Okay. Welcome back,
everyone. Hope you all had a nice and
relaxing spring break. We're into the
home stretch now, so only a few
more weeks left. Just a little bit
of housekeeping. Some of you may notice
I've kind of made an error on Brightspace
with the grades for this upcoming homework.
It's out of, what, like 200 points or
something like that? Yeah, and I can't edit
it or fix it. so it's out of four the current
setup is the grade doesn't count but the
percentage will let you know how you did
and once everyone's done it I'll manually
alter it so it's out of four points as per
all the other homework so don't worry too
much about that I just yeah made a little bit
error setting it up so the exam I've come
in in a few weeks time it's going to be topic
eight what we did before market structures
this topic here intro to game theory and
probably topic 10 the oligopoly stuff as well
and if you've noticed in the textbook
oligopoly comes before um the game theory stuff
which makes zero sense because oligopoly is
game theory so um i was speaking with jonathan
who teaches industrial organization here
which is all like the oligopoly stuff and he's
like i can't believe the textbook puts
that before the game theory so this is why
we're skipping to game theory and then going
back to do the oligopoly so so far in this course
we have considered individual decision
making and in these cases outcomes really
only depend on two things the action chosen by
the individual or a firm and the state of the
world so the constraints, the resources etc
so examples are how do consumers go about
maximising their utility these individual
decisions for firms how should they go about,
you know, producing, how many workers
should they hire, how many units of capital
should they use, et cetera. And
even when it comes to pricing or
producing quantity in markets, the markets
we've looked at so far have had no
strategic interaction. So in our perfect
competition situation, because every seller,
every firm is a price taker, it doesn't
really matter what anyone else does
because no one can individually affect
the price. And in a monopoly situation,
you're the only firm. you're not really
competing against anyone. So there aren't
these interactions with other agents,
you're just making your own individual
decisions to optimise whatever it is you're
trying to optimise. However, many outcomes are
not only affected by your own decisions
in life, but also the decisions of other
people or other firms. And we call these
types of interactions strategic interactions
when someone else's decision is going to
impact your payoff and your decision will
impact their payoff. So, to get us in
the mood, I thought we'd play a few games
to get us to go in and get into this
strategic mindset. So, I'm going to call up a bunch of volunteers. At first, I'd rather
people volunteer on their own than
me point out, but I will point
out eventually. So, can I get two
people? Yeah, Daniel? Yeah, come on up. Okay, so, can I get
one of you on this side of the table, one
of you on that side? Right. Brilliant. so
you're going to play a game called split
or steal this is a classic you've
probably seen something similar to this in in
econ 251 or even in in media so we have two
players our two players here and on the
count of three each player can choose an
action they can split by holding out a hand
handshake split or they can steal by
throwing up a cross like that you're not
allowed to talk to each other and coordinate
beforehand this is going to be at the
same time without any communication and
the outcomes are as follows. If you both
split, you can both choose one piece of candy,
whatever you want. No, don't, don't,
no, no. Turn around. I don't want any
contact whatsoever. If you both steal, you
get nothing. If one of you steals and
the other splits, the person who steals gets
two pieces of candy and the person who
splits gets nothing. We all understand?
Yeah? No more hidden
communication? Understand? Brilliant.
Okay, on the count of three, split
or steal, ready? One, two, three, go. Well done, guys. And you can take
a piece of candy. Okay, can I get two
more volunteers? Am I already going to
have to start pointing people up? Yeah, come
on down, brilliant. Want to have
a go as well? Yeah, what was your
name again? Clayton. Clayton. And what's
your name again? Emilia. Sorry? Emilia. Emilia
and Clayton, great. So, you're going to
play the exact same game, but I'm going
to give you 20 seconds before to talk about
what you want to do. Great. So, you
guys can have a chat, introduce
yourselves, and discuss what
you want to do. You don't need to say
anything else? Okay. On the count of three? One, two, three,
go. Well done. Okay, you can both take
a piece of candy. Where are my strategic
people? I thought you were going to have
a steal there for a second. I thought
about it. You thought about it? Yeah, I'm a
bad person. Okay, yeah. In front of the whole
class, it's probably not enough of a gain
to steal. I actually thought about that
in my head. There was more marginal benefit
to get no candy, but it seems like a
good candy. Yeah, yeah. No, that's very fair. Like, I was curious.
