Can you beat Piggy in this simple dice game? Can you figure out
Piggy's strategy, and is there a better one?
Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.
Is the age of this very old man statistically believable?
Published December 1999,July 1999,December 2011,February 2011.
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
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
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
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
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.
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
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
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.