Presentation on theme: "Decision Analysis Scott Ferson, 25 September 2007, Stony Brook University, MAR 550, Challenger 165 HANDOUT MASTER HAS HEADER AND FOOTER."— Presentation transcript:
Decision Analysis Scott Ferson, email@example.com 25 September 2007, Stony Brook University, MAR 550, Challenger 165 HANDOUT MASTER HAS HEADER AND FOOTER ANIMATED
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 A i (strategies, decisions, choices) Scenarios S j Payoffs X ij for action A i in scenario S j Probability P j (if known) of scenario S j Decision criterion S 1 S 2 S 3 … A 1 X 11 X 12 X 13 … A 2 X 21 X 22 X 23 … A 3 X 31 X 32 X 33 ….... P 1 P 2 P 3 …
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 A i so (P j X ij ) is biggest
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 buying selling
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
What’s a fair price? The expected winnings would be a fair price –The chance of ending the game on the k th toss (i.e., the chance of getting k 1 heads in a row) is 1/2 k –If the game ends on the k th toss, the winnings would be 2 k 1 cents 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 click No “solution” really resolves the paradox –Bankrolls are actually finite –Can’t buy what’s not sold –Diminishing marginal utility of money If the game is truncated, then a related paradox emerges that we might call the “Powerball paradox” in which people grossly overpay (relative to the EU) to play the game. Consider the expected winnings of the St. Petersburg game if the coin tossing is limited to 35 tosses. The maximum possible winnings (i.e., 35 heads) would be about 172 million dollars. The EU for such a game would be less that 20 cents, yet a lot of people would be willing to throw away substantially more than this, perhaps a dollar or two, for even a remote chance at 172 million. The reason is that people overweight low probability events (Kahneman and Tversky >).
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 AGamble B Get $100 if draw red Get $100 if draw black Gamble CGamble D Get $100 if red or yellowGet $100 if black or yellow R > B R + Y < B + Y HERO
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 P j = 1 Analyst can come up with A i, S j, X ij, P j A i and S j are few in number X ij are unidimensional A i not rewarded/punished beyond payoff Picking A i doesn’t influence scenarios Uncertainty about X ij 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 P j are inconsistent Getting even subjective P j can be difficult
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 A i that gives largest minimum payoff (across S j ) Important if “gambler’s ruin” is possible (e.g. extinction) Chooses action C 1234 A105155 B201005 C 2015 D056025 Scenario S j Action A i
Maximax Optimistic decision maker Loss-tolerant decision maker Examine max payoffs across S j Select A i with the largest of these Prefers action D 1234 A105155 B201005 C 2015 D056025 Scenario S j Action A i
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 A i with highest average –If h=1, it’s maximin –If h=0 it’s maximax Favors D if h=0.5 1234 A105155 B201005 C 2015 D056025 Scenario S j Action A i
Minimax regret 1234 A1054520 B006020 C1004010 D20500 (20)(10)(60)(25) minuends 1234 A105155 B201005 C 2015 D056025 RegretPayoff Several competing decision makers Regret R ij = (max X ij under S j ) X ij Replace X ij with regret R ij Select A i with smallest max regret Favors action D
Bayes-Laplace Assume all scenarios are equally likely Use maximum expected value Chris Rock’s lottery investments Prefers action D
Pareto Choose an action if it can’t lose Select A i if its payoff is always biggest (across S j ) Chooses action B 1234 A105 55 B20153025 C10 2015 D052025 Scenario S j Action A i
Why not Complete lack of knowledge of P j 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 (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 Wishful thinking –Overreaching conclusions 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 ) Steven Milloy is a Fox News columnist whose website junkscience.com was hosted by the Cato Institute until revelvations that he had accepted money from Exxon while criticizing global warming research and from the Tobacco Institute while criticizing the evidence about adverse health effects of second-hand smoke.
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 One might expect Milloy, who studied law, to be familiar with this idea from jurisprudence, where a Type I error (an innocent person is convicted and the guilty person escapes justice) is considered much worse than the Type II error (acquitting the guilty person). (Milloy’s criticisms would have merit if he discussed the power of tests that don’t show significance)
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” Wald (1939)
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 non- quantitative 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 Transparency Proportionality Non-discrimination Consistency Explicit examination of the costs and benefits of action or inaction Review of scientific developments (Science 12 May 2000)
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
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 P j P j +1 Extremize partial averages of payoffs, e.g. max ( X ik / j ) j j k=1
Difficult example 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 Scenario 1Scenario 2Scenario 3Scenario 4 Action A7.5-5159 Action B5.59-515
If probabilities are ranked 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) Scenario 1Scenario 2Scenario 3Scenario 4 Action A7.5-5159 Action B5.59-515 Partial184.108.40.2066.62 averages220.127.116.11.12 Most likely Least likely
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
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 f a : S R i.e., f a 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 These four notions are identical with the traditional and Bayesian approaches. The difference under IP is that we may not have precise estimates for the probabilities of each of the possible values of the future state of the world S, or perfect knowledge about what the rewards might be exactly under each possible pairing of S and decision a.
