Game Theory Lecture on Nash Equilibrium and Rationalizability

Name: Lecture 5: Nash Equilibrium
Uploaded: 2026-05-18T16:01:35+00:00
Duration: 1 h 19 min 26 s
Channel: MIT OpenCourseWare
Description: Summary and key takeaways on Lecture 5: Nash Equilibrium: Summary & Key Takeaways, covering Rationalizability Rationalizability assumes that each player is
MIT OpenCourseWare
May 18, 2026
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Rationalizability assumes that each player is rational and that rationality is common knowledge. It offers an explanation for observed behavior by iteratively eliminating strictly dominated strategies. The process can continue arbitrarily deep, providing a justification for any remaining strategy profile. Importantly, rationalizability does not require the beliefs players hold about one another to be correct; it only demands that those beliefs be internally consistent with rational choice.
Nash Equilibrium

A Nash equilibrium is a strategy profile in which every player’s chosen strategy is a best response to the strategies chosen by the others. This definition imposes consistency: each player’s belief about the opponents’ play must match the actual strategies employed. Because a Nash equilibrium predicts how rational players will act, it serves both as a recommendation and as a forecast of actual play. Note that an equilibrium is always a strategy profile, not a payoff value.
Relationship Between Solution Concepts

The hierarchy of solution concepts can be expressed as
Dominant‑Strategy Equilibrium ⊆ Nash Equilibrium ⊆ Rationalizable strategies (S∞).
Every dominant‑strategy equilibrium automatically satisfies the best‑response condition of a Nash equilibrium, and every Nash equilibrium survives the iterative elimination that defines rationalizability. Consequently, rationalizability is the least demanding requirement, Nash equilibrium is stricter, and dominant‑strategy equilibrium is the most restrictive.
Mixed‑Strategy Nash Equilibrium

When a game lacks a pure‑strategy equilibrium—such as the classic hide‑and‑seek scenario—players may randomize over their actions. A mixed‑strategy equilibrium exists only if each player is indifferent between the pure strategies they mix, meaning the expected payoffs from those strategies are equal. This indifference condition determines the probabilities each player assigns to their actions. Nash’s theorem guarantees that every finite strategic‑form game possesses at least one equilibrium, whether in pure or mixed strategies, a result proved using fixed‑point theorems (Brouwer or Kakutani).
Interpretations of Nash Equilibrium

Nash equilibria can be viewed as stable conventions, self‑enforcing agreements, or steady states of a strategic interaction. Because each player’s strategy is optimal given the others’, no unilateral deviation is profitable, making the equilibrium robust to individual incentives to change. This robustness underlies the wide applicability of Nash equilibrium across economics, biology, and social sciences.
Takeaways

Rationalizability provides an explanation for behavior based on common knowledge of rationality but does not require beliefs to be correct.
A Nash equilibrium is a strategy profile where each player's strategy is a best response to the others, ensuring consistency between beliefs and actual play.
Every dominant‑strategy equilibrium is also a Nash equilibrium, and every Nash equilibrium is rationalizable, establishing the inclusion DSE ⊆ NE ⊆ S∞.
Mixed‑strategy Nash equilibria arise when no pure‑strategy equilibrium exists, and players randomize only if they are indifferent between the pure strategies they mix.
Nash's theorem guarantees that every finite strategic‑form game possesses at least one equilibrium, pure or mixed, based on fixed‑point arguments.
Frequently Asked Questions

Why is rationalizability considered less demanding than Nash equilibrium?

Rationalizability is less demanding because it only requires that each player’s strategy be optimal given some belief about opponents, without requiring those beliefs to match actual play. Nash equilibrium adds the consistency condition that beliefs must be correct, making it a stricter requirement.
How does the indifference principle determine mixed‑strategy equilibria?

The indifference principle states that a player will mix over pure strategies only when the expected payoff from each mixed‑in strategy is equal. By setting the opponent’s mixed probabilities so that this equality holds, the player becomes indifferent, and the resulting probabilities constitute a mixed‑strategy Nash equilibrium.
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Helpful resources related to this video

If you want to practice or explore the concepts discussed in the video, these commonly used tools may help.
Game Theory Textbook For College Students Recommended
Provides comprehensive coverage of Nash equilibrium, rationalizability, and mixed strategies, reinforcing the lecture concepts with formal proofs and exercises.
Amazon →
A Beautiful Mind Biography John Nash
Offers historical context on John Nash and the development of the Nash equilibrium, humanizing the mathematical concepts discussed.
Amazon →
Graph Paper Notebook For Game Theory
Useful for drawing payoff matrices and mapping out strategy profiles to identify Nash equilibria visually.
Amazon →
Strategy Board Games For Logical Thinking
Provides a practical, physical environment to apply game theory concepts like mixed strategies and best-response analysis.
Amazon →
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Full Transcript YouTube

