
Title:
20. Subgame perfect equilibrium: wars of attrition

Description:
Game Theory (ECON 159)
We first play and then analyze wars of attrition; the games that afflict trench warfare, strikes, and businesses in some competitive settings. We find long and damaging fights can occur in class in these games even when the prizes are small in relation to the accumulated costs. These could be caused by irrationality or by players' having other goals like pride or reputation. But we argue that long, costly fights should be expected in these games even if everyone is rational and has standard goals. We show this first in a twoperiod version of the game and then in a potentially infinite version. There are equilibria in which the game ends fast without a fight, but there are also equilibria that can involve long fights. The only good news is that, the longer the fight and the higher the cost of fighting, the lower is the probability of such a fight.
00:00  Chapter 1. Wars of Attrition: The Rivalry Game
17:39  Chapter 2. Wars of Attrition: Real World Examples
24:04  Chapter 3. Wars of Attrition: Analysis
47:53  Chapter 4. Wars of Attrition: Discussion of SPEs
01:06:54  Chapter 5. Wars of Attrition: Generalization
Complete course materials are available at the Open Yale Courses website: http://open.yale.edu/courses
This course was recorded in Fall 2007.

Professor Ben Polak:
Last time we looked at how

to apply our new idea of
subgame perfect equilibrium to

a whole bunch of games,
and our general idea for how to

solve the subgame perfect
equilibrium is as follows.

We looked at each subgame.
We solved for the Nash

equilibrium in the subgame,
that is something we learned to

do long ago.
And then we rolled back the

payoffs: we rolled them back up
the tree.

And towards the end we learned
something interesting.

I'm not going to go back to it
todayI just want to emphasize

it.
We learned that strategic

effects matter.
So in that investment game we

looked at last time,
when you're considering whether

to rent a new piece of
machinery,

it made a very big difference
whether you considered how this

action would affect the actions
of the other side;

in this case,
how it affects your

competition.
This is a very general idea,

a very general point.
So just to give you a couple of

more examples,
when you're designing tax

systemsI mentioned this last
timewhen you're designing a

tax system,
to make some changes in the

U.S.
tax system, it's not good

enough to look at how people are
behaving in the old tax system

and just calculate in an
accounting manner how much more

money you're going to raise,
or how much money it's going to

cost you.
You have to take into account

how that's going to lead to
changes in behavior.

Once again, that's a strategic
effect, and in the homework that

you're handing in today,
all of you will have had a nice

example of that in the toll
booth problem.

So in the toll booth problem,
when you're putting tolls on

roadsor more generally,
when you're building new roads,

new bridges,
new flyovers,

new bypasses,
you need to take into account

how those new tolls,
how those new roads will affect

all of traffic flow.
Traffic flow down the tree will

form a new equilibrium and you
need to consider that in

designing your tolls and
designing your road system.

So that's another example of
SPE.

So today I want to do something
quite different,

a little bit like what we did
with duel,

I want to play a game today,
and probably spend the whole of

today analyzing this one game.
So it's quite a complicated

game, but it's quite a fun game.
So what's the game we're going

to look at?
The game is going to involve

two players, and each player in
each period, they chooseor

each chooses I should sayeach
chooses,

whether to fight or to quit.
So F means fight and Q means

quit, and they make this choice
simultaneously.

The game ends as soon as
someone quits.

So there's good news and bad
news for this game.

Let's do the good news first.
The good news is that if the

other player quits first you win
a prize.

Generally we'll call this prize
V, but we'll play for some cash

in a minute.
The bad news is,

each period in which both
fightso each period in which

both players choose to
fighteach player pays a cost,

so they pay C.
Just to keep things interesting

let's fill in the other thing
here which is if both quit at

onceso if both quit at once
then they get 0 that period.

So this is a game we've seen a
little bit before.

We saw a little bit under the
auspices of Hawk Dove.

Those people in the MBA class
saw a game a lot like this.

But we're going to analyze this
in much more detail then we did

before.
As I said, we're going to spend

the whole of today talking about
it.

So, to start this out,
let's actually play this game.

So I want two volunteers.
Let me just,

since this is college football
season, let me see if I can play

off the rivalries.
So do I have anybody here from

the great state of Texas?
A whole bunch of Texans,

keep your hands up.
I want to use you a second and

I guess the rivalry here is
Oklahoma.: Anybody from

Oklahoma?
No Oklahomans?

What we'll do is we'll pick two
Texans then.

We'll assume this is Texas and
Texas A&M.

So Texans raise their hands
again.

All right, we're going to pick
out two Texans and I'm going to

give you a mike each.
So your name is?

Student: Nick.
Professor Ben Polak:

Why don't you keep hold of
the mike and just point it

towards you when you speak,
but still shout because

everyone here wants to hear you.
So this is Nick,

where was my other Texan back
here?

Why don't I go for the closer
one.

We'll start here.
And your name is?

Student: Alec.
Professor Ben Polak:

Alec, so shout it out.
Student: Alec.

Professor Ben Polak:
That's better,

okay.
So the game is this.

They're going to have to write
down for the first period

whether they choose fight or
quit.

Each player will have a referee.
So the person behind Alec is

going to be Alec's referee to
make sure that Alec is actually

saying what he says he's going
to do.

And what happened to my other
Texan.

I've lost my other Texan.
There he is.

Your name again was?
Student: Nick.

Professor Ben Polak:
Nick is going to write down

fight or quit.
And to make this real,

let's play for some real cash.
So we'll make the prizewhy

don't we make the prize equal to
a $1 and the cost equal to $.75.

So I've got some dollars here.
Here we go.

What do we call this?
In Texas we call this a fist

full of dollars,
is that right?

So where are my players?
Why don't you stand up,

you guys.
So everyone can see you.

I made this difficult,
because now you are going to

have to write down your
strategy.

So you are going to have to
grab a pen.

I didn't make that easy for you.
Let me come down to make it

easier for the camera person.
So why don't you write down

what your strategy is going to
be, and tell your neighbor what

it's going to be,
and we'll see what happened.

Show your referee.
Have you shown your referee?

Nick, speaking into the
microphone what did you do?

Student: I quit.
Professor Ben Polak:

He quit.
Student: I quit as well.

Professor Ben Polak:
What happened to remember

the Alamo?
All right, so Texas didn't work

very well.
Let's try a different state.

