Transcript Decision criteria - RAMAS Books and Software by

Decision Analysis
Scott Ferson, [email protected]
25 September 2007, Stony Brook University, MAR 550, Challenger 165
Outline
• Risk and uncertainty
• Expected utility decisions
• St. Petersburg game, Ellsberg Paradox
• Decisions under uncertainty
• Maximin, maximax, Hurwicz, minimax regret, etc.
• Junk science and the precautionary principle
• Decisions under ranked probabilities
• Extreme expected payoffs
• Decisions under imprecision
• E-admissibility, maximality, -maximin, -maximax, etc.
• Synopsis and conclusions
Decision theory
• Formal process for evaluating possible actions
and making decisions
• Statistical decision theory is decision theory
using statistical information
• Knight (1921)
– Decision under risk (probabilities known)
– Decision under uncertainty (probabilities not known)
Discrete decision problem
•
•
•
•
•
Actions Ai (strategies, decisions, choices)
Scenarios Sj
Payoffs Xij for action Ai in scenario Sj
Probability Pj (if known) of scenario Sj
S1 S2 S3 …
Decision criterion
A1
A2
A. 3
..
X13 …
X23 …
X. 33 …
..
X11
X21
X.31
..
X12
X22
X. 32
..
P1
P2 P3 …
Decisions under risk
• If you make many similar decisions, then
you’ll perform best in the long run using
“expected utility” (EU) as the decision rule
• EU = maximize expected utility (Pascal 1670)
• Pick the action Ai so (Pj Xij) is biggest
Example
Action A
Action B
Action C
Action D
Probability
Scenario 1 Scenario 2 Scenario 3 Scenario 4
10
5
15
5
20
10
0
5
10
10
20
15
0
5
60
25
.5
.25
.15
.1
10*.5
20*.5
10*.5
0*.5
+ 5*.25
+ 10*.25
+ 10*.25
+ 5*.25
+ 15*.15
+ 0*.15
+ 20*.15
+ 60*.15
+ 5*.1
+ 5*.1
+ 15*.1
+ 25*.1
= 9
= 13
= 12
= 12.75
Maximizing expected utility prefers action B
Strategizing possible actions
• Office printing
–
–
–
–
Repair old printer
Buy new HP printer
Buy new Lexmark printer
Outsource print jobs
• Protection Australia marine resources
–
–
–
–
–
–
Undertake treaty to define marine reserve
Pay Australian fishing vessels not to exploit
Pay all fishing vessels not to exploit
Repel encroachments militarily
Support further research
Do nothing
Scenario development
• Office printing
– Printing needs stay about the same/decline/explode
– Paper/ink/drum costs vary
– Printer fails out of warranty
• Protection Australia marine resources
–
–
–
–
–
–
Fishing varies in response to ‘healthy diet’ ads/mercury scare
Poaching increases/decreases
Coastal fishing farms flourish/are decimated by viral disease
New longline fisheries adversely affect wild fish populations
International opinion fosters environmental cooperation
Chinese/Taiwanese tensions increase in areas near reserve
How do we get the probabilities?
• Modeling, risk assessment, or prediction
• Subjective assessment
– Asserting A means you’ll pay $1 if not A
– If the probability of A is P, then a Bayesian
agrees to assert A for a fee of $(1-P),
and to assert not-A for a fee of $P
– Different people will have different Ps for an A
Rationality
• Your probabilities must make sense
• Coherent if your bets don’t expose you to sure loss
– guaranteed loss no matter what the actual outcome is
• Probabilities larger than one are incoherent
• Dutch books are incoherent
– Let P(X) denote the price of a promise to pay $1 if X
– Setting P(Hillary OR Obama) to something other than the
sum P(Hillary) + P(Obama) is a Dutch book
If P(Hillary OR Obama) is smaller than the sum, someone could make
a sure profit by buying it from you and selling you the other two
St. Petersburg game
•
•
•
•
•
Pot starts at 1¢
Pot doubles with every coin toss
Coin tossed until “tail” appears
You win whatever’s in the pot
What would you pay to play?
.
First tail Winnings
1
0.01
2
0.02
3
0.04
4
0.08
5
0.16
6
0.32
7
0.64
8
1.28
9
2.56
10
5.12
11
10.24
12
20.48
13
40.96
14
81.92
15
163.84.