I was thinking about doing this for
this one instead of candy, doing it for
extra credit. Do you think you would
have stolen then? Probably not. Okay, interesting, interesting. Okay, two more players. DK, want to have a go? Brilliant. Brooke, do you
want to have a go? Okay, so this
one is called Right or Left 1,
which means there's another one
coming, obviously. So the way this
works is both of you can either choose
to put out your right hand or put
out your left hand. And the way this
works, also no talking aloud, no signals or
anything like that. Now, if you both put
out your left hand, you get one piece
of candy each. If you both put out
your right hand, you both get one
piece of candy each. But if one of you puts
out the left and one of you puts out the
right, you get nothing. We understand? On
the count of three. One, two, three,
go. Okay, you both put out the right,
you can shake and then take a piece
of candy. Well done. Let's go. Why did you
choose the right? I don't know. I feel
like most people are right-handed.
That's what I was thinking. Oh,
is that what you're thinking as well?
Okay, brilliant. Okay. This one's called
right-er or left -er. I need two
volunteers, actually, again. Anyone
want to volunteer? Yeah, brilliant.
Come on down. I've got someone
from this side of the room. What's
your name, sorry? Andrew. Andrew,
want to have a go? Brilliant. Okay, this one's
a bit interesting. So, okay. Okay. Andrew, you're going
to be the, you're on the left to everyone,
so you're going to be the left player.
Remind me your name. Kevin. Kevin, you're
going to be the right player. So you're
right, you're left. And the way this
works is, if you both put out your right
hand, Kevin gets two pieces of candy, and
you get one piece. However, if both
of you put out your left hand, then
Kevin gets one piece, and Andrew,
you get two pieces. But if one of you
goes left and one of you goes right, you
both get nothing. And in fact, I'm
going to allow you to talk beforehand for
like 30 seconds, try and figure out
what you want to do. So feel free to introduce yourselves and
have a chat. You're going to push back
against that at all? No, not at all. Okay, on
the count of three. One, two, three,
go. Okay, well done. Kevin, you get two.
Andrew, you get one. Andrew, why were
you so quick to acquiesce
there? Oh, I don't know. I feel
like negotiation. I'm more cool with
one piece of candy. Yeah, Kevin, you were like, okay, fair enough. I would have done
the same thing, but I mean, he beat me.
That's that's really interesting. That's
really interesting. Yeah Okay Okay, so I did
this one talking as loud, so you need
two more people Anyone want to have a go? Yeah, come down
have a crack Want to have a
go? Come on down Okay, what do I call
this one? This one is called I didn't think
of a good name for this game game so two
players on the count of three gladiator
style you can go thumbs up or thumbs down and
the way this works is if both of you
choose thumbs up you both get two pieces of
candy if you go thumbs down no matter what
the other person does you get a piece
of candy however if you go thumbs up and the
other person goes thumbs down you get
nothing make sense okay brilliant on the count
of Three, gladiator style, thumbs up or
thumbs down. Ready? One, two, three, go.
Both thumbs up, you both get two pieces
of candy. Well done. Okay. Okay, I think this is
the last one coming up, so this is your last
chance to volunteer. Who's competitive
in here? Who's got competitive
spirit? You've had a go already. Anyone who
hasn't had a go yet? You're wearing a,
you know, a football -like jacket.
That shows some straightness. Want
to come on down? Eric, what about you?
Want to have a go? You don't have to if
you don't want to. Okay, great. So, has anyone seen
Squid Game by any chance? So this is
kind of one from Squid Game. This is
called Odds or Even. So, you're going to
be the odds player, you're going to be
the even player, and on the count of
three, you either hold out zero fingers
or one finger. You're even, you're
odd. Odd, even. So the way it works is if
you're added up numbers equal one, if odd player
wins, if they don't equal one, so zero or
two, then even wins. And whoever wins gets
a piece of candy. Whoever wins? Yeah,
gets one piece of candy. Loser
gets nothing. Really? Understand?
Yeah, I'm odd. I'm odd, yeah. Yeah, you're odd,
you're even. You can speak about it
in 20 seconds if you want to psych
each other out or anything. I want
ten, you can have it. I'll put zero,
you go one, okay? Okay, that kind
of destroys my fun of the game.
For sure, if you want to do that,
you can do that. Well, if he's
a dog, I'd put one. Yeah, yeah,
sure, sure. Okay, on the count of three.