I have to say my wife's from
Texas and my wife's family is

from Texas, and I thought they
had more fight in them than

that.
Maybe that's why they are

sliding in the polls.
Let's try somebody from Ohio,

anyone from Ohio?
Nobody from Ohio in the whole

class, that's no good.
I was going to pick Ohio

against Michigan.
How about some people from some

of our own teams?
Are there any players on the

football team other than the two
I've picked on before?

There we go.
I need a different team,

anybody from the hockey team?
Anybody from the baseball team?

Okay good.
So our friend from the baseball

team, your name is?
Student: Chris.

Professor Ben Polak:
Chris and our new football

team player is?
Student: Ryland.

Professor Ben Polak:
Ryland.

Okay so Ryland and Chris are
going to play this.

And neither of you is from
Texas, I take it,

so we have some hope of
something happening here.

So write down what it is you're
going to choose.

Have you both written something
down?

Yeah, all right Ryland what did
you choose?

Student: Fight.
Professor Ben Polak:

Chris?
Student: I'm going to

quit.
Professor Ben Polak:

Well that was easy too.
So the football team is looking

pretty good here.
So we're not getting much in

the way of action going here.
Anyone else want to try here?

Another little state rivalry
here, I don't suppose I've got

anyone from Oregon,
that's asking too much,

anyone from Oregon?
You guys must be from somewhere.

There's got to be a state where
at least one of you is from.

Well let's try something like
New Jersey, how about that?

There's some players from New
Jersey that's good.

Here we go.
And we'll try New Jersey and

New York.
That seems like there's a bit

of a rivalry there.
Are you from New York?

Excellent, here we go,
and your name is?

Student: Geersen.
Professor Ben Polak:

Your name is?
Student: Andy.

Professor Ben Polak:
Andy.

Okay so Geersen and Andy.
So stand up so everyone can see

where you are.
Let's see if there's any fight

in New York and New Jersey.
So write down your strategies.


Andy what did you choose?
Student: I'm going to

fight.
Student: Fight.

Professor Ben Polak:
Here we go,

this is better now.
I was getting worried there for

a second.
I know it's near the

Thanksgiving break,
but there has to be some sort

of spark left in the class.
So we have both people fighting

which means right now they're
down $.75 but the prize is $1.

So the game goes on,
so write down again.

What you're going to do second
period.

The $.75 is gone so now we're
just looking at this game for

$1.
Let's go to New York,

what does New York?
Student: I'm going to

fight.
Student: Fight.

Professor Ben Polak:
Fight okay.

So you have to stay on the east
coast to get life.

There's no point going west is
there.

That makes sense.
So write down again what you're

going to do, and let's go the
other way around,

to New Jersey?
Student: Fight.

Student: Fight.
Professor Ben Polak:

Fight again all right,
so right now we're down three

$.75 whatever that is,
and there's still this prize of

$1, plus perhaps a bit of pride
here.

So write down again.
Let's try again,

so let's go with New York this
time.

Student: I'm going to
fight.

Student: Fight.
Professor Ben Polak:

Fight okay,
I'm guessing we could keep this

going for quite a while,
is that right?

Now it might make a difference,
by the way, if they're allowed

to talk to each other here,
so let's see if it does.

So let's allow New Jersey and
New York to talk to each other.

You can't insult each other
about bridges and tunnels,

just regular talk.
So anything you want to say to

your friend from New Jersey
here?

Student: I can't let New
Jersey win, that's just New York

pride.
You guys are just worse in

every realm so I'm sorry.
It's just pride.

Professor Ben Polak:
Anything you want to say in

reply?
Student: Well I'm going

to keep fighting,
so your best choice is to give

up.
Professor Ben Polak:

Let's see if that works.
Did they get anything out of

that?
So choose the strategies again.

New York?
Student: I just can't

let Jersey win:
fight.

Student: Bring it on:
fight.

Professor Ben Polak:
All right,

so it's clear that if we kept
this going for a while,

it would pay for my lunch,
is that right?

We'll hold it here for a second.
We'll talk about it a bit,

but thank you.
A round of applause for our two

feistier players.
So what's going on here?

So clearly we can see what can
happen in this game.

You can get people quitting
early, and it could be that one

side quits and the other side
doesn't quit.

That is also something that can
happen.

That can happen pretty quickly.
But it's possiblewe just saw

it happenit's possible that a
fight could go on quite a while

here.
Now why?

What's going on here?
I mean the prize here was what?

Was $1, and the cost was $.75.
I could have raised the stakes

maybe on these guys and see if
that made a difference,

but I think $1 and $.75 will
do.

And by the time they had fought
the second time,

they'd exhausted the possible
prize of $1.

So it's true that if you won
this in the first period then

that's fine because you just get
a $1 and it wouldn't cost you

anything.
And even if you won in the

second period you'd be okay,
you'd only cost yourself $.75

for fighting in the first
period,

but you're getting $1 so that's
good.

But there onand "there on"
went on for plenty of time in

this casethere on you're just
accumulating losses,

so what's going on?
There are various conclusions

possible here.
One is that people from New

York and New Jersey are crazy.
That's a possible thing.

But what else is going on.
Why did we get involved in this

fight.
What happened here?

Why do we tend to see fights
like this emerging?

I'm claiming this isn't such an
implausible situation.

Why do we see it emerging?
Let's talk to our friend from

New York, shout out.
Student: By the time she

fought with me on the second
round, I knew I was going to be

losing money anyway so why not
just keep going and then,

there was no reason,
I wasn't going to win anyway,

so I might as well just keep
fighting until she quit.

Professor Ben Polak:
All right,

I think there's two things.
There's two parts to that

answer.
Part of the answer is:

I have lost the money anyway,
let's hold that piece of it.

And the other part of it is
what?

The other part of it is I'm
really determined to win this

thing.
So there's two things going on

there, and they're quite
different.

Let's take the second one first.
It's possible that the reason

these fights emerge and can go
on for quite a while is that

even though the prize is only $1
in money,

it could be that the actual
thing that the players care

about is what?
What do the players actually

care about here?
Somebody just raise your hand,

I'll put you on the mike.
What do people tend to care

about in these situations?
Winning, they care about

winning, or they care about
pride.

Is that right?
That's why I started with

Texas, but I couldn't find any
pride in Texas,

so we had to go to New York.
So people care about winning

per se.
It's a pride thing.

So it could be that $1 simply
isn't a good description of the

true payoffs here.
It could be that the payoffs

are actually about winning.
It could also be that both of

these guys know that they're
going to be interacting with you

at other times in the class,
or other times at Yale.

And they want to establish a
reputation, both of them,

as being the kind of guys who
fight.

