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Minimax expected loss strategies need not be admissible, as
fig. 4 will show.
FigureJ4.
Here is illustrated a convex set of which si and S2 and all of their
mixtures are minimax because they have the same maximum
loss L. But it is evident that only Si is admissible.
Another criterion is the minimax risk criterion. For convenience,
we repeat the table of losses.
ai
ai
01
20
37
62
25
0
Losses of utility
If 0i is the state of nature, the loss is 20 when action a 1 is taken.
This is the most favourable action: if a gross error has been made
it will cost at least 20. But then it can be argued that if the
decision maker has chosen «2, he need only regret 37 20
ƒ17: the 20 are lost anyway when a gross error has been made.
According to this principle which was stated by L. J. Savage, one
should not base his reasoning on expected losses but on expected
regrets which are called risks (Savage uses the term "negative
income" for loss and "loss" for regret.) It is not difficult to show
that in our example the risks R(Q,s) are obtained by subtracting
1.(0!,si) 20 from all L(0i,s) and that f?(02,s) L(02,s) for all s
