Original subtitles

so we're going to do a new topic

today we're going to talk about

Evolution we're going to talk about the

connection between evolution and Game

Theory all right now there's going to be

a uh an extra I think I already emailed

you about this there's going to be an

extra reading on this for the people who

want it uh I'm G to uh uh there's a

reading packet that's available to

anybody who wants it uh I I emailed you

all the websites where you can order

that reading packet and find and figure

out where to pick it up it's not

compulsory you don't have to look at

that reading packet it's just it might

help in addition as with last Wednesday

since some of the material here is new I

have written a handout that goes with

this lecture and that handout will

appear magically this afternoon on the

website all right so things today are

fast don't worry there's a handout that

goes with it all right so why why look

at evolution in the context of Game

Theory there are really two reasons the

first reason is because of the

influence of Game

Theory on

biology right it turns out that in the

last few decades there's been an

enormous amount of work done in uh

biology in particular looking at animal

behavior

and using Game Theory to analyze that

animal behavior and just to give you a

loose idea about how this works uh the

idea is to relate uh strategies with

genes or at least the phenotype of those

genes and to relate payoffs to genetic

fitness all right so the idea is that

strategies are related to genes and the

payoffs in the game

are related to genetic

fitness and the big idea of course is

that strategies uh grow if they do well

all right so strategies that do well in

these games grow strategies that do less

well die out all right so we're going to

be visiting that today one uh thing to

bear in mind from the start is there's

an important difference between Game

Theory as analyzed in

uh animal behavior in biology and game

theory that we've been doing so far in

the class and that is that we're going

to think of these uh behaviors these

strategies played by animals not as

chosen by reasoning individuals but

rather as being hardwired so strategies

grow it isn't that that some lion or ant

has chosen that strategy it's simply

that the ant or lion who has that Gene

that corresponds to that strategy grows

and has many children all right so it's

important to i this this is a new idea

for us that these strategies are

hardwired all right so that's

interesting in itself but there's also

an important influence the other way so

a second reason for studying this

stuff is that there's been an enormous

influence

from

biology or from evolutionary biology in

particular back on the social sciences

right this is this this I this this

influence the other way uses Evolution

largely as a metaphor and you'll find

this if you're a political scientist if

you're a historian if you're an

anthropologist if you're a sociologist

let me give you an example from

economics since it's perhaps closer to

home so here's an example you might

imagine some firms in the marketplace

and you could think of these firms not

necessarily reasoning out what is the

most profitable strategy for them or

what is the most cost a reducing

strategy for them however uh does have

rules of thumbs to select their

strategies however in a competitive

Marketplace survival of the fittest

firms will lead to Us ending up with a

bunch of firms who have low costs and

high profits all right that makes sense

so here competition in the marketplace

uh substitutes for competition in the

jungle as it were all right and the the

analog the analog of uh of uh uh

a gene dying out is a firm going

bankrupt all right so this is the kind

of ideas we're going to explore these

are the ideas are going to be in the

background for today and Monday now

before I go on I want to put in a couple

of kind of important writers so the F

there are two things that I am not right

the first thing I am not is I'm not a

biologist I'm sure there are people in

this room who are biologists who know a

lot more about the genetic side of this

than I do right so I'm not going to

attempt to teach you biology here I'm

going to focus on the game theory part

right the second thing I'm not is I'm

not an American citizen and uh I I worry

slightly when we talk about Evolution

because I I realize it's very

controversial here and some of you might

be from places like say Kansas uh and if

you are from some place uh like Kansas

uh then please don't write to your

Senator about this all right or if they

see the movie just tell them it with

some other English guy giving the

lecture all right so what we're going to

actually do is we're going to look at a

a highly simplified model of evolution

at least for today right we might get on

to a more complicated model on Monday

but today it's going to be very highly

stylized so this is going to be our our

extremely simplified

model and this simplified model is going

to do tremendous violence to the biology

but it's really just a fix some ideas

okay so we're going to focus on what

we're going to call Within in species

competition so the lions are competing

against the Lions and the Ants are

competing against the ants and the way

we're going to think about this is we're

going to look at

symmetric two player games so we look at

very simple

games they're only going to involve uh

two players and they're going to be

symmetric games which means both players

have the same strategies both players

have the same payoffs all right

the way we're going to think of this is

we're going to imagine that there's a

large population out there each of whom

is playing a particular strategy and

we're going to assume that what happens

is we randomly uh pick uh two people

from that population and pair them up so

everyone in this large population will

be randomly paired with someone else

they'll play the strategy that they are

hardwired to play and then we'll see

what happens right so the idea here is

there a large population

of of uh if you like animals that are

hardwired to play particular strategies

and they we're going to have random

matching and the idea here is when we do

lots of these random matching we're

going to keep track of the average

payoffs all right so what we're going to

focus on are the average payoffs of

particular strategies when randomly

matched in these games all right

all right and the again the underlying

idea is that

relatively relatively

successful

strategies will

grow and relatively unsuccessful Ones

Will decline I'm not going to write that

that's obviously the other part of that

so relatively successful strategies are

growing and relatively unsuccessful

strategies are declining all right now

to keep things simple I'm not going to

do any Dynamics here that would take is

beyond the math that you could probably

a lot of you can do in the class so

we're just not we're not going to worry

about Dynamics here but this is the

underlying Dynamic the idea is that if

strategy is successful that strategy

will grow in the population if a success

strategy is unsuccessful it will decline

so this isn't this isn't horrible

violence to biology yet it's the next

bit that's the horrible violence all

right so to keep things simple today

we're going to assume that there is no

Gene redistribution

all right we're going to assume that

there's no Gene redistribution so in in

principle what we're looking at is

asexual

reproduction which is clearly not a good

model but it will do for

