All language subtitles for English (auto-generated)_en_0_7. Nash equilibrium_ shopping_ standing and voting on a line(360P)

All right, so last time

we started to study imperfect

competition.

We looked at the Cournot model,

but the bigger theme

the bigger theme is the study of how

firms compete outside of the sort of

easy cases that you study in 115. So,

outside of the example of monopoly,

where there is only one firm and so

there's not much competition, and

outside the case of perfect competition,

where there are so many firms it's as if

each firm is a price taker.

All right? We're going to We're going to

continue with that today, and in fact

you're going to continue with it on into

the homework.

So, last time we looked at Cournot, that

was the model we started with.

We looked at two firms competing in

quantities, and just to review a little

bit what we did last time,

uh kind of exercise there. So, I I think

Is this right? I think I probably went a

little fast last time. I was kicking

myself afterwards. I don't know if

people can nod or shake their heads or

they like I I think I went a bit fast.

So, I'm so I'm sorry for that, but if if

there if there was a time to go too

fast, it was probably last time, because

it turns out this is something that's

covered very well in the textbook. So,

it's covered in chapter six of the

textbook.

But, the kind of exercise we did last

time

is a useful exercise for those of you

who are econ majors.

The way we solved that Cournot model was

we did three kinds of things. We did

something very nerdy, namely we just

played with calculus and algebra.

All right? That was mathy.

We did something a little bit less

nerdy, that is we drew some pictures

that represented what what we'd found.

And we did a third thing, which was we

tried to match this up to the economic

intuition about monopoly and perfect

competition and demand curves and so on.

And this exercise of being able to work

in these three different modes, economic

intuition, graphs, and kind of nerdy

high school math, is a lot of what we

want to get you used to as economic

majors. Just to to to be able to

translate easily between those. Again, I

apologize for going too fast, but it's

it's still a useful exercise, I think.

So, have a look at it in the book.

Now, back to the lessons and away from

the nerdiness a second, what did we

learn?

Well, at the end of the day, we learned

that in the Cournot equilibrium,

things were as we perhaps might have

anticipated, things were sat naturally

between the extreme cases. So, the

amount of output produced by the

industry was somewhere between the case

that would be under monopoly and under

perfect competition. It was more than

under monopoly, less than perfect

competition. Prices in the industry, if

we'd gone back and checked, would sit

between. They would be lower than the

monopoly prices, but higher than the

perfect competitive prices. Industry

profits would be in between. Industry

profits would be less than monopoly,

they must be less than monopoly since

that's the highest they could ever be,

and they're higher than perfect

competition, which of course has zero

profits in this case. All right? And

consumer surplus, the benefits flowing

to consumers lie in between.

So, in fact, the model, you know, in in

addition to the sort of nerdiness of the

model, it ended up with a result we kind

of believe in. You have imperfect

competition, it's somewhere between

perfect competition and no competition.

But, what's the butt? The butt is that

there are other ways we could model

imperfect competition, and as we're

going to see today, they yield different

answers. So, the first thing we're going

to do today is look at a different form

of competition,

which is called Bertrand competition.

So, Cournot was competing in quantities,

and Bertrand is competing

in prices.

All right? And just a quick check, and

this is not for the camera, how many of

you have seen Bertrand competition

before?

So, a good many of you. So, this is

going to be a review still. Again, for

those of you who it isn't for whom which

it isn't review, don't worry. Don't

worry. We'll do this relatively quickly,

but I hope I won't rush it like I did uh

last time.

If it's any consolation, I get when I

feel I've rushed a lecture, I have a

sleepless night. So, if you had a

sleepless night cuz you were worried

about it, think of you know, I'm I'm

having a sleepless night, too, about

that.

All right.

So, there's two firms, just as there was

before,

and just as there was before, the So,

the players

are those two firms, and just as there

was last time, they're producing an

identical product.

All right? So, the products we mentioned

last time were Coke and Pepsi, uh but

you could think of other products that

are pretty much identical.

All right? And just as last time, we're

going to assume

that costs are constant marginal costs.

We'll assume that constant marginal

cost, just as it was last time, is equal

to C. Just to remind those people who

are not economics majors what that

means, it means if I produce one unit,

then it costs me C to produce. If I

produce two units, it costs me 2C, 100

units 100C, and so on.

All right? Now, what's different here

from from last time, I've already told

you, but I'll repeat it. This time,

however, instead of setting quantities,

instead of just deciding how much Coke

and Pepsi to produce and spewing it out

into the market and letting prices take

care of themselves, this time the firms

are going to set prices

and let quantities take care of

themselves.

All right? So, the the the strategy set

this time

are prices.

And again, so we don't get confused, you

know, normally we use S for strategy,

but let's use P since they're prices.

So, they're going to be P1 will be the

price of good one

of firm one, and P2 will be the price of

firm two. And the strategy set,

formally, let's just to simplify here,

let's assume that

for each firm I, they can set their

price anything bigger than zero

and anything less than one.

Just to keep life simple.

All right? So, this is a little bit more

realistic, perhaps. It's perhaps more

realistic to think of firms actually

thinking about what prices to set and

then adjusting the quantities they

actually ship based on demand. All

right? Depends I I I don't need to uh

rigorous about that, cuz you could

imagine the other way around, but this

you know, if you talk to to managers,

this sounds more realistic, at least for

retail goods like Coke and Pepsi.

All right? So, we know how prices are

set, they're set by the firms.

So, the next question is, where do the

quantities come from?

All right? So, how do Where do the

quantities come from? And they're going

to come from demand, and let me use a

big Q of P, a big Q of P, to be the

total quantity produced So, so so the

total quantity demanded in the market.

So, this will be the total quantity of

uh of goods produced by good uh goods

produced by firm one and goods produced

by firm two that are consumed. All

right? So, this is the total quantity of

Coke plus Pepsi.

All right? And we're going to and I I

notice there's no subscript on this P,

and you'll see why in a second.

So, we'll assume just to make life

simple

that the total quantity produced is 1 -

P,

and I'll say this and then write it more

carefully, where P is the lower of the

two prices.

All right? So, the total quantity

demanded is 1 - P, where P is the lower

of the two prices.

All right? Now, to make that lower of

the two prices come out a little bit

more rigorous, let's figure out what the

demand actually is for these firms.

So, what's going on here?

So, let's look at the demand

for

firm one.

All right? Which is going to end up

being the quantity that they sell.

So, that's going to be Q1.

And what's that going to be?

Well, it's going to be 1 - P1

if they're the low price firm.

So, if P1

is less than P2.

All right? So, if they're the low price

firm, if Coke is the low price firm,

only Coke sells in the market, there's

no Pepsi in the market, and 1 - P1

quantity of Coke is sold.

It's going to be zero

if P1 is bigger than P2.