One, two, three, go. So, you're even,
yeah? I can take it, yeah. I just want
to check this out. I don't want it. Anyone else want it? Anyone want? Okay, great. So we're going to
come back to all these games and hopefully
you'll follow it along and see how they all kind
of differ slightly. And this is really
important at the end of the day, how
the setup of each game can change what
decisions people make. So what actually
is game theory? Game theory has
been used to study interactions in
many contexts. And many of these
contexts are very different to what
the popular notion of a game is. So when
someone says game, you're probably
thinking either like a board game, like
Monopoly, or like chess, or sports, or
something like that. And while a lot of
these can be analysed with game theory, it
goes beyond that as well. So for economists,
what the word game actually implies is a
process or interaction with a prescribed
population of participants, so a well-defined
amount of people that are in this game that
has a strict set of rules that everyone knows
and a set of payoffs associated to every
possible outcome in the game so in all the
games we just played all of these things
helped you knew how many players everyone knew
the rules and you knew what each person
would get in each scenario so what game
theory is is a formal way to analyse these
strategic interactions among a group of rational
players or agents. Rationality is
very important here. Who behave
strategically. So some examples
are sports are a big one. Anyone who follows
sports can probably see these. They
like to call these chess battles between
the coaches, etc. This is essentially
a lot of game theoretic
stuff going on. International relations.
Pretty hot topic right now. there's a
lot of game theory when it comes to international
relations from things like going to
war with other countries to solving global
problems like mitigating negative externalities.
In managerial economics and in business
there's a ton of game theory going on. We're
going to spend a whole topic in Oligopoly
talking about pricing decisions and output
decisions when you and another firm are
competing directly against each other. Advertising
decisions as well is another situation
related to game theory. Lying versus truth
telling. We saw Eric before tell a bold faced
lie. This is part of game theory as well
and we're not going to go into it in this
course but above and beyond there's some
stuff called actually psychological game theory
that a new professor here kind of founded
that field and it talks about things
like speech can change the way people play
games. and finally the classic stuff is just
looking at cooperation and defection so we
looked at this a bit in our market failures
with our strategy of the commons and our
public goods but this is a very common theme
in a lot of games so what's in the game
how do we define it a game has to be
comprised of the following components the players
who is involved but also does nature play
a role so for example the the the payoffs
that people get may differ if nature's one
thing or if nature's another for example
the day could be you know really hot or really
cold you don't know in advance if it's
hot or cold and this could have different
outcomes based on decisions you make to fish or
not fish for example actions what can each
player actually do in this game what are
their choices strategies these differ from
actions each player has a strategy that
comprises a complete contingent plan of action
this complete aspect is really important
what this means is for any given scenario
or any action that another player may do
the strategy specifies the actions to be
taken by the player so going before in one
of the examples we had split or steal the
actions are split or steal but the strategy would
be what do you do if the other person
splits and what do you do when the other
person's feeling. They could be the same thing,
they could be different things, but those
two things combined, that is your
strategy, the complete plan of action. Finally, well, not finally,
penultimately, we have information,
what each player knows before
making a decision. We're going to assume
perfect information, but there's a whole
class of games where you don't have perfect
information. You don't know a person's true
type, if they're actually talented or not, if
they're actually nice or not. hidden information
is a huge component of game theory and
finally payoffs a complete summary of the
value to each player of each set of actions
and which is known to all players so you've
probably seen this one before but I want
to go back to it and give the proper
explanation hey some of you may be a little bit rusty
with game theory but also there's a reason
why the prisoners dilemma has been studied
for like 80 years and there's still papers
being produced on it it's one of i think
the the most beautiful um aspects of economics
and also and also mathematics as well in
its simplicity but also its implications so
the way the story is set up is that there
are two individuals one and two they've been
arrested on suspicion of jointly committed
a crime they're placed in separate cells and
they're essentially asked to rat out the
other person to dog them in and say they committed
the crime so the each prisoner has two
options two actions they can either stay
silent and not say anything to the cops or they
can defect and kind of rat out the other
person if they both defect on their partner
so if they both say the other person did
it providing evidence they both receive a long
sentence of 10 years if they both stay
silent and cooperate not providing evidence
there's little evidence to prosecute and they
both only receive a sentence of one year
in jail however if one defects and says the
other person did it but the other cooperates
and stays silent there's evidence against
the person who stayed silent so they go
to jail for 12 years and the person who dobbed
in the other person gets to go free zero
years in jail so that's the basic
setup and based on our criteria before we've got
two players the actions that cooperate or
defect this is what we call a simultaneous
game all the games you played up here are
simultaneous where players make their
decisions at the same time. There's no ordering to
decisions, everyone's making their decision
instantaneously. Payoffs. This is a
complete summary of the value to each player
of each set of actions. This can be represented
for simultaneous games in a game table
and we call this the normal form game.