..
..
.
28 1,342,177.28
29 2,684,354.56
30
5,368,709.12.
..
..
.
What’s a fair price?
• The expected winnings would be a fair price
– The chance of ending the game on the kth toss (i.e.,
the chance of getting k1 heads in a row) is 1/2k
– If the game ends on the kth toss, the winnings
would be 2k1 cents
1
1
1
1
1
EU  1   2   4   8  16  
2
4
8
16
32
1 1 1 1 1 1 1
       
2 2 2 2 2 2 2

So you should be willing to pay any
price to play this game of chance
St. Petersburg paradox
• The paradox is that nobody’s gonna pay more
than a few cents to play
• To see why, and for a good time, call
http://www.mathematik.com/Petersburg/Petersburg.html click
• No “solution” really resolves the paradox
– Bankrolls are actually finite
– Can’t buy what’s not sold
– Diminishing marginal utility of money
Utilities
• Payoffs needn’t be in terms of money
• Probably shouldn't be if marginal value of
different amounts vary widely
– Compare $10 for a child versus Bill Gates
– A small profit may be a lot more valuable than the
amount of money it takes to cover a small loss
• Use utilities in matrix instead of dollars
Risk aversion
• EITHER get $50
• OR get $100 if a randomly drawn ball is red
from urn with half red and half blue balls
• Which prize do you want?
$50
EU is the same, but most people take the sure $50
Ellsberg Paradox
• Balls can be red, black or yellow (probs are R, B, Y)
• A well-mixed urn has 30 red balls and 60 other balls
• Don’t know how many are black, how many are yellow
Gamble A R > B
Get $100 if draw red
Gamble B
Get $100 if draw black
Gamble C
Gamble D R + Y < B + Y
Get $100 if red or yellow Get $100 if black or yellow
Persistent paradox
• Most people prefer A to B (so are saying
R.> B) but also prefer D to C (saying R < B)
• Doesn’t depend on your utility function
• Payoff size is irrelevant
• Not related to risk aversion
• Evidence for ambiguity aversion
– Can’t be accounted for by EU
Ambiguity aversion
•
•
•
•
Balls can be either red or blue
Two urns, both with 36 balls
Get $100 if a randomly drawn ball is red
Which urn do you wanna draw from?
Assumptions

• Discrete decisions

• Closed world Pj = 1

• Analyst can come up with Ai, Sj, Xij, Pj

• Ai and Sj are few in number

• Xij are unidimensional
.
• Ai not rewarded/punished beyond payoff
• Picking Ai doesn’t influence scenarios

• Uncertainty about Xij is negligible
Why not use EU?
•
•
•
•
•
•
Clearly doesn’t describe how people act
Needs a lot of information to use
Unsuitable for important unique decisions
Inappropriate if gambler’s ruin is possible
Sometimes Pj are inconsistent
Getting even subjective Pj can be difficult
Decisions under uncertainty
Decisions without probability
•
•
•
•
•
•
Pareto (some action dominates in all scenarios)
Maximin (largest minimum payoff)
Maximax (largest maximum payoff)
Hurwicz (largest average of min and max payoffs)
Minimax regret (smallest of maximum regret)
Bayes-Laplace (max EU assuming equiprobable scenarios)
Maximin
• Cautious decision maker
• Select Ai that gives largest minimum payoff
(across Sj)
• Important if “gambler’s ruin” is possible
(e.g. extinction)
Scenario S
Action Ai
j
• Chooses action C
1
2
3
4
A
10
5
15
5
B
20
10
0
5
C
10
10
20
15
D
0
5
60
25
Maximax
•
•
•
•
Optimistic decision maker
Loss-tolerant decision maker
Examine max payoffs across Sj
Select Ai with the largest of these
Action Ai
1
• Prefers action D
Scenario Sj
2
3
4
A
10
5
15
5
B
20
10
0
5
C
10
10
20
15
D
0
5
60
25
Hurwicz
• Compromise of maximin and maximax
• Index of pessimism h where 0  h  1
• Average min and max payoffs weighted by
h and (1h) respectively
• Select Ai with highest average
Scenario S
• Favors D if h=0.5
Action Ai
– If h=1, it’s maximin
– If h=0 it’s maximax
j
1
2
3
4
A
10
5
15
5
B
20
10
0
5
C
10
10
20
15
D
0
5
60
25
Minimax regret
•
•
•
•
Several competing decision makers
Regret Rij = (max Xij under Sj)Xij
Replace Xij with regret Rij
Select Ai with smallest max regret
Payoff
1
2
3
4
A
10
