Choice Under Risk
This is a mapping from alternatives to numbers.
What is a utility function?
A choice situation is a choice under uncertainty in these conditions.
What is if the states of the world are relevant but not all states of the world have a probability assigned?
If two options, A and B, have the same prize, then you should choose A over B if and only if A has a higher probability than B of getting the prize.
What is the better odds condition?
A set of betting rates is coherent if and only if this condition holds.
What is not being vulnerable to a sure-loss contract?
The main objection to the maximin rule.
What is that maximin recommends choosing (A) $1 for sure over (B) a choice between $0 and $z, no matter the value of z?
This is the principle of choice for choice under certainty.
What is maximize expected utility?
An option, A, strongly dominates another option, B, if and only if this condition obtains.
What is the utility of A is greater than the utility of B in every state of the world?
This formula gives your betting rate, p, for some proposition, A.
These are the probability rules that sets of betting rates must satisfy to be coherent.
What are normality, certainty, and additivity?
Maz preferring (B) $10 if A is true over (C) $10 if a diamond is drawn at random from a standard deck of cards, for instance.
What is p > 1/4?
This is the general formula for maximizing expected utility.
EU(A) = Σ(pi * ui)
Choose the option whose worst outcome is at least as good as the worst outcome of any other option.
What is the maximin rule?
When p is a fair betting rate, you will be indifferent between these two options.
What is betting on A at rate p and betting on ~A at (1 - p)?
P2 of the argument for the basic rules of probability being rationally demanding.
What is that sets of betting rates should be coherent?
The rational choice between (A) a 1/2 chance at 10 utils and a 1/2 chance at -5 utils and (B) a 1/4 chance at 15 utils and a 3/4 chance at -2 utils
What is a(u(x)) + b?
An option, A, weakly dominates another option, B, if and only if this condition obtains.
What is the utility of A is at least as great as the utility of B in every state of the world and the utility of A is greater than B in at least one state of the world?
This table summarizes your possible gains and losses for a given betting rate and stake
What is a payoff matrix?
A contract with X such that X is guaranteed to lose no matter what happens.
What is a sure-loss contract?
The payoffs for the contract where C bets $5 to win $6 if A occurs and C bets $9 to win $4 if C does not occur.
What are -$1 and -$3?
A choice situation is a choice under risk in these conditions.
What is if the states of the world are relevant and each has a probability assigned?
Choose the option whose best outcome is at least as good as the best outcome of any other option.
What is the maximax rule?
There's even more than one way to express personal probabilities - this ratio between the opponent's bet and your bet is an example.
What are odds?
This well-read fellow makes you bet according to a sure-loss contract.
Who is the bookmaker?
The rational choice between (A) a 9/10 chance at 1 util and a 1/10 chance at 12 utils, (B) a 4/5 chance at 5 utils and a 1/5 chance at -2 utils, (C) a 1/4 chance at 8 utils, a 2/4 chance at 2 utils, and a 1/4 chance at -1 util, and (D) a 1/3 chance at 20 utils, a 1/3 chance at 0 utils, and a 1/3 chance at -10 utils.
What is B?