In particular,
when they got to talk about it,

both of them said:
"look I'm a fighter":

something which we've seen
before with Ale and his pizza

shop.
Both of them said:

"I'm a fighter.
You better back out."

So both of them tried to signal
the fact that they were going to

fight to try and get the other
side to quit.

So that's about reputation,
and that reputation could

extend beyond this game.
It could be that they're going

to be involved in this kind of
conflict later on in life.

So both of those things are
around.

There's another element to
this, and it's the other part of

what our friend from New York
said which is about the costs.

What's true about the costs in
this game as we move from period

to period?
Somebody said it.

Say it again.
Say it loudly.

Student: Sunk cost.
Professor Ben Polak:

It's a sunk cost.
So all of those costs that you

accumulate as the game goes on,
they're irrelevant looking

forward because they're sunk.
The fact I've played this game

for ten periods and hence lost
ten times $.75which even I can

do,
that's $7.50the fact that

I've lost $7.50 is irrelevant
because I've lost it anyway.

I can't get that back.
That's a sunk cost.

So the game ten periods through
looks exactly the same as the

game did at the beginning,
when fighting seemed a good

option.
So ten periods through the

game, you have the same view
about fighting as you did at the

beginning.
Now that's not quite true

because at some point you're
going to run out of money,

but if we ignore that,
basically, those sunk costs are

irrelevant.
So what we're seeing here is

reasons why people fight and
some of these reasons seem to be

for standard economic reasons
like sunk costs,

and some of them seem to be
about things that are outside

the game like pride or possibly
reputation.

It's certainly the case within
the real world,

we do see fights like this.
Let's just spell out what the

key feature of this is.
The key feature of this is,

in these fights over a period
of time, even though you may

only be losing a little piece in
each period,

over a period of time you could
lose a lot.

In fact, you could lose far
more than the prize that was

originally at stake.
So the losses you could

accumulateand our friend from
New Jersey and New York,

the losses that they
accumulated vastly outweighed

the prize that was at stake
after a while.

That's a worry.
So this can occur in real life

not just in the classroom.
What do we call these kinds of

fights;
these fights where they're

holding out for this possibly
small prize and incurring,

possibly smallbut they
accumulated to being

largecosts each period.
What do we call those fights?

Let's think about some examples.
Let's see if the word comes out

of examples.
So one examplelet's do some

examples here.
One example is what happened in

World War I.
So in World War I,

as I'm assuming most of you
know, on the western front at

least,
the German and allies to

Germany armies faced off with
the British and French and

allied armies,
for an extraordinarily long

time, fighting over
extraordinarily small patches of

land,
little pieces of northern

France and Belgium.
You could argue that these

pieces of northern France and
BelgiumI don't wish to offend

anyone French or Belgian
herebut you could argue that

those few acres of northern
France and Germany weren't worth

a whole lot anyway.
Nevertheless,

the two sides kept on fighting
from 1914 to 1918,

and enormous losses of life
were accumulated in that period.

So that was a really costly,
long term battle.

Neither side would quit.
Each year huge numbers of lives

were lost.
If you doubt that go and look

at the war memorial in Yale that
shows how many Yale American

lives were lost,
and America was only in that

war for about a year.
Okay, so that's an example.

Another examplea more
business example,

an example we talked about in
our MBA class but not in this

class so faris examples in
business where there's a market

and that market is really only
going to hold one firm.

That market is only going to
hold one firm.

You can end up in an extremely
long fight about who's going to

end up being the one firm in
that market.

So a famous exampleactually
it's a famous business school

caseis the fight that ocurred
to control satellite

broadcasting in Europe.
So there was a fight between

Sky Television and the British
Satellite Broadcasting Company

that went on for a number of
years.

And these companies were doing
things like charging zero

prices, and giving away
satellite dishes,

and this that and the other.
And over the course of the

fight they accumulated so many
losses that, if you did the

accounting,
the entire future projected

profit flow of winning this
fight was vastly outweighed by

the amount of money that they'd
lost during the fight.

That was a fight that involved
on one side Rupert Murdock and

you can argue maybe Rupert
Murdock is a little crazy and

had a reputation to keep up,
but still it looks like another

example of this.
So that example was British

Satellite Broadcasting versus
Sky.

So with those two examples
there, anyone think of a general

term we call these fights?
How do people refer to the

method of fighting in World War
Ior for that matter during the

American Civil War?
Somebody in the back,

shout it out.
Student: War of

attrition.
Professor Ben Polak:

It's a war of attrition.
So these are wars of attrition.

These are wars of attrition.
And what we know about wars of

attrition is that they can go on
a long time, and a lot can be

lost.
A lot of life can be lost in

the case of real wars.
A lot of money can be lost in

the case of business wars.
Actually it turns out,

a lot of games have this
structure of a war of attrition.

Let me give you one more
example.

Suppose that two companies are
competing for a market not in

the manner of BSB and Sky by the
advertising or whatever,

but in the form of paying
bribes.

So suppose there's a company
let's say in France and a

company let's say in Britain,
and these two companies are

trying to win a contract in some
country where paying bribes is a

successful strategy.
And here I'm going to be

careful about the film and not
mention any real companies,

so let's call this imaginary
country Freedonia,

which comes from a Marx
Brothers film.

So here's this French company
and this British company,

and they both want this
contract to build a bridge in

Freedonia.
And they start paying bribes to

the general who controls
Freedonia.

And what happens?
Well you're not going to get

the bribe back.
So both sides pay a few

thousand dollars to this
general, and then the general

comes back and says,
well you both paid $1,000.

Which of you wants to pay the
bridge?

So they go on,
and they put more money in and

more money in.
And you can see once again this

is a war of attrition.
Those bribes that they've paid,

they're never getting back,
but once you've paid them

they're a sunk cost.
Once you've paid that bribe,

good luck saying:
I paid you this bribe.

You didn't let me build the
bridge.

Give me my money back.
There isn't a court in the

world that's going to enforce
that.

So these bribery contests look
a lot like this.


There's a technical name for
these bribe contests,

they're sometimes called all
pay auctions.

So what do we want to establish
today?

We want to establish we want
to talk about why fighting

occurs here and we wanted to do
so in some detail.