All right? So, if Pepsi is the is the

low price uh drink in the market, then

no Coke is sold at all.

And it's going to be

1 - P1 over two

in the case

where Coke and Pepsi cost exactly the

same.

All right? You should already be

realizing this is not exactly realistic,

right? We're making a lot of very strong

assumptions here. But, nevertheless,

it's going to be instructive to look at

this model.

All right? So, I've got the firms,

they're the players, I know what their

strategies are. I know a little bit

about the structure of the market. I

still need to tell you what payoffs are.

And just as last time, the firms are

going to try and maximize profits. All

right? So, what's the profit for firm

one going to be? You know, I won't

bother to write it separately for firm

two. It's going to be the quantity it

sells

times the price

it gets for that quantity

minus that quantity it sells

times the cost it it cost it incurs in

producing that quantity. So, again, it's

for firm one, it's the quantity it it

sells times the price, this is its

revenue,

minus the quantity it sells times C, the

cost. This is its cost.

All right? I can rewrite that a little

bit more simply as Q1

brackets

P1 - C.

All right? Where Q1, where this Q1,

let's put it, let's keep it in square

brackets, this Q1 is this object here.

All right. Does everyone understand

what's going on? I mean, the algebra

doesn't really help the intuition here.

The intuition is pretty straightforward.

Coke and Pepsi are producing this stuff.

They're setting prices. If Pepsi is the

low price, then Coke sells nothing.

If If Coke is the low price, it sells

this amount, and its profits are

basically given by this equation. It's

basically just revenues minus costs.

All right. So, we What we want to do is

we want to figure out what the Nash

equilibrium looks like in this market.

And just just before we get any further,

notice that really this is the same

basic market we looked at before.

Right? Both times there's a demand curve

out there. In one model, we thought of

the firms setting quantities and the

market determining prices. And here we

have the firms setting prices and the

market setting quantities. But the basic

underlying economic structure of this is

very very similar.

All right.

All right. So, to find the Nash

equilibrium, we're going to have to find

best responses. All right. So, what what

we're going to do together is to figure

out what these best responses look like.

And let's try and do that.

So, I want to figure out the best

response of firm one

as a function of the price chosen by

firm two. Best price for firm one to

choose, given that firm two is choosing

P2. And I want to do this carefully and

slowly. And before I even start, let me

point out in passing that calculus is

not going to help us here.

All right. So, for those of you who are

nerdy enough to understand the next

sentence, let me just say it. This is a

discontinuous

This is a discontinuous payoff function,

so differentiating isn't going to get

you very far. All right. For For people

whom that didn't mean anything, doesn't

matter. We're not going to use calculus

today.

All right.

Okay. So, there are different cases to

think about. So, one case to think about

is what if the other firm, you're Coke,

what if Pepsi is pricing below cost?

What if Pepsi is pricing below cost and

you're Coke?

What should you price at if you're

trying to maximize your profits?

Someone try and wave their hands in the

air if they know the answer. Pepsi's

pricing below cost. What's your best

response?

Yeah, there's a a guy in there. You want

to uh

shout out? Get out of the market because

it's not worth it for you to sell

anything. Right. So, the answer So, So,

I've forgotten your name. I should know

it by now. Your Your name is Sudipta.

Sudipta says, "Get out of the market."

Right? If the If the other guy is

pricing below costs, you don't want to

sell anything. That's the right

intuition. How in this game do I quote

get out of the market?

You're absolutely right. I I'm

absolutely right. I don't I don't want

to be involved in this market if the

other guy is selling below cost. How do

I How do I operationalize that in this

game? How do I actually How do I

actually manage How do I address

somebody here?

So, just just wait for the mic a second,

and I'll shout it out. You could set

your price above his price. So, the

the answer was, "Set your price above

his price." So, if the other guy is

pricing below cost, the way in which I

avoid making losses is to set my price

above his price.

All right. Let's just make sure we

understand this. If the other guy is

selling below cost, the only way I can

make any sales is to price below his

price. I will then make sales,

but each of those sales I'm making a

loss on.

I don't want to make losses, so I just

quote get out of the market. And the way

I quote get out of the market is by

pricing above the other guy.

All right. Everyone understand that?

All right.

Okay. So, now

more interestingly,

let me leave a little gap here.

What if

the other guy is pricing above costs?

What do I want to do if the other guy is

pricing above costs? Can I can get the

guy with the the the beard here? Shout

it out because remember that mic isn't

on. You You set a price below his, but

above above or equal to cost. All right.

So, So, if the other guy is pricing

above costs,

I want to set prices below his so that I

steal the whole of the market and make

profits on those sales.

So, where is it? So, suppose he's

pricing at We said the price is between

zero and one. Suppose he's pricing at

at 0.8.

What will be a good price for me to set?

Same same guy, yeah?

As a small amount below his price as you

can do. A small amount as a small amount

below his price as I can. A small amount

below his price as I can. So, basically

what I'm going to do here is I'm going

to set my price

to equal his price minus a little bit.

Now, I'll use the letter epsilon to mean

just a little bit. I'll just undercut

him a little bit. And by just

undercutting him a little bit,

I'm going to get the whole of the

market, and I'll make as much money as I

can on those sales.

Now, that's almost right.

That's almost right.

That's kind of 90% of right, but it's

not quite right. I need to be a little

bit careful.

Now, why do I need to be a little bit

careful here? So, this is almost right,

but it's not quite right. It's same same

guy?

You want it the your price to be at or

above cost. Oh, oh, that's Okay. That's

certainly true. Okay. So, that's that's

true. So, I

Fair enough. So, I want to I want to

make sure this epsilon is not so big as

it pushes me below cost. That's

certainly true.

That's That's correct. There's another

little issue here.

So, here's the other issue. There's a

price in this market we might want to

think of as a kind of focal price.

There's an interesting price. And that's

the price at which the price I would

choose were I the monopolist.

All right. Suppose the other guy didn't

exist. Suppose Pepsi didn't exist. So,

Coke has the whole market.

Right? Then we we would solve out this

problem. We would solve out what the

monopoly price is. That's an exercise we

did in 115.

All right.

And notice that if Pepsi has priced

above the monopoly price,

I suppose Suppose Pepsi has priced this

good so high that it's above the

monopoly price,

then as the gentleman said, I can

capture the whole market by pricing just

below Pepsi.

However, however, then I then I have the

whole market. It's as if I'm a

monopolist, but suddenly I'm pricing

above the monopoly price.

That can't be right.

Right? I never want to price above the

monopoly price cuz I know the monopoly

price, when I'm a monopolist, is the

most profit I could ever make.

Is that right? Right? So, we need to be

a little bit careful. This answer that

says I just undercut Pepsi, that's true

provided provided that Pepsi

hasn't gone

above the monopoly price.

Right? If he's gone above

If If If If If he's gone above the

monopoly price, we need to be a little

bit more careful.