today all right so you want to think

about a a practical example of asexual

reproduction think about root vegetables

uh or judging from our experiments with

the dating game we better hope that's

true for e ecomic Majors as

well all right maybe the fact I'm

focusing on asexual reproduction will

get me will get me off uh with a guys in

Kansas maybe I won't lose my green card

after all all right so that's going to

be our basic

story and the basic idea we're going to

use is this it's an idea due to a guy

called mid Smith in the 70s although

obviously the big idea goes back earlier

so we're this is the big idea suppose

that we imagine that uh there's a

there's a particular game

and there's a large population so think

of yourselves of the large population

and suppose that the entire population

the entire population were all playing

the same strategy call it s so they're

all hardwired to play the same strategy

s and suppose now that there's a

mutation right so some small group start

playing some other strategy let's call

it S

Prime right what we want to ask is will

that small ation group the S Prime

strategy will it will it Thrive or will

it die out if it's true that for all

possible mutations all possible little

groups of of of mutation of people

playing S Prime they'll die out then

we'll say that the original strategy the

strategy s is evolutionarily

stable right and we're going to write

that up more formally later on but

that's just the basic idea again there's

a strategy out there that everyone's

playing call it s we're going to look at

a mutation that's S Prime so a small

group of people are going to start

playing S Prime they're going to go on

being randomly matched everyone's going

to be randomly matched and we're going

to ask if the S Prime group does well in

which case they'll grow or does badly in

which case they'll shrink and eventually

die out if they die out we'll say that s

was uh evolutionary stable that's true

for all possible mutations all right and

just notice when they're when we're

randomly matching them one thing to not

is since there's only a small mutation

to start with most of the time when

they're randomly matched they're going

to match against somebody who's still

playing s occasionally they're going to

meet one of the other mutants but most

of the time we're going to have to worry

about how the mutants do against the

incumbent

population all right that was too

abstract to really get one's head around

so let's try and do an

example all right so let's remove our

motivation and get down to actually

doing some

work all right so we're going to start

with a very simple example that you'll

all

recognize here's a game it's a 2X two

game and the payoffs are as follows 22

03 30 0 and

one1 all right we'll call these

strategies cooperate or

defect C for cooperate D for defect C

for cooperate D for defect all right and

you'll will recognize immediately that

this game is work this game is work this

is prisoners dilemma right so we're

going to start by imagining these

animals playing prisoners dilemma and to

put it into context imagine that these

are a group of lions all right again

leave aside the fact that a sexual

reproduction for a second imagine these

are a group of lions and cooperating

means cooperating on the hunt it means

using a lot of energy going after as you

corporate in a group while hunting and

defecting here would mean uh not uh

working hard on the hunt letting the

other Lions uh uh catch the antelope

whatever and they're just sharing in the

spiles free riding basically all right

or another example think about ants uh

with an ant nest uh and imagine this ant

nest has been attacked by I don't know

what some other creature you could

imagine that cooperating is joining in

in defending the nest at the risk of

being hurt uh and uh defecting is

running away all right all right and we

all know in this game roughly how to how

to analyze it I'm not going to uh go

over that now I want to ask a different

question today I want to ask in this

model of asexual

reproduction is

cooperation is

cooperation evolutionarily

stable is Corporation evolutionarily

stable all right

okay so to try and illustrate that let's

just think about this game being played

for real out there so what we're going

to do is we're going to start off by

imagining that you are all I don't know

ants Lions what do you want to be ants I

think all right all right so you're from

up here look like ants right you're all

ants and all of you have been hardwired

to play the strategy

C all right you'll been wired to play

the strategy C all right so life's going

on fine and Let's do let's do some

random matching here so suppose that's

going to get me eventually suppose that

we randomly match this ant right so Cent

per second uh with right so this this

ant gets randomly matched with uh with

this ant all right all right so they now

play this prisoners dilemma against one

another right but both of them have been

hardwired to cooperate right both been

hard to cooperate so they both uh they

both cooperate is that right right so so

this an who whose name is Lenor Lenor

the ant cooperates and and woo the ant

cooperates so they they both cooperate

uh that we can look what their payoff is

their payoff is Corporate against

corporate so they get they get two so

that's pretty good so they they they

they're doing pretty well in terms of

genetic fitness and suddenly there are

two ants right so this this ant produces

another ant well that's only without a

computer okay so so this ant produces

another ant all right and this ant

produces another ant sorry right all

right so so they're doing fine right and

the whole population is matching against

other Cooperative ants and more ants are

being produced and everything's fine and

dandy right everything looks good all

right right sit down a second sit down a

second right now imagine that there's a

mutation a small mutation and this small

mutation involves a group of rather

nasty uncooperative ants all right it's

rather scary mutation so here's our

scary mutation all right let's use the

Tas right so the Tas are my scary

mutation so all the Tas here are the

scary mutation there's another one over

there all right so there's there's a

everyone else is still playing uh

corporate there's this vast number of

corporate ants but now there's this

small number of ants with this mutation

that says Don't cooperate play D all

right so let's see what happens with

random matching so a lot of these uh

corporate evants are still matching with

each other and doing fine but that's not

the point what we're worried about is

what's going to happen to the mutants

all right so let's pick our

representative nasty mutant so that's

going to be Rahul all right right so rul

is our is our scary mutant all right and

let's go back and and randomly match him

against one of our our incumbent ants

our nice corporate events uh so let's

let's pick this one all right so y uh so

so it just happens he's randomly matched

with this uh corporate EV an and this

corporate an whose name is Nick Nick is

hardwired since his since he's a regular

ant he's hardwired to play cooperate so

he's going to play cooperate say

cooperate all right I should put the mic

of in mind uh and meanwhile do we have a

can I have a mic

handy thank you all right so he says

cooperate but unfortunately he's been

matched against the nasty uncooperative

and it's not that he's nasty he's just

been hardwired to say not cooperate not

Co to say defect all right so what's