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IAN BALL: So today we're going
to talk about Nash equilibrium.
So this is one of the
most important concepts
in game theory.
But I want to start
by reminding you
a little bit about
rationalizability
because I think that's
going to provide us
a good foundation for talking
about Nash equilibrium,
seeing how these
two concepts differ.
So as before and
throughout this class,
we're going to be given
a strategic form game.
And this notation is
going to stay the same.
So we have a
strategic form game.
Maybe I'll write
strat-form game,
which specifies the set
of strategies, the n
players, and then the
payoffs that each player gets
as a function of the strategy
profiles that are chosen.
And last time, we
defined this procedure,
which we called iterated
elimination of strictly
dominated strategies.
And this was an
iterative algorithm
that gradually narrowed down
or pared down the strategies
that the players had.
So we started with, if you
remember, the first stage
or stage 0, the
initialization stage
was just the set
of all strategies,
all strategy profiles.
And then we went
to big S, S1, S2,
and we kept going down until
maybe we had a generic SK,
and then, at the end, we had
something called S infinity.
And it's important
to understand--
maybe I'll write this up here--
the interpretation of
each of these sets.
So below, let me say what
is the interpretation?
The interpretation
is that, as we're
willing to make stronger and
stronger behavioral assumptions
about the way that
the players play,
we're able to make sharper
and sharper predictions
about the strategies that we
could observe in practice.
So we start with S0.
This imposed no assumptions.
If we make no assumptions
about how people play, then,
of course, anything is possible.
The set of strategy profiles
that the players may choose
is unrestricted, but
then we move to S1,
where, there, the
interpretation is rationality.
What we said is, what if we
assume that every player is
rational in the sense that
every player plays a best
response to some belief
about the opposing player's
strategies?
And if we assume that
every player is rational,
then we can narrow
down our prediction
in the game to the set S1.
So capital S1 is the
set of strategy profiles
that are consistent with
every player's rationality.
And just to remember-- maybe
I'll put it down here--
when I write S capital K,
this is really a product.
Like this.
So S capital K is a set
of strategy profiles
that contain a strategy for
player 1, in this set S1 K,
all the way up to a strategy
for player n in the set SN K.
And then we could go
beyond just rationality.
Remember, we went
to S2, and there, we
assumed not only that
the players are rational
but that the players know that
everyone else is rational.
So we might call this
rationality plus knowledge
of rationality.
I'll just write knowledge
of R, so rationality
plus knowledge of rationality.
But then, as we
keep going down, we
can get to higher- and
higher-order knowledge
of rationality.
So when we get to SK, the
interpretation of this is SK is
our prediction in the game
if we're willing to assume
rationality, knowledge
of rationality,
knowledge of
others' rationality,
knowledge of others' knowledge
of others' rationality,
and so on, all the way down to
maybe I'll say rationality plus
"K minus 1 order"
knowledge of rationality.
Why K minus 1?
Well, the first stage,
we had rationality
but no knowledge of rationality,
so the degree of knowledge
of rationality is always just
1 smaller than our number K.
And when we get up to K, this
means the players are rational.
They know everyone
else is rational,
they know that everyone else
knows that they're rational,
and so on.
And those statements
go on until I say
the word knows k minus 1 times.
And then if we want
to be really formal,
there's a word
for this last set.
So this last set,
these are the set
of rationalizable strategies.
And the interpretation
of this is
that these are the
strategy profiles that
are consistent with
rationality and arbitrarily
high degrees of
knowledge of rationality.
So not only does each
player everyone else
knows their rational, but
I can continue those chains
arbitrarily deep.
And we have a word
for that-- that's
called common knowledge
of rationality.
So I'll write-- just write R. So
rationalizable captures the idea
that not only are
the players rational,
but they have common knowledge
of rationality in the sense that
they have K-th order knowledge
of rationality for any K.
They have first order,
second order, third order,
fourth order, 100th
order, and so on.
So this is kind of a summary
of where we've gone so far.
And I want you to notice how, as
we continue to the right here,
we make stronger and stronger
behavioral assumptions,
and we get sharper and
sharper predictions.
But sometimes, even
this set S infinity
is not going to give us
a very sharp prediction.
And we want to go
a step further.
And that's what Nash equilibrium
is going to do today.
So to motivate that,
let's start with
or revisit an example that
we've done before, which
we call the Boston game, BOS.
And remember, the
story of this game
is we had two friends who
like each other's company
but have different
sports preferences.
So they're choosing whether to
go to a Celtics game or a Red
Sox game.
They're both happier if
they go together than apart,
but one person prefers
the Celtics and one person
prefers the Red Sox.
So if we write out
the game like this--
I think I'll use slightly
different numbers this time
to make some of the
math later a bit easier.
So let's go over it.
If we choose differently,
that's along this off diagonal,
then we both get a payoff
of 0 because we don't
get to be together at the game.
If we both go to
the Celtics game,
then we both get
a positive payoff.
But it's higher for
player 1 because player 1
likes the Celtics more.
And then conversely, if we
both go to the Red Sox game,
we both get a positive payout.
But here, it's the second
player who prefers the Red Sox
and therefore, gets
the higher payoff of 2.
So let's go through
our procedure
for calculating
best responses here.
If my opponent is going
to the Celtics game,
I certainly want to go
to the Celtics game.
And if they're going
to the Red Sox game,
I want to go to
the Red Sox game.
And then we can do
it the other way.
Now, I'll put myself in
the shoes of player 2.
I'm saying, if player 1 plays
C, well, I look at my payoffs.
I prefer C. If player one
plays R, I look at my payoffs,
I prefer this.
So now, if we look
at that, can anyone
see, what are the set of
rationalizable strategies
or strategy profiles
in this game?
So maybe we'll separate
it out just to be clear.
So what are the set of
rationalizable strategies
for player 1 in this game?
Well, here, it's
going to be just C
And R because, here,
if my opponent plays C,
I want to play C. If my opponent
plays R, I want to play R.
So it turns out
everything is going
to be rationalizable
in this game.
So we're going to get--
if we go through
the procedure C, R,
and if we go through the
procedure, we also get C, R.
And just to make sure we
understand the notation,
S infinity is the set of
rationalizable profiles.
So this is going to
have four elements.
It's going to have
CC, CR, RC, and RR.
Sometimes I'll write
with parentheses,
but it just gets a
little messy here.
So this says there's
four strategy profiles.
This is the profile where we
both go to the Celtics game.
This is the profile where we
both go to the Red Sox game.
And the order here
indicates who does what.
So the first player plays C,
the second player plays R,
and then, conversely,
the first player plays R
and the second player plays C.
So in this case, everything
is rationalizable.
So this is a case where
rationality and common knowledge
of rationality alone doesn't
give us a very good prediction
or doesn't rule out
many possibilities.
So we're not able to get
a very sharp prediction.
And let's see, so in particular,
one rationalizable strategy,
profile, I should say, is this.
So this is rationalizable.
But it may seem like
something is going wrong here.
So does this seem like
a reasonable prediction
of how people are playing?
Or would you say someone's
making a mistake here?
What seems strange
about this-- when
player 1 is going to the
Red Sox and player 2 is
going to the Celtics game?
Yeah.
AUDIENCE: [INAUDIBLE]
since they're both losing.
IAN BALL: They're both losing.
And in a sense, well, I'm
going to the Red Sox game,
but if I just
changed my behavior
and went to the Celtics
game, I'd be better off.
And similarly, the other person
who's alone at the Celtics game,
if they move to the Red Sox
game, they'd be better off.
So there's a bit
of this incongruity
that we say, OK,
this is consistent
with rationality and common
knowledge of rationality,
but it might feel somehow
that the players are
making a mistake.
So maybe I'll say this
is all a bit informal.
It feels like a mistake.
And let's try to reconcile this.
Why does this satisfy the
definition of rationalizability
but feel like a mistake?
And I think the
best way to think
about that is, what is
rationalizability really mean?
It really means
that a player has
an explanation for why they're
doing what they're doing.
So I think
rationalizability should
be thought of as it means that
a player has an explanation.
So what is player
1's explanation
for why they're going
to the Red Sox game?
They would say, well, it's
reasonable, it's rational for me
to go to the Red
Sox game because I
believe that my friend is also
going to the Red Sox game.
And then we can
go a step deeper.
We could say, well, why do you
believe that your friend is
going to the Red Sox game?
And you could say,
