Is Washington or
Olympia? Kentucky. Louisiana. Yes. Albuquerque. Alba. You're
picking all the ones that are impossible
to spell. You're the easiest
ones to remember. Iowa is Des Moines. Yes. D.S. Des Moines. Des Moines. Ohio? Providence, Rhode Island. Is it Atlanta, Georgia? Manchester. Tolteca. Kansas. Okay, you guys,
you Americans had to brush up on you
all a geogrape. I'm better at naming
the state that goes with the city
than the state. Atlanta. Yeah, that's easy. Atlanta's just
better at it. Ohio? Columbus. I take back what I said. You had enough fun. Okay, great. We'll do US presidents next time. Okay, great. So, we finished up
last class speaking about this idea of mixed
national equilibria, but I just wanted
to go back and reinforce this idea of
what national equilibria is. No matter what
you do, this is the last bit of gang
theory you ever take, to you actually
going on further and taking gang theory.
Everything's about this idea of
Nash Equilibria as a solution concept
in game theory. So it doesn't have
this normative property that Nash Equilibria
are desirable. It's just simply a
situation where no one on their own can make
themselves better off. So our definition
of it is a strategy profile such that each
strategy that players are playing is the
best response to all other players' strategies.
So if everyone's doing this, what it
essentially means is no one can change
on their own and make themselves better
off. That is the idea of equilibrium at
this constant point. So you can't do better
given what I and everyone else are
doing, and I can't do better given what you
and everyone else are doing. We're stuck
here for good or bad. And if you remember,
this idea of good or bad really takes hold
in the prisoner's dilemma, where we have
this Nash equilibrium here at defect, defect.
and the whole idea of national equilibrium
if you remember is best response if
everyone is playing their best response this
is the same thing as everyone's strategy is
bringing them to a point where no one can make
themselves better off given everyone
else's strategies so the simple heuristic I gave
you to find everyone's best response is to
say what everyone would do given another's
action so if player 2 is defecting player
1 can defect and spend 10 years in jail
or cooperating and spend 12 years in jail
Obviously they prefer to spend less time in
jail So they will defect This is the best
response for player 1 Given player 2 defects
And if player 2 cooperates Player 1 can
defect and spend 0 time in jail Or cooperate
and spend 1 year in jail 0 is better than
1 We do the same thing for player 2 as well
If player 1 chooses an action defect
Player 2 can defect And spend 10 years in jail
Or cooperate and spend 12 And choose 10 and
if player 1 cooperates, player 2 can defect,
spend 0, use each other will cooperate,
spend 1, let's use 0. So when doing this,
highlight the best response. And if
you get a quadrant that has every player's
action highlighted in it, that means
every player is playing the best
response to each other in that quadrant,
that scenario, this is what the national
equilibrium is. So use that heuristic,
it makes things a lot easier. A lot easier.
and the cool thing once again about the
prisoner's dilemma is we have these strong
incentives we in fact have a dominant strategy
it's always better for each player to
defect no matter what the other player is
doing which leads us to this defect national
equilibria where both players go to jail
for 10 years but if they had both cooperated
both are much better off but the incentives
are so strong towards defection it makes
cooperation really hard and the prisoner's
dilemma is ubiquitous you probably have heard
the name prisoner's dilemma said before
in tv shows and in sport i know if anyone
watched a survivor they bring it up all
the time incorrectly but they do bring up the
prisoner's dilemma but this idea of the
incentives leading towards a bad outcome where
we could cooperate and get a good outcome
is very common in a lot of the economic
decisions that we make we then looked at
other types of games as well if you remember
here the Nash Equilibria we have two here
if both are playing Lucys then if anyone
changes their strategy they get zero they
can't be made better off if both are
choosing Crestful it's the same thing if
anyone changes their strategy they get zero
they're not better off so these are both
Nash Equilibria. This doesn't tell you which
of the equilibria to select. It doesn't
tell you what your actual strategy
profile should be. All it says is if everyone's
playing the strategy then no one should
deviate from it. This is all the Nash
Equilibria is saying. And finally, actually not
finally, penultimately, we had this stag
hunt game as well with the payoff dominant
equilibria and the risk dominant equilibria.
