WEBVTT

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Professor Ben Polak:
So last time we were

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focusing on repeated interaction
and that's what we're going to

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continue with today.

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There's lots of things we could
study under repeated interaction

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but the emphasis of this week is
can we attain--can we

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achieve--cooperation in business
or personal relationships

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without contracts,
by use of the fact that these

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relationships go on over time?

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Our central intuition,
where we started from last

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time, was perhaps the future of
a relationship can provide

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incentives for good behavior
today,

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can provide incentives for
people not to cheat.

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So specifically let's just
think of an example.

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We'll go back to where we were
last time.

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Specifically suppose I have a
business relationship,

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an ongoing business
relationship with Jake.

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And each period I'm supposed to
supply Jake with some inputs for

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his business,
let's say some fruit.

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And each period he's supposed
to provide me with some input

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for my business,
namely vegetables.

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Clearly there are opportunities
here, in each period,

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for us to cheat.
We could cheat both on the

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quality of the fruit that I
provide or the quantity of the

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fruit that I provide to Jake,
and he can cheat on the

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quantity or quality of the
vegetables that he provides to

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me.
Our central intuition is:

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perhaps what can give us good
incentives is the idea that if

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Jake cooperates today,
then I might cooperate

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tomorrow, I might not cheat
tomorrow.

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Conversely, if he cheats and
provides me with lousy

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vegetables today I'm going to
provide him with lousy fruit

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tomorrow.
Similarly for me,

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if I provide Jake with lousy
fruit today he can provide me

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with lousy vegetables tomorrow.

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So what do we need?

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We need the difference in the
value of the promise of good

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behavior tomorrow and the threat
of bad behavior tomorrow to

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outweigh the temptation to cheat
today.

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I'm going to gain by providing
him with the bad fruit or fewer

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fruit today--bad fruit because
those I would otherwise have to

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throw away.
So that temptation to cheat has

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to be outweighed by the promise
of getting good vegetables in

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the future from Jake and vice
versa.

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So here's that idea on the
board.

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What we need is the gain if I
cheat today to be outweighed by

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the difference between the value
of my relationship with Jake

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after cooperating and the value
of my relationship with Jake

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after cheating tomorrow.

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Now what we discovered last
time--this was an idea I think

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we kind of knew,
we have kind of known it since

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the first week--but we
discovered last time,

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somewhat surprisingly,
that life is not quite so

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simple.
In particular,

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what we discovered was we need
these to be credible,

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so there's a problem here of
credibility.

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So in particular,
if we think of the value of the

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relationship after cooperating
tomorrow as being a promise,

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and the value of the
relationship after cheating as

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being a threat,
we need these promises and

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threats to be credible.

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We need to actually believe
that they're going to happen.

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And one very simple area where
we saw that ran immediately into

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problems was if this repeated
relationship,

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although repeated,
had a known end.

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Why did known ends cause
problems for us?

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Because in the last period,
in the last period of the game

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we know that whatever we promise
to do or whatever we threaten to

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do,
in the last period,

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once we reached that last
period, in that sub-game we're

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going to play a Nash
equilibrium.

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What we do has to be consistent
with our incentives in the last

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period.
So in particular,

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if there's only one Nash
equilibrium in that last period,

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then we know in that last
period that's what we're going

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to do.
So if we look at the second to

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last period we might hope that
we could promise to cooperate,

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if you cooperate today,
tomorrow.

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Or you could promise to punish
tomorrow if you cheat today,

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but those threats won't be
credible because we know that

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tomorrow you're just going to
play whatever that Nash

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equilibrium is.
That lack of credibility means

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there's no scope to provide
incentives today for us to

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cooperate and we saw things
unravel backwards.

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So the way in which we ensure
that we're really focusing on

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credible promises and credible
threats here is by focusing on

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sub-game perfect equilibrium,
the idea that we introduced

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just before the Thanksgiving
break.

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We know that sub-game perfect
equilibria have the property

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that they have Nash behavior in
every sub-game,

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so in particular in the last
period of the game and so on.

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So what we want to be able to
do here, is try to find scope

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for cooperation in relationships
without contracts,

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without side payments,
by focusing on sub-game perfect

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equilibria of these repeated
games.

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Right at the end last time,
we said okay,

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let's move away from the
setting where we know our game

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is going to end,
and let's look at a game which

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continues, or at least might
continue.

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So in particular,
we looked at the problem of the

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Prisoner's Dilemma which was
repeated with the probability

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that we called δ
each period,

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with the probability δ
of continuing.

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So every period we're going to
play Prisoner's Dilemma.

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However, with probability 1 -
δ the game might just end

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every period.
We already noticed last time

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some things about this.

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The first thing we noticed was
that we can immediately get away

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from this unraveling argument
because there's no known end to

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the game.
We don't have to worry about

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that thread coming loose and
unraveling all the way back.

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So at least there's some hope
here to be able to establish

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credible promises and credible
threats later on in the game

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that will induce good behavior
earlier on in the game.

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So that's where we were last
time, And here is the Prisoner's

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Dilemma, we saw this time,
and we actually focused on a

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particular strategy.

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But before I come back to this
strategy that we focused on last

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time let's just see some things
that won't work,

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just to sort of reinforce the
idea.

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So here's a possible strategy
in the Prisoner's Dilemma.

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A possible strategy in the
Prisoner's Dilemma would be

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cooperate now and go on
cooperating regardless of what

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anyone does.
So let's just cooperate forever

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regardless of the history of the
game.

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Now if two players,
if Jake and I are involved in

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this business relationship,
which has the structure of a

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Prisoner's Dilemma and both of
us play this strategy of

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cooperate now and cooperate
forever no matter what,

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clearly that will induce
cooperation.

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That's the good news.

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The problem is that isn't an
equilibrium, that's not even a

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Nash equilibrium,
let alone a sub-game perfect

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equilibrium.
Why is it not a sub-game

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perfect equilibrium?

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Because in particular,
if Jake is smart (and he is),

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Jake will look at this
equilibrium and say:

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Ben is going to cooperate no
matter what I do,

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so I may as well cheat,
and in fact,

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I may as well go on cheating.

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So Jake has a very good
deviation there which is simply

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to cheat forever.

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So the strategy cooperate now
and go on cooperating no matter

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what doesn't contain incentives
to support itself as an

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equilibrium.
And we need to focus on

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strategies that contain subtle
behavior that generates promises

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of rewards and threats of
punishment that induce people to

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actually stick to that
equilibrium behavior.

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So is everyone clear that
cooperating no matter what--it

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sounds good--but it isn't going
to work.

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People aren't going to stick
with that.

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So instead what we focused on
last time, and actually we had

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some players who seemed to
actually--they've moved now--but

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they seemed actually to be
playing this strategy.

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We focused on what we called
the grim trigger strategy.

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And the grim trigger strategy
is what?

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It says in the first period
cooperate and then go on playing

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cooperate as long as nobody has
ever defected,

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nobody has ever cheated.

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But if anybody ever plays D,
anybody ever plays the defect

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strategy, then we just play D
forever.

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So this is a strategy,
it tells us what to do at every

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possible information set.

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It also, if two players are
playing the strategy,

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has the property that they will
cooperate forever:,

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that's good news.

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And what we left ourselves last
time was checking that this

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actually is an equilibrium,
or more generally,

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under what conditions is this
actually an equilibrium.

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So we got halfway through that
calculation last time.

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So what we need to do is we
need to make sure that the

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temptation of cheating today is
less than the value of the

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promise minus the value of the
threat tomorrow.

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We did parts of this already,
let's just do the easy parts.

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So the temptation today is:
if I cheat today I get 3,

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whereas if I went on
cooperating today I get 2.

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So the temptation is just 1.

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What's the threat?

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The threat is playing D
forever, so this is actually the

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value of (D, D) forever.

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You've got to be careful about
for ever: when I say for ever,

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I mean until the game ends
because eventually the game is

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going to end,
but let's use the code for ever

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to mean until the game ends.

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What's the promise?

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The promise is the value of
continuing cooperation,

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so the value of (C,C) for ever.

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That's what this bracket is,
and it's still tomorrow.

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So let's go on working on this.

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So the value of cooperating for
ever is actually--let's be a bit

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more detailed--this is the value
of getting 2 in every period,

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so it's value of 2 for ever;
and this is the value of 0

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forever.

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So the value of 0 forever,
that's pretty easy to work out:

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I get 0 tomorrow,
I get 0 the day after tomorrow,

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I get 0 the day after the day
after tomorrow.

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Or more accurately:
I get 0 tomorrow,

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I get 0 the day after tomorrow
if we're still playing,

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I get 0 the day after the day
after tomorrow if we're still

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playing and so on.

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But that isn't a very hard
calculation, this thing is going

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to equal 0.
So this object here is just 0.

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This object here is 3 - 2,
I can do that one in my head,

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that's 1.
So I'm left with the value of

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getting 2 for ever,
and that requires a little bit

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more thought.
But let's do that one bit of

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algebra because it's going to be
useful throughout today.

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So this thing here,
the value of 2 for ever is

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what?
Well I get 2,

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that's tomorrow,
and then, assuming I'm still

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playing the day after
tomorrow--so I need to discount

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it--with probability of δ
I'm still playing the day after

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tomorrow--and I get 2 again.

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And the day after the day after
tomorrow I'm still playing with

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the probability that the game
didn't end tomorrow and didn't

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end the next day so that's with
probability δ²

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and again I get 2.

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And then the day after,
what is it?

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This is tomorrow,
the day after tomorrow,

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the day after the day after
tomorrow: this is the day after

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the day after the day after
tomorrow which is δ³

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2 and so on.
Everyone happy with that?

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So starting from tomorrow,
if we play (C,

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C) for ever,
I'll get 2 tomorrow,

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2 the day after tomorrow,
2 the day after the day after

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tomorrow, and so on.

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And I just need to take an
account of the fact that the

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game may end between tomorrow
and the next day,

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the game may end between the
day after tomorrow and the day

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after the day after tomorrow and
so on.

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Everyone happy with that?

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So what is the value,
what is thing?

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Let's call this X for a second.

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So we've done this once before
in the class but let's do it

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again anyway.
This is the geometric sum,

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some of you may even remember
from high school how to do a

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geometric sum,
but let's do it slowly.

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So to work out what X is what
I'm going to do is I'm going to

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multiply X by δ,
so what's δX?

