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### Agent Technology for e-Commerce

Appendix A: Introduction to Decision Theory

Maria Fasli

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

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

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

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

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]

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)

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

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

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

Non-probabilistic decision-making under uncertainty

- The maximin rule
- The maximax rule
- The minimax loss

Payoff table

Loss table

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

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

Preferences

- The concept of preference is used to indicate that we would like/desire/prefer one thing over another
- oo’ 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 oo’, or o’o, or o ~ o’
- Given any o, o’ and o’’, then if oo’ and o’o’’, then o’o’’
- If oo’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 oo’, then (pq [p,o;1-p,o’] [q,o;1-q,o’] )

Utility functions

- A utility function provides a convenient way of conveying information about preferences
- If oo’, 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

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

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

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

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

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)

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