count of three: paper draws with paper, is defeated by scissors (since scissors
can cut paper), but defeats stone (because it can wrap stone). If playing ‘paper’
the payoffs are therefore 0, −1, +1, which is the top row of our completed
payoff table.
There is no saddle point for this game and no obvious pure strategy to adopt.
If a player always chooses the same action, say paper, the opponent will detect
this and simply choose scissors to win every time. By von Neumann’s ‘minimax
theorem’ there is a ‘mixed strategy’ or a way of choosing different actions based
on probability.
According to the mathematics, players should choose randomly but overall the
choices of paper, scissors, stone should each be made a third of the time. ‘Blind’
randomness may not always be the best course, however, as world champions
have ways of choosing their strategy with a little ‘psychological’ spin. They are
good at second-guessing their opponents.
When is a game not zero-sum?
Not every game is zero-sum – each player sometimes has their own separate
payoff table. A famous example is the ‘prisoner’s dilemma’ designed by A.W.
Tucker.
Two people, Andrew and Bertie, are picked up by the police on suspicion of
highway robbery and held in separate cells so they cannot confer with each
other. The payoffs, in this case jail sentences, not only depend on their individual
responses to police questioning but on how they jointly respond. If Aconfesses
and B doesn’t then A gets only a one year sentence (from A’s payoff table) but B
is sentenced to ten years (from B’s payoff table). If A doesn’t confess but B does,
the sentences go the other way around. If both confess they get four years each
but if neither confesses and they both maintain their innocence they get off scot-
free!
If the prisoners could cooperate they would take the optimum course of action