decisions under ignorance
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DESCRIPTIONDecisions under Ignorance. Last Time. Last time we introduced decision theory, the scientific/ philosophical study of how best to make your decisions, given what your goals and aspirations are. . Problem Specification. - PowerPoint PPT Presentation
Decisions under IgnoranceLast TimeLast time we introduced decision theory, the scientific/ philosophical study of how best to make your decisions, given what your goals and aspirations are. Problem SpecificationSolving a decision problem begins with a problem specification, breaking down the problem into three components:Acts: the various (relevant) actions you can take in the situation.States: the different ways that things might turn out (coin lands heads, coin lands tails).Outcomes: What results from the various acts in the different states.ExampleFor example, we considered a simple decision problem involved in betting on a sports team:Acts: Bet on the Dragons, bet on the Cherry blossoms.States: Dragons win, Cherry Blossoms win.Outcomes: +$500 for successful bet on Dragons, +$300 for successful bet on Cherry Blossoms, -$200 for any lost bet.Decision TablesDecisions, after being analyzed by a problem specification, can be summarized in a decision table, with the Acts on the left, the States on the top, and the corresponding Outcomes for each Act, State pair in their appropriate locations.Example Table:Dragons WinCherry Blossoms WinI bet on Dragons+$500-$200I bet on Cherry Blossoms-$200+$300Independence of States on ActsOne important feature we learned about was that the states we specify must not depend on which actions we take. For instance, the state I win depends on whether I bet on the Dragons or the Cherry Blossoms. When states depend on acts, we run the risk of incorrectly solving the decision problem.Immediate ChoicesWhen we make a decision table, we have to make choices about how to describe the acts and states. And obviously decision theory cant help us with those choices. It cant help us with a lot of choices: if a bear is chasing you, you dont need or want a decision table to make a choice to run away. But for more difficult problems that require advance planning, decision theory can be useful.Right Choice vs. Rational ChoiceDecision theory is not concerned with making the right choice its concerned with making the rational choice when you dont know what the right choice is.The right choice is the choice that gets you the best outcome. If you are betting on sports, you typically wont know what the right choice is (who will win). Sometimes the rational choice can still be the wrong choice.Varieties of DecisionsDecision theorists classify decisions into three categories:
Decisions under certaintyDecisions under riskDecisions under ignoranceDecisions under CertaintySometimes you know for certain what outcome an act will have. At a restaurant you are faced with a decision: do I order the soup or the salad? You know that if you order the soup, the outcome will be that you get soup, and if you order the salad, the outcome will be that you get salad. Any decision where all the acts are like this you know their outcomes in advance is a decision under certainty.Hard DecisionsJust because a decision is made under certainty does not mean it is an easy or obvious decision.
Deciding which classes to take is a decision under certainty (often): you get what you sign up for. But should you take philosophy this semester or music? That can be a difficult decision.Formal ApproachesWe wont be talking about decisions under certainty in this class.
There are formal approaches to these problems (for example, in the mathematics of linear programming), but thats a little bit out of our range.Decisions under RiskSometimes it is possible to assign determinate (or approximate) probabilities to various states.
For example, when I flip a fair coin, I know that the probability that it lands heads will be 0.5 and the probability that it lands tails will be 0.5. So if I bet on a fair coin (or an unfair coin, when I know its probabilities) that is a decision under risk.IdealizationA lot of things are very difficult to assign exact probabilities to, but that doesnt mean they shouldnt be treated as decisions under risk.
Sometimes confronting a problem requires idealization. You dont know exactly the probability that your plane flight will crash, but you can estimate it by considering the number of crashes divided by the number of flights.Next WeekNext week well talk about decisions under risk. They require a little bit more of a mathematical approach.
The general idea is to calculate which outcome we expect to happen given each of our acts, and then to choose the act that has the best expected outcome.Decisions under IgnoranceA large number of decisions are decisions under ignorance (also known as a decision under uncertainty).
