Decisions under Uncertainty. Logic and Decision Making Unit 1

Transcription

1 Decisions under Uncertainty Logic and Decision Making Unit 1

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3 Topics De7inition Principles and rules Examples Axiomatisation Deeper understanding

4 Uncertainty In an uncertain scenario, the decision maker knows what the alternatives are knows what outcomes the alternative result in is unable to assign any probability to the states The most part of our decisions are of this kind

5 Example We are in a restaurant, next to choosing a main course. We have only two optional courses: monk7ish (that, if badly cooked has a terrible taste) and hamburger (any chef can make it eatable) good chef bad chef monk7ish good monk7ish terrible monk7ish hamburger edible hamburger edible hamburger no main course hungry hungry

6 Initial de7initions Let s de7ine an ordinal function v: A S Values such that v(a i, s k ) = 1 means that the value of doing a i in the case the state of the world is s k is 1 Let s try to de7ine the function so that the qualitative outcomes are somehow respected (we need, at least, an ordinal scale) v(monk0ish, good chef) = 4 (the best outcome) v(monk0ish, bad chef) = 1 (the worst outcome) v(hamburger, good chef) = 3, and so on Then: good chef bad chef monk7ish good 4 monk7ish terrible 1 monk7ish hamburger edible 3 hamburger edible 3 hamburger no main course hungry 2 hungry 2

7 Initial de7initions Let s de7ine a relation between acts: a i a j iff it is more rational to perform act a i rather than act a j (and we will say: a i is more rational than a j ) a i a j iff it is at least as rational to perform act a i than act a j (and we will say: a i is more rational than a j ) a i a j iff act a i is equally rational than act a j (and we will say: a i is equally rational as a j ) Note that: a i a j means that (a i a j ) AND NOT(a j a i ) a i a j means that (a i a j ) AND (a j a i )

8 Dominance principle If v(a i, s) v(a j, s) for every state s we say that the act a i dominates the act a j Weak dominance principle: a i a j iff v(a i, s) v(a j, s) for every state s Strong dominance principle: a i a j iff v(a i, s m ) v(a j, s m ) for every state s m, and there is some state s n such that v(a i, s n ) > v(a j, s n ) What are their meanings? (?) It is irrational to choose a state dominated by other states

9 Examples s 1 s 2 a a a s 1 s 2 a a a Does any act dominate the others? (?) 1) v(a 2, s) v(a 3, s) for every state, and v(a 2, s 1 ) > v(a 3, s 1 ) a 2 a 3 2) v(a 1, s) v(a 2, s) for every s, and v(a 1, s 1 ) > v(a 2, s 1 ) a 1 a 2 thus a 1 strongly dominates the other options Does any act dominate the others? (?) a 2 strongly dominates a 3 What about a 1 and a 2? The main argument against the dominance principle: it can be rarely applied, as acts often have good outcomes in some states, bad in others

10 The Maximin principle Let min(a i ) be the value of the worst outcome of a i Maximin principle: a i a j iff min(a i ) min(a j ) Examples: a 3 a i for every other state a i all the states are equally good. What can we do?

11 The Leximin principle Let min 1 (a i ) be the worst value of a i, min 2 (a i ) be the second worst value of a i, and so on. In general, let min n (a i ) be the nth worst value of a i Leximin principle: a i a j iff there is some positive integer n such that min n (a i ) > min n (a j ) and min m (a i ) = min m (a j ) for all m < n Example: a 3 a i for every other state a i

12 The Maximax principle With maximin and leximin we focus on the worst outcomes, but what about the best ones? Let max(a i ) be the value of the best outcome of a i Maximax principle: a i a j iff max(a i ) max(a j ) s 1 s 2 s 3 s 4 a a a a 3 a i for every other state a i

13 Optimism- pessimism rule Is there a possibility to consider both the best and worst outcomes? Let min(a i ) and max(a i ) be the values of the worst and best outcomes of any a i Let the real number α [0, 1] be the degree of optimism (α=0 means maximal pessimism, α=1 means maximal optimism) Optimism- pessimism rule: a i a j iff α max(a i ) + (1 α) min(a i ) > α max(a j ) + (1 α) min(a j ) Note that if α=0 te rule collapses into the maximin rule, while if α=1 it collapses into the maximax rule

