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Minimax Criterion by Paper 1
After defining a decision scenario, expected utility, and maximum expected utility, Wang et al. introduce how we model the uncertainty of the decision system about the utility function of the user. The authors say that this modeling is done by assuming a set of linear constraints $$C$$over the set of possible utility functions $$U=[0,1]^n$$. Indeed, they are assuming that constraints over unknown utility values are linear. We use $$C \in U$$to denote the subset of $$U$$satisfying $$C$$. Adopting the minimax regret decision criterion, Wang et al. define the optimal decision $$d^*_u$$with respect to utility vector to be

$$d^*_u = \underset{d_i}{\operatorname{arg\,max}}\, EU(d_i,u)$$.

If the utility function were known, $$d^*_u$$would be the correct decision. The regret of the decision $$d_i$$with respect to $$u$$is

$$R(d_i,u)=EU(d_u^*,u)-EU(d_i,u)$$.

Then, Wang et al. define maximum regret of decision $$d_i$$with respect to $$C$$as:

$$MR(d_i,C)=\underset{u \in C}{max} R(d_i,u)$$.

Then, the decision $$d_C^*$$with minimax regret with respect to $$C$$will be:

$$d^*_C=\underset{d_i}{arg\,min}MR(D_i,C)$$.