General Logic

In order to decide what we ought to do to obtain some good or avoid some harm, it is necessary to consider not only the good or harm in itself, but also the probability that it will or will not occur...

The Port Royal Logic, 1662

Decision theory

A decision involves a choice between two or more acts, each of which will produce one of (possibly) several outcomes. Divide decisions into three categories: 1. Decisions under certainty2. Decisions under risk3. Decisions under uncertainty (a.k.a. Decisions under ignorance)

DECISIONS UNDER CERTAINTY

Decisions Under Certainty

The agent is certain what outcome each act will produce.

They are the easiest decisions to make:

Choose the act that will produce the outcome you most prefer!  

Example

If you raise your right hand, I will give you $100. If you raise your left hand, I will give you $10. If you raise neither hand, I will not give you anything.

Example

If you raise your right hand, I will give you $100. If you raise your left hand, I will give you $10. If you raise neither hand, I will not give you anything.

Right $100

Left $10

Neither $0

Both $110

Example

Right $100

Left $10

Neither $0

Both $110

DECISIONS UNDER IGNORANCE

Decisions Under Ignorance

These are decisions where :

1.) at least one act will produce one of several outcomes, and

2.) for at least one act you do not know the probabilities of the the occurrence of the outcomes.

Picnic ExampleYou are in charge of organizing a dinner party that is being held for the sole purpose of raising money for a worthy cause. You must decide whether to schedule an outdoor picnic or an indoor buffet supper. This event must be planned so far in advance that you have no way assigning a probability of rain on the day of the party. On the basis of past events of this type, you have the following information:

If it does not rain the outdoor picnic will yield a profit of $500, and the indoor buffet supper will yield a profit of $170.

If it does rain the outdoor picnic will yield a profit of $80, and the indoor buffet supper will yield a profit of $440.

To construct a decision table:

Make a row for each act

Make a column for each state

Enter in each square the outcome corresponding to the act of that row and the state of that column.

Picnic Example

States

Rain No Rain

Acts

Schedule outdoor picnic

$80

$500

Schedule indoor buffet

$440

$170

Decision Strategies for Decisions under Ignorance

Principle of Insufficient Reason: Assume every state is equally probable and choose the act that, on this assumption, maximizes expected utility.

Dominance: If, no matter what happens, an act is never worse and sometimes better than every other, then choose it.

Maximax: Choose the best of the best.

Maximin: Choose the best of the worst.

Minimax Regret: Choose the act that minimizes the maximum amount of regret.

ExampleSuppose that just after graduating college you are offered three jobs.

First, the Exe Company offers you a salary of $40,000. Exe is well established and secure.

Your next offer comes from the Wye Company. Here, the salary is $60,000. But Wye is a new company and is less secure. You do not know how likely it is to last. It might go bankrupt and then you’ll be left without a job.

The final offer comes from the Zee Company, which is as well established and secure as Exe. Here, the salary is $65,000.

In all other respects, the jobs are similar. Which job do you take?

To construct a decision table:

Make a row for each act

Make a column for each state

Enter in each square the outcome corresponding to the act of that row and the state of that column.

Wye does not go bankrupt

Wye goes bankrupt

Take job at Exe $40,000 $40,000

Take job at Wye $60,000 $0

Take job at Zee $65,000 $65,000

Wye does not go bankrupt

Wye goes bankrupt

Take job at Exe $40,000 $40,000

Take job at Wye $60,000 $0

Take job at Zee $65,000 $65,000

Taking the job at Zee dominates the other acts.

Wye does not go bankrupt

Wye goes bankrupt

Take job at Exe $40,000 $40,000

Take job at Wye $60,000 $0

Suppose the job at Zee was a practical joke. You must choose between Exe and Wye. Which should you choose?

“The Gambler”

Wye does not go bankrupt

Wye goes bankrupt

Take job at Exe

$40,000 $40,000 max = $40,000

Take job at Wye

$60,000 $0 max= $60,000

Chooses the best of the best.

Chooses Wye.

“Cautious”

Wye does not go bankrupt

Wye goes bankrupt

Take job at Exe

$40,000 $40,000 min = $40,000

Take job at Wye

$60,000 $0 min = $0

Chooses the best of the worst.

Chooses Exe.

Who is right?

