PHPE 4000 Individual and Group Decision Making Eric Pacuit University of Maryland pacuit.org 1 / 30
Given ( P , I , N ) where: ◮ P ⊆ X × X represents a decision maker’s strict preference relation ◮ I ⊆ X × X represents a decision maker’s indifference relation ◮ N ⊆ X × X represents a decision maker’s non-comparability relation you can your weak preference relation R ⊆ X × X as follows: For all a , b ∈ X , aRb iff either aPb or aIb 2 / 30
Given R ⊆ X × X representing a decision maker’s weak preference relation, you can defined P , I , and N as follows: ◮ aPb iff aRb and it is not the case that bRa ◮ aIb iff aRb and bRa ◮ aNb iff it is not the case that aRb and it is not the case that bRa 3 / 30
Given some choices of a decision maker, in what circumstances can we understand those choices as being made by a rational decision maker? 4 / 30
Given a relation R ⊆ X × X and Y ⊆ X . Which option would an agent choose from Y (assuming her choices are guided by her preference R ). 5 / 30
Given a relation R ⊆ X × X and Y ⊆ X . Which option would an agent choose from Y (assuming her choices are guided by her preference R ). ◮ C ( Y ) = { y | y ∈ Y and there is no z such that zRy } (undominated/maximal) ◮ C ( Y ) = { y | y ∈ Y and for all z , yRz } (greatest/maximum) 5 / 30
Sen’s α Condition R : red wine W : white wine L : lemonade 6 / 30
Sen’s α Condition R : red wine W : white wine L : lemonade 6 / 30
Sen’s α Condition R : red wine W : white wine L : lemonade 6 / 30
Sen’s α Condition R : red wine W : white wine L : lemonade 6 / 30
Sen’s α Condition R : red wine R : red wine W : white wine W : white wine L : lemonade L : lemonade If the world champion is American, then she must be a US champion too. 6 / 30
Sen’s α Condition R : red wine R : red wine W : white wine W : white wine L : lemonade L : lemonade If the world champion is American, then she must be a US champion too. 6 / 30
Sen’s β Condition R : red wine W : white wine L : lemonade 7 / 30
Sen’s β Condition R : red wine W : white wine L : lemonade 7 / 30
Sen’s β Condition R : red wine W : white wine L : lemonade 7 / 30
Sen’s β Condition R : red wine W : white wine L : lemonade 7 / 30
Sen’s β Condition R : red wine R : red wine W : white wine W : white wine L : lemonade L : lemonade If some American is a world champion, then all champions of America must be world champions. 7 / 30
Revealed Preference Theory A decision maker’s choices over a set of alternatives X are rationalizable iff there is a (rational) preference relation on X such that the decision maker’s choices maximize the preference relation. 8 / 30
Revealed Preference Theory A decision maker’s choices over a set of alternatives X are rationalizable iff there is a (rational) preference relation on X such that the decision maker’s choices maximize the preference relation. Revelation Theorem . A decision maker’s choices satisfy Sen’s α and β if and only if the decision maker’s choices are rationalizable . 8 / 30
Choice Functions Suppose X is a set of options. And consider B ⊆ X as a choice problem. A choice function is any function where C ( B ) ⊆ B . B is sometimes called a menu and C ( B ) the set of “rational” or “desired” choices. 9 / 30
Choice Functions Suppose X is a set of options. And consider B ⊆ X as a choice problem. A choice function is any function where C ( B ) ⊆ B . B is sometimes called a menu and C ( B ) the set of “rational” or “desired” choices. A relation R on X rationalizes a choice function C if for all B C ( B ) = { x ∈ B | for all y ∈ B xRy } . 9 / 30
Choice Functions Suppose X is a set of options. And consider B ⊆ X as a choice problem. A choice function is any function where C ( B ) ⊆ B . B is sometimes called a menu and C ( B ) the set of “rational” or “desired” choices. A relation R on X rationalizes a choice function C if for all B C ( B ) = { x ∈ B | for all y ∈ B xRy } . Sen’s α : If x ∈ C ( A ) and B ⊆ A and x ∈ B then x ∈ C ( B ) 9 / 30
Choice Functions Suppose X is a set of options. And consider B ⊆ X as a choice problem. A choice function is any function where C ( B ) ⊆ B . B is sometimes called a menu and C ( B ) the set of “rational” or “desired” choices. A relation R on X rationalizes a choice function C if for all B C ( B ) = { x ∈ B | for all y ∈ B xRy } . Sen’s α : If x ∈ C ( A ) and B ⊆ A and x ∈ B then x ∈ C ( B ) Sen’s β : If x , y ∈ C ( A ) , A ⊆ B and y ∈ C ( B ) then x ∈ C ( B ) . 9 / 30