The normal form. So this is what the
normal form looks like. So C player 1 is our
row player and they can defect or cooperate
not cooperate they can cooperate and player
two is our column player they can defect or
cooperate as well so each player's two
actions and the way the payoffs work is when
you've got a two-player game the payoff on
the left is always the row player or player one
and the payoff on the right is going to be
the column player or player two so you can
see if they both defect they both go to jail
for 10 years if they both cooperate they
both go to JAL for one year. If PiO1 defects
and PiO2 cooperates, PiO1 goes free, PiO2 goes to JAL
for 12 years. And if PiO1 cooperates
and PiO2 defects, PiO1 goes to JAL
for 12 years, PiO2 goes free. So, some of you have
seen this before. Feel free to answer
what should H -Pi actually do in
this game and what? Who's seen this before? Who remembers it? I'll go up there. I'll come back to you. Yeah? I think this is one's best interest to detect. Why do you think it's in one's best
interest to detect? Because I think, if
I want to remember, let's say you're
player one, you kind of have to
assume that player two is going to be
a winning strategy of detecting.
Yeah. So if you cooperate, you're
going to get 12 years. So it's in your
best interest to detect within
10 years. And in case the other
player is naive, then you go home
free of time. Great, great. You've pretty
much hit the nail on the head. And I
pretty much mean you have hit the nail on
the head. I'm going to tease this out a
bit more so we can find out how people
will act in other types of games as
well. And then I'll bring it back to the
prisoner's dilemma. Okay. So the cornerstone
of game theory is a concept called Nash
Equilibrium. And it's important to clarify
that Nash Equilibrium doesn't necessarily
mean an outcome is good or bad or desirable
or non-desirable. it's just a solution of how
people will act given incentives that's
all it is so the way we define national
equilibria is that national equilibria this is
the formal definition and then i've got a
little bit more of a layman's definition
after is a strategy profile such that each
strategy in the profile is a best response to
all other strategies in the profile and
what this means is And national equilibria
is a strategy profile where no player
in the game has an incentive to
unilaterally deviate from their strategy. So
everyone's playing a strategy and no player
can be better off by changing their
strategy. And if that holds for everyone in
the strategy profile or for every strategy
a player chooses, then we're at national
equilibrium. We're at this constant
point where no one can change. In other words,
you can't do better given what I'm currently
doing I can't do better given what
you're currently doing we're at this stable
state where no one will change that's what
national equilibrium is here and importantly
we are stuck there for good or bad and
in the prisoner's dilemma you'll see we're
stuck there for bad ah so the way we
figure it out national equilibrium I'm going to
teach you the heuristic but Remind me your
name again, sorry? Thava? Ataba, yeah. As
Ataba said, it's kind of about your
own incentives. What makes you better
off? So what we want to figure out is what
would each person do if they knew the
other person's action? So player one,
we want to see what player one
will do given they know what player
two is doing. So player two
decides to defect, and player one can
defect and spend 10 years in
jail, or they can cooperate and spend
12 years in jail. Obviously, it's better
for them to spend two years less in
jail, so they're going to choose defect. So
we highlight this. Now, Player 2's other action is to cooperate. So if they cooperate, Player 1 can
defect and spend zero years in
jail or cooperate and spend one
year in jail. Zero years is better than one year. We
highlight zero. Now we want to do
the same thing for Player 2 as well.
what will player 2 do given player
1's action is set? So if player 1 defects, then player 2 can defect
and spend 10 years in jail or cooperate
and spend 12 years. 10 is better than 12,
we highlight this. And if they cooperate, player 2 can defect
and spend no years in jail or cooperate and
spend 1 year in jail. 0 is better than 1
year in jail, so we highlight this. So this
is what it will look like after we've figured
out each player's best response to the
other action. And the heuristic is that if
you have any quadrant like this that has two
things highlighted, this is going to be
a Nash Equilibrium. And now I want to
give you more of the intuition
behind it than the heuristic. So think
about it. If both players are currently
playing defense, our definition of Nash
Equilibrium is that no player can
unilaterally deviate from their current strategy
and do better. So if player player 1
deviates, that means they choose cooperate
instead of defect, even player 2 is
playing defect, they end up here and they
spend 2 extra years in jail, so this is
not good for them, and because this is
a symmetrical game, it's going to be the
same thing for player 2, if they cooperate,
they spend 2 years in jail, so this isn't
a Nash Equilibria, but the reason why