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So this 2 here will become a
2δ, and this δ2 here

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will become a δ²2,
and this δ²2 will

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become a δ³2,
and this δ³2 will

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become a δ^(4)2,
and so on.

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Now what I'm going to do is I'm
going to subtract the second of

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those lines from the first of
those lines.

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So what I'm going to do is,
I'm going to subtract

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X--δX.
So I'm going to subtract the

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second line from the first line.

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And when I do that I'm going to
notice I hope that this 2δ

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is going to cancel with this
2δ,

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and this δ²2 is going
to cancel with this

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δ²2,
and this δ³2 is going

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to cancel with this
δ³2 and so on.

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So what I'm going to get left
with is what?

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Everything's going to cancel
except for what?

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Except for that first 2 there,
so this is just equal to 2.

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Now this is a calculation I can
do.

14:52.900 --> 15:03.690
So I've got X = 2 / [1-δ].

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So just to summarize the
algebra, getting 2 forever,

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that means 2 + δ2 +
δ²2 + δ³2

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etc..
The value of that object is

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2/[1-δ].
So we can put that in here as

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well.
This object here 2/[1-δ]

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is the value of 2 forever.

15:31.100 --> 15:34.140
Now before I go onto a new
board I want to do one other

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thing.
On the left hand side I've got

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my temptation,
that was 1, I've got the value

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of cooperating forever starting
from tomorrow which is

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2/[1-δ]
and I've got the value of

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defecting forever starting from
tomorrow which is 0.

15:54.549 --> 16:00.079
However, all of these objects
on the right hand side,

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they start tomorrow,
whereas, the temptation today

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is today.
Temptation today happens today.

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These differences in value
start tomorrow.

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Since they start tomorrow I
need to discount them because we

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don't know that tomorrow is
going to happen.

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The world may end,
or more importantly the

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relationship may end,
between today and tomorrow.

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So how much do I have to weight
them by?

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By δ, I need to multiply
all of these lines by δ

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and so on.
Now this is now a mess so let's

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go to a new board.

16:37.870 --> 16:44.010


16:44.009 --> 16:51.639
Now let's summarize what we now
have, What we're doing here is

16:51.640 --> 16:58.770
asking is it the case that if
people play the grim trigger

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strategy that that is in fact an
equilibrium?

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That is a way of sustaining
cooperation.

17:07.259 --> 17:17.879
The answer is we need 1,
that's our temptation,

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to be less than 2/[1-δ],
that's the value of cooperating

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for ever starting from tomorrow,
minus 0, that's the value of

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defecting forever starting
tomorrow,

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and this whole thing is
multiplied by δ

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because tomorrow may not
happen.

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Everyone happy with that so far?

17:44.750 --> 17:49.200
I'm just kind of collecting up
the terms that we did slowly

17:49.196 --> 17:51.466
just now.
So now what I want to do

17:51.468 --> 17:55.158
is--question mark here because
we don't know whether it is--I'm

17:55.156 --> 17:57.176
going to solve this for δ.

17:57.180 --> 17:59.750
So when I solve this for δ
I'll probably get it wrong,

17:59.750 --> 18:00.770
but let's be careful.

18:00.769 --> 18:08.479
So this is equivalent to saying
1-δ <

18:08.477 --> 18:17.407
2δ and it's also
equivalent to saying therefore

18:17.410 --> 18:22.140
that δ >
= 1/3.

18:22.140 --> 18:27.360
Everyone happy with that?

18:27.360 --> 18:30.870
Let me just turn my own page.

18:30.870 --> 18:37.060
So what have we shown so far?

18:37.059 --> 18:40.299
We've shown that if we're
playing the grim trigger

18:40.303 --> 18:44.013
strategy, and we want to deter
people from doing what?

18:44.009 --> 18:47.519
From defecting from this
strategy in the very first

18:47.519 --> 18:51.799
period, then we're okay provided
δ is bigger than 1/3.

18:51.799 --> 18:56.369
But at this point some of you
could say, yeah but that's just

18:56.366 --> 19:01.006
one of the possible ways I could
defect from this strategy.

19:01.009 --> 19:03.899
After all, the defection we
just considered,

19:03.904 --> 19:08.014
the move away from equilibrium
we just considered was what?

19:08.009 --> 19:12.949
We considered my cheating
today, but thereafter,

19:12.953 --> 19:18.533
I reversed it back to doing
what I was supposed to do:

19:18.527 --> 19:23.047
I went along with playing D
thereafter.

19:23.049 --> 19:25.429
So the particular defection we
looked at just now was in Period

19:25.430 --> 19:27.350
1, I'm going to defect,
but thereafter,

19:27.351 --> 19:31.131
I'm actually going to do what
the equilibrium strategy tells

19:31.131 --> 19:35.571
me to do.
I'm going to go along with the

19:35.571 --> 19:41.201
punishment and play my part of
(D,D) forever.

19:41.200 --> 19:44.990
So you might want to ask,
why would I do that?

19:44.990 --> 19:46.220
Why would I go along?

19:46.220 --> 19:49.640
I cheated the first time but
now I'm doing what the strategy

19:49.635 --> 19:51.935
tells me to do.
It tells me to play D.

19:51.940 --> 19:54.640
Why am I going along with that?

19:54.640 --> 19:58.030
You could consider going away
from the equilibrium by

19:58.028 --> 20:00.438
defecting, for example in Period
1,

20:00.440 --> 20:04.110
and then in Period 2 do
something completely different

20:04.113 --> 20:05.433
like cooperating.

20:05.430 --> 20:17.280
So we might want to worry,
how about playing D now and

20:17.284 --> 20:28.024
then C in the next period,
and then D forever.

20:28.019 --> 20:29.689
That's just some other way of
defecting.

20:29.690 --> 20:32.530
So far we've said I'm going to
defect by playing D and then

20:32.530 --> 20:35.520
playing D forever,
but now I'm saying let's play D

20:35.515 --> 20:38.875
now and then play a period of C
and then D forever.

20:38.880 --> 20:42.310
Is that going to be a
profitable deviation?

20:42.309 --> 20:46.069
Well let's see what I'd get if
I do that particular deviation.

20:46.069 --> 20:48.289
What play is that going to
induce?

20:48.289 --> 20:51.599
Remember the other player is
playing equilibrium,

20:51.602 --> 20:55.332
so that player is going to
induce, in the first period,

20:55.328 --> 20:58.018
I'm playing D and Jake's
playing C.

20:58.019 --> 21:01.929
In the second period Jake's
going to start punishing me,

21:01.927 --> 21:06.047
so he's going to play D and
according to this deviation I'm

21:06.048 --> 21:10.128
going to play C.
So in the second period I'll

21:10.133 --> 21:15.363
play C and Jake will play D,
and in the third period and

21:15.364 --> 21:19.934
thereafter, we'll just play D,
D, D, D, D, D.

21:19.930 --> 21:22.620
So these are just some other
deviation other than the one we

21:22.622 --> 21:26.462
looked at.
So what payoff do I get from

21:26.455 --> 21:30.215
this?
Okay, I get three in the first

21:30.221 --> 21:35.021
period, just as I did for my
original defection,

21:35.018 --> 21:37.058
that's good news.

21:37.059 --> 21:42.109
But now in the second period
discounted, I actually get -1,

21:42.111 --> 21:47.421
I'm actually doing even worse
in the second period because I'm

21:47.423 --> 21:50.823
cooperating while Jake's
defecting,

21:50.819 --> 21:57.799
and then in the third period I
get 0 and in the fourth period I

21:57.802 --> 22:03.822
get 0 and so on.
So the total payoff to this

22:03.822 --> 22:07.522
defection is 3 - δ.

22:07.519 --> 22:14.259
Now, that's even worse than the
defection we considered to start

22:14.260 --> 22:16.920
with.
The defection we considered to

22:16.924 --> 22:21.014
start with, I got 3 in the first
period and thereafter I got 0.

22:21.009 --> 22:23.869
Now I got 3 in the first
period, -1 in the second period,

22:23.873 --> 22:25.103
and then 0 thereafter.

22:25.099 --> 22:28.069
So this defection in which I
defect--this move away from

22:28.068 --> 22:31.468
equilibrium--in which I cheat in
the first period and then don't

22:31.468 --> 22:36.108
go along with the punishment,
I don't in fact play D forever

22:36.109 --> 22:40.369
is even worse.
Is that right? It's even worse.

22:40.370 --> 22:42.010
So what's the lesson here?

22:42.009 --> 22:48.809
The lesson here is the reason
that I'm prepared to go along

22:48.806 --> 22:55.956
with my own punishment and play
D forever after a defection is

22:55.955 --> 22:58.375
what?
It's if Jake is going to play D

22:58.375 --> 23:00.045
forever I may as well play D
forever.

23:00.050 --> 23:02.900
Is that right?
So another way of saying this

23:02.899 --> 23:06.109
is the only way which I could
possibly hope to have a

23:06.108 --> 23:09.268
profitable deviation,
given that Jake's going to

23:09.268 --> 23:13.078
revert to playing D forever is
for me to defect on Jake once

23:13.078 --> 23:15.918
and then go along with playing D
forever.

23:15.920 --> 23:21.200
There's no point once he's
playing D, there's no point me

23:21.201 --> 23:24.691
doing anything else,
so this is worse,

23:24.690 --> 23:26.860
this is even worse.

23:26.860 --> 23:29.220
This defection is even worse.

23:29.220 --> 23:33.100
More generally,
the reason this is even worse

23:33.098 --> 23:37.328
is because the punishment we
looked at before,

23:37.329 --> 23:42.439
which was (D,
D) for ever,

23:42.438 --> 23:53.878
the punishment (D,D) forever is
itself an equilibrium.

23:53.880 --> 23:57.740
It's credible because it's
itself an equilibrium.

23:57.740 --> 24:03.400


24:03.400 --> 24:07.560
So unlike in the finitely
repeated games we did last time,

24:07.559 --> 24:12.009
unlike in the two period or the
five period repeated games,

24:12.009 --> 24:14.639
here the punishment really is a
credible punishment,

24:14.644 --> 24:17.694
because what I'm doing in the
punishment phase is playing an

24:17.693 --> 24:20.733
equilibrium.
There's no point considering

24:20.728 --> 24:25.298
any other deviation other than
playing D once and then just

24:25.304 --> 24:27.044
going on playing D.

24:27.039 --> 24:31.899
So that's one other possible
deviation, but there are others

24:31.900 --> 24:34.290
you might want to consider.

