Original subtitles

all right so last time we did something

I think substantially harder than

anything we've done in the class so far

we looked at mixed strategies and in

particular we looked at mixed strategy

equilibrium all right and there was a

big idea last time the big idea was if a

player is playing a mixed strategy in

equilibrium then every pure strategy in

the mix that's to say every pure

strategy on which they Place some

positive weight must also be a best

response to what the other side is doing

and then we used that trick we used it

in this game here to help us find Nash

equilibria the way we helped it the way

it allowed us to find Nash equilibria is

we knew that if in this case Venus

Williams is mixing between left and

right it must be this case that her

payoff is equal to that of right and we

use that to find serene mix and

conversely since we knew that Serena is

mixing again between little L and little

r we knew she must be indifferent

between little L and little r and we

used that to find Venus's mix so I want

to go back to this example just for a

few moments just to make one more point

and then we'll move on but we'll still

be talking about mix strategies

throughout today so this was the mix

that we found before we changed the

payoffs we found that Venus's

equilibrium mix was

7.3

and Serena's equilibrium mix was

6.4 and a reasonable question at this

point would be how do we know that's

really an equilibrium right we kind of

found it but we didn't kind of go back

and check so what I want to do now is

actually do that that missing step we

rushed it a bit last time because we

want to get through a lot of material

let's actually check that in fact P star

uh is a best response to qar so what I

want to do is I want to

check that Venus's mix P star is a best

response for Venus against serenas Mak

qar and the way I'm going to do that is

I'm going to look at payoffs that Venus

gets now she know she's now we know

she's playing against qar all right so

let's look at Venus's payoffs I'm going

to figure out her her payoffs for L her

payoffs for R and also her payoff what

she's actually doing p

all right so Venus's

payoffs if she chooses L against qar

then she gets very similar to what we

had on the board last week but now I'm

going to put in what qar is explicitly

she gets 50 *

6 this is qar and this is one minus qar

so she gets 50 * 6 and 80 Time 1 minus 6

which

is4 AC

time4 all right and we can work this out

and I worked it out at home but if

somebody has a calculator they can

please check me uh I think this comes to

62 all right somebody should just check

that all right and if Venus chose right

remember right here means shooting to

Serena's right to Serena's beforehand if

she chose right then her payoffs

are 90 qar so

90.6 plus 20 1 minus qar so

20.4 so

90.6 +

20.4 and again I work that out at home

and fortunately that also comes out at

62 all right so what Venus's payoff for

p

star we've got her payoff for both her

PE

strategies so her payoff from actually

choosing P star is what well P star is7

so s of the

time she she will actually be playing

left and when she plays Left she'll get

a payoff of

62

and

three of the

time three of the time she'll be playing

right and once again she'll getting a

payoff of 62 and have a

calcul yes sorry thank you so p p star

is seven I yes you're absolutely right

so this is p star and one minus P Star

right so let's let's make that clear I

was sh what the equilibrium is but P

star itself is 7 thank you all right so

when Venus plays plays left with

probability 7 then 7 of the time she'll

get an expected payoff of 62 and three

of the time she'll get a payoff again of

62 and that's the kind of math I don't

have to do at home that's going to come

out at

62 again assuming my math is correct all

right what I've really done here is

confirm what we did already last time we

knew that we we in fact chose serenas

mix Q to make Venus indifference between

left and right and that's exactly what

we found here getting going left gets

Point 62 going going right uh gets 62

and hence P star gets 62 but I claim we

can now see something a little bit a

little bit else we can now ask the

question is p star in fact the best

response well for it not to be a best

response for this not to be an

equilibrium there would have to be some

deviation that Venus could make that

would make her strictly better off let

me repeat that if this were not an

equilibrium there would have to be some

deviation for Venus that would make her

strictly better off right by playing P

star she's getting a return of 62 so one

thing she could deviate to is playing

Left all the time if she deviates to

playing Left all the time her payoff is

still 62 so she's not strictly better

off that's not a strictly profitable

deviation right another thing she could

deviate too is she could deviate to

playing right if she deviates to playing

right her payoff will be 62 once again

she's not strictly better off she's the

same as she was before so that's not a

strictly profitable

deviation right so what have I shown so

far I've shown that P star is as good as

playing left and P star is as good as

playing right in fact that's how he

constructed it so deviating to left is

not a strictly profitable

deviation and deviating to right is not

a strictly Prof profitable deviation but

at this point somebody might ask and say

okay you've shown me that there's no way

to deviate to a pure strategy in a

strictly profitable way but how about

deviating to another mixed

strategy all right so so far we've shown

we've shown just up here we can

see

that

Venus

has no strictly

profitable pure strategy

deviation pure

strategy

deviation she has no strict pure

strategy deviation because each of her

pure strategies yield the same payoff as

did her pive mix

strategy yield the same as P star but

how do we know that she doesn't have a

mixed strategy that would be strictly

better how do we know that

anybody how do we know

that no no no hands going oh there was a

hand up good good anyx between uh left

and right will still yield 62 good good

so any mix any mix that Venus deviates

to will be a mix between left and right

and any mix between left and right will

be a mix between 62 and .62 and hence

we'll yield 62 all right so we're going

to use again this fact we developed last

week the fact we developed last week was

that any mixed strategy yields a payoff

that that is a weighted average of the

pure strategy payoffs the payoffs to the

pure strategies in the mix any mix

strategy yields a payoff that is a

weighted average of the payoffs of the

pure strategy in the mix that was our

key fact last week so here if if we've

shown that there's no pure strategy

deviation that's strictly profitable

then there can't be any mixed strategy

deviation that's strictly profitable why

because the mix strategy deviations must

yield payoffs that lie among the pure

Strat

deviations right so this is a great fact

for us what it what's the lesson

here the lesson

is we only

ever have to

check

for strictly

profitable

pure

strategy

deviations that's a good job why because

if we had to check for mixed strategy

deviations one by one we'd be here all