cooperate cooperate isn't a Nash
Equilibria, is if both players are playing
cooperate cooperate, if player one has
an incentive to deviate from co
-operate to defect, given player
two's playing co -operate, they
go from one year in jail to zero
years in jail. So both players
in this case will actually have an
incentive to deviate. So you've got to look at what the current state is and whether
people actually, given the other
person's strategy, have an incentive to
deviate or not. That's how you figure out
both in terms of the heuristics and
the understanding. And in fact,
the prisoner's dilemma goes a
little bit further. So we have this national equilibrium of
defect-defect. So, in fact, in the
prisoner's armor, one of the two
interesting components here is that defect is a dominant strategy
for both players. What this means
is, no matter what the other
person does, you're always better
off defecting. If the other person defects,
as I said before, you should defect,
and if the other person cooperates,
you should also do it. So in this case, it
actually doesn't even matter what the
other person chooses, you know you're always
going to defect. so that's what a
dominant strategy is and it's um strictly dominant
if it always gives a higher payoff than
every other strategy and weakly dominant if it
doesn't do worse at any point than any
other strategy and there is an important difference
here the game we played at the start
split or steal isn't actually a prisoner's
dilemma for this reason and i'll come back
to that probably on wednesday okay so what
makes the prisoners only interesting is that
while we have this um the incentives and the
dominant strategy leading us to here there's
actually a better outcome for both players
if they both cooperated they both cooperated
they both only spend one year in jail compared
to 10 years in jail so they're both better
off by a fair margin but we know that no
player can unilaterally choose to cooperate
because the other player always has an
incentive to defect and vice versa as well so
the challenge of the prisoners dilemma in all
its variety is is how can we get people to
get to the good outcome to cooperate and it's
not always like this jail setup it can be
things like public goods games have this
kind of set up as well and this is what's been
not puzzling but has been i think um uh you
know and engaging a lot of thinkers like
this was huge in the 80s and 90s about figuring
out strategies that can do better at allowing
for better welfare and cooperation it's
been extended to many forms involving outside
things like institutions and contracts and
things like punishment and social norms a
whole bunch of different things to see how
we can move from the national equilibrium
the incentives of the game to get a better
outcome for both players so this is a really
rich field in terms of what people have done
in terms of research okay any questions
about that yeah I'll say like for the prisoners
you don't really want to think of like player
one and player two because you know there's
that aspect of it where like you don't
want to just if they committed the same crime
you're like friends or whatever you
wouldn't just want to flip on the person yeah
so this is the thing it's like um reputation
and punishment are really important so
we're We're currently looking at what we
call one-shot games, but as you'll see, later
in this week, we're going to look at
repeated games as well. Things like reputation
matter a lot. When you can punish people
that have wronged you in the past, that
matters as well. So there's a lot of
complexity to such a simple game, and
hopefully we'll be able to unfold a fair bit
of that later this week. That's why there's
a lot of beauty to it. There's just so
much to talk about. so there are many
different types of games so simultaneous games
the games we played and we've started
to look at so non -cooperative games like
the prisoner's dilemma where the incentives
the dominant strategy lead to this non
-cooperative outcome we have zero-sum games
so a zero-sum game is a game where one
person wins only if the other loses so sport
pretty much or playing a game rock-paper
-scissors with someone. Someone only wins
when the other loses. There are also
cooperative games though. Players can
agree on contracts, implement joint
strategies and these types of coordination
games as well that we played as
Austro in a second. Then on the other
hand something we'll look at
either on Friday or Monday are
sequential games. So in simultaneous
games everyone's making their decision at
the same time but in sequential games you
don't make your decision at the same time as
anyone else. It's ordered. So DK will
move first and I'll move second. DK could move
again or it could be Logan and so there's a
whole bunch of different setups for sequential
games. And the thing is the move order
may matter. Who moves first matters in
these games sometimes. So things like
bargaining. So bargaining over
something like salary for example. So you
get an offer from the employer and you can
negotiate that wage or just accept it or
reject the offer. So the ultimate admin
game is the workhorse game we use in this
case to talk about these bargaining situations.