5
15
5
B
20
10
0
C
10
10
D
0
5
Regret
1
2
3
4
A
10
5
45
20
5
B
0
0
60
20
20
15
C
10
0
40
10
60
25
D
20
5
0
0
(20)
(10)
• Favors action D
(60) (25)
minuends
Bayes-Laplace
• Assume all scenarios are equally likely
• Use maximum expected value
• Chris Rock’s lottery investments
10*.25 + 5*.25 + 15*.25 + 5*.25
20*.25 + 10*.25 + 0*.25 + 5*.25
10*.25 + 10*.25 + 20*.25 + 15*.25
0*.25 + 5*.25 + 60*.25 + 25*.25
• Prefers action D
= 8.75
= 8.75
= 13.75
= 22.5
Pareto
• Choose an action if it can’t lose
• Select Ai if its payoff is always biggest
(across Sj)
Action Ai
1
• Chooses action B
Scenario Sj
2
3
4
A
10
5
5
5
B
20
15
30
25
C
10
10
20
15
D
0
5
20
25
Why not
• Complete lack of knowledge of Pj is rare
• Except for Bayes-Laplace, the criteria
depend non-robustly on extreme payoffs
• Intermediate payoffs may be more likely
than extremes (especially when extremes
don’t differ much)
Junk science and the
precautionary principle
Junk science (sensu Milloy)
• “Faulty scientific data and analysis used to further a
special agenda”
– Myths and misinformation from scientists, regulators, attorneys, media, and
activists seeking money, fame or social change that create alarm about
pesticides, global warming, second-hand smoke, radiation, etc.
• Surely not all science is sound
– Small sample sizes
– Overreaching conclusions
 Wishful thinking
 Sensationalized reporting
• But Milloy has a very narrow definition of science
– “The scientific method must be followed or you will soon find yourself
heading into the junk science briar patch. … The scientific method [is] just
the simple and common process of trial and error. A hypothesis is tested until
it is credible enough to be labeled a ‘theory’…. Anecdotes aren’t data. …
Statistics aren’t science.” (http://www.junkscience.com/JSJ_Course/jsjudocourse/1.html)
Science is more
•
•
•
•
•
•
•
Hypothesis testing
Objectivity and repeatability
Specification and clarity
Coherence into theories
Promulgation of results
Full disclosure (biases, uncertainties)
Deduction and argumentation
Classical hypothesis testing
• Alpha
– probability of Type I error
– i.e., accepting a false statement
– strictly controlled, usually at 0.05 level, so false
statements don’t easily enter the scientific canon
• Beta
–
–
–
–
probability of Type II error
rejecting a true statement
one minus the power of the test
traditionally left completely uncontrolled
Decision making
• Balances the two kinds of error
• Weights each kind of error with its cost
• Not anti-scientific, but it has much broader
perspective than simple hypothesis testing
Why is a balance needed?
• Consider arriving at the train station 5 min
before, or 5 min after, your train leaves
– Error of identical magnitudes
– Grossly different costs
• Decision theory = scientific way to make
optimal decisions given risks and costs
Statistics in the decision context
• Estimation and inference problems can be
reexpressed as decision problems
• Costs are determined by the use that will be
made of the statistic or inference
• The question isn’t just “whether” anymore,
it’s “what are we gonna do”
It’s not your father’s statistics
• Classical statistics
addresses the use of sample information to make
inferences which are, for the most part, made
without regard to the use to which they’ll be put
• Modern (Bayesian) statistics
combines sample information, plus knowledge of
the possible consequences of decisions, and prior
information, in order to make the best decision
Policy is not science
• Policy making may be sound even if it does
not derive specifically from application of the
scientific method
• The precautionary principle is a nonquantitative way of acknowledging the
differences in costs of the two kinds of errors
Precautionary principle (PP)
• “Better safe than sorry”
• Has entered the general discourse,
international treaties and conventions
• Some managers have asked how to “defend
against” the precautionary principle (!)
