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
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).
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 .
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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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.”
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?
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.
Battle of the Sexes
Male
Female
Football
2,1
0,0
Football
Ballet
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?
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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Matching Pennies
You
Friend
Heads
1,-1
-1,1
Heads
Tails
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)?
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.
Best-Responses
Definition 1. A particular strategy a∗i is a best-response for player i to a−i if,
for any other strategy a0i that player i could choose,
ui (a∗i , a−i ) ≥ ui (a0i , a−i )
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:
u
m
d
l
r
3, ∗ −2, ∗
2, ∗ −5, ∗
2, ∗ −2, ∗
So u is a best-response to l, and u and d are both best-responses to r.
Strategy Dominance
Definition 2. A strategy ai ∗ dominates a strategy a0i for player i if, for any
a−i that player i’s opponents might use,
ui (ai ∗, a−i ) ≥ ui (a0i , a−i ).
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Strategy Dominance
Going back to our example,
u
m
d
l
r
3, ∗ −2, ∗
2, ∗ −5, ∗
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.
Dominant Strategies
Definition 3. A strategy a∗i is a dominant strategy for player i if, for any profile
of opponent strategies a−i and any other strategy a0i that player i could choose,
ui (a∗i , a−i ) ≥ ui (a0i , a−i ).
Prisoners’ Dilemma
Row
Silent
Confess
Column
Silent
-2,-2
-1,-5
Confess
-5,-1
-3,-3
Example
A
U
M
D
L
0, −1
2, 4
1, 2
C
2, −3
−1, 1
0, 2
B
R
1, 1
2, 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?
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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Example
Suppose it is first and ten. What should offense and defense should football
teams use?
Offense
Run
Pass
Defend Run
3, −3
9, −9
Defense
Defend Pass
Blitz
7, −7
15, −15
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.)
Example
This can even work for large, complicated games.
A
B
C
D
E
a
b
c
d
e
63, −1 28, −1 −2, 0
−2, 45 −3, 19
32, 1
2, 2
2, 5
33, 0
2, 3
54, 1 95, −1
0, 2
4, −1
0, 4
1, −33 −3, 43 −1, 39 1, −12 −1, 17
−22, 0 1, −13 −1, 88 −2, −57 −2, 72
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?
Quantity Competition
Then we get a strategic form:
1
2
3
1
4, 4
6, 3
6, 2
2
3, 6
4, 4
3, 2
5
3
2, 6
2, 3
0, 0
Many games aren’t dominance solvable
But recall the Battle of the Sexes game:
l
u 2, 1
d 0, 0
r
0, 0
1, 2
This game isn’t dominance solvable. What do we do now?
Pure-Strategy Nash Equilibrium
Definition 4. A strategy profile a∗ = (a∗1 , a∗2 , ..., a∗n ) is a pure-strategy Nash
equilibrium (PSNE) if, for every player i and any other strategy a0i that player
i could choose,
ui (a∗i , a∗−i ) ≥ ui (a0i , a∗−i )
A strategy profile is a Nash equilibrium if all players are using a “mutualbest 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.
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.
Example
Consider the following game:
A
A
U
D
B
L
2, 1
1, −1
R
1, 0
3, 3
U
D
B
L
2, 1
1, −1
R
1, 0
3, 3
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Nash Equilibria in our Classic Games
Prisoners’ Dilemma:
s
s −3, −3
c −1, −7
c
−7, −1
−5, −5
So our new tool — PSNE — agrees with our prediction from IDDS.
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.
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?
Everyone should guess 0. Why? Suppose my best estimate is wi and all players
guess “honestly”: I should expect the average to be wi (otherwise my best estimate
wouldn’t be very good, would it?). But then, my guess of the winning guess is
(1/2)wi , I should guess wi /2 to try to win. But then guessing honestly isn’t a Nash
equilibrium, since all players have an incentive to guess half their estimate.
If I want to guess half my estimate, so must everyone else. Then I should
actually guess 1/4 my estimate... But then everyone else wants to as well, and so
on. Eventually, we are all bidding zero, and the farmer barely has to give us any
pie.
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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?
Well, the payoff of the firm on the left is
xl + xr
xr − xl
=
2
2
πl (xl , xr ) = xl +
and the payoff of the firm on the right is
πr (xl , xr ) = (1 − xr ) +
xl + xr
xr − xl
=1−
2
2
Suppose I guess (1/2, 1/2) is a solution. Then I need to show that neither firm can
change its behavior and get a better payoff. For simplicity, let’s focus on firm a
(since the arguments will be the same for firm b). So consider increasing xa from
1/2 to 1/2 + ∆ when firm b is playing xb = 1/2:
π(1/2 + ∆, 1/2) = 1 −
1/2 + ∆ + 1/2
1+∆
1 ∆
1
=1−
= −
<
2
2
2
2
2
So for any ∆ > 0, this is not a profitable deviation. What about reducing xa from
1/2 to 1/2 − ∆?
π(1/2 − ∆, 1/2) =
1/2 − ∆ + 1/2
1 ∆
1
= −
<
2
2
2
2
So neither player has a profitable deviation from the strategy profile (1/2, 1/2), and
it is a PSNE.
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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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 5. In any game with a finite number of players and pure strategies, a (mixed-strategy) Nash equilibrium is guaranteed to exist.
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.)
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.)
As a quick example of the second point, consider this game: Every country has
to decide which side of the road to drive down. If drivers crash, they get a payoff
of −1, and if they don’t, they get a payoff of 1. Then we can write out a strategic
form as:
L
R
L −1, −1
1, 1
R
1, 1
−1, −1
So there are two PSNE: (L, L) and (R, R). Either can be a prediction. In the US,
it’s (L, R), while in the UK it’s (R, L).
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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?
• Plausibility of Equilibria: In practice, many people don’t confess in prisoners’ dilemma games.
As an example of the computability problem, notice that chess is a finite game
of complete information. It has been shown that (i) Either the first mover has a
winning pure strategy, or either player can force a draw, and (ii) Every finite game of
chess has a pure-strategy Nash equilibrium. In fact, the first mover wins 55 percent
of the time at the highest levels. But there are about 10120 different possible chess
games that might occur (the mass of an average star is 2 ∗ 1030 kg, and there are
1057 atoms in the universe.)
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∗ .
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