5 Easy Ways to Understand Game Theory (for Jerks)
Game theory is the study of how to choose the best move and to get the most stuff when other players are trying to take it. They should teach it after counting but before the alphabet. Too many people make decisions based on gut instinct, invisible sorcerers, Jenny McCarthy, and other things that mean "Hey, even I don't understand why I'm such a stupid asshole." I've studied game theory, because I will do anything that makes my resume look more like Captain N's, and because I count choosing the craziest Magic: The Gathering cards as a career skill. Now my best move for making rent is telling you about the basics.
Berto and Robert have been arrested for robbing a bank and failing to correctly use a stolen getaway car. The police can't prove they robbed the bank, but found them in the car. (They failed pretty hard.) They're separated and offered a deal: rat out the other guy and go free while the other serves 10 years. But if they both rat each other out, they'll both do seven years. If no one says anything, they'll both do two for stealing the car.
And for using GTA as a driving instructor.
Each prisoner is a player, and their rewards can be written as (what they get, what the other person gets). For example, if I kick you in the crotch, my reward matrix is (I get slapstick victory, you get awful crotch pain). Because each prisoner has two choices, we can represent the results in a two-by-two grid.
So if Berto is silent but Robert rats him out, Berto serves 10 years while Robert serves none.
Real-World Application: Jerk Detection
This is where we see game theory's primary use: identifying total sociopaths. Real game theory is a powerful analytical tool, but amateur game theory is often a red flag planted in an asshole. People who value arithmetic over empathy say you're better as a rat because it leads to a shorter prison term no matter what the other player does. Technically true, if you're a short-sighted asshole who values numbers over human lives. Which is why game theory is so popular in finance. You couldn't be more of a jerk without a red shell.
The real problem with the prisoner's dilemma is that it ignores data. Like how you might not be a psychotic asshole who'd throw someone under the bus the moment a matrix told you to. Or how you'll go free only to be stabbed by your partner's angry friends, family, or some random thug he just promised your half of the money to because he won't see it for 10 years.
"We, uh, weren't very good at robbing the bank, either."
Worst of all, everyone in the prisoner's dilemma suffers from Zombie Movie Syndrome: they act like they've never heard of the prisoner's dilemma. The best move is to keep silent, then enjoy the money with a much closer friend after two years of anticipating the sweetest daiquiris of all time.
The daiquiri dilemma is "Should I have another daiquiri?" versus "Why don't I already have another daiquiri?"
Strictly Dominant Strategy
A strictly dominant strategy isn't something that costs extra at Mister De Sade's Vigorous Exercise Emporium.
The last time we used a whip in game theory we killed Dracula.
It's a move that gives the best rewards no matter what your opponent does. No matter what happens, you did the right thing. That's why people like the prisoner's dilemma so much: betrayal leads to a "better" result no matter what the other person does. Well, that's one of the reasons. The other reason is that it's Chapter 1, Section 1 of every game theory textbook and ignores enough of reality to make things look super simple.
Most games we play don't have strictly dominant strategies, because they'd be terrible games. You'd only ever do that one thing. In Rock-Paper-Scissors, there is no dominant strategy. But if you were playing against someone wearing oven gloves, so they can only make rock or paper, now you have a strictly dominant strategy: paper. Your paper will beat their rock or draw with their paper, and you can't lose, because they can't make scissors. Now that you have a strictly dominant strategy, you'd be a fool to try anything else.
Be warned that "rolling pin" beats everything.
Real-World Application: Tolerance
Another example of strictly dominant strategies would be looking at the rewards of being homophobic versus not being a hateful asshole.
As you can see, it is strictly dominantly better to not be homophobic. Which is a phrasing we're sure the target audience will love.
Battle of the Sexes
Games are more interesting when they don't have a strictly dominant strategy. For example, the battle of the sexes. Anjali and Borislav are going on a date but can't decide between ballet and boxing. Anjali enjoys boxing because she's interested in commercial blood sacrifice, destroying the minds of athletes for baying crowds who claim to be civilized because they used a credit card to pay for someone else's brain damage.
Borislav wants to watch ballet, because he understands that ballet dancers go through more physical trauma, extreme training, and moments of knowing that a single physical injury could end everything than SAS squads infiltrating nuclear launch facilities. Ballet dancers are some of the ultimate athletes on Earth. A ballet dancer could kick your head in, but wouldn't, because your entire face isn't worth as much to the world as the curve of their instep.
He can kick your ass while showing you his.
They both prefer their own event but won't enjoy anything without their partner, so their rewards are: a large value for doing something they like, a small value for just being with the other person, and zero for being alone.
