##

9.3.3 Randomized Strategies

The fact that some zero-sum games do not have a saddle point is
disappointing because regret is unavoidable in these cases. Suppose
we slightly change the rules. Assume that the same game is repeatedly
played by
and
over numerous trials. If they use a
deterministic strategy, they will choose the same actions every time,
resulting in the same costs. They may instead switch between
alternative security strategies, which causes fluctuations in the
costs. What happens if they each implement a randomized strategy?
Using the idea from Section 9.1.3, each strategy is
specified as a probability distribution over the actions. In the
limit, as the number of times the game is played tends to infinity, an
expected cost is obtained. One of the most famous results in game
theory is that on the space of randomized strategies, a saddle point
always exists for any zero-sum matrix game; however, expected costs
must be used. Thus, if randomization is used, there will be no
regrets. In an individual trial, regret may be possible; however, as
the costs are averaged over all trials, both players will be
satisfied.

**Subsections**
Steven M LaValle
2020-08-14