If you remember when we played this game
ourselves and looked at it, you're like,
why wouldn't anyone ever play here? I have
an example coming up for you later, which
will kind of show why. Finally, we
ended up talking about the zero
-sum game here. So matching pennies, so
the even player wants the pennies to match.
The odd player wants them not to match, so
on the count of three, they both take out
their penny. If they match or if they don't
match, that's what results in the payoffs.
so if both of them show a heads the even
player wins the odd player loses and as you can
see in every quadrant one person wins and
gets one one person loses and gets zero
sorry minus one and this is why this is a zero
sum game the payoffs in each square add up
to zero so one only wins when the other
loses you could have a game like rock pack the
scissors which would be three strategies
each then you'd have a zero zero when they play
the same one that is still a zero sum game
and you can see from the best responses here
no quadrant has both players playing a best
response so if both players are choosing
the strategy heads the odd player has an
incentive to deviate from heads to tails
because they go from minus one to one and you can
look at that in every quadrant one person
can always do better by changing their strategy
so there's no pure Nash equilibria here.
But this isn't the end of our story. We
have something called the mixed strategy
Nash equilibria. So you don't have to play
heads 100% of the time. You don't have to go
to Lucy's 100% of the time. You don't have
to cooperate or defect 100% of the time. You
can mix the probability that you play a certain
strategy. So in this case you can assign
a probability P to playing heads and 1
minus P to playing tails. So the definition of
our mixed strategy Nash and equilibrium
are, is pretty much the same. It's a
stable state in a game where players
choose their actions with a probability
mixture, not with 100 % and 0%. And these
probability mixtures are the best responses
to each other. Once again, what this
means is no player can make themselves
better off or have a higher expected
outcome by unilaterally, so on their own,
changing their probability to show a
randomised strategy. So once again, I
want to go through a simple intuition
process here, and then we'll get
through the math today as well to show
you how to do it. So Kevin, you're a
goalkeeper in soccer, and I'm taking a
penalty kick. You have two actions. You can
choose to dive left. You can choose to dive
right. If you save the penalty, you win.
If I score, I win. So I can kick left
or I kick right. I have perfect
accuracy if I kick left or right, so
the only way to save it is to dive
the right way. Let's say you
know I kick the ball left 100
% of the time, which way are you
going to dive? Yeah, you save it
100% of the time, which means you win
100% of the time. Kevin's exploited my
strategy of always kicking left, which is
clearly a bad strategy. What about if I kick
the ball left 80 % of the time and
right 20% of the time? What's your optimal
strategy here? Yeah, left all the
time. If you dive left 100% of the time, you
win 80% of the time, you lose 20% of
the time, but it's your best
strategy. You're exploiting my
probability mixture. What about 51%
of the time? I kick left, 49
% of the time I kick right. What
are you doing? Exactly. So Kevin's
going to win 51% of the time, lose 49
% of the time, but he's still able to
exploit my strategy. But if I go a mixture
of kicking left with 50% chance and right
with 50% chance, what does that mean
for your strategy now? exactly it doesn't
matter if he chooses to dive left he's going
to save 50% of the time and let the goal in
50% of the time if he chooses right exact
same thing you'll win 50% lose 50% so
exactly what Kevin said he's indifferent between
these two strategies and any mixture
between them if he goes left 70% and right 30%
he's still going to win 50% of the time I've
made him indifferent between his actions
he can't exploit my strategy and if we do
the same analysis for me so if I know Kevin
dives left 100% of the time I'm going to
shoot right 100% we do the same thing Kevin
can make me indifferent between my decision
by diving left 50% of the time and diving
right 50% of the time if we're both playing
these actions making each other indifferent
we're at this point when no one can be made
better off by changing their strategy this
is going to be our This is the intuition
behind it and I'm going to take
you through how to actually prove this
mathematically. So I've changed up the
story a little bit. The reason is, one
of the reasons I fell in love with the NFL
20 years ago living in Australia was I really
like the cat and mouse game that is NFL.