If P2 is actually above

the monopoly price,

then what's my best response in that

case?

What's my best response in that case? Uh

yeah. Can we get Katie who's in green

there?

Shout out. Pricing at the monopoly

price. Pricing at the monopoly price.

So, if If Pepsi is dumb enough to price

above monopoly, sure I will undercut

Pepsi, but I won't undercut Pepsi by a

penny. I'll undercut Pepsi all the way

down to the monopoly price and make

monopoly profits. I'll be very happy.

I'll be wondering what on earth is Pepsi

up to, but that's fine. All right. So,

here my best response

is to price at the monopoly price.

All right. And there's one other

possibility here I've missed, which is

what if

Pepsi

prices at marginal cost itself?

That's the one missing case here.

What if Pepsi prices at marginal cost?

What's my best response in this case?

Pepsi's pricing exactly at marginal

cost. What's my best response?

Uh yeah. Can we get Wait for the mic.

This This is Henry, I think.

Shout out. To price at marginal cost as

well. I could price at marginal cost as

well. How much profit will I make if I

if I price at marginal cost as well?

Zero profits. I'll make zero profits.

So, that certainly is a best response,

pricing at marginal cost as well. What

else would be a best response?

That's That's correct. But what else

would be a best response?

Price above that. All right. It doesn't

really matter as long as I don't price

below it.

All right. As long as I don't price

below it,

I'm not going to make any money anyway.

Right? If I price below it, I'm going to

lose money.

All right. So, this best response

actually is a pretty complicated object.

And we could, if we're going to go like

we did last time, we could take the time

to draw this thing, but it's a bit of a

mess. So, I won't I won't worry about

doing that right now.

All right. The best response is actually

pretty complicated. Even though the best

response is pretty complicated, and by

the way, obviously the things are

symmetric for for for player two. Even

though these best responses are pretty

complicated, it turns out that there's

only one Nash equilibrium in this game.

What's the Nash equilibrium? Somebody

What's the Nash equilibrium? Raise your

hand. All you guys There's a huge number

of hands who said they'd seen it before.

So, uh

Who saw this model before?

Some Fewer hands go up cuz they know

there's a question coming, right?

There's a different incentive, all

right? Who remembers what the Nash

equilibrium is?

Yeah, there's a guy There's a guy right

by you there.

At C.

Shout out cuz even I can't hear you. At

C. At C. Okay. So, the the Nash

equilibrium here

The Nash equilibrium is for both firms

to set their prices

equal to marginal cost.

And you can check that both firms are

then playing a best response. So, that's

that's all right. If If firm two is is

charging C, then a best response for

firm one is to set uh Let's put the ones

in here.

is to set price equal to marginal cost.

And if firm one is setting pricing at

marginal cost, then conversely the best

response for firm two is to price at

marginal cost.

All right, slightly harder exercise is

to check that nothing else is in Nash

equilibrium.

Well, let's think about that a second.

Suppose firm one is pricing

at marginal cost

and firm two

is pricing at something higher at um

uh

uh C plus

I don't know, three epsilon, a little

bit higher.

All right, suppose firm one is pricing

at marginal cost

and firm two and firm

two is pricing at C plus three epsilon.

Is this a Nash equilibrium?

Let's think about firm two first of all.

Is firm two playing a best response to

firm one?

Is firm two playing a best response to

firm one?

So, I claim it is, right? Well, let's

just check carefully. Firm one is

is pricing at marginal cost. Right? The

best response if the other guy is

pricing at marginal cost is to price at

marginal cost or above.

And this is marginal cost or above, so

this is a best response. So, firm two is

playing a best response.

This is a best response for firm two

given that

P1 is at C. What's the problem here? Why

is it nevertheless I claim this is not a

Nash equilibrium. Why is it not a Nash

equilibrium? Who has an incentive to

deviate? Take him get the guy way at the

back

way way way at the back by the door. Go

ahead. Shout out. Firm one's going to

want to produce at C plus two epsilon.

Exactly. Exactly. Exactly. This is not a

best response. This is not

a best response.

And the reason is that the best response

for firm one if firm two is charging C

plus three little bits is to try is to

price at C plus

two little bits.

Thank you.

All right. So, it turns out it's pretty

easy to check that this is the only Nash

equilibrium in the game.

All right.

Now, what's the lesson we want to draw

from this?

Well, as an exercise in game theory that

really wasn't very hard. All right, as

an exercise in finding Nash equilibrium

by this stage in the course most of you

looking at that saying that wasn't hard.

So, what's the point here? The point is

that in this game in which firms

competed in prices

even though there were only two firms in

the market right, only one firm more

than monopoly

right, we get a dramatically different

result than we had last time. In

particular, we find that prices in the

market

are equal to marginal costs. We find

that profits

in equilibrium is equal to what?

Zero.

And we find there's lots of consumer

surplus cuz prices are really as low as

they ever could be. In fact, the outcome

here, this equilibrium here is for all

intents and purposes the same

equilibrium we would have had had there

been thousands of firms in the market

and had this been a perfectly

competitive market. All right, so even

though there's only two firms here with

price competition

identical products, we end up with an

outcome that looks exactly like perfect

competition except that the for the fact

there's only two firms.

All right, so the outcome

is like

perfect competition

even though

there's only

two

firms.

That's a pretty surprising result.

It's It would suggest Think about this

as a policy thing. So, if you believe

this model, if you think this is really

an accurate model of of of society and

you're a regulator working in the

Department of Justice or you're a judge

trying to judge some monopoly case or

you're a commissioner on the European

Court or whatever trying to judge some

some competition competition case, all

you'd worry about is getting one

competitor in each market, two firms in

each market and you'd be done.

All right, you wouldn't worry about

entry beyond two.

All right.

Now, my guess is we don't believe that.

We don't believe that.

All right.

We don't believe it. We'll come back to

in a second, but let me make a different

remark before we get there.

Is that me doing that? Let me move this

down.

All right.

Still doing it, okay. It's probably the

wire.

Sorry.

I Let me just remove it and I'll shout.

Okay, I'm going to shout now. Can people

still hear me? Can people still hear

hear me?

Yep, okay, thank you. All right. Uh so,

we'll have to turn down the other mic a

bit.

All right.

So, what's And I And I another thing to

remark here is even though we looked at

essentially the same market as we looked

at last time and we didn't make any

really significant change. We just said

instead of thinking of them pricing in

setting quantities, think of them

setting prices. I mean, these were just

thought experiments. We ended up with a

radically different outcome.

All right, so another lesson here is the

same setting

the same setting as last time as Cournot

but with

a different

strategy set

a different way of thinking about what

they're doing

led to a very different outcome.

And that's worrying.

Right, it's worrying because you can't

go away feeling comfortable about this.

