Fun With NUMBERS
CATHERINE JOHNSON
Last week, I introduced a vocabulary game to one of my classes at school. The students had to work in groups of three, with one pupil trying to describe a mathematical word without using certain other, related words, one trying to guess the mathematical word and the third student acting as timekeeper and judge.
The game was very useful for vocabulary acquisition, but I found myself more interested in another game that the students introduced to settle the question of who went first.
Rock, paper, scissors
The game that my pupils used to decide on the order of play was "rock, paper, scissors", a game I played when I was a student and which is enjoyed by children in countries all over the world. After a count of three, players simultaneously reveal a shape made by their right hands. A clenched fist is "rock", two fingers is "scissors" and a flat palm is "paper". Paper trumps rock by wrapping around it, rock beats scissors by blunting them, and scissors defeat paper by cutting.
Some people take this simple schoolyard game very seriously indeed. The annual world RPS (Rock Paper Scissors) Championship offers prize money of $50,000 (1.75 million baht), and the competitors have a variety of strategies and psychological ploys to beat the field.
Probably the most common tactic is to use statistical analysis to predict what an opponent is likely to do next. One previous champion, who chooses to be known by the name Master Roshambollah, likes to play against opponents who analyse the statistics of the game, claiming that their strategy is easy to predict and therefore beatable.
Game theory
RPS is of interest to mathematicians in the field of game theory, who analyse winning strategies in games as simple as noughts and crosses and as complex, psychological and lucrative as poker. Game theorists quickly realised that the most effective strategy to win against a perfect opponent should be to throw at random, and were able to prove this hypothesis. However, none of the champion players actually adopt this approach. Why not?
First, it is much more difficult to adopt a random strategy than most people imagine. For example, think of a random number between one and five. More people will pick three than any other digit. Now try picking a random number between one and 10. People tend to favour seven. Generally, a RPS player trying to play randomly will be susceptible to manipulation.
Second, it seems as though the optimal strategy may not win the prize money even if a player ensures he has a truly random strategy, perhaps by memorising a previously generated sequence of moves. The top RPS players understand that adopting a random strategy means that the other player cannot have an advantage over them, but also see that they cannot profit by it either, and so the scores will even out in the long run and they will not be crowned RPS champion. Psychology wins over mathematics, and players develop their own styles and techniques.
Biology
The RPS rules seem rather artificial and contrived because, in most systems, if A beats B and B beats C, it implies that A beats C. However, there are some examples in biology where it transpires that C beats A. For example, three morphs of lizards in California have different mating strategies. The polygamous orange-throats best the monogamous blue-throats, who in turn beat the yellows. However, the sneaker strategy of the yellow-throats bests the oranges.
Another example involves different strains of E. coli, where A grows more quickly than B, which in turn grows more quickly than C. However, C is armed with a toxin that allows it to overpower A, but B is resistant to the toxin.
It is possible to invent new games that force players to fall into the trap of assuming that, if A beats B and B beats C, then A must win over C. For example, stick new numbers over the faces of three dice. A has the numbers three, five and seven, repeated once each. B has two, four and nine and C has one, six and eight. When shown that, over repeated trials, A beats B 5/9 of the time and B beats C 5/9 of the time you should be able to convince an opponent to select die A. All you need to do is choose C, which will beat A 5/9 of the time.
If manipulating an RPS-style game to your advantage seems underhanded to you, you might approve of the old Roman proverb Dignus est quicum in tenebris mices, which describes a man of honesty and virtue by saying that he could be trusted to play the Roman version of RPS in the dark!
Catherine Johnson is the head of mathematics at Shrewsbury International School. For more information or comments, you may email her at
catherine.mathematics@gmail.com .
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