-1
-3,-3
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Example
B
L
C
R
U 0, −1 2, −3 1, 1
M 2, 4 −1, 1 2, 2
A D 1, 2
0, 2 1, 4
Market Design
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Game Theory
19 / 38
Example
B
L
C
R
U 0, −1 2, −3 1, 1
M 2, 4 −1, 1 2, 2
A D 1, 2
0, 2 1, 4
So no strategies are dominant for either player. But some strategies are
certainly dominated. Maybe we can simplify the game by removing those
strategies?
Market Design
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Game Theory
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Iterated Deletion of Dominated Strategies
Step 1: For each player, eliminate all of his dominated strategies.
Step 2: If you deleted any strategies during Step 1, repeat Step 1.
Otherwise, stop.
If the process eliminates all but one strategy profile s ∗ , we call it a
dominant strategy equilibrium or we say it is the outcome of iterated
deletion of dominated strategies. Think of it as a “group process of
elimination”.
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Game Theory
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Example
Suppose it is first and ten. What should offense and defense should
football teams use?
Defense
Defend Run Defend Pass
Blitz
Run
3, −3
7, −7
15, −15
Offense Pass
9, −9
8, −8
10, −10
(If you really like football, think of these numbers as the average number of
yards for the whole drive, given a particular strategy profile chosen above.)
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Game Theory
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Example
This can even work for large, complicated games.
a
A 63, −1
B 32, 1
C 54, 1
D 1, −33
E −22, 0
Market Design
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b
c
d
e
28, −1 −2, 0
−2, 45 −3, 19
2, 2
2, 5
33, 0
2, 3
95, −1
0, 2
4, −1
0, 4
−3, 43 −1, 39 1, −12 −1, 17
1, −13 −1, 88 −2, −57 −2, 72
Game Theory
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Quantity Competition
Suppose there are two firms, a and b, who choose to produce quantities qa
and qb of their common product. Each firm can choose to product either
1, 2, or 3 units. They have no costs, and the market price is
p(qa , qb ) = 6 − qa − qb . The firm’s payoffs, then, are
πA (qa , qb ) = p(qa , qb )qa = (6 − qa − qb )qa
and
πB (qb , qa ) = p(qa , qb )qb = (6 − qa − qb )qb
Does the game have a dominant strategy equilibrium?
Market Design
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Game Theory
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Quantity Competition
Then we get a strategic form:
1
2
3
1 4, 4 3, 6 2, 6
2 6, 3 4, 4 2, 3
3 6, 2 3, 2 0, 0
Market Design
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Game Theory
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Many games aren’t dominance solvable
But recall the Battle of the Sexes game:
l
r
u 2, 1 0, 0
d 0, 0 1, 2
This game isn’t dominance solvable. What do we do now?
Market Design
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Game Theory
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Pure-Strategy Nash Equilibrium
Definition
A strategy profile a∗ = (a1∗ , a2∗ , ..., an∗ ) is a pure-strategy Nash equilibrium
(PSNE) if, for every player i and any other strategy ai0 that player i could
choose,
∗
∗
ui (ai∗ , a−i
) ≥ ui (ai0 , a−i
)
Market Design
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Game Theory
26 / 38
Pure-Strategy Nash Equilibrium
Definition
A strategy profile a∗ = (a1∗ , a2∗ , ..., an∗ ) is a pure-strategy Nash equilibrium
(PSNE) if, for every player i and any other strategy ai0 that player i could
choose,
∗
∗
ui (ai∗ , a−i
) ≥ ui (ai0 , a−i
)
A strategy profile is a Nash equilibrium if all players are using a
“mutual-best response”, or no player can change what he is doing and get
a strictly higher payoff.
Market Design
(ND)
Game Theory
26 / 38
Pure-Strategy Nash Equilibrium
Definition
A strategy profile a∗ = (a1∗ , a2∗ , ..., an∗ ) is a pure-strategy Nash equilibrium
(PSNE) if, for every player i and any other strategy ai0 that player i could
choose,
∗
∗
ui (ai∗ , a−i
) ≥ ui (ai0 , a−i
)
A strategy profile is a Nash equilibrium if all players are using a
“mutual-best response”, or no player can change what he is doing and get
a strictly higher payoff. Notice that we’re thinking about the structure of
the game, and not the motivations of any individual player, as with
strategy dominance.
Market Design
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Game Theory
26 / 38
How to find PSNE’s in Strategic Form Games
Finding Nash equilibria in strategic form can done quickly:
Pick a row. Underline the best payoff the column player can receive.
Check all rows.
Pick a column. Underline the best payoff the row player can receive.
Check all columns.
If any box has both pay-offs underlined, it is a pure-strategy Nash
equilibrium.