You can't think that it really could

make so much difference in the real

world how prices and quantities and

welfare and profit is going to work out

depending on some thought experiment

about how I think about my strategy set.

All right. So, there's a little worry

going in about not just this example but

about the course

about the whole theme of learning game

theory.

All right. So, what's going to save us

here?

What's going to save us is

if we inject a little bit more reality

back in the model we're going to get

back a more sensible result.

So, what we're going to do now is we're

going to relax some of the assumptions

of this Bertrand model and it's going to

do two things for us. First, it's going

to give us an outcome that we believe.

The outcome we believe I think is that

imperfect competition should look

something between monopoly and perfect

competition. It shouldn't look like

perfect competition. I think I think

most of us don't believe that two firms

is enough to make for perfect

competition, that the regulator

shouldn't worry about the third firm.

All right. And the second thing it's

going to do from a more theoretical

point of view is it's going to suggest

to us that actually actually things

aren't quite as bad as we thought. In

fact, if we model this more carefully,

we'll get roughly the same prediction

either way.

All right. So, what's the assumption

we're going to change?

Well, before I say that, I should say

I'm not going to change it. You are.

All right. So, rather than go through

this again a whole third time, I've gone

through it once in quantities and once

in prices, I'm going to get you guys to

do it this time by having you do it on a

homework assignment.

And well, why? Well, partly cuz I think

it's a good exercise, but also I don't

want this class to be a class where you

sit there, you know, with your cup of

Willaby's coffee if needed to keep

awake, and you watch me solve models cuz

that's not how you learn. What I want

this class to be is a class where at

least sometimes, like in this homework

assignment, you take you know, I set up

a model for you or set up the set up the

story for you and then you have to

actually figure out how do I set this up

properly and how do I solve it out? Cuz

in the end of the day, if game theory is

going to be useful for you in your later

lives, whether it's in dating or in

running a company or whatever it happens

to be, you need to be able to go from

the story to the model.

All right. Having said that, I'll tell

you what the homework assignment's going

to be about. The assumption we're going

to change is the idea that products are

identical.

All right, so we're going to look at a

case

I need a new board.

where products are differentiated.

And I'm going to claim

that assuming that products are not

identical

is a pretty safe assumption for most of

the world, for most things that you're

going to see in the world. So, we're

going to look at what are called

differentiated

differentiated products.

And in particular, we're going to look

at a particular version of this that

we're going to call the linear

city

model.

All right, so what do we mean by

products being differentiated? So, I

started off with an example that's

pretty bad for this story, namely Coke

and Pepsi. Now, be honest, how many of

you in a blind taste test can taste the

difference between

ordinary Coke and ordinary Pepsi? Oh,

quite a few. Okay, that's good for my

case, all right. So, a number of you

think those are different products.

Right? So, a number of you even if their

price was just a tiny bit different

might have a preference for one rather

than the other. So, without getting in

trouble by putting the camera on you,

how many of you have would have a

preference If the prices were

essentially the same, how many of you

would have a strict preference for

Pepsi?

And how many would have a strict

preference for Coke?

And how many would have a strict

preference for diet diet Coke?

That's amazing. That stuff's earning

them money.

Okay.

Okay, but but you're making my point.

The point I'm trying to make is products

are not identical.

All right? Products are not identical.

Okay? For the most part, there's a

little bit of difference between

products and that's actually going to

That

If we inject that little bit of realism

into the world, it's actually going to

help us. So, what we would like What

would like? We'd like a model in which

firms set prices cuz for the most part

we think firms do set prices, not

quantities. It's not always, but for the

most part.

But we'd like a model that yields the

yields an outcome that looks that that

when you only have two firms, looks

somewhere between monopoly and perfect

competition. So, we'd make an outcome

that looks a little bit like Cournot,

but we'd like the strategy set to be

prices. And this is going to do the

trick. This is going to turn out to do

the trick as you'll find out in your

homework assignment.

So, how are we going to model this on

the homework? The way we're going to

model differentiated products is to

imagine, just to take a simple example,

imagine a city

and this city has one long straight road

through it. So, this city is not a city

in New England, it's a city in the

Midwest where everything's flat and the

roads just go completely straight.

All right?

And you can think of it being I don't

know a mile long, doesn't really matter.

All right? Let's just think of it as

being a mile long.

And we're going to assume that consumers

are evenly spread along this city.

All right? Consumers are evenly spread

along this city. So, there's basically

consumers everywhere. They're evenly

distributed.

And we're going to assume that one of

these firms, one of these firms, let's

call it firm one,

sits at point zero

and the other firm

firm two sits at point

one.

Now, this is what you're going to do in

the homework assignment, but let me just

make it make the argument that you could

also imagine firms sitting somewhere

between zero and one, right? You could

We could do a more general job if we

wanted to. But for now, let's assume

that one of these firms

has its shop at one end of the town and

the other one has its shop at

the other end of the town.

All right? So, let's think about a

particular consumer. So, suppose this

consumer

is here at point Y, say.

All right? So, notice that this consumer

is a distance Y

away from firm one.

So, if if if she consumes from firm one,

she has to walk a distance of Y.

All right? And she's a distance of 1 - Y

from firm two.

Is that right? So, if she consumes from

firm two, she has to walk

she has to walk 1 - Y.

All right?

All right? That's going to turn out to

be key in our model as we'll see in a

second.

All right? So, firms

as before

are going to set prices.

And I won't write everything down cuz

it's all written down on the homework

assignment, which is already on the web,

but in fact firms are going to maximize

profits, aim to maximize profits. Firms

are going to have constant marginal

costs. We'll make one other assumption

to keep life simple. We'll assume that

each consumer buys one and and only one

product.

All right? So, each consumer is going to

buy one product either from firm one or

from firm two.

All right? So, the issue's going to be

which firm does each consumer go and buy

their product from?

All right? Which firm does each consumer

choose? And we'll assume

that each consumer

chooses

the product

whose And I'll be careful here.

Whose total

cost to her

is smaller. What do we mean total cost?

Well, let's look at for the for the

consumer at at point Y.

For for example,

for example,

for the consumer at point Y,

if they buy from firm one,

if they buy from firm one,

then they pay the price P1, which is set

by firm one,

but they also have to pay a transport

cost. The cost to them of having to walk

all the way there and walk all the way

back.

All right? And we'll

we'll give a name to that transport

cost. We'll say it's T

Y squared.

All right? So, Y Y is the distance they

have to travel.

All right? And TY squared is their

transport cost.

All right? So, this object here we could

think of as a transport cost.

All right? If the same consumer buys

from firm two,

she pays

P2

plus T

times again the distance squared, so

that's going to be 1

Y squared.

And once again, this last term is a

transport cost.

All right?

All right? Notice that these transport

costs go up in the in the distance you

have to travel and they go up pretty

fast. They go up

at at at at rate squared.