well, I believe
my friend is going
to the Red Sox game
because I believe that my
friend believes that I'm
going to the Red Sox game.
And then you go a step deeper.
You can say, why do you believe
that your friend believes
that you're going
to the Red Sox game?
Well, I think my friend
believes that I believe
that my friend believes he's
going to the Red Sox game,
and that explains his belief,
and we can go on and on and on.
And that's precisely what
rationalizability captures.
It says you can provide
an explanation that
goes arbitrarily deep.
So think of this as
the answer to the child
who always says, but why this?
You can always go a step deeper.
Why do they have this belief?
Well, they have this belief
because of this other belief.
And that chain never stops.
If it weren't rationalizable,
eventually, we'd
reach a problem.
We'd reach a contradiction
in that chain.
So rationalizability says,
there exists some explanation
for your play.
The problem is your
explanation may not be correct.
In theory, it's a
good explanation,
but it might not actually
comport with the way
that people are playing.
So rationalizability means
you have an explanation.
You're behaving optimally,
given this explanation
or given these beliefs,
but there's no requirement
that the explanation is
correct, that your beliefs
or your explanation--
I'll use these
explanations a bit deeper.
Because if we're down here, my
explanation involved my belief
that my friend was also going to
the Red Sox game, but in fact,
my friend is not doing that.
My friend is going
to the Celtics game,
and that's why I'm doing badly.
So what Nash equilibrium
is going to capture
is it's going to go
a step beyond simply
providing an explanation.
It's going to require that
the explanations we provide
are also correct.
That is, they also
accurately describe the way
that our opponents or
that our fellow players
are actually playing.
So let's now give the
formal definition of this.
So if you learn one
thing about game theory,
this is often the
definition you see.
In the movie about Nash,
they talk about this.
Though, actually, their story
is kind of wrong in the movie.
It doesn't make
mathematical sense.
But anyway.
OK, so a strategy profile--
let me just stop
here and say, I think
this is really an important
aspect of the definition,
that the thing that a
Nash equilibrium will be
is a strategy profile.
And often, a mistake
students make
on exams is they're asked
to give an equilibrium,
and they often don't
give a strategy profile.
They might give the payoffs
that the players get if they
use that strategy profile.
But an equilibrium is
always a strategy profile
because it's a prediction
or recommendation for how
people will play in the game.
So this is the key first part.
It's a profile
specifying a strategy
for the first
player, all the way
up to a strategy
for the n-th player.
So this kind of strategy
profile is a Nash equilibrium.
So this is named after
John Nash who won the Nobel
Prize in economics.
He wrote this paper in '48, '49.
He wrote a few papers that
introduced this concept.
It's actually interesting
to read the original paper.
The original paper is
like two paragraphs long,
and he got the
Nobel Prize for it.
So if you have good
ideas, you don't
need to go on and on and on.
So a strategy profile
is a Nash equilibrium
if, for every player
i, the following holds.
So I'm going to put myself
in the position of player i,
and I'm going to compare the
payoff I get from my strategy
to the payoff I could
get if I deviated
to a different strategy.
So I'm going to compare ui of
si star on the left to ui of si
prime on the right.
So I'm comparing--
I'm player i, and I'm
saying if I choose
si star, what I'm
supposed to do in
equilibrium, I want
to do better weekly than if
I choose some other strategy
si prime.
But I still have
to fill this in.
And this is going to say, given
the way that the other players
are playing.
So what I'm going to fill
in here is s negative i star
and s negative i star.
I can't control how the
other people are playing.
So I take that as fixed.
I take that as given.
Given The way that my
opponents are playing,
I just do what I can control,
I compare the strategy
I'm supposed to choose
to any deviation,
and I require that the
strategy I'm supposed to choose
is weakly better for
all si prime in si.
In particular, si prime
could just be si star,
but then this is going to
be satisfied automatically
because both sides are
going to be the same.
So let's think of
another way to say this.
Another way to say this-- so
this is equivalent to this
statement here--
is that si star is a best
response to s negative i star.
So another way of defining
a Nash equilibrium
is to say that every
player I look at,
if I look at the strategy that
player's using, that strategy
better be a best response
to the strategies
that the other
players are using.
Now, we have to be
a bit careful here
because, when we first
defined a best response,
we defined a best
response to a belief,
not to a strategy
of the opponents.
But when I say a best response
to this strategy profile, what
I really mean, if I want
to be really formal,
is I mean to the belief
that assigns probability
1 to my opponents playing
this strategy profile, so to,
more formally, the
belief beta negative I
with beta negative i of s
negative i star equal 1.
So if my belief is my opponents
are certain to play s negative
i star, then the best
thing for me to do
or a best thing for me
to do is to play si star.
And here, you can immediately
see in the definition
the consistency that
we're imposing on beliefs.
So I think keeping in mind
this interpretation is crucial
because, when you first
see the definition,
you might say, where
are the beliefs?
Well, because we're requiring
consistency of beliefs,
the beliefs are the same as
what's actually happening.
So s negative i star is how
the other players are actually
playing.
And beta negative i
is my correct belief
that the players are playing
exactly how they are playing.
So sometimes we can lose track
of this distinction between play
and beliefs because
our definition
requires that they agree.
Maybe we can give a bit of a
less mathematical frame on this.
So another way we
might say it is we
might say that no player
has a strictly profitable
unilateral deviation.
So let's break down
what this means.
No player-- so let's
focus on a single player--
should have a strictly
profitable unilateral deviation.
So a deviation for a
player is a strategy
that deviates from or is
different from their conjectured
equilibrium strategy.
So we start with si
star for player i,
and we contemplate an
alternative strategy
for player i.
A unilateral deviation
means I'm the only one who's
changing my strategy.
And this is I think
the key, maybe
the most important idea
that Nash introduced
was that, when I'm reasoning
about other people,
I should take their behavior
and their strategies
as fixed because I can't
directly control what they do.
So when I contemplate
deviating, I only
consider a change
in my strategy.
And you can see that up
here, in this inequality,
that the way that my
opponents are playing
is the same on both
sides of the inequality.
The only difference
between the two sides
is how I'm playing as player i.
And this strategy cannot
be-- this deviation cannot be
strictly profitable.
It could be that you have
a deviation that gives you
the same payoff as your
equilibrium strategy,
but it can't be that you can
deviate and do strictly better.
And again, we're
using profitability
not in the sense of
a monetary payoff
but in the sense of utilities.
So a profitable
deviation is a deviation
that gives me a higher utility,
a strictly higher expected
utility.
So let's go back to
this game and see
if we can find the Nash
equilibria of this game.
Any guesses on what the Nash
equilibria of BOS this game
are going to be based
on the definition
or based on the
things we've written?
Yeah.
AUDIENCE: Both [INAUDIBLE]
IAN BALL: Right,
so these are going
to be the two Nash equilibria.
How can we see this?
Well, let's suppose we're both
going to the Celtics game.
Does anyone have a profitable
unilateral deviation?
No, because if I were
to unilaterally deviate,
we're together at the
Celtics game, if I deviate,
I'm by myself with
the other game,
so I'm certainly worse off.
And the same is true if
we go to the Red Sox game.
If we're both going
to the Red Sox game,
if I unilaterally deviate,
I'm going to be worse off.
Now, of course, if we're both
going to the Red Sox game
and I'm player 1, I would
much rather that we both
go to the Celtics game
because I'm a Celtics fan.
But moving from here to here
is not a unilateral deviation.
All I can control is
whether I go to the Red Sox
game or the Celtics game.
Of course, I would love to force
you to come to the Celtics game
with me, but that's not
something I can do in this game.
So that's the
intuitive explanation.
Mathematically, we can see that
both numbers are underlined.
And that's actually a great
way to compute Nash equilibria
in games like this.
So we go through for player 1.
We underline all of
their best responses.
Then we go through
player 2 and underline
all of their best responses.
And we look-- so I guess this is
an algorithm for finding them.
So step one is underline
best responses.
And it's crucial
that you underline
both players best responses and
underline all best responses.
So let me say all best
responses for both players.
And then step 2 is you
just check which cells--
or which boxes in the matrix
have both numbers underlined.
So look for cells--
what is a cell?
Well, a cell in this context
is just a strategy profile.
So let's remember that.
And if you think
through this carefully,
you'll see this is just another