The trust game. Should you invest in
something? How do you know that if you invest
and make a profit someone's not going
to just come along and steal it all so the the
trust game is another staple game in economics
that uses um that models like the
importance of institutions and trust between you
and others and you and society you and
government etc finally as we just talked about
the difference between one shot and repeated
games and whether the repeated game is finite
or infinite makes a big difference as
well so games can have a lot of different
components this is why there's like multiple
courses on game theory and we're only going to
really like touch the surface here um dip
our toes in the water of the um game theory
ocean so we've looked at the the classic um
prisoners alone but there's also other
classic games so this is a pure coordination
game so the story goes as follows you and
your study partner are planning to meet at noon
at one of two coffee shops lucy's and
crestwood unfortunately you didn't specify
which one and both your phones are dead so you
don't have any way of getting in touch with
it with each other before noon so if you
manage to meet each other at noon you both get
a utility or a payoff of one otherwise you
get a payoff of zero so you're on the left
here you're on the road player you can
choose to go to lucy's or crestwood your study
partner the column player can choose to go
to lucy's or crestwood if you both end up at
the same coffee place you both get a payoff
of one so lucy's lucy's one one crestwood
crestwood one one but if you both go to
different places if i go to lucy's and they go
to crestwood we both get nothing and same
if i go to crestwood and they go to lucy's
so what is the national equilibrium in this
game does anyone want to take a crack at what
we discussed before daniel do you want to just tell us
your thoughts okay so who going
to crestwood those thoughts are
like perfect like I get where you're
going but we can play this out in
the exact same way we did before so
what we want to figure out is what
will each person do given the other
person's action is held constant
so if your study partner
goes to Lucy's then you can go to
Lucy's and get a payoff of 1 or go to
Crestwood and get a payoff of 0 you're
going to go to Lucy's and if they go
to Crestwood you can go to Lucy's
and get a payoff of 0 or go to Crestwood
and get a payoff of 1 you're going to choose
to go to Crestwood so we've figured out
what's best for you the row player and
we can do the exact same thing for the
column player as well so if we go to
Lucy's then the starting partner can go to
Lucy's and get 1 or go to Crestwood
and get zero they're gonna choose one so
we highlight it we do the same thing if we
go to Crestwood it's better for them to
go to Crestwood then go to Lucy's so
remember how heuristic what does our heuristic
tell us here what are the national
equilibria Daniel sure sure you can also pass
if you don't want to answer as well you
can pass for now does anyone want to jump in
yeah okay that's that's really interesting
Like I get where you're going in terms of like
how would I know to choose Lucy's or Crestwood's
and we'll actually dig into that in a
second But think about it our definition of
national equilibrium is simply It's a situation
where no person has an incentive to
unilaterally deviate from the strategy Giving
everyone else's strategy So when both are
playing Lucy's does anyone have
an incentive to deviate from
playing Lucy's? No, because if you
play Crestwood instead of Lucy's you now
get zero instead of one you don't have an
incentive to deviate same if you're both
playing Crestwood no one has an incentive
to deviate from Lucie's so in fact,
in this situation we have two pure
Nash equilibria here two pure Nash equilibria
here so Lucie's is a Nash equilibria
and as Daniel said before, Crestwood is
a Nash equilibria so both are Nash equilibria
in this game and as people are probably
thinking what does this mean this doesn't
help me to decide whether to choose
Lucy's or Cresswood in this game. And you're
right, we have other things that we'll
actually talk about at the end of this lecture
to help us figure that out in a way of
how would you go about it. But what this is
saying is, given the other person's
strategy, you don't have an incentive to deviate
from that strategy. So both of these are
Nash equilibrium. And why is this a pure
coordination game? The reason why
it's pure is because you both
get the same pile. You both get one, and
you both get one. so you want to coordinate
with each other because you both do just
as well as each other on the other hand
um we have an impure coordination game so
this is historically known as the battle
the sexes game i i kind of want to get away
from that framing a little bit but this
is what it is here we have alice and bob
and they want to go on holiday together they
they you know get more enjoyment spending
time with each other however they have
different preferences So Bob has a stronger
preference for going to Florida,
and Alice, for some reason, wants to
spend time in Ohio. So what is the
national equilibrium of this game? So
as you can see, if Alice and Bob
both choose Ohio, Alice gets a
payoff of three, Bob gets a payoff of one. If they both
choose Florida, Alice gets a
payoff of one, and Bob gets a
payoff of three. So does anyone want to
take a guess at what the national equilibrium
are in this game? yeah DK yeah so you
don't think the payoffs change
anything here oh well I guess they're probably going to pick two different situations but the
national defense is the same exactly even
though one prefers one over the other the
core idea is still the same given they're
both playing Ohio does either player have
an incentive to debut obviously Alice doesn't
she's doing her best but Bob doesn't have
an incentive to deviate either if he chooses
Florida instead of Ohio when they're both
playing Ohio he gets zero instead of one
so he has no incentive to deviate so it's
the exact same as last time you have two
national equilibria both play Ohio or both
play Florida so this is an impure coordination
game for the reason that you kind of
stated DK if you could select which of the
equilibria were chosen Bob would prefer it
to be Florida florida and alice would prefer
it to be ohio ohio all right so this game
is called the stag hunt um this is the game
that we played where i said i didn't have
a good name for this game because i don't like
calling the stag hunt i'm vegetarian but
this is what it is in the literature so i'll
just run with it and the way this works is
there are two players deciding what animal
they hunt both hunters need to cooperate to
catch the stag otherwise the stag can't be
caught however on their own they can always catch
a hare by themselves however it provides
less meat than the stag so as you can see
here if they both choose stag they both get a
payoff of three however if you choose the hare
no matter what the other person chooses
you always get a payoff of one and if you
choose the stag and the other person chooses
to here you get nothing those are the playoffs
here so what are the Nash equilibria or what
is the Nash equilibria in this game yeah
Daniel so choosing both stags would be one
Nash equilibrium because given that the other
player jumps in that to stack you have to
choose stack you have to do something and then
so so just I'll just stop you there so if
you're both playing stag you're right you're
both getting three if Hunter 1 deviates
from fine stag to hair, given that hunter to
fine stag, they go from three to one. So they
still get one, but this is less than three.
