Dr. Bean, who is unsatisfactory which the four decision criteria for decision making under strict uncertainty (Wald's maximin, Hurwicz's maximax, Savage's minimax,Laplace's) proposes the following Minimum Mean Regret Criterion: Choose an action $\displaystyle a_k$ such that its mean regret $\displaystyle \frac{(\sum_{j=1}^{n}r_{kj})}{n}$ is as small as possible, i.e.

choose $\displaystyle a_k$ such that $\displaystyle \frac{1}{n} \sum_{j=1}^{n}r_{kj} $$\displaystyle =$$\displaystyle min_{1\le i \le m} \frac{1}{n} \sum_{j=1}^{n}r_{kj} $

where $\displaystyle r_{ij}$ denotes the regret of consequences $\displaystyle v_{ij}$.

a) Show that this criterion is essentially the same as Laplace's criterion, i.e. an action which minimizes the mean regret if and only if it maximizes the mean return.

b) Does this criterion agree with Savage's Minimax Regret Criterion for any decision table? If your answer is positive, prove your result rigorously; otherwise construct a counterexample and give full explanation.

So, just say for a) shall I just try to show that if Laplace has such and such critiera, then the minimum mean regret criterion must also have the same?