02. Preferences under Uncertainty

• no, they are frequently stochastic

• -> e.g. buy lottery ticket?

-> lottery:

• function that assigns probabiliteis to all alternatives

• where probabilities add up to 1

• simply assign probability to all alternatives

• if it puts probabiility 1 on one alternative

where

L1, L2, … are simple lotteries

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

=> like doing two rounds of 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

• when k > 1

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

• 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

• support of 2

1. Most likely outcome: L2 > L1

   1. ML in L1: b

   2. ML in L2: a

   3. a > b => L2 > L1

2. L1 > L2

   1. least desirable: c

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

   3. => L1 > L2

3. 
   

• continuity

• independence

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…

-> L1 hawaii

-> L2 carribean

-> L3 plane crash

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

• savages sure thing principle

• 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…

• preference relation >= on L(A) is

  • rational, continuous and independent

<=>

• there exists a utility function u on A

• such that for two lotteries

• 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

• 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…

• 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)

• 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

• risk-seeking

• -> increasing utility with higher values

• completeness and transitivity

• continuity and independence (in stochastic settings)

• 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)