 When you make a decision, what you choose and the way the world turns out together determine an outcome, to which you assign a utility that measures how desirable it is for you. In a decision under certainty, each action that you may perform has exactly one possible outcome. In that case, it seems that you should simply perform an action that has maximal utility. (Read on, however!) In a decision under risk, you assign probabilities to the various ways the world could turn out—the possible states. Suppose that there are various actions \(A_i\) that you could perform, and various states \(S_j\) to which you assign probabilities \(p_j\). Together they determine outcomes to which you assign utilities \(u_{ij}\). Classic decision theory says that you should maximize expected utility: you should perform an action that maximizes the weighted average of the utilities associated with that action, the weights given by your probabilities. Formally, you should maximize