These are decisions where we cannot assign probabilities to some of the states, or where our estimates are too wide (somewhere between a 20% and 80% chance). ExampleFor example, suppose you go on a first date with someone. What is the probability that you will wind up getting married to them and subsequently having children and grandchildren with them? You certainly cant get an estimate by dividing your total number of marriages by your total number of first dates you probably dont have a very high number of either.ExampleSo suppose your friend calls you and asks if you want to go see the new James Bond movie instead of going on this date. You really want grandchildren someday, but you cant assign a probability to the state that you will have grandchildren with your date. This is a decision under ignorance.Second ExampleSuppose that Susan is starting her career in business. She has been very successful so far, and has made lots of contacts, and thinks that if she keeps at it, she will be able to become very rich.Susans ChoiceSusan is recently married, and she wants to have children some day. She is faced with the following decision: should she have children now and postpone her career, or should she keep on the successful track shes on and have children later? She knows whatever she does now shell be good at, but she doesnt know what shell be good at later.Decision under IgnoranceSusan has read a lot of parenting books and studies done on parenting late in life, so she is pretty confident that she can assign a probability to being a good parent later vs. now. However, she has no idea whether she will be able to resume a successful business career after staying home for years to raise her children. She cannot assign that state a probability, so this is a decision under ignorance.Note[Note: Im not saying that women have to stay home when they have children. Susan just happens to be conservative, and she thinks that women should not work when their children are young. As decision theorists, we dont judge what other peoples goals are, we just help them make the best decisions to achieve those goals.]ActsThe acts here are clear:
A1: Susan has children now, and a career later.
A2: Susan has a career now, and children later.Relevant StatesS1: In 7 years, Susan will be able to be both a good mother and a good businesswoman.S2: In 7 years, she will be able to be a good mother, but not a good businesswoman.S3: In 7 years, she will not be able to be a good mother, but she will be able to be a good businesswoman.S4: In 7 years, she will neither be able to be a good mother nor a good businesswoman.OutcomesThe outcome for each act-state pair is pretty clear here. For example, for A2 and S1:A2: Susan has a career now, and children later.S1: In 7 years, she will be able to be a good mother but not able to be a good businesswoman.O1: She is a good mother later and a good businesswoman now.It will be easier if we can summarize things as follows:
M = Susan is a good mother.M = Susan is not a good mother.B = Susan is a good businesswoman.B = Susan is not a good businesswoman.Susans Decision TableS1S2S3S4A1: children nowM & BM & BM & BM & BA2: career nowM & B M & BM & BM & BPreferencesHow is Susan going to make this decision. Well, first, a decision theorist needs to know her preferences. Which outcomes does she prefer to which other outcomes.
For example, clearly Susan prefers M & B to M & B: she prefers being a good mother and having a good career to just being a bad mother and having a good career.PreferencesBut what about what happens in S4: if she chooses children now, shell be a good mother, but not a good businesswoman (M & B). If she chooses a career now, shell be a good businesswoman, but not a good mother (M & B). Which of these two does she prefer, being only a good mother or only a good businesswoman?IndifferenceSusan might have a definite preference, but we also allow her to be indifferent: she might think that (M & B) and (M & B) are equally good outcomes, and she would be equally happy with either of them.Rational PreferencesWere still not in a position to help Susan. In order to help her, the decision theorist requires that her preferences be rational. Rationality here has nothing to do with which things she prefers. Its OK if she prefers M & B to M & B. That would be strange, but its not irrational. Rational PreferencesRational preferences have the following features:
They are connected.They are (appropriately) asymmetrical.They are transitive.ConnectedYour preferences for outcomes O1, O2, O3, are connected if, for any two of those outcomes X and Y:
Either you prefer X to YOr you prefer Y to XOr you are indifferent between X and YAsymmetricalAdditionally, rational preferences have certain asymmetries. If X and Y are outcomes, then:
You cant both prefer X to Y and prefer Y to X.You cant both prefer X to Y and be indifferent between X and Y.NoteWe can combine connectedness and asymmetry and get:One and only one of the following is true for any two outcomes X and Y:
You prefer X to Y.You prefer Y to XYou are indifferent between X and Y.
TransitivityRational preferences are also transitive, which