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Last Class (Before Review). Last Class!. This is our last class on substantive, new material (next time is the review). For the past week and a half, we have been discussing the basic framework of decision theory, particularly applied to decisions under ignorance. Decision Theory.
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Last Class! This is our last class on substantive, new material (next time is the review). For the past week and a half, we have been discussing the basic framework of decision theory, particularly applied to decisions under ignorance.
Decision Theory The goal of decision theory is to devise a rule or set of rules that tell us what act is appropriate, given a problem specification and a set of rational preferences. There are some obvious principles we should abide by, even though they don’t help in every decision– for example, the dominance principle.
Dominance Principle The dominance principle says that if one act is better than or equal to every other act in every state, then you should take that act. Clearly, however, this doesn’t help with most decisions, because in most decisions different acts are better depending on the state.
Decision Rules: Maximin Here’s a brief look at the decision rules we’ve covered: 1. The Maximin Rule: For each act, find the worst possible outcome that could result from that act. Choose the act whose worst possible outcome is the best of all the acts. Maximize the minimum outcome.
Decision Rules: Minimax Regret 2. Minimax Regret: For each act, calculate the amount of “missed opportunity” in each of the states. That is, how much does the outcome for that act in that state fall short of the best possible outcome for that act in that state? That’s how much you’d regret that act, if that state obtained. Find the maximum amount of regret for each act, then choose the act with the smallest maximum amount of regret.
Decision Rules: Optimism-Pessimism Rule 3. Optimisim-Pessimism Rule: For each act, find the best possible outcome, and the worst possible outcome. Figure out how much you care about obtaining what’s best and avoiding what’s worst (figure out your optimism index). Then choose the act with the best weighted average of best and worst outcomes.
Assume Optimism Index of 50% Here are the OPN’s for the acts given O = 50%. OPN for A1 = 0.5 x 1 + 0.5 x 13 = 7 OPN for A2 = 0.5 x -1 + 0.5 x 17 = 8 OPN for A3** = 0.5 x 0 + 0.5 x 20 = 10
Ignorance vs. Risk These rules are all possible rules for how to make decisions under ignorance (where we don’t know how probable all of the states are). When we have a decision under risk, instead, there is only one rule that decision theorists take seriously: maximize expected utility.
Utilities vs. Values As a way of illustrating, I am going to replace utilities with dollar amounts, and the rule “maximize expected utility” with “maximize expected value” (dollar value). This rule says: take the action with the greatest expected dollar value.
Expected Values Suppose I’m going to flip a fair coin twice, and pay you the following amounts for the following outcomes: • HH: $20 • HT: $8 • TH: $4 • TT: $1 How much money do you expect to win? How much would you pay to play this game?
Expected Values Here’s what we know: each of the outcomes (HH, HT, TH, and TT) is equally probable at a 25% chance of happening. The expected value of this game is the sum of the probabilities of each outcome multiplied by the values of those outcomes: P(HH)x$20 + P(HT)x$8 + P(TH)x$4 + P(TT)x$1 = $5 + $2 + $1 + $0.25 = $8.25
Expected Values According to decision theory (if your utilities are linear in dollars) you should always pay less than $8.25 to play this game, be indifferent to paying $8.25 exactly and playing this game, and never pay more than $8.25 to play. This game is worth is expected value: paying less than the EV to play is a bargain, paying more than the EV is like paying $2 to get $1. It’s irrational.
Reasons for Valuing at the EV Now there are lots of complicated reasons for believing that acts and games are worth their expected values (or better: expected utilities). We can’t go into all of these reason here. Here’s one of the reasons: the law of large numbers says that if you play this game a large number of times, your average payout per game will be $8.25.
Decisions Under Risk How does this involve decision theory? Well, for decisions under risk, decision theory says: calculate the expected value (utility) for each act. Then take the act with the highest expected value (utility). Maximize expected value (utility).
Example: Maximizing EV Suppose that you work for a medical insurance company. Everyone who applies for insurance must fill out a complicated medical history questionnaire. Each policy lasts for 1 year and has a premium of $1,000. If someone dies during that year, they receive $250,000.
Example: Maximizing EV Now suppose that I come into your office and apply for insurance. I fill out the medical questionnaire, and your statisticians determine that I have a 5% chance of dying within the next year. Should you insure me? (Remember this question means: should you rationally insure me; it might be moral to insure me even if it’s not rational.)
Maximize EV According to the rule “maximize expected value,” we should calculate the expected value of each act and take the one with the highest EV. EV don’t insure = $0 x 5% + $0 x 95% = $0 EV do insure = -$250000 x 5% + $1000 x 95% = -$11,500
Utility vs. Value Decision theorists prefer to talk in terms of utility rather than (monetary) value. ‘Utility’ is just a special name for non-monetary value. How much something is “really worth” not in dollars, but in terms of personal satisfaction to you. You might think that this can be measured by how much you would be willing to pay– and sometimes it is– but sometimes it isn’t.
