 |
Welcome to Game Theory.
Where everything's made up and the points don't matter |
Unfortunately game theory isn't quite as entertaining as Whose Line, but it is an interesting way to model human behaviour. I took an entire advanced level economics course in my 4th year at University on this stuff, so it must have some value, right? (right!?) And yes, pretty much everything in game theory is made up.
The best way to get your feet all wet with Game Theory is to become familiar with the simplest game: The Prisoner's Dilemma. You can skip ahead if you already know it, but you'd be missing out on my incredible rendition. It's a little story about two guys who rob a bank, and unfortunately they get caught.
The Prisoner's Dilemma
They had a perfect plan, Slim would stuff the money in the bag while Brute kept everyone down with a loaded gun. Once they had the cash, Greasy would pull up outside to drive them to a getaway boat. But there was one problem - Greasy had a different plan.
"That wretched greaseball!!!" Slim shouted as they dragged him away in handcuffs. Brute stayed quiet as he was loaded into the cruiser.
Greasy had tipped the cops off so he could take all of the money for himself, and he was nowhere to be found.
Authorities put Slim and Brute into 2 separate rooms.
"Alright dirtbag," the cop pushes Slim into the chair. "I want to know who's really responsible for this crime."
The cops give both of them the same deal. If one of them rats the other out while the other stays quiet, the rat will go free and the other sucker will get 10 years in jail.
If they both rat each other out, they both spend 7 years behind bars.
If they both stay quiet, both of them are locked up for just 3 years each.
The game, or
payoff table as they like to call it, is modelled as follows:
It's laid out like this for analytical purposes. Here's how game theory works:
1. Take the opponent's move as a given, and then ask yourself what you should play given that move?
If Brute stays quiet, Slim's optimal play is to defect. If Brute defects, Slim's optimal play is to defect. Those choices are highlighted in red.
2. Let's do the same thing for Brute, highlighting his best plays in blue:
The so called "Nash Equilibrium" is the panel where all plays have their move highlighted, it's that simple for this game! In this case, the Nash Equilibrium is to rat each other out. This places them both in the slammer for 7 years. Fools!
If only they had cooperated!
In aggregate, we actually ended up with the worst outcome, a total of 14 years in prison between the two, with the other strategies totalling 10 years and 6 years.
Let's get some closure on our robbers:
Slim and Brute spend 7 years in jail. Both never speak to each other again since they know they narced on each other. Brute reforms himself, he goes off to trade school and learns how to fix cars. Slim reverts back to petty crimes; robbing liquor stores and stealing purses. Greasy gets hit by a Karma Tsunami while boating away with the cash, turning him into a relic of the ocean.
There's a fancy proof of the Nash Equilibrium for all games in general, with N = {i=1,i=2,...,i=n} players, S = {s1,s2,...,sn} strategies, and Ai outcomes - the simplest one starts with something like this:
I really want to break out the proof here but I think 99% of readers would blacklist this blog, so we can just trust the logic. Economics students waste a lot of time with this kind of thing.
Anyways, beyond all the math they throw at you in an academic setting, there's actually some cool implications. The standard that we learn in economics is price levels in different classes of markets. But there's more applications, here's a couple that I've thought of:
War
 |
Other's needs are getting in the way of our goals?
My serpent blood says Wipe Them Out. |
War will continue to be a constant of humanity given the payoffs. We could model a simple war table similar to the prisoner's dilemma as follows:
Nash Equilibrium says "This means War!" Despite it being one of the worst outcomes, the two pride-led nations end up fighting each other. If leaders would check their ego at the door and maybe think about the human race as a whole, we might actually be able to achieve world peace - the best outcome in aggregate. I won't hold my breath though.
There's actually a classic game theory model of the cold war as well, where the Nash Equilibrium tells us that both the USA and Russia will arm themselves to the maximum, despite the costs.
"From each side's point of view: Disarming whilst your opponent continues to arm would have led to military inferiority and possible annihilation. If both sides chose to arm, neither could afford to attack each other, but at the high cost of maintaining and developing a nuclear arsenal. If both sides chose to disarm, war would be avoided and there would be no costs. If your opponent disarmed while you continue to arm, then you achieve superiority.
Although the 'best' overall outcome is for both sides to disarm, the rational course for both sides is to arm. This is indeed what happened. Both sides poured enormous resources into military research and armament for the next thirty years."
Advertising
Advertising is a great example of game theory in action. Let's check a static 1 period model of advertising:
The top left cell is the Nash Equilibrium.
Of course it doesn't take into account time dynamics and opponent's previous moves (although you can do this with game theory), it's a simple model that demonstrates the outcome of advertising.
Good and Evil
Because we have N players (where N is the population of the world), I can't model this one on paper because I need N dimensions, so let's check it out in words.
If everyone was good, wholesome honest and respectful, one player could change his play to evil and stand to gain a gigantic payoff, since he would gain a monopolistic benefit to the rewards of being evil.
So what we're left with is, in aggregate, a balance of good and evil right at the margin where switching sides would have no personal effect in terms of raw self-benefit. This model doesn't take into account our inner morals and instincts to protect one another, but it's an interesting discussion.
I thought you could think of a similar model with lying. If everyone was telling the truth, one player could begin telling lies to gain a massive benefit. So more and more people end up telling lies until we reach an equilibrium of lies&honesty which is what exists in our world today.
So Game Theory is actually pretty cool. It helps us understand human psychology a little better and to predict the outcomes of Human Action. Cooperation usually has the best overall outcome, but players tend to make uncooperative moves in anticipation of others doing the same.
The points don't matter, but the implications do.