So there may be informal
reasons why fighting occurs and

we've talked about that.
It could be that one side is

crazy.
It could be that both sides are

crazy.
It could be that national or

regional pride is at stack.
All of these things could

affect why we get fighting.
But what I want to try and

establish today is that you can
get long fights emerging in

potential wars of attrition even
if everybody is rational,

even if the payoff is just that
one dollar, and even if there's

no reputation at stake.
So again, my goal today is to

try and convince you that you
can get huge loss of life in

World War I or huge loss of
money in these business contexts

without having to argue
something outside the model like

irrationality or reputation.
Even rational players can get

themselves in trouble in wars of
attrition.

So for the rest of today,
I want to try and analyze this.

To get us started I want to
look at a version of this game,

at a simplified version,
which only lasts for two

periods.
So we'll do a two period

version of the game we played
just now.

Eventually, by the end of
today, I want to look at the

infinite version.
So we'll start small.

We'll start with a two period
version.


The trick in analyzing these
things is to be able to come up

with a tree and to be able to
come up with payoffs,

and be able to apply the
analysis that we know about to

get us to where we want to be.
So here's the game I claim.

I claim that it has the
following tree.

So first of all Player A
chooses and Player A can either

Fight or Quit.
And let me put a [1],

and we'll see what the [1]
is in a second.

Then we're going to model
Player 2.

But of course this is a
simultaneous move game.

This is a simultaneous move,
so this is an information set.

Let's not call him Player 2.
Let's call him Player B.

So Player B doesn't know what A
has done that first period when

B is making her choice.
This is a simultaneous move.

And B is choosing between
fighting or quitting.

Just to distinguish them,
let me use small letters for B,

so once again fight or quit,
and fight or quit.

Now, if both sides fight then
the game continues.

And everyone knows that both
sides fought at that stage.

So at this point on,
we're actually at a singleton

information node and it's Player
A's turn again.


So here we go again.
So in the second period,

once again, we've got A
choosing whether to Fight or

Quit.
And this time we'll put a [2]

to indicate we're in the second
period.

And after Player 2 has moved
once againafter Player A has

movedonce again Player B is
moving.

That's a simultaneous move.
And once again they're choosing

fight or quit.
I'll put [2]

to indicate that we're in the
second stage.

So that's the structure of this
two period game.

And let's write down what the
payoffs are, starting with the

easy payoffs.
So if both people quit in the

first stage, they get nothing.
If A quits and B fights,

then A gets nothing and B gets
V.

If A fights and B quits,
then A gets V and B gets

nothing.
And if they both fight we go

into the second stage.
So let's write down the payoffs

in the second stage.
So in the second stage,

if they both quit in the second
stage then their payoffs are

going to be,
for A, C, the costs they

accumulated in the first stage,
plus 0.

And, for B, C + 0.
If A quits and B fights in the

second stage then the payoffs
are C + 0 and C + V.

If A fights and B quits then
the payoffs are C + V and C +

0.
And if they both fight for two

periods, we have a decision to
make about how we're going to

end the game in this two period
game,

but let's just assume that what
we'll get here is C C and C

C.
We'll assume if they both fight

the game ends and no one gets
the prize, just to make life

simple.
So this is a two period version

of the game and the only change
I've made, other than making it

two periods, is I had to put in
a payoff.

I had to see what happened if
the game didn't resolve.

And I've assumed that if the
game didn't resolve,

no one got the prize.
Now there's another assumption

I'm going to have to make before
we analyze this,

there are two possible cases to
consider here.

There's the case when V >
C which is the case we just

played in the class;
and there's also the converse

case when C >
V.

So today we'll focus on the
case V > C which is the case

we just played in class.
I'm going to leave you to

analyze the other case,
the case when the cost is

bigger than the prize,
as a homework assignment.

So V > C is our assumption.
So everyone okay with the tree?

This tree I hope describes the
game, at least a two period

version of the game.
Now the first thing I want to

point out here is,
if we look at the payoffs that

are incurred at the end of the
second period game,

we notice that they all contain
a –C.

There's a C everywhere.
What is that C?

It's the cost that you
accumulated from having fought

in the first stage.
But the observation I want to

make straight away is that this
costso here it is,

here it is, here it is,
here it is, here it is,

here it is, and here it is,
and here it isthis cost is

sunk.
These objects here are sunk

costs.
There's nothing you can do once

you're in the second period of
the game to get these sunk costs

back.
They're just gone.

They're there,
but the fact that they enter

everywhere is going to make them
strategically irrelevant.

Okay, so what we want to do
here is we want to find all of

the subgame perfect equilibria
of this little game.


Let me get rid of the rules.
We all know the rules by now.


So our goal here is to use our
solution concept,

which is subgame perfect
equilibrium, to try and analyze

this game.
So how are we going to analyze

this game in terms of subgame
perfect equilibria?

How are we going to start that
discussion?

We've got a lot of work to do
here, where are we going to

start in finding subgame
perfect equilibria?

What's the first thing we
should do?

Well, I claim the first thing
we should do is just figure out

what the subgames are.
Let's start with that.

So having just pushed it far
away I'm going to need to use

the pointer.
I claim that the obvious

subgame to analyze first is
this subgame.

It's the subgame if you should
end up in period [2].

And notice that it is a
subgame: it starts from a

singleton node;
it doesn't break up any

information set;
and it contains all of the

descendants of the node from
which it starts.

So that is genuinely a subgame.
So we're going to start our

analysis by considering the
second subgame.


So let's write down the matrix
that corresponds to that second

subgame.
And I'm going to write it down

in the following way.
So I claim, in this second

subgame, each player has two
choices, they can fight or quit.

And I'm going to write the
payoffs in a particular way.

I'm going to write the payoffs
as C plus this thing.

So rather than keep that C in
all the boxes,

which it's going to get boring
after a while,

I'm going to pull that C out
and just put it in front.

Is that okay?
So we had this sunk cost box

everywhere and I'm going to pull
out this sunk cost box and put

it in front.
So here it is.

If you get into the second
period of the game you've

incurred this sunk cost.
And your payoffs in this game,

if you both fight then you
incur C for the second time.

If A fights and B quits,
then A is going to win the

prize so they'll get V and
Player B will get nothing.

If B fights and A quits,
then conversely,

A gets nothing and Player B
gets the prize.

And if they both quit they just
get nothing.

So just notice what I did here,
I could have written out the

box with CC here;
CC here;

C+V here;
C+0 here;

C+0 here, etc..
But I just pulled out that C

because it's just distracting
everything.

So I pulled out that C and put
it in the front.

Okay, so now we can analyze
this little game,

and let's start off by talking
about pure strategy equilibria

in this game.
So again, our goal is to find

subgame perfect equilibria,
so the way in which we find

subgame perfect equilibria is
what?