Like the offense is trying to predict what
the defense will do, the defense is trying
to predict what the offense will do. If
they're able to predict correctly, you get a
huge advantage. Our resident Madden player
here, DK, if you're playing a guy online
and you know his tendencies and you're
going to exploit them, exactly. So if you know that
they blitz all the time, you're probably going
to throw some screens or something like that
or hot routes, etc. But for simplicity,
we're going to have an offenser able to pass
a run and the defense can either blitz
to stop the run or cover to stop the pass.
They can't do both. So if the offense
passes and the defense tries to stop the run
by blitzing, the offense wins, the defense loses.
but if the offence runs when the defence
blitzes the offence loses the defence wins
the defence chooses to cover the pass if the
offence decides to pass the offence loses
the defence wins but if they choose to cover
the pass and leave the running lanes open if
the offence runs they will win and the defence
will lose so what will happen I've given
you the preemption what should happen here
with these numbers is the mixed strategy
Nash equilibria should be the offense passing
the 50% probability and the defense blitzing
the 50% probability. So how do we do
this? So I'm going to switch over to
the dot cam here. Actually, I might
want to do dot cam on this one and we can
keep it up like that. Brilliant. So the offense passes with probability P runs with probability
1 minus p. So with these two
actions, it's just p and 1 minus p because if
you add it together, you get p plus 1
minus p. That just equals 1. So the
probability mixture needs to add up to
1 or 100%. If it's not that, something's
gone wrong. And as we said before, what the offense wants
to do is make the defense indifferent
between choosing their two actions, which is
blitzing and covering if they get more
expected payoff from one than the other
then they're going to exploit your strategy
by always playing that instead so you
want to make them indifferent so the expected
utility of blitzing equals the expected
utility of covering and then the expected
utility of blitzing is just if the
defense blitzes what do they get when
the offense passes which is minus one
and if we go back to our expected value
expected utility it's the same thing
here the outcome times the probability
of the outcome the probability is the
probability that the offense passes which here
is p so minus one p plus the other outcome
which is what happens when they blitz and
the offense runs the defense gets one
times the probability that the offense runs
which is one minus p and this just equals
one minus two p And we do the same thing,
the expected utility that the defense
covers, which equals 1, if the offense
passes, so 1 times the probability the offense
passes, which is P, plus minus 1, I should
just go minus 1, given the
probability that the offense runs,
which is 1 minus P. So this equals 2P minus
1. so the next step here to figure out the
mixed strategy that makes the defense
indifferent between their two actions is
just to equate these two expected utilities to
each other as we do up here so 1 minus 2p
equals 2p minus 1 we get that goes to 2
equals 4p p equals a half so going back to
the same example that we showed before with
Kevin and I in the penalty kicks, the way
the offense can act via defense in different
between the two strategies is to pass
a 50% probability which means they also run
with 50% probability. And if you put in P
here you can see the expected utility is
going to be 0.5 and the expected utility here
is going to be 0.5 as well. Actually no, that's
wrong. The expected utility is going to
be 0 for both of them. So as a result the
defense They don't care if they choose blitz or
if they choose cover. If the offense is
doing this probability mixture, they get
the same payoff they might not
want. And we do the exact same thing
for the defense. So the defense blitzes
with probability Q and covers with
probability 1-Q. And the exact same
thing occurs here. They want to
make the offense indifferent between
passing and running. so the expected utility
of passing equals what they get when
they pass one when the defense blitzes the
defense blitzes with probability q and they
get minus one when they pass and the defense