way of saying the definition.
If both cells are
underlined-- well,
if the first player's
number is underlined,
that precisely means
player 1 is playing
a best response to
player 2's strategy.
And if player two's
number is underlined,
that means player 2 is playing
a best response to player 1's
strategy.
And that's exactly our
definition of a Nash
equilibrium, at least
in a two-player game.
So this is a good way going
through and finding and checking
for Nash equilibrium.
So now I think
it's helpful to try
to understand the relationship
between some of the solution
concepts and some of the
definitions we've given so far.
So so far, we have three, I
think, important definitions.
One was rationalizability.
And we represented the set
of rationalizable strategy
profiles.
We called that s infinity.
Then another important
notion is Nash Equilibrium.
And let's call
this NE, so let me
think of NE as the set of Nash
Equilibrium strategy profiles
in a game.
So just to be clear,
this is a subset of S.
What kind of objects are these?
These are strategy profiles.
So rationalizable
strategy profiles
live in the set of
all strategy profiles.
Then the set of Nash equilibria
that's also a subset of big S
because a Nash equilibrium
is also a strategy profile.
And then we had another
notion, remember,
which was Dominant-Strategy
Equilibrium.
And we'll call that DSE.
And that's also a subset
of S. So we have three sets
of strategy profiles here.
Let's remember a
dominant-strategy equilibrium,
if you recall from
the dominance lecture,
was a strategy profile where
each player's strategy is
a weakly dominant strategy.
That was the definition.
So now what we'd
like to understand
is the relationship
between them.
If something's a dominant
strategy equilibrium,
is it a Nash equilibrium?
If it's rationalizable,
is it a Nash equilibrium?
And so on.
I think a good way
to get started,
when you're approaching
a question like this,
is to work through an
example and see if you can
get some inspiration from that.
So let's look at
our BOS example.
So here, the set of
rationalizable strategies
was all four strategies.
The set of Nash
equilibria in this game
over here with just two strategy
profiles right, CC and RR.
What about the set of
dominant-strategy equilibria?
Which profiles were
dominant-strategy equilibria
in this game?
Any thoughts?
Well, let's go
through for player 1.
Let's check.
Is C a dominant
strategy for player 1?
That is, does C always
do better than R
and sometimes do
strictly better?
Well, no, because if the
other player plays R,
then C does worse than R. C
gives me 0, and R gives me 1.
So C does not weakly dominate R.
And now let's ask
the other question.
Does R weakly dominate C?
Well, no, because if my opponent
goes to the Celtics game,
I'd rather go to
the Celtics game
than go to the Red Sox game.
So neither does C dominate
R, nor does R dominate C.
And that means that
neither of them
is weakly dominant because
if it was weakly dominant,
it would have to weakly
dominate everything else.
And therefore, we have no
dominant strategy equilibria
in this game.
So here, we have none.
Or we might say we
have the empty set.
So at least in this
example, we see
that Nash equilibrium
is more demanding
than rationalizability.
The Nash equilibria are a subset
of the rationalizable strategy
profiles, and
dominant-strategy equilibrium
is even more demanding still.
It's a bit weird to
say it in this case
because there are none of
them, but it's vacuously true
that every dominant strategy
equilibrium is a Nash
equilibrium.
They just aren't any of them.
But if there were one, it
would be a Nash equilibrium.
So that gives us
maybe a conjecture
based on this example about
the relationship between them.
And I chose the example, so
we wouldn't be led astray.
And indeed, this is a
result that, in general, we
have DSE subset of NE
subset of S infinity,
and maybe I'll write subset of
S just to remind us the space
that we're in over here.
So let's understand what these
subset relationships mean.
DSE is a set.
It's the set of all
dominant-strategy equilibria
which is a set of
strategy profiles.
This is another set,
and this is another set.
So this inclusion tells me every
dominant-strategy equilibrium
is necessarily a
Nash equilibrium.
The converse is not
necessarily true.
It could be that something's
a Nash equilibrium but not
a dominant-strategy equilibrium.
Can you come up with an
example of a Nash equilibrium
that's not a dominant-strategy
equilibrium, maybe in this game?
Yeah.
AUDIENCE: [INAUDIBLE]
is a Nash equilibrium,
but it's not a
dominant strategy.
IAN BALL: Right,
and same with R,
so here, it's easy
because there are
no dominant-strategy equilibria.
So both of the
Nash equilibria are
examples of Nash
equilibria that are not
dominant-strategy equilibria.
And then, similarly,
we have, there
are some rationalizable
strategy profiles
that are not Nash equilibrium.
In this example, CR
and RC are two profiles
that are rationalizable
but not Nash equilibrium.
Now, sometimes these
sets could agree,
but sometimes this set
could be strictly bigger.
But our claim here is just that
everything in here is in here,
and everything in
here is in here.
And now we'd like to go
through the reasoning for that,
but are there any questions on
what I mean by this statement
before we go through
the argument?
OK, so let's try to
think through it.
So here's the argument.
There's two different
things we want to show.
I mean, let's be clear,
this is just for context.
This relationship is obvious.
It's just, by definition, every
rationalizable strategy profile
is a strategy profile.
So we have two things to check.
So let's do the first one.
The first one is that
DSE is a subset of NE.
So that's a mathy
way of saying it,
but what we mean
is if you give me
a dominant-strategy
equilibrium of some game,
I know that it must be a Nash
equilibrium of that game.
So let's see how
we can show that.
So first, let's start with a
dominant-strategy equilibrium.
So suppose S star, which I'll
write S1 star up to SN star,
is a DSE.
That's a starting point.
We're just going to
take a strategy profile
and make the
assumption that it is
a dominant-strategy equilibrium.
And we want to
now reason through
why this strategy profile must
also be a Nash equilibrium.
Well, to show that it's
a Nash equilibrium,
we have to check
that no player has
a profitable
unilateral deviation.
So let's consider a player.
So that means that
for, each player i,
let's consider player
i, and let's also
consider a potential
deviation by player i.
So for each player i and
each strategy si prime,
what do we want to show?
We want to show that
this deviation is not
profitable for player i.
That is, we want to
show that, for player i,
it's better for them to play
si star than to play si prime,
given the way that their
opponents are playing.
But that's what we want to show.
All we know is that this is a
dominant-strategy equilibrium.
So what's something useful--
do you have any
ideas of how we could
use the fact that this is a
dominant-strategy equilibrium
to get us started?
What do we know
about si star, given
that s star is a
dominant-strategy equilibrium?
Yeah.
AUDIENCE: si star [INAUDIBLE]?
IAN BALL: Yeah, so let's
go through it in turn.
Because it's a
dominant-strategy equilibrium,
we know that si star
is weakly dominant.
And what does it mean
to be weakly dominant?
It means that si star weakly
dominates every other strategy
of player i.
But in particular, that
means si star weakly
dominates si prime
because that's
another strategy of player i.
So we know that si star
weakly dominates si prime.
We know more than that.
But here, we're just
throwing away information.
If si star weakly
dominates everything,
in particular, it weakly
dominates si prime.
And now can we reach
the conclusion we want?
Well, yeah, if si star
weakly dominates si prime,
that means si star always does
weakly better than si prime,
no matter how my
opponents are playing,
so if that's true and
sometimes strictly better.
So if that's true no matter
how my opponents are playing,
it's certainly true
in the particular way
my opponents are playing now.
So that tells me that ui
of si star si negative--
s negative i star is
greater than or equal to ui
si prime s negative i star.
So because si star weekly
dominance si prime,
this inequality would be true
whatever I put on both sides
here.
I could put any as long as I put
the same thing on both sides.
So in particular, if I put the
way my opponents are actually
playing, this is
going to be true.
And this precisely means
that si star is a best
response to s negative i star.
So I conclude si star is a best
response to s negative i star?
Well, I have reason
for player i,
but I can go through
the exact same argument
for all the other players.
So not only is player i
playing a best response
to the way everyone
else is playing,
but every single player
i is playing a best
response to the way
everyone else is playing.
So I'll say, therefore, s
star is a Nash equilibrium.
So if you're not so familiar
with writing proofs,
this kind of reasoning
may be a little tricky.
Let me pause there.
Are any questions on this?
So it's really just about
being organized and going step
by step.
None of the steps
is too difficult,
but you have to keep
in mind where you're
going, where you're
starting, and carefully
go through the logic of it.
But the basic idea
is pretty simple.
If what I'm doing is best no
matter how my opponents play,
it's certainly best,
given the particular way
that my opponents are playing.
And that's the
high-level idea of this.
OK, so now let's try to
show the second part.
So I have shown that every
dominant strategy equilibrium
is a Nash equilibrium.
