1. 1 INFORMATION and INSURANCE DEMAND under IMPRECISE RISK Jean-Yves JAFFRAY, LIP6, UPMC-Paris6 and Meglena JELEVA, GAINS, U.Maine and CES, U.Paris1

2. 2 Imprecise Risk (i) The Decision Maker: Believes that event E has a true objective (e.g. frequentist) probability However only feels able to locate it in a probability interval [P - (E), P + (E) ]

3. 3 Imprecise Risk (ii) The algebra of events is endowed with a family of probability distributions P={ P in L : P(E) in [P - (E), P + (E) ] pour tout E } The true probability P 0 is only known to belong to P

4. 4 Decision Making under Imprecise Risk : the HURWICZ criterion Expected Utility criterion under Risk : U P (  ) = E P  =  x u(x) P(  =x) Hurwicz’s criterion under Imprecise Risk : V (  ) =  inf  P in P  E P  +(1-  sup  P in P  E P   in [0,1], pessimism index

5. 5 Questions What about dynamic decision making under imprecise risk? Can Hurwicz’s criterion be used in a dynamic decision model in a: Dynamically consistent way? Consequentialist way? Both ways?

6. 6 DYNAMIC CONSISTENCY The strategy preferred in A generates a sub-strategy with root B which is itself the strategy preferred in B 10 25 0 0 20 A B Up Down 0

7. 7 CONSEQUENTIALISM Preferences in the subtree with root B do not depend on the rest of the tree (structure; data) 10 25 0 0 10 20 A B Up Down 0

8. 8 MIN+MAX as a particular case of HURWICZ’s criterion Hurwicz’s criterion : V (  ) =  inf  P in P  E P  +(1-  sup  P in P  E P  Set :  = 1/2 ; u(x)=2x ; P=L ; then, V (  ) =min e  e  + max e  (e)

9. 9 INCOMPATIBILITY between CONSEQUENTIALISM and DYNAMIC CONSISTENCY with MIN+MAX criterion : MIN + MAX If the criterion is used both in A and in B : CONSEQ. but not DYN. CONS. If the criterion is used only in A and induces choice in B : DYN. CONS. but not CONSEQ. 10 25 0 0 10 20 A B Up Down 0

10. 10 Any solutions? From the prescriptive (decision aiding) point of view, how should one cope with this inconsistency? Choose a strategy by rolling back the decision tree? Use a recursive model? In both cases one can select a dominated strategy. An alternative solution: Resolute Choice

11. 11 RESOLUTE CHOICE (i) McClennen (attended several FUR conferences.) [ 1990, p.260] "the theory of resolute choice is predicated on the notion that the single agent who is faced with making decisions over time can achieve a cooperative arrangement between his present self and his relevant future selves that satisfies the principle of intrapersonal optimality". [A self=a decision node]

12. 12 RESOLUTE CHOICE (ii) A particular case : dictatorship of the present self (Commitment) Since preferences of the present self prevail, the monotony properties of his criterion make him immune to money pumps, dominated choices, etc. He is economically rational.

13. 13 RESOLUTE CHOICE (iii) General case : cooperation between the present self and the future selves (Compromise) If the selves are flexible enough, they may be able to reach a compromise solution which is not dominated. Then the DM’s behavior is economically rational. For the construction of such a solution, dynamic programming can be used, despite the dynamic inconsistency of preferences (Jaffray-Nielsen)

14. 14 Information and Resolute Choice Non-consequentialism however, prevents the possibility of existence of an updating rule : Prior beliefs +observations  posterior beliefs So, is information processed correctly? Is there such a thing as learning?

15. 15 Bets of a Resolute DM (i) An urn contains red and black balls; the proportion of red balls is either p - or p+; the DM has to decide to bet or not on red (stake: m ; gain: M) at the (n+1) th draw, depending on the outcomes of the n first draws. The DM uses Hurwicz’s criterion and is resolute: he chooses a betting rule before the drawing starts (commitment).

16. 16 Bets of a Resolute DM (ii) 0 M B k red observed -m bet on red no bet Although there is no closed form updating rule, the fact that betting or not depends on the observations shows that some data processing takes place

17. 17 Bets of a Resolute DM (iii) Result : For n large, the DM bases his decision to bet or not on the proportion of red balls observed. k n =min{k: bet on red if k reds observed}. Lim k n =[1 +(ln p + - ln p - )/(ln(1- p - )- ln(1- p+) ) ] -1

18. 18 Insurance and the Resolute DM An individual with initial wealth W faces a risk with a unique amount of loss L E = {loss occurs}, E c = {there is no loss} P(E)  {p -, p + } 2 periods of time –In period 1, no insurance choice, deductible K; –In period 2, possibility to buy full coverage for a premium  Probabilistic independence of the successive events: if E i = {loss in period i}, for all p in [0,1], P(E 1 )=p => P(E 2 / E 1 )=p

19. 19 W – K – L W – K W – K –  d W –  W – L W W –  d - K W – K –  E1E1 E2E2 E2E2 E2E2 E2E2 Ec2Ec2 Ec2Ec2 Ec2Ec2 Ec2Ec2 Ec1Ec1

20. 20 Comparison of strategies –for any K  [0, 40000]: > for any   [0, 0.003[ V(dd) > for any   [0.003,1] > for any   [0,1], –Consequently, for any K  [0, 40000] and  [0,1], {, } are dominated by {, dd}

21. 21 Optimal strategy dependence on K  K 40000 dd 1 0.220.290.33 OPTIMISMPESSIMISM

22. 22 For  = 0.31, K * = 4570, K ** = 20095

23. 23 Conclusions Resolute choice leads to “intuitive” insurance decisions under imprecise risk; The most pessimistic individuals buy insurance in any case; The others adapt their decision to the information received. Counterfactuals may influence the optimal strategy, but only for few intermediate values of .