night because there's an infinite number

of possible mixed strategy deviations

all right but there aren't so many pure

strategy deviations we have to check

let's just repeat the idea suppose

Suppose there isn't any pure strategy

deviation that's profitable then there

can't be any mixed Strat deviation

that's profitable because the the

highest the highest expected return you

could ever get from a mix strategy is

one of the pure strategies in the mix

you've already checked that none of

those are

profitable all right so this simple idea

the simple idea we developed last time

not only helps us to find Nash

equilibria but also to check for Nash

equilibria all right now a lot of people

I ga G from feedback from sections were

left pretty confused last time it's a

hard idea actually I looked at the uh

tape over the weekend you know I can see

it I could see where it could be

confusing but it's actually I think

what's really confusing here wasn't so

much I think this wasn't too much that I

could have been clearer though I'm sure

I could have been it's that this is

really a hard idea this I idea of mix

strategies so we're going to work on it

again today but I think one of the ideas

that gets people confused is the

following idea they say look we found

Venus's equilibrium mix by choosing a p

and one minus P to make Serena

indifferent and we found Serena's

equilibrium mix by finding a q and a one

minus Q to make Venus indiffer and a

natural question you hear people ask

then is why is Venus quote trying to

make Serena indifferent and why is

Serena quote trying to make Venus indiff

and that's not really the point here it

isn't that Venus is trying to make

Serena indifferent it's that in

equilibrium she is going to make Serena

indiffer it isn't her goal in life to

make Serena indifferent between left and

right and it isn't Serena's goal in life

to make Venus indifferent between left

and right but in equilibrium it ends up

that they make each other

indiff all right and the way that we can

see that is that if Venus puts something

we said last time let repeat it if Venus

puts too much weight more than 7 on on

left then Serena Serena just cheats to

the left all the time and that can't

possibly be an equilibrium and if Venus

puts too much weight on right then

Serena cheats to the right all the time

and that can't be an equilibrium so it

has to be that what Venus is doing is

going to make Serena exactly IND

different and vice

versa all right now let's see that idea

in some other applications let's talk

about this a bit before we move on all

right so it turns out that some very

natural applications for mixed strategy

equilibria arise in games in in sport

all right so let's talk about a few now

uh can any can anybody suggest some

other places where we see randomization

or at least mixed strategy in equilibria

in sporting

events let me let me actually grab the

mic myself so come down a bit anybody

anybody here uh I know anybody anybody

here play play football for example and

we're talking American football now grid

iron game not not the Civilized type

right anyone play yes so some of you

play football right so so where is the

mixing where's the mixing involved in

playing

football where in equilibrium would we

expect to see mixed strategies yeah

there's somebody down there let me go

and get them all right so shout it out

um running game and passing game all

right the running game and the passing

game right so very simple idea whether

to run or whether to pass when you have

the ball is likely to end up as a mix

equilibrium the defense is also

randomizing between for example rushing

the passer or playing a run defense that

right this is not exactly a game I know

a lot about but I'm I'm hoping I'm

getting close enough here right right it

couldn't possibly be a pure strategy

equilibrium right other than very

extreme parts of the game like at the

end of the game perhaps but for most of

the game it's very unlikely to end up as

a pure strategy equilibrium much more

likely that the offense is mixing

between passing and running and for that

matter between going to the left going

to the right going to the center and the

defense is also mixing between uh over

types of Defense all right so we see

that for those people who were watching

yesterday we see that in football games

all right why do we see it else in sport

so other

sports I can't have a room full of non

sports fans how many of you ever watch

any sport let just raise some hands here

all right some of you do some of you do

it's all right so this is this is basic

Bas playoff season right right how many

of you been watching the baseball

playoffs raise your hands you watching

the baseball playoffs come I'll let you

off I you should been doing my homework

how many of you been watching the

playoffs instead how many watched the

Yankees game last night right quite a

few of you all right all right so they

havn't been very exciting yet but we're

hoping it's going to get more exciting

right so what in what when you're

watching baseball what kind of things do

you see where you just know that there

must be mixed strategies involved there

must be randomization involved now I got

a few more hands on good so you sir uh

choosing how to pitch the ball choosing

how to pitch the ball so go enlarge a

little bit more say a more uh fast ball

versus slider versus change up all sorts

of different all right so there's

different ways of throwing the ball all

right and there's going to be

randomization from the pitcher or at

least it's going to look like there's

randomization by the pitcher over

whether to throw a fast ball or a curve

ball or or whatever how is the hitter

randomizing there how's the hitter

randomizing is the hitter randomizing at

all what's the doing while this is going

on

anybody yeah he's choosing whether to

swing or not to swing oh he's okay he's

choosing whether to swing or not to

swing although presumably he can do that

just after the ball thrown right so you

sometimes hear the commentators say that

that guy that that hitter was looking

for a fast ball is that right or looking

for a curve ball the hit is trying to

anticipate the pitch is that right this

is not a game I played a lot of either

i' play a little bit right you're trying

to you're trying to anticipate where the

ball's going to be thrown okay so uh the

type of ball you throw in baseball and

the way in which the pitch being

anticipated by The Hitter is likely to

be a mixed strategy what else is likely

to be a mixed strategy in

baseball what

else anybody here on the Yale baseball

team ah okay okay got one got one

volunteer here all right all right all

right so what else I mean stand up a

second it say let's have our Yale

baseball team member all right all right

what's your name Chris and where do you

play p you're a picture okay so he's not

going to get on Bas notes he's not going

to to answer this suppose you did get on

Bas pict don't know get on base I don't

know let's assume let's assume that

happens right right where what might you

randomize there you are you're standing

on base what might you randomize about

whether a steal second or not right

whether to steal or not whether to try

to steal or not stay up a second stay up

