Are the cosequeces of a decision always deterministic?

no, they are frequently stochastic

-> e.g. buy lottery ticket?

How is a set of lotteries defined?

-> lottery:

function that assigns probabiliteis to all alternatives

where probabilities add up to 1

What are simple lotteries?

simply assign probability to all alternatives

When is a lottery degenerate?

if it puts probabiility 1 on one alternative

How are compound lotteries defined?

where

L1, L2, … are simple lotteries

=> probabiilities that we “get” certain lottery….

=> like doing two rounds of lotteries…

How can we simplify compound lotteries?

Apply consequentialist premise

-> states that we are only interested in the outcome(consequence) and not how we come about it

=> multiply probabilities

then treat as simple lottery

Example how to simplify compound lotteries

When is a set of lotteries infinite?

when k > 1

-> infinite ways of assigning probability combinations… (except trivial case k = 1 with p(x1) = 1…)

What are exemplary criterias on which preferences over lotteries can be based on?

most likely outcomes

look at alternative with highest probability in each lottery

which one is preffered?

most desirable and/or least desirable outcomes

what alternative do we desire most / least

which lottery has higher / lower probability for it?

uniformity of probabilities

prefer lotteries where probabilities for alternatives are very similar

size of support

set of alternatives with positive probabliity

expected utility

requires existence of utility funciton

What is the support for L1[0.5: x1, 0.5: x2]

support of 2

Provided with

preference a>b>c

and lotteries

L1[0.3:a, 0.7:b, 0:c]

L2[0.6:a, 0:b, 0.4:c]

Which would you prefer for

most likely outcome

least desirable outcome

Most likely outcome: L2 > L1

ML in L1: b

ML in L2: a

a > b => L2 > L1

L1 > L2

least desirable: c

L1(c) = 0; L2(c) = 0.4

=> L1 > L2

What additional axioms should lotteries fulfill?

continuity

independence

How is continuity defined?

i.e.

L1 flying to hawaii

L2 staying at home

L3 plane crash

=> if L3 is sufficient small (some epsilon)

=> you still prefer going to hawaii instead of staying home

=> (weird) prefer staying home over dying in plane crash with small residual probabliity of flying to hawaii…

How is independence defined?

-> L1 hawaii

-> L2 carribean

-> L3 plane crash

-> equal probability of plane crash for both cases wont affect preference

What is another name for independence?

savages sure thing principle

Provide an example where continuity wont work?

aggregaton of different parameters

-> if i.e. safety is of utmost priority and then hawaii over home

-> no matter how small probability is for plane crash (unless 0)

-> wont go…

What does the vNM (von Neumann & Morgenstern) theorem state w.r.t. preference relatins and utiliy functions?

preference relation >= on L(A) is

rational, continuous and independent

<=>

there exists a utility function u on A

such that for two lotteries

Explain the vNM theorem

preference relation on lottery is rational, continuous and independent

equivalent

there exists utility functoin u on A (alternatives)

so that when we prefere one lottery over the other

the sum of probabilites times utility of the alternative in lottery 1

is greater equal than in lottery 2

How can we transform vNM utility functions so that they still are vNM utility functions?

every positive affine transformatino

f(x) = ax+b, a>0

f(u(.)) is new vNM utility fcuntion

representing the same preference relation…

compare to celsius, fahrehneit, kelvin…

What to keep in mind w.r.t. utility and monetary values?

monetary value != utility!!!

-> i.e. expected utility of

L1[1: 1 Mio] -> 1 Mio

L2[0.5: 2 Mio, 0.5: 0] -> 1 Mio

=> Utility is most certanily different…!!! (i.e. risk aversion)

What does a concave utility in value represent?

risk aversion

-> the higher the value (on x), the less the utility increases

-> decreasing utility per euro for the 2 mio alternative than the 1 mio alternative…

in picture:

green: utility for first million

blue: utility for second million

What does a convex utility in value represent?

risk-seeking

-> increasing utility with higher values

What do we assume in general in the lecture w.r.t. preferences of rational agents?

completeness and transitivity

continuity and independence (in stochastic settings)

Explani the dice game

also called efron’s dice

non-transivive, as for each dice, there is another that wins more often on average

although having different expected utilites

-> only interested in individual higher outcomes

Max winnint probabiility with n dices -> approaches 3/4

max winning probability with 3 non-transitive lotteries approaches 0.62 (goldener schnitt)

Last changed5 months ago