• Must it mean no risks, no progress?
Two essential elements
• Uncertainty
– Without uncertainty, what’s at stake would be clear
and negotiation and trades could resolve disputes.
• High costs or irreversible effects
– Without high costs, there’d be no debate. It is these
costs that justify shifting the burden of proof.
Proposed guidelines for using PP
(Science 12 May 2000)
•
•
•
•
•
Transparency
Proportionality
Non-discrimination
Consistency
Explicit examination of the costs and
benefits of action or inaction
• Review of scientific developments
But consistency is not essential
• Managers shouldn’t be bound by prior risky
decisions
• “Irrational” risk choices very common
– e.g., driving cars versus pollutant risks
• Different risks are tolerated differently
– control
– scale
– fairness
Take-home messages
• Guidelines (a lumper’s version)
– Be explicit about your decisions
– Revisit the question with more data
• Quantitative risk assessments can overrule PP
• Balancing errors and their costs is essential
for sound decisions
Ranked probabilities
Intermediate approach
• Knight’s division is awkward
– Rare to know nothing about probabilities
– But also rare to know them all precisely
• Like to have some hybrid approach
Kmietowicz and Pearman (1981)
• Criteria based on extreme expected payoffs
– Can be computed if probabilities can be ranked
• Arrange scenarios so Pj  Pj +1
• Extremize partial averages of payoffs, e.g.
j
max (  Xik / j )
j
k=1
Difficult example
Scenario 1 Scenario 2 Scenario 3 Scenario 4
Action A
Action B
7.5
5.5
-5
9
15
-5
9
15
• Neither action dominates the other
• Min and max are the same so maximin,
maximax and Hurwicz cannot distinguish
• Minimax regret and Bayes-Laplace favor A
If probabilities are ranked
Scenario 1 Scenario 2 Scenario 3 Scenario 4
Action A
7.5
-5
15
9
Action B
5.5
9
-5
15
Most likely
Partial
averages
•
•
•
•
7.5
5.5
Least likely
1.25
7.25
5.83
3.17
6.62
6.12
Maximin chooses B since 3.17 > 1.25
Maximax slightly prefers A because 7.5 > 7.25
Hurwicz favors B except in strong optimism
Minimax regret still favors A somewhat (7.33 > 7)
Sometimes appropriate
• Uses available information more fully than
criteria based on limiting payoffs
• Better than decisions under uncertainty if
– several decisions are made (even if actions,
scenarios, payoffs and rankings change)
– number of scenarios is large (because standard
methods ignore intermediate payoffs)
Extreme expected payoffs
• Focusing on maximin expected payoff is
more conservative than traditional maximin
• Focusing on maximax expected payoff is
more optimistic than the old maximax
• Focusing on minimax expected regret will
have less regret than using minimax regret
Generality
• Robust to revising scenario ranks
Mostly a selected action won't change by inversion
of ranks or by the introduction of a new scenario
• Can easily extend to intervals for payoffs
Max (min) expected values found by applying partial
averaging technique to all upper (lower) limits
When it’s useful
• For difficult payoff matrices
• When you can only rank scenarios
• When multiple decision must be made
• When the number of scenarios is large
• When facing identical problem a small number
of times (up to 10 or so)
When you shouldn’t use it
• If probability ranks are rank guesses
• If you actually know the risks
• When you face the identical problem often
– You should be able to estimate probabilities
Imprecise probabilities
Decision making under imprecision
• State of the world is a random variable, S  S
• Payoff (reward) of an action depends on S
• We identify an action a with its reward fa : S  R
i.e., fa is the action and its entire row from the payoff matrix
• We’d like to choose the decision with the largest
expected reward, but without precisely specifying
1) the probability measure governing scenarios S
2) the payoff from an action a under scenario S
Imprecision about probabilities
• Subjective probability
– Bayesian “rational agents” are compelled to either sell
or buy any bet, but rational agents could decline to bet
– Interval probability for event A is the range between the
largest P such that, for a fee of $(1P), you agree to pay
