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*Decision under certainty =
you know what relavant states will occure
*Decision under ignorance =
dont know the probablities that certain states will occure
*Decision under risk =
you know the probabilities of the relavant states
*Conditional Probablity =
Probablity that E will occure given D occures
*Unconditional Probablity =
probablity that E will occure
*State-act independence =
state is independent of act if P(conditional) = P (unconditional)
*state-act dependence =
state is dependent of act if its not the case that P(conditional = P(unconditional).
*Normative Decision Theory =
how we should make decisions if ideally rational
*Descriptive Decision Theory =
How we make decisions
what are 2 attributes of states in a decision table?
- (i) mutally exclusive
- (ii) collectively exaustive
act A dominates act B if in a state-by-state comparison, A yeilds outcomes that are at least as good as thouse yeilded by B and in some states are better.
when won't P.Dominance hold?
if the probablity is affected by actions
Normative D.Theory assumes...
ideally rational agent
decision theory is concerned with what is rational not what is ...
we cant have S1 = 'I made right choice' and S2 = 'i made wrong choice because ...
you dont want your state descriptions to pickout outcome depending on action chosen
*ordering Condition =
minimal set of conditions that preferences of ideally rational agent must satisfy
when two preferences have equal utility
preferences are unknown
*Indifference class =
agent is indifferent between items in same class but prefers one item over another if in different class
number that reflects relative importance of preference to agent
*ordinal utility function =
scale where only order of agents utility prefrence is reflected
*ordinal transformations =
a transformation that preserves ordering of agents preferences
*Maximin Rule =
tells agent to compare minimum utilites for each act and choose act whose minimum is maximum for all minimums (i.e. maximize the minimum)
*Minimax Regret rule =
pick act whose maximum regret is minimal (regret score: R = MAX - U)
*Optimism-Pessimism Rule =
calculate aMAX + (1 - a)min for each act (where 'a' is an optimism index) and pick the act that is maximal.
*Positive Linear transformation =
when a > 0 and transfromation takes the form u' = au + b
*interval scales =
utility scales that admit only positve linear transformatios
*principle of insuff. reason =
calculate expected utility for all acts and assume that each state is equally probable (A= u1 + u2+...un/n). pick maximal result.
*expected utility =
evaluation of act in terms of its utility ( u1p + u2(1-p) for two state problem)
*Mixture condition =
if agent is indifferent between two acts, agent will be indifferent between them and the thrid act of flipping a fair coin and doing the first on heads and second on tails
*irrelevant expansion condition =
the addition of a new act, which is not regaurded as better then the orginal ones, will not change a rational agent's ranking of the old acts
*Rawls veil of Ignorance =
prevents ppl from knowing who they are, their social status, ancestry and talents
the 8 ordering conditions are...
- (O1) if xPy, then not yPx
- (O2) if xPy, then not xIy
- (O3) if xIy, then not xPy and not yPx
- (O4) xPy or yPx or xIy
- (O5) if xPy and yPz, then xPz
- (O6) if xPy and xIz, then zPy
- (O7) if xPy and yIz, then xPz (O8) if xIy and yIz, then xIz
What 3 conditions must ordinal utility functions satisfy?
- (a) u(x) > u(y) iff xPy
- (b) u(x) = u(y) iff xIy
- (c) w >= v iff t(w) >= t(v) for all w and v on the u-scale
Problems w/ maximin are...
prohibits opportunties with slight loss and great gains
problems with Minimax regret are...
addition of act that will not be chosen can change the decision
formula for O.P. rule is
aMax + (1 - a)min, where 0<= a <= 1 and a is the level of optimism
formula for Minimax Regret
- R - regret number
- U - Utility number in each square
- MAX - Maximum number in column
- R = MAX - U
what are problems with optimism pessimism rule??
- - optimism index is arbitrary
- (i.e.) agents can change their minds by changing optimism index
PRINCIPLE OF INSUFFICIENT REASON formula for 2 state problem:
u1p + u2(1-p)
problems with Principal of insufficient reason are ...
- 1) if there is no reason to assign one set of probablities rather then another, then there is no justification for assuming the states are equiprobable
- 2) could lead to bad results - one state with terrible outcomes or produces large regret could have greatest chance of being true
principal of insufficient reason requires utility scale to be ...
invarient under linear transformation
O.P. presupposes an interval...
the common ground in the Rawls vs Harsnayi debate is that ...
- principals of social justice must pass test of fairness
- principals must be accepable by agents with:
- -> rational
- -> self-interest
- -> under *veil of ignorance (prevents them from knowing how things will pan out)
Rawls argues for ...
- maximin rule
- - principals of social justice that protect intrests of ppl at the bottem
- -> difference principal: one society is better than another if worst
- -off members of former do better then worst of in the latter
Harsanyi argues for ...
