Games:Stones

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Stones is a mini-game within the Improbable Island MUD. It is a single-player gambling game with a single decision between two choices, and three possible results (win, lose, draw).

Premise

The game can be played by going to the Raven Inn (which you can go to directly from the Jungle if you have an appropriate mount) and talking to the Old Man:

The old man explains his game, "I have a bag with 6 red stones, and 10 blue stones in it. You can choose between 'like pair' or 'unlike pair.' I will then draw out pairs of stones two at a time. If they are the same color as each other, they go to which ever of us is 'like pair,' and otherwise they go to which ever of us is 'unlike pair.' Whoever has the most stones at the end will win. If we have the same number, then it is a draw, and no one wins."

You can choose whatever initial stake you wish to, up to a limit of however many credits you are carrying. Selecting the correct outcome will double your stake. If the outcome is the opposite to your choice then you lose your stake. If an equal number of Like and Unlike pairs are drawn from the bag then the game is a draw and your stake is returned.

Statistical analysis

(Where decimal figures are provided in this section, they are quoted to six significant figures.)

This analysis and the conclusions drawn from it assume that the pairs of stones are drawn purely randomly, with any remaining stone in the bag having an equal probability of being drawn next.

The state space can be defined as a probability tree with nodes for each possible result of drawing a pair of stones (2 red / 2 blue / 1 of each colour). There are a total of 784 paths in the state space, and each path has one of four different probabilities. Without giving an exhaustive list of all paths and how their probabilities are calculated, here is a summary of the path probabilities (if you do want the exhaustive list of all paths and their probabilities, there is a link to a spreadsheet containing that information in the references section below):

  • Result: more Like pairs
    • 420 paths each with probability 4 in 8008.
    • 56 paths each with probability 1 in 8008.
    • Total probability of result is 31 in 143 (= 0.216783).
  • Result: equal Like and Unlike pairs
    • 280 paths each with probability 16 in 8008.
    • Total probability of result is 80 in 143 (= 0.559441).
  • Result: more Unlike pairs
    • 28 paths each with probability 64 in 8008.
    • Total probability of result is 32 in 143 (= 0.223776).

The rational decision is therefore to always bet on there being more Unlike pairs than Like pairs.

However, due to the distribution of probabilities, the standard deviation of the result of a single bet is much higher than the mean return. If we give the values of 1 to a win, 0 to a draw, and -1 to a loss, the mean is 0.00699301 and the standard deviation is 0.663710 (variance 0.440511).

The mean result increases in proportion to the number of games played, whereas the standard deviation increases in proportion to the square root of the number of games played. Therefore the more games are played, the less variance there will be comparing the actual to expected result. This table gives the statistics for an increasing number of games, assuming that the same stake is used for each game:

Number of games played Mean result Variance in result Standard deviation of result
1 0.00699301 0.440511 0.663710
10 0.0699301 4.40511 2.09883
100 0.699301 44.0511 6.63710
1000 6.99301 440.511 20.9883
10,000 69.9301 4405.11 66.3710
100,000 699.301 44,051.1 209.883

(The point at which the mean and standard deviation are equal is at 9008 games, when they both have the value 62.9930.)

Practical application

One can play Stones within the Improbable Island MUD, always betting on Unlike, and expect to make in-game money over the long term. However the rate at which the money is earned is slow due to the high probability of a draw and the number of clicks required to play the game. Additionally, due to the small spread between winning and losing, one would need to have a relatively high initial wealth in order to reduce the probability of ruin to acceptable levels. Note that given a high initial wealth, it is valid to use the Martingale strategy since there is no fixed upper limit on the size of the bet.

In conclusion, the mini-game makes a interesting diversion, but it is not a viable source for building funds in comparison to other methods. The lack of strategic decisions in playing the game and the high number of clicks required to play it make it interesting for only a brief amount of time.

References

For a detailed breakdown of the interim calculations (so that you can explore the model and see how it works) download a copy of the spreadsheet model from:

Input fields are marked with a yellow background (so that you can see the effects of changing the starting conditions). The spreadsheet also includes detailed comments throughout to explain how the individual cell calculations work and what they mean.