covers with probability one minus q so this
just equals two q minus one because the the
game is symmetrical we're going to actually
get the exact same thing going on here but
i do want to take you through the process and
the expected utility of running equals
minus one times the probability of blitzing
minus one q plus running when the defense
covers which gives you one with the probability
that the defense covers one minus q and
this equals one minus two q And as you can
probably already see, 1 minus 2q equals 2q
minus 1, 2 equals 4q, q equals 1 half. So, the mix strategy, Nash Equilibria equals p equals 1 half,
q equals 1 half. so when both players
are mixing in this way no player can
change their strategy and be better off if
you remember as we said before it
doesn't matter what probability mixture the
defense will do if the offense is playing
p equals a half i'll get the same outcome
no matter what and the same as before
as well if we look at it from the defense's
perspective okay finally i know it's a
weird shape it's not a swastika i had the
other class yeah like like go off their rock
a little bit um this is a reaction function
and what this tells us i know this one is
blurry i've got to clear a one later we have
the probability p here so passing the probability
q here blitzing and what this does
is showing the best strategy of each player
of each player so what this says is that if
the probability of q is less than a half then
player one the offense should always pass
should always pass as a result if sorry
actually these numbers are the wrong way around
i need to fix that um but the idea basically
is here is if the strategy is on either
side of a half the other player will choose
either p or one minus p with a hundred percent
probability and only at a half will they
be indifferent so this shows they're
indifferent between any mixture of p then it
goes out like this and we do the same thing for
q as well when p is equal to zero as you
can see or below a half then q will always be
equal to zero and when p is as you can see
up here greater than a half then the q will
always be at a hundred percent it's only when
p equals a half is the player indifferent
between their mixtures and it's when these
two reaction functions cross, we get the
mixed strategy Nash equilibria to half a half
here. So this is really important, because
when we talk about Cournot equilibria in
the next lecture, which is how firms decide how
much output to produce when it's them and
one other competitor, this is the exact way
they solve it. Look at each other's
reaction functions, finding the point
where they cross, and this is going to be
the Nash equilibria. So this is really cool. so MixNash may seem like relatively complex
before you learn about it it's like a lot
of words and stuff but there's really
cool evidence showing that professionals
actually conform to the MixStrategyNash
equilibrium so this is a paper by Gary O. Page
and Wooders and what they do here is they
actually get hundreds of thousands of
observations from professional tennis
games and these are from the serves so when
you serve in tennis If you serve from the
right baseline, you've got to land it in
this quadrant here. And as you can see,
here are the observed serves, the blue
dots, and the red serves are the
imputed observations. So these are balls
that hit the net and didn't go over,
so they didn't land. But using
Hawkeye, you can track where they
would have landed. And they care about
this because they're interested in the
strategies that people use. So if you're playing
the mixed strategy Nash equilibria
correctly in tennis, you want to serve on the
left hand side 50% of the time and on the
right hand side 50% of the time as well and
the data is really cool here it's showing
this is exactly what's happening these two
sizes are essentially the same so the pros
and mix in their strategies in the predicted
manner in the predicted manner and it's even
cooler than that so that as I say in
the paper utilizing modern ball tracking
tech we can buy an even larger data set
encompassing nearly 500,000 serves from more than
3,100 matches, our findings revealed
that the directions in which players serve were
remarkably consistent with mixed Nash
equilibria predictions. Players seemed to closely
equalize the winning probability in each
serve direction in each match they played.
