Simultaneous GamesSam HwangFebruary 12, 20161 13.docx

Simultaneous Games
Sam Hwang
February 12, 2016
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Simultaneous games?
I Simultaneous games are the ones in which players move only
once and at the same time, therefore do not observe other
players’ choices
I Example 1: Matching Pennies
I Example 2: Rock paper scissors
I Why simultaneous games?
I What we want to learn is how to predict the outcome of a
game
I It is easier to learn that in simultaneous games
I We will start with simultaneous games and learn dynamic
games eventually
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Let’s Play a Example of Simultaneous Game: Prisoner’s
Dilemma
I Two criminals are captured and put in a separate cell; no
communication possible between the two
I The district attorney offers to each of them
I ”if you are the only one who confess, then your sentence will
be 1 year and the other will go to jail for 10 years”
I ”but if you do not confess but the other guy does, then he will
serve 1 year and you will serve 10 years”
I ”if both of you confess, then some mercy will be shown to you
and both of you will serve 5 years”
I ”if neither of you confess, then I will still convict both of you
and each will serve 2 years”
I In terms of total number of years served, both of you not
confessing is the best option
I In terms of individual number of years served, you (and only
you) confessing is the best option
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Normal Form Representation of Prisoner’s Dilemma
I C: cooperate, not confess (cooperate with your inmate)
I D: defect

4 / 131
Extensive Form Representation of Prisoner’s Dilemma
5 / 131
Let’s play Prisoner’s Dillema
I This time, let’s think carefully about what you want to do
before we begin
6 / 131
Let’s make a prediction
I Everyone will cooperate
I Everyone will defect
I You will randomize
I What else?
7 / 131
Prisoner’s Dillema

I If we stare at the game long enough, we will notice that
I when the other player cooperates, your payoff is 14 from
defecting, 10 from cooperating
I when the other player defects, your payoff is 6 from defecting,
2 from cooperating
I Regardless of what the other player does, your payoff is
always
greater when you defect!!!!!
I It makes sense if you only care about your payoff, then you
choose to defect
I You can check that that is how the other player feels, too
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I We are ready to make our prediction based on this reasoning
I We predict that both prisoners will defect because ”defecting”
provides them with higher payoffs than ”cooperating”,
regardless of what the other prisoner does. Therefore, the
outcome of the game will be ”defect,defect”
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Results of the Game
I Was our prediction correct?

I Tell me about how you played
10 / 131
Recap of the Lecture on Jan. 28th
Making ”nature” a player in the game
I Some aspect of the game is not known with certainty to any
of the players
I Example: coordination of vacation destination
I You and Sam were going to meet at a resort tomorrow
I But neither of us can communicate with each other; cannot
coordiate which resort
I We haven’t finalized our destination; we just know it is going
to be either
1. A ski resort
2. A resort with an indoor hot spring
I The chance of rain tomorrow is 40%; both places are far away,
so both you and Sam have to leave tonight (without knowing
tomorrow’s weather nor where the other is headed)
I In this example, ”weather tomorrow” is not known to any of
the players with certainty; just the probability of rain
I We make ”nature” choose rain with probability of 0.4 and not
rain with 0.6
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Move By Nature
I Are we done?
I No, information sets!!
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Move By Nature
I Are we done?
I No, information sets of YOU!!
13 / 131
Move By Nature
I Are we done?
14 / 131
Recap of the Lecture on Jan. 28th
I We started learning simultaneous game
I Players move only once; do not observe others’ moves when
they make a move

I Prisoner’s Dillema
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Let’s Play One Round of Prisoner’s Dillema
I Last time we played it for five rounds
I This time we are playing it only once
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Results of the Game
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I Information set for each player?
I Set of pure strategies for each player?
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Strictly Dominant Strategies
I Strategies such as Defect in PD are said to be strictly
dominant

I A pure strategy is a strictly dominant strategy for a player if
the payoff from the strategy is greater than all the other pure
strategies of that player, REGARDLESS OF what strategies
other players use
I Alternatively, with notation we can define strictly dominant
pure strategies more precisely
I Some might feel not certain whether ”greater” means ”strictly
greater (¿)” or ”greater than equal to (≥)”
I Given a game, let Si denote the set of pure strategies of
player i
I For example, in PD, Srow player =
{(Play C at his information set), (Play D at his information set)}
I Then a pure strategy si is a strictly dominant strategy if
ui (si,s−i ) > ui (s
′
i ,s−i ) for all s
′
i ∈ Si such that s
′
i 6= si and for all s−i ∈ S−i
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Definition of Strictly Dominant Pure Strategies
I A pure strategy si is a strictly dominant strategy if

′
′
′
I Let’s not panic...let’s look at the definition one term at a time
I ui (si,s−i ) : payoffs (in utils) for player i when player i plays
si
and the other player(s) play s−i
I For example, in PD, urow player(D,C ) : payoff for row player
when the row player plays Defect and the other player plays
Cooperate
I ucolumn player(C,D) : payoff for column player when the
column
player plays Cooperate and the other player plays Defect
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I For another example, consider rock paper scissors played by
THREE players
I In such game, ui (si,s−i ) : player i’s payoff when player i
chooses strategy si and the other two players choose a profile

of strategies s−i
I For example, what does uplayer 1(Rock, (Rock,Paper)) mean?
I i = player 1
I si : Rock
I s−i = (Player 2’s strategy, Player 3’s strategy) =
(Rock,Paper)
I So uplayer 1(Rock, (Rock,Paper)) denotes player 1’s payoff
when he plays Rock and player 2 plays Rock and player 3
plays Paper
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I What does uplayer 2(Rock, (Scissors,Paper)) mean?
I i = player 2
I si : Rock
(Scissors,Paper)
I So uplayer 2(Rock, (Scissors,Paper)) means Player 2’s payoff
when he plays Rock, player 1 plays Scissors, and player 3 plays
Paper
22 / 131

I What does uplayer 3(Scissors, (Rock,Scissors)) mean?
I i = player 3
I si : Scissors
(Rock,Scissors)
I So uplayer 3(Scissors, (Rock,Scissors)) means Player 3’s
payoff
when he plays Scissors and Player 1 plays Rock and Player 2
plays Scissors
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I Then a strategy si is a strictly dominant strategy if
′
′
′
I So we know what ui (si,s−i ) > ui (s
′
i ,s−i ) means: it means
player i’s payoff from playing si is strictly greater than when
he uses an alternative strategy s′i when other players play a

profile of strategies s−i
I And this inequality holds
1. regardless of what alternative strategy s′i is considered,
except
for si (Why? What happens to the inequality if the alternative
strategy is equal to si ?); and
2. For ALL profiles of strategies that can be played by other
players (”for all all s−i ∈ S−i ”)
I What is S−i ?: the set of all possible profiles of strategies that
players other than i can play
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S−i In Different Games
I S−i : set of profiles of pure strategies with player i removed
(hence the negative sign, -, in front of i)
I For example, in PD, S−row player : the set of all possible
profiles
of strategies that player(s) other than row player can play
I Since there are only two players in PD,
S−row player = Scolumn player = {Cooperate,Defect}
I In a game of RPS with three people, S−player 1?
I Player 2 can play R,P or S AND player 3 can play R,P, or S
I Therefore, S−player 1 =