24:34.289 --> 24:37.499
So far all we've considered is
what?

24:37.500 --> 24:40.760
We've considered the deviation
where I, in the very first

24:40.760 --> 24:44.080
period, I cheat on Jake and then
I just play D forever.

24:44.079 --> 24:45.659
But what about the second
period?

24:45.660 --> 24:58.010
Another thing I could do is how
about cheating not in the first

24:58.010 --> 25:05.780
period of the game but in the
second.

25:05.780 --> 25:09.340


25:09.339 --> 25:11.539
So according to this strategy
what am I going to do.

25:11.539 --> 25:16.659
The first period of the game
I'll go along with Jake and

25:16.658 --> 25:21.868
cooperate, but in the second
period I'll cheat on him.

25:21.869 --> 25:25.279
Now how am I going to check
whether that's a good deviation

25:25.280 --> 25:28.650
or not?
How do I know that's not going

25:28.648 --> 25:30.918
to be a good deviation?

25:30.920 --> 25:34.390
Well we already know that I'm
not going to want to cheat in

25:34.386 --> 25:36.236
the first period of the game.

25:36.240 --> 25:40.070
I want to argue that exactly
the same analysis tells me I'm

25:40.067 --> 25:44.157
not going to want to cheat in
the second period of the game.

25:44.160 --> 25:47.840
Why?
Because once we reach the

25:47.841 --> 25:49.671
second period of the game,
it is the first period

25:49.671 --> 25:51.811
of the game.
Once we reach the second period

25:51.808 --> 25:54.238
of the game, looking from period
two onwards,

25:54.240 --> 25:59.920
it's exactly the same as it was
when we looked from period one

25:59.922 --> 26:02.112
initially.
So to say it again,

26:02.110 --> 26:05.490
what we argued before was--on
the board that I've now covered

26:05.490 --> 26:11.430
up--what we argued before was,
I'm not going to want to cheat

26:11.433 --> 26:18.383
in the very first period of the
game provided δ

26:18.376 --> 26:21.466
> 1/3.
I want to claim that that same

26:21.468 --> 26:24.428
argument tells me I'm not going
to want to cheat in the second

26:24.434 --> 26:26.724
period of the game provided
δ > 1/3.

26:26.720 --> 26:31.820
I'm not going to want to cheat
in the fifth period of the game

26:31.821 --> 26:34.081
provided δ
> 1/3.

26:34.079 --> 26:35.159
Because this game from the
fifth period on,

26:35.157 --> 26:36.027
or the five hundredth period
on,

26:36.029 --> 26:39.799
or the thousandth period on
looks exactly the same as is it

26:39.799 --> 26:41.489
does from the beginning.

26:41.490 --> 26:56.920
So what's neat about this
argument is the same analysis

26:56.921 --> 27:11.211
says, this is not profitable if
δ > 1/3.

27:11.210 --> 27:18.610


27:18.610 --> 27:22.020
So what have we learned here?

27:22.019 --> 27:24.299
I want to show you some nerdy
lessons and then some actual

27:24.299 --> 27:25.459
sort of real world lessons.

27:25.460 --> 27:30.030
Let's start with the nerdy
lessons.

27:30.029 --> 27:34.339
The nerdy lesson is this grim
strategy works because

27:34.342 --> 27:39.252
both--let's put it up again so
we can actually see it--this

27:39.247 --> 27:44.337
grim strategy,
it works because both the play

27:44.336 --> 27:51.106
that it suggests if we both
cooperate and the play that it

27:51.112 --> 27:57.772
suggests if we both defect are
themselves equilibria.

27:57.769 --> 28:03.039
These are credible threats and
credible promises because what

28:03.044 --> 28:07.974
you end up doing both in the
promise and in the threat is

28:07.967 --> 28:10.777
itself equilibrium behavior.

28:10.780 --> 28:17.490
That's good.
The second thing we've learned,

28:17.486 --> 28:22.236
however, is for this to work we
need δ >

28:22.241 --> 28:28.111
1/3, we need the probability
continuation to be bigger than

28:28.109 --> 28:31.969
1/3.
So leaving aside the nerdy

28:31.969 --> 28:38.779
stuff for a second--you have
more practice on the nerdy stuff

28:38.776 --> 28:45.916
on the homework assignment--the
lesson is we can get cooperation

28:45.923 --> 28:52.053
in the Prisoner's Dilemma using
the grim trigger.

28:52.049 --> 28:53.429
Remember the grim trigger
strategy is cooperate until

28:53.427 --> 28:54.537
someone defects and then defect
forever.

28:54.539 --> 28:59.919
So you get cooperation in the
Prisoner's Dilemma using the

28:59.917 --> 29:04.537
grim trigger as a sub-game
perfect equilibrium.

29:04.539 --> 29:10.679
So this is an equilibrium
strategy, that's good news,

29:10.679 --> 29:17.999
provided the probability of
continuation is bigger than 1/3.

29:18.000 --> 29:21.630


29:21.630 --> 29:26.280
Let's try and generalize that
lesson away from the Prisoner's

29:26.275 --> 29:28.505
Dilemma.
So last time our lesson was

29:28.513 --> 29:31.173
about what in general could we
hope for in ongoing

29:31.168 --> 29:34.238
relationships?
So let's put down a more

29:34.243 --> 29:39.183
general lesson that refines what
we learned last time.

29:39.180 --> 29:52.820
So the more general lesson is,
in an ongoing relationship--let

29:52.817 --> 30:04.667
me mimic exactly the words I
used last time--so for an

30:04.666 --> 30:17.856
ongoing relationship to provide
incentives for good behavior

30:17.857 --> 30:23.297
today,
it helps--what we wrote last

30:23.301 --> 30:29.121
time was--it helps for that
relationship to have a future.

30:29.119 --> 30:45.449
But now we can refine this,
it helps for there to be a high

30:45.451 --> 30:59.531
probability that the
relationship will continue.

30:59.529 --> 31:02.819
So the specific lesson for
Prisoner's Dilemma and the grim

31:02.818 --> 31:05.698
trigger strategy is we need
δ, the probability

31:05.702 --> 31:08.012
continuation,
to be bigger than 1/3.

31:08.009 --> 31:11.399
But the more general intuition
is, if we want my ongoing

31:11.401 --> 31:14.731
business relationship with me
and Jake to generate good

31:14.731 --> 31:18.181
behavior--so I'm going to
provide him with good fruit and

31:18.184 --> 31:21.704
he's going to provide me with
good vegetables--we need the

31:21.698 --> 31:24.968
probability that that
relationship will continue to be

31:24.966 --> 31:28.566
reasonably high.
I claim this is a very natural

31:28.567 --> 31:30.317
intuition.
Why?

31:30.319 --> 31:34.599
Because the probability that
the relationship will continue

31:34.603 --> 31:37.783
is the weight that you put on
the future.

31:37.779 --> 31:44.429
The probability that the
relationship will continue,

31:44.429 --> 31:51.599
this thing, this is the weight
you put on the future.

31:51.600 --> 31:56.180


31:56.180 --> 32:00.920
The more weight I put on the
future, the easier it is for the

32:00.921 --> 32:05.031
future to give me incentives to
behave well today,

32:05.029 --> 32:08.199
the easier it is for those to
overcome the temptations to

32:08.198 --> 32:10.888
cheat today.
That seems like a much more

32:10.892 --> 32:14.912
general lesson than just the
Prisoner's Dilemma example.

32:14.910 --> 32:19.310
Let's try to push this to some
examples and see if it rings

32:19.309 --> 32:21.739
true.
So the lesson we've got here is

32:21.738 --> 32:25.228
to get cooperation in these
relationships we need there to

32:25.229 --> 32:28.329
be a high probability,
a reasonably high probability

32:28.328 --> 32:30.058
that they're going to continue.

32:30.059 --> 32:31.799
We know exactly what that is
for Prisoner's Dilemma but the

32:31.799 --> 32:32.639
lesson seems more general.

32:32.640 --> 32:33.870
So here's two examples.

32:33.870 --> 32:35.680
How many of you are seniors?

32:35.680 --> 32:37.860
One or two, quite a few are
seniors.

32:37.860 --> 32:39.360
Keep your hands up a second.

32:39.359 --> 32:41.869
All of those of you who are
seniors--we can pan these guys.

32:41.870 --> 32:43.690
Let's have a look at them.

32:43.690 --> 32:49.340
Actually, why don't we get all
the seniors to stand up:

32:49.336 --> 32:52.156
make you work a bit here.

32:52.160 --> 32:57.010
Now the tricky question,
the tricky personal question.

32:57.009 --> 32:59.389
How many of you who are seniors
are currently involved in

32:59.388 --> 33:01.558
personal relationships,
you know: have a significant

33:01.555 --> 33:03.495
other?
Stay standing up if you have a

33:03.504 --> 33:04.474
significant other.

33:04.470 --> 33:08.230
Look at this, it's pathetic.

33:08.230 --> 33:10.750
What have I been saying about
economic majors?

33:10.750 --> 33:13.700
All right, so let's just think
about, stay standing a second,

33:13.700 --> 33:16.160
let's get these guys to think
about it a second.

33:16.160 --> 33:20.660
So seniors who are involved in
ongoing relationships with

33:20.662 --> 33:24.602
significant others,
what do we have to worry about

33:24.602 --> 33:27.342
those seniors?
Well these seniors are about to

33:27.335 --> 33:29.405
depart from the beautiful
confines of New Haven and

33:29.408 --> 33:31.978
they're going to take jobs in
different parts of the world.

33:31.980 --> 33:35.740
And the problem is some of them
are going to take jobs in New

33:35.741 --> 33:39.001
York while their significant
other takes a job in San

33:39.000 --> 33:41.320
Francisco or Baghdad or
whatever,

33:41.319 --> 33:45.179
let's hope not Baghdad,
London shall we say.

33:45.180 --> 33:50.110
Now if it's the case that you
are going to take a job in New

33:50.108 --> 33:54.448
York next year and your
significant other is going to

33:54.452 --> 33:58.742
take a job in Baghdad or London,
or anyway far away,

33:58.741 --> 34:02.721
in reality, being cynical a
little bit, what does that do to

34:02.720 --> 34:06.700
the probability that your
relationship is going to last?

34:06.700 --> 34:09.010
It makes it go down.