So that's what the data shows us, but
then they also broke it down by seniority, and
they actually found that junior players
deviated more from the Nash equilibria than
senior players. So junior place we're getting
their strategies exploited and they can't
define the cause the way this works but
essentially they're saying one of two things
must happen over time as you go from junior
to professional you learn that you have to
mix your strategy in a certain way or you'll
get exploited but the other channel is
maybe the juniors don't work then they just
lose so much they drop out of the tool so
we don't know if it's selection effects or
learning effects But essentially, as people
become pro, they will become consistent in
the way that they serve. So in sports and in
other places as well, people utilize the mixed national equilibria. So this is the exact
same situation as before. It's no longer
technically zero -sum, but I wanted to
show you with different numbers in the same
format. You can just go through the
same process again. And the reason why
I wanted to do this is if anyone likes to
hire analytic stuff, in every play in
American football, they're usually given
an expected points or gains or EPA,
expected points added. So each play is not
necessarily worth the same, and you can
actually calculate what strategy you should use
based on the outcomes times the probabilities
that you use. And in fact, there's a
really cool video that I'll try and send
out tonight that does exactly this to determine
the optimal rate that a team should run
the ball in football. So, I am not going
to write again. I've already
written this, and to be honest, my hand
kind of hurts, but we'll go
through it all. So, great. Yeah. So, as you can see,
like before, we define the offense's
two actions probabilistically.
so they pass with probability P and they
run with 1 minus P and the next thing
they want to do remember is to make
the other player indifferent between
their two actions so the expected utility
that the defense blitzes should
equal the expected utility that they
cover they shouldn't be able to get more payoff
from one than the other otherwise
they'll do that 100% of the time. As soon
as it goes over the threshold they do
it 100% of the time we can calculate the
expected utility of blitzing by taking the
payoff of blitzing where the offense passes,
which is 2, times the probability that the
offense passes is P, plus the other
outcome, the payoff of blitzing is 9, when
the offense decides to run with
probability 1 minus P, and we get 9
minus 7P, is the expected utility
of blitzing. And we do the same
thing here, the expected utility
of covering, they get 6 when
the offense passes, so 6 times the
probability of pass is P, plus 4 times
the probability that the offence
runs, which is 1 minus P, which
equals 4 plus 2P. We equate them to,
and when you do the math and put it all
together, you get P equals 5 divided by 9. 5
divided by 9. That's the probability
mixture, which means 1 minus P is 4 divided
by 9, of how much they should pass and run
to make the defence indifferent between
their two strategies. And we do the same
thing for the defense as well. So the defense
can blitz with probability Q, cover with
probability 1 minus Q. And they want to
make the offense indifferent between
their two actions. So the expected
utility of the offense passing is they
get 8 when the defense blitzes, which
is probability Q. They get 4 when the
defense covers with probability 1 minus Q,
which is 4 plus 4 Q. and then yeah the
expected utility of of running is one times
the probability that the defense blitzes which
is q plus six times the probability that the
defense covers which is one minus q which
gives six minus five q and then we have six
minus five q equals four plus four q two equals
nine q q equals two divided by nine the
defense should blitz with probability two
divided by nine and cover with probability 7
divided by 9. And that gives us our mixed
strategy, Nash Equilibria, where no player has an
incentive to deviate. No player has an
incentive to deviate, no one is better off.