going to happen now all right so

unfortunately unfortunately for Nick uh

he The Cooperative ant uh was matched

against a not very nice ant he was

playing C and rul was playing D so

Nick's payoff is zero so we went from

one Nick to being no Nick right he got

wiped out all right and Rahul meanwhile

he got a payoff of what what was rahul's

payoff r p three so suddenly there's not

just raal but there's

there's limitation is growing all right

all right and now we go on matching each

each each period some of the some of you

are going to be matched against

Cooperative other Cooperative ants

you're going to be doing fine but every

now and then and actually increasingly

often you're going to be matched against

one of our mutants and those mutants are

going to grow those mutants are going to

go on growing now it's true that every

now and then a mutant's going to meet

another mutant is that right right but

we don't have to worry about that what

we have to worry about is are they going

to die out uh uh early on it's going to

be very rare for them to meet meet

another mutant at least early on right

so the strategy cooperate is

evolutionary stable if this small

mutation there our small mutation

disappears and doesn't turn into a

bigger mutation all right and of course

in this case as we just saw the mutation

didn't die out it actually

grew all right so be formal about this

and be careful the strategy cooperate

for the strategy cooperate to be

evolutionary stable it needs to be the

case that the mutation actually dies out

on average and here far from dying out

it grew we now have lots of lots of

these nasty tats right people understand

all right let's just thanks all right so

let's just try and do that a little bit

more formally on the

board right that was the basic

idea basic Graphics of the idea but

let's look at it a little bit more

formally so we basically have co uh

Cooperative ants and a few mutant

Defector ants out there a mutant not

cooperate ants and what we want to keep

track of are their average payoffs in

random matches all right so the

incumbent ants The Cooperative ants

they're playing interested in their

paying in their payoff and they're

playing a population that's mixed almost

everybody in the population is

cooperative so let's say one minus

Epsilon of the people they're playing

against where Epsilon is a small number

small very small number are also

Cooperative but every now and then like

our poor friend Nick every now and then

they're going to come across a nasty

mutant they're going to come come across

rul all right so what's their payoff

going to be on average their average

payoff okay so 1 minus Epsilon of the

time we can actually think of this mix

here so it's one minus Epsilon and

Epsilon one minus Epsilon of the time

they're going to meet another

Cooperative ant and get a payoff of two

that right but Epsilon of the time

they're going to meet Rahul and get a

payoff

of zero is that correct so that's their

average payoff now how about the mutant

payoffs so the mutants that aren't many

of them around but they are also playing

against the same mixed population one

minus Epsilon of the time they're going

to be matched against a Cooperative ant

and Epsilon of the

time the two mutants the two Tas are

going to play off against each other all

right uh let's have a look at what their

their payoff is so their payoff is 1

minus Epsilon they get a payoff of three

that's what we saw when Rahul met Nick

and the other Epsilon of the time Rahul

meets Jake or somebody and has to make

do with the payoff of

one all right but if we keep track of

this what do we have this equals 2 1 -

Epsilon and this equals 3 1 - Epsilon

plus Epsilon is that right all right and

clearly I hope this is clear the payoff

to the mutant is bigger everyone happy

with that the PA of the mutant is

bigger which means the mutation is not

going to die out in fact it's going to

grow mutation is not going to die out so

we can

conclude that c is

not evolutionarily stable I'm going to

start using ES for evolutionarily

stable all right right so in this

particular game

evolution is not evolutionary stable so

we might ask what is evolutionary St

evolutionarily stable in this game well

it's not going to be hard to figure it

out but since there's one only one

choice I claim that D is going to be

evolutionarily stable but let's just

prove

it so is it going to be the case that

these mutants will eventually take over

the whole

population and everyone everyone will

end up looking like a TA and being

uncooperative all right so they're

busily asexually reproducing all the

time they get bigger and bigger and

isn't in fact the case that once they've

conquered everything that they're

evolutionarily

stable so is defect evolutionarily

stable in this game all right to figure

that out we have to do exactly the

revers experiment so the experiment now

is Imagine everyone in the population

all of of you guys all of the students

in the room are nasty non-cooperative

defecting ants all right right and

you're going along and you're most of

the time you're meeting you're meeting

each other all the time for now and

you're getting a payoff of one and now

there's mutation but this time the

mutation isn't the scary uh um

non-operative mutation that we saw

before with rul uh it's going to be MTO

over there all right and merto is a nice

is a nice mutation who cooperates all

right so MOS are nice Cooperative

mutation all right let me just get the

camera to pan on poor little MTO all

right so there's there's poor little M

who's the only little Cooperative

mutation in the room wave all right

there we go okay all right all right

what's what's going to happen to our

Cooperative mutation all right so most

of the time you non-cooperative people

are matching up against each other we

can figure out what your payoff is so

the the uncooperative incumbents they're

playing a population that is 1us Epsilon

non

cooperative and Epsilon Epsilon of the

time they meet MTO all right so what's

their average pay off so let's be

careful let's switch things around up

here just to make sure we can see what

happened so switching things around on

our chart up here we're now looking at

the case where there's one minus Epsilon

non- corporators non-cooperators and

Epsilon Cooperators

all right so this these these you guys

you non-cooperative ants are most of the

time meeting each other and when you

meet each other you're getting a payoff

of one so one minus Epsilon of the time

you're getting a payoff of one but

Epsilon of the time you're doing great

because you're meeting meru and

unfortunately you're you're beating up

on

meru all right and how's she doing well

she's a cooperator and one minus Epsilon

of the time she's meeting you guys and

Epsilon of the time she meets another

nice Cooperative ant like Jake all right

so her payoff is one minus Epsilon of

the time she gets nothing and Epsilon of

the time she does pretty well and gets

two but it's only Epsilon of the time is

that right so far so far so good so

what's going to happen to this mutation

well the incumbent population their

average payoff comes down to being 1 -

Epsilon plus 3

Epsilon and mto's payoff the defect the

the mutant uh ends up being 2

Epsilon all right so

unfortunately unfortunately these nasty

ants are thriving and motor gets wiped

out so you can sit down

again all right so what have we shown

here we've shown that defect not Cooper

is evolutionally stable in this

game right any mutation isn't one

possible mutation any mutation gets