The only winning move is not to ... wait, wrong game.
Some people propose bull-headed brinkmanship: if you make it absolutely clear that you're doing your thing no matter what, the other person has to match your choice or lose everything. Like I said, simplified game theory is an excellent asshole detector.
Real-World Application: Avoid the Brink
Of course, that strategy has significant problems. The first is that if you refer to your dates as the "battle of the sexes," then jeez, it's not working. Break up so you can both go find someone you like. But the second problem is that in this scenario they're so desperately insecure that they can't do that. They can't enjoy an ultimate expression of the human body's ability, or the destruction of that ability, without their partner clinging on their arm. That kind of desperation is how you end up rationalizing a partner's sweet habit of skinning road-kill.
"The swearing as she flays their skulls is adorable!"
The real winning strategy is for each to do what they want and then meet for a drink afterward. Or the next day, when they're not busy. Or they take turns, alternating boxing and ballet, before revolutionizing the world of physical entertainment by inventing thunder-ballet-boxing.
"Thunder" appears automatically when a sport gets awesome enough.
The Nash equilibrium isn't a way of building nice bridges or a gritty Bourne-style reboot of the '90s police series that would make sense of that joke. The Nash equilibrium is a set of moves where no one wishes they'd done something differently after the fact. And if we can get that working, game theory will replace all philosophy, religion, and finance on the planet, because "wishing they hadn't screwed up" is a more powerful driving force for humanity than fire.
And sometimes simultaneous.
Quick, let's share $100. You and I decide how much of the hundred we claim and announce the amounts simultaneously. If our total is less than a hundred, we both get what we wanted. If the total is more than a hundred, whoever asked for the least gets what they wanted, and the greedier person gets whatever's left over. If we ask for the same amount we get $50 each. How much do you ask for? How do you divide the money? There is a single best move.
And a single worst move.
Asking for $51 gives you the maximum amount no matter what your opponent chooses. If they ask for more, you get $51. If they ask for $50 or $51, you get $50. And if they ask for less than $50, you get $51. In every case there is no other choice that definitely gives you more money. The Nash equilibrium is where we both choose $51.
Real-World Application: Think First
This is the whole point of game theory. Not necessarily to win, and definitely not to screw over other players, but to make the best move for yourself no matter what the world throws at you. And if that's also the best move for other players, that's even better. This is the kind of math that could rewrite a society.
An interesting variant on this idea is drinking, which is the time-dependent Nash equilibrium. When you're drinking properly you don't care what other people's moves are no matter what they do, but the day after, you absolutely wish you'd done something differently.
"I wish I'd bought a toilet that had Netflix on the inside rim."
In matching pennies, Player 1 and Player 2 each has a coin, but they don't have names, because this game is so staggeringly boring they'd fall unconscious before they could ever use them. Each player simultaneously chooses heads or tails. If they match, P1 gets P2's penny. If they don't, P2 gets P1's. It's a lot less boring if you imagine it as a Borg orgy.
Oooh, hot stuff.
The reward matrix is simple ...
... as is the optimum strategy: play completely at random. Which is harder than you'd think, because it has to be absolutely random. If you have the least preference for heads or tails your opponent can use that to take your money.
Of course, the real problem here is that they'd be better off whipping one penny back and forth at each other's faces. Which would result in the same net profits, and the physical trauma might help people so desperately bored that they're playing this game to feel something again. Or eventually put them out of their misery. Because this is the worst game ever. And it's a perfect model for penalty shootouts.
BUT I'M TERRIFIED OF DISCS!
Real-World Application: Penalties
In soccer, hockey, and many other games, the tie-breaker is a penalty shootout. And they'd be better off basing the result on how many cartwheels the players can do in full uniform, because that would at least be based on physical ability and be fun to watch. Goalies can't actually react after seeing the ball/puck/sportsthing start to move, because our sports are still sadly free of robotic warriors. They have to guess left or right and hope they match the opponent taking the shot. Which turns it into the matching pennies game. Which sucks.
Note that it's not a perfect game of matching pennies, as the goalie might not save it even if they dive the right way, or the shooter might not score even if they guess right. Which means it's not "not just" matching pennies -- it's not even matching pennies. Both players must be at the perfect pinnacle of their abilities just to achieve pointlessly random results.
So, our game theory conclusion? Ball games should end with a "multiball" mode, where an extra ball/puck is released every minute until somebody scores. Which would be based on actual game skill, instead of what's effectively a random coin toss, and would turn the world's most pointless stress test into fun.
After all, game theory should be used to make games smarter. And therefore better.