Lirong Xia

Utilities and decision theory

Friday, Feb 28, 2014

• Midterm Mar 7

– in-class

– open book and lecture notes

– simple calculators are allowed

– cannot use smartphone/laptops/wifi

– practice exams and solutions (check piazza)

• Project 2 deadline Mar 18 midnight

2

Reminder

• Given random variables

Z1,…Zp, we are asked

whether X Y|Z⊥ 1,…Zp

– dependent if there exists a

path where all triples are

active

– independent if for each path,

there exists an inactive triple3

Checking conditional independence from BN graph

• Compute a marginal probability p(x1,…,xp) in a

Bayesian network

– Let Y1,…,Yk denote the remaining variables

– Step 1: fix an order over the Y’s (wlog Y1>…>Yk)

– Step 2: rewrite the summation as

Σy1 Σy2

…Σyk-1 Σyk

anything

– Step 3: variable elimination from right to left 4

General method for variable elimination

sth only

involving

X’s

sth only

involving Y1

and X’s

sth only

involving Y1,

Y2 and X’s

sth only

involving Y1,

Y2,…,Yk-1

and X’s

• Utility theory

– expected utility: preferences over lotteries

– maximum expected utility (MEU) principle

5

Today

Expectimax Search Trees

6

• Expectimax search– Max nodes (we) as in minimax search– Chance nodes

• Need to compute chance node values as expected utilities

• Next class we will formalize the

underlying problem as a Markov

decision Process

Expectimax Pseudocode

7

• Def value(s):If s is a max node return maxValue(s)If s is a chance node return expValue(s)If s is a terminal node return evaluations(s)

• Def maxValue(s):values = [value(s’) for s’ in successors(s)]

return max(values)

• Def expValue(s):values = [value(s’) for s’ in successors(s)]weights = [probability(s, s’) for s’ in successors(s)]

return expectation(values, weights)

Expectimax Quantities

8

Maximum expected utility

9

• Why should we average utilities? Why not

minimax?

• Principle of maximum expected utility:– A rational agent should chose the action which

maximizes its expected utility, given its (probabilistic) knowledge

• Questions:– Where do utilities come from?– How do we know such utilities even exist?– Why are we taking expectations of utilities (not, e.g.

minimax)?– What if our behavior can’t be described by utilities?

Inference with Bayes’ Rule

10

• Example: diagnostic probability from causal probability:

• Example:– F is fire, {f, ¬f}

– A is the alarm, {a, ¬a}

– Note: posterior probability of fire still very small– Note: you should still run when hearing an alarm! Why?

Causep Effect Cause p Cause

p Effectp Effect

p(f)=0.01

p(a|f)=0.9p(a)=0.1

11

0.009

stay

run out

0.991

100%

Utilities

12

• Utilities are functions from outcomes

(states of the world, sample space) to

real numbers that represent an

agent’s preferences

• Where do utilities come from?– In a game, may be simple (+1/-1)– Utilities summarize the agent’s goals

-10100

100

-100

Preferences over lotteries

13

• An agent chooses among:– Prizes: A, B, etc.– Lotteries: situations with

uncertain prizes

• Notation:

L

A

B

p

1-p

A is strictly preferred to B

In difference between A and B

A is strictly preferred to or indifferent with B

• How many lotteries?

– infinite!

• Need to find a compact representation

– Maximum expected utility (MEU) principle

– which type of preferences (rankings over

lotteries) can be represented by MEU?

14

Encoding preferences over lotteries

Rational Preferences

15

• We want some constraints on preferences before we call them rational

• For example: an agent with intransitive preferences can be induced to give away all of its money– If B>C, then an agent with C would

pay (say) 1 cent to get B– If A>B, then an agent with B would

pay (say) 1 cent to get A– If C>A, then an agent with A would

pay (say) 1 cent to get C

Rational Preferences

16

• Preference of a rational agent must obey constraints– The axioms of rationality: for all lotteries A, B, C

Orderability

Transitivity

Continuity

Substitutability

Monotonicity

• Theorem: rational preferences imply behavior describable as maximization of expected utility

MEU Principle

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• Theorem:– [Ramsey, 1931; von Neumann & Morgenstern, 1944]– Given any preference satisfying these axioms, there exists a

real-value function U such that:

• Maximum expected utility (MEU) principle:– Choose the action that maximizes expected utility– Utilities are just a representation!

