Let Z be a set of prizes. Assume that >, a preference relation over L(Z), satisfies the independence property. Let a and b be two prizes with [a] > [b] and let a and 8 be two numbers between zero and one. In class we showed that a>b=a-ae(1-a) -b> 8 -ae(1- 8) -b Show that ( -a@ (1-a) -b>8 -de(1-B) -b=a>f Problem 2 Arnold's preferences over I(Z) are defined by Show that Arnold's preferences satisfy the continuity and independence properties. Remark: You may not appeal to Proposition 3.1 in your proof, but you can see how the Proposition was proved so that you can figure out how to do the problem yourself.preliminary # 3 a set of probes IVIN Leuma 3.1 preference relation over 2 (2 ) satistics independence a , b in t such that [ a ] 7 [ b ] 12 6 (0, 1 ) BELO, 1 thin 1 7 3 67 C . a + ( 1 - ( ) b ? B . a + ( 1 - B) b *l puter lottenes for thin MJJ You like ) monotoncity Proof : Start W/ [ a ] > [b ]) > & . 6 6) ( 1 - Q ) b & . a O ( 1 - d ) b > I b ] Let Q 3 B turetur B double independence . ola 21 ( 1 - B ) . a . . B ( 1 - x ) b 2 1 6 ) [b] Ka + ( 1 - c ) b ? P / g L ux a + ( 1 - d b ) + 1 - B / B a o ( 1 - B ) b