All right? So, what you're going to do

on the homework assignment is solve out

this market. You're going to assume that

firms set prices to maximize profits.

You're going to know what firms' costs

are. You're going to work out what

firms' demands are going to look like

for each possible prices they could set

and you're going to solve out the whole

Nash equilibrium.

And then we're going to look at that

Nash equilibrium and you're going to

think

how does that compare to what I saw in

the Cournot case and so on.

All right? So, that's on the homework

assignment.

But before I leave this, let me just

point out that this is a little bit more

general than it might appear.

So, here I've treated what makes

products different as being

where the where the shops are located.

All right? So, here I've interpreted

these terms here as transport costs and

I've interpreted what makes the products

different is the fact that one of them

is selling at one end of the town and

the other one is selling at the at at

the at the at the right end of town.

All right? But actually, we could

consider this model more generally.

Let's just do so briefly here.

So, let me just redraw the town. Here's

my town again.

All right? I don't have to regard this

this product this this this line as

being distance along the the high street

of the town.

It could be something else about the

product. So, for example, in whatever it

is you guys imagine makes Coke and Pepsi

different, it could be that thing. I

don't know what that is, but whatever it

is whatever whatever that is. Let me

take an example that I understand better

than I understand Coke and Pepsi. So,

imagine we're talking about beer.

All right? Think about beer.

This is the beer market.

All right? This distance here could be

something like the alcohol content or

the flavor of the beer.

All right? And you can imagine different

products therefore that are on the

market positioning themselves or being

positioned at different points on the

line. So, for example, you know, up here

I'm talking about beer flavor, this

might be Guinness.

All right? This might be Guinness, which

I can't even spell, but you know what I

mean.

All right? And if you think about the

drinks industry more generally here

rather than just beer, this would be

Guinness, this would be

Poland Spring.

All right? This is water.

All right? And everything else would be

in between here, all right? So, if we

just go a tiny distance in here,

all right? This is Bud Light.

All right? And so on, right? Right? You

you you you could put everything on this

line, all right? All right? I think that

there may be

I'm not really allowed to do this on

this model. The one thing I can't do is

Part of the truth is that Bud Light

might be down here somewhere, but we're

not allowed to do that.

All right?

All right? So,

leaving aside the the specific example

of beer, you think about some product

that has some dimension on which it

varies and we can use this model We can

use this model to see how competition is

going to work in that market. But now,

notice that this transport cost

is going to be a different

interpretation. Now, instead of being

the cost of traveling that distance to

go and buy the product, what's it going

to be? It's going to be that if my if my

preferred beer flavor is here, say, this

is me,

my preferred beer flavor is here, then

if I end up having to consume Guinness,

I have to pay the price of Guinness and

I also incur some costs because Guinness

isn't the perfect beer for me. It's a

little bit too strong.

All right? And if I consume Bud Light,

then I have to pay for the Bud Light, I

think typically costs less at the price

the raw price is less than Guinness, but

I also pay an additional cost from the

fact that the Bud Light when I drink it

um

uh causes me disutility, to put that

politely like that.

All right? All right? All right? So,

basically, that transport cost is now

the the the lack of pleasure caused by

drinking a product that isn't perfect

for me.

All right?

So, everyone understand the story here.

All right? So, you're going to figure

out what happens in this story very

generally uh well, not very generally,

in the particular case, actually, uh on

the homework assignment, but I want you

to understand that there's this much

more general story underlying this. And

this is a pretty good model uh of how uh

of a lot of markets. It's the kind of

model that you'd see again if you went

on to graduate school.

All right.

Now, I want to spend the rest of today

doing something quite different.

But, you'll see it isn't all that

different.

I'm going to go away from studying

imperfect competition and go back and

visit something we studied almost the

first day or maybe the second day, and

that's elections. I want to go back to

elections, and why do I want to go back

at this point? Well, one reason I want

to go back is you'll notice I've been

putting up these lines on the board, and

when we visited elections that uh on the

second day, we said that you could think

of that line on the board as being not

just uh

not just uh

left-wing, right-wing politics, but also

some dimension of of products. So, I

want to go back again. Now, we're

considering the line on the board as

being

flavor in beer

or location in a town.

And I want to go back to politics now

and go back to the interpretation we

started with, so that left and right

will end up being left-wing politics

versus right-wing politics. All right?

So, we're going to take the same basic

idea back to the politics model.

And we'll spend the rest of today in

politics. So, we're going to be doing

political science

or

political science as it meets game

theory uh for the rest of the day.

We're going to study something called

the candidate

voter

model.

Candidate voter model.

All right? And this model is going to

look a lot like the models we looked at

already a few weeks ago.

In particular,

as advertised, there will be a line,

and this line will go from a zero to

one,

and the left-hand side of this line will

represent the left wing, and the

right-hand side of the line will

represent the right wing. So, that's the

same as before.

And as before, as in the Downs or

Hotelling model we discussed already a

few weeks ago, we're going to assume

that voters are evenly spread along the

line.

All right? So, we're going to assume

even distribution

of voters on the line.

And we're also going to assume, just as

we did last time, that voters are going

to vote for the closest candidate.

So, voters vote

for the closest

candidate.

So, all of these assumptions are the

same assumptions we studied 2 weeks ago

or 2 and 1/2 weeks ago, and which you

actually studied in your first homework

assignment. Is that right?

But, I want to go back there, cuz what I

want to do now is I want to change some

critical assumptions of that model

and see

that by making those making those

changes, we're going to get some very

different outcomes.

So, in particular, the new things here,

two things. One,

the number of candidates

the number of candidates

is not

is not fixed.

So, the number of candidates in this

model is going to be endogenous.

All right? Previously, we looked at

models where there were two candidates

or on your homework assignment, three

candidates. Now, we're going to allow

the number of candidates to adjust

itself.

And the second assumption we're going to

make, which is new,

is that we're going to assume that

candidates cannot choose their position.

Right? So, the idea is any candidate who

stands in this election, you know who

that candidate is. You know whether

they're right-wing or left-wing. So,

they can't tell you they're something

else.

Right? So, candidates

can not

choose

their position.

All right? So, this is a this is a

subtle thing, right? Because you think

about the current election, there's a

debate about whether Hillary Clinton,

for example, can choose right now to be

at the center of the Democratic Party,

given her past history of votes, for

example, on the Iraq war. Or on the

other side, there's a debate about

whether Mitt Romney can choose to be I

guess he's trying to choose himself to

be on the right-hand wing of the

Republican Party, given he has a record

of government as being governor of

Massachusetts when when he provided

state health care, for example. All

right? So, it's quite difficult in the

real world for candidates to position

themselves. Voters tend to know that

those candidates have track records.

That's something we mentioned already 3

weeks ago, 2 weeks ago. We're going to

take that fixed now. We're going to

assume you know who these candidates

are.

