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Making Simple Decisions
• Utility Theory• MultiAttribute Utility Functions• Decision Networks• The Value of Information• Summary
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 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
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
Making Simple Decisions
• Utility Theory• MultiAttribute Utility Functions• Decision Networks• The Value of Information• Summary
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
Making Simple Decisions
• Utility Theory• MultiAttribute Utility Functions• Decision Networks• The Value of Information• Summary
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
Making Simple Decisions
• Utility Theory• MultiAttribute Utility Functions• Decision Networks• The Value of Information• Summary
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.
The Value of Information
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?
The Value of Information
• 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.
The Value of Information
Value of info:
Expected improvement in utility compared with making a decision without that information.
Making Simple Decisions
• Utility Theory• MultiAttribute Utility Functions• Decision Networks• The Value of Information• Summary
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.