teoria podejmowania decyzji relacja preferencji

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Teoria podejmowania decyzji Relacja preferencji. Agenda. Binarny relations – properties Pre-orders and orders , relation of rational preferences Strict preference and indifference relation. Today – another approach to decision making. Preferences : - PowerPoint PPT Presentation

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Teoria podejmowania decyzji

Relacja preferencjiBinarny relations propertiesPre-orders and orders, relation of rational preferencesStrict preference and indifference relationAgenda2Preferences:capability of making comparisonscapability of deciding, which of two alternatives is better/is not worse

Mathematically binary relations in the set of decision alternatives:X decision alternativesX2 all pairs of decision alternativesRX2 binary relation in X, selected subset of ordered pairs of elements of Xif x is in relation R with y, then we write xRy or (x,y)RToday another approach to decision making3Examples of relations:Being a parent of is a binary relation on a set of human beingsBeing a hat is a binary relation on a set of objectsx+y=z is 3-ary relation on the set of numbersx is better than y more than x is better than y is a 4-ary relation on the set of alternatives.

ExampleX={1,2,3,4}R a relation denoting is smaller thanxRy means x is smaller than y

Thus:(1,2)R; (1,3)R; (1,4)R; (2,3)R; (2,4)R; (3,4)R1R2, 1R3, 1R4, 2R3, 2R4, 3R4eg. (2,1) doesnt belong to R

Binary relations example #1512341234ExampleX={1,2,3,4}R a relation with no (easy) interpretationR={(1,2), (1,3), (2,3), (2,4), (3,2), (4,4)}Binary relations example #2612341234complete: xRy or yRxreflexive: xRx (x)antireflexive: not xRx (x)transitive: if xRy and yRz, then xRzsymmetric: if xRy, then yRxasymmetric: if xRy, then not yRxantisymmetric: if xRy and yRx, then x=ynegatively transitive:if not xRy and not yRz, then not xRz equivalent to: xRz implies xRy or yRz acyclic:if x1Rx2, x2Rx3, , xn-1Rxn imply x1xnBinary relations basic properties7R1R2R3R4R5R6completereflexiveantireflexivetransitivesymmetricasymmetricantisymmetricnegatively transitiveR1: (among people), to have the same colour of the eyesR2: (among people), to know each otherR3: (in the family), to be an ancestor of R4: (among real numbers), not to have the same valueR5: (among words in English), to be a synonymR6: (among countries), to be at least as good in a rank-table of summer olympicsExercise check the properties of the following relations8R1R2R3R4R5R6completereflexiveantireflexivetransitivesymmetricasymmetricantisymmetricnegatively transitiveR1: (among people), to have the same colour of the eyesR2: (among people), to know each otherR3: (in the family), to be an ancestor of R4: (among real numbers), not to have the same valueR5: (among words in English), to be a synonymR6: (among countries), to be at least as good in a rank-table of summer olympicsExercise check the properties of the following relations9Preferences capability of making comparisons, of selecting not worse an alternative out of a pair of alternativeswell talk about selecting a strictly better (or just as good) alternative later on

Depending on its preferences well use one of the relations:preorderpartial ordercomplete preorder (rational preference relation)complete order (linear order)Preference relation10R is a preorder in X, if it is:reflexivetransitive

We do not want R to be:Complete we cannot compare all the pairs of alternativesAntisymmetric if xRy and yRX, then not necessarily x=yPreorder11Micha is at a party and can pick from a buffet onto his plate: small tartares, cocktail tomatoes, sushi (maki), chunks of cheeseA decision alternative is an orderd four-tuple, denoting number of respective pieces, there can be at most 20 pcs on the plateMicha preferes more pcs than fewer. At the same time, he prefers more tartare than less. Micha cannot tell, if he wants to have more pcs if it mean less tartare.Preorder an example12ElementFormal notationSet of alternatives XX={x=(x1,x2,x3,x4)N4: x1+x2+x3+x420}Relation R at least as good asxRy x1y1 x1+x2+x3+x4y1+y2+y3+y4Is R reflexive? transitive? complete? antisymmetric?YesYesNo (why?)No (why?)Zapisz formalnie zbir wariantw dopuszczalnych: (a,b,c,d), a,b,c,d in N, a+b+c+d= (a2,b2,c2,d2) jeli a1>=a2 i a1+b1+c1+d1>=a2+b2+c2+d2Zwrotny takPrzechodni takZupeny nie, np. (1,0,0,0) vs (0,1,1,1)Antysymetryczny nie, np. (0,1,0,0) vs (0,0,1,0)12Micha is at a party and can pick from a buffet onto his plate: small tartares, cocktail tomatoes, sushi (maki), chunks of cheeseA decision alternative is an orderd four-tuple, denoting number of respective pieces, there can be at most 20 pcs on the plateMicha preferes more pcs than fewer. At the same time, he prefers more tartare than less. Micha cannot tell, if he wants to have more pcs if it mean less tartare.Preorder an example13ElementFormal notationSet of alternatives XX={x=(x1,x2,x3,x4)N4: x1+x2+x3+x420}Relation R at least as good asxRy x1y1 x1+x2+x3+x4y1+y2+y3+y4Is R reflexive? transitive? complete? antisymmetric?YesYesNo (why?)No (why?)Zapisz formalnie zbir wariantw dopuszczalnych: (a,b,c,d), a,b,c,d in N, a+b+c+d= (a2,b2,c2,d2) jeli a1>=a2 i a1+b1+c1+d1>=a2+b2+c2+d2Zwrotny takPrzechodni takZupeny nie, np. (1,0,0,0) vs (0,1,1,1)Antysymetryczny nie, np. (0,1,0,0) vs (0,0,1,0)13R is a partial order in X, if it is:reflexivetransitiveantisymmetric (not needed in the preorder)

