An Important First
He’s an often quoted icon of American sports, an Italian American great, known for his quirky, folksy, and seemingly ignorant yet poignant wit. He has appeared in more World Series games than any other player, was a fifteen time all-star, and voted American League MVP three times. He was a leader, too, not just a player, managing seven teams into the World Series, and winning it three times.
Who could have known? The sports writers clearly didn’t know. The fans didn’t know, or the other players. He himself didn’t even know it, at least not consciously, and he probably still doesn’t. But Lawrence Peter “Yogi” Berra may have been the first and most important climatoleconomist, ever.
The future ain’t what it used to be. — Yogi Berra
People love games. Sports, video games, board games, card games, parlor games, and gambling with small or large stakes. Baseball itself is known as The Great American Pastime. It would be hard to find someone that didn’t enjoy a game of some sort.
If people don’t want to come out to the ball park, nobody’s gonna stop ’em. — Yogi Berra
Okay, people also like to win at games. But frustration with losing teams aside, people love games.
Which is good, because we learn a lot from games. We learn how to compete, how to manage our resources and our future, and we learn better how to survive.
Sociologists, economists, political scientists and even philosophers make use of a fascinating branch of applied mathematics called Game Theory. Game theory attempts to structure the parameters and outcomes of competitive and even cooperative strategic situations – a complicated way of saying “games” – with the goal of studying the decision making processes of the players, and if possible defining the best strategies for each player.
A simple grid, a payoff matrix, is designed with a box for each possible outcome. In a two player game, one player’s choices are listed in the columns, while the opposing player’s choices are listed in the rows. Numbers are placed in each intersecting cell to represent the value of the outcome, as in the meaningless and random example below:
|Player 2||Action A||Action B|
Let’s examine a simple real world example to demonstrate more clearly – baseball, or rather, one specific part of a baseball game.
On the off chance that you’re not familiar with baseball, both pitching and batting are something of a guessing game. Visualize the duel between a baseball pitcher and the batter. The pitcher wants to select the best pitch to get an out, or at least a strike, while the batter wants to guess the pitch that is coming in order to improve his chances of getting a hit.
A curveball comes in more slowly than a fastball, and it actually moves from one side to the other, seemingly approaching from one direction, then veering away in another. If a batter isn’t ready for it, and swings, he will either miss or just might make contact, but any hit will be weak and off balance, probably resulting in an out. But if the batter is ready for the pitch, because it’s coming in more slowly than a fastball, then he can get a hit, but because of the curve in the path of the ball, he’s not likely to get a really good hit. A single (a chance to take one base) isn’t out of the question, though.
On the other hand, if the batter is looking for a curveball and a fastball whizzes by, he’s pretty much going to swing late or not at all, and take a strike. If he’s instead looking for the fastball, which comes in straight and hard, and if he can connect, then he has a chance of hitting it equally hard in return and getting more than a single out of it.
We’ll assume for our little game, for simplicity, that the hitter’s skill is very good, almost infallible, and that the pitcher always throws strikes (i.e. balls that the batter must swing at to avoid a strike, and that the batter might hit if he tries).
In game theory we could represent this pitcher-batter-baseball guessing game like this:
|Batter looks for…||Curve||Fastball|
Except that this is mathematics, so we translate things into numbers, like this:
|Batter looks for…||Curve||Fastball|
Here, a 1 is a single, a 2 is a double, a -1 is a strike, and a -3 is an out (the equivalent of 3 strikes). Positive numbers are good for the batter, while negative numbers are good for the pitcher.
There’s only one number for each result because this is what is known as a zero-sum game, meaning that what is good for one player is bad in exactly equal measure for the other player. Most games that are played for fun, and most competitive games, are zero-sum games. For instance, in chess, the loss of any piece is exactly as bad for the player losing the piece as it is good for his opponent. In betting, a winner wins exactly as much money as the loser wagered (unless a bookie is involved, but then that’s not a simple game anymore, it becomes economics, which is very interesting but is not our topic – yet).
For simplicity, we’ll avoid probabilities and the chance that different things may happen different times, like the chance that the batter will squeak a hit out of an unexpected curveball, or get a triple or fly out instead of a double when connecting with a fastball. We’re interested in the decision making process, so in our game there’s only one, definite outcome for any combination of pitcher/batter choices.
Partly because it is a zero sum game, and because each choice has good and bad returns, there is no optimal strategy for this particular game, and no reason or tendency for the players to keep making the same choices (known as a Nash Equilibrium) in repeated play.
Play the game against yourself a few times, by playing both sides, or using the roll of a die or a coin toss to decide the actions of the other player. Once you have a feel for it, read on.
Studying the payoff matrix, you can see that a conservative pitcher could choose to throw more curve balls, figuring that his risk is lower (giving up singles instead of doubles when the batter guesses correctly) and his return is higher (getting an out instead of a mere strike when he guesses correctly). But a batter is likely to take the same approach, risking only strikes instead of an entire out by looking for curve balls. In this event, the batter would get repeated hits, so the pitcher can’t afford to continuously, thoughtlessly throw curve balls. Once the pitcher starts to throw some fastballs, the batter can’t simply keep guessing curve, and the pitcher likewise can’t afford to continuously throw fastballs (or the batter will adjust, and hit a lot of doubles, scoring runs).
Fun Games to Play
Of course, the real fun for the mathematician (or sociologist, or climatoleconomist) comes from considering unusual games, and using game theory to determine what the best strategy would be in perplexing situations.