Notice, I didn't show
that there exists
a dominant-strategy equilibrium.
My starting point
here was let's take
something that's a
dominant-strategy equilibrium.
Maybe there's no dominant
strategy equilibrium.
That's OK.
But I'm saying if there
is one, it must also
be a Nash equilibrium.
And now let's go see if
we can do this next step.
So part 2 is to
show that every Nash
equilibrium is rationalizable.
So if I start with something
that's Nash equilibrium,
it better be a rationalizable
strategy profile.
Now this seems a bit
hard because the way
we defined S infinity
was really tricky.
We went through this
whole iterative procedure.
We had an algorithm.
We had to go through
infinitely many steps.
So do you have any ideas
about how we might show this?
Any mathematical
technique that might be
useful for this kind of proof?
Yeah.
AUDIENCE: Proof
by contradiction.
IAN BALL: We could--
I think a proof by
contradiction would work, yeah,
but I saw another hand.
Yeah.
AUDIENCE: Induction.
IAN BALL: Induction
is a good idea.
Contradiction would work.
A lot of things, you
could convert it.
But I think induction
is a really natural idea
because we've defined
S infinity inductively.
So we'd like to use induction.
And if you haven't heard
of induction, that's OK,
I'll go through it.
OK, but let's first
just understand
what we're trying to do.
We're going to follow the
same procedure as before.
So suppose s star--
this is a strategy profile--
is a Nash equilibrium.
What do we want to do?
We want to show that s
star is rationalizable.
So we want to show that
s star is in S infinity.
But doing that directly
is hard because S infinity
is only the result of
this iterative procedure.
But remember, from
over here, the way
we got S infinity is we just
kept making the set smaller
and smaller and smaller.
So one way-- or what's
equivalent to showing that s
star is in S infinity--
so I'll say, or equivalently
and easier to show
S star is in Sk for all k.
So let's just say this in words.
We want to show
that s star survives
every round of elimination
and makes it till the end.
Well, the way we
show that is we just
show it makes it to the first
step and the second step
and the third step and the
fourth step and the fifth step,
and it never gets thrown out.
And therefore, it must survive
all the way to the end.
So we want to show that this
strategy profile must survive
every round of elimination.
And since we're showing
something for every k,
this is where we're
going to use induction.
So what do we want to show?
We want to show that, first--
so there's a base case of k
equals 0.
We want to show first
that s star is in there
after the zeroth round.
And then we have
an inductive step
that says if it's made
it to the k-th round,
It's also going to make it
to the k plus first round.
And then we can just
tie it all together.
If we know it makes it
to the zeroth round,
then it must make
it to the first.
And if it makes it the first,
it must make it to the second
and so on.
And we can just chain
our reasoning together.
So for the base case of k
equals 0, what we need to show
is that s star is in S0.
And maybe I'll put
a question mark
because this is not a claim.
I'm not saying it's true.
It's what we're trying to show.
So is s star in S0?
Well, yeah, because S0 is
just the set of all strategy
profiles.
And s star is a strategy
profile, so that's pretty easy.
This is just by definition.
The heart of the proof
is the inductive step.
So here's the inductive step.
What we want to
show is, in general,
is if s star makes it
through the first k rounds,
then it must make
it one more round.
It must also make it to
the k plus first round.
So what we Want To
Show-- and maybe I'll
say WTS because it's
always important
it proves to be clear
about what you're assuming
and what you're trying to show.
So WTS is just an
abbreviation for Want To Show.
What I Want To Show is
that if s star is in Sk,
then s star is in Sk plus 1.
So if I haven't thrown
it out by the k-th round,
I'm not going to throw it out
at the k plus first round.
I think if we just
look at it like this,
it can be a little
tricky because, remember,
the way we defined
this procedure was we
had to reason player by player.
So let's write that
more explicitly.
What does it mean for this
strategy profile to be in Sk?
It means the i-th
component of that profile,
player i's strategy is in si
k for each of the players i.
And I think that's
a bit more concrete.
So that means that si star--
that is, remember, s star
is a profile of strategies.
si star is the strategy that
player i is actually using.
That must be in si k.
And this is true for all i.
And now, symmetrically,
what we're trying to show
is that si star is in
si k plus 1 for all i.
Now, when we go through
proofs and arguments,
we only go through the
right way of doing it,
and I think you
can sometimes lose
sight of how hard
it is because you
don't show all the wrong paths.
And there's an easy
mistake people make here.
What's really
tempting at this point
is to say, OK, let me
reason player by player.
So I want to look
at player i, and I
want to say if player i's
strategy has survived up
to round k, then player i
strategies should survive up
to round k plus 1.
That's not going to
get you very far.
And the reason is that which
strategies for player i
survive up to round
k plus 1 depend
on which strategies of
player i's opponents
have survived up to round k
because, when I'm choosing,
when we go through
the algorithm, what's
the best response for player i?
It depends on what
strategies of my opponents
have survived so far.
So we really have to
reason simultaneously
about all the players.
We can't just reason
through it separately
because which strategies
survive for player i depend
on which strategies of the other
players have survived so far.
So let's try to go
through this argument.
OK, so here's the argument.
So let's consider player i.
So we know that si
star is in si k.
But I've argued that that's
actually not so useful for us.
What's more important from
the perspective of player i?
What do I care about
as I'm player i?
I care about what strategies
my opponents might play.
So the important thing is
that we know that s negative i
star is in s negative i k.
This is the key part of our--
so our inductive assumption
is that everyone's strategy
has survived so far?
But the relevant aspect
of that assumption
is the fact that all of
my opponent strategies
have survived so
far because that's
what's going to
determine what strategies
are best responses for me.
So this is what we know.
And what we want to show is
that si star is in si k plus 1.
So this is what we know.
This is what we want to show.
I think this is often a good way
to try to get through proofs.
So I think the first
thing we should do
is just write down
some definitions.
What is si k plus 1?
What is the definition
of si k plus 1?
Yeah, over here.
In the Boston sweatshirt, yeah.
AUDIENCE: It's like all--
it's the set of strategies that
are a best response to something
in si k.
IAN BALL: Exactly right.
And we have a notation for that.
That's exactly what
it is in words.
And in math-- we'll just
convert that into math,
is s negative i k.
So which strategies survive
up to the k plus first round?
It's all the
strategies for player i
that are best
responses for player i
to some belief over the
strategies in s negative i k?
So these are the strategies
I consider it possible
that my opponents might play.
And I want to think about
all best responses to that.
So now we just have to
put things together.
We know that s negative i
star is in s negative i k.
We want to show that si star
is a best response to something
in s negative i k.
What's the key step?
What's the leap here?
So why is this true?
Yeah.
AUDIENCE: So si star is by
definition a best response
to s negative i star,
and that's what we just
showed because their
game best [INAUDIBLE].
IAN BALL: Exactly, right.
So by the definition
of Nash equilibrium--
let me explain this.
By the definition
of Nash equilibrium,
we know that si star is a best
response to s negative i star.
And remember, this is a
small s, this is a big S,
this is a small s.
So we know that
si star is a best
response to s negative i star.
And we know by our
inductive hypothesis
that s star is a strategy
in s negative i k.
So it just follows by
definition that, therefore,
s negative i star is a best
response to some belief
about things in s negative i k.
Specifically, it's
a best response
to the belief that puts all the
probability on this strategy.
So we're done.
We've shown that si
star is in si k plus 1.
And if we reason exactly the
same way for all the players,
we'll show that this holds
for every single player i,
and now we've completed
our inductive step.
So again, I think each
individual step is not too hard.
But piecing it together
and thinking through this
is a bit tricky.
So again, are there any
questions on this argument?
So now let's go
a little farther.
Let's go to another example.
This is another
classic game, and this
is called hide-and-seek.
So think of this as
the childhood game.
Someone hides, the
other person seeks,
and the hider doesn't
want to get caught,
and the seeker does want
to catch the other person.
Of course, there's more
substantive examples.
You might think of someone
cheating on their taxes,
trying to hide, and
then the IRS seeking,
trying to figure out which line
of their taxes they cheated on.
So I think it has
broader implications,
but we'll keep with the
hide and seek story.
So let's say player
1 is the hider,
and there's two
places they can hide.
And player 2 is the seeker,
and there's two places
they can look for the person.
So the seeker is going to win if
they look where the person hid.