wiped out all right think about these

two different models here the first we

not different model but different

metaphors so we started out with this

population of relatively nice people and

we had a nasty mutation so think of the

movie alen actually don't it's a

horrible movie but all right right and

that that that as if you remember that

horrible movie uh Alien or or or species

is not think of the movie Species an

even worse movie right think of the

movie species that was that was uh R and

he grew all bit asexually very rapidly

that's a pretty scary movie right right

conversely when we had this population

of nasty people and we tried to have a

nice little invasion of a nice mutant so

think of the movie ET unfortunately ET

got

squished all right all right so that's

that's the two extremes all right so

what is the lesson that we can actually

draw from

this right

so there's a lesson

here uh and the lesson I guess there's

going to be two lessons here but let's

start with one the first lesson here is

that well let's just put it on the

forward first of all

nature the outcome of evolution

nature can

suck all right so again we could use a

more formal term but this is an

important lesson there's a tendency for

some people to think that if something

arises as a consequence of evolution if

something is natural if something is in

nature it must therefore be good in some

moral or other way efficient or

something right and what we're seeing

here is in this game uh the consequence

of nature is a horrible

consequence right for those people who

doubt that nature can be pretty

unpleasant have a look at yesterday's

science page of the New York Times and

read the piece about how baboons

basically kill kill baby baboons the

biggest cause of death among baby

baboons is infanticide right this is

this is evolutionary stable it turns out

and they explain why in the times it

isn't Pleasant all right so nature can

suck nature can be pretty inefficient

okay now this raises a question in this

particular game because look at the

examples we started with we started with

lions who are who are thinking about or

not think they weren't think about they

were hardwired to go after Antelope and

either cooperate or not and we thought

about ants who are hardwired to defend

the nest or not and we all know from

watching endless nature shows as

children on TV that actually Lions do

cooperate when they go after Antelope

and actually ants do defend defend the

nest when it's invaded by a spider or

something is that right looking looking

for some nodding is that right yeah okay

good so so what happened what what's

what's wrong with our model here we've

just we've argued here that in this

model nature is going to produce this

non-cooperative Behavior we know there

are examples of cooperation in nature

what's going on what what are we missing

yeah can I get a can I get a mic in in

in it's it's here let me do it I'll do

it yeah so what's going on um different

communities might Comm compete against

each other so one community that doesn't

have any of these mutations and stays

Cooperative might succeed so communities

in general don't have as many mutations

okay so so that's an interesting idea so

part of this is because we're focusing

on within species uh competition rather

than AC cross species competition that

turns out to be a complicated idea and

and large literature on it and it's

actually a little difficult I was

looking for but it's good it's a good

suggestion I was something something

simpler actually let me let me come back

here so yeah um do do uh these species

have ways to detect cheaters and punish

them ah okay that might be a possibility

and that certainly that is some evidence

of that but not much again we looking at

pretty high primates by time you get to

that something simpler something you

should all have on mind all the time if

you're normal

teenagers the payoffs could change as

the proportion of cheaters increases so

you're not going to get the payoff of

three if everyone's cheating all right

but okay but it's true that you're going

to get a different expected payoff but

that isn't the payoff the I think it's

really simple that let me repeat I'm

guessing is on all of your minds as

teenagers let's see that hint going to

get get the right answer yeah uh answer

not sexual right right the point the the

Assumption that's really driving this

here is is the Assumption of that of of

asexual reproduction right sexual

reproduction with Gene exchange is going

to make a difference why I don't want to

delve too much into the biology here but

the the uh the main why is what matters

is survival of the gene not survival of

the individual ant or individual lion so

if you have if you have sexual

reproduction you get Gene redistribution

among your among children and cousins to

a less extent all right so provided the

other ants in the nest are closely

enough gen genetically related to you

and provid the lions are Clos enough

related to you then it may turn out to

be uh uh it may turn out that that you

will cooperate and that in fact does

change the payoffs of the game doesn't

change the payoffs for the for the

individual lion or ant it changes the

payoffs for the gene all right now if we

have time we'll come back and look at

that on Monday and even if we don't have

time in the reading I've left for you in

the reading packet it goes into that in

some detail all right it's much more

complicated model but that's where we're

going to get for the most part that's

where we're going to get cooperation

from rather than from the the other

points that people made all right so so

sex can make a difference here but we're

going to stick with a sexual

reproduction because it's easier to

analyze and stick with it for for the

minute and what we've learned the second

lesson we've learned here I'm going to

generalize from this lesson we learned

that a strategy that was strictly

dominated was not evolutionary stable

right right so I've only shown an

example here but we'll extend from the

example all right so if a

strategy is strictly dominated

if strategy is strictly dominated and of

course cooperate is strictly dominated

here then it is not evolutionary stable

at least in this simple game of just

asexual

reproduction and even though I haven't

proved this here the idea is exactly the

idea of this example so let's just try

and talk it through suppose a strategy

was strictly dominated how do we know

how can we see that it will not be

evolutionarily stable somebody where are

my where are my pre-made Majors I not

premade me but my pre-made people

they're all hiding now uh I know you're

out there because I've seen your your

your forms but never mind okay so why is

why is this dominated strategy not

evolutionary stable try

job

yes the strategy that that dominates

will be a successful mutation good good

so the strategy that dominates the

strategy that does the mination of this

dominated strategy would be a successful

mutation if it enters it does well not

just against this strategy but against

any mix involving itself and this

strategy all right so uh the strictly do

the the strictly Dominator strategy will

invade all right so we can't have uh we

can't have uh uh strictly dominated

strategies surviving in evolution all

right let's do another example

and we'll see if we can learn some

more so this is going to be a slightly

more complicated more complicated

example let's have a 3X3

game so here's our 3x3 game we'll just

label the strategies a b c and ABC and

