Teoría de juegos: Pareto y Nash - YouTube

In this video we're gonna see a last batch of interesting concepts in game theory,

starting with the Pareto efficiency.

The Pareto efficiency is a concept that refers to the bettering of a variable

that doesn't worsen any of the others.

Let's imagine a simplified version of the Mario Kart racing game, in which each character

has two variables: speed and weight.

Speed is how fast you drive and weight is how much strength it takes to accelerate

so the more speed and weight, the better.

In this version of the game we have 4 characters:

Mario, who is the most well-rounded and has 5 points in speed and 5 in weight.

Bowser, who is heavier but also the slowest, with 6 points in weight and 4 in speed.

Peach, who is the opposite and thus lighter and faster, with 6 points in speed and

4 in weight.

And lastly Luigi, who is like Mario but minus one point in weight.

Out of these characters, Mario, Bowser, and Peach are what we would call Pareto optimal because

you can't switch to another character increasing a variable without diminishing the other, the

only thing you can do is trade speed for weight.

All of them being Pareto optimal they also form the Pareto frontier, because everything

that goes over the frontier we know is automatically suboptimal.

For example we have Luigi who is dominated by Mario because Mario is exactly like him

except with a higher weight, so from a competitive point of view there's no

reason to pick Luigi.

Now let's go onto Nash equilibrium.

Nash equilibrium is a situation that happens when none of the two players can

better their situation by changing their strategy if the rest keep the same strategy they've been

following, as such all players are "forced" to maintain their strategy

even if they know they're gonna lose.

Imagine a war game in which your opponent is attacking your castle with their army,

but you don't have an army, so you simply keep repairing your walls.

This setting would be a Nash equilibrium because the moment you stop repairing

your walls the enemy will conquer you over, and if your rival stops attacking

they'd give you an opportunity to build an army, so neither of you can switch up your

strategies without making your situation worse.

One of the classic games that is used to study cooperation strategies

and that has an interesting Nash equilibrium is the prisoner dilemma.

The situation is as follows: Two prisoners are caught and put into separate cells

to be interrogated.

The prisoners have two options: One is to collaborate with their partner and not

say anything to the police, with which both would land a year in prison.

The other option is to betray your partner, with which this prisoner would go free

and their partner would land 3 years in prison.

But if both betray each other, then they get 2 years in prison each.

This situation is interesting because the optimal strategy is to betray your partner

since it's the only option in which you'd go free and it also stop you from being betrayed.

Nonetheless, your partner will reach the same conclusion and if both of you betray each other

the end result is worse than if you'd have worked it together.

Furthermore, the dynamic of the game changes if you play one round or multiple rounds,

in which case we could switch the strategy up depending on what our partner picked

on the previous round.

In this game, Nash equilibrium is obtained if both of the prisoners betray each other because,

at that point, none of them can better their situation on their own, because if

only they switch their strategy, their situation worsens.

As such, both will keep betraying each other each round even though collaborating

would be a Pareto optimal option for both.

It's interesting because even when acting rationally they're both getting the worst result.

This situation might seem too theoretical, but in reality you see it constantly in

non-cooperative games with more than 2 players.

Imagine, for example, that we play a fighting game with 3 players.

If you and I cooperate and we attack the third player, we take a rival out of the

game and increase our chances of winning from 33% to 50%, so it's logical.

But once you've started cooperating your partner will lower their

guard towards you so it's the perfect moment to stab them in the back

and take them out of the game while taking the least damage.

But of course, if we both do this, we're gonna end up fighting among us, with which

we're dooming each other because the third player is just gonna stand there waiting for us to

finish each other to then come and finish the survivor off.

Then what is the best strategy for the prisoner dilemma? Well, it depends on

if we know how many rounds we're gonna play or not.

If we know how many rounds we're gonna play, the ideal thing is to collaborate in all of them

and then betray each other on the last round, because your your partner won't be able to take revenge since there's no more rounds.

But then, your partner will think the same, so they too will betray you in

the last round, and as such, you should betray them in the second to last round.

But, again, your partner will also think the same and then we follow this

logic until we reach the conclusion that we have to always betray them

from the beginning. This is the rational strategy.

However, there also exists the concept of superrationality which is obtained when

you're perfectly rational, your partner is perfectly rational and both of you

know each other to be perfectly rational and assume that you'll reach the same conclusion,

and therefore you pick the same option.

The superrational strategy then is to always collaborate, given that both of you will

reach the same conclusion and thus will both pick the same, there is no possibility

of betrayal. Either both of you betray each other, or both work together, and

since collaborating is the best result, you will choose to collaborate.

This changes when we play a lot of rounds but we don't know exactly how many, which we see

constantly in politics between countries, for example.

In this case, the strategy of an eye for an eye is the best, which consists of

initial cooperation until your partner betrays you, then you betray them too and don't switch

your strategy until your partner chooses to collaborate again, showing good faith.

That way you're letting the message across that taking advantage of you is not a good option

in the long run, so, if your partner isn't dumb, they'll have to realize

that collaborating is always the better option, because if they betray you, they'll end up

screwing both over, not just you.

Same thing happens with day to day friendship and in many other situations,

so think twice before betraying a friend to win just the one round if you wish to

keep collaborating with them in the future.