Jump to ContentJump to Main Navigation
Risk and Rationality$

Lara Buchak

Print publication date: 2013

Print ISBN-13: 9780199672165

Published to Oxford Scholarship Online: April 2014

DOI: 10.1093/acprof:oso/9780199672165.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 25 February 2017

Appendix C. Tradeoff Equality and Utility Differences

Appendix C. Tradeoff Equality and Utility Differences

Source:
Risk and Rationality
Publisher:
Oxford University Press

Claim

For REU maximizers, xy ~*(C) zw holds just in case u(x) − u(y) = u(z) − u(w).

Proof

Assume REU maximization, and assume xy ~*(C) zw. Then xEf ~ yEg and zEf ~ wEg 
for some xEf, yEg, zEf, wEg that are comonotonic and E non-null on the comoncone. Therefore, there is some set of states {E1, ...  , En} and some rank-ordering permutation Φ such that xEf(EΦ(1)) ≥ ...  ≥ xEf(EΦ(n)), yEg(EΦ(1)) ≥ ...  ≥ yEg(EΦ(n)), zEf(EΦ(1)) ≥ ...  ≥ zEf(EΦ(n)), wEg(EΦ(1)) ≥ ...  ≥ wEg(EΦ(n)) and:

  1. (1) EΦ(i) = E for some i.

  2. (2) Appendix C. Tradeoff Equality and Utility Differences

  3. (3) Appendix C. Tradeoff Equality and Utility Differences

See part one and three of the proof in Appendix A.

Since xEf and zEf only differ on EΦ(i), and yEg and wEg only differ on EΦ(i), subtracting eqn. (3) from eqn. (2) yields:

[r(p(E Φ(1) ∪ ...  ∪ E Φ(i))) − r(p(E Φ(1) ∪ ...  ∪ E Φ(i−1)))][u(x) − u(z)] 
= [r(p(E Φ(1) ∪ ...  ∪ E Φ(i))) − r(p(E Φ(1) ∪ ...  ∪ E Φ(i−1)))][u(y) − u(w)]

This simplifies to u(x) − u(z) = u(y) − u(w).

Claim

For EU maximizers, xy~*(Appendix C. Tradeoff Equality and Utility Differences ) zw holds just in case u(x) − u(y) = u(z) − u(w).

Proof

Assume EU maximization, and assume xy ~*(Appendix C. Tradeoff Equality and Utility Differences ) zw. Then xEf ~ yEg and zEf ~ wEg for some xEf, yEg, zEf, wEg in Appendix C. Tradeoff Equality and Utility Differences  and E non-null in Appendix C. Tradeoff Equality and Utility Differences . Therefore, there is some set of states {E1, ...  , En} such that:

  1. (p.249) (1) Ei = E for some i.

  2. (2) Appendix C. Tradeoff Equality and Utility Differences

  3. (3) Appendix C. Tradeoff Equality and Utility Differences

Since xEf and zEf only differ on Ei, and yEg and wEg only differ on Ei, subtracting eqn. (3) from eqn. (2) yields:

p(Ei)[u(x) − u(z)] = p(Ei)[u(y) − u(z)].

This simplifies to u(x) − u(z) = u(y) − u(w).