So that means we're
going to have--
give myself a
little more space--
negative 1, 1, negative 1, 1.
Because here, person
1 hid in spot A,
person 2 looked
for them in spot A.
That means person 1
gets caught, so they
get a payoff of negative
1, and the other player
gets a payoff of 1.
And the same is true if they
both hide and seek in spot B.
But then, over here, we're
going to have the opposite.
So here, if i, as player 1,
hide in spot B and player 2
looks for me in A, then
they're not going to find me.
And therefore, I
get a payoff of 1
and the other player gets
a payoff of negative 1.
You could also think as the
hider as the penalty taker
and the seeker as the goalie.
There's a lot of different
stories you could tell.
So notice a difference
between this game and the game
we discussed before.
So this is a game of pure
conflict of interest.
In the Boston game,
there were both.
There was some
conflict of interest
because one person liked
the Red Sox and one person
liked the Celtics.
But there was also
some common interest
because they both liked
each other's company,
and they wanted to be together.
Here, it's purely a
conflict of interest.
I want us as player 1.
I want us to hide
in different places.
So maybe I'll say that.
So what does player 1 want?
They want them to look and
hide in different places, maybe
different locations.
And player 2 wants them to
be in the same location.
Of course, the part is you
can only choose your location.
If you chose the other person's
location, then it would be easy.
So as a first step
for this, let's go
through our standard procedure
where we compute player 1's best
responses to player 2 and player
2's best responses to player 1.
So let's do it slowly.
If I'm player 1, if player 2
guesses A, then I want to hide.
So I want to do B. And
if player 2 guesses B,
I want to hide and be
in a different place.
So I would like to be A.
Now let's look at
it the other way.
Now let's look at the
perspective of player 2.
If player 1 plays A, then player
2 does better if they also go A
because they want
to find the person.
And if player 1 goes B, then
player two wants to go B.
So we've computed
the best responses.
We can look, and if we
follow our algorithm,
where are the Nash
equilibria in this game?
We want to look for a cell where
both numbers are underlined.
But here, we have a problem
because none of the cells
are underlined.
So what this tells us is
that this game has no--
and this will be important in
the second-- pure-strategy Nash
equilibrium.
And we can think through
it reasonably right.
In order for us to have
a Nash equilibrium,
neither player can have
a profitable deviation.
But if we're in the same
place, then someone certainly
has a profitable deviation.
The hider would rather deviate
if we're in the same place.
And if we're in
different places,
then, again, someone
would like to deviate,
the seeker would like to
deviate and change places
so that they find the
person who's hiding.
So because of this immense
conflict of interest,
we can't have a pure-strategy
Nash equilibrium
because someone's losing,
and whoever's losing
would rather change their
behavior so that they could win.
This is a zero-sum
game in a formal sense
that we'll talk
about more later.
But we'd still like
to make a prediction
about play in this game.
If we just use
rationalizability,
well, anything could
happen because I
can hide an A if I think the
other person's going to B,
and I can hide in B If I
think they're going to A,
and we can explain everything.
So I don't know, how
would you play this game?
If you were taking penalty
kicks in the World Cup,
how would you-- what
strategy would you use?
Yeah.
AUDIENCE: Mixed-strategy.
IAN BALL: You'd use a
mixed strategy, right?
So you'd randomize.
And in this case, how do you
think you would randomize?
Would you play A
99% of the time?
AUDIENCE: 1/2 and 1/2.
IAN BALL: Probably 1/2 and 1/2
because you don't want yourself
to be predictable.
If you look at the data on
World Cup penalty takers,
that's a pretty good prediction.
I mean, sometimes
right-footed players
have a bit of a left
advantage, and there's
all these subtleties.
But roughly, people mix
50/50 because if I develop
a reputation, if
everyone knows that--
if you think about playing
hide and seek with a young kid,
sometimes they just always
hide in the same place.
It's pretty easy to beat them.
And even if you don't always
hide in the same place,
if you hid in the same
place 95% of the time,
you'd also be
pretty easy to find.
So it seems pretty intuitive
that you'd want to do 1/2, 1/2.
So that's a good guess.
And that's right.
So it turns out it will have a
mixed-strategy Nash equilibrium.
But 1/2, 1/2, meaning
player 1 randomizes 1/2,
1/2 between A and B, let
me write this a bit more
explicitly.
1/2 A plus 1/2 B comma
1/2 A plus 1/2 B.
This means player 1 mixes
50/50 and player 2 mixes 50/50.
This is going to be a Nash
equilibrium in mixed strategies.
So I'd say this is--
maybe I'll use the
acronym Mixed-Strategy.
NE, MSNE, mixed-strategy
Nash equilibrium.
But I haven't even
defined what that is.
So let's now define that.
I think it's exactly
what you'd think.
So definition, a
mixed-strategy profile--
and now we'll use
sigma instead of s
to remind us that this is mixed.
So sigma star
equals sigma 1 star,
sigma n star is a
Nash equilibrium--
and we'll give exactly the same
definition, for each player i,
what happens?
Well, I'm comparing now what
I'm supposed to do as player i
to a deviation, which I'll
maybe call sigma i prime.
So this is the mixed-strategy
I'm actually playing.
This is the mixed-strategy
I could deviate to.
And I want this to be optimal
given the way my opponents are
playing.
So this is some new notation.
We've been using capital SI to
denote all the pure strategies
of player i.
It's often convenient to use,
this, it's the summation symbol,
and this is capital
Sigma, and this
is the set of all mixed
strategies by player i.
So we use lowercase
s for pure strategy,
uppercase S for a set
of pure strategies,
lowercase sigma for
a mixed strategy,
and uppercase Sigma for a
set of mixed strategies.
So it's the same idea, given the
way my opponents are playing,
my strategy is a best response.
My mixed strategy
does weakly better
than any other mixed
strategy I could use.
But if you look at
this definition,
it's pretty messy because
there's so many different mixed
strategies I could deviate to.
Even in a game with
just two strategies,
I could deviate to any
probability between 0 and 1.
That's a lot of
deviations to deal with.
But fortunately,
we don't actually
have to worry about that.
So it turns out, even if I were
to check every mixed strategy,
it's actually enough just
to check my pure strategy
deviations.
So it turns out this
is the same as--
the same thing, but with
si prime for all si prime.
So when I use the
two dashes, I'm
just saying everything else
in the definition is the same.
But instead of saying,
it must do better
than any mixed
strategy, I'm saying
it must do weakly better
than any pure strategy.
And we could give a
formal math proof,
but let's just think
of the intuition.
If I'd rather do what
I'm doing than deviate
to any pure strategy,
then by choosing
to flip a coin and mix
over those pure strategies,
I can't do strictly better
because if I mix over
those pure strategies,
my payoff is just
going to be an
average of my payoff
from all the pure deviations.
So if each of the
pure deviations
makes me weakly worse off,
then a mixture of them
must also make me
weakly worse off.
That's the intuition.
You could write down a
formal mathematical proof.
But the point is, it's
really important here
that what I am
doing can be mixed.
So I want people to
really understand this.
This is crucial.
It's crucial that we're
allowing my equilibrium strategy
to be mixed and not pure.
But when I contemplate
the deviations,
it's enough to only
look at pure strategies.
And that's a key
difference here.
So indeed, how can we
tell that mixtures matter?
Well, already in
this game here, we
saw that it didn't have a
pure-strategy Nash equilibrium.
But it does have a
mixed-strategy Nash equilibrium.
So the fact that we allow the
equilibrium strategy to be mixed
is essential.
But when you're
contemplating deviations,
you only have to worry
about pure strategies.
And let's take
that randomization
idea little further.
Why would I ever want to
play a mixed strategy?
I mean, why am I willing to flip
a coin and sometimes go to A
and sometimes go to B?
I mean, if A is better, wouldn't
I just always want to go to A?
And if B is better, wouldn't
I just always want to go to B?
Why would I ever
want to flip a coin?
Or why would I ever be
willing to flip a coin
and play A some of the time
and B some of the time?
What must be true?
If I'm-- yeah.
AUDIENCE: A is better
in some circumstances
while B is better [INAUDIBLE].
IAN BALL: OK, that's
true, but what
about given my belief, given the
way my opponents are playing,
what must be true?
If I strictly
preferred A to B, given
the way my opponents
were playing,
would I want to mix
between A and B?
No.
Yeah.
AUDIENCE: Does your
opponent also have
to be playing a mixed strategy?
IAN BALL: So that's related, but
maybe not quite where we are.
So if I'm willing to
mix between A and B,
it must be that I'm
indifferent between A and B.
That's the key thing.
If I strictly
preferred A to B, I
would never play B with
positive probability at all.
And if I strictly
preferred B, I would never
play A with positive
probability?
So I'm only willing
to flip a coin
and sometimes play A and
sometimes play B if I'm exactly
indifferent between A and B.
Now this gets to your point.
Why will I be indifferent
between A and B?
It'll be precisely because