once again we're going to focus on

symmetric games so this game will be

symmetric

220 0

z0 0 Z lots of zeros in this 0 0 0 0 1

1 0 0 1

1 z0 right so this is a symmetric

game okay and if we look at this game a

little bit I don't want to look at all

of it I want to ask the question

is strategy

C evolutionarily

stable and strategy C evolutionary

stable let's go back to the experim

what does it involve imagine everyone in

the room is hardwired to play C could

that population be invaded by a

mutation so what do we think what do we

think is C evolutionary stable who

thinks it is

stable who thinks it's not stable so

most you think it's not stable so you

must have some idea why why do we think

it's not stable let me just try and C Co

a little bit anybody

let me let me sample somebody who if

it's not stable it must be it's going to

get invaded so who's going to invade it

can we try this this gentleman have have

a guess who's going to invade

it uh strategy B strategy B okay that's

correct okay so your name is Greg so

Greg is saying strategy B might invade

here let's have a look let's see what

happens all right

so suppose there's an invasion of B so

again the Tas are playing B everyone

else is playing C and let's see how

these incumbent incumbent uh genes do so

C whoops so C is playing against one

minus Epsilon of the population who are

playing C and Epsilon who are playing B

and it's average payoff well one minus

Epsilon of the time it'll be matched

against essentially itself and get a

payoff of

zero all right but Epsilon of the time

uh it will be matched against it'll be

matched against a b and get a payoff of

one right everyone agree with that right

so one minus Epsilon of the time it

meets itself and gets nothing Epsilon of

the time it gets lucky and it meets a TA

and get to payoff of one how about those

B Invaders the TA in the class so one

minus Epsilon of the time they're going

to meet C's and Epsilon of the time

they're going to meet B's and 1us

Epsilon of the time therefore the B is

meeting a c and getting a payoff of

one right so oneus Epsilon time the TA

is meeting one of you and Epsilon of the

time the TA is meeting another ta

playing B against B and getting

zero all right so this works out as

Epsilon and this works out as 1 minus

Epsilon all right but notice that if

Epsilon is small

if there are just a small number of

these mutations 1 minus Epsilon is

bigger than

Epsilon is that right and that means

that this mutation will not die

out right it probably go on we can

actually say a bit more it'll probably

go on growing until it's roughly half

the

population all right but in particular

it won't die out since it won't die out

since the mutation doesn't die out we

can conclude that c is not evolutionary

stable

right everyone okay with that everyone

okay with that that idea now this idea

this example is a little bit more

complicated than the previous example

because it turns out that the invading

mutant population the population B is

itself not evolutionarily

stable everyone see that right C wasn't

evolutionary stable because it got

invaded by B who's going to invade

B who's going to invade B

C is going to invade B turns out

everything's exactly symmetric here so

even though even though we're arguing

that c is not evolutionary stable

because it's invaded by B it doesn't

have to be the case that the IM it

doesn't have to be the case that the

successful mutant is itself

evolutionarily stable right so this

particular

Example The

Invader Invader namely B is itself

not evolutionary stable nevertheless it

doesn't die out when it invades a

population of

seeds right now there's a second

observation I want to draw off this

one give some better chalk

here not finding it all right there's a

second uh observation I want to draw

here what do we what do we observe what

what else is true about this about both

C and B so what's true about everyone

playing C or everyone playing

B what else just going back to what

we've learned in the course so far what

do we think about everyone playing

C so if everyone playing C we'd be

looking at an object like this C comma C

right and we might ask the question is C

comma C what is it a nation

equilibrium right so

is is CC

an Nash equilibrium here we asked was it

evolutionary stable we found it

wasn't now let's ask a different

question and forget Evolution for a

second is CC a n

equilibrium that's a question you all

should be able to answer for the midterm

so uh somebody uh tell me the

answer it's not it's not an ash

equilibrium how do we know it's not an

ash equilibrium what we have to what we

have to do to show it's not an natural

equilibrium we have to show that there's

a profitable deviation right right so

what's the shout it out what's the

profitable deviation here the profitable

deviation is B right it's

not

no because B is

a strictly

profitable

deviation right B is a strictly

profitable deviation notice that the

thing that was a strictly profitable

deviation was the same thing that would

have invaded the population of seas in

the context of our ants nest in context

of this classroom being an ants nest and

looking at Evolution that right right so

what have we just learned here well this

idea turns out to be

General the idea is if a strategy

S I should make this a

lesson

lesson if a strategy s is not

Nash so in other words SS is not a Nash

equilibrium it's not an ash

equilibrium then s is

not evolutionarily

stable right if s is not nash then s is

not evolutionary

stable all right a little bit of logic

we flip that around

and we will say what that actually tells

us is it's equivalent to saying if s is

evolutionary stable then

SS

is a Nash

equilibrium that's the equivalent way of

saying

that all right so what's the idea here

let's look at the first line rather than

the second line it's pretty easy to

understand the idea here is if a

strategy is not nash what that means is

is is there's some other

strategy that would be a strictly

profitable deviation and that's enough

to tell us that s cannot be evolutionary

stable why because take that same

strategy that was a strictly profitable

deviation B in this case and that

strategy can be thought of as the

mutation that's going to invade the

strategy we started from right everyone

see that say once more supposed strategy

is strategy s s is not an ash

equilibrium right that means there some

other strategy S Prime say that is a

strictly profitable deviation now think

of that strictly profitable deviation S

Prime as a mutant invasion so now Rahul

is playing is hardwired to play S Prime

right since it was strictly profitable

it's going to be a strictly profitable

as a deviation it's going to be

successful as an

Invader all right again I haven't

formally proved it but that's exactly

how the proof would run

all right so let's pause for a second

and just see where we are all right what

we've managed to

show is there's a connection here

between dominance and evolutionary

stable we've said if a strategy is not

dominated it cannot be evolutionary

stable and we've also begun to show a

connection between Nash equilibrium and

evolutionary stability so two of the

ideas we've developed the last 6 weeks

are reemerging in this completely other

context of animal behavior the idea of

dominance and the idea of Nash and so

far how far we got we've said that if

something's going to be evolutionary

stable it better be the case that it's

also

Nash right and that raises I think a

natural question which is is the

opposite