And this is actually a cleaner reaction
function here. So, I should have made
the first one myself. So this is the
probability the offense passes with P,
and here is the probability the
defense blitzes with Q. The blue is the
offense's reaction function, and the
red is the defense's reaction function,
their best response to the other player's
probability mixture. So as you can see,
if the offense passes with a probability
less than 5 over 9, the defense is going
to blitz with 100% probability. This
essentially means the offense is running too
much, and the defence can blitz to, on
average, do better overall. They're
exploiting the tendencies. At this point, 5 over
9, we get the vertical shift here, which
means for the defence, it doesn't matter
if they blitz with 100%, 99, 98, 97,
so on and so forth, they get the
exact same payoff. The exact same payoff. Finally, if the
offence passes with more than 5 divided
by 9 probability, then the defence
will always cover. They'll play blitz
with 0% probability. and it's the same thing
for the offense as well if the defense
is playing blitz with less than 2 over 9 then
we see the offense is always going to run
this is the point of indifference when they
blitz at 2 divided by 9 the offense doesn't care
what they do they're indifferent and then
if they blitz higher than 2 over 9 then the
offense will always pass so where these
two reaction functions meet is where the mixed
Nash equilibrium is no player can do better
off by changing their strategy, no strategy
is being exploited. Does that make
sense to everyone? Great. Okay, let's go on a
slight tangent here. Okay, so let's
say I wanted to be nefarious. I could
announce the following and I know this is
purely hypothetical. Okay, for the next
exam, if If no one turns up to
the exam in the whole class,
everyone gets 100%. However, if someone or multiple
people rock up to the exam, they get
what they actually score on the exam and
everyone who didn't rock up to the exam
gets zero on the exam. What do you think
will happen? It's more difficult
to escape that way. I'm going to personally
email everyone, I'll announce it
on Brightspace as well, I'll start in
all the classes, etc. I think out of fear
some students are going to think one
person is going to forget or one person
is going to show up. They're going to think
like, oh, I can't risk getting this
here, so I need to go. This is going
to be someone who goes there. It's an
interesting point. Let's do a quick
hypothetical. Everyone close their
eyes and raise your hand if you would walk
up to the exam in this hypothetical scenario.
Raise your hand. Okay, keep your hands open and open your eyes. If you're hands
down, you just got a zero because half
the class had their hands up in this
hypothetical scenario. So I think Ivan's
thoughts are really interesting here.
As soon as you worry that someone else
might rock up, not rock up, then you're
like, I'm going to get a zero if
someone's too nervous about it, then I
should rock up as well. So this is a very
common result in this situation. Do
you have a comment, Ivan? No, I'm
just so surprised. So the question is, is there a national
equilibria in this guy? I haven't drawn
up the pile or anything yet, but
I've seen a couple of nodding heads.
I'll come over here. What's the national
equilibria? It's either everyone
goes or nobody goes. Because if
you're assuming people know that
either people are going to show
up or you know there won't be an
instance where you know people will go
and you will choose not to or where
you will choose to go when you know no
one else exactly, so like anything
else I've done a few tweaks here but we
can draw it up in a game table so we
have you and everyone else so both players
have two actions go to the exam or
skip the exam if both go, they both
get 100 on the exam if you skip the exam
and everyone else goes you get nothing and
everyone else gets what they score on the exam
xj if you go to the exam everyone skips
you get what you score xi everyone else get
zero and if everyone um goes to the exam you
get xi everyone gets everyone score xj and
let's just assume xi is a number between zero
and 100 not inclusive so what this essentially
means is if everyone else skips the exam
then you should skip the exam too because
100 is higher than whatever xi is and if
everyone else goes to the exam if you skip you
get zero if you go you get xi which is greater
than zero and we can do the same thing
for everyone else as well if you skip the
exam everyone else will skip the exam as well
and if you go to the exam everyone else will
go to the exam as well x-ray is higher than
zero so as pointed out we have two Nash
equilibria here we have either everyone skips
the exam or everyone goes to the exam that's
the best response from the situation
does this kind of seem familiar to anyone this
is not a prison not pure coordination it's it's
impure coordination so this has a specific
property that I may have brought up at the
start of the lecture this is a stag hunt
game this is a stag hunt game so think about it
we have two equilibria this is the payoff
dominant equilibrium this is the highest
both players can possibly get and this
is our risk dominant equilibrium if you choose
to go to the exam no matter what everyone
else does you get XI so you don't have to
worry about what others are doing and remember
when we looked at the original stag hunt
we played it in class everyone was probably
thinking like which idiot would choose
here everyone should just choose stag that
makes sense well I just showed you through a
couple of tweaks by making the the risk
higher and even adding more people to it, you
can get half the class to raise their hands
and say they're playing here. And this
was just for this class, not even the entire
course. If I had 200 people in the room,
probably everyone would have raised their hands.
