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

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Appendix A Agent Technology for e-Commerce 2 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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Appendix A Agent Technology for e-Commerce 3 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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Appendix A Agent Technology for e-Commerce 4 Probability space The sample space S={e 1,e 2,…,e n } 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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Appendix A Agent Technology for e-Commerce 5 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=[p 1,o 1 ;p 2,o 2 ;…;p n,o n ]

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Appendix A Agent Technology for e-Commerce 6 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(a b)=P(a|b)P(b) Independence P(a|b)=P(a) and P(b|a)=P(b) or P(a b)=P(a)P(b)

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Appendix A Agent Technology for e-Commerce 7 Bayes’ rule The product rule can be written as: P(a b)=P(a|b)P(b) P(a b)=P(b|a)P(a) By equating the right-hand sides: This is known as Bayes’ rule

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Appendix A Agent Technology for e-Commerce 8 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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Appendix A Agent Technology for e-Commerce 9 An alternative representation of payoffs – tree diagram

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Appendix A Agent Technology for e-Commerce 10 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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Appendix A Agent Technology for e-Commerce 11 Non-probabilistic decision-making under uncertainty The maximin rule The maximax rule The minimax loss Payoff table Loss table

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Appendix A Agent Technology for e-Commerce 12 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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Appendix A Agent Technology for e-Commerce 13 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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Appendix A Agent Technology for e-Commerce 14 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 (p q [p,o;1-p,o’] [q,o;1-q,o’] )

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Appendix A Agent Technology for e-Commerce 15 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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Appendix A Agent Technology for e-Commerce 16 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

17 Appendix A Agent Technology for e-Commerce 17 To determine the value of u(R) consider: L 1 : Receive R for certain L 2 : Receive R + with probability p and R - with probability 1-p Expected utilities: EU(L 1 )=u(R) EU(L 2 )=(p)u(R + )+(1-p)u(R - )=(p)(1)+(1-p)(0)=p If u(R)>p, L 1 should be selected, whereas if u(R)

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Appendix A Agent Technology for e-Commerce 18 Utility and money The value, i.e. utility, of money may differ from person to person Consider the lottery L 1 : receive £0 for certain L 2 : 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 L 2 The agent also assess u(-£50)=0.4 and u(£50)=0.9

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Appendix A Agent Technology for e-Commerce 19

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Appendix A Agent Technology for e-Commerce 20 If p is fixed, the amount of money that an agent would need to receive for certain in L 1 to make it indifferent between two lotteries can be determined. Consider: L 1 : receive £x for certain L 2 : 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 L 2 The amount of £30 is called the risk premium – the basis of insurance industry

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Appendix A Agent Technology for e-Commerce 21 Utility function of risk-averse agent

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Appendix A Agent Technology for e-Commerce 22 Utility function of a risk-prone agent

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Appendix A Agent Technology for e-Commerce 23 Utility function of a risk-neutral agent

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Appendix A Agent Technology for e-Commerce 24 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)=w x u(x)+w y u(y)+(1-w x -w y )u(x)u(y)

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