true it would be really pretty great if

it were true the other way around if if

I could show that if a strategy was Nash

then it would necessarily also be

evolutionary stable that would be pretty

cool right say this idea we've spent a

lot of time developing in the class

turns out to be the key idea

unfortunately life isn't quite so neat

let's see

why all right so what I'm going to do is

I'm going to show you an example and

what this example is going to illustrate

is that there's an is we can find Nash

strategies that are not evolutionarily

stable and the example is embarrassingly

simple here's a game it's a two player

two strategy game and the payoffs are 1

one0 Z

z0

z0 all right this is kind of an

embarrassingly simple game all right

what are the Nash equilibria in this

game well let's go through slowly all

right if uh uh if player row is playing

a then column's best response is to pick

a and if row is playing B then column's

best response is either to play A or

B right if column is playing a then

Row's best response is to choose a if

column is playing B then Row's best

response is either A or B all right this

is kind of exercise that is almost

second nature for you guys now

all right so we can conclude by looking

where the Nash where these best

responses

coincide that the Nash equilibria here

are a a

and

BB

okay everyone happy with that should

should should be looking like this is

yeah this is easy right all right so all

right but let's look at this second Nash

equilibrium the BB Nash equilibrium the

one that's down

here right is B Evolution

stable is be evolutionarily

stable well again let's think about it

let's just think through the exercise

suppose the entire class were playing B

all right wake up the guy in the middle

and tell him he's playing B too him yeah

okay all right whole class is playing B

all right and suppose there's an

invasion could The Invasion be the

invasion better be an invasion of

A's right what's going to be the

expected payoff of the average payoff of

the incumbent of all of you in the class

who are playing B well without doing it

too laboriously one minus Epsilon at a

time you're going to meet another

student in which case your payoff which

case you'll be playing B against B and

your payoff will be zero and Epsilon of

the time you're going to meet a TA and

these Tas are playing a but again you'll

get zero so your average payoff will be

what zero all right zero all right so so

if you're incumbent and you uh against

this mix your payoff will be zero and if

you're an Invader so how's Rahul doing

this time rahul's playing a oneus

Epsilon of the time this a is playing

against a b right so one minus one minus

Epsilon of the time he playing against a

b right so one minus Epsilon of the time

Rahul meets a student and gets a payoff

of

zero but Epsilon of the time Rahul hits

it lucky and Rahul meets another ta

Rahul meets

Jake and when he meets Jake his payoff

is

one is that correct is that correct so

his total average payoff is Epsilon

which is bigger than

zero all right so indeed it's turns out

that Rahul is rahul's uh uh Gene is

going to grow the bees are going to

shrink BB was Nash but it's not

evolutionarily stable it can be

invaded right everyone see that everyone

see how how that Invasion worked so the

key to that invasion was when Rahul met

another incumbent student he did no

better than the students did against

students but on those rare occasions

when Rahul met another

TI he made hay or not hey asex

production he he got a payoff of

one right and that those rare occasions

were enough to make Rahul grow and

Thrive whereas the bees relative

speaking shrink all right everyone happy

with

that yeah okay so what we have here is

an example of something that is Nash but

is not evolutionary stable can anyone

say what's special about this example

how did I rig this

example there's something really kind of

knife edge and rigged about this example

much rigged about this

example

anybody I want a c call on

this no takers everyone's kind of in

that Prem midterm scared

mode yeah yeah yeah this guy's rescuing

this guy is saving the class today

someone else has to get in okay what's

your name again uh Steven so Steven is

saving the rest of you but go ahead

stepen uh B is only a weak best response

to anything good so what's true about

this example is that although B is a

best response against B it only is so

weakly is that right it's only that so

weakly right if in fact we got rid of

Nash equilibria that relied only on weak

best responses then we wouldn't be able

to produce this

example right so in particular if we

looked at Nash equilibria where the Nash

strategies was strictly a best response

it was strictly better than playing any

other pure strategy those cases would be

evolutionarily stable right anyone

remember what we call Nash equilibria

where uh the the Nash strategy is a

strict best response we call them strict

Nash all right so what is in fact true

is that if if

if if s s is a strict

Nash a strict Nash equilibrium by which

I means s is a strict best response to

S then s

is a nashy equ that's right n s is

evolutionary

State all

right okay so we've been pretty we've

been only semi-formal so far I haven't

really put up a formal definition and

what I've been grouping towards slowly

is a connection between an idea growing

out of biology namely evolutionary

stability

and an idea growing out of economics and

Mathematics namely Nash equilibrium what

I want to do now is I want to show you

formally how are these connected I'm not

going to prove this part of the class

because a proof is a bit lengthy but I

will include a proof uh on the handout

that's going to be available this

afternoon I'm just going to argue it for

now there we go that's some all right so

what I want to do first is I want to

write down a formal definition of

evolutionary

stability and it's a definition that

comes out of biology and it's a little

bit of a you're going to see it's a

little bit of a a mouthful this

definition right so for those people who

are who are a little bit sleepy you need

to wake up a little bit because this

going to be a little bit of a difficult

definition all right but this is the

definition that came out of

biology this is a formal definition

came comes out of biology and in

particular it's due to a guy called

meard

Smith who wrote this in

1972 so obviously the earlier idea is

due to Darwin but this formal idea is

due to Mayor

Smith All right so here's our formal

definition so in a

symmetric two player game so everything

we've been looking at so far

the pure

strategy the pure strategy s let's call

it s hat to give it a name s hat is

evolutionarily stable and again I'll

just use es for that but let me be a

little bit nerdy here I'm going to say

inure strategies it's evolutionary

stable in pure strategies I'm putting

this in because we're going to come back

on Monday and consider mixed strategies

right so it's evolutionary stable in

pure

strategies

if and just leave a little bit of a

space here because I'm going to need a

little bit of space between between

these next two lines leave a little

space here all right so here's our big

mouthful what we need is one minus

Epsilon of the payoff of playing s hat

against s

hat plus Epsilon of the payoff of

playing s hat against S

Prime has to be strictly bigger than one

minus Epsilon of the payoff of playing S

Prime against s

hat plus Epsilon of the payoff of

playing S Prime against

itself right this has to be

true for

all possible

[Music]