a second all right so the decision

whether to try and steal or not is is

likely to end up being random right and

if you're the pitcher if you're the

pitcher what can you do in response to

he is a pitcher right what can you do in

response to that you can either choose

to try to pick him off or not all right

what else can you so one thing you can

try and pick him off what else you be

quicker to the plate you can quicker to

the plate what else you can pitch out

you can pitch out what

else at least those three things right

at least those three things okay okay

thank you thank you all right I have an

expert here I'm glad I had an expert all

right so in this case in this case we

can see there's randomization going on

from the from the runner whether he

whether he attempts to steal the base or

not and by the picture on whether he

throws the pitch out or whether he tries

to throw to get to the plate faster all

right okay so we see this in sport we

don't see it well anticipated by sports

commentators let me give this down a

second so in baseball for example you'll

sometimes see see quite sophisticated

statistical analyses of baseball in

which uh somebody will have looked at

base dealers across the major leagues

and they'll look at all the instances in

which a player was on first base and had

in a position where you think they might

steal and they'll look at what happened

on every attempt to steal whether they

were in fact caught stealing or not all

right and they'll try and measure the

value of these things and they'll see

the conclusion they'll come to is

something like this they'll conclude

that whether the guy stole or not

whether the guy attempted to steal or

not sorry or whether they just sat in

first base doesn't seem to make much

difference they'll say they say the

payoff for even great base beers of a

ing to steal or not all right when you

take into account the

pickoffs versus just staying put turns

out the payoff in terms of the impact on

the game is roughly equal and then

they'll draw that these analysts will

then draw the following conclusion

they'll say oh look speed or the ability

to steal bases is therefore overrated in

baseball how have they made a mistake

what's the mistake they made

there right so the the premise was let's

give them the premise the premise was

that when a base dealer is attempting to

steal or not the expected return in

terms of outcome of the game is roughly

equal whether they attempt to steal or

don't attempt to steal the conclusion is

therefore stealing doesn't seem such a

big deal what's the mistake they've

made yeah let me borrow it again

sorry yeah um the pitcher the pitcher

has to react differently um in pitching

when he knows that there's a fast guy on

good so our pitcher has to rect

it's talk to our pitcher again right so

one thing our pitcher said was he wants

to get to the plate faster right right

what does that mean getting to the plate

faster that's just it means just shout

out people can hear you getting the ball

to the catcher as fast as possible so

that he has the best chance to throw out

the runner right right so you're going

to you're going to pitch from what's

you're not going to do that funny wup

thing right you're going thank you

you're going to pitch from the stretch I

knew there's a term there somewhere I'm

learning American by being here right

and you're more likely to throw a fast

ball right there there's some advantage

in throwing a fast ball rather than a

curve ball all right but both of both

both actions of which both having to

move more towards fast balls and

pitching the stretch are actually costly

for the pitcher but let's we'll get

there in a second let's just back up a

second so that was good that's right but

let's just back up a second the premise

of these commentators was what it was

that the return to stealing attempting

to steal seems to be roughly a wash

seems to be that the expected return

when this great base runner attempts to

steal the base is roughly the same as

the return when they don't attempt to

steal the base but I claim we knew that

was going to be the case we didn't have

to go and look at the data why do we

know that was going to be the

case how do we know that you were bound

to find a

return in that in that analysis that

finds those things roughly

equal Well

yeah if randoming that means the returns

will be equal if they wer just do one or

the other all the time good good

excellent excellent since we're in a

mixed strategy equilibrium since he's

randomizing it must be the case that the

returns are equal all right that that

clear right that's the big idea here

right that's the thing we learned last

time if this player and these a

professional baseball players doing this

they've been very well trained a lot of

money has been spent on getting the

tactics right all right there's people

sitting there who are paid to get the

tactics right right if it was the case

that the return to base stealing wasn't

roughly equal when you attempt to steal

or didn't attempt to steal right then

you shouldn't be randomizing since you

are randomizing it must be the case that

the returns are roughly equal all right

so that's the first thing to observe and

the second thing to observe is what we

just pointed out in fact the the the

value of having a fast base speler on

the team doesn't show up in the expected

return on the occasions on which he

attempts to steal or which he does not

attempt to steal it shows up where it

shows up in the fact that the pitching

team changes their behavior to make it

harder for this guy to steal by going

faster to the plate or throwing more

fast balls and where will that show up

in the statistics if you're just a

statistician like me you just look at

the data where will that show up I mean

suppose I can't keep track of every

single pitch I don't I can't actually

observe all these fast balls where will

I see the effect of all these extra fast

balls and pitching in the from the

stretch in the

data

somebody it's going to show up in the

batting average of the guy who's hitting

behind the base dealer

all right the guying behind the base

dealer is going to have a higher batting

average because he's going to get more

pitches which are fast balls to hit and

more pitches out of the stretch right so

if you ignore that effect you're going

to be in trouble but we know we Analyze

This properly using game we know we're

in a mixed strategy equilibrium we know

in fact the pitching team must be

reacting to it we know there must be a

cost in doing that and the cost turns up

in the hit or behind all right so when

you're watching the playoffs in the in

the last no now I'm giving you

permission to watch a bit of TV at night

after you've done my homework assignment

right before anyone else's homework

assignment you can have a look at these

uh these baseball games and have have

have a go at being a little bit better

than the commentators who who are

working on all right so one application

for mixed strategies is in sport but not

the only application let's just talk

about another application a slightly

more scary application so after 991

there was a lot of talk in the US about

the placement of uh uh baggage checking

machines at airports all right actually

