Monday, April 28, 2014

Learning to play.

In 2007, the Swedish gambling agency ran a simple gambling game called LIMBO. Gamblers were invited to stake 10 kronor on a number of their choice between 1 and 99,999. The person choosing the smallest number that no-one else chose is the winner: taking home a prize of around 100,000 kronor. What number would you choose?

Poisson-Nash equilibrium for an average of 53,783 players
When I played the game at Erik Mohlin's interesting and engaging seminar at the Institute for Futures Studies last Friday, I chose '1'. A bit naive, maybe, and quite a few others thought the same way. The winning number in an audience of about twenty of us was '5'. While the game is not straightforward, it is possible to determine a probabilistic equilibrium strategy. If all players pick numbers according to the distribution on the right, then no player can improve their performance by changing away from this distribution. Note that the probability of choosing large numbers is not zero, but it is very very small. 

The question Erik asked was how people learn to play the game. It is unlikely that everyone worked out the best strategy using stochastic game theory. Indeed, in the first week in Sweden people didn't play so well. Like me, they clustered around very low numbers, maybe not realizing that everyone else would do the same. But over time the distribution stretched out and they collectively took a strategy close to the optimal. Winning numbers after two months of play were between 162 and 3590.  

How do people get collectively better at games line LIMBO? This question was tested on data from a lab experiment, where smaller numbers of people played a similar game. Erik and his co-workers found that the explanation that best fit the data was a form of imitative learning, where the players would look at previous winners' numbers, and increase their probability of choosing a number the same or near to the winners.  Through this learning they eventually arrive at the equilibrium shown above, both in the model and in the data. Imitative learning is quick way of finding out how to play a game well.

The fact that imitative learning works so well for game playing is an important insight for basic economic theory. Economist often argue about if and when the equilibrium of a game is useful for predicting the behavior of real people. In this case it seems to be very useful. The equilibrium is quickly reached through a simple process of imitation. 

But…….. The Swedish gambling agency closed LIMBO about 3 months after it started because they believed people were using syndicates to cheat. It isn't hard to see why. The winning numbers were usually in an interval 1000 to 3500. So an investment of 25,000 kronor by a consortium was almost guaranteed to win the 100,000 kronor first prize. If there was more than one consortium then this would drive the winning numbers up and maybe the consortiums would start to lose. But it is clear that the game becomes unfair for honest, single players. If you can read Swedish, it is quite amusing to read the press releases from Svenska Spel as they first introduce the game as the result of 5 years research, then provide some tips on how to pick your numbers, then finally admit that the game was against gaming laws all along.   

I think this real-life outcome raises just as many interesting economic questions as the original game. As soon as the game was introduced, independent gamblers built a syndicate to make sure they took home a profit. This is where the collective behavior comes in. Groups of independent actors quickly self-organizing to manipulate the market. Understanding how groups form and manipulate these games is a much harder problem to study scientifically. But if related to the behavior of economic agents such as banks and other financial institutions over the last few years, it is certainly a no less important a question to answer. 


  1. I played this game (unsuccessfully) during a seminar at the Niels Bohr Institute. Sebastian Bernhardsson studied the game during his PhD in Umeå. Here are some references:

  2. Thats really interesting. The test they did on the UK data seems a bit similar to the results in Sweden. People really seem to stick to lower guesses than the equilibrium.