decision making under uncertainity.pdf

Decision Making Under Risk and Uncertainty

Decision Making Under Uncertainty: Classi�cation

We distinguish between:

decision-making under risk

the decision maker is informed about the probabilities of di¤erentevents;the objects of choice are probability distributions / lotteries

and

decision-making under uncertainty

the decision maker has no information about the probabilities ofdi¤erent events;the objects of choice are state-contingent acts.

Ani Guerdjikova Cornell Univeristy Decision Theory II Spring 2010 32 / 126


Questions

1 How can we model uncertainty?2 How should one make decisions under uncertainty?3 Are probabilities objective or subjective?4 Can we measure probabilities?5 How does new information change our perception of probabilities?



Axiomatizations

Von Neumann and Morgenstern (1947)

Anscombe and Aumann (1963)

Savage (1956)



Choice over Lotteries: The Von Neumann-MorgensternTheory

Consider a �nite set of outcomes (e.g. monetary outcomes)

Z = fz1...zng

The set of probability distributions / lotteries on Z is given by:

∆n =

((p1...pn) j pi � 0 for all i 2 f1...ng and

n

∑i=1pi = 1

)

Preferences % are de�ned on ∆n.



St. Petersburg Paradox

Consider the following game:

A coin is tossed.

If the coin lands heads after the n-th toss, you receive $2n and thegame stops.

If the coin lands tails, the game continues, until heads comes up.

How much would you be willing to pay to participate in this game?

The expected value of this game is...

∞

"...the mathematicians estimate money in proportion to its quantity,and men of good sense in proportion to the usage that they maymake of it", Gabriel Cramer.



The Expected Utility Representation

We are looking for the following representation:

U (p) =n

∑i=1piu (zi )



Compound Lotteries

How can we interpret convex combinations of lotteries?

Consider the object αp + (1� α) q.

Intuitively, you �rst use a randomization device, which gives youlottery p with probability α and lottery q with probability (1� α).

In a second step, either p or q is actually played and the outcomes aredetermined according to the corresponding probability distribution.

Our representation suggests that:

αp + (1� α) q � (αp1 + (1� α) q1...αpn + (1� α) qn)



The Von-Neumann-Morgenstern Axioms

Axiom A1 (Completeness) % on ∆n is complete.

Axiom A2 (Transitivity) % on ∆n is transitive.

Axiom A3 (Independence)For all p, q and r 2 ∆n, and for all α 2 [0; 1],

p % q i¤

αp + (1� α) r % αq + (1� α) r

Axiom A4 (Archimedian Axiom / Continuity)For all p, q and r 2 ∆n, such that p � q � r , there exist numbers α andβ 2 (0; 1) such that

αp + (1� α) r � q � βp + (1� β) r



The Expected Utility Theorem

TheoremThe following two statements are equivalent:

(i) The preference relation % on ∆n satis�es Axioms A1 � A4

(ii) There exists a utility function u : Z ! R such that for all pand q 2 ∆n,

p % q i¤n

∑i=1piu (zi ) �

n

∑i=1qiu (zi )

Furthermore, u is unique up to a positive-a¢ netransformation.



Lemma 1

Assume A1, A2, A3 and A4.

(i) If p � q and 0 � α < β � 1, then

βp + (1� β) q � αp + (1� α) q.

(ii) If p % r % q and p � q, then there exists a unique α� suchthat:

α�p + (1� α�q) � r .

(iii) If p � q, then for all r 2 ∆n and all α 2 [0; 1],

αp + (1� α) r � αq + (1� α) r .



Lemma 2

Assume A1, A2, A3 and A4. Then, there exist z0 and z0 2 Z such that forall p 2 ∆n,

δz 0 % p % δz0 .



Uniqueness of the Von-Neumann-Morgenstern UtilityFunction

If function u : Z ! R can be used to represent a preference order % on∆n, then so can any function v given by:

v (z) =: au (z) + b

for some positive a > 0.


decision making under uncertainity.pdf

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