We made too many wrong mistakes. — Yogi Berra
The most common introductory example used in game theory is known as the Prisoner’s Dilemma. The idea is that two criminals have been arrested, separated and interrogated. Each of them is being made the same offer; rat out the other guy, and you’ll get a reduced sentence. Of course, without a confession, there’s not much evidence, so if both stay quiet, they’ll only do 1 year in prison. If they both confess, all deals are off and they’ll get 5 years each. But if one squeals on the other, while his buddy stays loyal, then the rat goes scott free while his buddy gets 10 years.
Here’s the payoff matrix:
In this matrix there are two payoff numbers. The first is the number of years that the first prisoner gets, the second is the number of years that the second prisoner gets. This is not a zero-sum game, because what happens to one player is really irrelevant to the other. They can each win or lose in different amounts. The goal for each player is to minimize his own sentence*.
*After all, there is no honor among thieves, lawyers, and politicians, right?
Again, use a coin or dice to play it yourself.
This game has a surprising, dominant strategy, which is to squeal, even though both players would be far better off keeping silent. Since one player can’t be sure what the other player is going to do, it would always be best to squeal. If the other guy squeals and you don’t, you’d do ten years. But if you squeal and the other guy squeals, you’d only get 5 years, and if he’s stupid enough not to squeal, you’ll go free. Your own result is always better if you squeal.
So squealing is always without question the best strategy. Assuming both players have thought this out, they can expect to spend five years in prison, and if they are smart they’ll use the time to brush up on their game theory.
We’ll look at two other games of note, before we begin to apply game theory to current events.
In theory, there is no difference between theory and practice. In practice, there is. — Yogi Berra
The first is called Global Thermonuclear War. This one covers the Cold War between the U.S.A. and the U.S.S.R. Each country had thousands of nuclear armed missiles aimed at the other. If one were to launch, and the other didn’t, the reluctant nation would be annihilated and the other would rule the world. But if both launch, the world would be destroyed. Finally, if neither launches, there is peace, in which no one wins a war, but no one loses.
For the sake of this game, it is assumed that if one side gets in a first strike, the other will not have time to launch, so the decision must be made before you are entirely certain as to whether the other side has launched their missiles or not. Imagine that it’s 3 AM, when General Jack Ripper calls. He says it’s too early to tell for sure, but he thinks the commies have fired their missiles in a massive strike. Does he have permission to fire back, and blow the commies all to hell?
Here is the payoff matrix, where the first number is the payoff for the U.S.A. and the second number is the payoff for the U.S.S.R.
If neither side launches and peace is maintained, the status quo remains and neither side gets any points (0,0). If one side launches and the other doesn’t, the other is destroyed (-1) while the winner is left in control (+1). But if both sides launch, in the combined nuclear holocaust the entire world is destroyed (represented by a -∞ for both players).
This is not a zero-sum game, because while in the event of war one side wins and the other loses in equal measure, and in the event of peace nothing changes, but in the event of Armageddon, both sides lose everything.
In this game, there is no obviously dominant strategy. The question becomes one of how much you are willing to risk. You could argue that the only way to win is to launch, but you’d be risking everything.
Consider your strategy in a scenario where you get to play this game repeatedly. Imagine that even if you lose one nuclear war, over time you rebuild your country and so get to play again. Then your average return is better if you do not launch (-0.5 per play, instead of -∞). You’d have to play forever to be as badly off as if you both launched just once, so seemingly the best plan, if you go with best average return, is to wait. If both sides use this strategy, then the best possible outcome for both sides is achieved and peace is continually maintained.
Interestingly, there is a Nash Equilibrium in both sides launching. That is, if you could somehow play the game repeatedly, once one player decides to launch, the other player should anticipate that the player will launch again, and should decide to launch as well. Once both players are launching, it would be a mistake to change strategies. The best you could do would be to break even, but only if the other player decided not to launch at the same time. Of course, as the game is structured now, once both players launch at the same time, the game ends, but this is all just theory, and the point is to think about thinking.
We could modify the game somewhat.
If you instead declare that any nuclear war is final, and your concern is solely for your own country, one could argue that the payoff (loss) in losing a nuclear war is also -∞. That is, there’s no difference between losing a nuclear war and the end of the world if you don’t care about the entire world. In that event, the payoff for waiting while the other country launches is not merely -1 but -∞.
That payoff matrix looks like this:
This would change your approach to the game in that your fate is entirely in your opponent’s hands. You will lose everything if he launches, no matter what you do. Given that fact, you might as well launch. If he doesn’t launch, he loses and you win. If he does launch, you both lose. This is much like the prisoner’s dilemma, where you might as well do the “bad” thing, because you can’t trust your opponent to cooperate.
Of course, in real life this game never played out any way but peacefully, because world leaders never saw, understood and used game theory, and this payoff matrix in particular. World leaders are always smart enough to take easier courses in their college days than applied mathematics, or if they did, they cut class a lot*.
* Actually, it’s because of a lot of reasons, including the fact that after you’ve played the game of Global Thermonuclear Warfare once, you don’t get to go for best two out of three. Whatever the ultimate reason, we’re thankfully here to talk about it today. Humanity successfully dodged the first deadly bullet that it aimed at itself after inventing the nuclear bomb, and then decided it was such a good idea that a lot of different, competing countries should have a lot of nuclear bombs, too, all at once.