So they don't have an incentive to deviate,
like you said, but they still get something
if they deviate. That was just a
clarification I didn't make. Continue going? With hair, I
guess that's an interesting case
because the person on the right that
shows hair will still get one,
despite deviating. so yeah it's an
interesting case yeah so okay daniel if both
players are playing the strategy hair does
either one unilaterally on their own have an
incentive to deviate from hair to stack
not really i mean i mean it's one of them's
a different the other one's like just going
to pick hair because okay so so this is an
important clarification you're almost right
but both of them aren't indifferent both
don't actually have an incentive to deviate
so if we look at hunter one if they're
both plain hair, if they now play stag,
given Hunter 2 is plain hair, they end up
here and get nothing. For Hunter 2, if
Hunter 1 is plain hair, and they now play stag,
they get 0 instead of 1. So they're
both actually worse off if they switch
from hair to stag when both are plain hair.
So neither have any incentive to deviate.
So we once again actually have two
Nash equilibria here, stag, stag,
and hair, hair. and we call these equilibria
slightly different things so this one
here is what we call the payoff dominant
equilibria both players get their best payoff
here but this is also equilibria and we
call this the risk dominant equilibria
because it doesn't matter what the other person
chooses you always get the same payoff no
matter what now you're probably thinking like
who in the hell would ever choose hair in
this situation obviously you should both choose
stack and i can't remember who played this
one here but you both like kind of chose a
stag one and got two pieces of candy each
um who played that one remind me yeah and
yeah okay so you're probably thinking in what
situation would anyone choose hair well
i've got a situation coming up in the next
lecture where i promise you none of you would
choose the equivalent of stag in that
situation um i can change the payoffs in a certain
way where you just don't want to take the
risk essentially okay finally we have um
matching pennies so this was our like squid
game odd even game um it involves two players
even and odd who each have a penny each
player must select one side heads or tails
and at the same time show the penny to each
other if they match the even player wins
they don't match the odd player wins so as
you can see the even player wants them to
match so heads heads they get one tails tails
they get one and when they get one the other
person gets minus one so they win the
other person loses and vice versa so when it's
heads and tails the odd person wins the
even person loses tails heads the odd player
wins the even player loses okay is there
an equilibrium in this game no exactly there
isn't so as you can see what I've done
here is if player two who's odd plays heads
then player one would rather play heads and
tails one is better than minus one winning
is better than losing if they play tails and
player one will play tails and so so forth
and as you can see none of the quadrants
have both of the payoffs highlighted so
what this means is let's say we're at heads
heads then the odd player has an incentive
to deviate from heads to tails and any
quadrant if you know what the other player
is doing and you're currently losing you're
gonna have an incentive to change your
decision and vice versa so there is no pure
Nash equilibrium here. And we call this a zero
-sum game. Remember, if you add up the
payoffs in each quadrant, it adds to
zero. So one player wins only when the other
player loses. And in situations like rock,
paper, scissors, you can draw, but you
both get zero in that situation. So the
payoff is zero. And one person can only win
when the other loses. Okay. However, there are
some strategies that go into whether you play
heads or tails in our coordination games
there are some other things that go into
making decisions do you choose loosies or do
you choose westwards and something that can
help us here is something called mixed strategy
Nash Equilibria so strategy profiles
don't have to always be pure strategies
you don't have to play heads 100% of the time
you don't have to go to loosies 100% of
of the time. You can choose 60% of the time
you go to Lucy's, 40% of the time you go to
Westwood for example. So essentially you can
assign a probability P to play in one action,
in this case heads, and one minus P to play
in the other action tag. So you can mix
your probabilities. Some games have no
pure strategy but they have a mixed strategy
in our Shea Equilibria. and this is a
kind of a lengthy definition but
I'll try and explain a little
bit more in terms of intuition on
the next page so a mixed strategy
Nash equilibria is once again this
stable state of a game where players choose
their actions
probabilistically so you're not
playing something necessarily with a 100%
chance you're mixing how likely you are to play
between your actions and these probabilistic
choices are the best possible responses to
each other, meaning no player can improve