there's still quite a lot of talk but

there was a lot of talk then about the

placement of machines to to search the

luggage that goes on that goes on board

the hand luggage was being searched

anyway but to search luggage going into

the cabins right and it was pointed out

at the time this has changed since there

weren't actually enough machines in the

US uh on the day after 911 to search

every single bag that went into the hold

and you'd hear discussions of the

following type you'd hear these experts

on

night line or whatever and they'd say

look there's no point trying to do this

because if we put all our baggage

searching machines at uh Logan Airport

in Boston for example then the

terrorists will simply move their attack

to oare and if we put them at oare then

they'll move their attack to Logan if we

put if we have enough to do both Logan

and O'Hare then they'll move their

attack to some third

airport right so that there was a sense

of Doom in the air it was kind of a

depressing time anyway there was a sense

of Doom in the air

saying that when you if you put your uh

your uh baggage searching machines

somewhere all you do is cause the the

attempted terrorists terrorists

attempting to blow up the planes to go

elsewhere and you hear the same things

today about searching individuals as

they go on the plane for example you

hear the discussion says if we only

search uh uh uh men traveling alone

let's say then you'll quickly end up

with all the people carrying bombs being

coupl or or women or women right and

again there's this sense of Doom this

sense that says it's hopeless whatever

we do we're just going to force the

terrorist to do something else we won't

have gained anything so once again

that's wrong right what's wrong about

that

one what what should we be doing in that

setting maybe come come down again what

should we be what what should they have

done in fact they did do with those

luggage bagging baggage uh searching

machines when they were in short supply

after 911 what do they do with searching

people as they get on

planes what do they

do well here what they didn't do they

didn't just put them at certain airports

and announce they're just at these

airports right that would have been a

crazy thing to do that would have been

hopeless entirely hopeless but not you

know not wise what should they have done

what did they

do anybody want a

guess

yeah

in name they randomized who they were

checking right right so when they're

checking passengers they're going to

randomly check passengers right when

they're checking when they think about

the baggage machines a sensible thing to

do is to put a big metal box at every

single airport and say we're not going

to tell you which of these boxes

actually have uh have have baggage

checking machines which effectively is

randomizing right from the point of view

of the terrorists they don't they're not

going to know where the B where the

baggage checks are going on all right

right that's worth doing it doesn't it

isn't going to perfectly eliminate well

unfortunately it isn't probably going to

perfectly eliminate all terrorist

attacks but it does make it harder for

the terrorists all right so

randomization there whether it's

literally randomizing over who is

checked or whether it's as it were

randomizing by concealing where in fact

you have placed those machines can be

very effective the hard thing both in

sport and in these military examples is

really uh mimicking randomization it's

very hard for us as humans to do it and

there's a famous story about a uh

military commander actually an English

military commander during an insurgent

war in I think it was Malaysia uh uh I

think it was Malaysia after World War II

where again he had to worry about

randomizing which convoys to protect and

the way in which he uh he figured out

that randomizing was was the right thing

to do to try and protect these convoys

as well as he could with small numbers

of of troops and the way in which he r

randomized was he literally randomized

every morning he put a bit of paper in

his hand and he had somebody had one of

his sergeants pick which hand the bit of

paper was in all right so we do actually

see these random strategies used that

the reason we have to literally

randomize is because it's very difficult

to do so unless you're a professional

sports

player

okay but it turns out that mixed

strategy equilibria mix strategies in

general are relevant Beyond just these

contexts in which you think of people

literally random rizing and I want to

look at a different context

now all right so I want to go back to a

game uh we started a few weeks ago uh

this isn't the same game it's a sequel

it's a followup in our exciting

adventure of our deting couple in the

classroom who were our deting couple do

we still have them

here that was the guy who is who is

the yeah there they are look they're

even sitting closer what a success here

all right all right can we get the

camera on them a second uh stand up a

second uh yeah stand up a

second thank you and your name was uh

shout out I'll just repeat

it David yeah can you actually and look

at this is this romantic or what David

and and and your name is Nina Nina Nina

and David okay and I think I think we

just I think we pretty much figured out

last time that Nina's player one and

David's player two is that right

that's right and and you as we remember

last time I'll pick on you in a second

you can sit down a second all right so

we'd figured out last time that they

were trying to go on a date and uh they

they had arranged to go to the movies

they picked out two in fact three movies

but two that remained viable and the

problem was being typical economics

Majors who are you both economics Majors

I didn't figure that out they are look

at that so being typical economics M

just hopeless at dating they'd forgotten

to tell each other which movie they're

going to so that you know that that work

I don't know if that work worked out

well or not but now the life has moved

on they're going to try it again but

this time taking advantage of fall in in

New England rather than go to a movie

they've decided on some new activities

so they might either go uh apple

picking or they might go to the Yale

rep all right oops rep War rep and see a

play all right and uh so apple picking

is has it advantages in the fall weather

its local flavor it's uh uh you know it

has certain uh undertones about the

Garden of Eden or something it

has I don't know if we can use the term

flavor local or otherwise we American

apples but mind that's right and the

Yale rep you know Yale rep is a good

thing to do in New Haven go to a play I

think it's Richard II is showing now is

that it probably not a great date play

but you know economists are trying to

show they have culture so uh you know

there It Go

and uh let's assume the payoffs are are

like

this much as they were before where by

we mean that that nah wants to meet

David uh but she would you know given a

choice she'd rather meet David uh in the

Apple field all right and David who's a

who's a dark personality likes the sort

of darker side of Shakespeare and uh he

would he also wants to meet Nina but he

would rather meet at the Yale R all