We start at the last subgames,
we look for Nash equilibria in

those last subgames,
and then eventually we're going

to roll those back.
So there's our last subgame.

There's the matrix for it.
Let's just find the Nash

equilibria.
So if Player B is fighting then

Player A's best response is to
quit and if Player A is fighting

then Player B's best response is
to quit.

Conversely, if Player B is
quitting, Player A's best

response is to fight,
and if B is fightingsorry:

if A is quitting then Player
B's best response is to fight.

I didn't say that right.
Let me try again.

So if A is fighting,
if the other side is fighting

you want to quit.
If the other side is quitting

you want to fight.
Is that clear?

So there are actually
twolet's be careful herepure

strategy equilibria,
there are two pure strategy

Nash equilibria in this
subgame.

What are they?
They are (Fight,quit) and

(Quit, fight).
So if we get into the second

subgame and if we know we're
going to play a pure strategy in

the second subgame,
this is what's going to happen,

that's our claim.
Now notice that it didn't

matter, the sunk cost didn't
matter there.

I could have included the sunk
cost in the payoffs,

but I would have found exactly
the same thing with or without

the sunk costs.
So, as we'd expect,

the sunk cost is irrelevant.
So we've got both the

equilibria in the subgame.
Let's roll these back into the

first stage of the game.


The payoffs associated with
this one are V and 0,

and the payoff associated with
this one is 0 and V.

Everyone okay with that?


Okay, let's revisit the first
stage of this game.

Now, things get a little bit
more complicated,

so what I'm going to do is I'm
going to redraw the first stage

of this game.
So here it is.


A can fight or quit just as
before.

And, following this,
B can fight or quit just as

before.
So this is a picture of the

first stage of the game,
but I'm going to chop off the

second stage.
Let's put the payoffs in.

So the payoffs down here are
the same as they were before:

(0,0);
working up, (0, V);

if A fights and B quits then
its V,0.

But what about the payoff if
they both fight?

So what I want to do now is I
want to look at the payoff when

they both fight by considering
what they would get,

if they both fight,
in the second period of the

game.
So our idea is find the Nash

equilibrium in the second period
of the game and roll back these

possible payoffs.
So here the payoffs are going

to be C plus stage [2]
Nash equilibrium payoffs for

Player A;
and C plus the same thing,

stage [2]
Nash equilibrium payoffs for

Player B.
So just to understand what I've

written here then,
I put the same payoffs as

before but I've replaced that
enormous thing that was the

second stage of the game,
just with the payoffs that we

know that we're going to get in
the second stage of the game,

if we play Nash equilibrium in
the second stage.

So these objects have a name,
and the name is continuation

payoffs.
These objects are the

continuation payoffs.
They are the payoffs I'm going

to get tomorrowand possibly
forward in a more complicated

caseif,
in this case,

if we both fight in the first
period.

Now what we want to do is we
want to draw up the matrix that

corresponds to this first stage
game, and we're going to have to

do so twice.
We're going to have to do so

once for the case where the
continuation payoffs are (V,

0), where the equilibrium we're
playing tomorrow is (Fight,

quit).
And we're going to have to do

again for the case where the
continuation payoffs tomorrow

are (0, V), namely the
continuation play is (Quit,

fight).
So we have to do it twice.

[So, you've got the
continuation payoffs down so

let's delete this to give
ourselves a little room.]

So the matrix is going to look
as follows: a nice big matrix,

2x2.
Player A is choosing Fight or

Quit, and Player B is choosing
fight or quit.

That much is easy.
It's what we put in here that

matters.
So let's do all the easy cases

first.
So (Quit, quit) is (0,0)'

(Quit, fight) is (0,V);
(Fight, quit) is (V, 0);

and in here it's going to
depend which of these two

gameswhich of these two
equilibria is being played

tomorrow.
So what we're going to do here

is we're going to do the case
for the equilibrium (Fight,

quit) in stage two.
We're going to write out the

matrix for the case where we're
going to play (Fight,

quit) tomorrow.
[Thank you,

try and keep me consistent
today, because it is very easy

to slip up.
So once again I'm going to use

capital letters for Player A and
small letters for Player B.

]
So what happens if we both

fight?
We both incur costs of C from

fighting, and tomorrow we're
going to get the payoffs from

the equilibrium (Fight,
quit).

That's this equilibrium,
so we're going to get V and 0

tomorrow.
So let's add those in.

So this will be +V and this
will be +0.

And let's just put some chalk
around here to indicate that

these are going to be
continuation payoffs.

So this is the matrix we're
going to use to analyze the

first stage of the game in the
case where the equilibrium we're

playing tomorrow is (Fight,
quit).

And as I promised,
we have to do this twice.

So the other case,
of course, is if we're in the

other equilibrium tomorrow.
So let's just do that.

So once again we have Fight
[1], Quit [1];

fight [1], quit [1];and the
payoffs aresame as we

had(0,V); (0,0);
(V, 0);

and then here,
this time, we're going to have

C + 0 and C + V.
And the reason for the change

is that we're now looking at the
continuation game,

the continuation play where
it's Player A who quits in the

second stage.
So this is for the case (Quit

[2], fight [2]) in period 2.
So let's just pause,

let's make sure everyone's got
that down, everyone okay?

So what we've done here is we
started off by analyzing what's

going to happen in period 2,
and that really wasn't very

hard, is that right?
That was a pretty simple game

to analyze, pretty easy to find
the equilibria.

Then what we did was we rolled
back the equilibrium payoffs

from period 2 and we plunked
them on top of the relevant

payoffs in period 1.
So in particular,

if you both fight and you know
you're going to play the (Fight,

quit) equilibrium tomorrow,
then you're payoffs will be C

+ V and C + 0.
If you both fight and you know

you're going to play the (Quit,
fight) equilibrium tomorrow

then you're payoffs will be C +
0 and C + V.

And just to emphasize once
again, these four boxes we

created correspond to the stage
2 Nash equilibrium payoffs,

so the continuation payoffs of
the game.

Okay, so now we're ready to
analyze each of these games.

So this isn't going to be too
hard.

Let's try and find out the Nash
equilibrium of this game.

So let's start with the left
hand one.

This is the case where Player A
is going to fight and win in

period 2.
So if Player B is going to quit

in period 1 then,
if Player A fights,

she gets V;
if she quits,

she gets 0: so she's going to
want to fight.

