Lecture 7 Flashcards
What kind of probabilities are involved in decision under risk (DUR)?
Ones that are objectively given, known and agreed upon
See
Notation top of notes
What do prospects under risk do?
They map probabilities to outcomes (deals) which are called probability-contingent prospects
Define DUR?
In DUR, an objective probability measure P is given on the state space S, assigning each event E with its probability P(E) and R is the outcome set
Define prospects?
Probability distributions over outcomes
What would make different state-contingent prospects preferentially equivalent?
If they generate the same probability distribution over outcomes (eg. If different states/situations result in same outcomes w same probabilities)
See notes
Copper example - outcomes not state dependent
When can you apply De Finetti’s theorem to DUR?
When objective and subjective probabilities agree
See
Slide 11 objective vs subjective probability
Define certainty equivalent?
Guaranteed amount of return (cash) that would yield the exact same expected utility as a given risky asset
See
Notes st Petersburg paradox
Explain st Petersburg paradox?
The prospect of the game given by EV equals infinity but the CE is below £5 tf EV doesn’t hold.
What was Bernoulli’s solution to the st Petersburg paradox?
He realised that the value of an item depends on the utility it yields rather than the price tf proposed (see notes) leading to EU(g) = ln2
Utility function ln
How do you then get the CE equivalent of prospect g for st Petersburg paradox?
Assume CE is α, and tf α~g, using EU as the representation function of our preference for risk:
EU(α)=EU(g)
U(α)=ln2
Tf α=2=CE (less than 5, this agrees with empirical findings)
When does expected utility hold for DUR?
If and only if there is a strictly increasing utility function u, mapping the outcome to R, such that preferences maximise the EU of prospects defined by: p1u(x1)…pnu(xn)
