Hello everybody, Santiago here. I made

this video so we could dive into unit

eight a little bit more so we're gonna

explore what I mean by tension in games

by looking at the tension between

individual level and group level

incentives in the prisoner's dilemma

so some brief highlights of the unit so

far we talked about an important

assumption in game theory being that

individuals act in order to maximize

their own utility and utility remember

is this numerical measure of the outcome

of a game which could be profit

happiness reproductive success we saw

all of this earlier in the unit so we're

taking this to mean that the individuals

are selfish they're right if they act in

order to maximize their own utility

they're only looking at their personal

gain then we're calling them selfish

agents and a big question in the unit is

can selfish individuals cooperate so

let's go back to the prisoner's dilemma

the numbers in this prisoner's dilemma

are a little bit different than those

Melanie used but the game is still

essentially the same if you want to

compare this one with the other one and

see go through that process of

deliberation that each of the players

goes through to make their decision I

encourage you to do that now to make

everything even clearer in the attempt

to make things clearer I color-coded the

agents and the utilities that they

receive so player one is in blue and he

can choose between the blue cooperate

and effect player two is in green and

they can choose between the green

cooperate and effect and depending on

what they choose if player 1 chooses

cooperate player 2 chooses the fact then

they're in this cell this is the outcome

and player one gets the utility on the

Left player two gets the utility on the

right so to look at this tension let's

decompose let's rewrite the game as the

sum of cooperative and selfish forces

and to do this I'm going to use a

technique that was developed in this

paper I'm citing below strategic and

behavioral decomposition of games and

I'm adding this paper to the

supplementary materials page of the

website and I encourage you to take a

look at it unfortunately I can't go

through all the details of the paper

everything they do there's a lot of good

stuff but I gotta keep it simple keep it

short and point you where to find more

so alright let's keep going so if I

apply the decomposition to this

prisoner's dilemma game I get these

three components so if you want to make

sure pause the video add these three

matrices down here add them up and see

that you get this one and yeah so I so

let's see what's going on what are these

components telling us what do you think

they're telling us what what kind of

information is contained in each of

these components and why is it that we

can separate these information so feel

free to pause the video analyze each of

these components see what you think is

going on and because I'm about to

extenuate this first component we

interpret it to be this kind of fluff it

contains no real information it's not

really doing anything because it

contributes the same value to all the

utility's of each of the players so it

doesn't have to be the same value for

both players right we see here that both

players all have to

it could be different but I'm keeping it

simple and so this this component of the

game contains no info because since it's

the same value for all of the outcomes

well in game theory were interested in

comparing the results of different

outcomes so anyway we compare the

outcomes is gonna cancel out this common

value so saying it's just fluff there's

no real information here the authors of

the paper called us the kernel now the

second component we say contains the

game's selfish information now why is

that well so check out the other

components can you gain anything from

changing your own strategy alone no

right so check out the kernel if you

let's say player 2 is fixed playing

cooperate or defect

if you switch between cooperate out or

defect can you is there any incentive

for you to change your strategy no so

all of that information all of that

individual level incentive is contained

in this middle component in this selfish

info component which the authors called

the Nash or strategic component and the

reason they call it the Nash component

is that this component contains all the

necessary information in a game to

compute Nash equilibria which we haven't

really talked about so I won't really

get into more but right in this entire

game we can decompose it into these

three layers these three components but

to compute Nash equilibrium we only need

this it's pretty fascinating so if

you're curious if you're liking this

check out the paper study more game

theory

but anyway the next component this last

component doesn't have any selfish

information but something's going on

it's not trivial like the kernel it's

not just the same value right so what do

you think is going on here I'm calling

it left over info to keep it mysterious

we haven't figured out what this

component does so maybe pause the video

think about it a little bit here let me

blow it up

there's the leftover info component what

is happening with it okay pause the

video okay so look in this component

each agent is powerless to change their

own utility alone but what happens when

the other agent changes their strategy

so let's say okay start over let's say

your player 1 and player 2 is playing

cooperate you have a choice between

cooperate and getting a - or defect and

getting it - so if that's all you're

looking at you don't care right you're

not this isn't going to sway your

decision either way but what happens

when you play cooperate and what happens

when you play defect when you play

cooperate agent 2 is receiving a payoff

of positive - no matter what they do if

you play defect agent 2 is getting a

payoff of negative - no matter what they

do so it's like your decision is

affecting the other person so this idea

they call it externalities in X in

economics so an externality is a utility

imposed on another agent on another