So this is why the Stag Hunt game can
be really interesting when you tweak the
payoffs for risk. And you can tweak the number
of people playing. So the assumption is,
let's just say it was Brooke and Ivan playing. No one else in the class, if I did this
for you two. Do you think you
would play, Hey, skip the exam
or go to the exam in that situation. You know what,
close your eyes. Raise your hands
if you'd go to the exam. One, two,
three, open your eyes. You've both got 100%,
congratulations. So when there's less
people, it's easier to coordinate because
you're less worried about one person
flaking which tears everything apart. But
if we add more people to it, this will
increase the probability that people start
flaking out. So it's all about your beliefs
of other people. and we can do the
same thing in the stag hunt game as we did
with our more like competitive versions of
games, so remember mix match equilibria and
match equilibria isn't really telling us
anything other than if everyone's playing
these strategies no one can change their strategy
to be made better off unilaterally, so
if we go back to the so we have hunter
1 and we have hunter 2. They
can play stag or hare and these
are the payoffs. So to find the mixed
strategy Nash equilibrium in this case, the
hunters like don't want to try and make the
other person indifferent. Just making each
other indifferent is how we find the mixed
strategy Nash equilibrium. It has no other
normative content other than that. So they play
stag with probability p and hare with
probability 1 minus p. And the whole idea
here is that the second hunter has the same
payoff from playing stag and the same
payoff from playing here if this is the case
i don't have any incentive to deviate to
one strategy or another so we do the same
thing as before so the expected utility of
playing stag is three when player one or
hunter one plays stag with probability p so
3p and they get zero otherwise so this whole
thing just disappears and becomes 3P, and
the expected utility of Hunter 2 playing
here is they get 1 with the probability
that Hunter 1 plays Stag, which is P, plus
1, the probability times the probability
that Hunter 1 plays here, which is 1 minus
P, which equals 1. Equate them together
and we find the probability mixture for
player 1, 3P equals 1, P equals the third.
That is the probability mixture to make the
other player you do. Play stag with 33
.3%. And you can plug this back in. If p
equals a third, then the payoff for player
two playing stag is one, and the payoff for
playing here is also one. So, no matter
what they play, they get a payoff of one.
They're indifferent. And we do the exact
same thing for player two as well.
So, player two, hunter two, plays stag
with probability q, and here with
probability 1 minus q. They want to make
hunter one indifferent between playing stag
and playing here so because this game's
symmetrical whenever you have a symmetrical
game you don't actually have to do it for
both it's going to be the exact same but
for posterity let's just go through the
process one more time so the expected utility
of hunter one playing stag is i get three
when hunter two plays stag with probability
q three q plus i get zero when hunter
two plays here so zero times one minus
q which just gives 3q. And the expected
utility of playing here is I get 1
if Hunter 2 plays stag with
probability q, so 1q. Plus I also get 1
when Hunter 2 plays hair with probability
1 minus q, which equals 1. We
do the same thing here. So there's
actually a third Nash equilibria here
in the stag hunt game which is both
players mixing to play stag one
third of the time. And the interesting
thing here is, in almost all these
games that are finite there's an
odd number of Nash Equilibria, both
pure and mixed. So our coordination
games that were mentioned before our
Lucys and Cresswood you'll be able
to find a mixed strategy Nash Equilibria
there as well. In our zero-sum games
where there's no pure strategy Nash
Equilibria, there will be a mixed strategy
Nash Equilibria. In the Prisoner's
Dilemma you won't be able to find a
mixed strategy Nash Equilibria. There's
no reason for anyone to play anything
else besides the So this is
something important to keep in
mind, especially for these
coordination games. I don't want to get
into this today, we're going to next look
at what happens in these types of games
when they're repeated. We've looked at one
shot only so far, and the context of
these games can change a lot more. They're
repeated finitely, infinitely, and
stochastically as well. So we'll get into that
on Friday's class. all get out of here
early. Well I can't because I'm teaching
the next class but you can all get
out of here early.