All right? So, what are we actually

going to assume?

We're going to assume that each voter in

the model

is a potential candidate.

So, this is I think a nice idealized

version of American politics. Each of

you who is above 18 and is an American

citizen and was born in America and

you have to be more than above 18. Above

whatever it is, whatever the rule is,

whatever the Constitution says, each of

you can potentially stand up now and say

you're going to run for president.

All right? I'm guessing all of you are

ruled out by age, actually. There's some

rule in the Constitution, but never

mind, let's pretend that's not there.

All right? So, the idea of the model is

each voter

is

a potential

a potential candidate.

All right? That's a very appealing

assumption, I think,

in a democracy.

So, what I want to do is I want to lay

this out a little bit more formally as a

game, and then, since we haven't done

anything like this for for a week or so,

we're going to play the game.

All right? So, you're about to play this

game, so pay attention, please.

All right? So, who are the voters? Who

are the players?

The players are the the voters.

Right? The voters or candidates,

whatever you want to call them.

Candidates, candi- dates.

I don't know where they stand. In this

game, they're going to be you.

And the key strategy here,

essentially, the strategy is going to be

very simple. The strategy is do you run

or not?

The reason that's going to be the

strategy is that voting's not going to

be difficult. You're always going to end

up voting for the candidate who's

closest.

All right? So, the only really relevant

strategy is

to run

or not

to run. To stand or not to stand.

All right?

So, just to make that clear,

voters

vote

for the closest

running candidate.

First, and second, what does it mean to

win? We'll assume that we're in a

plurality election here. So, the winner

is the person who gets a plurality.

You win

if you get the most votes, in other

words.

You win if you get the most votes. And

we'll assume that if there's a tie,

that we flip a fair coin

or a Supreme Court judge, whatever you

want to say it. You get the picture.

All right? So,

flip if tie.

All right?

And the payoffs in this game

are as follows.

We'll assume that there's a prize for

winning.

So, if you win the election, you get a

prize equal to B.

All right? We'll also assume

that there's a cost of running.

So, if you enter this election, win or

not, you incur a cost of C.

And we'll assume

that B is greater than equal greater

than or equal to 2C.

And actually, for today for today, let's

just keep it simple and assume it's

actually equal to 2C.

All right?

And but that's not the only part of the

payoffs. There's also a part of the

payoffs that's analogous to forcing me

to drink Bud Light or forcing the Pepsi

drinkers to drink Coca-Cola.

And what's that? That's if regardless of

of

uh of whether I run or not, if some

other candidate is elected other than

me,

then they're not going to have my ideal

policies.

All right? So, it's going to cause me

unhappiness. It's going to cause me

disutility

having a candidate win who's far away

from me.

All right? So, there's going to be a an

extra thing,

and

So if

if your position is X, if you are

at X on that line,

and

the winner of the election

is at Y,

then you pay a cost

of minus

X minus Y, the absolute distance between

between you and the winning candidate.

So again, if you're at X

and the winner's at Y,

it hurts you minus the distance between

X and Y

in terms of your unhappiness at having a

winner who isn't who's far away from you

winning.

So let's do an example.

So make sure we understand the payoffs.

So example one,

if you're at X,

let's let's let's call you Mr. X.

Let's call him Mr. X to make it clear.

So Mr. X, person who's at position X, if

he enters

the election and he wins the election,

then his payoff is B because he entered

minus C, sorry, B because he won minus C

cuz he had to pay his election expense.

Right? But the winning candidate is him,

so he gets no disutility from the

winning candidate being someone else.

All right?

Second possibility,

if Mr. X

enters,

but

Mr. Y

wins,

then X's payoff is what? He's He's still

incurred minus C cuz he ran,

and he also

has a cost of X minus Y because he

doesn't like Mr. Y winning.

All right? And

third,

if Mr. X

stays out,

but Mr. Y wins,

then Mr. X's payoff he doesn't win, he

doesn't get B, he doesn't lose C cuz he

didn't run, but he still has this

disutility

minus X minus Y cuz he doesn't like Y

winning.

All right?

Does everyone understand this game?

Everyone understand the game?

Everyone not understand the game at this

point?

All right, so we can in principle we

could play this with the whole class,

but let's

single out

a particular row of the class. So I'm

going to

come down here and I guess eventually

while grabbing a minute.

All right? So I'm going to use this row,

I think. This row, okay?

This row, does everyone in this row

stand up a second?

And this row stand up.

All right. This is the row of potential

voters. So they're they're they're the

voters, they're also potential

candidates.

All right? So we need to we need to have

a convention here where we're where

where we're viewing this from the

perspective of people behind them or

people in front of them. Let's view it

from the perspective of people in front

of them. So so

I got an R. Okay, I'll do it the other

way around. People from behind them have

the perspective, okay? So this gentleman

here whose name is

Andy. And Andy, thank you. This So Andy

is our crazy right-wing guy.

All right? And

Sudipto? Sudipto is our crazy left-wing

guy. All right? Everyone else is in

between. There's 1 2 3 4 5 6 7 8 9 10 11

12 13 14 15 16 people here.

Let me let me add

Let me make it 17 people so we have an

odd number. So go go and go and sit in

that row.

Yeah. Yeah. Yeah, let's go and sit in

that row. Okay, now we have 17 people. A

bit easier. All right?

Just want to avoid obvious ties, okay?

All right. So how's this going to work?

Everybody sit down again, but don't put

your Keep your books handy cuz you're

going to you're going to this is going

to be pretty aerobic. All right? So

these are these are the potential

voters, and each of them is in the

position that Fred Thompson was in 2

weeks ago in the Republican primary when

he's deciding whether to run or not.

All right? But in the real world, that's

a sequential game. People decide to run

in some order. We're going to assume

this is all simultaneous.

All right, so what we're going to do is

we're going to hold essentially this

game.

All right, we're going to ask everybody

in this row, right, whether they want to

enter or not. Let's let's make some real

numbers. Let's assume that B is equal to

$2. So the spoils of government are $2.

And entering costs you $1, so C is $1.

And we'll assume that the amount of

disutility you get from being from being

far away from the winner is 1/17 of the

Each of these Each place is worth 1/17

of a dollar.

All right, everyone understand So

whatever 1/17 of a dollar

roughly 5 cents, roughly 5 cents for

each position away.

Okay, everyone in this row understand

what we're doing?

I'm not getting any doubts. You guys You

guys over here, you understand what

we're doing? All right? So on the count

of three on the count of three, anybody

who's entering this election is going to

stand up. Ready?

One.

Two.

Okay, what?

That's the opposite of what's happening

with the Florida primary. Yeah.

Question. I'm sorry.

Question is up on the count of You guys

have to decide whether you Okay, so if

you're going to If you're running, you

stand up. Okay? If you're standing, you

stand.