Everyone okay with that?
If Player B fights in period 2

[error; 1]
then, if Player A fights,

she gets C + V and if she
quits she gets 0,

and here's where our assumption
is going to help us.

We've assumed,
what did we assume?

We assumed V is bigger than C,
just like we played in class.

So because V is bigger than C
this is going to be the best

response: again fighting is
going to be the best response.

So we know that,
in fact, Player A here has a

dominant strategy,
the dominant strategy is to

fight in period 1 in this
analysis of the game.

And since A is fighting,
not surprisingly,

we're going to find that B is
going to quit.

So B's best response is to quit.
So here is our Nash equilibrium

in this subgame.
This game has a Nash

equilibrium, it only has one,
and the equilibrium is (Fight

[1], quit [1]).
Now let's just talk about it

intuitively for a second.
Intuitively,

if I know, if I'm playing Jake,
and I know that Jake is going

to fight tomorrowsorry,
other way roundI know that

Jake's going to quit tomorrow
and I'm going to fight

tomorrowI know that tomorrow
I'm going to win.

So that prize is there for me
tomorrow, so why would I want to

quit today?
I'm going to get $1 tomorrow if

I just fight in this period.
So why would I want to quit

today when, at worst case
scenario, I'm only going to lose

$.75 today.
So if I know Jake is quitting

tomorrow I'm going to stay on
and fight now.

And conversely,
if Jake knows he's quitting

tomorrow, hence,
he knows I'm going to fight

now, he may as well just quit.
So what we're learning here is

in this particular example,
if we know that tomorrow I'm

going to win the war,
I'm actually going to win it

today.
Say it again,

if we know that tomorrow I'm
going to win the war,

I'm actually going to win it
today.

The converse is true for the
case where I'm the quitter and

Jake's the fighter tomorrow.
So once again it's pretty quick

to see that from Jake's point of
view if I'm going to fight he's

going to want to fight.
If I'm going to quit,

he's going to want to fight.
So in either case he's going to

want to fight.
So I'm going to want to quit.

So the Nash equilibrium in this
game is (Quit [1],

fight [1]).


So at this stage,
we found all of the pure

strategy subgame perfect
equilibria in the game.

Let's describe them before I
write them up.

One pure strategy Nash
equilibrium has me fighting in

period 1 and Jake quitting in
period 1;

and if we got to period
2which in fact we won'tthen

I fight again and he quits
again.

So let's write that equilibrium
up and we'll do it here.

We'll do it on the top board
actually.


So let me get it right,
when I write it up as a whole

equilibrium.
So I claim that I've now found

all of the pure strategy SPE in
this game.


One of them involves my
fighting in the first period and

fighting in the second period,
and Jake quitting in the first

period and quitting in the
second period.

The other one just flips it
around, I quit in the first

period, and if I got there I
would quit in the second period

and Jake fights in the first
period and,

if he got there,
he would also fight in the

second period.
So these are perfectly natural

equilibria to think about.
If you want to get it

intuitively, each of these
equilibria involves a fighter

and a quitter.
The fighter always fights,

the quitter always quits.
If I know that I'm playing a

quitter, I'm always going to
fight, so that's a best

response.
If I know I'm facing a fighter,

I'm going to want to quit,
so that's a best response and

those are two very simple
equilibria.

That's the good news.
What's the bad news here?

The bad news is we haven't
achieved our goal.

Our goal was to argue that
rational players might get

involved in a fight,
and notice that in each of

these two pure strategy subgame
perfect equilibria,

in each of them,
no real fight occurs.

Is that right?
In each of them one person

fights for the first period,
but the other person just runs

away.
That isn't much of a fight.

Let me say it again.
In each of these equilibria,

one side is willing to fight,
but the other side isn't,

so no fight occurs.
In particular,

no costs are incurred in either
of these equilibria.

But I claimed at the beginning,
I wanted to explain how we

could have costs occur in
equilibrium.

Rational players are going to
incur costs.

So what am I missing here?
What should I do to try and

find a more costly equilibrium?
I claim I'm still missing some

equilibria here.
What kind of equilibria am I

missing?
I'm missing the mixed strategy

equilibria.
So far, all we've done is solve

out the purestrategy equilibria
but we need to go back and

reanalyze the whole game
looking now for mixed strategy

equilibria.


So we're going to do the
entiretake a deep breath,

because we're going to take the
whole analysis we just did,

we're going to repeat the
entire analysis we just did,

but this time we're going to
look at mixed strategy

equilibria.
Everyone happy with what we're

doing?
So first of all,

we're going to go back to the
second subgame.

Here's the second subgame,
and we already found the pure

strategy equilibria,
so let me get rid of them,

and in your notes you probably
want to rewrite this matrix.

But I'm not going to rewrite it
here because we're a little bit

short of time.
This is exactly the same payoff

matrix we saw before,
but now I want to look for a

mixed strategy equilibrium in
this game.

How do I go about findingit's
good review thishow do I go

about finding a mixed strategy
equilibrium in a game like this?

What's the trick for finding
mixed strategy equilibria?

Should we try our guys from New
Jersey and New York?

Let's try our guys from New
Jersey and New York.

Where's my New Yorker?
We'll have the true battle here

between New York and New Jersey,
how do we find a mixed strategy

equilibrium?
Student: You use the P's

and Q's and set them equal to
one another.

That's a very crude explanation.
Professor Ben Polak:

That's a crude thing okay.
So the answer was we find the

P's and Q's and "set them equal
to one another."

What is it we're actually
setting equal to what?

Let's try and get some response
on this.

Did our New Jersey guy flee?
Where's my New Jersey person?

They fled.
We could give our Texans

another chance.
Where's our Texan?

Tthere was a Texan down here
somewhere, what is it they set

equal to what?
Student: I guess the

chances that one would quit and
the other would fight.

Professor Ben Polak:
Not quite.

The remark about using P's and
Q's was right.

This is good review for the
final.

What is it I'm going to do with
those P's and Q's?

Shout it out.
Student: You use the

other player's payoffs.
Professor Ben Polak:

Use the other player's
payoffs and?

Student: Make them
indifferent between their

strategies.
Professor Ben Polak:

Good.
I'm going to choose Player B's

mix in such a way as to make
Player A indifferent between

choosing fight and quit.
So as to make it plausible that

A is actually mixing.
So again the intuition is for A

to be mixing they must be
indifferent between fight and

quit.
So I'm going to choose the mix

of B to make A indifferent.
So that's good review.

Let's do that.
So here I've usually used the

letter Q but to avoid confusion
here, let me use the letter P.