{(R,R), (P,P), (S,S), (R,P), (R,S), (P,R), (P,S), (S,R), (S,P)}
I So in a game of RPS with three players, Rock would be
strictly dominant for player 1 if
u1(Rock,s−1) > u1(s
′
1,s−1) for all s
′
1 ∈ S1 that is not Rock and for all s−1 ∈ S−1
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Strictly Dominated Pure Strategies
I Strategies such as Cooperate in PD are said to be strictly
dominated
I A pure strategy is strictly dominated for a player if the payoff
from the pure strategy strategy is less than that from another
strategy, REGARDLESS OF what strategies other players use
I With notation: given a game, si is a strictly dominated pure
strategy if for some s′i ∈ Si,
ui (s
′
i ,s−i ) > ui (si,s−i ) for all s−i ∈ S−i
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Strictly Dominant Pure Strategy in a game of RPS
I We have seen that in PD, a strictly dominant pure strategy
exists (Defect)
I Assuming that every player would like to maximize his payoff,
we predicted that both players would choose to Defect
I So whenever there exist a strictly dominant pure strategy in a
game, we would like to predict that the player will play it
I We know how to predict the outcome of a game (players will
play strictly dominant pure strategyes)
I Are we done with game theory then?
I We shall see...
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Strictly Dominant Pure Strategy in a game of RPS
I Let’s see if there is a strictly dominant pure strategy in a
game of RPS with three players
I The set of pure strategie={Rock,Paper,Scissors}
I Is Rock a strictly dominant strategy?
I No. It feels like the right answer. But how do we actually
show that Rock is not strictly dominant?
I Let’s go back to the definition: Rock would be strictly
dominant for player 1 if for any profile of strategy player 2 and
3 can play, Rock is better than all the other pure strategyes of

player 1
I To show that this is false, we need to show: there is a profile
of strategies player 2 and 3 can play that Rock is worse than
Scissors or Papers
I It is easy:
ui (Rock, (Paper,Paper)) < ui (Scissors, (Paper,Paper));
therefore Rock is not strictly dominant
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Strictly Dominant Pure Strategy in a game of RPS Does
Not Exist
I What about Scissors?
I ui (Scissors, (Rock,Rock)) < ui (Paper, (Rock,Rock)) : no,
Scissors is not a strictly dominant pure strategy
I What about Paper?
I ui (Paper, (Scissors,Scissors)) <
ui (Rock, (Scissors,Scissors)) : no, Paper is not a strictly
dominant pure strategy
I NO STRICTLY DOMINIANT PURE STRATEGY EXISTS IN
A GAME OF RPS
I Then we cannot make any prediction for a game of RPS if we
assume that players play strictly dominant pure strategies in
all games because there are games in which no pure strategies
are strictly dominant
I The reason that we do not have any strictly dominant pure

strategies in RPS might be that we focus only on pure
strategies
I There might be strictly dominant mixed strategies in RPS 29 /
131
Strictly Dominant Mixed Strategy
I Definitions
I A mixed strategy is strictly dominant if the expected payoff
from it is greater than any other mixed strategies regardless
which profile of mixed strategies other players play
I A mixed strategy is strictly dominanted if the expected payoff
from it is less than another regardless of which profile of mixed
strategies other players play
I A mixed strategy is a function that maps a set of pure
strategies to probabilities that each pure strategy is played,
and the sum of the probabilities is 1
I In a game of RPS, an example of mixed strategy is
I σ(Rock) = 1
3
I σ(Paper) = 1
3
I σ(Scissors) = 1
3
I Is this a mixed strategy?
I σ(Rock) = 1

I σ(Paper) = 0
I σ(Scissors) = 0
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Strictly Dominant Mixed Strategy
I All pure strategies are mixed strategies
I We want to find a mixed strategy that is strictly dominant in
a game of RPS with three players
I We do not know what it is yet (and we do not know whether
such strategy exists), so let σ denote a mixed strategy such
that
I σ(Rock) = prock
I σ(Paper) = ppaper
I σ(Scissors) = pscissors
I The goal is to figure out whether there is a strictly dominant
mixed strategy, and if there is, then what the values for
prock,ppaper,pscissors are
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Strictly Dominant Mixed Strategy for RPS
I Given the mixed strategy σ
I σ(Rock) = prock

I We can calculate the expected utility from σ
I What is the expected utility from σ when player 2 and 3 play
(Rock,Rock)?
I ui (Rock, (Rock,Rock)) = 0 with probability prock
I ui (Paper, (Rock,Rock)) = 1 with probability ppaper
I ui (Scissors, (Rock,Rock)) = −1 with probability pscissors
I So when player 2 and 3 play Rocks, σ induces the lottery
(0,prock ; 1,ppaper ;−1,pscissors )
I So the expected utility from σ, ui (σ, (Rock,Rock)) is
ppaper −pscissors
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Strictly Dominant Mixed Strategy for RPS
I σ be strictly dominant, the expected utility from σ when the
other players play (Rock,Rock) should be strictly greater than
the utility from all the other strategies
I In particular, the expected utility from σ has to be greater
than that from a pure strategy (Paper)
I ui (σ, (Rock,Rock)) = ppaper −pscissors
I ui (Paper, (Rock,Rock)) = 1
I ui (σ, (Rock,Rock)) can NEVER be greater than
ui (Paper, (Rock,Rock)) = 1 (Why?)
I Probabilities are greater than or equal to zero
I Probabilities are less than or equal to one
I So the largest ppaper −pscissors can be is...when ppaper is...

and
pscissors is...
I So ui (σ, (Rock,Rock)) can NEVER be greater than
ui (Paper, (Rock,Rock))
I Therefore, NO MIXED STRATEGIES ARE STRICTLY
DOMINANT
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No Strictly Dominant Strategy for RPS
I So in a game of RPS,
I No strictly dominant pure strategy
I No strictly dominant mixed strategy
I If we adhere to the principle that players play strictly
dominant strategy in a game, then we are unable to make any
prediction in a game of RPS because there are no strictly
dominant strategies
I So we are not done with game theory...
I Some of you might say: strict dominance might be too strong
a requirement. No wonder in some games there are no strictly
dominant strategies!
I Strict dominance is indeed quite strong: it requires a strategy
provides higher expected utility than any other strategies at
any strategy profile of others
I If we relax such a strong requirement just a little bit, we

might be able to make a prediction in RPS
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Relaxing ”Strict” in Strictly Dominant Strategy
I Strict dominance is a strong requirement
I So we define ”weak” dominance”
I A pure strategy si weakly dominates another pure strategy s
′
i
if the utility from si is not smaller than that from s
′
i at all
strategy profiles of other players, AND greater at some
strategy profile
I With notation, a pure strategy si weakly dominates another
pure strategy s′i if
ui (si,s−i ) ≥ ui (s′i ,s−i ) for all s−i ∈ S−i AND
′
i ,s−i ) for some s−i ∈ S−i
I Compare this to the definition for strict dominance: a pure
strategy si strictly dominates another pure strategy s
′
i if the

utility from si is greater than that from s
′
i at all strategy
profiles of other players
I A strategy is weakly dominant if it weakly dominates all the
other strategy
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Relationship between Strict and Weak Dominance
I Is strictly dominant strategy weakly dominant?
I A strategy is strictly dominant if the utility from it is greater
than all the other strategy at any strategy profile
I So yes
I Is weakly dominant strategy necessarily strictly dominant?
I A strategy is weakly dominant if the utility from it is not
smaller than all the other strategy at any strategy profile and
greater at SOME OF THE STRATEGY PROFILES
I So no, weakly dominant strategies are not necessarily strictly
dominant
I This means that in the game of RPS, even though we were
not able to find a weakly dominant strategy, it might be
possible to find a weakly dominant strategy
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Weakly Dominant Pure Strategy in RPS
I Is Rock a weakly dominant strategy?
I −1 = ui (Rock, (Paper,Paper)) <
ui (Scissors, (Paper,Paper)) = 1
I At some strategy profile, Rock is WORSE than scissors
I So Rock is not weakly dominant
I Is Scissors a weakly dominant strategy?
I −1 = ui (Scissors, (Rock,Rock)) < ui (Paper, (Rock,Rock)) =
1
I At some strategy profile, Scissors is WORSE than Paper
I So Scissors is not weakly dominant
I Is Paper a weakly dominant strategy?
I −1 = ui (Paper, (Scissors,Scissors)) <
ui (Rock, (Scissors,Scissors)) = 1
I At some strategy profile, Paper is WORSE than Rock
I So Paper is not weakly dominant
I So none of the pure strategies are weakly dominant
I Is any mixed strategy weakly dominant?
37 / 131
Weakly Dominant Mixed Strategy in RPS
I Again, we do not know whether a mixed strategy is weakly
dominant, and if it is, then what it is

I First, let σ denote a mixed strategy such that
I σ(Rock) = prock
I Again, the expected utility from σ when other players play
(Rock,Rock) is ppaper −pscissors
I The utility from a pure strategy (Paper) is 1
I For σ to be weakly dominant, its expected payoff should not
be worse than 1, i.e.,
ppaper −pscissors ≥ 1
I What should ppaper and pscissors be so that this inequality
holds?
I ppaper = 1 and pscissors = 0, then the inequality holds (they
are
in fact equal)
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Weakly Dominant Mixed Strategy in RPS
I So for σ to be weakly dominant strategy, ppaper = 1 and
pscissors = 0
I What does it imply about prock ?
I prock = 0 because...
I However, then what has become of σ?

