7
lottery is evaluated according to the expected value of u.
1.2 Axioms on preferences over gambles
1. Completeness. 2. Transitivity. 3. Given Axioms 1 and 2, we can rank all outcomes in A. (ai can be viewed as a degenerate gamble that yields outcome 1 with probability one.) Without loss of generality, we assume a1 a2... an.
jectively defined. 3. A compound gamble is a gamble whose prizes are gambles (simple or compound). For any compound gambles, we can calculate the probability of each outcome. (I use the term “prize” to refer to what one may receive from a gamble. A prize may be another gamble or an outcome in A.) 4. Example: suppose A = {+1, −1}. g1 = (p1 ◦ 1, (1 − p1) ◦ −1) g2 = (p2 ◦ 1, (1 − p2) ◦ −1) g3 = (q ◦ g1, (1 − q ) ◦ g2)
9
to x is also better than x . 5. The Independence Axiom. For any gambles g , g , g , and α ∈ (0, 1), g g ⇔ αg + (1 − α) g αg + (1 − α) g .