right I if that's backwards I apologize

to their preferences all right all right

and but once again because they're still

in competent economics Maes they've

again forgotten to tell each other where

they're

going all right they forgot to tell each

other where they're going so let's

analyze this game again we we figured

out this was a coordination game last

time or several weeks ago and we know in

this game we know what the pure strategy

Nash equilibria are all right so no

prizes for be able to spot them uh one

of the Nash equilibria in pure

strategies

let's put this in pure

strategies so one of the pure strategy n

equilibrium equilibria is for them both

to go apple picking and meet up in

Bishop's Orchard or whatever and another

pure strategy equilibrium is them both

to choose the

rep all right and we' figured out that

if they were able to communicate there's

really a pretty good chance of them

managing to coordinate at one of these

equilibria but we suspect I think that

this is not all that's going on here it

looks quite likely that come you know

next Saturday afternoon when we send

these guys out on their date they're

going to fail to meet it's at least

plausible right and to test the

plausibility of that let's ask them have

you been have you managed to meet on a

date

yet see no no haven't managed to meet on

Dat see so I'm I'm proving the point I'm

proving the point that in fact they

haven't managed to this coordinated

equilibrium yet all right all right so

it it seems at least plausible that

they're going to fail to coordinate Le

plausible they're going to fail to

coordinate and we'd like to sort of

capture that idea and the way we're

going to capture that idea is let's see

if there's another equilibrium in this

game well there certainly isn't another

pure strategy equilibrium in this game

is that we know that so if there's

another equilibrium it better be mixed

so let's try and

find a mixed

Nash equilibrium in this game and

remember this game is called Battle of

the

Sexes it's a famous

game his battle the sex is

Revisited all right so how are we going

to go about finding this mixed Nash

equilibrium in the game we we'll we'll

interpret it later but let's just work

on finding it so in particular I'm going

to

postulate the idea that Nina is going to

mix P1 minus p and David is going to

mix

Q 1 minus

Q so how do we go about finding David's

equilibrium mix q1 minus Q it's our

trick from last week should been a CO

call at this point but it's not have to

how am I going to find Q the equilibrium

Q somebody

ah thank you thank you use Venus's

payoffs good good good so to find it

isn't Venus's payoffs it's Nina payoffs

okay uh but fair enough to to

find sorry to

find the Nash equilibrium Q to find the

mix that David's using we use Nina's

payoffs we use Nina's

payoffs all right let's do

that

all right so in particular for Nina if

she goes apple picking then her payoff

is

two with probability Q if she meets

David

and zero

otherwise and if she goes to the

rep then her payoff is one if she meets

David uh sorry that's going be careful

uh start again it she goes to the rep

her payoff is zero if David Goes apple

picking with probability q and her

payoff is one if she meets David at the

rep which happens with probability 1

minus q that

correct all right so this is her payoff

from apple picking and this is her

payoff from seeing Richard II and what

do we know if Nina is in indeed mixing

what do we know about what do we know

about these two

payoffs they must be equal right if Nino

is in fact

mixing then these two things must be

equal and that means what we're saying

is 2 Q = 1 1 minus Q

or uh Q equals uh uh 2/3 I guess it is

no it's one3

sorry

one3 one3 sorry that right here is one3

okay so our guess is that if there's a

mixed strategy equilibrium it must be

the case that David is assigning a

probability a third to going apple

picking which means he's assigning

probability

2/3 to his more favored activity which

is going to see Richard

II right what

about I'm going to pull these both

down okay how do we find Nina's mix all

right so to

find

to find the Nash

equilibrium P to find Nina's mix what do

we do what's the trick

somebody use David's payoffs use David's

payoffs so David's payoffs if he goes

apple

picking then he gets a payoff of

one if he meets Nina there and

zero otherwise and if he goes to the

rep he gets a payoff

of zero if Nina's gone apple picking and

he gets a payoff of two if he meets Nina

at the

rep all right and once again if David is

indifferent it must be these are

equal so if these are if David is in

fact mixing between apple picking and

going to the rep it must be that these

two are equal and if we set this now

carefully we'll get a c we'll get 1 p =

2 1 minus

P which is p equal

2/3 and 1 minus

P equals

13 so here we have Nina assigning 2/3 to

going apple picking which in fact is her

more favored thing and 1/3 to going to

the rep all

right okay so we just Ed the same trick

as last time let's check that this is in

fact an equilibrium all right so in

particular let's check that it is in

fact an equilibrium for Nina to choose

2/3 one CH all right let's

check so

check that p = 2 thir is in fact a best

response for Nina all right let's go

back to Nina's

payoffs for Nina if she chose to go

apple picking her payoff now is two * q

that Q is equal to a

thir plus 0 one minus uh oneus

q and if she choses chooses to go to the

rep then her payoff is zero with

probability a thir and one with

probability now 2/3 all I've done is

I've taken the lines I had before and

substituted in now what we know must be

the correct q and 1 minus

q and this gives her a

payoff of 2/3 in either

case all right if she chooses P her

payoff to p

P will be 2/3 of the time she'll get the

payoff from apple

picking which is 2/3 and one3 of the

time she'll get the payoff from going to

the

rep which is

2/3 for a total of

2/3 all right so Nina's payoff from

either of her pure strategies is 2/3 her

payoff from our claimed equal ium mixed

strategy is 2/3 so neither of her

possible pure strategy deviations were

profitable didn't didn't lose or

anything either but they weren't

profitable and by the lesson we started

the class with that means there cannot

be any strictly profitable mixed

deviation either so indeed for Nina p is

a best response to Q right and we could

do the same for David but let's not

bother it's AET it's a all right so in

this game we found another

equilibrium the other equilibrium the

new equilibrium is Nina

mixed

2/3

1/3 and David

mixed

1/3

2/3 and we also know the payoff from

this equilibrium the equilibrium from

this

payoff for both

players was

2/3 there are three equili in this game

they managed to meet at apple picking in

which case the payoffs are two and one

they managed to meet at the rep that's

the pure the second pure strategy

equilibrium in which case the payoffs

are one and two or they mix both of them

mix in this way and their payoffs are

2/3

2/3

right why is the payoff so bad

in this mixed strategy

equilibrium everyone agree this is a

pretty lousy payoff right the other

equilibrium payoff the worst you got was

one and you sometimes got two but now

here you are playing a different

