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

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.

Example: Dominance Principle

S1 S2A1** 5 7

A2 3 7

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.

Example: Maximin

S1 S2 S3A1** 1* 14 13

A2 -1* 17 11A3 0* 20 6

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.

Example: Minimax Regret

S1 S2 S3A1 1 14 13

A2** -1 17 11A3 0 20 6

Example: Minimax Regret

S1 S2 S3A1 0 6* 0

A2** 2 3* 2A3 1 0 7*

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.

Example: Optimism-Pessimism Rule

S1 S2 S3A1 1 14 13A2 -1 17 11

A3** 0 20 6

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: $1How 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.)

Decision Table

Michael Dies Michael LivesInsure Michael -$250000

0.05+$1000

0.95Don’t Insure Michael

$00.05

$00.95

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.

Example from Last Time

S1 S2 S3 S4 S5 S6 S7 S8 S9A1 $0 $99 $99 $99 $99 $99 $99 $99 $99A2 $100 $0 $0 $0 $0 $0 $0 $0 $0

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…

Gambling with the FutureS1: The start-up is wildly successful.

S2: The start-up fails when Google engineers find a way to do everything it does, but better.

A1: Invest your life savings in a promising, but unproven start-up company.

You make hundreds of millions of dollars.

You lose your life savings.

A2: Play it safe, and invest a conservative stock portfolio with a modest, but guaranteed payout.

You pay for your retirement.

You pay for your retirement.

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 = $675,000EV for A2: ½ x .15M + ½ x .15M = $150,000

JUSTICE

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.

Cage Homes

Luxury Home

Bedroom

Dining Room

Kitchen

Living Area

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%).

John Rawls

John Rawls is a social philosopher whose most famous work is A Theory of Justice. Rawls argues there that one society is more just than another on the basis of the “difference principle”: x is more just than y = x treats the least well-off members of its society better than y treats the least well-off members of its society. ‘Least well-off’ here means poor, but also disabled, or elderly, or mentally handicapped.

HK vs. Denmark

So, according to Rawls, when evaluating Hong Kong and Denmark, we’d ignore the rich people, and focus on the least well-off people. And since government policies and high taxes on the rich provide a strong social safety net for the poor (unemployment insurance, high minimum wages) in Denmark, whereas so many people in Hong Kong live in terrible conditions, Rawls would say that Denmark is a more just society than Hong Kong.

John Harsanyi

John Harsanyi is a decision theorist and a utilitarian, someone who believes that the right action is the one that brings about the greatest average happiness. A just society for Harsanyi would be one that had the greatest average happiness. If we assume for a moment that happiness can be measured in money, we find that Hong Kong is more just than Denmark: #7 in GDP per capita vs. #14.

This is a critical thinking class and not a social philosophy class, so we won’t try to answer this question (“which is more just?”).

But it is interesting to look at how both authors argue for their views: using different decision rules that we have discussed.

The Veil of Ignorance

Rawls introduced the idea of the “veil of ignorance.” The goal is to come up with the principles or laws that will govern a just society. You are supposed to imagine that you are in charge of writing all the laws for this new society. Importantly, you will be a member of this new society. But, you do not know who you will be: man or woman, sick or healthy, rich or poor, smart or dumb, etc.

Self-Interest

Rawls wants you to choose purely on the basis of self-interest. He thinks that even if you act completely selfishly behind the veil of ignorance, all of your principles will be just. This is because you will never accept a principle like “put the poor in zoos and make them dance for our amusement” because for all you know, in this new society, you might be one of the poor in the zoo made to dance.

Decision Theory

This is where decision theory comes in. Decision theory is all about how to make decisions in your own self-interest. We consider different acts (“enact this law,” “enact that law”), different states (“I am rich, healthy, and smart,” “I am poor, sick, and dumb”), and the various outcomes that result (“I live in a cage home,” “I live in a decent-sized, government provided flat with food and air-conditioning”).

Maximin

Rawls thinks that everyone will decide on the basis of the maximin principle. They will consider the people who would be hurt most by each law, and choose the law that would result in the best outcomes for those hurt most– because they might turn out to be one of these people who are most hurt. You’ll want a world where the disabled are taken care of at an expense to the abled, because you might be disabled, and you would need that.

Harsanyi: Maximize Expected Utility

Harsanyi accepts the basic set-up involving the veil of ignorance, but he denies that people would choose laws based on the maximin principle. Harsanyi maintains that we should decide based on the principle: maximize expected utility under the principle of insufficient reason: assume that I’m equally likely to be any person in the society.

Example

For example, suppose we could set up society of 1,000 people where 100 did all the work, and 900 just had fun and did whatever they liked.

Under the principle of insufficient reason, you have a 100/1,000 = 10% chance of being a worker and a 900/1,000 = 90% chance of being a fun-haver.

Expected Utility

And suppose the workers only get 1 unit of happiness (utility) while the fun-havers all get 90 units. Then the expected utility of choosing this society is:

EU: 10% x 1 + 90% x 9 = 81.1 utils

Unequal but Just

Compare this to another society where everyone works a little on a rotating schedule, and averages 56 utils. Harsanyi would say that you should choose the unequal society, because that has the greatest expected utility. And he would say that that society is just, because what it is for a society to be just is for people to selected it for self-interested reasons from behind the veil of ignorance.

Lots at Stake

We can’t decide the case here, but there are two things that recommend Rawls’ solution:

First, as Rawls points out, there is a lot at stake. If you choose laws that allow people to live in cage homes with cockroaches and barely enough food to eat, you might wind up being one of those people. Would you really choose this society just b/c the average wealth is great?

Happiness is Cheap

Second, as Rawls also points out, you don’t need lots of money to be happy. This means that we won’t run into lots of regret by choosing to help the least well-off. If I wind up rich in Denmark, I won’t say, “oh no! I could have been so much more rich if I had chosen the laws of Hong Kong for my society!” Rich is good enough. But if I wind up poor, and I haven’t chosen to help the poorest, I will regret that a lot.

Do You Need $ for Happiness?

This second consideration is questionable. According to a survey by Royal Skandia (an investment firm), Hong Kong ranked third (after Dubai and Singapore) in the amount of money people said they would need to be happy. This was higher than countries in Europe, for example (HKers need HKD$1.5m/ year vs. $0.67 m/ year in Germany). However, the surveyors speculated that this was because of the lack of a social safety net in the East.

SUMMARY

Summary

Decisions are hard to make even when we’re certain of the outcomes. When we’re not certain, but can assign definite probabilities, decision theory recommends that we maximize the expected utility of our actions. With decisions under ignorance, there are lots of different rules to consider, but considerations (like the possibility of regret, or the severity of bad choices) can recommend certain rules for certain decisions.