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Making Simple Decisions

CSci 5512: Artificial Intelligence II

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February 17, 2022

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Making Simple Decisions

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HW2 due Feb 24

Making Simple Decisions

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Preferences

A lotter is a situation with uncertain prizes Lottery L = [p, A; (1 − p), B]

A ≻ B A ∼ B A≽B

A is preferred to B indifference between A and B B notpreferredtoA

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Rational Preferences

Preferences of a rational agent must obey constraints Constraints

Orderability: (A≻B)∨(B ≻A)∨(A∼B)

Transitivity: (A≻B)∧(B ≻C) =⇒ (A≻C)

Continuity:A≻B≻C =⇒∃p[p,A;1−p,C]∼B

Substitutability: A ∼ B =⇒ [p,A;1−p,C] ∼ [p,B;1−p,C]

Monotonicity:

A≻B =⇒ (p≥q ⇐⇒ [p,A;1−p,B]≽[q,A;1−q,B])

Decomposability:

[p,A;1−p,[q,B;,1−q,C]] ∼ [p,A;(1−p)q,B;(1−p)(1−q),C]

Violating the constraints leads to self-evident irrationality

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Rational Preferences

Example of irrationality when violating transitivity Suppose agent has A ≻ B ≻ C ≻ A

If agent has A then offer to trade C for A+$1 which the agent will accept

Then offer to trade B for C + $1 which the agent will accept again

Finally trade A for B + $1

We are back where we started except now the agent has $3

Continue until agent has no money at all

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Maximizing Expected Utility

Theorem (Ramsey, 1931; von Neumann and Morgenstern, 1944): Given preferences satisfying the constraints there exists a real-valued function U such that

U(A) ≥ U(B) ⇔ A ≽ B U([p1,S1;…;pn,Sn]) = piU(Si)

MEU principle

Choose the action that maximizes expected utility

Rational agent need not use utilities and probabilities Still can be consistent with MEU

Example: A lookup table for perfect tic-tac-toe

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Utilities map state to real numbers. Which numbers? Standard approach to assessment of human utilities:

Compare a given state A to a standard lottery Lp that has “best possible prize” u ̄ with probability p

“worst possible catastrophe” u with probability 1 − p

Adjust lottery probability p until A ∼ Lp Normalized utilities: u ̄ = 1.0, u = 0.0

Behavior is invariant w.r.t. positive linear/affine transformation

U ̃(x) = k1U(x) + k2 where k1 > 0

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Multi-attribute Utility

How to handle utility functions of many variables X1 , . . . , Xn ? What is U(Deaths, Noise, Cost)?

How to get (complex) utility functions from preferences? Two key ideas

Dominance structures

Decisions can be made without knowing U(x1, . . . , xn) Independence structures

Independence in preferences yields canonical forms for U(x1,…,xn)

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Strict Dominance

Typically define attributes such that U is monotonic in each Strict dominance:

Choice B strictly dominates choice A iff

∀i Xi (B) ≥ Xi (A) (and hence U(B) ≥ U(A))

Strict dominance seldom holds in practice

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Stochastic Dominance

Distribution p1 stochastically dominates distribution p2 iff

∀t p1(x)dx ≤ p2(t)dt

⇔ ∀tP1(X≤t)≤P2(X≤t)

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Stochastic Dominance (cont.)

Let U be monotonic in x

Action A1 has outcome distribution p1 Action A2 has outcome distribution p2 A1 stochastically dominates A2

p1(x)U(x)dx ≥ p2(x)U(x)dx

⇔ EP1[U] ≥ EP2[U]

Multi-attribute case: stochastic dominance on all attributes

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Preference structure: Deterministic

X1 and X2 preferentially independent (PI) of X3 iff preference between ⟨x1, x2, x3⟩ and ⟨x1′ , x2′ , x3⟩ does not depend on x3 Example: ⟨Noise,Cost,Safety⟩

⟨ 20,000 suffer, $ 4.6 billion, 0.06 deaths/mpm ⟩ ⟨ 70,000 suffer, $ 4.2 billion, 0.06 deaths/mpm ⟩

Theorem (Leontief, 1947): If every pair of attributes is PI of its complement, then every subset of attributes is PI of its complement ≡ Mutual PI

Theorem (Debreu, 1960): Mutual PI =⇒ there exists additive value function:

V(S) = Vi(Xi(S)) i

Assess n single-attribute functions

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Preference structure: Stochastic

Need to consider preferences over lotteries

X is utility-independent (UI) of Y iff preferences over lotteries

in X do not depend on Y

Mutual UI: Each subset is UI of its complement Mutual UI =⇒ ∃ multiplicative utility function:

U =k1U1 + k2U2 + k3U3 + k1k2U1U2 + k2k3U2U3 + k3k1U3U1 + k1k2k3U1U2U3

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Decision Networks

Add decision (action) nodes and utility nodes to belief networks

Enable rational decision making

Algorithm:

For each value of action node, compute expected value of

utility node given (action, evidence) Return MEU action

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Value of Information

Compute value of acquiring each possible piece of evidence Can be done directly from decision network

Example: Buying oil drilling rights

Solution: Compute expected value of information

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General Formula

Current evidence E, current best action α

Possible action outcomes Si , potential new evidence Ej Expected value of best action without information

EU(α|E) = maxU(Si)P(Si|E,a)

Suppose we knew Ej = ejk , then we would choose αejk s.t.

EU(αejk |E,Ej = ejk) = maxU(Si)P(Si|E,a,Ej = ejk)

Ej is a random variable whose value is currently unknown

Must compute expected gain over all possible values

Value of perfect information

VPIE(Ej) = P(Ej = ejk|E)EU(αejk |E,Ej = ejk) −EU(α|E) k

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Properties of VPI

Non-negative (in expectation, not post hoc)

∀j,E VPIE(Ej) ≥ 0 Non-additive (consider, e.g., obtaining Ej twice)

VPIE (Ej , Ek ) ̸= VPIE (Ej ) + VPIE (Ek ) Order-independent

VPIE(Ej,Ek) = VPIE(Ej)+VPIE,Ej (Ek) = VPIE(Ek)+VPIE,Ek (Ej)

When more than one piece of evidence can be gathered, Maximizing VPI for each to select one is not always optimal Evidence-gathering becomes a sequential decision problem

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Qualitative Behaviors

(a) Choice is obvious, information worth little

(b) Choice is non-obvious, information worth a lot (c) Choice is non-obvious, information worth little

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Making Simple Decisions

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