34:09.010 --> 34:12.780
It lowers the probability that
your relationship's going to

34:12.775 --> 34:15.465
continue.
So what is the

34:15.474 --> 34:20.014
prediction--let's be mean here.

34:20.010 --> 34:22.390
These are the people with
significant others who are

34:22.392 --> 34:25.152
seniors, how many of you are
going to be separated by a long

34:25.147 --> 34:27.667
distance from your significant
others next period?

34:27.670 --> 34:29.710
Well one of them at the back,
okay one guy,

34:29.706 --> 34:31.396
at the back,
two guys, honesty here,

34:31.403 --> 34:32.813
three, four of you right?

34:32.810 --> 34:34.540
So what's our prediction here?

34:34.539 --> 34:37.629
What does this model predict as
a social science experiment.

34:37.630 --> 34:39.430
What does it predict?

34:39.429 --> 34:42.899
It predicts that for those of
you who just raised your hands,

34:42.896 --> 34:46.356
those seniors who just raised
their hands who are about to be

34:46.363 --> 34:49.543
separated by large distances,
those relationships,

34:49.542 --> 34:52.722
each player in that
relationship is going to have a

34:52.724 --> 34:54.574
lower value on the future.

34:54.570 --> 34:57.820
So during the rest of your
senior year, during the spring

34:57.821 --> 35:01.191
of your senior year what's the
prediction of this model?

35:01.190 --> 35:04.800
They're going to cheat.

35:04.800 --> 35:08.100
So we could actually do a
controlled experiment,

35:08.098 --> 35:12.238
what we should do here is we
should keep track of the people

35:12.238 --> 35:14.928
here,
the seniors who are going to be

35:14.933 --> 35:19.263
separated--you can sit down now,
I'm sorry to embarrass you all.

35:19.260 --> 35:22.000
We could keep track of those
seniors who are about to be

35:22.001 --> 35:24.541
separated and go into a long
distance relationships,

35:24.544 --> 35:25.894
and those that are not.

35:25.889 --> 35:29.009
The people who are not are our
control group.

35:29.010 --> 35:32.070
And we should see if during the
spring semester the people who

35:32.070 --> 35:35.130
are going to be separated cheat
more often than the others.

35:35.130 --> 35:38.850
So it's a very clear prediction
of the model that's relevant to

35:38.849 --> 35:40.109
some of your lives.

35:40.110 --> 35:43.660
Let me give you another example
that's less exciting perhaps,

35:43.660 --> 35:45.140
but same sort of thing.

35:45.139 --> 35:50.339
Consider the relationship that
I have with my garage mechanic.

35:50.340 --> 35:54.360
I should stress this is not a
significant other relationship.

35:54.360 --> 35:58.710
So I have a garage mechanic in
New Haven, and that garage

35:58.706 --> 36:00.566
mechanic fixes my car.

36:00.570 --> 36:03.590
And we have an ongoing business
relationship.

36:03.590 --> 36:06.340
He knows that whenever my car
needs fixing,

36:06.339 --> 36:09.809
even if it's just a small thing
like an oil change,

36:09.809 --> 36:12.729
I'm going to go to him and have
him fix it, even though it might

36:12.726 --> 36:15.176
be cheaper for me to go to Jiffy
Lube or something.

36:15.179 --> 36:17.279
So I'm going to take my car to
him to be fixed,

36:17.284 --> 36:19.854
and he's going to make some
money off me on even the easy

36:19.846 --> 36:22.146
things.
What do I want in return for

36:22.153 --> 36:24.413
that?
I want him to be honest and if

36:24.406 --> 36:27.886
all I need is an oil change I
want him to tell me that,

36:27.889 --> 36:30.459
and if what I actually need is
a new engine,

36:30.458 --> 36:32.368
he tells me I need new engine.

36:32.369 --> 36:35.889
So my cooperating with him,
is always going to him,

36:35.894 --> 36:39.964
even if it's something simple;
and his cooperating with me,

36:39.955 --> 36:42.465
is his not cheating on fixing
the car.

36:42.469 --> 36:44.589
He knows more about the car
than I do.

36:44.590 --> 36:47.550
But now what happens if he
knows either that I'm about to

36:47.546 --> 36:50.076
leave town (which is the example
we just did),

36:50.079 --> 36:53.249
or, more realistically,
he kind of knows that my car is

36:53.251 --> 36:56.071
a lemon and I'm about to get rid
of it anyway.

36:56.070 --> 36:58.790
Once I get a new car I'm not
going to go to him anymore

36:58.788 --> 37:01.508
because I have to go to the
dealer to keep the warranty

37:01.506 --> 37:03.156
intact.
So he knows that my car is

37:03.155 --> 37:05.505
about to break down anyway,
and he knows that I know that

37:05.513 --> 37:07.033
the car is about to break
anyway,

37:07.030 --> 37:12.380
so my lemon of a car is about
to be passed on--probably to one

37:12.383 --> 37:17.213
of my graduate students--then
what's going to happen?

37:17.210 --> 37:19.860
So I'm going to have an
incentive to cheat because I'm

37:19.855 --> 37:22.895
going to start taking my useless
car to Jiffy Lube for the oil

37:22.900 --> 37:25.450
changes.
And he's going to have an

37:25.449 --> 37:27.009
incentive to cheat.

37:27.010 --> 37:30.280
He's going to start telling me
you know you really need a new

37:30.279 --> 37:33.329
engine or a new clutch--it's a
manual so I have a clutch:

37:33.331 --> 37:36.661
it's a real car--so I'm going
to need a new clutch rather than

37:36.656 --> 37:38.396
just tightening up a bolt.

37:38.400 --> 37:41.900
So once again the probability
of the continuation of the

37:41.895 --> 37:43.605
relationship,
as it changes,

37:43.611 --> 37:45.901
it leads to incentives to
cheat.

37:45.900 --> 37:49.050
It leads to that relationship
breaking down.

37:49.050 --> 37:52.810
That's the content,
that's the real world content

37:52.805 --> 37:54.835
of the math we just did.

37:54.840 --> 37:59.930
Let's try and push this a
little further.

37:59.929 --> 38:05.499
Now what we've shown is that
the grim trigger works provided

38:05.500 --> 38:09.280
δ > 1/3,
and δ being bigger than

38:09.279 --> 38:13.809
1/3 doesn't seem like a very
large continuation probability.

38:13.809 --> 38:16.739
So just having a probability of
1/3 that the relationship

38:16.736 --> 38:18.876
continues allows the grim
trigger to work,

38:18.879 --> 38:21.439
so that seems good news for the
grim trigger.

38:21.440 --> 38:24.580
However, in reality,
in the real world,

38:24.580 --> 38:28.630
the grim trigger might have
some disadvantages.

38:28.630 --> 38:30.680
So let's just think about what
the grim trigger is telling us

38:30.684 --> 38:31.374
in the real world.

38:31.369 --> 38:35.939
It's telling us that if even
one of us cheats just a little

38:35.939 --> 38:40.899
bit--I just provide one item of
rotten fruit to Jake or he gives

38:40.903 --> 38:45.323
me one too few branches of
asparagus in his provisions to

38:45.316 --> 38:50.276
me--then we never do business
with each other again ever.

38:50.280 --> 38:51.150
It's completely the end.

38:51.150 --> 38:53.030
We just never cooperate again.

38:53.030 --> 38:54.970
That seems a little bit drastic.

38:54.969 --> 38:57.439
It's a little bit draconian if
you like.

38:57.440 --> 39:00.050
So in particular,
in the real world,

39:00.050 --> 39:04.450
there's a complication here,
in the real world every now and

39:04.450 --> 39:08.030
then one of us going "to cheat"
by accident.

39:08.030 --> 39:10.870
That day that I didn't have my
glasses on and I put in a rotten

39:10.871 --> 39:12.751
apple in the apples I supplied
to Jake.

39:12.750 --> 39:15.430
In the fruit,
he was counting out the

39:15.425 --> 39:19.735
asparagus and he lost count at
1,405 and he gave me one too

39:19.735 --> 39:22.365
few.
So we might want to worry about

39:22.366 --> 39:26.436
the fact that the grim trigger,
it's triggered by any amount of

39:26.440 --> 39:30.320
cheating and it's very drastic:
it says we never do business

39:30.317 --> 39:33.837
again.
The grim trigger is the analog

39:33.838 --> 39:35.888
of the death penalty.

39:35.889 --> 39:39.009
It's the business analog of the
death penalty.

39:39.010 --> 39:41.150
It's not that I'm going to kill
Jake if he gives me one too few

39:41.150 --> 39:42.770
branches of asparagus,
but I'm going to kill the

39:42.773 --> 39:45.213
relationship.
For you seniors or otherwise,

39:45.214 --> 39:47.904
who are involved in personal
relationships,

39:47.899 --> 39:50.999
it's the equivalent of saying,
if you even see your partner

39:50.998 --> 39:52.908
looking at someone else,
let alone sitting next to them

39:52.905 --> 39:55.025
in the class,
the relationship is over.

39:55.030 --> 39:58.070
It seems drastic.

39:58.070 --> 40:00.930
So we might be interested
because mistakes happen,

40:00.925 --> 40:05.155
because misperceptions happen,
we might be interested in using

40:05.163 --> 40:09.683
punishments that are less
draconian than the grim trigger,

40:09.679 --> 40:11.409
less draconian than the death
penalty.

40:11.410 --> 40:14.610
Is that right?
So what I want to do is I want

40:14.611 --> 40:17.911
to consider a different
strategy, a strategy other than

40:17.909 --> 40:21.389
the grim trigger strategy,
and see if that could work.

40:21.390 --> 40:25.970
So where shall I start?

40:25.969 --> 40:31.019
Let's start here,
so again what I'm going to

40:31.023 --> 40:38.193
revert to is the math and the
nerdiness of our analysis of the

40:38.191 --> 40:44.891
Prisoner's Dilemma but I want
you to have in mind business

40:44.889 --> 40:48.009
relationships,
your own personal

40:48.007 --> 40:49.947
relationships,
your friendships and so on.

40:49.949 --> 40:52.659
More or less everything you do
in life involves repeated

40:52.658 --> 40:55.118
interaction, so have that in the
back of your mind,

40:55.120 --> 40:56.450
but let's be nerdy now.

40:56.449 --> 41:07.729
So what I want to consider is a
one period punishment.

41:07.730 --> 41:10.470
So how are we going to write
down a strategy that has

41:10.473 --> 41:12.693
cooperation but a one period
punishment.