All right, if you're standing, you

stand. All right, let's try again. One.

Two.

Three.

Come on.

All right, stay standing. Stay No, no,

stay standing. Stay Stay standing. Stay

standing.

And I'll come around a second. All

right, so here's our here's our

election. Here's our election, and it's

between A Which way do I want to do it?

This is our left-wing candidate. This is

our moderate left-wing candidate whose

name is Alex. Alex. And our kind of

centrist candidate, maybe even the

centrist candidate whose name is

Beatrice. Beatrice. All right? All

right? So this is the election. Who's

going to win this election?

Right? Beatrice is going to win this

election, right? Let us check that.

Beatrice is going to get

I thought for me, too. Beatrice is going

to get all of these right-wing votes,

right?

Beatrice is also going to get

uh one and a half of the votes in the

middle. So what right? Right? So So this

vote and half the person guy's vote,

right? And I've forgotten your name

already again. And Alex is going to get

all the left-wing votes, but there

aren't enough left-wing votes here,

right? So So Beatrice is going to win.

All right, so Beatrice won this

election. Is this array

a Nash equilibrium?

Is this an equilibrium?

How do we know it's not an equilibrium?

I'm not telling you. Katie, how do we

know this is not an equilibrium? Wait,

with the mic with the mic. How do we

know this is not an equilibrium? Because

Alex

best response to her running is to not

run. Absolutely. So Alex here Alex ended

up with a payoff of what? Alex ended up

with a payoff of minus C, actually more

than that, minus C minus the distance

between him and Beatrice. So that's

another

$1.05, $0.10, $0.20, whatever it is,

right? And he would have done better to

not to run. If he hadn't run, Beatrice

would still have won. It would have

still cost him the $0.20 of unhappiness

at Beatrice winning, but at least he

would have saved himself the C of

running. Everyone see that? Let's try it

again. Let's try it again, okay? Okay,

same row, same row, but don't hesitate

so much this time, right?

Come on at three, you can just go or

not. Everybody in this row close their

eyes. You can't look around, you? Close

your eyes. Close your eyes. We can do

all sorts of nasty things to you now.

Close your eyes. All right?

Uh most of you are still in this row,

okay?

All right? All right? So ready? All

right? So one, two, three.

All right, you can open your eyes again.

All right, so now

All right, so now what happened? We end

up with two centrist candidates. That's

the what happened. So we have Beatrice

and your name is

Clarice. All right, and Clarice is going

to get her own vote plus one one two

three four five six seven eight eight

votes. And Beatrice is getting one two

three four five six seven How did you

end up with eight? You had an odd

number. One two three four

five six seven. It must have been an odd

number four. So So Clarice is going to

win this election. All right? All right,

cuz you got one more vote. Now here we

ended up with two centrist candidates,

which is a result pretty close to what

we saw in the Hotelling model. Is that

right?

Is this an equilibrium?

Let's check I I counted right. One two

three four five six seven eight. And so

you one

two three four five six seven. I guess I

I guess I did count right. So Clarice

turned out to win this. I counted that

correct.

Uh yeah, okay, so so let's try again. So

one two three four five six seven eight.

I apologize. Beatrice won this election.

I apologize. Beatrice won this election.

All right? Is this an equilibrium? Is

this an equilibrium?

How do we know this isn't an

equilibrium?

Cuz Clarice would rather not have run.

Is that right?

All right, let's try once more. All

right, down again. Close your eyes

again. Let's

uh let's just just keep life

interesting. Let's switch to these to

this row. Can we switch to row, okay?

All right, so this is our row now.

Everyone understand this? Close your

computer so you can get up without

destroying thousands of dollars of

technology. All right, this is our row.

And at the count of three

we're going to see who's going to run,

okay? Everyone understand the game?

All right.

Close your eyes.

Close your eyes, madam. 1 2 3.

All right.

All right. So

He's Okay, so it's obvious who's going

to win this election, is that right?

All right.

Uh and just just uh ask the question, is

this an equilibrium?

Is this an equilibrium?

Is he in this Let's count. 1 2 3 4 5 6 7

8 9

Uh he's going to get 10 votes. Uh so he

he There are 10 votes including himself.

There are 10 votes here. I can't count

correctly. 1 2 3 4 5 6 Is he actually

the center guy here? I don't think he is

the center guy here. All right. So I

claim, if I've counted correctly, that

that this actually isn't an equilibrium.

Right? Who can deviate and do better

here?

Yeah, that this guy. Stand up a second.

Whose name is John. So if had John

entered, if I've counted correctly,

let's pretend I have, John would

actually have won this election cuz John

is actually further to the center.

All right.

All right.

Okay. So let's Thanks for a second,

guys. I'll I'll come back to you in a

second. All right.

All right. Let's try and figure out

how to analyze this game. We've played

it three times. Let's try and figure out

what the equilibria look like in this

game. And this game is going on more or

less right now in the primaries.

Well, I guess the entrance stage entry

stage of it is gone now.

All right.

So first of all, is there a Nash

equilibrium

in which no candidates stand?

Is there a Nash equilibrium in which

nobody stands?

No, how do we know that's not a Nash

equilibrium? Somebody?

Somebody? Yeah. How how do we know

that's not a Nash equilibrium?

Shout it out. Cuz then somebody's

definitely

uh better off by standing up. Right. If

nobody stands, each and every possible

candidate would do better individually.

So any any particular

uh voter would do better standing. They

would win the election.

Is that right?

All right. So clearly it's not an

equilibrium for nobody to stand. That's

good cuz we don't see that very

often. All right. Is there a Nash

equilibrium in which one and only one

candidate stands?

Yeah, so so so Yeah, so If they're

directly in the center.

All right. So given that we've chosen

odd number of an odd number of people in

the row, if exactly one candidate stands

and that candidate is the center

candidate,

then that's an equilibrium.

How do we know that's an equilibrium?

Somebody give me the argument of why

that's an equilibrium. So that's that's

it's a it's a good it's in fact a

correct guess, but how do we know that's

an equilibrium? That's the guess part.

How do we check that's an equilibrium?

Yeah, go ahead. Um if

Shout out. Okay.

If anybody on either side of them chose

to run, they would lose.

lose. Exactly. So if anybody on either

side of them runs,

right, that person would simply lose. It

wouldn't make any difference to the

outcome.

Right? If a person on the right stood

Let's just try it again. As we we

figured out that

uh Tell tell that if if Beatrice stands

up a second, right? So here's our

potential equilibrium with just Beatrice

standing, right? And if somebody to the

which way do we do it? Somebody to the

right of Beatrice was to stand, so

madam, if you stand a second.

