# 5 Mathematical Reasons Why Things Suck

For those who already know math, have fun pointing out what I got wrong in the comments section.

Most movies would have us believe that shitty situations arise because some malevolent force comes into town, like an evil wizard or anyone played by Christoph Waltz. In reality, suck is caused by deeper, less exciting forces you can't solve by smashing an amulet, shooting some Nazis, or blowing up a mainframe. There are some situations where reasonable people, all behaving reasonably to attain their reasonable goals, end up creating an outcome that is shitty for all of them.

Examples of this aren't hard to find, and they're such glorious clusterfucks of rational self-interest that they should be the preface to every copy of The Fountainhead. And when you use math to explain them, you realize that, given the right conditions, they are as inevitable as your parents turning your old bedroom into a sex dungeon.

"You don't even want to know what we've done with Mr. Cuddles."

First, consider a little toy example: those loyalty cards for cafes. You bring your card in to get punched every time you get a coffee, and the 10th coffee is free.

For the first coffee shop that did this, it was a great idea. For a relatively small cost (1/10th of a free cup of coffee and however much printing the cards cost), you would get a little bump in customer loyalty. Basically any bump in customer loyalty would justify that small cost, so this was the kind of brilliant plan that could make you Harvard Business School's "Dickwad of the Year," or whatever their award is.

At Yale, the title is "Douche Regent."

But, once one place started doing it, their competitors had to as well. If any of them doesn't print these loyalty cards, they'd be giving an advantage to the other coffee shops (which we know is the only business more cutthroat than dealing smack in Baltimore).

That leads us to our present situation, in which every single fucking cafe has these loyalty cards. Of course, since every shop has them, none actually inspire any kind of loyalty, because "loyal to all" is exactly equivalent to "loyal to none." So they're more like little frustration cards for people who like carrying around garbage. Because after every business is on the same dumb punch card system, we're back to square one, except that everyone has to go through the hassle of printing and keeping track of these stupid little punch cards.

The only winning move is to not play.

Can't the cafes just all agree that was a dumb idea, get rid of the cards altogether, and all be in a better position? The answer is "not as long as the universe functions as we understand it" and the reason is sub-optimal Nash equilibria. In order to explain that, I'm going to have to teach you some math. For that, I deeply apologize. For those who already know math, have fun pointing out what I got wrong in the comments section! I guarantee it won't be the take-home message that our lives are unavoidably terrible!

## 5 Game Theory

We've written about game theory before -- specifically, it's usefulness in detecting complete assholes. But we're going to talk about it's more common use: mathematically modeling situations in which the decisions of two parties jointly determines the outcome. An example of this is you and a friend deciding where to go eat together. Both of your actions have an impact on the outcome, so that's an interaction you could model with game theory, if you love modeling things and never want to eat with anyone ever again.

The main thing you study in game theory is called a payoff. Like so many things in math, it's much simpler than it sounds: It's just what result you'd get from each pair of decisions (one from you, one from the other person), and how good or bad that outcome would be for each of you.

"For instance, I choose to send my right fist rocketing toward your face, instead of my left."

So suppose you and your friend are deciding where each of you will have dinner. You want to eat together, but you want different cuisines. You really want Italian, while she wants English food (it's terrible, but she only eats cuisine from countries we have defeated in war). You could a) go your separate ways and eat what you want, but you wouldn't be together, b) one of you could cave and suck it up like a civilized human, or c) you could split up and each go where you don't want to go as some kind of sadistic punishment to yourselves. You might represent your payoffs like this:

You eat Italian, she eats Italian-- +5 for you, +2 for her

You eat Italian, she eats English -- +1 for you, +1 for her

You eat English , she eats Italian -- +0 for you, +0 for her

You eat English, she eats English -- +2 for you, +5 for her

An easier way to lay that information out is in a payoff matrix. Don't go to sleep because I made you read the word "matrix!" It's just a way to lay out the exact same information above in a way that will be useful later:

Quit clawing at the back of your head, it's not that kind of matrix.