$1 if A is doesn’t occur, and the smallest Q such that,
for a fee of $Q, you agree to pay $1 if A occurs
Lloyd Dobler
• Frequentist probability
– Incertitude or other uncertainties in simulations may
preclude our getting a precise estimate of a frequency
Comparing actions a and b
Strictly preferred
Almost preferred
Indifferent
Incomparable
where Ep( f ) =
a>b
ab
ab
a || b
Ep( fa) > Ep( fb)
Ep( fa)  Ep( fb)
Ep( fa) = Ep( fb)
Ep( fa) < Ep( fb)
Eq( fa) > Eq( fb)
for all p M
for all p M
for all p M
and
some p,q M
 p(s) f (s), and
sS
M is the set of possible probability distributions
E-admissibility
• Fix p in M and, assuming it’s the correct
probability measure, see which decision
emerges as the one that maximizes EU
• The result is then the set of all such decisions
for all p  M
Alternative: maximality
• Maximal decisions are undominated over all p
a is maximal if there’s no b where Ep(fb)  Ep(fa) for all p  M
•
• Actions cannot be
linearly ordered,
but only partially
ordered
•
•
•
•
•
•
•
•
•
•
•
•
Another alternative: -maximin
• We could take the decision that maximizes the
worst-case expected reward
• Essentially a worst-case optimization
• Generalizes two criteria from traditional theory
– Maximize expected utility
– Maximin
Interval dominance
_
• If E(fa) > E(fb) then action b is inadmissible
because a interval-dominates b
dominant
overlap
inadmissible
• Admissible actions are those that are not
inadmissible to any other action
Several IP decision criteria
-maximin
maximal
-maximax
E-admissible
interval dominance
Example
(after Troffaes 2004)
• Suppose we are betting on a coin toss
– Only know probability of heads is in [0.28, 0.7]
– Want to decide among seven available gambles
1: Pays 4 for heads, pays 0 for tails
2: Pays 0 for heads, pays 4 for tails
3: Pays 3 for heads, pays 2 for tails
4: Pays ½ for heads, pays 3 for tails
5: Pays 2.35 for heads, pays 2.35 for tails
6: Pays 4.1 for heads, pays 0.3 for tails
7: Pays 0.1 for heads, pays 0.1 for tails
Problem setup
p(H)  [0.28, 0.7]
f1(H) = 4
f2(H) = 0
f3(H) = 3
f4(H) = 0.5
f5(H) = 2.35
f6(H) = 4.1
f7(H) = 0.1
p(T)  [0.3, 0.72]
f1(T) = 0
f2(T) = 4
f3(T) = 2
f4(T) = 3
f5(T) = 2.35
f6(T) = 0.3
f7(T) = 0.1
M
• M is all Bernoulli probability distributions (mass at
only two points, H and T) such that 0.28  p(H)  0.7
p=0
0
p = 1/3
1
0
1
p=½
0
1
p = 2/3
0
p=1
1
0
1
• It’s a (one-dimensional) space of probability measures
0
p
1
Reward
3
2
3
5
2
1
4
6
1
Action 1
Action 2
Action 3
Action 4
Action 5
Action 6
Action 7
7
0
0.2
0.3
0.4
0.5
p(H)
0.6
0.7
0.8
E-admissibility
Probability
p(H) < 2/5
p(H) = 2/5
2/ < p(H) < 2/
5
3
p(H) = 2/3
2/ < p(H)
3
Preference
2
2, 3 (indifferent)
3
1, 3 (indifferent)
1
Criteria yield different answers
-maximin
{5}
-maximax
{2}
maximal
{1,2,3,5}
E-admissible
{1,2,3}
interval dominance
{1,2,3,5,6}
So many answers
• Different criteria are useful in different settings
• The more precise the input, the tighter the outputs
•  criteria usually yield only one decision
•  criteria not good if many sequential decisions
• Some argue that E-admissibility is best overall
• Maximality is close to E-admissibility, but might be
easier to compute for large problems
Traditional Bayesian answer
• Decision allows only one action, unless
we’re indifferent between actions
• Action 3 (or possibly 2, or even 1); different
people would get different answers
• Depends on which prior we use for p(H)
• Typically do not express any doubt about
the decision that’s made
IP versus traditional approaches
• Decisions under IP allow indecision when
your uncertainty entails it
• Bayes always produces a single decision
(up to indifference), no matter how little
information may be available
• IP unifies the two poles of Knight’s division
into a continuum
Comparison to Bayesian approach
• Axioms identical except IP doesn’t use completeness
• Bayesian rationality implies not only avoidance of
sure loss & coherence, but also the idea that an agent
must agree to buy or sell any bet at one price
• “Uncertainty of probability” is meaningful, and it’s
operationalized as the difference between the max
