 Suppose we asked the fugitive to pick, from the available set of outcomes, a best one and a worst one. ‘Best’ and ‘worst’ are defined in terms of expected payoffs as illustrated in our current zero-sum game example: a player maximizes his expected payoff if, when choosing among lotteries that contain only two possible prizes, he always chooses so as to maximize the probability of the best outcome—call this W—and to minimize the probability of the worst outcome—call this L. Now imagine expanding the set of possible prizes so that it includes prizes that the agent values as intermediate between W and L. We find, for a set of outcomes containing such prizes, a lottery over them such that our agent is indifferent between that lottery and a lottery including only W and L. In our example, this is a lottery that includes being shot and being crushed by rocks. Call this lottery T . We define a utility function q = u(T) from outcomes to the real (as opposed to ordinal) number line such that if q is the expected prize in T , the agent is indifferent between winning T and winning a lottery T* in which W occurs with probability u(T) and L occurs with probability 1 − u(T). Assuming that the agent’s behaviour respects the principle of reduction of compound lotteries (ROCL)—that is, he does not gain or lose utility from considering more complex lotteries rather than simple ones—the set of mappings of outcomes in T to uT* gives a von Neumann—Morgenstern utility function (vNMuf) with cardinal structure over all outcomes in T. 