Market Design
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Game Theory
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Example
Consider the following game:
B
L
R
A U 2, 1 1, 0
D 1, −1 3, 3
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Game Theory
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Nash Equilibria in our Classic Games
Prisoners’ Dilemma:
s
c
s −3, −3 −7, −1
c −1, −7 −5, −5
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Game Theory
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Nash Equilibria in our Classic Games
Prisoners’ Dilemma:
s
c
s −3, −3 −7, −1
c −1, −7 −5, −5
So our new tool — PSNE — agrees with our prediction from IDDS.
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Game Theory
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Nash Equilibria in our Classic Games
Battle of the Sexes:
F
B
F 2, 1 0, 0
B 0, 0 1, 2
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Game Theory
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Nash Equilibria in our Classic Games
Battle of the Sexes:
F
B
F 2, 1 0, 0
B 0, 0 1, 2
There are two PSNE: (F , F ) and (B, B). So PSNE can make useful
predictions where dominance solvability does not.
Market Design
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Game Theory
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Guess Half the Average
At the county fair, a farmer proposes the following game: The
townspeople all guess the weight of a large pumpkin pie, and the person
who is closest to half the average of the guesses gets her guess in pounds
of pumpkin pie, and no one else gets anything. No one is quite sure how
large the pie is, but they all have an estimate.
Market Design
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Game Theory
31 / 38
Guess Half the Average
At the county fair, a farmer proposes the following game: The
townspeople all guess the weight of a large pumpkin pie, and the person
who is closest to half the average of the guesses gets her guess in pounds
of pumpkin pie, and no one else gets anything. No one is quite sure how
large the pie is, but they all have an estimate. More formally,
Each townsperson i = 1, 2, ..., N has a best estimate wi of the pie’s
weight. They each get to submit a guess gi > 0.
The average guess is
ḡ =
1
(g1 + g2 + ... + gN )
N
The townsperson with the guess gi closest to ḡ gets a payoff of gi .
Everyone else gets nothing
What is the pure-strategy Nash equilibrium of the game?
Market Design
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Game Theory
31 / 38
Hotelling’s Main Street Game
Suppose there are customers uniformly distributed along Main Street,
which is one mile long. Then on the interval [0, 1], whenever
1 ≥ b ≥ a ≥ 0, there are b − a customers in the subinterval [a, b]. There
are two gas stations, a and b trying to decide where to locate their gas
stations in [0, 1]; call these locations xa and xb . All customers visit the
closest gas station, and buy an amount of gasoline that gives the gas
station profits of 1 per customer. Do the players have weakly dominant
strategies? What are the Nash equilibria of the game?
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Game Theory
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Nash Equilibria in our Classic Games
Rock-Paper-Scissors:
R
P
S
R 0, 0 −1, 1 1, −1
P 1, −1 0, 0 −1, 1
S −1, 1 1, −1 0, 0
Market Design
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Game Theory
33 / 38
Nash Equilibria in our Classic Games
Rock-Paper-Scissors:
R
P
S
R 0, 0 −1, 1 1, −1
P 1, −1 0, 0 −1, 1
S −1, 1 1, −1 0, 0
And we have at least one “class” of games that don’t have pure-strategy
Nash equilibria: No strategy profile is underlined twice, so there are no
pure-strategy Nash equilibria.
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Game Theory
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Mixed-strategy Nash equilibria
For games like RPS, where there is no pure-strategy Nash equilibrium,
there will exist a “mixed-strategy” Nash equilibrium where players
behave randomly
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Game Theory
34 / 38
Mixed-strategy Nash equilibria
For games like RPS, where there is no pure-strategy Nash equilibrium,
there will exist a “mixed-strategy” Nash equilibrium where players
behave randomly
We won’t cover it, since we won’t want people to behave randomly in
our markets (exception: auditing)
Market Design
(ND)
Game Theory
34 / 38
Mixed-strategy Nash equilibria
For games like RPS, where there is no pure-strategy Nash equilibrium,
there will exist a “mixed-strategy” Nash equilibrium where players
behave randomly
We won’t cover it, since we won’t want people to behave randomly in
our markets (exception: auditing)
Some games with PSNE also have MSNE: Battle of the sexes
Market Design
(ND)
Game Theory
34 / 38
Mixed-strategy Nash equilibria
For games like RPS, where there is no pure-strategy Nash equilibrium,
there will exist a “mixed-strategy” Nash equilibrium where players
behave randomly
We won’t cover it, since we won’t want people to behave randomly in
our markets (exception: auditing)
Some games with PSNE also have MSNE: Battle of the sexes
Nash is important for proving this:
Theorem
In any game with a finite number of players and pure strategies, a
(mixed-strategy) Nash equilibrium is guaranteed to exist.