41:12.690 --> 41:16.370
So here's the strategy.

41:16.369 --> 41:27.329
It says--it's kind of weird
thing but it works--play C to

41:27.329 --> 41:39.459
start and then play C if--this
is going to seem weird but trust

41:39.464 --> 41:49.844
me for a second--play C if
either (C, C) or (D,D) were

41:49.837 --> 41:55.727
played last.
So, if in the previous period

41:55.733 --> 42:00.073
either both people cooperated or
both people defected,

42:00.070 --> 42:03.590
then we'll play cooperation
this period.

42:03.590 --> 42:11.210
And play D otherwise:
play D if either (C,

42:11.205 --> 42:17.145
D) or (D, C) were played last.

42:17.150 --> 42:18.500
Let's just think about this
strategy for a second.

42:18.500 --> 42:22.010
What does that strategy mean?

42:22.010 --> 42:24.990
So provided people start off
cooperating and they go on

42:24.985 --> 42:28.235
cooperating--if both Jake and I
play this strategy--in fact,

42:28.237 --> 42:29.777
we'll cooperate forever.

42:29.780 --> 42:33.850
Is that right?
So I claim this is a one period

42:33.852 --> 42:35.482
punishment strategy.

42:35.480 --> 42:36.910
Let's just see how that works.

42:36.909 --> 42:40.409
So suppose Jake and I are
playing this strategy.

42:40.409 --> 42:42.419
We're supposed to play C every
period.

42:42.420 --> 42:49.040
And suppose deliberately or
otherwise, I play D.

42:49.039 --> 42:53.419
So now in that period in which
I play D, the strategys played

42:53.422 --> 42:55.542
were D by me and C by Jake.

42:55.539 --> 42:59.679
So next period what does this
strategy tell us both to play?

42:59.679 --> 43:04.879
So it was D by me and C by
Jake, so this strategy tells us

43:04.884 --> 43:08.064
to play D.
So next period both of us will

43:08.064 --> 43:10.504
play D.
So both of us will be

43:10.496 --> 43:15.766
uncooperative precisely for that
period, that next period.

43:15.769 --> 43:17.359
Now, what about the period
after that?

43:17.360 --> 43:20.650
The period after that,
Jake will have played D,

43:20.653 --> 43:22.303
I will have played D.

43:22.300 --> 43:26.060
So this is what will have
happened: we both played D,

43:26.063 --> 43:29.033
and now it tells us to
cooperate again.

43:29.030 --> 43:33.010
Everyone happy with that?

43:33.010 --> 43:35.580
So this strategy I've written
down--it seems kind of

43:35.579 --> 43:38.749
cumbersome--but what it actually
induces is exactly a one period

43:38.754 --> 43:41.724
punishment.
If Jake is the only cheat then

43:41.722 --> 43:46.002
we both defect for one period
and go back to cooperation.

43:46.000 --> 43:48.930
If I'm the only person who
cheats then we both defect for

43:48.934 --> 43:51.034
one period and go back to
cooperation.

43:51.030 --> 43:53.820
It's a one period punishment
strategy.

43:53.820 --> 43:56.260
Of course the question is,
the question you should be

43:56.264 --> 43:57.914
asking is, is this going to
work?

43:57.910 --> 44:00.290
Is this an equilibrium?

44:00.290 --> 44:02.770
So let's just check.

44:02.770 --> 44:11.700
Is this an SPE.
Is it an equilibrium?

44:11.700 --> 44:13.780
So what do we need to check?

44:13.780 --> 44:22.470
We need to check,
as usual, that the temptation

44:22.473 --> 44:33.623
is less than or equal to the
value of the promise--the value

44:33.624 --> 44:44.964
of the promise of continuing in
cooperation--the value of the

44:44.964 --> 44:53.284
promise minus the value of the
threat.

44:53.280 --> 44:59.270
And once again we have to be
careful, because the temptation

44:59.274 --> 45:05.984
occurs today and this difference
between values occurs tomorrow.

45:05.980 --> 45:08.370
Is that right?
So this is nothing new,

45:08.372 --> 45:09.712
this is what we've always
written down,

45:09.714 --> 45:10.884
this is what we have to check.

45:10.880 --> 45:15.970
So the temptation for me to
cheat today, that's the same as

45:15.965 --> 45:18.415
it was before,
it's 3 - 2.

45:18.420 --> 45:26.970
The fact that it's tomorrow is
going to give me a δ

45:26.971 --> 45:29.571
here.
Here's our square bracket.

45:29.570 --> 45:31.780
So what's the value of the
promise?

45:31.780 --> 45:35.040
So provided we both go on
cooperating, we're going to go

45:35.040 --> 45:38.060
on cooperating forever,
in which case we're going to

45:38.063 --> 45:41.043
get 2 for ever.
Is that right?

45:41.039 --> 45:46.949
So this is going to be the
value of 2 forever starting

45:46.951 --> 45:53.421
tomorrow (and again for ever
means until the game ends).

45:53.420 --> 45:56.570
The value of the threat is what?

45:56.570 --> 45:59.820
Be a bit careful now.

45:59.820 --> 46:05.010
It's the value of--so what's
going to happen?

46:05.010 --> 46:10.740
If I cheat then tomorrow we're
both going to cheat,

46:10.736 --> 46:14.626
so tomorrow,
what am I going to get

46:14.631 --> 46:16.431
tomorrow?
0.

46:16.434 --> 46:24.444
So it's the value of 0
tomorrow: we're both going to

46:24.440 --> 46:30.250
cheat, we're both going to play
D.

46:30.250 --> 46:33.150
And then the next period what's
going to happen?

46:33.150 --> 46:36.640
We're going to play C again,
and from thereon we're going to

46:36.636 --> 46:42.306
go on playing C.
So it's going to the value of 0

46:42.312 --> 46:49.902
tomorrow and then 2 forever
starting the next day.

46:49.900 --> 46:54.510
That's what we have to evaluate.

46:54.510 --> 46:59.360
So 3 - 2, I can do that one
again, that's 1.

46:59.360 --> 47:03.500
So what's the value of 2
forever, well we did that

47:03.499 --> 47:05.949
already today,
what was it?

47:05.950 --> 47:07.030
It's in your notes.

47:07.030 --> 47:10.150
Actually it's on the board,
it's the X up there,

47:10.147 --> 47:12.567
what is it?
Here it is, 2 for ever:

47:12.565 --> 47:15.735
we figured out the value of it
before and it was

47:15.742 --> 47:17.232
2/[1–δ].

47:17.230 --> 47:27.290
So the value of 2 forever is
going to be 2/[1–δ].

47:27.290 --> 47:30.430
How about the value of 0?

47:30.429 --> 47:35.769
So starting for tomorrow I'm
going to get 0 and then with one

47:35.771 --> 47:39.601
period delay I'm going to get 2
for ever.

47:39.599 --> 47:42.109
Well 2 forever,
we know what the value of that

47:42.108 --> 47:45.058
is, it's 2/[1–δ],
but now I get it with one

47:45.062 --> 47:47.532
period delay,
so what do I have to multiply

47:47.532 --> 47:49.902
it by?
By δ good.

47:49.900 --> 47:56.920
So the value of 0 tomorrow and
then 2 forever starting the next

47:56.920 --> 48:01.110
day is δ
x 2/[1–δ].

48:01.110 --> 48:03.090
And here's the δ
coming from here which just

48:03.085 --> 48:05.755
takes into account that all this
analysis is starting tomorrow.

48:05.760 --> 48:09.650
So to summarize,
this is my temptation today.

48:09.650 --> 48:13.230
This is what I'll get starting
tomorrow if I'm a good boy and

48:13.233 --> 48:15.163
cooperate.
And this is the value of what

48:15.161 --> 48:16.291
I'll get if I cheat today.

48:16.289 --> 48:19.139
Starting tomorrow I'll get
nothing, and then I'll revert

48:19.139 --> 48:20.279
back to cooperation.

48:20.280 --> 48:24.730
And since all of these values
in this square bracket start

48:24.730 --> 48:28.010
tomorrow I've discounted them by
δ.

48:28.010 --> 48:30.980
Now this requires some math so
bear with me while I probably

48:30.976 --> 48:33.836
get some algebra wrong--and
please can I get the T.A.'s to

48:33.842 --> 48:36.962
stare at me a second because
I'll probably get this wrong.

48:36.960 --> 48:40.480
Okay so what I'm going to do
is, I'm going to look at my

48:40.476 --> 48:44.116
notes, I'm going to cheat,
that's what I'm going to do.

48:44.119 --> 48:48.579
Okay, so what I'm going to do
is I'm going to have 1 is less

48:48.575 --> 48:52.385
than or equal to,
I'm going to take a common

48:52.394 --> 48:57.624
factor of 2 / [1–δ]
and δ, so I'm going to

48:57.623 --> 49:03.333
have 2δ/[1–δ],
and that's going to leave

49:03.331 --> 49:08.911
inside the square brackets:
this is a 1 and this is a

49:08.907 --> 49:09.977
δ.

49:09.980 --> 49:14.070


49:14.070 --> 49:16.440
So this δ
here was that δ

49:16.438 --> 49:19.498
there, and then I took out a
common factor of

49:19.503 --> 49:22.363
2/[1–δ]
from this bracket.

49:22.360 --> 49:23.320
Everyone okay with the algebra?

49:23.320 --> 49:25.970
Just algebra,
nothing fancy going on there.

49:25.969 --> 49:33.519
So that's good because now the
1-δ cancels,

49:33.517 --> 49:41.547
this cancels with this,
so this tells us we're okay

49:41.545 --> 49:48.285
provided 1/2 <= δ:
it went up.

49:48.289 --> 49:50.749
So don't worry too much about
the algebra, trust me on the

49:50.753 --> 49:52.613
algebra a second,
let's just worry about the

49:52.611 --> 49:56.051
conclusion.
What's the conclusion?

49:56.050 --> 50:00.990
The conclusion is that this one
period punishment is an SPE,

50:00.994 --> 50:04.784
it will be enough,
one period of punishment will

50:04.778 --> 50:08.848
be enough to sustain cooperation
in my Prisoner's Dilemma

50:08.850 --> 50:12.050
repeated business relationship
with Jake,

50:12.050 --> 50:15.410
or in the seniors'
relationships with their

50:15.414 --> 50:18.144
significant others,
provided δ

50:18.138 --> 50:19.988
> 1/2.
What did δ

50:19.988 --> 50:21.818
need to be for the grim
strategy?

