Wanna play a game? It’s an easy game. All you have to do is decide whether or not to push a button. But either way, you must face the consequences.
At the beginning of this game, I give you $100 and a button. Imagine that 100 viewers in 100 rooms across the country are reading this, and that each has been given $100, just like you, and a button, just like you. In a moment, I’m going to ask you to decide whether to push your button. That’s the only decision that you’ll have to make, and in doing so, you’ll be deciding upon a strategy.
In every game, every player has a strategy. If you’re a rational player, you’re going to try to adopt the strategy that will maximize your expected payoff given what you know—or think you know—about the other players in the game.
In every game, every player has a strategy. If you’re a rational player, you’re going to try to adopt the strategy that will maximize your expected payoff given what you know—or think you know—about the other players in the game. But you don’t know enough yet to know whether to push. What does it do? Pushing this button has two effects: one that affects you and one that affects everybody else. Actually, if nobody’s actions affected anyone else, it wouldn’t be a game. Games are interactive.
When you push your button, the first thing it does is to take $2 away from every other player. You push your button, and just like that, everyone else is down to $98. You still have your $100. Sounds rather vicious on your part. But other people may press their buttons, too, you know, and every time they do, you lose $2 along with everybody else. If 60 other people pushed their buttons, you’re going to lose 2 × 60, or $120. Given that I gave you only $100 to start with, you’re going to end up $20 in debt. You’ll have to pay up. Except that there’s a way out of this for you.
I said that pressing your button has two effects, and the second one targets you. If other people press their buttons and cause you damage, pushing your button will cut that damage in half. A moment ago, I said that if 60 people push, you lose $120, but if 60 other people push and you do, too, then you only lose $60. You’re still $40 to the good. You’ve done $2 damage to everybody else, but you’ve saved yourself $60. Are you going to push?
You probably made some assumptions about this game, reasonable ones, as it turns out. You’ve assumed that everybody else’s button works the same way that yours does. It does because this game is symmetric; everyone is in the same boat. Also, you’ve probably assumed that everybody else has the same information that you do, that is, that the structure of the game is common knowledge to everyone. Actually, being common knowledge means quite a bit more than that. It’s not just that everybody knows the rules of the game; it’s that everybody knows that everybody knows the rules of the game, and everybody knows that everybody knows that everybody knows the rules of the game, and … you get the idea. And you’re right: Everyone knows the same information that you do.
I want you to think carefully now and decide what you’re going to do: push or not push.
Five Lines of Reasoning
A hundred people, I don’t know. Some of them are going to push. No matter what the other people do, I’m at least as well off pushing as not pushing.
You’re probably entertaining several different lines of thought right about now. One line of reasoning is this: We all know how the game works; it’s obvious. If nobody pushes the button, everybody gets $100. I might not even be concerned about being a nice person, but I don’t have to be. We can all get $100. I’d be crazy to push. That’s a good argument. A second line of argument, maybe even more compelling, is this: A hundred people, I don’t know. Some of them are going to push. No matter what the other people do, I’m at least as well off pushing as not pushing. If I don’t push, I could end up $100 in debt. If I push, at least I end up breaking even. Heck, I’m a good person, and if I’m thinking about pushing, I can imagine what the other people will do—I have to push in self-defense.
Here’s the third line: I’m not going to push. I’m not pushing because it’s the right thing to do in a moral sense. I could lose up to $100. I could go $100 in debt, but it’s worth it for the sake of my ethics.
Or you may decide: Eh, $100; it’s not that much money. It would be too much fun to just stir things up and see what happens. Push the button. Or you may have a competitive streak, and you know that if you don’t push, everybody who does will end up ahead of you. Maybe you don’t have much of a taste for being a chump. Of these five lines of reasoning, it’s interesting to know which, if any, actually are rational. That’s a question that we’ll be visiting over and over as this course goes on.
Okay, it’s time to decide. I really wish that I could tally the votes that are coming in in real time, but of course, I can’t. What I can and will do is tell you, after you make your choice, the results of similar games that I’ve played with other people. Make your choice and, please, state it out loud; keep yourself honest. Push or don’t push. Three, two, one; done.
With groups of strangers who have no training in game theory, generally somewhere between 30% and 70% of the people push the button. That’s a pretty wide range, but if you take the average, you get 50%.
If you didn’t push, that means that you’re now broke. If you pushed, you still have $50.
More than Just a Game
This might not make that much of an impact on you; after all, this was just a game. No, that’s the wrong way to say that. What I mean to say is, of course, this was just pretend, but the game is real. It’s not pretend. We’re not talking about child’s play here. We defined a set of possible moves by which players interacted with each other; they had common knowledge of the structure of the game; and they made rational decisions about strategies that led to their best expected payoff. These components—players, strategies, payoffs, and common knowledge—are what make a game a game in the game-theoretic sense. And if you change the context of this game by replacing the players with countries and by changing, pushing the button to being willing to engage in military conflict, then we have something that is much more than just a diversion.
Later in this course, we’ll find out how game theory says this game should be played. But at the moment, what we know is how it is played. The variety of responses that we’ve seen in this game—between 30% and 70% pushing—show that one of two things must be the case: Either the theory of game theory isn’t sufficiently common knowledge that people are comfortable choosing rightly, or maybe this game is an inherently dangerous one. Maybe we need to find a way to keep pushing the button from being so tempting an option, because if 30% to 70% of the people in the nuclear version of this game decide to press the button, we’re all in for a very, very bad time.
In any case, the name “game theory” may be an unfortunate one. A more descriptive name would be “strategic interaction decision making.” Game theory sounds like child’s play, and it’s not.