The upper-left box is both of you going for Italian, which means you get a full five points for eating where you want and also bullying your friend into submitting to your will, while she gets two for the pleasure of your company (ha!). The lower-right box is the reverse situation, the upper-right is each of you eating separately where you want and the lower-left is each of you being miserable in some kind of lame Gift Of The Magi scenario.

## 4 Nash Equilibria

Now is when you get to understand Nash equilibrium in a deeper way than "when that guy from A Beautiful Mind took his meds." Notice that you don't actually get to control which box you end up in. You can only control which row you end up in. Your friend's decision determines which column you end up in.

You'd be surprised how often "best for them" and "screws you the most" are the same decision.

A Nash equilibrium is an outcome wherein you couldn't have done better if you'd picked the other decision, and the same is true of the other player. So in the example above, there is a Nash equilibrium where you both go for Italian. Given that she's going for Italian, you're glad you aren't choking down bangers and mash alone. And given that you went for Italian, she's glad she didn't go for the Yorkshire pudding served with a side of spite. (In this scenario, I've made you care more about eating together than what you eat. Which may be unrealistic, given that you're reading this article rather than being around humans.) Similarly, there is a Nash equilibrium where you both go for English "food" for the exact same reasons: Neither of you has a reason to change your move if the other person hasn't changed theirs.

So here we saw two Nash equilibria at the "best" two outcomes -- the ones where both players were happiest. Are Nash equilibria always these warm, fuzzy, best options? You may suspect not if you've read the title of this piece, have existed a day in this goddamn world, or are familiar with ...

## 3 Credit Card Cashback

Most credit cards these days give you a little perk for using them instead of cash: about 1 percent cash back. So even though using your card isn't necessarily more convenient, you might tip the scales slightly toward using plastic anyhow because you're incentivized to do so, and also it makes you look like a grownup. Plus, you get to play with the buttons on the little machine thingy, which is fun.

But the Konami code does not give you unlimited money, and the people at Walmart get really pissed when you try.

Of course, merchants would prefer that you don't use a credit card, because they get charged a fee. The credit card companies are very, very good at making money, so we can safely assume that in order to give you 1 percent cash back, they're charging over 1 percent more to the merchants than they'd have to otherwise. Say, for argument's sake, that because of this feature, they are charging merchants an extra 3 percent on average, just to make the numbers nice. The merchants, of course, are also businesses. They can't shoulder an extra 3 percent fee on their own, so they have to pass on at least some of the cost to their customers. That means you. Say after Visa hikes their rates 3 percent, they charge you an extra 2 percent, and they eat the other 1 percent, because capitalism is a bitch.

Now you are paying an extra 2 percent at the register, and the merchant is paying an extra 1 percent. All so that you can get a measly 1 percent cashback and your credit card company can make commercials telling more people about their 1 percent cashback policy. It's like a handjob: inefficient, frustrating, and it would have left everyone happier if there were fewer people involved. But we're never getting out of it, because it's a Nash equilibrium and The Man isn't exactly known for his empathy.

Which explains why handjobs aren't illegal in the first place.

Why is it a Nash equilibrium, the thing that gave us the warm fuzzies above? Consider a model of this situation where the players are consumers as a whole and merchants as a whole. The merchants' possible moves are to ban credit cards or to accept them (and increase prices accordingly). The consumers' possible moves are to pay exclusively with cash (boycotting credit cards) or to use credit cards whenever they want. Here's a payoff matrix:

Also known as "Fucked if you do, fucked if you don't"

If consumers use cash and the merchants are priced for cash (they haven't bumped up prices because of credit card fees), then everything is as expected -- no loss, no gain for either merchants or consumers. If the merchant is priced for credit cards and the customer pays with a credit card, they both net a 1 percent loss, as described above (but cool commercials!). But if customers can get pre-fee prices while paying with a credit card, they net themselves a cool 1 percent, while the merchant picks up the 3 percent fee all on their own. Finally, paying cash to a merchant who is priced for credit cards leaves the customer down 2 percent because of the rate hike, just leaving money on the table.

So the only equilibrium state is both sides using credit cards. Note that this is worse for both players (but better for the credit card companies) than if they'd just never started this 1 percent cashback ordeal at all, or everyone just decided to make their own products from hemp. I heard you can make anything out of hemp. Maybe even cool stuff like commercials for credit cards!

