Axioms:
It is possible to construct a utility index which can be used to predict choice in uncertain situations if the consumer conforms to the following five axioms:
• Axiom of Complete-ordering: For the two alternatives A and B one of the following must be true: the consumer prefers A to B, she prefers B to A, or she is indifferent between them. The consumer's evaluation of alternatives is transitive: if she prefers A to B and B to C, she prefers A to C.
• Axiom of Continuity: Assume that A is preferred to B and B to C. The axiom asserts that there exists some probability P, 0
• Axiom of Independence: Assume that the consumer is indifferent between A and B and that C is any outcome whatever. If one lottery ticket L_{1} offers outcome A and C with probability P and 1-P respectively and another L2 the outcomes B and C with the same probabilities P and 1-P, the consumer is indifferent between the two lottery tickets. Similarly, if she prefers A to B, she will prefer L1 to L_{2}.
• Axiom of Unequal-probability: Assume that the consumer prefers A to B. Let L_{1} = (P_{1}, A, B) and L_{2} = (P_{2}, A, B). The consumer will prefer L_{2} to L_{1} if and only if P_{2}>P_{1}.
• Axiom of Compound-lottery: Let L_{1} = (P_{1}, A, B), and L_{2} = (P_{2}, L_{3}, L4), where L_{3} = (P_{3}, A, B) and L_{4} = (P_{4}, A, B), be a compound lottery in which the prizes are lottery tickets. L2 is equivalent to L1 if P_{1} = P_{2}P_{3} + (1-P_{2}) P_{4}. Given L_{2} the probability of obtaining L_{3} is P_{2}. Consequently, the probability of obtaining A through L_{2} is P_{2}P_{3}. Similarly, the probability of obtaining L_{4} is (1-P_{2}), and the probability of obtaining A through L_{4} is (1-P_{2}) P_{4}. The probability of obtaining A with L2 is the sum of the two probabilities. The consumer evaluates lottery tickets only in terms of the probabilities of obtaining the prizes, and not in terms of how many times she is exposed to a chance mechanism.
These axioms are very general in nature, and it may be difficult to object to them on the grounds that they place unreasonable restrictions upon the consumer's behaviour. However, they rule out some types of plausible behaviour. Consider a person who derives satisfaction from the share of gambling. It is conceivable that there exists no P other than P=1 or P=0 for such a person, so that she is indifferent between outcome B with certainty and the uncertain prospect consisting of A and C; she will always prefer the "sure thing" to the dubious prospect. This type of behaviour is ruled out by the continuity axiom and the compound lottery axiom.