We're going to choose P to make
Player A indifferent.

So if A fights then their
payoff is what?

Let's have a look.
It's C with probability P,

and V with probability of 1 
P.


This should be coming back now.
This is before the midterm,

but you guys were alive before
the midterm so you should

remember this.
So if they fight,

they get C P + V [1P].
If they quit then they get 0

with probability P,
and 0 again with probability 1

 P.
So we know that if A is mixing,

B must be mixing in such a way
as to make these two numbers

equal.
So we know these two must be

equal to one another.
Since they're equal we can now

solve for P, so what's that
going to give us?

It's going to give us V [1P]
= P .

C and that I think is P = V /
[V + C].

Is that right?
Someone just check my algebra.

If you remember the game of
Hawk Dove that we saw just

before the midtermit was a
game we looked at when we looked

at evolutionthis is
essentially the same game more

or less as that game,
and that you'll notice it's the

same kind of mixture we've got
here.

So P = V / [V + C]
which means 1  P = C / [V +

C].
I'm leaving it up there for a

bit hoping that one of the
T.A.'s is just going to do my

algebra for me.
I think that's right though.

So this game is symmetric so we
could do the same for B but

we'll find the same thing:
it's a symmetric game.

So the mixed strategy
equilibrium, the mixed Nash

equilibrium in this game has
both mix, both fight with

probability equal to V / [V +
C].

Now this is good news,
because at least we are getting

some fighting going on,
but we need to do something.

We need to take this Nash
equilibrium we've just found,

which is a Nash equilibrium in
the second subgame.

It's a Nash equilibrium in the
subgame way up here.

And we need to roll back the
payoffs from this subgame,

the equilibrium payoffs from
this subgame into the first

stage.
That's our method.

How are we going to do that?
Well we better first of all

figure out what those payoffs
are.

So what are the payoffs in this
equilibrium?

The payoffs in this mixed Nash
equilibrium are what?

Anyone see what the payoffs are
going to be if they're both

playing this mix?
Well presumably the payoff from

fight and the payoff from quit
must be the same,

is that right?
So we may as well choose the

easier one.
So I claim that the payoff from

quit is 0 x P + 0 x 1  P which
is 0.

V / [V + C]
+ 0 .

C / [V + C]
but that's equal to what?

0, okay good.
So it's got to be the case

(kind of conveniently) that if
they do play this mixed strategy

equilibrium in stage 2,
the payoff they'll get from

playing it is 0.
That's going to make life a

little bit easier later on.
That's our new equilibrium in

the second subgame.
Now let's roll that back to the

first game.
Here's our first game again,

and everything about this is
correct except what's down here.

So let's get rid of what's down
here.


Our analysis from before is
more or less still intact.

It's still the case,
that if they both quit they'll

get 0.
If they (Quit,

fight) they'll get (0,
V);

or (V, 0) and it's still the
case if they both fight they'll

both incur costs of C and
they'll both then get stage 2

continuation Nash payoffs.
Is that right?

But now instead of those
continuation Nash payoffs being

(V, 0) or (0,
V), those continuation Nash

payoffs are going to be what?
They're going to be 0.

So what we're going to do here
is we're going to backward

induct, or roll back,
those zero payoffs and come up

with the corresponding matrix to
describe the first stage of the

game.
Here it is.

Fight, QuitI'll try to get it
right without Jake having to

correct me this timelittle f,
little q.

This is A.
This is B.

And the payoffs here are (0,0)
here;

(0, V);
(V, 0) just as before;

and, in this box now,
we've got C + 0 and C + 0.

So it's exactly the same box we
saw before, but now the

continuation payoffs are just 0.
Again, what is this?

This is for the Nash
equilibriumlet's just say for

the mixed Nash equilibrium in
period 2.

Now what I want to do is I want
to find the mixed equilibrium in

period 1.
We found the mixed equilibrium

in period 2.
Now I want to find the mixed

equilibrium in period 1.
So what I could do here is I

could spend a lot of time.
I could put in a P and a 1  P

and I could work out what mix of
Player B will make A

indifferent.
I could work out what mix of

Player A would make B
indifferent.

But has anybody noticed
something about this matrix?

What do you notice about this
matrix?

Somebody help me out?
Somebody's got to help me out

here.
Tell me something about this

matrix.
What's true about this matrix?

Student: It's the same
as the one above.

Professor Ben Polak:
It's the same as the one

above.
The matrix I just drew,

when I rolled back the payoffs
is exactly the same matrix that

I had here.
It's exactly the same matrix.

So we already know what the
mixed strategy equilibrium is in

this.
The mixed Nash equilibrium in

this matrix is both fight with
probability P = V / [V + C]

So now we're ready to show our
new subgame perfect

equilibrium,
let's drag it down.

Here's our whole game,
we found the pure SPEs but now

we're ready to find the mixed
SPE.

The mixed subgame perfect
equilibrium has Player Abefore

I do that let me just give this
P a name.

Let me call this P, P*.
So V / [V + C],

let's call it P*.
So the mixed subgame perfect

equilibrium has Player I mixing,
fighting with probability of P*

in the first stage;
and in the second stage,

again mixing,
fighting with probability of

P*.
So this is Player 1 and Player

2 does exactly the same thing.
While we're here,

what's the expected payoff for
each player if they're playing

this mixed subgame perfect
equilibrium?

It's 0the payoff from
thisthe expected payoff is 0.

So now we're actually getting
somewhere, now we're really

getting somewhere.
So let's just take a deep

breath and see where we are.
We broke this game down,

this complicated game we played
in class, that conceivablyfor

example,
when New York is playing New

Jerseyconceivably it could go
on all night.

Apparently not when the Texans
are playing each other or the

football team is playing the
baseball team,

but when we have New York and
New Jersey it could go on all

night.
We curtailed it to a two period

game, but in a minute,
we're going to go back to the

infinite game.
In this two period game,

I tried to argueI'm trying to
convince youthat you could get

fighting occurring just in
equilibrium with absolutely

standard rational players:
nothing to do with pride,

nothing to do with reputation,
nothing to do with the fact

that these guys are crazy guys
who drunk the water in New York

and New Jersey,
God help them.

You can get fighting with
ordinary people in equilibrium.

What we've shown is the way in
which you can get fighting is in

a mixed strategy equilibrium.
In each period of the game,

people fight with probability
P.