I σ(Rock) = 0
I σ(Paper) = 1
I σ(Scissors) = 0
I This is just a pure strategy of playing Paper with probability 1
I We have already seen that the pure strategy of Paper is not
weakly dominant because
−1 = ui (Paper, (Scissors,Scissors)) < ui (Rock,
(Scissors,Scissors)) = 1
I This means no mixed strategy is weakly dominant...
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Still No Prediction For RPS...What To Do?
I In the game of RPS, no player has
I strictly dominant pure strategies
I strictly dominant mixed strategies
I weakly dominant pure strategies
I weakly dominant mixed strategies
I Maybe the result of RPS is just not meant to be predicted, so
we should not be discouraged by our inability to predict the
results of RPS
I Let’s forget about RPS now; let’s play a game to refresh our
RPS-soaked mind!
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Let’s Play a Game: Beauty Contest
I Players: all of you
I The rules of the game
I Choose an integer among 0, 1, 2, . . . , 100
I No one is able to observe others’ numbers before she can
choose hers
I Outcome: Whoever chooses the number closest to the 1
3
of
the average of all numbers chosen by all of you wins; if there
are multiple students who chose the winning number, the
winner will be chosen randomly among them
I Payoffs: The winner gets 20 utils
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43 / 131

Let’s Talk About How You Chose Your Numbers
I Anyone want to talk about how they chose their numbers?
44 / 131
Beauty Contest
I Let’s think about the game
I The rule is to ”guess” the 1
3
of the average of the number
that everyone chose
I What is the largest the average can be? When can that
happen?
I When everyone chooses 100, then the average is...
I If the average is 100, then 1
3
× (Average) = 33.33
I If you have thought this much, what should you conclude?
I The largest the target (that is, 1
3
×(Average)) can be is 33.333
I Therefore, I should not choose a number that is greater than
or equal to 34 (because the closest integer to 33.333. is 33)

45 / 131
Recap of the Lecture on Feb 2nd.
I We learned what a strictly dominant pure strategy is
I A pure strategy of a player strictly dominates another pure
strategy of hers if the payoff from the former is greater than
that from the latter regardless of which strategy profile other
players use
I Let’s try writing this down with notation on your own
I ”a player”: i
I ”A pure strategy of a player”: si
I The set of all strategies of the player: Si
I ”Another pure strategy of” the player: s′i
I The payoff function for the player: ui
I A profile of strategies of other players?
I s−i
I The set of all possible profiles of pure strategies of other
players?
I S−i
46 / 131
Recap of the Lecture on Feb. 2nd
I A pure strategy si strictly dominates another pure strategy s
′

i
if
′
i ,s−i ) for all s−i ∈ S−i
I A pure strategy is strictly dominated if there is another
strategy (mixed or pure) that strictly dominates it
I A pure strategy is strictly dominant if it strictly dominates all
the other strategies (mixed or pure)
I Our prediction about games is that whenever there exists a
strictly dominant strategy for a player in a game, she will play
it
I In Prisoner’s Dillema, our prediction works beautifully
because
there is a strictly dominant strategy for each player, so we
predict a unique outcome of (Defect,Defect)
47 / 131
I But why our prediction is not enough? Why should we
continue studying game theory?
I Because not every game has strictly dominant strategies
I An example of such game is Rock Paper Scissors

I We verified that there are no strictly dominant pure strategies
or strictly dominant mixed strategies
I After failing to predict anything based on strictly dominance
in RPS, we lowered the bar and looked for weakly dominant
strategies
48 / 131
I A pure strategy weakly dominates another pure strategy if the
payoff from the former is not smaller than that from the latter
regardless of the strategy profile of others and greater at some
strategy profile of others
I Again, let’s try stating this definition with notation on your
own
I ”a player”: i
I ”A pure strategy of a player”: si
I The set of all strategies of the player: Si
I ”Another pure strategy of” the player: s′i
I The payoff function for the player: ui
I A profile of strategies of other players: s−i
I The set of all possible profiles of pure strategies of other
players: S−i
49 / 131

I A pure strategy si weakly dominates another pure strategy s
′
i if
ui (si,s−i ) ≥ ui (s′i ,s−i ) for all s−i ∈ S−i AND
′
i ,s−i ) for some s−i ∈ S−i
I A pure strategy is weakly dominated if there is another
strategy (pure or mixed) that weakly dominates it
I A pure strategy is weakly dominant if it weakly dominates all
the other strategies (pure or mixed)
I However, even though we lowered the bar, there is no weakly
dominant strategy in RPS, either pure or mixed
I We gave up on RPS and moved on to Beauty Contest
50 / 131
I Beauty Contest
I Players: all of you
I Choose an integer among 0, 1, 2, . . . , 100
I No one is able to observe others’ numbers before she can
choose hers

I Outcome: Whoever chooses the number closest to the 1
3
of
the average of all numbers chosen by all of you wins; if there
are multiple students who chose the winning number, the
winner will be chosen randomly among them
I Payoffs: The winner gets 20 utils
51 / 131
Beauty Contest
I What is the set of the possible strategy profiles?
I There are 48 of you
I A strategy profile
(s1,s2, . . . ,s48)
I s1 can be any number between 0 and 100
I . . .
I So there can be 10148 possible strategy profiles (number of
stars in galaxy≈400)

I A player i can calculate the payoffs for herself for each of
these strategy profiles (assuming she understood the rule)
ui (s1,s2, . . . ,s48)
52 / 131
Beauty Contest
I If player i calculated her payoffs for each of the possible
strategy profiles, she will notice something
I ”Wait a minute, payoff for when I choose 33 is always greater
than or equal to when I choose a number 34 or larger!!”
I That is, 33 ( ) dominates 34 or higher
I But calculating payoffs for such a large number of strategy
profile is impossible
I So we discussed the following logic that seems like true
1. ”The largest the average can be is 100 when everyone
chooses
100”
2. If the average is 100, then 1
3
× (Average) or your target, is
33.33
3. So the target will always be smaller than 33.33
4. 33 is always closer to numbers smaller than 33.33 than 34 or

any other numbers, and my payoff when I choose 33 should be
greater than or equal to 34 or higher
I Let’s think carefully about this logic
53 / 131
Beauty Contest
I We are claiming that if you choose 33, then it is always closer
to the target than when you choose 34 or larger numbers
I But whether this is correct or not is not very obvious, because
the target depends on the number you choose as well
I But this is true; it will be a part of your pset 3 to verify this
I So 33 is always closer to the target
I This means that whenever you win with number 34 or larger,
you win with 33 as well
I But this does not make 33 weakly dominate all the numbers
34 or larger
I What else do you need for 33 to weakly dominate 34?
I 33 weakly dominates 34 if
1. ui (33,s−i ) ≥ ui (34,s−i ) for all s−i ∈ S−i ; and
2. ui (33,s−i ) > ui (34,s−i ) for some s−i ∈ S−i
I The payoff is either 20 (win) or 0 (lose); we have just shown
that whenever ui (34,s−i ) = 20, ui (33,s−i ) = 20 as well
I But we have not shown the second part of the definition
54 / 131

