47 Game theory
Some said Johnny was the smartest person alive. John von Neumann was a child prodigy
who became a legend in the mathematical world. When people heard that he arrived at
a meeting in a taxi having just scribbled out his ‘minimax theorem’ in game theory, they
just nodded. It was exactly the sort of thing von Neumann did. He made contributions to
quantum mechanics, logic, algebra, so why should game theory escape his eye? It didn’t
- with Oskar Morgenstern he coauthored the influential Theory of Games and Economic
Behavior. In its widest sense game theory is an ancient subject, but von Neumann was
key to sharpening the theory of the ‘two-person zero-sum game’.
Two-person zero-sum games
It sounds complicated, but a two-person zero-sum game is simply one ‘played’
by two people, companies, or teams, in which one side wins what the other
loses. If A wins £200 then B loses that £200; that’s what zero-sum means. There
is no point in A cooperating with B – it is pure competition with only winners and
losers. In ‘win–win’ language A wins £200 and B wins −£200 and the sum is 200
- (−200) = 0. This is the origin of the term ‘zero-sum’.
Let’s imagine two TV companies ATV and BTV are bidding to operate an extra
news service in either Scotland or England. Each company must make a bid for
one country only and they will base their decision on the projected increased size
of their viewing audiences. Media analysts have estimated the increased
audiences and both companies have access to their research. These are
conveniently set down in a ‘payoff table’ and measured in units of a million
viewers.
If both ATV and BTV decide to operate in Scotland then ATV will gain 5
million viewers, but BTV will lose 5 million viewers. The meaning of the minus