equilibrium and this different

equilibrium you're only getting 2/3 why

why are you only what what happened why

has these payoffs got pushed down so far

what's happening to our poor hapless

couple or not Happ us I don't know

what's happening to our

coule yeah uh here

Ash okay yeah sometimes they don't meet

yeah they're failing to meet right the

reason what what's forcing these payoffs

down is they're not meeting very often

how often are they actually

meeting how often are they

meeting let's have a look let's go back

to the previous

board here it is all right so they meet

when they end up in this box or this box

is that right all right so what's the

probability of them ending up in those

boxes well ending up in this box is

probability

2/3

1/3 and ending up in this box is

probability

13 2/3 is that right you end up in you

end up meeting apple picking the 2/3 of

the time when Nina goes there times the

1/3 of the time when David Goes goes

there and you end up meeping at the rep

the 13 of the time Nina goes there times

the 2/3 of the time that David goes

there all right so this is the total

probability of meeting this is the

probability of

meeting and it's equal

to can't

right sorry uh 49ths

49ths is that way

four

n all right so four nths of the time

they're meeting but 59th of the time and

more than half the time more than half

the time they're screwing up and failing

to meet this is why I call them hopeless

dating couple right all right all

right right so this is a very bad

equilibrium but it captures something

which is true about the game what is

surely true about this game is that if

they just played this game they wouldn't

meet all the time in fact what we're

arguing here is they'd meet less than

half of the

time all right but certainly this idea

that we given from the pure strategy

equilibria that they would magically

always manage to meet seems very

unlikely so this does seem to add a

little bit of realism to this analysis

of the

game however it leads to a bit of an

interpretation problem you might ask the

question why on Earth are they

randomizing in this way why are they

doing this it's bad for every why are

they doing

this right and this leads us to think

about a second interpretation for what

we think mixed strategy equilibria are

rather than thinking of them literally

as randomizing it's probably better in

this case to think about the following

idea we need to think about David's

mixture as being a statement about what

Nina believes David's going to do David

may not be literally

randomizing but his mixture q1 minus Q

we could think of as Nina's belief about

what David's going to do and

conversely Nina may not literally be

randomizing but her P1 minus P we could

think of as David's belief about what

Nina's going to

do and what we've done is we've found

the beliefs we found the beliefs such

that these players are exactly

indifferent over what they do we found

the beliefs for for David over what Nina

is going to do such that David doesn't

really quite know what to do and we

found the beliefs that Nina holds about

what David's going to do such n doesn't

quite know what to

do that makes

sense all right so it's probably better

here to think about this not as people

literally randomizing but these mix

strategies being a statement about what

people believe in

equilibrium all right we'll come back

and look at this game some more later on

so our our couple I'm afraid are not

quite out of the woods

yet but I want to spend the rest of the

day looking at a yet another

interpretation of mixed strategy

equilibrial so so far we have two we

have people people are literally

randomizing we have thinking of these as

Expressions about what people believe

believe in equilibrium rather than what

they're literally doing and now I'm

going to give you a third

interpretation so for now we can get rid

of the Venus and Serena

game so to motivate this third idea here

I want to think about taxx

audits all right so none of you here

have ever probably ever had to P out a

tax form except for the fact that there

seems to be a lot of parents in the room

today is it parents weekend is that

what's going on all right so the the

where are the parents W wave your arms

in the air your parent here yeah so okay

so at least these guys are probably some

point in their life filled out a tax

form all right right so come tax day uh

the parents in the room face a choice

and the choic is are are they going to

honestly fill out their taxes or are

they going to cheat I'm not going to ask

them what they well maybe I will but for

now I won't ask them what they did all

right

and so they can choose one of two things

they can choose to pay their taxes

honestly we'll call it H or to

cheat this is the the tax player the

parent all right and at the same time

the audit office

the auditor can has has to make a choice

and the auditor's choice is whether to

audit you or not and this is not

literally true because literally the

auditor can wait till your tax returns

comes in and then decide whether to

audit you all right but for now let's

think of these choices being made

simultaneously we'll see why that makes

it more

interesting all right so let me put down

some payoffs here and then I'll explain

them

so

20 4 - 10 40 and

04 all right so how do we interpret this

let's look at the auditor's payoffs

first of all so the auditor is very

happy not having to audit your parents

and having your your parents pay taxes

so we'll give that a payoff of

four and it'll turn out in this game we

we've decided uh in the payoffs that the

auditor is equally happy if she actually

audits your parents in the year that

they

cheated all right that we'll get the

we'll say that makes the auditor equally

happy all right now the auditor is not

so happy if she audits your parents when

they're honest because audits are

costly all right and uh the auditor is

really unhappy if she fails to audit

when the Parents cheat

it all right let's look at the uh I keep

calling parents I talk parents let's

call them taxpayers right so the for the

taxpayers what are their payoffs well

we'll normalize things so if they're

honest we'll give them a payoff of

zero right that means they correctly

fill in their tax form and pay what

they're supposed to pay but if they

conceal some of their income they

pretend to have a whatever it is a third

child all right all right then then they

might be in trouble if they're audited

if they're audited they're going to have

to pay a big fine maybe even go to jail

so that's - 10 of course if they're not

audited they get to keep a chunk of

money so we'll call out

four all right everyone understand the

basic idea of this game right in reality

we could add more complications we can

think of different ways to cheat in your

taxes but I don't want to give tutorials

on how to cheat your tax on cheating

your taxes here all right so it's not

going to take long staring at this game

to figure out that there are no pure

strategy equilibri in this game let's

just do that so from the uh taxpayers

point of view if they're going to be

audited then they'd rather pay their

taxes than not and if they're not going

to be audited then according to these

