What is the best decision for the robot, given that it is engaged in a game against nature? This depends on what information the robot has regarding how nature chooses its actions. It will always be assumed that the robot does not know the precise nature action to be chosen; otherwise, it is pointless to define nature. Two alternative models that the robot can use for nature will be considered. From the robot's perspective, the possible models are
Assume first that Formulation 9.3 is used and that and are finite. Under the nondeterministic model, there is no additional information. One reasonable approach in this case is to make a decision by assuming the worst. It can even be imagined that nature knows what action the robot will take, and it will spitefully choose a nature action that drives the cost as high as possible. This pessimistic view is sometimes humorously referred to as Murphy's Law (``If anything can go wrong, it will.'')  or Sod's Law. In this case, the best action, , is selected as
Worst-case analysis may seem too pessimistic in some applications. Perhaps the assumption that all actions in are equally likely may be preferable. This can be handled as a special case of the probabilistic model, which is described next.
Under the probabilistic model, it is assumed that the robot has gathered enough data to reliably estimate (or if is continuous). In this case, it is imagined that nature applies a randomized strategy, as defined in Section 9.1.3. It assumed that the applied nature actions have been observed over many trials, and in the future they will continue to be chosen in the same manner, as predicted by the distribution . Instead of worst-case analysis, expected-case analysis is used. This optimizes the average cost to be received over numerous independent trials. In this case, the best action, , is
Under the nondeterministic model of nature, , which results in in the worst case using (9.14). Under the probabilistic model, let , , and . To find the optimal action, (9.15) can be used. This involves computing the expected cost for each action:
If the probability distribution had instead been , then would have been obtained. Hence the best decision depends on ; if this information is statistically valid, then it enables more informed decisions to be made. If such information is not available, then the nondeterministic model may be more suitable.
It is possible, however, to assign as a uniform
distribution in the absence of data. This means that all nature
actions are equally likely; however, conclusions based on this are
dangerous; see Section 9.5.
In Formulation 9.4, the nature action space depends on , the robot action. Under the nondeterministic model, (9.14) simply becomes