deviations S

Prime but we also this is why need the

extra line

and for

all

mutation

sizes Epsilon less than some Epsilon

bar right now this is where I need to go

back and just be a bit more careful so

I'm going to write in this extra line

there

exists an Epsilon

bar all right that's why I left the

extra line there so this looks like very

nerdy things let's talk our way through

it all right what's it saying it's

saying that s hat is evolutionary stable

if against all possible mutations so all

possible versions of raahul S

Prime all right the payoff of s hat

against the sub subsequently mixed

population is bigger than the payoff of

S Prime against the subsequently mixed

population let's just see why that's so

so this on the left is the payoff of s

hat against a population in which one

minus Epsilon of the population like it

is playing s hat and Epsilon of the

population is like Rahul right one minus

Epsilon of the time it meets something

like itself and Epsilon of the time it

meets a

TA all right conversely on the right

hand side we have the payoff to the

mutation the payoff to the mutation is 1

minus Epsilon of the time the TA meets a

student and gets the payoff of S Prime

against s hat Epsilon of the time the TA

meets itself and gets a payoff of S

Prime against S Prime and we have this

inequality we had before it says the

says the mutation The Invader has to do

worse so it dies out all right so the

nasty bit of this definition the part

that's a little bit if it takes a while

to get your head around is is this

qualifier about the size of

mutations and all it's saying we ignore

the math of it all it's saying is this

better be true for all small

mutations right you could think of that

saying this has to be true for all small

mutations so this definition from

biology is exactly it exactly mimics the

argument we we went through several

times now both using the class and using

the board and figuring out average pays

all right this is an ugly definition

everyone agree this is kind of ugly all

right so now what I'm going to do is I'm

going to give you an entirely different

definition it's also going to be ugly

but a little bit less

ugly all right so think of that as

definition one and it came from biology

it came from this paper in 1972 I think

it was in nature by Main and Smith and

co-authors

and think of this other definition

that's coming out of

Economics so definition

two a

strategy s

hat or I should have the same qualify in

a symmetric two-player game okay same

same thing to start off with a strategy

s hat

is

es in pure strategies so same as we had

before

if right if two things thing number one

let's call it

a

if SS if s hat s

hat is a Nash equilibrium is a symmetric

obviously is a

symmetric Nash equilibrium of the

game we're not done yet

let me just write down what it means to

be a symmetric National equilibri the

game what does that mean that means IE

the payoff of s hat against s hat must

be at least as big as the payoff of S

Prime against s

hat for all S Prime right that's a

standard thing we SE many many times it

says s hat is a best response against s

hat all right not quite done yet

and and

B what else do we need we

need

if this weak inquality I wrote above is

actually an

equality if if the payoff of s hat

against itself is actually equal to the

payoff of S Prime against s hat

then the payoff of s hat against S Prime

must be bigger than the payoff of S

Prime against

itself this is still still a bit of a

mouthful but I claim it's going to end

up being it end up being a little simple

to think to think

about right so what it says again it

says s hat is evolutionally stable if

it's an ash

equilibrium right that's basically this

and if it's it's only a weak Nash

equilibrium it's only a weak Nash

equilibrium if there's a tie then it

better beat up on the

mutant right better meet up beat up on

the mutant when it meets the

mutant all

right so before I'm going to try and

give you an intuition as to why this is

true in a minute but first I want to

tell you why you should care about this

so without getting too religious I want

to tell you why this why I think this is

an exciting result why am I dragging you

through more algebra than usual so I

think there are two reasons why this is

an important result the first is it's

going to turn out that when we analyze

games it's very easy to check definition

two it's going to turn out it's very

easy to check definition two it's really

rather a pain to check definition one

why because you got to keep track of

these epsilons and so on

but the fact that definition two is

equivalent to definition did I say that

that's the point the fact that

definition one and two are

equivalent so one is equivalent to two

the fact that these definitions are

equivalent means we only have to check

the second definition and that's easy so

when biologists are setting up

experiments involving wasps or ants or

lions or chimpanzees provided they can

check this in the game they're done and

that turns out to be

and we'll see that on Monday all right

that's the sort of instrumental

reason all right now I want to give you

the religious reason why you should care

about this all right so let me just try

and I often find that this this appeals

to like a third of the students and the

other two-thirds of the students think

I'm completely Bonkers at this point

okay so that's fine you can think I'm

Bonkers I'm fine I want to guess for the

third of you who are nerdy like me I

want you to of see appe of this here we

have two ideas one idea arises out of

biology I I I said it from M Smith but

it really comes from Darwin so it's a

19th century idea in biology it's

probably the most important idea in

biology in the 19th century it may be

the most important idea in biology for

200 years is that is that a fair

judgment right the notion of

evolutionary

stability right the other idea comes out

of Economics it's looking at Nash

equilibria and strict Nash equilibri and

so on right up there we've got biology

down here we have economics this is an

idea that emerged in economics in the in

the 20 in the Century uh in in the 1950s

right so roughly uh 80 years after

Darwin right this is a big idea in

economics this is a big idea in biology

so what appeals to me because I'm kind

of a nerdy kind of guy is I think it's

kind of wonderful that those two ideas

are almost the

same I think it's kind of it's a

beautiful thing that those two ideas are

kind of the same thing and now you are

looking at me I'm Bonkers okay so we

push it harder all right so think think

back to the think about the great

intellectual coincidences of earlier

periods think about the 17th century in

the 17th century people figured out that

the mechanical laws that governed the

rotation of the planets were at least

approximately the same as those that

governed a clock right the basically the

basic laws of neonian physics were the

same and they were they thought this was

a wonderful thing right they had these

completely different areas of

intellectual Pursuit and they turned out

to be the same the Scientific Revolution

said they were the same right and for

those people who chose to continue

believing in God at that point they

thought of God now as being what as

being a a Heavenly clock maker right it

was a wonderful moment so here in our

own time or perhaps just before it we

see a similar thing right here we see

you know one of the most important ideas

in science coinciding with one of the

most important ideas in economics right

where does this lead us well well now I

guess I the bad news is this may make a

lot of people doubt the existence of God