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Game Theory
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Dominant Strategy, Pure-Strategy, and Mixed-Strategy
Nash Equilibria
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(ND)
Game Theory
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Dominant Strategy, Pure-Strategy, and Mixed-Strategy
Nash Equilibria
If a game is solvable by iterated deletion of dominated strategies, the
outcome is a pure-strategy Nash equilibrium, but not all pure-strategy
Nash equilibria are the result of iterated deletion of dominated
strategies
Market Design
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Game Theory
35 / 38
Dominant Strategy, Pure-Strategy, and Mixed-Strategy
Nash Equilibria
If a game is solvable by iterated deletion of dominated strategies, the
outcome is a pure-strategy Nash equilibrium, but not all pure-strategy
Nash equilibria are the result of iterated deletion of dominated
strategies
For any game with a finite number of players and strategies, a
mixed-Nash equilibrium exists. (This result is what Nash won the
Nobel prize for.)
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Game Theory
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How to interpret Nash Equilibria
Market Design
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Game Theory
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How to interpret Nash Equilibria
The Outcome of Strategic Reasoning: The logical end result of each
player trying to reason about what their opponents will do, knowing
the others are doing the same thing.
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Game Theory
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How to interpret Nash Equilibria
The Outcome of Strategic Reasoning: The logical end result of each
player trying to reason about what their opponents will do, knowing
the others are doing the same thing.
Norms and Conventions: The strategies that can be predicted as
stable “norms” or “conventions” in society, where — given that a
particular norm has been adopted — no single person can change the
convention.
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Game Theory
36 / 38
How to interpret Nash Equilibria
The Outcome of Strategic Reasoning: The logical end result of each
player trying to reason about what their opponents will do, knowing
the others are doing the same thing.
Norms and Conventions: The strategies that can be predicted as
stable “norms” or “conventions” in society, where — given that a
particular norm has been adopted — no single person can change the
convention.
The Outcome of “Survival of the Fittest”: Suppose we have a large
population of players, and those who get low payoffs are removed
from the game, while those who get high payoffs remain. As this
game evolves, the stable outcomes of the dynamic process are Nash
equilibria. (This is one foundation for evolutionary biology.)
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Game Theory
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Criticisms of Nash and Dominant Strategy Equilibria
Market Design
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Game Theory
37 / 38
Criticisms of Nash and Dominant Strategy Equilibria
Multiplicity of Equilibria: If a game has multiple equilibria, how do
the players know which one to use?
Market Design
(ND)
Game Theory
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Criticisms of Nash and Dominant Strategy Equilibria
Multiplicity of Equilibria: If a game has multiple equilibria, how do
the players know which one to use?
Computability of Equilibria: In very large or complicated games, how
can players do IDDS or find pure-strategy Nash equilibria?
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Game Theory
37 / 38
Criticisms of Nash and Dominant Strategy Equilibria
Multiplicity of Equilibria: If a game has multiple equilibria, how do
the players know which one to use?
Computability of Equilibria: In very large or complicated games, how
can players do IDDS or find pure-strategy Nash equilibria?
Plausibility of Equilibria: In practice, many people don’t confess in
prisoners’ dilemma games.
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Game Theory
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Quick Preview: Mechanism design
Mechanism design is the “inverse” of game theory
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Game Theory
38 / 38
Quick Preview: Mechanism design
Mechanism design is the “inverse” of game theory
Game theory asks, given a game {Ai , ui (a)}N
i=1 , what are reasonable
predictions as to what the players will do?
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Game Theory
38 / 38
Quick Preview: Mechanism design
Mechanism design is the “inverse” of game theory
Game theory asks, given a game {Ai , ui (a)}N
i=1 , what are reasonable
predictions as to what the players will do?
Mechanism design asks, given the players’ preferences ui (x) over
∗
outcomes x, {ui (x)}N
i=1 , what games cause a particular outcome x
to arise as a (dominant strategy or Nash) equilibrium? How do we
design the actions for the players to induce them to choose actions
that result in x ∗ ?
Market Design
(ND)
Game Theory
38 / 38
Quick Preview: Mechanism design
Mechanism design is the “inverse” of game theory
Game theory asks, given a game {Ai , ui (a)}N
i=1 , what are reasonable
predictions as to what the players will do?
Mechanism design asks, given the players’ preferences ui (x) over
∗
outcomes x, {ui (x)}N
i=1 , what games cause a particular outcome x
to arise as a (dominant strategy or Nash) equilibrium? How do we
design the actions for the players to induce them to choose actions
that result in x ∗ ?
If a game exists which induces the players to select actions leading to
x ∗ in a (dominant strategy or Nash) equilibrium, we say the game
implements x ∗ .
Market Design
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Game Theory
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