my opponent is mixing.
So this is a key observation.
Maybe I'll start a new board.
So this is the key observation.
So mixing is optimal.
Only if you're indifferent
between the things
you're mixing over.
And in fact, we can say
something even stronger.
It's not just enough
that I'm indifferent,
but I'm only willing to mix
over pure strategies that
are best responses.
So it must be--
and in fact, it's only optimal
to mix over pure best responses.
So what does that mean?
Well, it's just a silly example.
What if I had A,
B, and C and given
the way my opponents were
playing, A gave me 1,
B gave me 1, and C gave me 2.
Do I want to mix over A and B?
Well, no, even though I'm
indifferent between A and B,
that's necessary, but
it's not sufficient.
I certainly don't want
to mix over A and B
because C is strictly better.
So if I'm willing to mix,
I must be indifferent.
But sometimes
people get confused,
and they get it backwards and
they say, if I'm indifferent,
I must be willing to mix.
But no, it's not just
enough to be indifferent.
It also has to be optimal
what you're mixing over.
And that's precisely
what I'm saying here.
It's only optimal to mix
over pure best responses.
So if I added D at
2, now, indeed, I'd
be willing to mix
between C and D.
So now let's try to check--
going back to this game--
that 1/2, 1/2 is actually
a Nash equilibrium.
So I think the best
way to do it is to say,
well, let's take the
perspective of player 2.
And let's suppose that
player 1 mixes 1/2, 1/2.
What payoff does player 2 get
from playing A and playing B?
So what I'm going to imagine is
I'm going to add a new strategy
here, 1/2 A, 1/2 B. So I'm
adding a new row to this matrix
that's going to represent the
payoffs if player 1 chooses this
mixed strategy.
So in here, what I
want to write is,
what are the expected
payoffs if player 1
uses this mixed strategy
and player 2 plays B?
Sorry, plays A.
And then, over here,
what am I going to write?
I'm going to write the payoffs--
if player 1 plays this mixed
strategy and player 2 plays
B. And it turns out, this--
well, let's go through it.
If player 2 plays and player
1 plays A with probability 1/2
and B with probability 1/2,
then what is the payoff of--
what are the payoffs going
to be for the players?
Well, player 1 is going to get
negative 1 with probability 1/2
and 1 with probability 1/2,
which just gives them 0.
And player 2 is also going
to get 1 with probability 1/2
and negative 1 with
probability 1/2.
So we just get 0, and
if we go to this side,
we're actually just
going to get 0's again.
And mathematically, the way
we form this row of the matrix
is we just take this
row, multiply it by half,
take this row, multiply it by
half, and add them together.
And when we do that, we
exactly get 0, 0, 0, 0.
So in particular, the
key thing about this
is that these
numbers are the same.
So what this tells me is if
player 1 plays 1/2 A plus 1/2 B,
then player 2 gets a payoff of 0
if they play A and a payoff of 0
if they play B. And that means
that player 2 is indifferent
between A and B, and in fact,
A and B are both pure best
responses.
So if I'm indifferent
between A and B,
then I'm also willing to mix
over A and B. And therefore,
it's a best response for
player 2 to mix 1/2, 1/2.
So technically, what I've shown
is if player 1 is mixing 1/2,
1/2, it's a best response
for player 2 to mix 1/2, 1/2.
So technically, if I want
to finish the argument,
I need to show the
other direction,
that if player 2 mixes 1/2,
1/2, then it's a best response
for player 1 to mix 1/2, 1/2,
but it's going to be entirely
symmetric.
I could add another
column here and go 0, 0,
but we can check that--
I'll just put a check mark.
We checked it.
It's an equilibrium.
Don't do that on your homework,
but that's all we need.
Now, let's return
to the Boston game.
I don't know if we have it.
I may have to write it again.
You can write it again.
That's OK.
So now let's go to the BOS game.
This is a bit trickier.
And we have Celtics, Red Sox,
Celtics, Red Sox 2, 1, 1, 2, 0,
0, 0, 0.
So this game, we already
found some pure equilibria.
We found two of them.
We said there's one equilibrium
where we both play C,
and there's one equilibrium
where we both play R.
But it turns out, there's also
a mixed-strategy equilibrium.
So here, what we'd like to do
is find a mixed-strategy Nash
equilibrium of this game.
Now, of course, we could say the
pure-strategy Nash equilibria
we already found are
technically mixed-strategy Nash
equilibria where people's mixing
probabilities are just 1 and 0.
But what I mean is something
that's actually mixed.
So what I really mean is a
mixed-strategy Nash equilibrium
that's not a pure-strategy
Nash equilibrium.
OK, well, I think the
first step is let's
just write some notations.
So we're going to look
for an equilibrium.
We're going to look
for some mixing.
So let's look for an
equilibrium of the form-- well,
player 1 is going to put,
say, probability P1 on C.
And let's say player 2 is going
to put probability P2 on C.
So our goal is to find a Nash
equilibrium of the form player
1 plays P1 times C plus
1 minus P1 times R.
So what this means is,
with probability P1,
player 1 goes to
the Celtics game,
and with probability
1 minus P1, player 1
goes to the Red Sox game.
And then, similarly
for player 2,
but we have potentially
a different number here.
And we want to find an
equilibrium of this form
where maybe what we'd like is
we want it to actually involve
mixing.
So let's say P1--
0 is less than P1 is less
than 1, and 0 is less than P2
is less than 1.
This is what we want.
We want something where each
player is actually mixing.
They're not just playing one
strategy with probability 1.
So I guess the trick
is to think, well,
what must be true here?
So if we want this to be a
mixed-strategy Nash equilibrium,
it must be that player 1 is
willing to mix between C and R,
given the way that
player 2 is playing.
And based on what we said
before, what does that mean?
What must be true if player 1 is
willing to mix between C and R?
Yeah.
AUDIENCE: Player 1
[INAUDIBLE] C, R [INAUDIBLE].
IAN BALL: Exactly, and
then the other direction.
Player 2 must also be
indifferent between C
and R, given player 1 strategy.
So the way that
we as the analyst
often solve for an
equilibrium like this is we
look for mixing
probabilities that
make the players indifferent.
So let's look at this,
there's two steps.
So the mixture P1 must
make player 2 indifferent.
And this is a key point.
You might think I made a typo
here, but I didn't actually.
It turns out that the
probability with which player
1 mixes is what determines
whether player 2 is indifferent
because player 1's mixing
probabilities in equilibrium
determine the beliefs
that player 2 has.
So whether player
2 wants to play
C or R depends on their
beliefs about player 1
and the frequency
with which player
1 plays C or R, so it's,
actually, player 1's
mixing probability that is
responsible for determining
whether player 2 is indifferent.
And conversely, it's
player 2's mixing
probability that determines
whether player 1 is indifferent.
So when you try to solve
for these equilibrium,
it all looks nice.
We have two indifference
conditions and two numbers
to solve for.
Mathematically, it's pretty
easy, but the thing to remember
is that player 2's indifference
condition is actually,
in this case, going to
be determined by P1,
and player 1's going
to be determined by P2.
So player 2's
indifferent condition,
we need to make sure that player
2 gets the same payoff from C
as from R. So let's
look at player 2.
If player 2 plays C, well,
they get a payoff of 1
if player 1 plays C and a
payoff of 0 if player 1 plays R.
So what is their
expected payoff?
Their expected
payoff from playing C
is going to be P1 times 1
plus 1 minus P1 times 0.
But the 0 disappears,
and we just
get P1 in the left-hand side.
So here, P1 is the utility,
the expected utility
for player 2 from playing C.
So maybe I'll put a
C here and an R here
just to keep track of
what we're comparing.
And this makes sense.
If I'm player 2, and I
go to the Celtics game,
I know that if my opponent
goes to the Red Sox game,
I get a payoff of 0.
So my only utility
comes from if they also
go to the Celtics game, that
happens with probability P1.
And if it does happen,
I get a payoff of 1
because I like
the Red Sox more--
I only get a payoff of
one from the Celtics.
What about the other case?
If we have the R--
if I play r, what is
my payoff going to be?
Yeah.
AUDIENCE: [INAUDIBLE]
2 times 1 minus R.
IAN BALL: Exactly, right.
So here, again, I get no payoff
if they go the other way.
1 minus P1 is the probability
that my friend also
goes to the Red Sox game.
But now there's a 2 here because
I really like the Red Sox.
Or if they go to the
Red Sox game with me,
I get a payoff of
2 rather than 1.
And now let's do the
same thing for player 1.
I'm player 1.
If I play C, given that
my opponent is playing
C with probability
P2, what is my payoff
going to be as player 1?
Well, it's going to be 2.
I get 2 with probability P2 and
0 with probability 1 minus P2,
so I just get 2 P2.
And then if I play R, again,
0 with probability P2, 1
with probability 1 minus
P2, so I get 1 minus P2.
And we can just do
some algebra here.
And if we pull this over, we're
going to get P2 equals 1/3.
And then P1 equals 2/3.
So I want to caution
here against, I think,
a common
misinterpretation of this.
What a lot of people say is
they say, OK, this makes sense.
Player 1 goes to the
Celtics game more often.
P1 is 2/3, more than 1/2,
because player 1 likes
the Celtics.
And player 2 goes to
the Red Sox game more.
Player 2 goes to Celtics game
only with probability 1/3,
which means they go to the Red
Sox game with probability 2/3