50:21.820 --> 50:26.070
1/3, so what have we learned
here?

50:26.070 --> 50:34.900
We learned--nerdily--what we
learned was that for the grim

50:34.898 --> 50:40.628
strategy we needed δ
> 1/3.

50:40.630 --> 50:44.250
For the one period punishment
we needed δ

50:44.254 --> 50:48.044
> 1/2, but what's the more
general lesson?

50:48.039 --> 50:52.859
The more general lesson is,
if you use a softer punishment,

50:52.859 --> 50:57.759
a less draconian punishment,
for that to work we're going to

50:57.762 --> 50:59.842
need a higher δ.

50:59.840 --> 51:04.570
Is that right?
So what we're learning here is

51:04.565 --> 51:08.925
there's a trade off,
there's a trade off in

51:08.926 --> 51:13.236
incentives.
And the trade off is if you use

51:13.237 --> 51:16.647
a shorter punishment,
a less draconian

51:16.648 --> 51:22.088
punishment--instead of cutting
people's hands off or killing

51:22.088 --> 51:24.748
them,
or never dealing with them

51:24.747 --> 51:28.957
again, you just don't deal with
them for one period--that's okay

51:28.957 --> 51:32.427
provided there's a slightly
higher probability of the

51:32.432 --> 51:34.372
relationship continuing.

51:34.369 --> 51:44.999
So shorter punishments are okay
but they need--the implication

51:44.999 --> 51:54.579
sign isn't really necessary
there--they need more weight

51:54.583 --> 51:58.943
δ on the future.

51:58.940 --> 52:01.060
I claim that's very intuitive.

52:01.059 --> 52:04.799
What its saying is,
we're always trading things off

52:04.802 --> 52:06.302
in the incentives.

52:06.300 --> 52:11.550
We're trading off the ability
to cheat and get some cookies

52:11.548 --> 52:15.978
today versus waiting and,
we hope, getting cookies

52:15.983 --> 52:18.033
tomorrow.
So if, in fact,

52:18.030 --> 52:21.340
the difference between the
reward and the punishment isn't

52:21.337 --> 52:24.227
such a big deal,
isn't so big--the punishment is

52:24.225 --> 52:27.845
just, I'm going to give you one
fewer cookies tomorrow--then you

52:27.854 --> 52:31.314
better be pretty patient not to
go for the cookies today.

52:31.309 --> 52:33.359
I was about to say,
those of you who have children.

52:33.360 --> 52:35.910
I'm probably the only person in
the room with children.

52:35.909 --> 52:38.379
That cookie example will
resonate for the rest of

52:38.377 --> 52:40.997
you--wait until you get
there--you'll discover that,

52:40.999 --> 52:43.209
in fact, cookies are the right
example.

52:43.210 --> 52:46.820
So shorter punishment,
less draconian punishments,

52:46.822 --> 52:51.172
less reduction in your kid's
cookie rations tomorrow is only

52:51.172 --> 52:54.332
going to work,
is only going to sustain good

52:54.331 --> 52:58.121
behavior provided those kids put
a high weight on tomorrow.

52:58.119 --> 53:00.539
In that case,
it isn't that the kids will

53:00.541 --> 53:03.081
worry about the relationship
breaking down,

53:03.084 --> 53:06.114
you're stuck with your kids,
it's just that they're

53:06.111 --> 53:09.061
impatient.
Okay, so we've been doing a lot

53:09.061 --> 53:12.961
of formal stuff here and I want
to go on doing formal stuff,

53:12.960 --> 53:16.650
but what I want to do now is
spend the rest of today looking

53:16.649 --> 53:17.899
at an application.

53:17.900 --> 53:22.090
An application is,
I hope going to convince you

53:22.091 --> 53:26.011
that repeated interaction really
matters.

53:26.010 --> 53:31.240
So this is assuming that the
one about the seniors and their

53:31.239 --> 53:35.049
boyfriends and girlfriends
wasn't enough.

53:35.050 --> 53:38.830
Okay, so the application is
going to take us back a little

53:38.834 --> 53:43.154
bit because what I want to talk
about is repeated moral hazard.

53:43.150 --> 53:47.050


53:47.050 --> 53:51.800
Moral hazard is something we
discussed the first class after

53:51.801 --> 53:55.371
the mid-term.
So what I want you to imagine

53:55.368 --> 53:59.188
is that you are running a
business in the U.S.

53:59.190 --> 54:03.210
and you are considering making
an investment in an emerging

54:03.205 --> 54:06.075
market, and again,
so as not to offend anybody who

54:06.078 --> 54:08.718
watches this on the video,
let's just call that emerging

54:08.719 --> 54:11.919
market Freedonia,
rather than give it a name like

54:11.917 --> 54:15.957
Kazakhstan, a name like
something other than Freedonia.

54:15.960 --> 54:18.740
So Freedonia,
for those of you who don't

54:18.735 --> 54:22.005
know, is a republic in a Marx
Brothers film.

54:22.010 --> 54:25.760
So you're thinking of
outsourcing some production of

54:25.760 --> 54:29.070
part of what your business is to
Freedonia.

54:29.070 --> 54:31.050
The reason you're thinking of
doing this outsourcing,

54:31.045 --> 54:33.395
what makes it attractive is
that wages are low in Freedonia.

54:33.400 --> 54:35.390
So you get this outsourced in
Freedonia.

54:35.389 --> 54:38.349
You think you're going to get
it done cheaply.

54:38.349 --> 54:42.129
The down side is because
Freedonia is an emerging market,

54:42.126 --> 54:45.426
the court system,
it doesn't operate very well.

54:45.429 --> 54:48.419
And in particular,
it's going to be pretty hard to

54:48.422 --> 54:52.212
enforce contracts and to jail
people and so on in Freedonia.

54:52.210 --> 54:55.010
So you're considering
outsourcing.

54:55.010 --> 54:56.920
The plus is,
from your point of view,

54:56.924 --> 54:59.964
the plus is wages are cheap
where you're going to get this

54:59.956 --> 55:02.666
production done.
The down side is it's going to

55:02.674 --> 55:05.714
be hard to enforce contracts
because this is an emerging

55:05.708 --> 55:07.878
market.
So what you're considering

55:07.882 --> 55:11.692
doing is employing an agent and
you're going to pay that agent

55:11.689 --> 55:14.309
W, so W is the wage if you
employ them.

55:14.309 --> 55:16.419
I'll put this up in a tree in a
second.

55:16.420 --> 55:22.810
Let's assume that the "going
wage" in Freedonia is 1:

55:22.813 --> 55:26.013
we'll just normalize it.

55:26.010 --> 55:28.890
So the going wage in Freedonia
is 1, and let's assume that to

55:28.889 --> 55:31.579
get this outsourcing to work
you're going to have to send

55:31.576 --> 55:37.046
some resources to your agent,
your employee in Freedonia.

55:37.050 --> 55:38.920
And let's assume that the
amount you're going to have to

55:38.920 --> 55:40.450
send over there is equivalent to
another 1.

55:40.449 --> 55:44.959
So the going wage in Freedonia
is 1 and the amount you're going

55:44.955 --> 55:49.525
to have to invest in giving this
agent materials or machinery is

55:49.532 --> 55:52.782
another 1.
Let's assume that this project

55:52.780 --> 55:55.160
is a pretty profitable project.

55:55.159 --> 55:59.979
So if the project succeeds,
if the project goes ahead and

55:59.980 --> 56:04.800
succeeds, it's going to generate
a gross revenue of 4.

56:04.800 --> 56:08.680
Of course you have to invest 1
so that's a net revenue of 3 for

56:08.678 --> 56:12.368
you, but nonetheless there's a
big potential return here.

56:12.369 --> 56:18.219
The bad news is that your agent
in Freedonia can cheat on you.

56:18.219 --> 56:21.279
In particular,
what he can do is he can simply

56:21.282 --> 56:23.802
take the 1 that you've sent to
him,

56:23.800 --> 56:27.930
sell those materials on the
market and then go away and just

56:27.929 --> 56:30.169
work in his normal job anyway.

56:30.170 --> 56:34.030
So he can get his normal wage
of 1 for just going and doing

56:34.028 --> 56:36.218
his normal job,
whatever that was,

56:36.223 --> 56:39.153
and he can steal the resources
from you.

56:39.150 --> 56:41.370
So let's put this up as a kind
of tree.

56:41.369 --> 56:44.289
This is a slight cheat,
this tree, but we'll see why in

56:44.286 --> 56:52.426
a second.
So your decision is to invest

56:52.434 --> 56:57.484
and set W.
So if you invest in Freedonia,

56:57.477 --> 57:01.247
you'll invest and set W,
set the wage you're going to

57:01.251 --> 57:04.301
pay him.
The going wage is 1 but you can

57:04.303 --> 57:08.013
set a different wage or you
could just not invest.

57:08.010 --> 57:11.490
If you don't invest you get
nothing and your agent in

57:11.489 --> 57:14.299
Freedonia just gets the going
wage of 1.

57:14.300 --> 57:17.990
If you do invest in Freedonia
and set a wage of W,

57:17.987 --> 57:20.317
then your agent has a choice.

57:20.320 --> 57:28.130
Either he can be honest or he
can cheat.

57:28.130 --> 57:30.100
If he cheats,
what's going to happen to you?

57:30.099 --> 57:33.609
You had to invest 1 in sending
it over there,

57:33.610 --> 57:37.840
you're going to get nothing
back, so you'll get -1.

57:37.840 --> 57:42.250
And he will go away and work
his normal job and get 1,

57:42.248 --> 57:45.988
and, in addition,
he'll sell your materials so

57:45.991 --> 57:48.821
he'll get a total of 1 + 1 is?

57:48.820 --> 57:52.600
2, thank you.
So he'll get a total of 2.

57:52.599 --> 57:55.049
On the other hand,
if he's honest,

57:55.051 --> 57:59.291
then you're going to get a
return of 4 minus the 1 you had

57:59.285 --> 58:02.995
to invest minus whatever wage
you paid to him.

58:03.000 --> 58:10.340
So your return will be 3 minus
the wage you pay him.

58:10.340 --> 58:14.560
You're only going to pay him
once the job's done,

58:14.559 --> 58:17.459
3 - W, and he's going to get W.

