You have a utility function, which assigns real numbers to outcomes, propositions, states or affairs. Ordering Conditions
• (1) If xPy, then not yPx.
• (2) If xPy, then not xIy.
• (3) If xIy, then not xPy, and also not yPx.
• (4) xPy or yPx or xIy, for all relevant outcomes x and y. • (5) If xPy and yPz, then xPz.
• (6) If xPy and xIz, then zPy.
• (7) If xPy and yIz, then xPz.
• (8) If xIy and yIz, then xIz.
Continuity axiom states if you prefer A to B and B to C, there must be a lottery between A and C that has an indifference between that lottery and B. The axioms that do not mention gambles or probabilities are called ordinal and the ones that do mention gambles or probabilities are called cardinal. If axioms are always followed, expected utility will also be maximised. Axioms of decision theory are supposed to be requirements of rationality, because it is plausible to show how a perfectly rational person could fail to satisfy one or another of the axioms.
The continuity axiom states if you prefer A to B and B to C (and thus A to C) there must be some lottery which has exclusive prizes A and C such that you will be indifferent between that lottery and B. Counter: Someone who is risk averse may not like any lotteries. If A = £1, B = 50p and C = 10p, you would prefer A to B to C, but you may prefer 10p to any lottery with...