because player 2 likes
the Red Sox more.
Is that right?
Do you see a problem
with that reasoning?
I don't know, what do you think?
Does that seem right to you?
So it turns out that
the reason, if we really
look through the
equilibrium, the reason
that player 1 goes
to the Celtics a lot
is not because they
like the Celtics more
but because their opponent
likes the Celtics less.
Player 1's mixing
probabilities have
to make player 2 indifferent.
And because player 2 doesn't
like the Celtics very much,
they're only willing to
go to the Celtics game
if they believe that player 1 is
very often going to the Celtics
game.
And let's do the
reverse reasoning.
Why does player two go to
the Red Sox game a lot?
It's true that player 2
likes the Red Sox a lot,
but the real reason
for this equilibrium
is that they have to
go to the Red Sox game
a lot to make player 1 willing
to go to the Red Sox game
because player 1
prefers the Celtics,
so they're only going to
go to the Red Sox game
if they think it's
more likely than not
that player 2 goes
to the Red Sox game.
So I think that's
just a key point
here about what's driving the
structure of this equilibrium.
And I think that
should make you wonder
a bit because I shouldn't
care about making
my opponent indifferent.
So I think an issue,
maybe a question,
is if I'm player 1 here, given
the way my opponent is playing,
I am indifferent
between C and R.
So why should I mix with exactly
the right probability that
makes my opponent indifferent?
It doesn't really seem like,
if I'm indifferent, I mean,
why go through the difficulty
of doing all the randomization?
And I think that's a question
we'll just leave for now.
We'll talk a bit more throughout
the course about this,
but I think it's
something to keep in mind,
that it's true, I'm willing to
mix, but why do I actually mix?
We do see it in practice.
I don't know.
I will open it up.
Are there any thoughts?
I mean, why do you think
poker players actually do mix?
They're indifferent, they
should be indifferent
if everyone else is
mixing just right.
Why do they mix?
Why would you mix if
you took penalties?
What would go wrong
if you didn't mix?
So yeah.
AUDIENCE: They would know
what you were going to do,
and they would update their
strategy based on that.
IAN BALL: Right,
so often, the story
we tell about mixed
strategies is often really
a story about a dynamic process
where the game is repeated.
And for this reason, we might
think that a mixed-strategy Nash
equilibrium is a more compelling
prediction if the game is
repeated over time.
You don't want to
have a reputation
as someone who always shoots
left because then your opponents
would update on that.
Now, then we have to
really think about, well,
what does it mean to develop
a reputation for something?
And that takes us
down a whole new path
that people have tried to
formalize and think through.
It's not clear
exactly what it means.
So that's one story.
I guess, another
story I would say
is sometimes mixing
is not literal mixing,
but what mixing means is
you're making a decision based
on something that the other
person isn't aware of.
So it may be that--
let's take, I don't know,
how someone dresses.
You may think, oh,
if someone just
randomizes the way they
dress, but really, they
wear certain colors
on certain weather
because that makes
them feel happy.
But to you, not knowing
their careful algorithm
for how they choose
their clothes
as a function of the weather,
it just appears random to you.
Or similarly, maybe
a penalty kicker,
when they come to the
spot, feels a direction.
One of their feet feels
a little different,
they just feel a
direction to go to,
and then they make a
deterministic choice about which
direction to shoot, but
because you, as the opponent,
can't see that signal that
they're basing their shot on,
it appears random to you.
So sometimes we literally
observe randomization,
and sometimes behaviors
appear random because they're
based upon things that we
can't actually observe.
So just one thing
to keep in mind.
And I think that's good.
Let me finish with one last--
two last things.
So notice that, in the game
that we looked at here,
this game did not have a
pure-strategy Nash equilibrium.
But when we moved to
mixed-strategy Nash equilibrium,
it did have a mixed
strategy Nash equilibrium.
And the question
is, could we come up
with some really weird game
where there's no equilibrium.
That maybe wouldn't be so
very good for our theory
because then we wouldn't
have a prediction about how
people play.
And fortunately, it turns
out-- and this is really
Nash's key result in 1948-- that
that's never going to happen.
Every game, within reason, has a
mixed strategy Nash equilibrium.
So more precisely, let me say
every finite strategic-form
game--
would be a little more precise,
I'll say-- has at least one.
So at least one.
It might have only one.
It might have two.
It might have three.
It may have infinitely many,
but it has at least one.
And when I say mixed strategy,
that includes pure strategy
as a special case.
So I might say mixed
or pure, but remember,
we're going to think
of a pure-strategy Nash
equilibrium as a special
case of a mixed-strategy Nash
equilibrium where
the players mix
with degenerate probabilities.
Now, this is not something we're
going to prove in the course.
This is actually quite a
deep result. So remember,
a couple of weeks
ago, last week, we
talked about this
dominance/best response duality,
and I also didn't prove that.
That is somewhat deep.
This is much deeper.
So this is maybe the
mathematically deepest result
in the course.
And it uses something called
a fixed-point theorem.
And if you want
to google it, it's
either sometimes attributed
to Brouwer or Kakutani.
And these are, actually, quite
deep mathematical results.
So one of the
mathematical results
was actually proven,
I think, in 1944.
And then Nash proved
this result in 1948.
So it was really using
cutting-edge math
to prove this new result.
Maybe not so cutting edge
anymore in 1944, but at the
time, the gap between the math
frontier and the econ frontier
was actually very, very close.
If you want to
see the proof, you
can look in the lecture notes.
The lecture notes give a proof
building upon these theorems.
The real depth is proving
these theorems, which
we're not going to talk about.
We're not going to
talk about this proof
either, but just want
you to keep that in mind.
And before we go, we
have three more minutes.
So let me just conclude with
a few interpretations of Nash
equilibrium.
I think we talk about
Nash equilibrium
as the crowning
achievement of game theory.
But over the last
seven years, people
have always been searching
for justifications
of Nash equilibrium.
And I would actually argue
that Nash equilibrium does not
have as compelling of a
justification as, say,
rationalizability does.
Of course, it makes
stronger assumptions,
and it gets sharper predictions,
but whether we actually
find it compelling, I think,
is not totally obvious.
Again, on exams, if I ask you
to solve for a Nash equilibrium,
you have to solve it.
You can't just say, I don't
believe in Nash equilibrium,
but I do think it's helpful to
think about how compelling this
is.
So I want to present a few
interpretations of this.
So one interpretation is
about a stable convention.
So the idea of Nash equilibrium
is really about stability.
It's that given the way
people are behaving,
no one wants to deviate.
So a great, I think, example
of this is the side of the road
that we drive on.
Why do we all drive
on the right in the US
and drive on the
left in England?
You might say, oh, you get a
ticket if you drove on the left,
but that's not
really the reason.
It's really because you're
afraid you'd get hit by a car.
So if everyone drives on
the right, it's best for you
to drive on the right.
If everyone drives on the
left, it's best for you
to drive on the left.
These are both
stable conventions,
and we don't really
need to ticket people
for driving on the
wrong side of the road.
I mean, we do, but that's not
really what motivates them,
I think, to drive on the
correct side of the road.
Another is you can think of it
as a self-enforcing agreement.
Again, self-enforcing
because no one
wants to unilaterally
deviate from the agreement.
And I think this
explains the success
of a lot of multilateral
international agreements.
When countries sign
treaties, there's
no court that can adjudicate
a treaty between countries,
or generally, there's not.
So why do people
follow the treaty?
Well, often when treaties
are set up in the right way,
people don't have an incentive
to unilaterally deviate.
And even though there's
not some court that
would throw a country
in prison, they often
are still motivated to
follow these agreements.
And then another interpretation
is as a steady state.
And this gets to a
discussion of penalty kicks.
If you were to
always shoot right,
the goalies would start
always guessing that you
were going to shoot right.
So then you might start shooting
left, and then, over time,
we think that you might
converge to the steady state.
Whether there's actually
a real justification
for dynamics converging is
actually a bit more subtle.
But the point is, if you are in
a Nash equilibrium, it's steady,
and there's no reason
to move away from that.
And that's the final
interpretation,
and I'll leave it there.
So I'll see everyone next week.
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Lecture 11: One-Shot Deviation Principle and Bargaining

MIT OpenCourseWare
May 18, 2026
Nash Equilibrium

Relationship Between Solution Concepts

Mixed‑Strategy Nash Equilibrium

Interpretations of Nash Equilibrium

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