Beauty Contest
I We need to show that there is some profile of strategies
played by other players at which
1. 33 wins; but
2. 34 does not;
I We only need to find one such strategy profile
I 48 of you; 46 chooses 100; 1 choose 33
I If you choose 33, then target is 32.402, and you have 50%
chance of winning
I Gets expected payoff of...
I If you choose 34, then target is 32.409, and you lose
I So 33 gives higher payoffs than 34 at some strategy profile of
others, therefore weakly dominates 33 and by similar logic, all
numbers larger than 34
55 / 131
Beauty Contest
I Now we know that 33 weakly dominates 34 and larger
numbers
I So if you want to maximize your payoff, or in other words if
you are rational, you should not choose a number ≥ 34
I But even after ruling out weakly dominated strategies, you

still have 33 options to choose from
I What about everyone else other than you?
I Let’s assume that
I Everyone else understands the game and calculated payoffs for
each strategy profile
I Everyone else is rational, i.e., wants to maximize his/her
payoff
I If we are willing to assume these two things, then what can
we conclude?
I That is, everyone else also finds out that 34 or larger numbers
are weakly dominated by 33
I Therefore, they will not choose 34 or larger
I Now this is getting interesting!
56 / 131
Beauty Contest
I Again, if we assume that
payoff
I Then we can conclude that everyone (you and everyone else)
will choose a number 33 or less

I That is, with the following four assumptions
1. Your rationality
2. Everyone else’s rationality
3. You understand the game and calculated payoffs for each
strategy profile
4. Everyone else understands the game and calculated payoffs
for
I We can make some predictions: that everyone will choose a
number between 1 and 33
57 / 131
Some Notes About Assumption
I We already made these assumptions (although not explicitly)
when we predict that in a game with strictly dominant
strategy, players will play it
I Because
I You need to understand the game and calculate the payoffs for
each strategy profile to figure out which is strictly dominant
strategy
I You actually choose to play the strictly dominant strategy if
you are rational (i.e. you maximize your payoff)
I But as we have seen, that did not get us too far in terms of
predicting the outcome of RPS

I Here we did a little better than RPS, but still have a long way
to go
I With a stronger assumption (that everyone is rational), we
might be able to make a sharper prediction
58 / 131
More Assumptions
I Are we happy with the prediction?
I It is certainly better than our prediction in the RPS, where we
were not able to produce any predictions
I But still, according to our prediction, the possible strategy
profiles that can occur as the outcome of the game is too
many
I We predict that everyone will choose a number between 1 and
33
I There are 48 of you
I So the number of possible outcomes of the game is 3348 (still
larger than the number of stars in the galaxy)
I Maybe if we are willing to make more assumptions, it might
give us a sharper prediction
59 / 131

Keeping Track of the Assumptions
I Here are the assumptions we have made so far
strategy profile
2. You are rational, want to maximize your payoff
for
4. Everyone else is rational, i.e., wants to maximize his/her
payoff
I Now let’s assume that
5. You know that everyone else understands the game and
calculated payoffs for each strategy profile
6. You know that everyone else is rational, i.e., wants to
maximize his/her payoff
60 / 131
Implication of Additional Assumptions
I If we assume that
5. YOU KNOW that everyone else understands the game and
6. YOU KNOW that everyone else is rational, i.e., wants to

I Recall that
payoff
I means that everyone else will choose a number less than or
equal to 33
I This means that the additional assumption, you KNOW that
everyone else will choose a number less than or equal to 33
61 / 131
Implication of Additional Assumptions
I Now that you know that everyone else will choose a number
less than or equal to 33, what should you conclude?
I We assumed that
strategy profile
I This means that you again can figure out that 11 will be
closer to the target than 12 or larger
I And since we assumed that
2. You are rational

I You will not choose a weakly dominated strategy and choose
a number between 1 and 11
62 / 131
Additional Assumptions for Everyone Else
I What about everyone else?
I Let’s make the same additional assumptions about everyone
else
7. EVERYONE ELSE KNOWS that everyone else understands
the game and calculated payoffs for each strategy profile
8. EVERYONE ELSE KNOWS that everyone else is rational,
i.e.,
wants to maximize his/her payoff
I Then everyone else KNOWS that everyone else will choose a
I This means that, because we assumed that
for
4. Everyone else is rational, i.e., wants to maximize his/her
payoff,
I Everyone else will also know that 11 weakly dominates 12 and
larger

63 / 131
Even More Assumptions?
I FANTASTIC!!! By adding the following assumptions
5. YOU KNOW that everyone else understands the game and
6. YOU KNOW that everyone else is rational, i.e., wants to
7. EVERYONE ELSE KNOWS that everyone else understands
the game and calculated payoffs for each strategy profile
8. EVERYONE ELSE KNOWS that everyone else is rational,
i.e.,
wants to maximize his/her payoff
I We sharpend our prediction; now everyone will choose a
I The number of possible strategy profiles that can occur under
such prediction is now 1148, still larger than the number of
stars in the galaxy
I Why stop now with assumptions? Let’s make more to sharpen
our prediction
I Can you guess what assumptions we want to make?
64 / 131

Beauty Contest
I To see what’s going on more clearly, let’s make this reasoning
more visual; and to make things simple, let’s consider two
players A and B
65 / 131
Beauty Contest
66 / 131
Beauty Contest
67 / 131
Beauty Contest
68 / 131
Beauty Contest
69 / 131
Beauty Contest

70 / 131
Beauty Contest
71 / 131
Beauty Contest
72 / 131
Beauty Contest
73 / 131
Beauty Contest
74 / 131
Beauty Contest
75 / 131
Beauty Contest

76 / 131
Unique Prediction!!!!
I We finally reached a unique prediction!!!!!
I Everyone will choose 0
I But it comes at a price; we made many assumptions. So our
prediction is not without qualitifiction
I If our assumptions are correct, then everyone will choose 0
I Let’s look at the results
77 / 131
Results of the Game
78 / 131
Results of the Game
I The prediction was so grossly wrong
I It seems like most of you figured out that you should not
choose a number greater than 33
I Out of 31 people, only 4 people (13%) chose a number greater
than 33

I However, 16 people (52%) chose a number greater than 11,
which you would not have chosen
1. if you are rational and understands the game
2. if you knew that everyone else is rational and understands the
game
I 27 people (87%) choose a number greater than 4, which you
would not have chosen
1. if you are rational and understands the game
2. if you knew that everyone else is rational and understands the
game
3. if you knew that everyone else knew that everyone else is
rational and understands the game
79 / 131
Why Was The Prediction So Wrong?
I Our assumption might have been wrong
I It is usually a common assumption that we make in game
theory
I Such assumption is called ”common knowledge”
I In a game of two players A and B, if
1. all players know X
2. A knows that B knows X; B knows that A knows X
3. A knows that B knows that A knows X; B knows that A

knows that B knows X
4. . . .
5. repeat this infinitely
I Then X is said to be common knowledge
80 / 131
Common Knowledge
I In the Beauty Contest we analyzed above, there were two
things that we assumed to be (sort of) common knowledge
1. That everyone is rational (i.e., maximizes one’s payoffs)
2. That everyone understands the game and so is able to
calculate payoffs for each possible strategy profile
I This was (sort of) common knowledge because we did not
need these to be common knowledge for us to reach at a
unique prediction
I But one can show that instead of choosing integers, one were
to choose real numbers, we need them to be common
knowledge in order to conclude that everyone will choose 0
I At any rate, assuming
1. rationality of everyone
2. that everyone understands the structure of the game
I is common in game theory, so we will make the same
assumptions from now on

81 / 131
Iterated Deletion of Weakly Dominated Strategies
I In the Beauty Contest, essentially what we did to make the
unique prediction is
1. Step 1. assume that the rationality of every player and that
everyone understands the structure of the game a common
knowledge
2. Step 2. Eliminate weakly dominated strategies for each player
3. Step 3. Further eliminate weakly dominated strategies
assuming that none of the players play strategies eliminated in
Step 2
4. Step k. Further eliminate weakly dominated strategies
assuming that none of the players play strategies eliminated in
Step 1, 2, . . . ,k − 1
5. Stop when there is no weakly dominated strategies
I We call this iterated deletion of weakly dominated strategies
82 / 131
applying iterated deletion of weakly dominated strategies
to rps
I now we have learned another way to make a prediction in a

game, let’s apply this to rps with three players
I is there any pure strategy that is weakly dominated by other
pure strategy?
I we know that there is no weakly dominant strategy in rps; but
there may be weakly dominated strategies
I is rock weakly dominated?
I paper does not weakly dominate rock because...
I ui (rock, (scissors,scissors)) > ui (paper, (scissors,scissors))
I scissors do not weakly dominate rock because...
I ui (rock, (scissors,scissors)) > ui (scissors, (scissors,scissors))
I by the same reasoning, none of the pure strategies are weakly
dominated
I so still there is nothing we can do with rps
83 / 131
Recap of the Lecture on Feb. 4th
I We learned what it means for something to be common
knowledge
I When we discussed strictly/weakly dominated dominant
strategy, we did not need to assume anything is common
knowledge
I A player i only needs to know HIS OWN PAYOFFS for each
strategy profile to figure out what is his strictly dominant

