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[ EDITORS NOTE: The Mean Game involves players predicting the mean of all the numbers predicted by all the players so it involves a lot of strategic play and some group psychology. NRICH organises such games by email or on the webboard from time to time with the additional rule that players have to declare their strategy and give some explanation for it. Edward conducted his own investigation using simplified versions of the game. ]

Introduction

The mean game is a game played with numbers. Many players are involved. Each picks 2 numbers between 1 and 1000 inclusive. The winner is the player who has picked a number closest to the mean of all the numbers picked.

Having played the game on NRICH, I decided to make a statistical study of it for some coursework. This article is an explanation of my findings.

In order to make the game easier to study, I modified the original version so that only two players are involved, each of whom picks 4 numbers between 1 and 1000 inclusive. The winner is still the player who has picked a number closest to the mean of all the numbers picked.

This game is interesting because of the mechanism by which the mean is determined by numbers which are also targeted at the mean, i.e. players who are trying to find the mean also determine the mean. This has certain similarities to the behaviour of markets, where traders aim to "win" on the market by buying and selling while at the same time their transactions affect the state of the market. The statistical analysis of behaviour in this game may therefore shed some light on the behaviour of traders in a market.

I played a game against a friend, and another game was played with one random computer player set against a player who always picked the same numbers (450, 480, 520 and 550). This was to see the success in gameplay of a random player against a midrange-centred predictable player, hopefully gaining more insight into strategies within the game.

Each pairing played 50 games against each other. The two human players played all 50 games in one session. The results of all previous games were visible to the players, but the 4 numbers they picked each game were only revealed to each other after both players had picked their numbers. This was expected to produce results which depended on the previous game.

One game was also played with 8 players involved, each player picking 2 numbers each. This was to gain a glimpse of the behaviour of larger groups of players, where each player has less ability to affect the average. These 8 players were a sample taken from my maths group.

What was found

The distribution of averages for human players does not approximate to a normal distribution whereas the distribution of averages for random players does. This agrees with the central limit theorem, which states that if $X_1, X_2, \ldots, X_n$, is a random sample of size $n$ from any distribution with mean $\mu$ and finite variance $\sigma^2$ then, for large $n$, the distribution of the sum of the $n$ random variables is approximately normal with mean $n \mu$ and variance $n \sigma^2$.

The distribution of picked numbers for random players was approximately uniform whereas the distribution of picked numbers for human players was trimodal with peaks at about 2, 500 and 998. This difference in the distribution of picked numbers has obviously led to the difference in the distribution of the means. Another factor worth noting is that the human players tended to pick many high numbers at a time, or many low numbers at a time, meaning that the mean of a particular game was skewed higher or lower respectively.

The game averages for human players in run 1 did not settle down to a constant or cyclical mean. There appears to be no long- or short-term underlying pattern to the behaviour of the mean with increasing game numbers. The game averages for random players obviously have no pattern.

The game averages for the 8-player game appear to converge to a point somewhere around 655, although it is not possible to tell the long-term behaviour of this system because of the small number of games played. It is interesting that in the 8-player games nos. 7-10 the standard deviation varied wildly even though the mean stayed within a small distance.

There appears to be little or no correlation between the means and standard deviations in any of the games played.

Conclusion

The human players do not play using random numbers. The differences between the distributions of numbers in the human vs human game compared to the random number game was so great that human behaviour can clearly not be seen as random. This has two consequences. Firstly, picking random numbers is not a suitable simulation for a human. Neither is picking random numbers a strategy for success, since any human opponent would realise this and play like the predictable opponent in the predictable vs random game, who won 80% of the time. However, although human behaviour is not random, it is also not predictable. There are some areas of short-term order within picked numbers for particular players but these do not produce order in the means of the games.

These differences occur because the human players play with winning in mind. Specifically, they are aware that they are trying to engineer the mean to be in a particular place so that they control the game and therefore win. This does not always work, but it is the only strategy that consistently works when played against itself, i.e. if player 1 is using this strategy player 2 can only realistically compete if he uses the same strategy. It is obvious that if one player always picks numbers with the same mean the other player can place their numbers very close to this mean value or consciously move the mean somewhere else. If a player plays randomly, a situation such as that in predictable vs random game would occur. The distributions occur as they are because the players pick numbers at the extremes of the population so as to move the mean as much as possible. This is somewhat similar to bearish and bullish traders in markets -- those who try to bid the market either up or down in order to make money at the new market value.

It should not be difficult to produce a computer player with the same statistical distribution of picked numbers as a human player but whether this would be successful at winning the game is difficult to tell. What this means is that it is difficult to produce a mechanical agent which reliably wins, and that unpredictable behaviour is a good strategy for winning. This has implications for game theory.

In the 8-player game, which is closer to a "real" market, the average appeared to settle to a reasonably constant value. This was because each player could only have a small effect on the average by himself, and because they expected the mean to be near to the mean for the previous game. This is reminiscent of an iterative formula, e.g. that for the Feigenbaum fractal, where the number of players determines whether the mean settles down to a relatively constant average or not. However, in the many-player mean game there appears to be no way to tell where the mean will settle down to.

I'd like to investigate this sort of game more: it's more complex than the basics of game theory such as chicken, but simple enough to be relatively easy to study. Playing many more games, and with different size groups, is needed to get a better picture of behaviour in The Mean Game.