◮ What is the relationship between choice and preference? ◮ Should a decision maker’s preference be complete and transitive? ◮ Are people’s preferences complete and transitive? 10 / 30
Weak preference is transitive: for all a , b , c if a R b and b R c then a R c . ◮ strict preference is transitive: for all a , b , c if a P b and b P c then a P c ◮ indifference is transitive: for all a , b , c if a I b and b I c then a I c Weak preference is complete: for all a , b if a R b or b R a . ◮ For all a , b , either a P b , b P a or a I b Non-comparability is transitive: for all a , b , c if a N b and b N c then a N c . 11 / 30
Transitivity Indifference: For all x , y , z ∈ X , if x I y and y I z , then x I z . ◮ You may be indifferent between a curry with x amount of cayenne pepper, and a curry with x plus one particle of cayenne pepper for any amount x . But you are not indifferent between a curry with no cayenne pepper and one with 1 pound of cayenne pepper in it! 12 / 30
Transitivity Indifference: For all x , y , z ∈ X , if x I y and y I z , then x I z . ◮ You may be indifferent between a curry with x amount of cayenne pepper, and a curry with x plus one particle of cayenne pepper for any amount x . But you are not indifferent between a curry with no cayenne pepper and one with 1 pound of cayenne pepper in it! Incomparibility: For all x , y , z ∈ X , if xNy and yNz , then xNz . ◮ You may not be able to compare having a job as a teacher with having a job as lawyer. Furthermore, you cannot compare having a job as a lawyer with having a job as a teacher with an extra $1,000. However, you do strictly prefer having a job as a teacher with an extra $1,000 to having a job as a teacher. 12 / 30
Strict preference: For all x , y , z ∈ X , if x P y and y P z , then x P z . 13 / 30
Strict preference: For all x , y , z ∈ X , if x P y and y P z , then x P z . Cycle: x 1 P x 2 · · · P x n , yet x n P x 1 13 / 30
Cyclic Preferences I do not think we can clearly say what should convince us that a man at a given time (without change of mind) preferred a to b , b to c and c to a . The reason for our difficulty is that we cannot make good sense of an attribution of preference except against a background of coherent attitudes...My point is that if we are intelligibly to attribute attitudes and beliefs, or usefully to describe motions as behaviour, then we are committed to finding, in the pattern of behaviour, belief, and desire, a large degree of rationality and consistency. (Davidson 1974: p. 237) D. Davidson. ‘Philosophy as psychology’ . In S. C. Brown (ed.), Philosophy of Psychology, 1974. Reprinted in his Essays on Actions and Events. Oxford: OUP 2001: pp. 229–244. 14 / 30
Money-Pump Argument M C P ( M ) = ⇒ ( C , − 1 ) = ⇒ ( P , − 2 ) = ⇒ ( M , − 3 ) = ⇒ ( C , − 4 ) = ⇒ · · · 15 / 30
Money-Pump Argument M C P ( M ) = ⇒ ( C , − 1 ) = ⇒ ( P , − 2 ) = ⇒ ( M , − 3 ) = ⇒ ( C , − 4 ) = ⇒ · · · 15 / 30
Money-Pump Argument M C P ( M ) = ⇒ ( C , − 1 ) = ⇒ ( P , − 2 ) = ⇒ ( M , − 3 ) = ⇒ ( C , − 4 ) = ⇒ · · · 15 / 30
Money-Pump Argument M C P ( M ) = ⇒ ( C , − 1 ) = ⇒ ( P , − 2 ) = ⇒ ( M , − 3 ) = ⇒ ( C , − 4 ) = ⇒ · · · 15 / 30
Money-Pump Argument M C P ( M ) = ⇒ ( C , − 1 ) = ⇒ ( P , − 2 ) = ⇒ ( M , − 3 ) = ⇒ ( C , − 4 ) = ⇒ · · · 15 / 30
Assumptions ◮ Ann prefers x to y , written x P y , iff Ann always takes x when y is the only alternative. ◮ If x P y , then x + $ w P y + $ w ◮ If x P y , then there is some v > 0 such that for all u , x − $ u P y iff u ≤ v . ◮ x + $ w P x + $ z iff w > z . Note : x − $ w means that you keep item x and pay $ w 16 / 30
◮ A P B P C P A ◮ Decision maker is faced with a choice over three days. ◮ “ I will give you C for A , B for C , or A for B at a charge of $1” ◮ Each day, the decision maker can either accept ( a ) or reject ( r ) the offer. 17 / 30
A − $ 3 a t 3 r a t 2 B − $ 2 a r a t 1 t 3 r r a t 2 C − $ 1 r a t 3 r A t 1 t 2 t 3 t 4 time 18 / 30
A − $ 3 a t 3 r a t 2 B − $ 2 a r a t 1 t 3 r r a t 2 C − $ 1 r a t 3 r A t 1 t 2 t 3 t 4 time 18 / 30
A − $ 3 a t 3 r a t 2 B − $ 2 a r a t 1 t 3 r r a t 2 C − $ 1 r a t 3 r A t 1 t 2 t 3 t 4 time 18 / 30
A − $ 3 a t 3 r a t 2 B − $ 2 a r a t 1 t 3 r r a t 2 C − $ 1 r a t 3 r A t 1 t 2 t 3 t 4 time 18 / 30
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