strategy
84 / 131
I Example: Prisoner’s Dillema
I What does row player need to know to find his strictly
dominant strategy?
1. urow (D,C ) > urow (C,C )
2. urow (D,D) > urow (D,C )
I She did not need to know anything about
ucolumn(C,C ),ucolumn(C,D),ucolumn(D,C ),ucolumn(D,D)
85 / 131
I By definition, si is a strictly dominant strategy for player i if
′
′
′
i 6= si

for all s−i ∈ S−i
I So for player i to check whether a strategy si is strictly
dominant or not, he only needs to know
I ui (si,s−i ) for all s−i ∈ S−i
I ui (s
′
′
−i ∈ Si such that s
′
i 6= si and s−i ∈ S−i
I Player i does not need to know anything about other players’
payoff functions to find a strictly dominant strategy
I In predicting that players will play their strictly dominant
strategy in games that have them, we are implicitly assuming
that
1. Players know their own payoff functions
2. Players are rational (i.e., want to maximize their payoffs)
86 / 131
I However, such assumptions were not enough for us to produce
meaningful predictions in games without strictly dominant

strategy (for example, Rock Paper Scissors)
I In Beauty Contest, a rational player i who knows her own
payoff function
I ui (0,s−i ),ui (1,s−i ), . . . ,ui (99,s−i ),ui (100,s−i ) for all s−i
∈
Si
will not choose numbers greater than 33 because they are ...
I Without further assumptions, we could not sharpen our
predictions that everyone will choose a number between 0 and
33
I So we started making more assumptions
87 / 131
I First set of assumptions that we made were
3. Player i knows player -i’s payoff function and that player -i
knows his payoff function
4. Player i knows that player -i is rational
I With these assumptions what does player i know?
I Player -i will only choose a number between 0 and 33
I Then player i will not choose a number greater than 11
I If we make the same assumptions about player -i

5. Player -i knows player i’s payoff function and that player i
knows his payoff function
6. Player -i knows that player i is rational
I Then player -i will not choose a number greater than 11 either
88 / 131
I With these additional assumptions we can predict that all
players will choose a number between 0 and and 11
I We continue making more assumptions...
5. Player i knows that player −i knows
5.1 player i’s payoff function;
5.2 that player i knows his payoff function; and
6. Player i knows that player −i knows that player i is rational
I Then eventually we reach the conclusion that all players will
choose 0
I In general, X is said to be common knowledge if
I Every player knows X
I Every player knows that everyone else knows X
I Every player knows that everyone else knows that everyone
knows X ...
89 / 131

I In game theory, we usually assume that
1. Payoff function for each player
2. Each player’s rationality (i.e., each player maximizes her
expected payoff)
are common knowledge
I What we did to reach a unique prediction in Beauty Contest is
referred to as Iterated Deletion of Weakly Dominated
Strategies
1. Make the above common knowledge assumption
2. Delete weakly dominated strategies for each player (in
Beauty
Contest, numbers from 34 to 100)
3. Delete weakly dominated strategies for each player when
other
players do not play already deleted strategies (in Beauty
Contest, numbers from 12 to 33)
4. Continue this way until there is no weakly dominated strategy
to delete
90 / 131
Clarification/Correction About Strictly Dominance

I A strategy strictly dominates another if the payoff from the
former is greater than the latter regardless of the strategy
profile of others
I When we restrict players’ strategy sets so that players are
allowed to play only pure strategies and no mixed strategies,
then
1. a pure strategy si is strictly dominant if it strictly dominates
all
the other pure strategies
2. a pure strategy si is strictly dominated if there is another pure
strategy that strictly dominates si
I But if we let players play mixed strategies as well, then
1. a pure strategy si is strictly dominant if it strictly dominates
all
the other strategies, pure or mixed
2. a pure strategy si is strictly dominated if there is a strategy
(mixed or pure) that strictly dominates it
91 / 131
Clarification/Correction About Strictly Dominance
Player 2
L R
Player 1
U 10,1 0,4

M 4,2 4,3
D 0,5 10,2
I If player 1 can play pure strategies only, is M strictly
dominated?
I Now suppose that player 1 can play mixed strategies
I Consider a mixed strategy σ(U) = 1
2
and σ(D) = 1
2
I What is the expected payoffs from σ when player 2 plays L?
I What is the expected payoffs from σ when player 2 plays R?
I σ strictly dominates M
92 / 131
Let’s Play Another Game: Cournot Competition
I The players: two players; both of you are producing
homogeneous goods
I The set of possible quantity = 0 ≤ q ≤ 4
I Choose quantity ≥ 0 to produce simultaneously
I No collusion!
I The outcome: the market price is determined according to the
market demand function p = 8 − (q1 + q2)
I The payoff: the profit function is (price) × (quantity

produced) (marginal cost is 0)
93 / 131
Let’s Play Another Game: Cournot Competition
I Write down your quantity
94 / 131
Cournot Competition
I We want to make a prediction about the outcome of this game
I Suppose you are firm 1; all you can choose is q1
I If q2 is the quantity produced by firm 2, what is firm 1’s
profit?
(8 − (q1 + q2)) q1
I FOC
q1 = 4 −
q2
2
I Whatever q2 is, if firm 1 knew what q2 is, then firm 1 would
want to produce q1 = 4 − q22
95 / 131

Cournot Competition
96 / 131
Cournot Competition
I Similarly, if firm 2 knew q1, then firm 2 would want to
produce q2 = 4 − q12
I By common knowledge assumption, firm 1 knows that firm 2
is rational
I Therefore, firm 1 knows that firm 2 will produce quantity
equal to 4 − q1
2
for some expectation q1
I Firm 1 may not know what firm 2 will expect q1 to be, but
firm 1 knows that firm 2 knows that q1 ≥ 0 (according to the
rule of the game), so firm 1 knows that q2 = 4 − q12 ≤ 4
97 / 131
Cournot Competition
I When firm 2 produces 4, then firm 1 wants to produce 2
I Therefore, firm 1 will produce a quantity greater than 2

98 / 131
Cournot Competition
I Similarly, firm 2 knows that firm 1 is rational
I Therefore, firm 2 knows that firm 1 will produce a quantity
less than 4
I Therefore, firm 2 will produce a quantity greater than 2 as
well
99 / 131
Cournot Competition
I By common knowledge assumption, firm 1 knows that firm 2
knows that firm 1 is rational
I Therefore, firm 1 knows that firm 2 will produce a quantity
greater than 2
I When firm 2 produces 2, then firm 1 wants to produce 3
I Therefore, firm 1 will produce a quantity greater than 2 and
less than 3
100 / 131
Cournot Competition
I The same is true for firm 2; firm 2 will produce a quantity
greater than 2 and less than 3

I We can continue this forever, because the rationality of each
player is common knowledge, so
I Firms are rational
I Firm i knows firm -i is rational
I Firm i knows firm -i knows that firm i is rational
I . . .
I At each step we delete some quantity level for each firm
I What is the quantity that remains after all the deletions?
101 / 131
Iterated Deletions in Cournot Competition
102 / 131
103 / 131
104 / 131

105 / 131
106 / 131
107 / 131
Iterated Deletion in Cournot Competition
I It seems like the interval of the quantities that survive
deletions get narrower and narrower around the intersection of
the two lines
I The intersection is at q1 = q2 =
8
3
I Anybody got the quantity 8
3
?
108 / 131