Utility < EV Suppose you’ve been saving up to put a down payment on a flat– your dream flat, the one you plan on living in for the rest of your life. A down payment is HKD$150,000. Yesterday, you just finished saving enough money, and tomorrow, you plan on purchasing the flat. Then your stock broker calls you with a hot tip: an 80% at a $500,000 return for a $150,000 investment. But a 20% chance of losing everything.
Utility < EV The EV of the investment is: $500,000 x 80% - $150,000 x 20% = $370,000 Clearly that’s worth paying a measly $150,000 for! But it’s rational not to accept the deal. You are right now certain to be able to purchase the home of your dreams. If you gamble here, there’s a 20% chance you’ll never get it.
Utility > EV Suppose you really want to go see a once-in-a-lifetime sporting match, and tickets are only $200. However, you only have $100. A suspicious man comes up to you on the street. He offers you a gamble: roll two dice, and if you roll two 1’s, you get $200, otherwise, you pay him $100. The EV is -$75, but you might rationally take the bet, because otherwise you have no chance of seeing the game!
Principle of Insufficient Reason Can we use the rule “maximize expected value/ utility” be used to solve decisions under ignorance? It seems not: to calculate expected values/ utilities, you need to assign probabilities to the different states. But the defining feature of decisions under ignorance is that you cannot assign probabilities to the states.
Principle of Insufficient Reason However, according to the principle of insufficient reason, since you have no reason to assign any particular probability to any state, you should assign each state the same probability. Then you should calculate the expected value of each act, and choose the act with the highest expected value.
Maximin Gets the Answer Wrong In this example from last time, the intuitive correct act is A1. The Maximin Rule says to pick A2. The worst possible outcome for both A1 and A2 is 0. So for a tie-breaker we consider the second-worst possible outcome, which is $99 in A1 and $100 in A2. So we maximize and choose A2.
Minimax Regret Wrong Too The minimax regret principle gets the answer wrong too. A1 has a maximum regret of $100 (if state S1 obtains) whereas A2 has a maximum regret of $99. If we minimize the maximum regret, we pick A2 again.
Optimism-Pessimism Rule Wrong The optimism-pessimism rule also gets the wrong answer. Remember that this rule says to compare a weighted average of the best and worst outcomes of each action. But the worst outcomes are both $0 for A1 and A2, so the optimism-pessimism rule just says: pick the one with the best best outcome, and that’s A2 again.
Maximize Expected Value However, if we assign each state equal probability (1/9) under the principle of insufficient reason, and calculate expected values, we get: EV A1 = $0 x (1/9) + $99 x (8/9) = $88 EV A2 = $100 x (1/9) + $0 x (8/9) = $11.11
Is the PIR Correct? The biggest objection to the principle of insufficient reason is that it is based on a faulty assumption. Just because we don’t have enough information to assign probabilities to the states does not mean that we should assign them equal probability– that’s as unjustified as assigning them any other probabilities.
Example For example, it’s hard to assign probabilities to the states “in the next 30 years there is a nuclear holocaust that brings about an apocalyptic future” and “things are pretty normal 30 years from now.” Surely that doesn’t mean we should treat these as equally likely, and spend half our money preparing for nuclear winter and half for retirement.
Disaster Additionally, the principle of insufficient reason may lead us to a disaster. For example, consider this problem from last time…
Expected Values If we assume the two states are equally likely (50%), and that your life savings is $150,000 and the possible payout for the investment is $1.5 million, we get: EV for A1: ½ x -.15M + ½ x 1.5M = $675,000 EV for A2: ½ x .15M + ½ x .15M = $150,000
Just Society I want to finish with an application of decision theory to philosophical views of social justice. We as public policy makers, voters, or citizen activists make choices to affect the nature and structure of the societies we live in. Sometimes we are motivated by self-interest, but often our goal is a fair and just society.
Hong Kong Different societies are obviously different. In Hong Kong, an estimated 100,000 people (1.5% of the population) live in “inadequate housing” (cage homes, rooftops, subdivided spaces), 1.15 million (16.5%-- and 33% of the elderly) live in poverty– less than HKD$13,350/mo. for a family of four. But then, the wealthiest are very wealthy: the top 10 per cent of earners have 40% of the wealth, for Asia’s largest Gini score.
Denmark Compare Denmark, which spends a lot of time taking care of the least well off: the highest minimum wage in the world, high unemployment payments, the lowest Gini coefficient in the world, and high taxes on the highest earners (45-55% on people making more than HKD$1 million). The percentage of USD millionaires in HK is about 5 times more than that of Denmark (8.6% to 1.7%).