- -principal of insufficient reason
- -principals that promote the average level of well-being in a society
- -> Don't look at worst off, we should compare societies in terms of average amounts of utilities
*expected monetary value =
multiply monetary value in each square by probablity in the square and then sum across row
*probability calculus =
mathmatical theory that enables us to calculate additional probablities
*absolute probability =
probablity that takes the form of P(S) = a
*conditional probablity =
probablity that takes the form P(S|W) = a
*probabilistic independence =
p is independent of q iff P(p) = P(p|q)
*Mutual exclusiveness =
p and q are mutually exclusive iff it is impossible for them both to be true
in situations where you don't know prior probablity, take best hunch as prior probability and use it to apply Bayes's theorm
*Objective interpertation =
see probablity as measuring something independent of human judgments, and consistent from person to person (further classified as logical or empirical)
*Subjective interpertation =
see probablity as measuring an individuals belief or confidence in a statement and permit it to vary from person to person
*propensity interpertation =
probability as a physical propensity or disposition, or tendency of a given type of physical situation to yield an outcome of a certain kind, or to yield a long run relative frequency of such an outcome
*relative frequency =
probablities are proportions or relative frequencies of events of one kind to those of another
*Degree of Belief Interp. =
probabilities are seen as a 'degree' to which one's confidence is on a certain statement
*Duch Book argument =
arg. for probabilism (namely the view that an agent's degrees of belief should satisfy the axioms of probability)
if agents subjective probablities are coherent, they obey the probability calculas
Suppose that D is the conjunction of statements describing new data. For each statement p, take PN(p) = PO(p|D).
V1(P1) + V2(P2) +...Vn(Pn)
Problems with EMV are ...
- - Neglects relevant factors that are non-monetary (e.g. pleasures, wellbeing)
- - EMV is supposed to be an average expected value of actions. What if we are only preforming action once?!
Example of absolute probablity ...
- probablity of drawing a hear at random from fair deck
- P(heart) = 1/4
example of conditional probablity...
- probablity of drawing a heart GIVEN that the card to be drawn is red
- P(heart|red) = 1/2
- P(red|heart) = 1 (every heart is red)
p is independent of q iff ...
P(p) = P(p|q)
p and q are mutually exclusive iff...
it is impossible for both to be true
if p is certain, then P(p) =...
if p and q are mutually exclusive, then P(p v q) =...
P(p) + P(q)
if p and q are equivalent, then P(p) =...
disjunction that works in all cases: P(p v q) =...
P(p) + P(q) - P(p & q)
P(p & q) =...
P(p) * P(q|p)
if ~(P(p) = 0), then P(q|p) =...
P(p & q)/P(p)
if q is independent of p, then P(p & q) =...
P(p) * P(q)
(inverse probablity law) if ~(P(q) = 0), then P(p|q) =...
[ P(p) * P(q|p) ] / P(q)
(Bayes's theorem) if ~(P(q) = 0), P(p|q)=...
[P(p) * P(q|p)] / [ P(p) * P(q|p) ] + [ P(~p) * P(q|~p)]
a satisfactory interpertation of probablity must ...
satisfy the probablity calculas
problem with relative frequencey...
- does not have firm grip on "true" probablity
- - no guarantee that frequencies to date are close to long-run frequencies
Duch Book Theorem:
suppose that no duch book can be made against an agent sing the odds he posts on the Dfinetti closure of set of statements. Then his betting quotients for DeFinetti closure in question satisfy the probablity calculas.
*interval scale =...
tells us about order of data points, and size of intervals inbetween data points (suffice for decisions under risk)
*Von Neumann theory of Utility =
measuring the strength of a person's prefrence for a thing by the risk he or she is willing to take to recieve it
a defining feature of an interval scale is that ...
zero rep. additional pt of measurement (i.e. zero is not absolute lowest val
ordinal utility scales do not suffice for making decisions under...
interval scale is all you need when making decison under...
risk in addition to ranking ofoutcomes
interval scale must satisfy what 2 principals ?
- (a) xPy iff u(x) > u(y)
- (b) XIy iff u(x) = u(y) pref interval between x and y is greater or equal to that bewtween z and w iff |u(x) - u(y)| >= |u(z) - u(w)|
decision strategy that attemps to meet an acceptability threshold and is contrasted with optimal decision making
3 problems with satisficing are ...
- 1) one ought to take into account decisions and info costs in deciding what to do
- 2) its hard to detrmine where to set the acceptability threshold
- 3) definitly choosing the lesser of two options -> IRRATIONAL!
*causal decision theory =
requires a causal connection between your actions and the desirable outcome.
*evidential decision theory =
takes observation at face value and uses those probablities to calculate EU
3 relationships between emotions and decisions?
- 1) emotions help decide pref.
- 2) emotions can affect assignment of probablities
- 3) emotions can cause you to fail to do the rational act (fail to max utility)
*Hot prefrence =
associated with strong feelings and emotions
*cold prefrence =
little to no emotions associated
why do we have emotions? what stratigic role might they play?
- *reputation: while being quick to anger has costs, it also prevents others from crossing you.
- *commitment problems: a problem where material facts in short run tempt you to do what what wold be against your long term intrest.