Agent Technology for e-Commerce Appendix A: Introduction to Decision Theory Maria Fasli .

Agent Technology for e-Commerce

Appendix A: Introduction to Decision Theory

Maria Faslihttp://cswww.essex.ac.uk/staff/mfasli/ATe-Commerce.htm

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Decision theory

Decision theory is the study of making decisions that have a significant impact

Decision-making is distinguished into: Decision-making under certainty Decision-making under noncertainty

Decision-making under risk Decision-making under uncertainty

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Probability theory

Most decisions have to be taken in the presence of uncertainty Probability theory quantifies uncertainty regarding the

occurrence of events or states of the world Basic elements of probability theory:

Random variables describe aspects of the world whose state is initially unknown

Each random variable has a domain of values that it can take on (discrete, boolean, continuous)

An atomic event is a complete specification of the state of the world, i.e. an assignment of values to variables of which the world is composed

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Probability space The sample space S={e1,e2,…,en} which is a set of atomic events The probability measure P which assigns a real number between

0 and 1 to the members of the sample space

Axioms All probabilities are between 0 and 1 The sum of probabilities for the atomic events of a probability

space must sum up to 1 The certain event S (the sample space itself) has probability 1,

and the impossible event which never occurs, probability 0

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Prior probability

In the absence of any other information, a random variable is assigned a degree of belief called unconditional or prior probability

P(X) denotes the vector consisting of the probabilities of all possible values that a random variable can take

If more than one variable is considered, then we have joint probability distributions

Lottery: a probability distribution over a set of outcomes

L=[p1,o1;p2,o2;…;pn,on]

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Conditional probability

When we have information concerning previously unknown random variables then we use posterior or conditional probabilities: P(a|b) the probability of a given that we know b

Alternatively this can be written (the product rule):

P(ab)=P(a|b)P(b)

Independence

P(a|b)=P(a) and P(b|a)=P(b) or P(ab)=P(a)P(b)

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Bayes’ rule

The product rule can be written as:

P(ab)=P(a|b)P(b)

P(ab)=P(b|a)P(a)

By equating the right-hand sides:

This is known as Bayes’ rule

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Making decisions

Simple example: to take or not my umbrella on my way out

The consequences of decisions can be expressed in terms of payoffs

Payoff table

Loss table

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An alternative representation of payoffs – tree diagram

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Admissibility

An action is said to dominate another, if for each possible state of the world the first action leads to at least as high a payoff (or at least as small a loss) as the second one, and there is at least one state of the world in which the first action leads to a higher payoff (or smaller loss) than the second one

If one action dominates another, then the latter should never be selected and it is called inadmissible

Payoff table

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Non-probabilistic decision-making under uncertainty

The maximin rule The maximax rule The minimax loss

Payoff table

Loss table

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Probabilistic decision-making under uncertainty

The Expected Payoff (ER) rule dictates that the action with the highest expected payoff should be chosen

The Expected Loss (EL) rule dictates that the action with the smallest expected loss should be chosen

If P(rain)=0.7 and P(not rain)=0.3 then:

ER(carry umbrella) = 0.7(-£1)+0.3(-£1)=-£1

ER(not carry umbrella) = 0.7(-£50)+0.3(-£0)=-£35

EL(carry umbrella) = 0.7(£0)+0.3(£1)=£0.3

EL(not carry umbrella) = 0.7(£49)+0.3(£0)=£34.3

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Utilities

Usually the consequences of decisions are expressed in monetary terms

Additional factors such are reputation, time, etc. are also usually translated into money

Issue with the use of money to describe the consequences of actions:

If a fair coin comes up heads you win £1, otherwise you loose £0.75, would you take this bet?

If a fair coin comes up heads you win £1000, otherwise you loose £750, would you take this bet?

The value of a currency, differs from person to person

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Preferences

The concept of preference is used to indicate that we would like/desire/prefer one thing over another

o o’ indicates that o is (strictly) preferred to o’ o ~ o’ indicates that an agent is indifferent between o and o’ o o’ indicates that o is (weakly) preferred to o’

Given any o and o’, then o o’, or o’ o, or o ~ o’ Given any o, o’ and o’’, then if o o’ and o’ o’’, then o’ o’’ If o o’ o’’, then there is a p such that [p,o;1-p,o’’] ~ o’ If o ~ o’, then [p,o; 1-p,o’’] ~ [p,o’; 1-p, o’’] If o o’, then (pq [p,o;1-p,o’] [q,o;1-q,o’] )

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Utility functions

A utility function provides a convenient way of conveying information about preferences

If o o’, then u(o)>u(o’) and if o ~ o’ then u(o)=u(o’) If an agent is indifferent between:

(a) outcome o for certain and

(b) taking a bet or lottery in which it receives o’ with probability p and o’’ with probability 1-p

then u(o)=(p)u(o’)+(1-p)u(o’’) Ordinal utilities Cardinal utilities Monotonic transformation

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Assessing a utility function

How can an agent assess a utility function?

Suppose most and least preferable payoffs are R+ and R- and

u(R+)=1 and u(R-)=0

For any other payoff R, it should be:

u(R+) u(R) u(R-) or 1 u(R) 0

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To determine the value of u(R) consider: L1: Receive R for certain

L2: Receive R+ with probability p and R- with probability 1-p

Expected utilities: EU(L1)=u(R)

EU(L2)=(p)u(R+ )+(1-p)u(R- )=(p)(1)+(1-p)(0)=p

If u(R)>p, L1 should be selected, whereas if u(R)<p, L2 should be selected, and if u(R)=p then the agent is indifferent between the two lotteries

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Utility and money

The value, i.e. utility, of money may differ from person to person Consider the lottery

L1: receive £0 for certain

L2: receive £100 with probability p and -£100 with (1-p)

Suppose an agent decides that for p=0.75 is indifferent between the two lotteries, i.e. p>0.75 prefers lottery L2

The agent also assess u(-£50)=0.4 and u(£50)=0.9

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If p is fixed, the amount of money that an agent would need to receive for certain in L1 to make it indifferent between two lotteries can be determined. Consider:

L1: receive £x for certain

L2: receive £100 with probability 0.5 and -£100 with 0.5

Suppose x=-£30, then u(-£30)=0.5 and -£30 is considered to be the cash equivalent of the gamble involved in L2

The amount of £30 is called the risk premium – the basis of insurance industry

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Utility function of risk-averse agent

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Utility function of a risk-prone agent

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Utility function of a risk-neutral agent

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Multi-attribute utility functions

The utility of an action may depend on a number of factors Multi-dimensional or multi-attribute utility theory deals with

expressing such utilities Example: you are made a set of job offers, how do you decide?

u(job-offer) = u(salary) + u(location) +

u(pension package) + u(career opportunities)

u(job-offer) = 0.4u(salary) + 0.1u(location) +

0.3u(pension package) + 0.2u(career opportunities)

But if there are interdependencies between attributes, then additive utility functions do not suffice. Multi-linear expressions:

u(x,y)=wxu(x)+wyu(y)+(1-wx-wy)u(x)u(y)

Agent Technology for e-Commerce Appendix A: Introduction to Decision Theory Maria Fasli .

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