58:17.460 --> 58:20.390
He's done his job--he hasn't
exercised his outside option,

58:20.387 --> 58:23.107
he hasn't sold your
materials--so he'll just get W.

58:23.110 --> 58:26.530
Now, I'm slightly cheating here
because this isn't really the

58:26.533 --> 58:29.963
way the tree looks because I
could choose different levels of

58:29.957 --> 58:32.077
W.
So this upper branch where I

58:32.078 --> 58:35.698
invest and set W is actually a
continuum of such branches,

58:35.699 --> 58:38.239
one for each possible W,
I could set.

58:38.239 --> 58:40.569
But for the purpose of today
this is enough.

58:40.570 --> 58:43.120
This gives us what we needed to
see.

58:43.119 --> 58:47.699
So let's imagine that this is a
one shot investment.

58:47.699 --> 58:50.949
What I want to learn is in this
one shot investment,

58:50.950 --> 58:52.480
I invest in Freedonia.

58:52.480 --> 58:56.090
I hire my agent once,
what I want to learn is how

58:56.088 --> 59:00.898
much do I have to pay that agent
to actually get the job done?

59:00.900 --> 59:02.350
Remember the starting position.

59:02.349 --> 59:08.059
The starting position is it
looks very attractive.

59:08.059 --> 59:11.269
It looks very attractive
because the returns on this

59:11.274 --> 59:15.594
project are 4 or 4 - 1,
so that the surplus available

59:15.588 --> 59:21.068
on this project is 3 minus the
wage, and the going wage was

59:21.065 --> 59:23.125
just 1.
So it looks like there's lots

59:23.128 --> 59:25.208
of profit around to make this
outsourcing profitable.

59:25.210 --> 59:27.300
I mumbled that so let me try it
again.

59:27.300 --> 59:30.610
So the reason this looks
attractive is the going wage is

59:30.614 --> 59:34.114
just 1, so if I just pay him 1
and he does the project then

59:34.110 --> 59:37.730
I'll get a gross return of 4
minus the 1 I invested minus the

59:37.726 --> 59:40.856
1 that I had to pay him for a
net return of 2.

59:40.860 --> 59:44.710
It seems like that's a 100%
profitable project,

59:44.709 --> 59:47.219
so it looks very attractive.

59:47.220 --> 59:48.800
What's the problem?

59:48.800 --> 59:52.810
The problem is if I only
set--this is going to give us

59:52.813 --> 59:57.663
backward induction--if I set the
wage equal to the going wage,

59:57.659 --> 1:00:01.919
so if I set W = 1 what will my
agent do?

1:00:01.920 --> 1:00:04.860
He's going to cheat.

1:00:04.860 --> 1:00:15.120
The problem is if I set W = 1,
which is the going wage,

1:00:15.119 --> 1:00:24.999
the going wage in Freedonia,
the agent will cheat.

1:00:25.000 --> 1:00:27.200
If he cheats I just lose my
investment.

1:00:27.199 --> 1:00:31.369
So how much do I have to set
the W to?

1:00:31.370 --> 1:00:32.930
Let's look at this.

1:00:32.930 --> 1:00:36.010
So we have to set W.

1:00:36.010 --> 1:00:40.900
What I need is I need his wage
to be big enough so that being

1:00:40.902 --> 1:00:45.552
honest and going on with my
projectoutweighs his incentive

1:00:45.550 --> 1:00:51.930
to cheat.
I need W to be bigger than 2.

1:00:51.930 --> 1:00:55.830
Is that right?
I need W to be at least as big

1:00:55.833 --> 1:01:00.013
as 2.
So in setting the wage,

1:01:00.009 --> 1:01:06.639
in equilibrium,
what are we going to do?

1:01:06.639 --> 1:01:11.049
I'm going to set a wage,
let's call it W* = 2 (plus a

1:01:11.046 --> 1:01:13.076
penny), is that right?

1:01:13.079 --> 1:01:18.039
So this is an exercise which we
visited the first day after the

1:01:18.039 --> 1:01:20.579
mid-term.
This is about incentive design.

1:01:20.579 --> 1:01:27.029
In this one shot game,
which we can easily solve by

1:01:27.032 --> 1:01:33.482
backward induction,
I'm going to need to set a wage

1:01:33.484 --> 1:01:38.134
equal to 2, and then he'll work.

1:01:38.130 --> 1:01:42.560


1:01:42.559 --> 1:01:44.309
So in a minute,
we're going to look at the

1:01:44.309 --> 1:01:46.699
repeated version of this,
but before we do let's just sum

1:01:46.698 --> 1:01:47.848
up where we are so far.

1:01:47.850 --> 1:01:51.040
What is this telling us?

1:01:51.039 --> 1:01:55.169
It's telling us that when you
invest in an emerging market,

1:01:55.172 --> 1:01:59.522
where the courts don't work so
they aren't going to be able to

1:01:59.518 --> 1:02:03.008
enforce this guy to work
well--in particular,

1:02:03.010 --> 1:02:05.350
he can run off with your
investment--even though wages

1:02:05.351 --> 1:02:07.871
are low, so it seems very
attractive to do outsourcing,

1:02:07.869 --> 1:02:12.529
if you worry about getting
incentives right you're going to

1:02:12.529 --> 1:02:17.269
have pay an enormous wage
premium to get the guy to work.

1:02:17.269 --> 1:02:23.749
So the going wage in Freedonia
was 1, but you had to set a wage

1:02:23.754 --> 1:02:29.824
equal to 2, a 100% wage premium,
to get the guy to work.

1:02:29.820 --> 1:02:38.910
So the wage premium in this
emerging market is 100%,

1:02:38.909 --> 1:02:47.819
you're paying 2 even though the
going wage is 1.

1:02:47.820 --> 1:02:52.010
By the way, this is not an
unreasonable prediction.

1:02:52.010 --> 1:02:55.260
If you look at the wages payed
by European and American

1:02:55.260 --> 1:02:58.030
companies in some of these
emerging markets,

1:02:58.030 --> 1:02:59.790
which have very,
very low going wages,

1:02:59.789 --> 1:03:02.599
and if you look at the wages
that are actually being paid by

1:03:02.595 --> 1:03:05.255
the companies that are doing
outsourcing you see enormous

1:03:05.258 --> 1:03:08.208
wage premiums.
You see enormous premiums over

1:03:08.213 --> 1:03:09.953
and above the going wage.

1:03:09.949 --> 1:03:17.109
Now what I want to do is I want
to revisit exactly the same

1:03:17.110 --> 1:03:24.640
situation, but now we're going
to introduce the wrinkle of the

1:03:24.642 --> 1:03:27.472
day.
What's the wrinkle of the day?

1:03:27.469 --> 1:03:30.529
The wrinkle of the day is
you're not only going to invest

1:03:30.531 --> 1:03:33.681
in Freedonia today,
but if things go well you'll

1:03:33.684 --> 1:03:36.764
invest tomorrow,
and if things go well again

1:03:36.762 --> 1:03:40.842
you'll invest the day after at
least with some significant

1:03:40.843 --> 1:03:43.183
probability.
So the wage premium we just

1:03:43.179 --> 1:03:44.919
calculated was the one shot wage
premium.

1:03:44.920 --> 1:03:50.590
It was getting this job--this
single one shot job--outsourced

1:03:50.592 --> 1:03:53.532
to Freedonia.
Now I want to consider how much

1:03:53.532 --> 1:03:56.332
you're going to have to pay,
what are wages going to be in

1:03:56.331 --> 1:03:58.591
Freedonia in the foreign
investment sector,

1:03:58.590 --> 1:04:01.930
if instead of just having a one
shot, one job investment,

1:04:01.925 --> 1:04:04.125
you're investing for the long
term.

1:04:04.130 --> 1:04:06.300
You're going to be in Freedonia
for a while.

1:04:06.300 --> 1:04:19.440
So consider repeated
interaction with probability

1:04:19.435 --> 1:04:25.725
δ of continuing.

1:04:25.730 --> 1:04:27.720
So we don't know that you're
going to go on in Freedonia.

1:04:27.719 --> 1:04:30.409
Things might break down in
Freedonia because there's a

1:04:30.414 --> 1:04:31.924
coup.
It might break down in

1:04:31.923 --> 1:04:35.073
Freedonia because the American
administration says you're not

1:04:35.069 --> 1:04:37.009
allowed to do outsourcing
anymore.

1:04:37.010 --> 1:04:38.410
All sorts of things might
happen, but with some

1:04:38.407 --> 1:04:39.807
probability δ
the relationship is going to

1:04:39.805 --> 1:04:42.605
continue.
So repeated interaction with

1:04:42.610 --> 1:04:44.590
probability of δ.

1:04:44.590 --> 1:04:49.680
Let's redo the exercise we did
before to see what wage you'll

1:04:49.678 --> 1:04:56.518
have to charge.
Our question is what

1:04:56.519 --> 1:05:09.749
wage--let's call it W**--what
wage will you pay?

1:05:09.750 --> 1:05:14.130


1:05:14.130 --> 1:05:17.170
The way we're going to solve
this, is exactly using the

1:05:17.170 --> 1:05:19.310
methods we've learned in this
class.

1:05:19.309 --> 1:05:32.529
So what we're going to compare
is the temptation to cheat

1:05:32.529 --> 1:05:46.219
today--and we better make sure
that that's less than δ

1:05:46.220 --> 1:06:00.620
times the value of continuing
the relationship minus the value

1:06:00.620 --> 1:06:08.410
of ending the relationship.

1:06:08.410 --> 1:06:12.970
Let's call this tomorrow.

1:06:12.969 --> 1:06:16.249
So what's happening now is,
once again, I'm employing my

1:06:16.248 --> 1:06:19.148
agent in Freedonia,
and provided he does a good

1:06:19.150 --> 1:06:22.150
job, I'll employ him again
tomorrow, at least with

1:06:22.145 --> 1:06:23.485
probability δ.

1:06:23.489 --> 1:06:26.269
But if he doesn't do a good
job, if he runs off with my

1:06:26.271 --> 1:06:29.261
investment and doesn't do my
job, what am I going to do?

1:06:29.260 --> 1:06:32.430
What would you do?

1:06:32.430 --> 1:06:38.110
You'd fire him.
So the punishment--it's clear

1:06:38.114 --> 1:06:41.094
what the punishment's going to
be here--the punishment is,

1:06:41.090 --> 1:06:43.440
if he doesn't do a good job,
you fire him.