Comparison between Cournot Outcome and Monopoly
Outcome
I A monopolist facing the market demand function 8 −q = p
maximizes its profit
(8 −q)q
I The FOC
−2q + 8 = 0
I The monopolist’s quantity produced is 4
I The price under monopoly is 4
I With two firms in the market, the price falls to 8
3
≈ 2.67
I Marginal cost is 0; in a competitive industry, the price would
be?
I Under duopoly the price is still closer to the monopoly level
than competitive level
109 / 131
But Firms Usually Choose Price Rather Than Quantity
I In Cournot competition, we let the firms choose quantity, not
the price
I But it might be more natural to let firms choose the price of

its own product and sell at the chosen price; the quantity sold
will be determined by the market demand function
I The high profit margin in the Cournot Competition might
have something to do with the fact that they are choosing
quantities, not price
I So let’s
110 / 131
Let’s Play Another Game: Bertrand Competition
I The players: two players (your classmates next to you); both
of you are producing homogeneous goods
I The set of possible prices = {0, 1, . . . , 8}
I Choose the price of your product simultaneously
I No collusion!
I The outcome
I The one who chooses the lower price serves the entire demand
of the market 8 −p
I If choose the same price, then split the demand of the
marketk; serve 8−p
2
I The payoff: the profit function is (price) × (quantity sold)
(marginal cost is 0)
111 / 131

Let’s Play Another Game: Bertrand Competition
I We could represent this game in a normal form (i.e., 81 by 81
table)
I That would be time consuming; there might be an easier way
I What is the largest profit a firm can get from this market?
I If there is only one firm, i.e. monopoly, then the maximized
profit is
maximize p(8 −p) when p ∈ {0, 1, . . . , 8}
I The FOC
−2p + 8 = 0 ⇒ p = 4
I So the largest profit a firm can make in this market is 16,
when the price is 4
I When the other firm is charging a price 5, 6, 7, or 8, which
price would you rather charge, 4 or 5 or 6 or 7 or 8?
I 4, because...
112 / 131
Bertrand Competition
I When the other firm is charging a price 1, 2, 3, what is your
payoff from charging 4, 5, 6, 7, or 8?

I 0, because...
I When the other firm is charging a price 4, what is your payoff
from charging 4, 5, 6, 7, or 8?
I When you charge 4, then your payoff will be 8 because...
I When you charge 5 or higher, then your payoff will be 0
because...
I We have just shown that charging the price of 4 ( ) 5, 6, 7, or
8
I WEAKLY DOMINATES!!!
I So we have deleted 5, 6, 7, 8; is there any other we can
delete?
113 / 131
I If you charge 0, your profit is always 0, even if you serve the
entire market
I When the other firm charges a price 1, 2, . . ., 8, which would
you rather charge, 0 or 1? Why?
I When the other firm charges 0, your payoff from charging 1
is...
I When the other firm charges 0, your payoff from charging 0
is...

I We have just shown that charging the price of 1 ( ) 0
I WEAKLY DOMINATES!!!
I So we have deleted 0 as well; now we have 1, 2, 3, 4 for each
firm
I Not obvious if there is any dominance relationship; let’s
represent this reduced game in a normal form
114 / 131
Player 2
1 2 3 4
Player 1
1 3.5,3.5 7,0 7,0 7,0
2 0,7 6,6 12,0 12,0
3 0,7 0,12 7.5,7.5 15,0
4 0,7 0,12 0,15 8,8
I Look at price 3 and 4; is there any dominance relationship
between 3 and 4?
I 3 weakly dominates 4!
I Let’s delete 4
115 / 131

Player 2
1 2 3
Player 1
1 3.5,3.5 7,0 7,0
2 0,7 6,6 12,0
3 0,7 0,12 7.5,7.5
I Look at price 2 and 3; is there any dominance relationship
between 2 and 3?
I 2 weakly dominates 3!
I Let’s delete 3
116 / 131
Player 2
1 2
Player 1
1 3.5,3.5 7,0
2 0,7 6,6
I Is there any dominance relationship between 1 and 2?
I 1 strictly dominates 2!
I Delete 2 and we have only 1 left
I The iterated deletion of weakly dominated strategies results in

each firm charging price of 1, the smallest price over marginal
cost (of zero)
I This is good because
I We worry about monopoly and the welfare loss from it
because
monopolistc firm prices over marginal cost
I We only need one more firm to return to (close to)
competitive price (marginal cost ≈ price)
117 / 131
Results of the Game
I How many of you chose (1,1)?
118 / 131
Summary of What We Learned So Far
I We are interested in predicting the outcome of a game
1. ”Players will play strictly/weakly dominant strategy”
I Only need to assume that each player knows her payoff
functions and is rational
I Drawback?
2. ”Players will iteratively delete dominated strategies”
I Assumed that all players’ payoff functions/rationality is

common knowledge
119 / 131
Iterated Deletion is not Panacea
I So far we have seen three games for which the iterated
deletion of strictly/weakly dominated strategies produced a
unique prediction
1. Beauty contest
2. Cournot competition
3. Bertrand competition
I But for some games iterated deletion may not work as well
I Rock paper scissors...
I Now we move on to the next step, which involves sharpening
our predictions further by making additional assumptions
120 / 131
Let’s Play a Game: Battle of the Sexes
I Players: two players
I Rules of the game
I Choose a venue for a date simultaneously, concert or play
I Outcome: each player will go to the venue they choose
I Payoffs: if both players end up at the same place, the player

who ended up at the place she prefers less gets 2 utils and the
other gets 3. If the two players end up at a different place,
both get zero util
121 / 131
Battle of the Sexes
I Is there a strictly dominant strategy?
I Is there a weakly dominant strategy?
I Is there a strictly/weakly dominated strategy?
122 / 131
Let’s Play a Game: Battle of the Sexes
123 / 131
Best Response
I To faciliate the dicussion to follow, we introduce a definition
I A mixed strategy σi is a best response for player i to the
strategy profile of others σ−i if the payoff from σi is greater
than or equal to those from other strategies when others are
playing σ−i
I With notation: σi is a best response for player i to σ−i if

ui (σi,σ−i ) ≥ ui (σ′i,σ−i ) for all σ
′
i ∈ ∆(Si )
I Please do not confuse best response with weakly dominant
strategy: a strategy σi is a weakly dominant strategy for
player i if the payoff from σi is greater than or equal to those
from each of other strategies regardless of what others are
doing, and greater at some strategy profile of others
I What are the two differences?
1. Best response: for one particular σ−i ; weakly dominant
strategy: for all σ−i
2. Best response: ≥; weakly dominant stratey: ≥ AND
sometimes >
124 / 131
Best Responses in Battle of the Sexes
I What is player 1’s best response when player 2 goes to
concert?
I What is player 1’s best response when player 2 goes to play?
I Similarly, going to concert is player 2’s best response when
player 1 goes to concert
I At the outcome (concert,concert) and (play,play), both
players are playing best responses to each other
125 / 131

Let’s Play a Game: Battle of the Sexes Ver. 2
I Two players (a couple)
I Go to a concert or a play (without communicating to each
other)
I None of the strategies are dominated
I Both prefer to be at concert; the row player gains much more
from being at concert than the column player
126 / 131
Let’s Play a Game: Battle of the Sexes Ver. 2
127 / 131
I We played two more games where iterated deletion of
strictly/weakly dominated strategies result in a unique
prediction
1. Cournot competition
2. Bertrand competition
I In Cournot competition, two firms
I Identical products
I Choose positive quantity to produce simultaneously
I Market price p = 8 − (q1 + q2)

I Marginal cost = 0
I We made a prediction of the outcome of this game by iterated
deletion of strictly dominated strategies
128 / 131
I If firm 1 knows that firm 2 will produce q2, then the profit
maximizing q1 is 4 − q22
I From the plot we know that whatever q2 is, the profit
maximizing quantity for firm 1 will be less than or equal to 4
I But this does not mean quantity greater than 4 is strictly
dominated
129 / 131
I For the quantity greater than 4 to be strictly dominated, what
has to be true?
I There has to be a quantity less than or equal to 4 that yields
higher profit than q1 > 4 REGARDLESS OF what q2 is
I Such quantity; 4
I Let’s check 4 in fact strictly dominates any quantity greater
than 4