their expected outcome or their expected
payoff by unilaterally changing their
probability mixture. So if both players are
playing a probability mixture of their strategies
when no player can do better by changing
their own mixture, we're at a mixed
strategy Nash Equilibria. So I think it's more
intuitive to talk about this idea of
mixed strategy Nash Equilibria in zero
sum games but these extend to all types
of games as well so we'll be able to see
it in the stag hunt of the coordination
games as well which aren't zero sum the
reason why is the idea behind mixed strategy
Nash Equilibria is you don't want
the other player in this case zero sum your
opponent to exploit your strategy to
exploit your strategy you want to find
a place where no matter what option your
um opponent chooses it gives them the
same expected pale so you want to make
them indifferent between their pure strategies
indifferent between heads and indifferent
between tails so think about sports why mix
your strategies so um any any soccer fans
in here dick i know you're a soccer fan you're
taking a penalty and you know this goalkeeper
dives left 100% of the time, where are
you kicking the ball? You're exploiting
their strategy. If you know this
goalkeeper dives left 80 % of the time, where
are you kicking the ball? To the right,
yeah. So we can lower this percentage all
the way down to 51% and it's still always
in DK's interest to exploit this strategy
by choosing one of his strategies,
kicking right over the other kicking left.
He's not indifferent between them. So that's
a simple one there. In NFL, if a team
blitzes every single play, you're
probably going to be able to exploit that
strategy as well. A big one is in tennis. When you serve, you
can choose to serve to the left or to the
right. And if you know the server always
serves to the left, as the receiver, you're
going to stand, or to your right, to their
left, essentially. So this is why, and
we'll see, there's really cool data on
this, that tennis players want to mix where they
serve to the point where their strategy
can't be exploited by the receiver a
simple one as well is this one here in rock
paper scissors Logan if we were playing and
you knew I played rock a hundred percent of
the time what would you choose paper if I
played rock 70 percent of the time what would
you choose paper every time if I chose rock
I'm 50 percent of the time what would you
choose paper still yeah my strategy has been
exploited the expected payoff that Logan gets
from plain paper is always going to be higher
than the other ones so what I need to
find is a probability mixture of rock paper
and scissors or in the penalty kicks diving
left or diving right as the keeper as a server
in tennis serving left or serving right that
makes the other person indifferent between
their choices and we'll go into the mathematics
of that in the next lecture and this
will help us also think about these coordination
decisions as well. That's all I have
for you today. I'll see you all
on Wednesday. Yeah. Hi, how are you? Good. How can I help? So, you know
the last part we talked about like
the goalkeeper is it like a double
block kind of thing and then that can
keep going on the goalkeeper also
could be knowing that you know yeah so that's
there's something a little weird
but as you'll see thanks Kevin there are some games
like when we have it quite simple there are
stable states so this is the thing I don't
know if you're an NFL fan at all but there
are cycles in strategy like every few years
like the meta changes because a strategy by
offense exploits the defense and then the
defense all changes to exploit this new like
type of offense and this is changing back and
forth so what you're saying in terms of
things like double fluffy and like in cricket
when like someone walks across their stumps
and walks back things like that so this is
like you've got to just think what all the
pure strategies are what people can do and
how would you mix your strategies to make them
indifferent between staying or walking
across their stance or walking up the pitch or
it almost sounds like like because you kind
of have to assume that they are like you
know there's so much data yeah you kind of
have to assume they know what you're attempting
yeah yeah it's almost like you have you
should have no strategy well this is the thing
so when you're when you're when we talk
about mixed strategy national equilibrium
your own strategy doesn't matter your own aim
is to make the other player indifferent
between their strategy if they're indifferent
between their strategies they can't exploit you
and then if they're doing the same thing
they're trying to make you indifferent between
like you go york or a bouncer etc and if
both like players are optimally doing this
then they were at a point where no player
has an incentive to change their strategy
in any way yeah so it's not the best outcome
for yourself but it's like no no no it's not
the best outcome for yourself it's making
it so think about like So if you're the
goalkeeper and you dive left 50% of the time,
dive right 50% of the time, they've got to
score 50% of the time.