making simple decisions utility theory multiattribute utility functions decision networks the value...

Making Simple Decisions

• Utility Theory• MultiAttribute Utility Functions• Decision Networks• The Value of Information• Summary

Beliefs and Uncertainty

Utility Function

Outcome Probabilities

Expected Utility

Maximum Expected Utility

EU(A|E) = Σ P(Resulti(A) | E) U(Resulti(A))

Principle of Maximum Expected Utility:

Choose action A with highest EU(A|E)

Example

Robot

Turn Right

Turn Left

Hits wall (P = 0.1; U = 0)Finds target (P = 0.9; U = 10)

Fall water (P = 0.3; U = 0)Finds target (P = 0.7; U = 10)

Choose action “Turn Right”

Notation Utility Theory

A > B A is preferred to BA ~ B indifferent between A and BA >~ B A is preferred to or indifferent to B

Lottery (or random variable)

L = [p1, S1; p2, S2; …, pn, Sn]

where p:probability and S: outcome

Utility Principle

Principle

U(A) > U(B) A > B

U(A) = U(B) A ~ B

Utility Functions

Television Game Show:Assume you already have won $1,000,000 Flip a coin:

Tails (P = 0.5) $3,000,000

Head (P = 0.5) $0

Utility Functions

EU(Accept) = 0.5 U(Sk) + 0.5 U(Sk + 3M)

EU(Decline) = U(Sk + 1M)

Assume: Sk = 5 Sk + 1M = 8 Sk + 3M = 10

Utility Functions

Then EU(Accept) = 0.5 x 5 + 0.5 x 10 = 7.5

EU(Decline) = 8

Result: Decline offer in view of assigned utilities

Risk-Averse

Positive part: slope decreasing.

Utility is less than expected monetary value

$

U

Risk-Seeking

Negative part: desperate region.

$U

Linear curve:risk neutral

$U

Connection to AI

• Choices are as good as the preferences they are based on.• If user embeds in our intelligent agents :

• contradictory preferencesResults may be negative

• reasonable preferencesResults may be positive

Assessing Utilities

Best possible outcome: Amax

Worst possible outcome: Amin

Use normalized utilities: U(Amax) = 1 ; U(Amin ) = 0

MultiAttribute Utility Functions

Outcomes are characterized by more thanone attribute: X1, X2, …, Xn

Example:

Choosing right map successful tripFinding right equipment unsuccessful tripAcquiring food supplied

Simple Case: Dominance

Assume higher values of attributes correspondto higher utilities.

There are regions of clear “dominance”

Stochastic Dominance

Plot probability distributions against negative costs.

Example:

S1: Build airport at site S1S2: Build airport at site S2

Decision Networks

• It’s a mechanism to make rational decisions

• Also called influence diagram

• Combine Bayesian Networks with other nodes

Types of Nodes

• Chance Nodes.Represent random variables (like BBN)

• Decision NodesChoice of action

• Utility NodesRepresent agent’s utility function

Decision Nodes

Chance Nodes

Utility Nodes

The Value of Information

Important aspect of decision making:What questions to ask.

Example:

Oil company. Wishes to buy n blocks of ocean drilling rights.


Exactly one block has oil worth C dollars.The price of each block is C/n.

A seismologist offers the resultsof a survey of block number 3.

How much would you pay for the info?


• With probability 1/n the survey will indicate there is oil in block 3. Buy it for C/n dollars to make a profit of C – C/n = (n-1) C / n

• With probability (n-1)/n the survey will show no oil. Buy different block. Expected profit is C/(n-1) – C/n = C/n(n-1) dollars.

Expected Profit

The expected profit given the info is

1/n x (n-1)C / n + (n-1)/n x C / n(n-1) = C/n

The info. is worth the price of the block itself.


Value of info:

Expected improvement in utility compared with making a decision without that information.

Summary• Decision theory combines probability and utility theory.• A rational agent chooses the action with maximum expected utility.• Multiattribute utility theory deals with utilities that depend on several attributes• Decision networks extend BBN with additional nodes• To solve a problem we need to know the value of information.

Video

Rover Curiosity explores Mars (decision makingis crucial during navigation)

https://www.youtube.com/watch?v=W6BdiKIWJhY

making simple decisions utility theory multiattribute utility functions decision networks the value...

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