1:06:43.440 --> 1:06:44.930
The value of ending the
relationship.

1:06:44.929 --> 1:06:52.669
This is firing and this is
continuing.

1:06:52.670 --> 1:06:56.010


1:06:56.010 --> 1:06:59.490
So let's just work out what
these things are.

1:06:59.489 --> 1:07:05.259
So his temptation to cheat
today: if he cheats today,

1:07:05.264 --> 1:07:08.044
he doesn't get my wage.

1:07:08.039 --> 1:07:10.589
But he does run off with my
cash, and he does go and do his

1:07:10.586 --> 1:07:11.636
job at the going wage.

1:07:11.639 --> 1:07:15.609
So if he cheats today he gets
2, he stole all my cash,

1:07:15.609 --> 1:07:19.429
and he's going off and working
at the going wage,

1:07:19.429 --> 1:07:29.279
but he doesn't get what I would
have paid him W** if the job was

1:07:29.283 --> 1:07:32.403
well done.
We need this to be less than

1:07:32.400 --> 1:07:34.260
the value of continuing the
relationship.

1:07:34.260 --> 1:07:35.440
Let's do the easy bit first.

1:07:35.440 --> 1:07:38.320
What does he get if we end the
relationship?

1:07:38.320 --> 1:07:43.240
He's been fired,
so he'll just work at the going

1:07:43.243 --> 1:07:49.583
wage for ever.
So this is the value of 1 for

1:07:49.579 --> 1:07:56.969
ever, or at least until the end
of the world.

1:07:56.970 --> 1:08:03.010
This is the value of what?

1:08:03.010 --> 1:08:05.420
As long as he stayed employed
by me what's he going to get

1:08:05.423 --> 1:08:06.273
paid every period?

1:08:06.270 --> 1:08:09.320
What's he going to get paid?

1:08:09.320 --> 1:08:26.500
W**.
So the value of W** for ever.

1:08:26.500 --> 1:08:30.020
Let me cheat a little bit and
assume that the probability of

1:08:30.022 --> 1:08:33.612
some coup happening that ends
our relationship exogenously is

1:08:33.605 --> 1:08:37.425
the same probability of the coup
happening and ending his ongoing

1:08:37.426 --> 1:08:40.986
wage exogenously,
so we can use the same δ.

1:08:40.989 --> 1:08:44.389
So let's just do some math
here, what's the value of W**

1:08:44.391 --> 1:08:46.431
forever?
So remember the value of 2

1:08:46.434 --> 1:08:47.424
forever was what?

1:08:47.420 --> 1:08:53.940
2/[1-δ].
So what's the value of W**

1:08:53.938 --> 1:08:58.998
forever?
So this is going to be

1:08:58.995 --> 1:09:05.325
W**/[1-δ].
What's the value of 1 forever?

1:09:05.330 --> 1:09:14.910
1/[1-δ].
The whole thing is multiplied

1:09:14.914 --> 1:09:18.794
by δ and this is 2-W**.

1:09:18.789 --> 1:09:24.229
Now I need to do some algebra
to solve for W**.

1:09:24.230 --> 1:09:27.460
So let's try and do that.

1:09:27.460 --> 1:09:43.510
So I claim that this is the
same as [1--δ]

1:09:43.510 --> 1:09:54.680
2--[1--δ]
W** < W**δ

1:09:54.676 --> 1:10:02.486
- δ 1.
Everyone okay with that?

1:10:02.489 --> 1:10:05.609
One more line:
let me just sort out some terms

1:10:05.605 --> 1:10:15.805
here.
So taking these on the other

1:10:15.812 --> 1:10:35.262
side, I have [1–δ]
2 + δ1 <= W**δ

1:10:35.258 --> 1:10:46.368
+ [1–δ]
W** = W**.

1:10:46.369 --> 1:10:48.869
So someone should just check my
algebra at home,

1:10:48.866 --> 1:10:50.296
but I think that's right.

1:10:50.300 --> 1:10:54.020
So the last two steps were just
algebra, nothing fancy.

1:10:54.020 --> 1:10:55.880
What have we learned?

1:10:55.880 --> 1:11:01.890
We have learned that the wage I
have to pay this guy,

1:11:01.891 --> 1:11:08.481
the wage I have to pay him lies
somewhere between 2 and 1,

1:11:08.480 --> 1:11:13.220
but we can do a bit better than
that.

1:11:13.220 --> 1:11:16.670


1:11:16.670 --> 1:11:18.220
Let's just delete everything
here.

1:11:18.220 --> 1:11:24.110


1:11:24.109 --> 1:11:29.089
So in particular,
if δ = 0,

1:11:29.091 --> 1:11:36.001
what's W**?
If δ = 0,

1:11:36.002 --> 1:11:43.582
W** is equal to what?

1:11:43.580 --> 1:11:49.030
Somebody?
Equal to 2 and that's what we

1:11:49.030 --> 1:11:52.260
had before.
In the one shot game,

1:11:52.258 --> 1:11:56.478
there it is up there,
where there was no possibility

1:11:56.480 --> 1:12:00.040
of continuing the relationship
tomorrow,

1:12:00.039 --> 1:12:03.229
I had to pay him a wage of 2,
or if you like,

1:12:03.234 --> 1:12:05.054
a wage premium of 100%.

1:12:05.050 --> 1:12:08.760
If there's no probability--if
there's no chance of continuing

1:12:08.758 --> 1:12:11.598
this relationship,
if δ = 0--we find again

1:12:11.601 --> 1:12:13.951
that I'm paying 100% wage
premium.

1:12:13.950 --> 1:12:15.980
Let's take the other extreme.

1:12:15.979 --> 1:12:18.489
If δ = 1,
so I just know this

1:12:18.488 --> 1:12:21.678
relationship's going to
continue--if δ

1:12:21.681 --> 1:12:24.731
= 1,
so there's no probability of

1:12:24.727 --> 1:12:29.687
the world ending or there being
a coup--then what's W**?

1:12:29.690 --> 1:12:33.630
It's equal to 1.
What's that?

1:12:33.630 --> 1:12:36.800
What's 1?
It's the going wage.

1:12:36.800 --> 1:12:39.060
So this is the going wage.

1:12:39.060 --> 1:12:43.950
If I know for sure we're going
to continue forever I can get

1:12:43.951 --> 1:12:48.511
away with paying the guy the
going wage, at least in the

1:12:48.510 --> 1:12:52.560
limit.
If we know we're not going to

1:12:52.560 --> 1:12:57.890
continue then I have to play the
one shot wage.

1:12:57.890 --> 1:13:02.940
But let's look at a more
interesting intermediate case.

1:13:02.940 --> 1:13:05.330
Suppose δ = ½.

1:13:05.329 --> 1:13:08.439
There's just a 1/2
probability--that's pretty

1:13:08.438 --> 1:13:11.898
low--there's 1/2 probability
that your company,

1:13:11.899 --> 1:13:15.099
American Widgets,
is going to stay in Freedonia:

1:13:15.095 --> 1:13:18.965
with probability 1/2 it's going
to be done next period,

1:13:18.970 --> 1:13:21.230
with probability 1/2 it's going
to stay.

1:13:21.230 --> 1:13:23.770
What does that do to the wage?

1:13:23.770 --> 1:13:28.960
What happens to the wage in
this case in which there's a

1:13:28.955 --> 1:13:34.795
probability of 1/2 of American
Widgets staying in Freedonia?

1:13:34.800 --> 1:13:38.500
It's a 1/2 between 2 and 1,
which is therefore one and a

1:13:45.794 --> 1:13:54.034
is, the wage premium is now only
50%.

1:13:54.029 --> 1:13:57.549
What have we learned from this
example?

1:13:57.550 --> 1:14:00.090
Just an example of using
repeated games.

1:14:00.090 --> 1:14:03.220
Well the first thing we've
learned is it's going to be

1:14:03.219 --> 1:14:06.729
easy, once we get used to it,
it's easy to use this

1:14:06.732 --> 1:14:10.262
technology of comparing
temptations to cheat,

1:14:10.258 --> 1:14:15.148
with values of continuing in a
cooperative relationship versus

1:14:15.145 --> 1:14:21.185
the value of the punishment,
which is in this case was just

1:14:21.194 --> 1:14:24.674
firing the guy.
But more specifically in this

1:14:24.667 --> 1:14:28.147
example we've learned that even
a relatively small probability

1:14:28.154 --> 1:14:31.134
of this relationship
continuing--so this is good news

1:14:31.126 --> 1:14:34.556
for those of you who are seniors
and are about to move to San

1:14:34.556 --> 1:14:38.096
Francisco and your significant
other is going to London--even a

1:14:38.099 --> 1:14:40.789
small probability of the
relationship continuing

1:14:40.786 --> 1:14:43.526
drastically reduces the wage
premium.

1:14:43.529 --> 1:14:46.939
The amount you have to "pay"
your significant other not to

1:14:46.939 --> 1:14:50.289
cheat on you as they go off to
London or San Francisco is

1:14:50.289 --> 1:14:53.279
drastically lower if there's
some probability,

1:14:56.880 --> 1:14:58.940
Before you leave,
one more thought okay.

1:14:58.940 --> 1:15:00.450
So how did this all work?

1:15:00.449 --> 1:15:04.669
Just to summarize,
to get good behavior in these

1:15:04.671 --> 1:15:10.421
continuing relationships there
has to be some reward tomorrow.

1:15:10.420 --> 1:15:14.760
That reward needs to be higher,
if the weight you put on

1:15:14.764 --> 1:15:18.794
tomorrow, if the probability of
continuing tomorrow,

1:15:18.793 --> 1:15:21.773
is lower.
The less likely tomorrow is to

1:15:21.768 --> 1:15:24.948
occur the bigger that reward has
to be tomorrow.

1:15:24.949 --> 1:15:28.549
We're going to have to charge
wage premia to employ people in

1:15:28.547 --> 1:15:32.027
Freedonia but those premiums
will come down once we realize

1:15:32.025 --> 1:15:35.435
that we're in established
relationships in Freedonia--once

1:15:35.443 --> 1:15:38.683
the American firms are
established and not fly by night

1:15:38.681 --> 1:15:40.601
operations in Freedonia.

1:15:40.600 --> 1:15:43.760
Whether that's good news or bad
news for Freedonia we'll leave

1:15:43.756 --> 1:15:46.996
there.
On Monday, totally new topic.