buying price and min selling price
• If you know all the probabilities (and utilities)
perfectly, then IP reduces to Bayes
Why Bayes fares poorly
• Bayesian approaches don’t distinguish ignorance
from equiprobability
• Neuroimaging and clinical psychology shows
humans strongly distinguish uncertainty from risk
– Most humans regularly and strongly deviate from Bayes
– Hsu (2005) reported that people who have brain lesions
associated with the site believed to handle uncertainty
behave according to the Bayesian normative rules
• Bayesians are too sure of themselves (e.g., Clippy)
IP does groups
• Bayesian theory does not work for groups
– Rationality inconsistent with democratic process
• Scientific decision are not ‘personal’
– Teams, agencies, collaborators, companies, clients
– Reviewers, peers
• IP does generalize to group decisions
– Can be rational and coherent if indecision is
admitted occasionally
Take-home messages
• Antiscientific (or at least silly) to say you
know more than you do
• Bayesian decision making always yields one
answer, even if this is not really tenable
• IP tells you when you need to be careful and
reserve judgment
Synopsis and conclusions
Decisions under risk
How?
– Payoffs and probabilities are known
– Select decision that maximizes expected utility
Why?
– If you make many similar decisions, then you’ll
perform best in the long run using this rule
Why not?
–
–
–
–
–
Needs a lot of information to use
Unsuitable for important unique decisions
Inappropriate if gambler’s ruin is possible
Getting subjective probabilities can be difficult
Sometimes probabilities are inconsistent
Bayesian (personalist) decisions
• Not a good description of how people act
– Paradoxes (St. Petersburg, Ellsberg, Allais, etc.)
• No such thing as a ‘group decision’
– Review panels, juries, teams, corporations
– Cannot maintain rationality in this context
– Unless run as a constant dictatorship
Multi-criteria decision analysis
• Used when there are multiple, competing goals
– E.g., USFS’ multiple use (biodiversity, aesthetics, habitat, timber, recreation,…)
– No universal solution; can only rank in one dimension
• Group decision based on subjective assessments
• Organizational help with conflicting evaluations
– Identifying the conflicts
– Deriving schemes for a transparent compromise
• Several approaches
– Analytic Hierarchy Process (AHP); Evidential Reasoning;
Weight of Evidence (WoE)
Analytic Hierarchy Process
• Identify possible actions
– buy house in Stony Brook / PJ / rent
• Identify and rank significant attributes
– location > price > school > near bus
• For each attribute, and every pair of actions, specify
preference
• Evaluate consistency (transitivity) of the matrix of
preferences by eigenanalysis
• Calculate a score for each alternative and rank
• Subject to rank reversals (e.g., without Perot, Bush beat Clinton)
Decision under uncertainty
How?
– Probabilities are not known
– Use a criterion corresponding to your attitude about risk
(Pareto, Maximin, Maximax, Hurwicz, Minimax regret, Bayes-Laplace, etc.)
– Select an optimal decision under this criterion
Why?
– Answer reflects your attitudes about risk
Why not?
– Complete ignorance about probabilities is rare
– Results depend on extreme payoffs, except for Bayes-Laplace
– Intermediate payoffs may be more likely than extremes
(especially when extremes don’t differ much)
Why IP?
•
•
•
•
•
•
Uses all available information
Doesn’t require unjustified assumptions
Tells you when you don’t know
Conforms with human psychology
Can make rational group decisions
Better in uncertainty-critical situations
– Gains and losses heavily depend on unknowns
– Nuclear risk, endangered species, etc.
Policy juggernauts
• “Precautionary principle” is a mantra
intoned by lefty environmentalists
• “Junk science” is an epithet used by rightwing corporatists
• Yet claims made with both are important
and should be taken seriously, and can be
via risk assessment and decision analysis
Underpinning for regulation
• Narrow definition of science?
– Hypothesis testing is clearly insufficient
– Need to acknowledge differential costs
• Decision theory?