payoffs they'd rather

cheat all right and from the auditor

point of view if they knew everyone was

going to pay taxes then uh they wouldn't

bother

auditing and if they knew everyone was

going to cheat then uh they'd of course

ought

it all right so you can quickly see that

there's no box in which the best

responses coincide there's no pure

straty Nation

equilibrium right for those people who

think who this is seeming otherworldly

uh you will have to pay tax taxes in a

couple of years and trust me your

parents are paying taxes

now all right so what we want to do here

is we're going to solve out and find a

mix strategy equilibrium but we're going

to give it a different interpretation to

the the equilibria we've found so far

but the basic initial exercise is what

we're going to find we're going to try

and find the equilibrium

here to find the Nash equilibrium here

we know it's going to be

mixed all right so to find the

probability with which taxpayers pay

their

taxes and let me already start getting

ahead of myself and to say to find the

proportion of tax players who are going

to pay their taxes

what do we do what must what must be

true of that equilibrium proportion Q of

taxpayers who pay their taxes how am I

going to find that Q shout it out

somebody yeah look at the auditor's

payoffs all right so from the auditor's

point of

view if the auditor audits their payoff

is two

Q + 4 1 minus Q

and if they don't audit their payoff is

4 Q

Plus

0 1 minus Q all right every once I did

this this is 2 Q + 4 1us q this is 4 q +

01 minus q and if indeed the auditor is

mixing then these must be

equal this must be equal and if they're

equal let's just do a little bit of

algebra here and we'll find that 2 Q

equals

4 1 minus

Q that's right so

Q equals uh boom 2/3 is that right four

yes

2/3 qal

2/3 all right so our claim is to make

the auditor exactly indifferent between

whether to audit or not it must be the

case that 2/3 of the parents of of the

the parents of the kids in the room are

going to be paying their paying their

taxes honestly which is a third an which

is kind of worrying never mind all right

let's have a look at the taxpayer to

find to find sorry sorry we found the

taxpayer we found the proportion of

taxpayers who are paying the taxes now I

want to find out the probability of

being audited how do I figure out the

equilibrium probability of being audited

in this

model how do I work out the equilibrium

probability of being

audited shout it

out the equilibrium probability of being

audited I'm going to use p and one minus

P so p is going to be the probability of

being audited how do I find

P yeah I'm going to look at the

taxpayers payoffs so from the taxpayers

point of

view if the taxpayer pays their taxes

their payoff is just zero and if they

and if they cheat their payoff is

-1

P plus

4 1 minus

P all right and if

indeed the taxpayers are mixing or other

words saying that is not all taxpayers

are cheating and not all taxpayers are

honestly paying their taxes then these

must be equal so if these are equal I'm

going to get 4 P

equals

14 no it doesn't I'm going to get 4al 14

P let's try again 4al 14 P that's bit

woring 4 14 P which is the same as

saying P equals

27 someone can just check my algebra I

think that's right hope that's

right all right so I my claim is that

the equilibrium here is for

23 of the taxpayers to pay their taxes

and for the audits for the auditor to

audit 27th of the

time all right now we could go back in

here and we could check I could do what

I did before I could plug the p's and

q's in here and check that in fact this

is an equilibrium but trust me that I've

done that right trust me that it's okay

okay so here we have an equilibrium

let's just write down what it

is from the auditor's point of view it

is that they audit 27th of the time or

27th of the

population and from the taxpayers point

of view it's that they pay their taxes

honestly 2third of the time and not

otherwise all right now without focusing

too much on these exact numbers for a

second I want to focus for for a minute

on how do we interpret this mixed

strategy

equilibrium right so from the point of

view of the auditor we're really back

where we were before with the base

dealer or the person who's search

searching baggage on the air at the

airport we can think of the auditor

literally as

randomizing right in fact there's some

truth to that it actually is the case

that by law that the Auditors literally

have to

randomize all right so this 27 57 this

has the same interpretation we had

before this is really a

randomization right but this 2/3 1/3 has

a different interpretation and a

potentially exciting

interpretation it isn't that we think

that your parents get to tax day work

out what their taxes would be and then

toss a

coin right they may be doing that I'm

looking at the parents here I don't

think that's what they're doing right

the interpretation here is that the

parents some parents are paying their

taxes and some parents aren't paying

their taxes there's a lot of parents out

there a lot of potential taxpayers at in

the population in equilibrium if these

numbers were true 2third of parents

would be paying their taxes and onethird

would be cheating so this is a a

randomization by a player and this is a

mixture in the population

the new interpretation here is we could

think of the mix strategy not as players

randomizing but as a mix in a large

population in which some people are

doing one thing and the other group are

doing the other it's a proportion of

people paying

taxes right so I don't know if this 2/3

1/3 is an accurate number for the US

it's probably not very far off actually

for Italy I'm ashamed to say uh that

number uh the number of people who pay

taxes is more like 40% maybe even lower

now uh and there are countries I think

uh where it gets as high as 90% I think

the US rate when they end up auditing is

a little higher than this but not

much all right okay so again we're going

to think of this not as randomization

but as a prediction of the proportion of

American taxpayers who are going to pay

their

taxes all right now I want to use this

example in the time we have left to

actually actually think about a policy

experiment all right so let's put this

up somewhere we can see

it and let's think about a new tax

policy so suppose that Congress gets fed

up with all these newspaper reports

about how two-thirds of Americans don't

pay their taxes whatever whatever the

true proportion is I think it's actually

a little higher than that but never mind

they get fed up with all these reports

and they say this isn't fair we should

make people pay their taxes so we're

going to change the law and instead of

paying instead of being in jail for

instead of being in jail for 10 years or

the equivalent of a fine of minus 10 if

you're Court cheating we're going to

raise the fine or the time in jail so

that it's now minus 20 right so the

policy experiment

is

let's raise the

fine the fine the

cheating to minus 20 and the aim of this

policy is to try to deter cheating

right right seems a plausible thing for

for government want to want to

do all right let's redraw The Matrix so

here's the game

20 4 -

20

[Music]