"That's how Benjamin Franklin made his credit card commercials, man."

It's important to note that nobody is evil here. The credit card companies are charging what it makes sense for them to charge, the store is doing the only rational thing for them, and you giving up that sweet, sweet 1 percent isn't going to change anything for the better, so you might as well take it. Yet it adds up to less than the sum of its parts. You and the store optimizing for yourselves leads you away from the best solution for both of you.

I know we're only talking about small percentages here, but we're talking about small percentages of ludicrously large sums of money (In 2014, just Visa in just the U.S. had a credit purchase volume of \$1.2 trillion). That's money that could possibly be spent on Guns4Toys or whatever the charity of your choice is (toys are much cheaper than guns, so they really need your help, guys). Or if the loss of huge amounts of money doesn't strike you as important, how about ...

## 2 The Death Of Investigative Journalism

What are the incentives of newspapers to staff investigative reporters and actually sink their teeth into original stories versus ripping off stories that have been investigated elsewhere? Sure, you get a little bump for being the first news outlet to report a story, but it very well may not be enough to cover the expense of investigative reporting and making sure the story wasn't just some bullshit an internet troll made up.

If The Times goes to all the trouble of investigating a piece and The Post can reap nearly all the benefit with none of the cost by just copying their story, it's not hard to see what they're going to do and when Rupert Murdoch will buy them out.

Guess which one means he'll hack your phone? Hint: It's all of them.

So we end up with everyone printing the same ripped-off articles and settling into that behavior, unable to do something better without losing out to their competitor. Of course, it would be better for us as readers if the cost/benefit analysis worked out to favor investigation. But what's stable and what's "best" are two different things. Which is how we get modern-day CNN.

Now that you understand the basic concept, you'll see these Nash equilibria everywhere. You can use it to model all kinds of shitty situations where, despite everyone rationally pursuing their non-shitty interests, the way the payoffs are set up leads everyone into mutually shitty outcomes. It explains nuclear stockpiling just as well as it explains why smart people who love movies will keep pirating them, even though that leads studios to stop making movies for cinephiles and instead make movies for people who are too dumb to figure out how to pirate them.

"I AM THE INEVITABLE CONCLUSION OF YOUR FLAWED HUMAN BRAINS!"

But of course, now that we know this is happening, we can change it, right?

## 1 Nope

The thing about Nash equilibria is that they're pretty darn stable by definition. Unless the model you're using has left out some really important factors, there isn't any reason to think things will change anytime soon. They do make a couple of assumptions about the players, however.

"The First Law clearly states that players shall only hate the game, unless hated upon by another player."

First, they assume that the players aren't cooperating. Despite what most of human history has to say about it, people sometimes do cooperate for a mutually beneficial outcome. This happens with things like weapon bans. A large number of countries got together to say, "Chemical weapons aren't just bad; they're as terrible as Forrest Gump. So none of us are ever going to use them, because history cannot bear another Forrest Gump." (You know the Pentagon has locked away somewhere a copy of Forrest Gump 2: Lil' Gump, Big City, just in case.)

Don't take too much hope from that, though, because just a few non-cooperative individuals can throw you right back into a shitty Nash equilibrium. If two countries were fighting and one started using chemical weapons, turning the tide of the war, how long do you think it would take the other side to serve everyone on the battlefield a big helping of mustard gas?

"We're only using it because they did! It's just a coincidence that we had it here, ready to go!"

Second, they assume that all the players are rational actors pursuing their own interests. While people certainly pursue their interests, you know that rational actors are few and far between if you've ever seen how excited people get to meet celebrities they don't even like. So irrationality might take a chunk out of some of this thinking, but that thought somehow isn't particularly comforting either.

Barring those caveats, the above logic holds as long as 1+1=2. We're surrounded by systems which drive participants -- making the "right" move every step of the way -- toward outcomes that suck for all of them.

"Told ya."

"The game, that's what happened here today." -- The Wire

For more, check out 5 Scientific Reasons Powerful People Will Always Suck and 5 Hollywood Secrets That Explain Why So Many Movies Suck.

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