8. Mix your plays
Let us return for a moment to the world of sport. In boxing, both opponents
mix right- and left-handed jabs, which might or might not be successfully
anticipated and deflected. In tennis, Lleyton the server might aim at the
forehand or the backhand of Pete the receiver, while Pete, in turn, can try to
return crosscourt or down the line. In these examples, each side has an idea
of its own strong points and of its opponent’s weaknesses. It will have a
preference for the choice that exploits these weaknesses, but not exclusively.
It is well understood, by players and barrackers alike, that one should mix
one’s plays, randomly throwing in the unexpected move. The point is that if
you do the same thing all the time, the opposition will be able to counter you
more effectively by concentrating its resources on the best response to your
one strategy.
Mixing your plays does not mean rotating your strategies in a
predictable manner. Your opponent can observe and exploit any systematic
pattern almost as easily as he can the unchanging repetition of a single
strategy. it is unpredictability that is important when mixing.
Imagine what would happen if there were some known formula that
determined who would be audited by the ATO. Before you submitted a tax
return, you could apply the formula to see if you would be audited. If an
audit was predicted, but you could see a way to “amend” your return until the
formula no longer predicted an audit, you probably would do so. If an audit
was unavoidable, you would choose to tell the truth. The result of the ATO
being completely predictable is that it would audit exactly the wrong people.
All those audited would have anticipated their fate and chosen to act honestly,
while those spared an audit would have only their consciences to watch over
them. When the ATO audit formula is somewhat fuzzy, everyone stands some
risk of an audit; this gives an added incentive for honesty.
There are similar phenomena in the business world. Think of
competition in the market for razors. Imagine that Gillette runs a coupon
promotion on a regular schedule — say, the first Sunday of every other month.
Bic can pree¨mpt Gillette by running a competing coupon promotion the week
before. Of course, Bic’s move is then predictable and Gillette can pree¨mpt the
week before. This process leads to cutthroat competition and both make less
profit. But if each uses an unpredictable or mixed strategy, together they
might reduce the fierceness of the competition.
The importance of randomized strategies was one of the early insights
of game theory. The idea is simple and intuitive but needs refinement if it is
to be useful in practice. It is not enough for a tennis player to know that he
should mix his shots between the opponent’s forehand and backhand. He
needs some idea of whether he should go to the forehand 30% or 64% of the
time and how the answer depends on the relative strengths of the two sides.