I First we are going to calculate the difference between
1. Firm 1’s profit when q1 = 4
2. Firm 1’s profit when q1 > 4
I And see how the difference changes as q1 increases from 4
130 / 131
I Firm 1’s profit when q1 = 4 (at some level of q2)
(8 − (4 + q2))4 = 16 − 4q2
I Firm 1’s profit when q1 > 4
(8 − (q1 + q2))q1 = −q21 + 8q1 −q1q2
I Difference D
D = (16 − 4q2) − (−q21 + 8q1 −q1q2)
= q21 − 8q1 + q1q2 − 4q2 + 16
I Now to see how D changes as q1 changes, what do we do?
I differentiate D with respect to q1
131 / 131

I ∂D
∂q1
= 2q1 − 8 + q2
I When q1 = 4,
∂D
∂q1
= q2
I Then ∂D
∂q1
≥ 0 when q1 = 4 (Why?)
I Then ∂D
∂q1
> 0 when q1 > 4 (Why?)
I ∂D
∂q1
= 2q1 − 8 + q2
I q2 ≥ 0 and 2q1 − 8 > 0 when q1 > 4
I And of course D = 0 when q1 = 4
I So if we were to plot D as a function of q1...
132 / 131

I So D > 0 for all q1 > 4
I Therefore, regardless of what q2 is, 4 strictly dominates q1 >
4
133 / 131
I So firm 1, just by being rational, will not produce q1 > 4
because they are strictly dominated
I In addition, we want to show that none of the quantities
q1 ≤ 4 are strictly dominated
I That is, q1 ≤ 4 survive the first round of deletion
I Take q1 = 3 for example. For q1 = 3 not to be strictly
dominated, what has to be true?
I There is some level of q2 at which 3 is the profit maximizing
quantity for firm 1
I Is there?
I When q2 = 2, q1 = 3 is the profit maximizing quantity for firm
1
I Similarly, for each q1 ≤ 4, we can find q2 at which q1 is profit
maximizing
I This means that q1 ≤ 4 is not strictly dominated
134 / 131

I So firm 1, just by being rational, will produce q1 ≤ 4
I Similarly, firm 2, just by being rational, will produce q2 ≤ 4
I We can continue to delete more quantities because we
decided to assume that rationality is...
I COMMON KNOWLEDGE
I Firm 1, knowing that firm 2 is rational, knows that q2 ≤ 4
135 / 131
I The profit maximizing quantity for firm 1 when q2 = 0 is 4
I The profit maximizing quantity for firm 1 when q2 = 4 is 2
I So it seems like firm 1 does not want to produce less than 2
I However, this does not mean that q1 < 2 is strictly dominated
I Showing this will be on problem set 3
136 / 131

I Similarly, firm 2, knowing that firm 1 is rational, will produce
2 ≤ q2 ≤ 4
I We can continue this forever...
I It turned out that the quantity for each firm that survives the
iterated deletion of strictly dominated strategies is 8
3
I Geometrically, 8
3
is the intersection of the two lines
137 / 131
I Cournot competition resulted in the market price of 8
3
I Monopoly price is 4
I We wondered what the price would be in a game where
I Two firms produce identical product
I They choose the ”price” of their respective products instead of
”quantity”
I So we played Bertrand Competition
I The resulting price in Bertran Competition is closer to the
competitive price than Cournot Competition

138 / 131
I We introduced a new definition, best response
I A mixed strategy σi is a best response for player i to the
strategy profile of others σ−i if the payoff from σi is greater
than or equal to those from other strategies when others are
playing σ−i
I With notation: σi is a best response for player i to σ−i if
ui (σi,σ−i ) ≥ ui (σ′i ,σ−i ) for all σ
′
i ∈ ∆(Si )
I Not to be confused with strict/weak dominance
139 / 131
Best Response Vs. Weak Dominance
Player 2
L C R
Player 1
U 0 0 1
M 1 0 0
D 0 0 0

I Which pure strategies weakly dominates C?
I What is player 2’s best response when player 1 plays
I U?
I M?
I D?
I C is a weakly dominated strategies; but C is a best response
to D
140 / 131
Best Response Vs. Weak Dominance
Player 2
L C R
Player 1
U 0 0 1
M 1 0.5 0
D 0 0.5 1
I Is there any strategy that weakly dominats C?
I What is player 2’s best response when player 1 plays
I U?
I M?
I D?
I C is not weakly dominated; but it is never a best response

I Strategies like C are called strategies that are never a best
response; regardless of what the other players do, they are
never a best response
141 / 131
Never-a-best-response Strategies and Weakly Dominated
Strategies
I Weakly dominated strategies may not be
never-a-best-response strategies
I Never-a-best-response strategies may not be weakly
dominated
I Of course, never-a-best-response strategies may be weakly
dominated strategies
Player 2
L R
Player 1
U 0 1
M 0 0.5
D 0 0.5
I Is L weakly dominated?
I Is L ever a best response?
142 / 131

Never-a-best-response Strategies and Weakly Dominated
Strategies
143 / 131
Never-a-best-response Strategies vs. Strictly Dominated
Strategies
I Are strategies that are never a best response strictly
dominated?
Player 2
L C R
Player 1
U 0 0 1
M 1 0 0
D 0 0 1
I Is C ever a best response?
I Is C strictly dominated?
144 / 131
Never-a-best-response Strategies vs. Strictly Dominated
Strategies
I Are strictly dominated strategies ever a best response?
I No

I By definition, a strategy is strictly dominated by another
strategy if the latter results in higher payoffs than the former
at all strategy profiles of others
I Therefore, if a strategy is strictly dominated, then there is
always a better response than it, regardless of what others do
I So strictly dominated strategies are never a best response
145 / 131
Never-a-best-response Strategies and Strictly Dominated
Strategies
146 / 131
Cournot Competition Revisited: Never-a-best-response vs.
Strictly Dominated strategies
I In Cournot Competition, we plotted firm 1’s profit maximizing
quantity when firm 2 produces a quantity q2
I In other words, we plotted firm 1’s BEST RESPONSES to
each level of q2
I We checked earlier that q1 > 4 was strictly dominated
strategies for firm 1
I q1 > 4 is NEVER a best response because...
1. We know that just by looking at the plot
2. Strictly dominated strategies are never a best response, by

definition
147 / 131
Iterative Deletion of Strategies That Are Never a Best
Response
I Similarly to strictly/weakly dominated strategies, strategies
that are never a best response are not very good strategies
I If a strategy si is never a best response, then it means that
whatever other players’ strategies are, there is always a better
strategy than si
I So, just as we deleted strictly/weakly dominated strategies,we
can iteratively delete strategies that are never a best response
I We can do this for Cournot Competition
148 / 131
Response In Cournot Competition
I Firm 2 produces q2 ≤ 4 because q2 > 4 is never a best
response
149 / 131

I Firm 1 produces 2 ≤ q1 ≤ 4 because other quantities are
never a best response if 0 ≤ q2 ≤ 4
150 / 131
I Firm 1 produces 2 ≤ q1 ≤ 3 because other quantities are
151 / 131
I Firm 1 produces 5
2
≤ q1 ≤ 3 because other quantities are
152 / 131
I We can continue this forever

I Again, there is only one quantity that survives all the deletion
of strategies that are never a best response
I 8
3
, the same as before
153 / 131
Iterated Deletion of Never-a-Best-Response Strategies in
the Battle of the Sexes
I Is there any Never-a-Best-Response Strategies?
I All strategies survive the iterated deletion of
Never-a-Best-Response Strategies
154 / 131
Battle of the Sexes
I Now players’ choice will depend on where they believe the
other player will go
I If player 1 believes that player 2 will go to concert, then
player
1 will go to the concert
I Player 1’s such belief is reasonable
I Player 1’s belief would be unreasonable if she beliefs that

player 2 will choose a never-a-best-response strategy
I But both of player 2’s strategies are a best response to some
of player 1’s strategies
155 / 131
Correctness of Beliefs
I Even though such belief might be reasonable, they might be
incorrect
I If players’ beliefs about what others will choose can be
incorrect, then any outcome is a possibility
I What are the outcomes when both players’ beliefs about the
other’s choice are correct
I (Concert,Concert): player 1 believes that player 2 will go to
the concert and vice versa (and they did)
I (Play,Play): player 1 believes that player 2 will go to the play
and vice versa (and they did)
I What are the outcomes when both players’ beliefs about the
other’s choice are incorrect
I (Concert,Play): player 1 believes that player 2 will go to the
concert (but he did not)
I (Play,Concert): player 1 believes that player 2 will go to the
play (but he did not)