I guess that's going to get me in

trouble I didn't say that but uh at

least if you if you do go on believing

in God you're going to have to believe

God's in the Economist that's a pretty

good thing all right all right so so

that's my little religious piece on this

all right now I want to spend the last

10 minutes having told you this

wonderful thing trying to convince you

that it's true and again there's a proof

there's a proof on the hand out that you

can read on the web I just want to give

you an idea of why it's true okay so

here we

go I need some okay all right so what I

want to convince you of is that this

definition this kind of game theory

definition this econ definition is

implies the one above the one above

we've kind of already argued several

times today corresponds to the notion of

evolution stability we talked about it

several times we wrote down these kind

of equations several times so I want to

convince you that this one implies that

one all

right okay so what do I have to convince

you so let's let's

imagine let's

fix should be I

okay there's no way of we just need to

see that one

without no never mind it was a bad idea

never mind okay let's bring it back

here the embarrassment of playing with

heavy boards here all right let's try

and Sho this one

up if you ever think of asking your Dean

to about some new technology for Yale we

might think about having some slightly

lighter

blackboards Just For The evolutionary

stability of the average weakling

Economist all

right all right so what we're going to

do is let's

fix a strategy s hat and

suppose s hat s hat is in fact

Nash all right so that's that's what I

want to convince you of is that s hat is

going to be evolutionally stable so I

claim there's two

possibilities there are two

cases right the two cases are either

it's the case that s hat against s hat

does strictly

better than S hat against S

Prime for all S Prime right so let must

be careful since we know it's Nash we

already know that the payoff of s hat

against itself is at least weekly better

than any possible deviation that's what

it means to be Nash right so everyone

agree with that we know that we know

that s hat is the best response to S hat

so it must be weakly better than any

possible

deviation all right so take the first

case where actually it's strictly better

and let's go back to our our classroom

example and suppose that the let's go

back to the first definition by going

back to our metaphor of the class so you

guys are all ants all right and suppose

that every student in the class is

playing s hat right you're all playing s

hat and suppose it's the case that in

fact the payoff of s hat against s hat

is bigger than the payoff of s hat

against S Prime and suppose that a

mutation arises sorry rul Here We Go

Again suppose the mutation arises here's

Rahul our mutation right and he's going

to be randomly matched against one of

you guys right so let's compare rahul's

payoff against uh this Gent

payoff up a seconds your name is Pat all

right so Pat is our typical s hat

incumbent ant doesn't look like an ant

but never mind you know stretch your

imagination a bit and most of the time

most of the time what's happening most

of the time is Pat is being matched

against one of the rest of you and when

he's matched against one of the rest of

you he's getting the payoff of s hat

against s hat which is US hat as hat all

right that's getting most of the time

right and every now and then every now

and then he's meeting a mutin okay fine

okay fine so every now and then he's

getting a slightly different payoff but

most of the time he's getting a payoff

comes which is US hats hat right what

about Rahul so Rahul okay every now and

then he's going to be lucky and meet

another ta but most of the time almost

all the time he's meet he's being

matched against one of you for example

he's matched against Pat right and when

he's matched against Pat his payoff is

US hats Prime which is lower

right so Pat's payoff almost all the

time is US hats

hat and rol's payoff almost all the time

is US hat S Prime but our assumption is

US hat S Prime is lower so rahul's going

to die

out okay okay so in this case in this

case rul dies out because most of the

time he's playing an incumbent and just

doing

horribly okay sorry

okay okay everyone convinced by that so

in this case in this Cas the mutant dies

out so the

mutant dies

out

because she or

he meets s

hat often right the mutant meets the

incumbent often and just does horribly

against the incumbent

that's the first way that's the first

case that's case one and case two is it

could be the case that in fact rul does

pretty well against the

incumbent rul does pretty well against

the incumbents but unfortunately for

rul could be the case in but sorry but

in that case according to definition B

rul doesn't do so well against other

Tas so this is case two case one case

one was Rahul get gets beaten up

whenever he meets one of you sorry rul

getting beaten up a lot here but rul

gets beaten up when he meets one of you

and here's the case where Rahul is doing

fine against you but he does horribly

against incumbent so let's just that's

the harder case let's talk through that

case so back up again so R sorry again

you're up again all right so once again

you you guys are all playing s hat

there's a mutation of rahul's playing S

Prime all right and let's get Pat up

again all right so when Pat meets one of

you guys so let's let's pick out one of

you guys so let me pick up uh sorry your

name is Christine Christine so when Pat

makes uh meets Christine which is most

of the time he does okay and when Rahul

meets Christine which happens most of

the time he does exactly the same so if

you just compared them in their random

matches against Christina or the rest of

you they're the same

all right but every now and then thanks

Christine every now and then instead of

every Epsilon of the time every rarely

Rahul is gonna meet Jake own Jake's

writing he's gonna meet Kai all right

right second all right and on those

occasions on those occasions let's see

what happens so when Pat when Pat meets

Kai he does really pretty well he gets

some big Bonanza he gets a much bigger

payoff than when Rahul meets Kai they

just do horri horribly against each

other right and the fact that they do

horribly against each other causes them

to die

out all right let's talk this through

there were two ways in which Rahul could

die out one is he dies out because he

does horribly against you and the second

way he can die out is if he does equally

well against you as you do against

yourself but he does horribly when he

meets the other Tas right in either case

you guys are evolutionary stable and

Raul's in

trouble all right was that convincing

enough an argument there's a formal

proof thanks guys there's a formal proof

in the handout but this will be enough

to get us started all right so in this

case in this

case the

mutant does

okay against s sorry against s

hat but gets

clobbered does badly let say

against S Prime right so mutants that

are going to die out are those that do

badly against incumbents or do badly

against themselves in either case s will

be evolutionarily stable right now next

time I want to take this further in two

directions I want to see what happens

about evolutionary stability in in in

more complicated games for example in

cooperation games and we're going to see

that evolution doesn't do great in

corporation gains right and we're also

if we have time going to look at sexual

reproduction which is probably what you

guys are all interested in anyway all

right I'll see you all on Monday