– Decision theory only optimal for unitary decision maker
(group decisions are much more tenuous)
– Gaming the decision is rampant
• Maybe environmental regulation should be modeled
on game theory instead of decision theory
Game-theoretic strategies
• Building trust
– explicitness
– reciprocity
– inclusion of all stake holders
• Checking
– monitoring
– adaptive management
– renewal licensing
• Multiplicity
– sovereignty, subsidiarity
– some countries take a risk (GMOs, thalidomide)
References
•
Foster, K.R., P. Vecchia, and M.H. Repacholi. 2000.Science and the precautionary principle.
Science 288(12 May): 979-981.
•
Hsu, M., M. Bhatt, R. Adolphs, D. Tranel, and C.F. Camerer. 2005. Neural systems responding
to degrees of uncertainty in human decision-making. Science 310:1680-1683.
Kikuti, D., F.G. Cozman and C.P. de Campos. 2005. Partially ordered preferences in decision trees:
computing strategies with imprecision in probabilities. Multidisciplinary IJCAI-05 Workshop on
Advances in Preference Handling, R. Brafman and U. Junker (eds.), pp. 118-123.
http://wikix.ilog.fr/wiki/pub/Preference05/WebHome/P40.pdf
Kmietowicz, Z.W. and A.D. Pearman.1976. Decision theory and incomplete knowledge: maximum
variance. Journal of Management Studies 13: 164–174.
Kmietowicz, Z.W. and A.D. Pearman. 1981. Decision Theory and Incomplete Knowledge. Gower,
Hampshire, England.
Knight, F.H. 1921. Risk, Uncertainty and Profit. L.S.E., London.
Milloy, S. http://www.junkscience.com/JSJ_Course/jsjudocourse/1.html
Plous, S. 1993. The Psychology of Judgment and Decision Making. McGraw-Hill.
Sewell, M. “Expected utility theory” http://expected-utility-theory.behaviouralfinance.net/
Troffaes, M. 2004. Decision making with imprecise probabilities: a short review. The SIPTA
Newsletter 2(1): 4-7.
Walley, P. 1991. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London.
•
•
•
•
•
•
•
•
•
Exercises
1.
2.
3.
If you have a bankroll of $100, how many tosses could you allow in a
finite version of the St. Petersburg game (if you were compelled to pay
the winnings from this bankroll)? What are the expected winnings for
a game limited to this many tosses? How much do you think your
friends might actually pay to play this game? What are the numbers if
the bankroll is $100 million?
Demographic simulations of an endangered marine turtle suggest that
conservation strategy “Beachhead” will yield between 3 and 4.25 extra
animals per unit area if it’s a normal year but only 3 if it’s a warm year,
and that strategy “Longliner” will produce between 1 and 3 extra
animals in a normal year and between 2 and 5 extra animals in a warm
year. Assume that the probability of next year being warm is between
0.5 and 0.75. Graph the reward functions for these two strategies as a
function of the probability of next year being warm. Which strategy
would conservationists prefer? Why?
How are -maximin and maximin expected payoff related?
End
Dutch book example
Horse
Danger Spree
Windtower
Shoeless Bob
Offered odds
Evens
3 to 1 against
4 to 1 against
Probability
0.5
0.25
0.2
0.95 (total)
A gambler could lock in a profit of 10, by betting
100, 50 and 40 on the three horses respectively
http://en.wikipedia.org/wiki/Dutch_book
Knight’s dichotomy bridged
Decisions under risk
Probabilities known
Maximize expected utility
Decisions under imprecision
Probabilities somewhat known
E-admissability, partial averages, et al.
Decisions under uncertainty
Probabilities unknown
Several possible strategies
Bayesians
• Updating with Bayes’ rule
• Subjective probabilities (defined by bets)
• Decision analysis context
• Distribution even for “fixed” parameter
• Allows natural confidence statements
• Uses all information, including priors
Prior information
• Suppose I claim to be able to distinguish
music by Haydn from music by Mozart
• What if the claim were that I can predict the
flips of a coin taken from your pocket?
• Prior knowledge conditions us to believe the
former claim but not the latter, even if the
latter were buttressed by sample data of 10
flips at a significance level of 1/210
Decision theory paradoxes
•
•
•
•
•
•
•
St. Petersburg Paradox
Ellsberg Paradox
Allais Paradox
Borel Paradox
Rumsfeld’s quandry (unknown unknowns)
Open worldness
Intransitivity