person another group of people it's an

is utility imposed on others from your

actions so a simple example if I play

music really loud and it bothers my

neighbor then they're receiving a

negative externality from something I'm

doing right so maybe I'm making a

decision that gives me

a nice private payoff a utility but it

gives my neighbor a negative utility

that's the idea of externalities so this

left over information component

I'm calling the author's called the

behavioral component

I'm calling the externalities component

sorry for all the names I hope this is

not confusing yeah so I talked about

this externalities you do something the

result creates utility the result is

utility for yourself and for others so

externality is this utility that gets

imposed on others now just this

so let's look at the prisoner's dilemma

again ignoring the kernel so when I

cooperate I'm not getting the best for

myself no matter what you do right

look at that negative one if you

cooperate and I cooperate I'm getting a

negative one if you defect and I

cooperate I get a negative one remember

we're only looking at the selfish info

component so looking at the selfish info

I'd rather always defect I always get a

better individual payoff by defecting

following if I'm selfish then I'm always

gonna defect and that is shown right

here because being selfish I don't care

about a 2 or a 2 or a negative 2 and a

negative 2 right I only care about this

individual I want to maximize

individually

so following our individual interest we

choose to defect but that hurts each

other because one may defect we impose a

payoff of negative two on one another

right so here player 1 is defecting

player 2 gets a negative 2 here player 2

is defecting player 1 gets a negative 2

so following our individual interest we

hurt each other if we both cooperated

we'd supply each other with this

positive 2 utility going against that

negative one selfish utility but it'd be

better right if we look at the full game

we'd be better off both cooperating them

both defecting so the prisoner's dilemma

exemplifies this tension between

cooperating and defecting between what

does our selfish what does the

individual level pull us towards what

does it want us to do and what does the

group level want us to do right so it's

better if we cooperate but individualism

sort of unravels it and we ends up and

we end up defecting

now let me just go back so we talked

about in the previous units we talked

about tit for tat and so how does

Tiferet tat make corporation work so

Tiferet at actually changes the game one

way it changes the game as the game

becomes repeated and another way is that

the strategies change so if you play tit

for tat you're not simply playing

cooperate or defect right you're playing

this it's almost like this higher level

strategy in this repeated game and so

what this ultimately ends up doing is it

brings the externalities into the

selfish information and it makes

cooperating attractive to someone

selfish that's the only way right we're

assuming the individuals are selfish so

at the end of the day no matter how many

hoops we jump through the agents are

gonna behave selfishly so the only way

for them to cooperate at least with

these assumptions is to make cooperating

simple is to make cooperation something

a selfish agent would play so I can

barely get into that the stuff I think

is super cool and I should I suggest you

check out that cited paper they talked

about tit for tat and they show with

more detail how how things change how

things move around how some of the

externalities end up affecting a selfish

level choice which is how tit for tat

and the repeated game makes cooperation

possible so let me lastly tell you the

quote unquote rules to create the

components of these 2x2 games so you can

play around and in fact in fact I'm

gonna make an additional homework which

is going to be optional for you to

explore all of this a little bit more so

in it I'll ask you to follow these rules

and create different kinds of 2x2 games

where cooperation is aligned with

selfish incentives or it's not like we

just saw

so on all kinds of interesting examples

for you to play around with and explore

so let me tell you these rules and these

quote-unquote rules are actually

mathematical symmetries that make this

whole thing work but I'm not going to

get into the details of it but so with

The Selfish component every pair of

alternative individual choices must be

negatives of each other sorry I forgot

some words here it must be negatives of

each other anyway so here we see I put

an example player 1 chooses strategy a

and player 2 choose a strategy a so

player 1 gets a payoff of a if they

switched they'd get a payoff of negative

a and so we can see this idea repeats

throughout this selfish info component

the externality component on the other

hand remember that there's no individual

power so here if player 2 is choosing a

your player one you're just getting an M

no matter what the only way you can get

something different is if the other

agent chooses changes their strategy so

here when the other agent changes their

strategy you get the negative of what

you were getting right so player 2 by

choosing a imposes M a utility of M and

then if they changed if they played B

they impose negative N and so just to be

clear M could be three so this would be

three three negative three negative

three

but M could be negative three so then

over here we'd have negative three

negative three and then over here we'd

have negative negative 3 which would be

pot so we'd have positive 3 positive 3

and then finally the kernel was the same

value for all the utilities of each

agent so agent one would have the common

value J whatever J is an agent 2 would

have whatever value K is throughout and

so here it is here are the rules

and thanks for watching everybody

I hope this made sense if anything send

me an email post questions on the forums

I'll be happy to explain any of this a

little more and yeah thanks for watching

have a great day bye

you