That's just enough fight to
give the other side an incentive

to quit and just enough
probability of the other side

quitting to give the other side
an incentive to fight;

just exactly enough.
If they play that equilibrium

in every period,
there's some chance of the game

ending but with probability P,
the game goes forward to the

next period.
So you could potentially have

fights for two periods.
By the way, with what

probability would there be a
fight in both periods?

That's a good homework
question, I won't answer it

here, you can work it out at
home.

Should we do it here?
Anybody want to tell me?

So okay, with what probability
do we get a fight in the first

period;
a real fight,

a fight involving both players?
We need both players to fight,

each are fighting with
probability of P,

so the probability of both of
them fighting is what?

P².
So to get a fight in the first

period, the probability is
P².

To get a fight in both periods
is what then?

P^(4).
But we get a fight with

probability of P^(4) going
through.

We get fighting in equilibrium.
Moreover, we get some very

intuitive things that we already
learned in the HawkDove game.

The probability of fightso in
this equilibriumthe

probability of fights occurring
goes up as V goes up.

So the prize gets bigger,
you're more likely to see

fights occur:
that seems right.

It goes down in C.
So the probability of fights

occurring goes up in the size of
the prizethat seems

intuitively rightand down in
the cost of fighting.

Now, okay, that's reasonable,
but I claimed I could show you

this not in a two period game,
but in an infinite period game.

So let me spend the last five
minutes taking us to infinite

period games.
So everybody take a deep breath.

We're now going to consider
something, we've never done

before.
We're going to consider a game

that could go on forever.
It could go on forever.

The way we're going to do that
is to use the following idea,

the following picture.


So I can't really draw a true
tree for the infinite period

game.
The reason I can't draw a true

tree for the infinite period
game is: (1) I would run out of

chalk;
and (2) I'd run into lunch time.

But you can imagine what it
looks like.

It looks like this,
roughly speaking.


The infinite period game looks
something like this.

And then it goes again,
and then it goes again,

and so on, and it would go
right the way through the board

and work right the way across
whatever street that is,

right across campus.
That's what the infinite tree

would look like,
so I clearly can't really

analyze that object.
But what I want to show you is

that we can still solve this
game even though it's an

infinite period game.
How are we going to do that?

Let's look at a particular
stage.

Let's call the stage Stage
4,503 whatever that number was:

4,503, whatever it was.
So here is the stage,

this arbitrary stage,
and the tree for this arbitrary

stage looks like this.
What I'm going to do is:

this is Stage 4,503,
and what I'm going to add to

this is that before you get into
this stage,

you're going to incur sunk
costs.

If you go on playing after this
stage then you're going to get

continuation values.
So going into the beginning of

the game, you've incurred some
sunk costs and if you come out

on the other side and go on
playing,

you're going to play some
equilibrium and get continuation

values.
But otherwise everything else

is the same.
We still have (0,0) here,

we still have (0,
V) here, we still have (V,

0) here and here we still have
C plus continuation values and

C plus continuation values.
This is something we've seen

before.
This little box is something

we've seen before.
Essentially we've got sunk

costs in front,
but they're irrelevant.

We've got continuation values
at the end but we know how to

handle them, we just put them
into the payoffs.

So suppose now that in the
continuation game,

people play the mixed strategy
that we just found.

Suppose that in the
continuation game people mixed

with probability P:
so they fight with P* and quit

with probability 1  P*.
Suppose in the continuation

game they're playing a mixed
strategy.

In that case,
what is the continuation value

of the game?
What is it?

It's 0 right.
If they're mixing in the

future, they always have the
option to quit so it must be

that the continuation value is
0.

So if they mix in the future
then the continuation value is

(0,0).
So now let's go back to this

board.
To make this board equivalent

to the board above,
all I need to do is one thing.

I need to add on some sunk
costs at the front.

I've got sunk costs at the
front.

I'm going to play the game.
And then I'm going to get,

instead of stage 2 values,
I'm going to get stagewhat

was it?4,503 and all stages in
the future values in here and

here, but otherwise it's the
same thing.

And what's convenient is:
all of those are 0 anyway.

Since they're all 0 anyway,
this matrix is still correct,

the continuation values are 0
and 0, these are now the

continuation values.


And so if I look for a
mixedstrategy equilibrium in

this game, it's something I've
solved already.

What's the mixed strategy
equiliibrium in this game?

Anybody?
It's exactly what we found

before.
Just as before,

I'm going to mix with
probability of V / [V + C].

Let's summarize,
we did something todayjust

nowthat we've never done
before.

We've looked at an infinite or
at least potentially infinite

period game: a game that could
go on forever.

The way in which we handled the
game that could go on forever

was what?
We noticed two things.

We noticed that part of the
game that comes before,

that part of the game that's
passed already,

anything that happened there is
just a sunk cost.

It's irrelevant.
It hurts if it's a cost.

It's nice if it's a gain.
But it's sunk,

you can't affect it now.
Anything in the future can be

summarized by the value,
the payoff I'm going to get in

the future by playing the
equilibrium in the future.

In this case,
the future meant mixing.

Mixing gave me the value of 0.
So here I am getting 0 from the

future.
Then I can just analyze the

game in the stage in which I'm
in, just as if it was an

ordinary, bogstandard,
simultaneous move game.

When we did so,
in this particular

examplewe're going to see more
examples like this after the

breakbut in this particular
example,

we found out something quite
surprising.

This thing we found out was,
in these war of attrition

settings, there is an
equilibrium with rational

playersmore than that,
common knowledge of

rationality: everybody's
rational, everyone knows

everyone else is rationalthere
are equilibria in which,

not only people fight but they
could fight forever.

In every period they fight with
some probability and we got an

extra prediction out of it,
a prediction that we weren't

expecting.
Let me just give you that

prediction, and then we'll leave
the class with that.

The extra prediction is this,
if we look at these wars of

attrition, and we keep track of
the time in which thehang on

guys don't rush to the back
yetone more thing.

If we look at the time in which
the games have gone on and keep

track of the probability that a
war will end at that time.

So imagine this is World War I.
You could imagine World War I

going for one year,
or two years,

or three years,
or 20 years or whatever.

The probability distribution in
this war of attrition is going

to look like this.
In every period,

the probability of continuing
is just P*².

So, in every period,
the chance thatas you get

further into the future there's
a greater chance the war will

end.
You can get very long,

very costly wars;
that's the bad news.

The good news is it doesn't
happen very often.

I guess we're all involved in a
rather large and costly war

right now, so I'll leave you
with that pleasant thought over

Thanksgiving.
Have a good break and we'll see

you afterwards.