The Six Stages of Player Rationality

Have you ever heard of the "2/3 Game"? A large group of people is told to guess a number between 0 and 100 and write it on a piece of paper. No talking; no looking at other people's guesses. The people are then told that their papers will handed in, and the average of the guesses will be calculated. A prize will be given to the person whose guess is closest to 2/3 of this average.

The 2/3 game has been researched a lot, and I run it in my classes all the time. The winning guess is always the same. Can you predict what it is? Hint below; spoiler after that.

Hint: It is one of these numbers: 0, 24, 33, 50.

Spoiler: In my classes, the winning number is always 24.

I use the game to teach game design students about player rationality. In order to understand the impact of your mechanics, you need to know how people will react to them. This means that you have to have some kind of mental model of player behavior. The question is, in your model, how much rationality do you assume?

Philosophers and social scientists generally say that someone is either rational or irrational. But in the 2/3 Game, we can see SIX STAGES of PLAYER RATIONALITY. The stages represent how deeply a player thinks about what he and everyone else is doing. Lower stages involve very little thinking. Higher stages involve sophisticated calculation and reflection.

Stage 1. Nonrational. No goal. No reaction to incentives. Choosing without thinking. Example reasoning in the 2/3 game: None. Player chooses randomly.

Stage 2. Simple Rationality. Low-effort attempt to understand incentives. Assumes others are not rational, then reacts rationally to that assumption. Example reasoning: Who knows what people will guess. It will probably average out to the middle, which is 50. To win I should guess 2/3 of 50, which is 33.

Stage 3. Tactical Rationality. The player assumes everyone else is a Stage 2 reasoner, and reacts rationally to that. However, he doesn't go any further; he assumes that few people will get to Stage 3. The player is not thinking globally about the game, only about being one step ahead of the others. Example reasoning: Other people are going to figure out that 2/3 of the average guess is 33, so they will guess 33, so I will outsmart them and guess 2/3 of that, which is 22.

Stage 4. Hyper-rationality. The player reasons as far as Stage 3 and then continues iterating on the logic, eventually seeing that the logical, self-interested guess in a room of logical, self-interested people is 0. This is homo economicus: Assumes unlimited cognition and self-interest among all the players. This stage is the first to look at the game from a global perspective, thinking carefully about what other players should do. Example reasoning: the game is a recursive algorithm leading to zero, which means everyone will guess 0, which means I should as well.

Stage 5. Exploitive Rationality. Seeing the game from a global point of view, the player designs his own goals and seeks to exploit the entire situation to suit his own interests and desires. Creative effort to assess the entire incentive structure, including meta-game aspects. Player is unconcerned with or consciously opposes the intent of the design, and looks for ways to break it. Example reasoning: I am unlikely to win, but I enjoy messing things up, so I will guess whatever. How about 100. That should be interesting.

Stage 6. Collaborative Rationality. Player takes into account all the stages of rationality that may be present among the players. He assesses the likelihood of each of the stages and the resulting patterns of behavior. However, unlike the Stage 5 player, who sees the game as a forum for enjoyment, the Stage 6 player is interested in the game's mechanics and outcomes, and so he consciously supports the design, reacting to the game in a way that he thinks the designer intended. He collaborates with the game designer, in that his goal is not to win but just to see how the game plays. Example reasoning: It will be interesting to see how people guess. I'm sure a lot of them don't care and their guesses will average out to about 50. The winning guess would be 33 in that case. However, the hyper rational answer here is zero, and I know there are a bunch of very rational people in here; their guesses are going to drag that average down. There are also some people in here who just enjoy breaking games, and if they have reasoned as far as I have, they are going to guess big numbers just to mess things up. But there probably won't be too many of those, nor will there be too many of the rational types, and they cancel each other out more or less. The main mix will be between people who guess 33 and those who guess randomly, which will put the average somewhere above 33. If that's the case, the winning guess will be something north of 22, like 25 or so. Now, should I guess that? Sure! The game designer is trying to show us all a good time, so I will put in the number that I sincerely think will win.

Having run the 2/3 Game many times in class, my sense is that the winners are generally operating at Stage 3 rationality, and almost everyone else is at Stage 2. If they all were at Stage 2, the winning answer would be 22. But enough are at Stage 1 and/or stage 5, both of which pull the average up from 33 to 36, that 24 is the winning pick. Only rarely are people operating at Stage 4.

If you repeat the game, there is a time when more people are doing Stage 4, balanced out by a few doing Stage 5. But soon everyone gets to Stage 6, at which point everyone is guessing how much plain irrationality and/or griefing there will be.

Takeaway: Academic notions of rationality do not map very well onto player behavior. Most player populations will be a mix of Stage 1, 2, and 3, with the Stage-3 people doing better than the others. Stage 4 types will min-max and theorycraft but will not be representative of the way most people play and will not win. Players who think more wisely about what's rational will at times use their thinking to exploit the game (Stage 5), and at other times collaborate with the design intent (Stage 6).

Predicting player behavior depends on the mix of stages in the player base, and each one is different. Some games may have a funnel, where the longer people play, the more likely they are to be in Stage 5 or 6.