404 audit not

audit

and pay honestly or

cheat all right so here's our new

payoffs and let's ask the question with

this new fine in place now we've raised

the

fine to being caught not paying your

taxes in the long run once things have

worked their way back into equilibrium

again

right after a few years do we expect

American taxpaying compliance to go up

or to go down or what do we

expect what do we think's going to

happen so who thinks it's going to go

up who thinks it's going to go

down who thinks going to stay the

same who's abstaining here I Noti the

parents are abstaining not really meant

to abstain you have to have to to vote

here well how are we going to figure

this out how we going to figure out

what's going to happen to compl

liance what

happens to tax

compliance tax compliance remember that

that was our that was our p no it wasn't

sorry it was our it was our Q it was our

Q right well only way we're going to

figure this out is to work it out so

let's work out the new Q in

equilibrium all right let's do this

all right so to find out the new q and

equilibrium once again we're going to

have to look at the auditor's payoffs

and the auditor's payoffs if they audit

they're going to get 2 Q + 4 1 - q and

if they don't order it they're going to

get 4 q + 0 1 - q 4 q + 0 1 - Q 2 Q + 4

1 - q and if the auditor is indifferent

if they're mixing it must still be the

case that these are equal and I want to

ask you a question where have you seen

that equation

before yeah it's still there right I

didn't delete it it's the same

equation as sits up there is that right

from the auditor point of view given the

payoffs to the Auditors nothing has

changed so the tax compliance rate that

makes the auditor exactly indifferent

between auditing your parents and not

auditing your parents

is still exactly the same as it was

before at 2/3 in

equilibrium tax compliance hasn't

changed at

all say that again right the policy was

we're going to double the

fines for being caught

cheating and in equilibrium it made

absolutely no difference

whatsoever to the equilibrium tax

compliance rate

all right now why did it make no

difference why did it make no difference

well let's let's have a techy answer and

then and then a better a more intuitive

answer the techy answer is

this what determines the equilibrium tax

compliance rate what determines the

equilibrium mix for the column

player is what is the row

payoffs right what determines the

equilibrium mix for the column player

are the rows payoff row players payoffs

we didn't change the row players

payoffs so we're not going to change the

equilibrium mix for the column

player all right say again we changed

one of the payoffs with a column player

but the columns players equilibrium mix

depends on the row players payoffs and

we haven't changed the roow players

payoffs so we won't change the

equilibrium compliance rate the

equilibrium Mak by the column

player what will have changed here what

will have changed in the new

equilibrium so we've pretty much

established that people are cheating as

much as they were before an equilibrium

R can I get Henry here wait for the mic

say again the probability of audit will

have changed the probability of audit

will have changed what's going to change

is not the Q but the P the probability

with which you're audited is going to

change in this model all right let's

just check it to find the new P I need

to look at the taxpayers payoffs and the

tax payers payoffs are

now zero uh if they sorry if they pay

their taxes

honestly then they get zero and if they

cheat they get minus 20 with probability

p and four with probability 1 minus

p

if they're mixing if some of them are

paying and some of them not this must be

the same and being more careful than I

was last time I hope uh this gives me uh

24 p is equal to 4 or P equals a

6 all right so the audit rate has gone

down from

27th

to6 I'm guessing that probably wasn't

the goal of the policy isn't necessarily

a bad

thing right there is some there is some

benefit for society here because audits

are costly both to do for the auditor

and they're unpleasant to be audited so

the fact we've managed to lower the

audit rate from 27ths to a 6th is a good

thing all right but we didn't manage to

raise the compliance

rate all

right so I don't want to take this model

too literally because it's you know it's

just it's just a toy model but

nevertheless let's try and draw out some

lessons from this

model all right so here what we did was

we changed the payoff to cheating we

made it made it worse right but a

different kind of change is we could

have

changed sorry we changed the payoff

negatively to being Court

cheating right but a different change we

could have done is we could have left

the minus 10 in place and we could have

raised the payoff to cheating a not

getting caught we could have left this

10 in place and changed this four let's

say to a six or an eight right we could

increase the benefits to cheating if

you're not caught what would that have

done in the

equilibrium what would that have done in

the equilibrium so I claim once again

that would have done nothing in

equilibrium to the probability of people

paying their taxes but that would have

done what to the audit

rate the audit rate would have gone up

the equilibrium audit rate would have

gone up right let's tell that story a

second right so rich people people who

are well paid have a little bit more to

gain from cheating on their taxes if

they're not caught there's more money at

stake all right so my colleagues who are

Finance professors in the business

school have more money on their on their

tax returns than I do so in principle

they gain more if they cheat all right

does that mean that they cheat more than

me in

equilibrium

no it doesn't mean that they cheat more

than me equilibrium what does it

mean it means they get audited more

often all right in equilibrium richer

people aren't necessarily going to cheat

more but they are going to get audited

more and that's true right the the the

federal audit rates are designed so they

audit the rich more than they audit the

poor again it's not because they think

the rich are inherently less honest or

the poor are inherently more honest or

anything like that it's simply that the

gains to cheating and not getting caught

or bigger if you're rich so you need to

audit more to push back into

equilibrium all right now suppose we did

in fact want to use the policy of

raising

fines to push down to push up the

compliance rate to push down

cheating how would we change the

law supposing we want to we want to get

we want to raise the fines for cheating

we don't like people cheating so we

raise the findes but we're worried about

this result we're worried about this

result that didn't push up compliance

rates how can we change the law or

change the incentives in the game so it

actually would change compliance rates

what could we

do

yeah um if we change the payoffs um of

auditing um to four from two to four

good good good if we want to change the

compliance rates we should change the

payoffs to the

auditor all right the problem with the

way the auditor is paid here is that the

auditor is paid more if they manage to

catch people right but audits are costly

all right and the problem with that is

when you raise the fine on the other

side all all that happens is is the

Auditors audit less often in equilibrium

so if you want to get a higher

compliance rate one thing you could do

is RA is change the payoffs to the

auditor to make auditing less costly for

them or making or making catching people

uh uh uh nicer for them give them a

reward or you could simply take it out

of game three altogether you could

enforce you could have a congressional

law that sets the audit rates outside of

equilibrium right and that's been much

discussed in Congress over the last five

years simply setting audit rates as it

were exogenously by Congress why might

that not be a great idea leaving aside

economic theory for leaving aside Game

Theory why might it not be a great idea

to have Congress set the audit rates

rather than rather than some office yeah

most members in Congress have a lot of

money so they're gonna lower the audit

rates right so so the lady in the front

is saying that a lot of congressmen are

rather Rich so maybe they have

particular incentives here I I don't I

don't I don't I don't want to take a a

particular political stance here but it

could be whatever side of the political

Spectrum you guys sit on all right it

could be that you might not trust

Congress to get this right right you

might think they're going to be

political considerations going on in

Congress other than just having an

efficient tax system all right okay so

what do I want to draw out as lessons

here the big lessons from this class are

there are three different ways to think

about randomization and equilibrium or

out of equilibrium one is it's genuinely

randomization another is it could be

something about people's beliefs and a

third way in a very important way is it

could be telling us something about the

proportion of people who are doing

something in society in this case the

proportion of people who are paying

tax all right and a second important

lesson I want to draw out here Beyond

just finding

equilibria right two other things we

draw out today one lesson was when

you're checking

equilibria checking mixed strategy

equilibria you only have to check for

Pure strategy deviations be careful you

have to check for all possible pure

strategy deviations not just the pure

strategies that were involved in the mix

if the guy has seven strategies and is

only mixing on two you have to remember

to check the other five

right and the third lesson I want to

draw out today is because of the way

equilibria work this mix strategy

equilibrial work if I change the column

players payoffs it changes the row

players equilibrium mix and if I change

the row players payoffs it changes the

column players equilibrium mix next time

we're going to pick up this idea that

mixed strategies can be about

proportions of people playing things and

take it to a totally different setting

namely Evolution so on Wednesday we'll

start talking about

evolution