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 Frequentist probability –Incertitude or other uncertainties in simulations may preclude our getting a precise estimate of a frequency Interval probability is the range between the largest buying price and the smallest selling price s/he accepts “I don't want to sell anything, buy anything…” (Lloyd Dobler [John Cusack] in the film Say Anything…) Lloyd Dobler
Comparing actions a and b Strictly preferreda > b E p ( f a ) > E p ( f b ) for all p M Almost preferreda b E p ( f a ) E p ( f b ) for all p M Indifferenta b E p ( f a ) = E p ( f b ) for all p M Incomparablea || b E p ( f a ) < E p ( f b ) and E q ( f a ) > E q ( f b ) some p,q M where E p ( f ) = p(s) f (s), and M is the set of possible probability distributions s Ss S
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 E p ( f b ) E p ( f a ) 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(f a ) > E(f b ) then action b is inadmissible because a interval-dominates b Admissible actions are those that are not inadmissible to any other action _ inadmissible dominant overlap
E-admissible Several IP decision criteria -maximax maximal -maximin interval dominance
Example 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 (after Troffaes 2004)
Problem setup p(H) [0.28, 0.7]p(T) [0.3, 0.72] f 1 (H) = 4 f 1 (T) = 0 f 2 (H) = 0 f 2 (T) = 4 f 3 (H) = 3 f 3 (T) = 2 f 4 (H) = 0.5 f 4 (T) = 3 f 5 (H) = 2.35 f 5 (T) = 2.35 f 6 (H) = 4.1 f 6 (T) = 0.3 f 7 (H) = 0.1 f 7 (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 It’s a (one-dimensional) space of probability measures 0101010101 p = 0p = 1p = ½p = 1 / 3 p = 2 / 3 01 p
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
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 right- wing 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. Cosmides, L., and J. Tooby. 1996. Are humans good intuitive statisticians after all? Rethinking some conclusions from the literature on judgment under uncertainty. Cognition 58:1-73.
Exercises 1.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? 2.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? 3.How are -maximin and maximin expected payoff related? ////////////////////////////////////////////////////////////////////////////////////////////////////// Answer to #1 Bankroll = $100 Max allowable tosses = 14 Liability if all tosses show heads = $81.92 Expected winnings from game limited to 14 tosses = 7 cents Bankroll = $100 million Max allowable tosses = 34 Liability if all tosses show heads = $85,899,345.92 Expected winnings from game limited to 14 tosses = 17 cents Judging by the popularity of “Power Ball” lotteries, a lot of people would likely spend between a dollar and five dollars to potentially win $85M ////////////////////////////////////////////////////////////////////////////////////////////////////// Answer to #2 Bw =[3,4.25] // reward for Beachhead under warm year Bn =3 // reward for Beachhead under normal year Lw = [1,3] // reward for Longliner under warm year Ln = [2,5] // reward for Longliner under normal year p = [.5,.75] // probability of a warm year for i = 0 to 25 do begin ii = i/25 P = left(p) + ii * width(p) B = P*Bw + (1-P)*Bn L = P*Lw + (1-P)*Ln say P, tab, left(B),tab,right(B),tab, left(L),tab,right(L) end 0.5 3 3.625 1.5 4 0.51 3 3.6375 1.49 3.98 0.52 3 3.65 1.48 3.96 0.53 3 3.6625 1.47 3.94 0.54 3 3.675 1.46 3.92 0.55 3 3.6875 1.45 3.9 0.56 3 3.7 1.44 3.88 0.57 3 3.7125 1.43 3.86 0.58 3 3.725 1.42 3.84 0.59 3 3.7375 1.41 3.82 0.6 3 3.75 1.4 3.8 0.61 3 3.7625 1.39 3.78 0.62 3 3.775 1.38 3.76 0.63 3 3.7875 1.37 3.74 0.64 3 3.8 1.36 3.72 0.65 3 3.8125 1.35 3.7 0.66 3 3.825 1.34 3.68 0.67 3 3.8375 1.33 3.66 0.68 3 3.85 1.32 3.64 0.69 3 3.8625 1.31 3.62 0.7 3 3.875 1.3 3.6 0.71 3 3.8875 1.29 3.58 0.72 3 3.9 1.28 3.56 0.73 3 3.9125 1.27 3.54 0.74 3 3.925 1.26 3.52 0.75 3 3.9375 1.25 3.5 //See graph at left // It seems likely that a conservation biologist would favor the “Beachhead” strategy // because it has about the same optimistic performance as the “Longliner” strategy // but its pessimistic performance would be a lot better. 0 1 2 3 4 0.50.60.70.8 Beachhead Longliner
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 Dutch book example HorseOffered oddsProbability Danger SpreeEvens0.5 Windtower3 to 1 against0.25 Shoeless Bob4 to 1 against0.2 0.95 (total)
Knight’s dichotomy bridged Decisions under risk Probabilities known Maximize expected utility Decisions under uncertainty Probabilities unknown Several possible strategies Decisions under imprecision Probabilities somewhat known E-admissability, partial averages, et al.
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 1/2 10
Decision theory paradoxes St. Petersburg Paradox Ellsberg Paradox Allais Paradox Borel Paradox Rumsfeld’s quandry (unknown unknowns) Open worldness Intransitivity