14 Improvements Some improvements have been introduced with the optimism- pessimism rule: We pay attention, at the same time, to both the best- case and the worst- case scenarios We have introduced an index (α) for describing the relative importance of outcomes The rule operates in an interval scale (while in the previous principles we needed only ordinal scales) A problem remains: what about the outcomes in between? Example:

15 The Minimax Regret Idea: minimising the maximum amount of regret Let max{r 1, r 2,, r n } be a function that returns the maximum value from the set of real numbers Let max(s i ) be the maximum outcome that is achievable through one of the alternative acts Given a state of the world, let s de7ine the regret as the difference between the outcome under the chosen act and the best outcome obtainable in such state Minimax Regret Principle: a i a j iff max{(v(a i,s 1 ) max(s 1 )), (v(a i,s 2 ) max(s 2 )), } > max{(v(a j,s 1 ) max(s 1 )), (v(a j,s 2 ) max(s 2 )), }

16 Example 1) Max of each column (set {M j }) 2) Subtract M i from each v ij 3) Min of each act (set {m i }) 4) Max of the set {m i } (act a i ) s 1 s 2 s 3 s 4 M 1 =30 M 2 =15 M 3 =30 M 4 =20 Regret matrix m 1 = 18 m 2 = 20 m 3 = 9 m 4 = 11 max{ 18, 20, 9, 11}= 9 à a 3

17 Remarks The minimax regret rule urges to choose an alternative which minimises the maximum regret value The minimax regret rule is parallel to the maximin rule (both involve the same decision: minimising the maximum regret is equivalent to applying the maximin rule to the regret matrix) Objection: irrilevant alternatives may alter the choice between better alternatives. This can be problematic, but can also be disputed and accepted as rational or not?

18 Example a 5 has been added to the previous alternatives M 1 =30 M 2 =15 M 3 =30 M 4 =39

19 Principle of Insuf7icient Reason The rules considered so far focus on maximums and minimums while ignoring the intermediate values and the number of states A solution is to use the Principle of Insuf7icient Reason: if there is no reason to think that one state is more probable than the others, then all states should be considered equally probable and then calculate the expected value, i.e. Principle of Insuf@icient Reason: a i a j iff!!!!! 1!!!!,!! >!!!! 1!!!!,!!!

20 Remarks Is it really rational to assume equal probability between the states (without any other information)? For any set of alternatives? Simmetric acts seem to be required Is it rational to assign greater probability to one of the states?

21 Remarks Decision rules implement different approaches Decision rule Maximin Leximin Maximax pessimism Minimax Regret Insufficient Reason Character of the rule pessimism pessimism varies with index depends on

22 Axiomatic analysis Which rule is the best one? They yield con7licting prescriptions, coming from their different approaches Arguing that one rule is better by purely intuitive considerations seems not enough (intuitions may vary across agents) One possible answer: providing axiomatic justi7ication Formulation of a set of fundamental principles (having intuitive justi7ication) Derive the decision rule that satis7ies them

23 Randomised acts Instead of choosing one option, why not relying on chances? We could introduce a new act, obtained by averaging on the other acts. Example: a 3 would be chosen by? (in class exercise) maximin, leximin, minimax regret and also by optimism- pessimism if α > ½, hence it is the best option but what is its meaning?

24 Axioms Ordering: is transitive and complete transitivity: if a 1 a 2 and a 2 a 3 then a 1 a 3 completeness: any two acts can be ordered Symmetry: the ordering imposed by is independent of the labelling of acts Strict Dominance: better outcomes imply better ranking of one act Continuity: if one act weakly dominates another in a sequence of decision problems under ignorance, then this holds true also in the limit decision problem under ignorance Interval scale: the ordering imposed by remains unaffected by a positive linear transformation of the outcome values

25 Axioms Irrelevant alternatives: the ordering does not change if new alternatives are added Column linearity: if a constant is added to a column, the ordering imposed by remains Column duplication: the ordering imposed by remains if an identical state (column) is added Randomisation: if two acts are equally valuable, then every randomisation between the two acts is also equally valuable Special row adjunction: adding a weakly dominated act does not change the ordering of old acts.

26 Relation between axioms and rules Axioms with necessary and sufficient for the rule Axioms with are Axioms with are