Lecture 8: choice under uncertainty
Lecture 8:choice under uncertainty
Content Lotteries and expected utility Risk aversion · Metric Subjective probability theory
Content • Lotteries and expected utility • Risk aversion • Metric • Subjective probability theory
Lotteries and expected utility A lottery:L=(P,…,)P≥0∑P=1 A compound lottery: L=(L 1k:1 va20∑P=1andL=(P…,P) A simplified lottery of(L, . Lk;a, ak) 1sL=ax1L1+…+ a,l or l=(P;…P) P=a,P1+…+a,PR See the fig
Lotteries and expected utility • A lottery: • A compound lottery: and . • A simplified lottery of is or See the fig 1 ( , , ) L P P = N 1 , 0 1 N i i i i P P = = 1 ( , , ) k k L P P k N = 1 1 ( , , ; , ) L L L = k k 1 , 0 1 N k i i k P = = 1 1 ( , , ; , ) L Lk k L L L = + + 1 1 k k 1 ( , ) L P P = k 1 1 k P P P n n k n = + +
Lotteries and expected utility The preference of lotteries Continuous: LL'L"e@ the set is closure {a∈[0,1]:aL+(1-a)L%L"}c[0,1 {a∈[0,1]:L"%aL+(1-a)L}c[0,1 Independence axiom aL+(1-a)L”%aL+(1-a)L”冷→L%L 连续性和不相关性
Lotteries and expected utility • The preference of lotteries: – Continuous: the set is closure – Independence axiom: L L L , , L { [0,1]: (1 ) } [0,1] + − L L L % { [0,1]: (1 ) } [0,1] + − L L L % L L L L L L + − + − (1 ) (1 ) % % 连续性和不相关性
Expected utility V N-M expected utility function U(D)=4P+…+RN Proposition 1: a utility function U: @->R is an expected utility function if and only if it's liner, that is vk L∈and(a,…a)>0.∑ar=1 We have:UC∑aL)=∑AaU(L)
Expected utility • v.N-M expected utility function: • Proposition1: a utility function is an expected utility function if and only if it’s liner, that is we have: 1 1 ( ) U L u P u P = + + N N U :L → 1 ( , ) 0, 1 k k i = k L and L 1 1 ( ) ( ) K K k k k k k k U L U L = = =