We do not want it to be:Complete we cannot compare all the pairs of alternativesPartial order14Micha is at a party Decision alternatives are ordered pairs : # of pcs, # of tartares

Partial order an example15ElementFormal notationSet of alternatives XX={x=(x1,x2)N2: x2x120}Relation R at least as good asxRy x1y1 x2y2Is R reflexive? transitive? complete? antisymmetric?YesYesNo (why?)Yes (why?)15Micha is at a party Decision alternatives are ordered pairs : # of pcs, # of tartaresConclusion different structure (of the same problem), different formal representationPartial order an example16ElementFormal notationSet of alternatives XX={x=(x1,x2)N2: x2x120}Relation R at least as good asxRy x1y1 x2y2Is R reflexive? transitive? complete? antisymmetric?YesYesNo (why?)Yes (why?)Complete: Np. (3,0) oraz (1,1)16R is a complete preorder in X, if it is:transitivecomplete

Completeness implies reflexivityWe do not want it to be:antisymmetric equally good alternatives are allowed to differ

In our example if Micha didnt value tartare (and just wanted to eat as much as possible)Complete preorder rational preference relation17ElementFormal notationSet of alternatives XX={x=(x1,x2,x3,x4)N4: x1+x2+x3+x420}Relation R at least as good asxRy x1+x2+x3+x4y1+y2+y3+y4Is R reflexive? transitive? complete? antisymmetric?YesYesYesNoComplete preorder rational preference relationR is a complete order in X, if it is:transitivecompleteantisymmetric

In our example:Micha wants to eat as much as possiblewe represent alternatives as # of pcsComplete order (linear)19ElementFormal notationSet of alternatives XX={x=x1N: x120}Relation R at least as good asxRy x1y1Is R reflexive? transitive? complete? antisymmetric?YesYesYesYesComplete order (linear)Preference relations21PreorderPartial orderComplete preorderComplete orderreflexivetotaltransitiveantisymmetricLet R be a complete preorder (transitive, complete)xRy means x is at least as good as y

R generates strict preference relation P:xPy, if xRy and not yRxxPy means x is better than y

R generates indifference relation I:xIy, if xRy and yRxxIy means x just as good as yPreference and indifference relation22X={a,b,c,d}R={(a,a), (a,b), (a,c), (a,d), (b,a), (b,b), (b,c), (b,d), (c,c), (c,d), (d,d)}

Find P and I

P={(a,c), (a,d), (b,c), (b,d), (c,d)}I={(a,a), (a,b), (b,a), (b,b), (c,c), (d,d)}R=PIAn exercise23Let P and I be generated by R a complete preorder

P is:asymmetricnegatively transitiveantireflexiveacycylictransitive

I is an equivalence relation:reflexivetransitivesymmetricProperties of P and I (of previous slides)24reflexive (xIx)obvious using reflexivity of R we get xRx

transitive (xIy yIz xIz)predecessor means that xRy yRx yRz zRyusing transitivity we get xRz zRx, QED

symmetric (xIy yIx)predecessor means that xRy yRx, QEDProof of the properties of I (xIy xRy yRx)25Logical preliminarypqpqp qp qq p0011111100100001101111100111 pqpqp q(p q)q p0011011100110001101001100100R is complete iff P is asymmetric

R is transitive iff P is negatively transitiveP vs R (xPy xRy yRx)271. Prove thatxRy yPz xPz

2. Show thatx,y: xPy xIy yPx

Homework28Lets start with relation P:asymmetricnegatively transitive

Then we say thatxIy, if xPy yPxxRy, if xPy xIy

Homework. Prove that with such definitions:I is an equivalence relationR is a complete preorderAnother definition of rational preferences29X={a,b,c,d}P={(a,d), (c,d), (a,b), (c,b)}

Find R and I

I={(a,a), (a,c), (b,b), (b,d), (c,a), (c,c), (d,b), (d,d)}R=PIExercise30Can we start with I?reflexivesymmetrictransitive

No we wouldnt be able to order the abstraction classes

Another definition of rational preferences31Is it enough to use P?asymmetricacyclic (not necessarily negatively transitive)

No lets see an exampleAnother definition of rational preferences32Mr X got ill and for years to come will have to take pills twice a day in an interval of exactly 12 hours. He can choose the time however.

All the decision alternatives are represented by a circle with a circumference 12 (a clock). Lets denote the alternatives by the length of an arc from a given point (midnight/noon).

Mr X has very peculiar preferences he prefers y to x, if y=x+p, otherwise he doesnt care

Thus yPx, if y lies on the circle p units farther (clockwise) than xP from the previous slide an example33What properties does P have?asymmetrynegative transitivitytransitivityacyclicity

P generates weird preferences:1+2p better than 1+p, 1+p better than 1, 1+2p equally good as 11 equally good as 1+p/2, 1+p/2 equally good as 1+p, 1 worse than 1+pExercise34What if we take P?asymmetrictransitive (not necessarily ne