• an agent can be entirely rational (consistent with MEU) without ever representing or manipulating utilities and probabilities

– Utilities are NOT money

18

0.009

stay

run out

0.991

100%

What would you do?

-10100

100

-100

19

Common types of utilities

• State of the world: money you earn– Money does not behave as a utility function, but we can

talk about the utility of having money (or being in debt)• Which would you prefer?

– A lottery ticket that pays out $10 with probability .5 and $0 otherwise, or

– A lottery ticket that pays out $3 with probability 1

• How about:– A lottery ticket that pays out $100,000,000 with probability .5 and

$0 otherwise, or– A lottery ticket that pays out $30,000,000 with probability 1

• Usually, people do not simply go by expected value

20

Utility theory in Economics

Risk attitudes

• An agent is risk-neutral if she only cares about the expected value of the lottery ticket

• An agent is risk-averse if she always prefers the expected value of the lottery ticket to the lottery ticket– Most people are like this

• An agent is risk-seeking if she always prefers the lottery ticket to the expected value of the lottery ticket

Decreasing marginal utility• Typically, at some point, having an extra dollar

does not make people much happier (decreasing marginal utility)

utility

money$200 $1500 $5000

buy a bike (utility = 1)

buy a car (utility = 2)

buy a nicer car (utility = 3)

Maximizing expected utility

• Lottery 1: get $1500 with probability 1– gives expected utility 2

• Lottery 2: get $5000 with probability .4, $200 otherwise– gives expected utility .4*3 + .6*1 = 1.8– (expected amount of money = .4*$5000 + .6*$200 = $2120 > $1500)

• So: maximizing expected utility is consistent with risk aversion

utility

money$200 $1500 $5000

buy a bike (utility = 1)

buy a car (utility = 2)

buy a nicer car (utility = 3)

Different possible risk attitudes under expected utility maximization

utility

money

• Green has decreasing marginal utility → risk-averse• Blue has constant marginal utility → risk-neutral• Red has increasing marginal utility → risk-seeking• Grey’s marginal utility is sometimes increasing,

sometimes decreasing → neither risk-averse (everywhere) nor risk-seeking (everywhere)

Example: Insurance

25

• Consider the lottery [0.5, $1000; 0.5, $0]

– What is its expected monetary value (EMV)? ($500)

– What is its certainty equivalent?

• Monetary value acceptable in lieu of lottery

• $400 for most people

– Difference of $100 is the insurance premium

• There is an insurance industry because people will pay to reduce their risk

• If everyone were risk-neutral, no insurance needed!

Example: Insurance

26

• Because people ascribe different utilities to different amounts of money, insurance agreements can increase both parties’ expected utility

Amount Your Utility UY

$0 0

-$50 -150

-$200 -1000

You own a car. Your lottery:LY = [0.8, $0; 0.2, -$200]

i.e., 20% chance of crashing

You do not want -$200!

Insurance is $50

UY(LY) = 0.2*UY(-$200)=-200

UY(-$50)=-150

Example: Insurance

27

• Because people ascribe different utilities to different amounts of money, insurance agreements can increase both parties’ expected utility

You own a car. Your lottery:

LY = [0.8, $0; 0.2, -$200]

i.e., 20% chance of crashing

You do not want -$200!

UY(LY) = 0.2*UY(-$200)=-200

UY(-$50)=-150

Insurance company buys risk:

LI = [0.8, $50; 0.2, -$150]

i.e., $50 revenue + your LY

Insurer is risk-neutral:

U(L) = U(EMV(L))

UI(LI) = U(0.8*50+0.2*(-150))

= U($10) >U($0)

• Finite number of periods:– Overall utility = sum of rewards in individual periods

• Infinite number of periods:– … are we just going to add up the rewards over infinitely many

periods?– Always get infinity!

• (Limit of) average payoff: limn→∞Σ1≤t≤nr(t)/n– Limit may not exist…

• Discounted payoff: Σtϒt r(t) for some ϒ < 1– Interpretations of discounting:

• Interest rate r: ϒ= 1/(1+r)• World ends with some probability 1-ϒ

• Discounting is mathematically convenient– We will see more in the next class

28

Acting optimally over time