156 / 131
Correctness of Beliefs
I If we are willing to assume that players’ beliefs have to be
correct, we might be able to sharpen our prediction
I Let’s not worry about how reasonable such assumption is; it is
tempting to assume the correctness of beliefs
157 / 131
Our Assumptions So Far
1. Each player is rational and each knows his/her own payoff
function
I Our prediction: ”players will not play strictly/weakly
dominant
strategies”
2. Rationality and players’ payoff functions are common
knowledge
I ”Players will only play strategies that survive iterated
deletions
of strictly/weakly dominated strategies or
never-a-best-response strategies”
3. Players have a correct belief about what strategies others will
choose

I ”The outcome of the game will be...”
158 / 131
Nash Equilibrium
I A strategy profile is said to be a Nash equilibrium if everyone
is best responding to other players at the strategy profile
I With notation (and for simplicity for now let’s assume that
players only play pure strategies): a strategy profile
(s1,s2, . . . ,sI ) is a Nash equilibrium if for all i,
ui (si,s−i ) ≥ ui (s′i ,s−i ) for all s
′
i ∈ Si
159 / 131
Nash Equilibria in the Battle of the Sexes
I What are Nash equilibria in pure strategies in the Battle of
the Sexes?
1. (concert,concert): Because when player 2 goes to the concert,
it is player 1’s best response to go to the concert as well
2. (play,play): Because when player 2 goes to the play, it is
player
1’s best response to go to the play as well
I So our prediction will be (under the assumption that players’

beliefs are correct) that the results of Battle of the Sexes are
either of the two Nash equilibria
I You played the game; so let’s look at how you did
160 / 131
Results of the Game
I The frequency of equilibrium outcomes vs. the rest
Round 1 35% vs. 64%
Round 2 54% vs. 46%
Round 3 63% vs. 36%
Round 4 45% vs. 54%
Round 5 54% vs. 37%
I Our prediction is not very accurate; even after five rounds of
playing the same game with the same player, your beliefs do
not seem to be correct
I This is a bit disappointing; let’s look at the result of the
version 2 and see if our new way of prediction (Nash
equilibria) worked better
161 / 131
Battle of the Sexes Version 2
I Again, none of the strategies can be deleted because they are
best response to some strategy of others
I Especially, going to the concert is not a dominant strategy for

anyone
I But (Concert,Concert) is a ”better” Nash equilibrium than
(Play,Play) because both players get higher payoffs
162 / 131
Results of the Game
I This time your beliefs were correct more often than before
I Some games with multiple equilibria might have an obvious
equilibrium to coordinate on
1. Two UBC students who agreed to meet with each other for
lunch; forgot to decide where
I It can be any restaurants/eateries on campus (each of which
can be a Nash equilibrium)...
I But there might be some obvious choices...(Places these
friends usually meet for lunch?)
2. Sidewalk is just wide enough for two people; you are walking
on
it and another is coming toward you; which side do you walk
on (both simultaneously decide; no one wants to collide)?
I Nash equilibria will be (left,right) and (right,left)
I But there are social conventions (walk on the right side)
163 / 131

Not Very Happy With Nash Equilibria/Corret Beliefs
Assumption?
I But not all games are this obvious
I Is it really reasonable to assume that players’ beliefs are
correct?
I In other words, is it reasonable to predict that the outcome of
games will be Nash equilibria?
I Very interesting question to ask
I I don’t think there is a definitive answer to this question yet
I Outside the scope of this class
164 / 131
Nash Equilibria of the Games We Have Played
I Now that we learned what Nash equilibria are, we can figure
out what it is in games we already played
I Games we have played so far
I Matching Pennies
I Rock-paper-scissors
I Hide-and-seek
I Bank run
I Prisoner’s Dillema
I Beauty Contest
I Cournot competition
I Bertrand competition
I Some games are easier to find equilibria than others

165 / 131
Nash Equilibria of Prisoner’s Dillema
I Our previous prediction of this game (when we made much
less assumptions): (D,D)
I We added assumption to sharpen our predictions; but with
less assumptions our prediction is already as sharp as it can be
(it’s unique)
I But we want to verify that Nash equilibria as a prediction
makes sense by
I checking if there is a Nash equilibrium
I if there is, whether it is unique
I if it is unique, whether it is (D,D) 166 / 131
Nash Equilibria of Prisoner’s Dillema
I In Nash equilibrium, both players have to best respond to the
other
I But there is only one ”best response” in this game (why?)
I D is strictly dominant, so the best response regardless of what
your opponent plays
I So both player has to play D is the only equilibrium
I So it exists, unique and equal to (D,D)

167 / 131
Nash Equilibria of Matching Pennies
I Row player wants to choose the same side; Column player
wants to choose the opposite side
I (H,H) a Nash equilibrium?
I No, at (H,H), column player is not best responding
I (H,T) a Nash equilibrium?
I No, at (H,T), row player is not best responding
168 / 131
Nash Equilibria of Matching Pennies
I (T,H) a Nash equilibrium?
I No, at (T,H), row player is not best responding
I (T,T) a Nash equilibrium?
I No, at (T,T), column player is not best responding
I No Nash equilibrium?
I No Nasn equilibrium where both players play a PURE strategy
I There might be a Nash equilibrium where both players play a
MIXED strategy
169 / 131

Nash Equilibria of Matching Pennies Ver. 2
Player 2
(H,H) (H,T) (T,H) (T,T)
Player 1
H -1,+1 -1,+1 +1, -1 +1,-1
T +1,-1 -1, +1 +1, -1 -1, +1
I Can you find Nash equilibria in this game?
I (H,(H,T)) and (T,(H,T))
I (H,T) is a weakly dominant strategy for player 2 (who
observes player 1’s move)
170 / 131
Nash Equilibrium in Rock Paper Scissors
I (R,R) a Nash equilibrium?
I (R,P) a Nash equilibrium?
I (R,S) a Nash equilibrium?
I Is there any Nash equilibrium where both players play a Pure
strategy?
I No
171 / 131

Problem 2.1.
Recall Matching Pennies version 2, where player 2 observes
what side of the
coin player 1 has chosen before he chooses.
(a) Player 1, sensing this is quite unfair for her, demand that
they revise the rule of the
game. Player 2 agreed, but he demanded that he still be able to
observe player 1’s
choice before he chooses. What can be changed about and/or
added to the game so that
the game is just as fair as when player 2 does not observe player
1’s choice? Anything
about the game can be changed except that
1) player 1 moves first then player 2 moves;
2) player 1 and 2 only may choose head or tail at each of his/her
information sets; and
3) that player 2 observe player 1’s choice. Represent the
modified game in extensive
form
(b) Represent the game in normal form. Is there a Nash
equilibrium where both players
play pure strategies?
Problem 2.2. Recall the Cournot competition discussed in class.
Show that when firm 1
knows that firm 2 is rational and that firm 2 knows his profit
function, 0 ≤ q1 < 2 is strictly
dominated for firm 1 by q1 = 2 and also show that 2 < q1 ≤ 4 is
not strictly dominated.
Problem 2.3. Represent the Hide-and-Seek game where there are
only two cups and two
players in normal form. If you hide a coin under a cup and seek
where you hid, you do not
get any payoff (unless your opponent happens to hide where you
seek (and hide) as well). Is
there any Nash equilibrium where both players play a pure
strategy?

Problem 2.4. Recall the Bertrand competition, but now players
can choose any price, not
just the integer price. The market demand function is p = 8− q
and only the firm that offers
lowest price gets to earn revenue. When both firms offer the
same price, say p, then they
each sell the quantity 8− 2 p at the price of p (so of course firms
will not want to charge a price
p > 8) For simplicity, let the marginal cost be equal to zero for
both firms. Show that
(a) both firms offering difference prices, both of which are
greater than 0, is not a Nash
equilibrium;
(b) both firms offering the same price higher than zero is not a
Nash equilibrium; and
(c) both firms offering the price of zero is a Nash equilibrium

Simultaneous GamesSam HwangFebruary 12, 20161 13.docx

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