Soliciting strategies

I once was a research assistant on a project that called on participants in an experiment to make a decision that depended on the expected value of a randomly drawn object. The scenario and instructions were printed on a bit of paper and we asked players to put a checkmark next to their choice. So far, so good.

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More interesting games than the prisoner’s dilemma

I think game theory is awesome. First of all, it is the most ingeniously named field in the history of thought. Who doesn’t want to study something called “game theory”? Marketing!

Second of all, it forces a student—it certainly forced me—to confront kinds of logic, problem-solving, and epistemological questions that are quite different to the norm. Certainly quite different to the norm elsewhere in economics, at least. It is thinking about thinking, which for a certain type of person is a fun rabbit hole.

Game theory is about how we can try to understand situations in which the outcome of my choice depends also on what you choose. These are situations with strategic interaction. I can’t just pick the thing I like best, as in the standard rational choice model. I have to think about what you want, what you might do, how you might respond, what you think about me, what you think I might do—and you have to think all of those things right back at me. It’s gnarly!

So I saw this at the Browser today:


How often do you see a reference to game theory that isn’t about the prisoner’s dilemma? Very seldom, if ever, I think, and in my opinion game theory is worse for it.

I include in that references to game theory in academic courses too, by the way. I’ve taught game theory for undergraduates many times (in the context of applications to economics), and it’s remarkable how many students come to those classes under the impression that game theory is the prisoner’s dilemma. I don’t think they are wholly to blame for the mistake.

I’m not here to tattle on anyone, but I’ve had smart students show up at game theory class and tell me that game theory is about how you should rat out your friends. Uh oh!

Now, I come not to praise the prisoner’s dilemma, but nor do I want to bury it. First, it swiftly and usefully undermines the notion that decentralized decision-making necessarily leads to mutually beneficial outcomes. Second, as with any simple game, it can be extended and varied in endless ways to tell many rich and useful stories—most obviously by thinking about a repeated prisoner’s dilemma with inducements, rewards, and punishments. Third, it’s a useful parable for a variety of real-world situations.

There are more qualities of the game, of course. For a dive into the depths of what it can teach us, I recommend its entry in the Stanford Encyclopedia of Philosophy.

But here’s the problem: the vanilla prisoner’s dilemma is the least interesting game in game theory. I’m not going to rehash the setup again here (trust me, it’s been done enough). The reason it’s the least interesting game is that the very point of the prisoner’s dilemma is that its setup gives each of the two players in the game a unilateral incentive to make a given choice.

The vanilla prisoner’s dilemma boils down to the smallest variation on the rational choice model. I don’t particularly need to think about what you might do, or what you might be thinking. No matter what you do, one of my options is always better. Yeah, it carries a non-trivial lesson, but to me it just seems so flat. I don’t really get much of a hint of the mind-bending joys of game theory from it. If this is game theory, then game theory isn’t much at all.

And so often, in my experience, students see the vanilla prisoner’s dilemma, see it called game theory, and then move on to something else. Nothing else is being imprinted on them about the example or the field. I think that’s a shame.

So in that spirit I call on those of us who would like to touch on game theory in non-game theory course, or those of us who write game theory into introductory economics texts, to consider showing, instead of or as well as the prisoner’s dilemma, some different interesting games. Here are a few of my suggestions:

  1. The Keynesian beauty contest is another very famous game. I think it is super useful in a classroom setting. It gets very quickly at many of the “big questions” in the study of games and it’s a lot of fun to play and argue about in a group. It also unleashes “One, Two, (Three), Infinity, … : Newspaper and Lab Beauty-Contest Experiments” by Rosemarie Nagel et al, which is one of my all-time favorite papers to teach, especially its peerless appendix.
  2. The centipede game is not a million miles away in spirit from the prisoner’s dilemma, but it is (i) way more fun for students to play in the classroom, and (ii) teaches the power and weirdness of backward induction. Also an introduction to the thorniest problem for the typical game theory student, that “a strategy” is a pre-play complete contingent plan, not an on-the-fly decision. Bonus: the “crazy centipede” variant: what if there’s a tiny chance that my opponent is irrationally cooperative?
  3. Rock-paper-scissors. Seriously. Everyone knows it, and the soul of the concept of Nash equilibrium lives here. So is the distinction between playing a game in real life versus “a strategy” in game theory that I mentioned a second ago in #2. See also Wolfers on the Patriots-Seahawks Super Bowl.
  4. The Hotelling location game. Readily graspable, hugely widely applicable, has a ton of interesting variations, and teaches many of the same lessons as the prisoner’s dilemma. It also opens the door to location problems more generally, which are always fun.

I’m not saying we have to retire the prisoner’s dilemma’s number just yet. But I think we can do a little more when we give our students their first taste of the rich, fun field of game theory.

If you have any thoughts on my game suggestions, do let me know!

Game theory often looks silly

From Tim Harford’s blog:

“Game theorists know all about the centipede game:

One instance of the centipede game is as follows. A pile of $4 and a pile of $1 are lying on a table. Player I has two options, either to “stop” or to “continue.” If he stops, the game ends and he gets $4 while Player II gets the remaining dollar. If he continues, the two piles are doubled, to $8 and $2, and Player II is faced with a similar decision: either to take the larger pile ($8), thus ending the game and leaving the smaller pile ($2) for Player I, or to let the piles double again and let Player I decide. The game continues for at most six periods. If by then neither of the players have stopped, Player I gets $256 and Player II gets $64. Figure 1 depicts this situation. Although this game offers both players a very profitable opportunity, all standard game theoretic solution concepts predict that Player I will stop at the first opportunity, getting just $4.

Except, nobody really thinks this is the way players would behave in reality. The optimal strategy seems sociopathic; isn’t it worth playing cooperatively in the hope that the other player will do the same thing? (Unlike much real human interaction, standard game theory does not accomodate the “hope” that someone else will play suboptimally: optimal play is to be expected at all times. )”

Game theory is very clever and very useful, but often seems very naive. When it’s used in economics, it’s arguably the part of economics most hamstrung by the scattershot application of the “money=utility” fallacy. If you want your game theoretic result to be predictive or descriptively powerful, you must (must must) try really hard to make the payoffs reasonably accurate; in Harford’s quoted example the assumption is that the players care only about cash and that, as Harford says, they aren’t willing to take a shot on the other player prolonging the game. At the risk of being tautologically critical: can you read the setup of that game and not entertain the idea of waiting? I remember being taught the centipede game in David Myatt’s excellent game theory course as an undergrad; he showed us the ‘crazy centipede’ variant, which wondered exactly that: what chance of you choosing to continue the game is enough to make me also want to continue?

The kicker to me is that ‘game theoretic predictions’ are overwhelmingly often not as successful for the players as alternative strategies, even when we’re just measuring ‘success’ in the same cash-payoff terms as the theory. This is just what Harford goes on to describe:

But Ignacio Palacios-Huerta (best known to Undercover Economist readers as discovering that strikers and goalkeepers play optimal strategies in penalty-taking) and Oscar Volij gave the centipede game to skilled chess players. They found that the chess players were far more likely to play optimally; grandmasters always played optimally and took the $4. Hyper-rationality can be a disadvantage. (Or did the experiment discover something else: that chess grandmasters are sociopaths?) Palacios-Huerta and Volij don’t speculate. My guess is that they have discovered something about the rationality rather than morality or empathy of chess players, but I may be wrong.

It really does just beg for the ‘behavioral economics’ explosion: if predictions aren’t great, and in any case are less profitable than reality, we’re up the creek without a paddle or a boat.