And your name is Stacy. So Stacy stands,

it doesn't make any difference. She just

loses. Beatrice wins anyway. It just

costs uh Beatrice some money. Right? And

conversely, if Beatrice doesn't stand

and uh Say say say say again a second

and the woman who's one in from the side

there. Sit Sit Sit Sit down a second,

Stacy. If this person stands and your

name is

If Sarah stands, it doesn't help Sarah

at all. She just loses.

Is that right? So there is an

equilibrium with exactly one candidate,

the center candidate.

Now that's looking a little bit like the

median voter model.

Right? It says there is an equilibrium

with only one candidate standing and

that it and that candidate would be the

median candidate.

Right? Just like it was in the model we

saw on your homework assignment and also

uh um

in class.

However,

we're not done yet.

There could be other equilibria.

There could be other equilibria.

All right. So let's just try and think

about whether there are other

equilibria. So for example, suppose now,

just to make life a little bit more

interesting, suppose that the voters

were actually two rows and just just

allow me the the

suspension of disbelief that these rows

have the same number of people in them.

I know they don't really. All right. So

now what I want you to think of is that

at every at every political position,

there are two voters and hence two

possible candidates.

Right? There's two people at every

position. These two people are at this

position. These two people are at this

position. Right? Everyone understand

that? Let's assume that the rows are the

same even though they're not. And let's

examine the following thing. Suppose

Beatrice stands again. Sorry, Beatrice.

And the gentleman in front of him in

front of her whose name is Robert.

Robert stands as well. So stand a

second.

All right. So now we have two candidates

standing who are identical.

Uh politically they're identical.

Right? They're right on top of each

other.

All right. Now, is that an equilibrium?

Is that an equilibrium?

After all, I mean that looks a lot like

the Downs to Telling model, right? We've

got two candidates exactly at the

middle.

Is that an equilibrium?

Let me try the guy up here.

Uh

no, because if someone on either side of

them were to stand up, that person on

the side would win. Good. Good. This

can't be an equilibrium because look

what happens if Clarice stands a second.

All right. Clarice is going to win all

of the right-wing votes, right, ranging

from crazy to moderate.

All right. And these guys are going to

split the left-wing votes.

Right? So the the total sum of

right-wing votes is going to beat out

half of the left-wing votes.

All right.

All right. So that So it can't be an The

exact prediction of the Downs model, two

guys right on top of each other, is not

an equilibrium. Sit down again, guys.

All right.

All right. Let's go back to one row

again cuz it's easy to work with.

All right. Let's try a slightly

different pattern. So suppose, assuming

I counted right, uh suppose that the

following candidates enter. So uh

Clarice enters

and I'm sorry, I don't know your name.

Uh

Jean. Jean enters. So let's let's let's

have a look at this. All right.

All right. And assume I got it right and

Beatrice is actually the center.

All right. Now, is that an equilibrium?

Is that an equilibrium?

Who thinks that is an equilibrium?

Who thinks it's not an equilibrium?

Who's waiting to see how how how other

people vote?

Well, let's check. Let's check. All

right. So there's three possible types

of deviation here

that we can need to check. We need to

check another entrant from the outside,

left wing or right wing.

We need to check another entrant in the

middle. There's only one possible one

there.

And there's a third kind of deviation we

should check. What's the third type of

deviation?

One of these might choose not to run.

One of you might try to might choose not

to run.

Let's do them in turn. So suppose

Suppose that we consider a deviation in

which and again I've forgotten your

name.

In which Stacy stands as well. Stacy,

just stand a second.

All right. So is this a profitable

deviation for Stacy?

No, it's not, right? It's not. Why isn't

it a profitable deviation for Stacy? So

two reasons. And also at this end. One

is that she loses,

right? She's not going to win this

election by standing.

But there's a second reason why this is

a really bad deviation. Why is it What's

the second reason? Yeah, sir.

The winner is going to be further away

from her. Good. Good. So by standing,

not only does she not win the election,

but she actually causes the election to

be won for sure by Jean who's further

away from her. Is that right? So Stacy's

going to pick up

the right-wing votes. She's going to

split the crazy right-wing votes she's

going to pick up. The moderate

right-wing votes she's going to split

with with Clarice. Right? And Jean's

going to have all the left-wing votes,

extreme and moderate. And so from

Stacy's point of view, this is a double

bad. She doesn't get to be president,

sorry, and you end up with a with a

left-wing president which you didn't

like.

All right. Everyone see that?

So clearly that's not Thank you. That's

not a profitable deviation. Hang on a

minute, you guys. Okay. And it's also

not a profitable deviation for people on

the on the left. Hang on a second. Hang

on a second. All right. So I think it's

probably obvious why it's not of it's

not a good strategy for Beat for

Beatrice to enter here. If Beatrice

enters, she loses.

All right. Everyone can see that. The

person in the middle, if they enter

here, lose loses. So they're not going

to want to enter.

All right. So the other possible

deviation we have to consider is a

deviation of the form of one of these

guys dropping out.

All right. So let's consider it. So So

come back to to

Clarice. Is Clarice doing better in or

out?

Is Clarice doing better in this election

or out?

All right. Let's think it through a

second. If she stays in, what's her

payoff?

She wins B

with probability of half.

She costs C for sure.

That's a wash, right? Because B over The

way we worked things out, B was $2, C

was $1, so B over 2 minus C is is a

wash.

All right? In terms of the of So, in

terms of the the benefits of of the

spoils of government versus the cost of

running, it's a wash for Clarice whether

she enters or not.

All right? But by entering, she has a

half chance of winning the election.

All right? So, what we think about it is

is the expected distance from her of the

winning candidates is with probably a

half it's herself, that's nothing, no

distance, and with probably a half it's

two places away.

All right? So, on expectation, it's one

place away.

If she drops out, Jean wins for sure.

All right? So, the expected distance

away from her of the winning candidate

is two away.

All right? Let's let's say it again. So,

the cost and direct benefits of

government are a wash for her, but by

dropping out, she ensures that somebody

further away from her wins for sure.

That's bad, so she's going to stay in.

And then the same argument goes the

other way. So, this is an equilibrium.

Now, we're not done, and of course we're

out of time. So, what are we going to do

when we come back next week? What are we

going to don't pick up yet. What are we

going to do?

We're going to come back to this row,

and we're going to figure out not just

this equilibrium, but all equilibria.

Before you go though, let's just think

about it a second. Hang on. Just you can

think about it and talk about it at

home.

So, what we just saw was an election in

which we saw two candidates, Jean and

Clarice, who both stood very close to

each other.

But what a different question is, hang

on before you go, before you go, stand

up my very left my very left-wing guy.

Yeah, it it it's it's a different way at

the end. Way there.

Yep, up up.

Yep. And stand up my very right-wing

guy.

Here's a candidate. Here's an outcome

with two entrants in it, an extreme

right-wing guy and extreme left-wing

guy. Is that an equilibrium?

Think about it until we come back on

Monday.