Maximax: The Gambler

This strategy tells us to maximize the maximum. To apply maximax we first need to find the maximum outcome for each act. Simply read across a row and at the end of the row write down the maximum value; then choose the act with the maximum of the maximum values.

According to maximax, you should choose A1.

S1 S2

A1 + $4- $1,000

max = + $4

A2 + $2 - $1 max = + $2

Criticism of Maximax

Too risky!

Prohibits us from avoiding huge losses where there are only slight gains.

According to maximax, you should choose A1.

S1 S2

A1 + $4- $1,000

max = + $4

A2 + $2 - $1 max = + $2

Maximin: The Cautious Player

This strategy tells us to maximize the minimum. To find the minimum for each act, read across the row and write down the minimum value; then choose the act that has highest of lowest values

According to maximin, you should choose A2.

S1 S2

A1 + $4- $1,000

min = -$1,000

A2 + $2 - $1 min = - $1

Criticism of Maximin

Too conservative.

Prohibits us from taking advantage of huge gains where there are only slight losses.

S1 S2

A1 $1.50 $1.75 min = $1.50

A2$1.00

$10,000

min = $1.00

Principle of Insufficient Reason

When faced with a decision under ignorance, treat all outcomes as equiprobable. In effect, you chose in terms of expected utility.

Exe: ($40k x .5) + ($40k x .5) = $40,000Wye: ($60k x .5) + ($00k x .5) = $30,000

Wye does not go bankrupt

Wye goes bankrupt

Take job at Exe

$40,000 .5

$40,000 .5

Take job at Wye

$60,000 .5

$0 .5

Criticism of the Principle of Insufficient Reason

ARBITRARY! Gives no reason for preferring one probability distribution over another.

Wye does not go bankrupt

Wye goes bankrupt

Take job at Exe

$40,000 .5

$40,000 .5

Take job at Wye

$60,000 .5

$0 .5

Minimax Regret

This strategy tells us to minimize the maximum of regret. To apply this strategy we need to transform our decision table (left) into a regret table (right). To do this:

1.) construct a table of the same size2.) find the maximum in each column3.) subtract each outcome from the maximum of its column4.) enter that number into its corresponding square in the regret table.

Minimax Regret

Wye does not go bankrupt

Wye goes bankrupt

Take job at Exe

$40,000 $40,000

Take job at Wye

$60,000 $0

$20,000 $0 max = $20,000

$0 $40,000 max = $40,000

Take job at Exe!

Regret Table

Summary: Decision Strategies for Decision under Ignorance

Principle of Insufficient Reason: Assume every state is equally probable and choose the act that, on this assumption, maximizes expected utility.

Dominance: If, no matter what happens, an act is never worse and sometimes better than every other, then choose it.

Maximax: Choose the best of the best.

Maximin: Choose the best of the worst.

Minimax Regret: Choose the act that minimizes the maximum amount of regret.

DECISIONS UNDER RISK

Decisions Under Risk

These are decisions where: 1.) at least one act will produce one of several outcomes, and

2.) you know the probabilities of the occurrence of each outcome, given that you take its corresponding action, and

3.) you can not only rank each outcome in terms of your preference but can also say, more or less, how much you prefer one outcome

to another.

Picnic Example

Suppose you know that there’s a 1/3 chance of rain on that day. Now what?

States

Rain No Rain

Acts

Schedule outdoor picnic

$80

$500

Schedule indoor buffet

$440 $170

Picnic Example

States

Rain No Rain

Acts

Schedule outdoor picnic

$80 1/3

$500 2/3

Schedule indoor buffet

$440 1/3

$170 2/3

Expected Utility

The most widely accepted strategy for making decisions under risk is to choose the action that maximizes expected utility. To calculate expected utility for an act, multiply each outcome by its associated probability and sum the results:  Picnic = ($80 x 1/3) + ($500 x 2/3) = $356.40Buffet = ($440 x 1/3) + (170 x 2/3) = $257.40 Then, choose the act with the highest expected utility: Picnic.

Rain No Rain

Schedule outdoor picnic

$80 1/3

$500 2/3

Schedule indoor buffet

$440 1/3

$170 2/3

What would Pr(bankrupt) have to be to make taking the job at Wye the rational choice?

Wye does not go bankrupt

Wye goes bankrupt

Take job at Exe $40,000 $40,000

Take job at Wye $60,000 $0