1. Decision Theory Lecture 4

2. Decision Theory – the foundation of modern economics Individual decision making – under Certainty Choice functions Revelead preference and ordinal utility theory Operations Research, Management Science – under Risk Expected Utility Theory (objective probabilities) Bayesian decision theory Prospect Theory and other behavioral theories Subjective Expected Utility (subjective probabilities) – under Uncertainty Decision rules Uncertainty aversion models Interactive decision making – Non-cooperative game theory – Cooperative game theory – Matching – Bargaining Group decision making (Social choice theory) – Group decisions (Arrow, Maskin, etc.) – Voting theory – Welfare functions

3. Individual decision making – under Certainty Choice functions Choice Choice function Weak axiom of revealed preference (WARP)

4. 4 Pick the cheapest (e.g. public tenders) Pick the second cheapest (wine for a party) Maximize the IRR (investment projects) Pick whoever gets majority of votes (Talent shows on TV) … Exemplary choice functions

5. 5 Choice functions – some intuition (1) A B Out of the gray set, A was chosen (a unique choice) good 1. good 2. Out of the blue set, B was chosen (a unique choice) Do we find these choices confusing? (when considered collectively)

6. 6 Choice functions – some intuition (2) A B good 1. good 2. Out of the gray set, A was chosen (a unique choice) Out of the blue set, B was chosen (a unique choice) Do we find these choices confusing? (when considered collectively)

7. 7 Choice functions – some intuition (3) A B good 1. good 2. Out of the gray set, A was chosen (a unique choice) Out of the blue set, B was chosen (a unique choice) Do we find these choices confusing? (when considered collectively)

8. B 8 Choice functions – some intuition (4) Good 1. Good 2. A C Out of the gray set, A was chosen (a unique choice) Out of the blue set, B was chosen (a unique choice) Do we find these choices confusing? (when considered collectively) Out of the golden set, C was chosen (a unique choice)

9. 9 1.Can we, using only linear budget constraints, construct such an example for two goods, that there is a „consistency problem” when considering more than two alternatives, and no problem when considering only each two alternatives separately? 2.And when considering three goods? Homework

10. 10 Notation: (Technical) properties: If C(B) contains a single element  this is the choice If more elements  these are possible choices (not simultaneously, the decision maker picks one in the way which is not described here) Choice functions – a formal definition always a choice out of a menu set of decision alternatives available menus (non-empty subsets of X) choice function, working for every menu

11. 11 Let X={a,b,c}, B =2 X Write down the following choice functions: – C 1 : always a (if possible), if not – it doesn’t matter – C 2 : always the first one in the alphabetical order – C 3 : whatever but not the last one in the alphabetical order (unless there is just one alternative available) – C 4 : second first alphabetically (unless there is just one alternative) – C 5 : disregard c (if technically it is possible), and if you do disregard c, also disregard b (if technically possible) An exercise

12. 12 BC 1 (B)C 2 (B)C 3 (B)C 4 (B)C 5 (B) {a} {b} {c} {a,b}{a} {b}{a,b} {a,c}{a} {c}{a} {b,c} {b} {c}{b} {a,b,c}{a} {a,b}{b}{a} The solution

13. 13 BC 1 (B)C 2 (B)C 3 (B)C 4 (B)C 5 (B) {a} {b} {c} {a,b}{a} {b}{a,b} {a,c}{a} {c}{a} {b,c} {b} {c}{b} {a,b,c}{a} {a,b}{b}{a} The solution

14. 14 Sometimes an internal consistency is postulated Why so? – positive approach – non-consistent will go bankrupt – normative – in order not to go bankrupt We’ll discuss the following: – weak axiom of revealed preferences –  property –  property –  property Desirable properties

15. 15 Definition (WARP): A pair( B,C()) satisfies WARP, if the following holds: if for some B from B, s.t. x,y  B, we have x  C(B), than for every B’ from B, s.t. x,y  B’, if y  C(B’), then x  C(B’). Intuitively: if x was shown to be at least as willingly picked as y (for a menu B), then for every menu B’ containing x,y, if y is picked, so does x have to be. WARP – weak axiom of revealed preferences

16. 16 WARP – an intuition A B Out of the gray set, A was chosen (a unique choice) good 1. good 2. Out of the blue set, B was chosen (a unique choice) Do we find these choices confusing? (when considered collectively)

17. 17 WARP – an intuition A B good 1. good 2. Out of the gray set, A was chosen (a unique choice) Out of the blue set, B was chosen (a unique choice) Do we find these choices confusing? (when considered collectively)

18. 18 WARP – an intuition A B good 1. good 2. Out of the gray set, A was chosen (a unique choice) Out of the blue set, B was chosen (a unique choice) Do we find these choices confusing? (when considered collectively)

19. 19 Check which functions C 1 -C 5 do not fulfill WARP, prove by giving exemplary menus An exercise BC 1 (B)C 2 (B)C 3 (B)C 4 (B)C 5 (B) {a} {b} {c} {a,b}{a} {b}{a,b} {a,c}{a} {c}{a} {b,c} {b} {c}{b} {a,b,c}{a} {a,b}{b}{a}

20. 20 C 1 – fulfils C 2 – fulfils C 3 – doesn’t! b picked from {a,b,c} and not from {a,b} C 4 – doesn’t! b picked from {a,b,c} and not from {b,c} C 5 – doesn’t! b picked from {a,b} and not from {a,b,c}, while a picked The solution

21. 21 Definition (  property): Assume B =2 X. C() meets , if the following holds: if for some B out of B we have x  C(B), then for every B’  B, s.t. x  B’, we have x  C(B’). Intuitively: if x picked from menu B, then shall be picked from each smaller menu B’ (if present in it).  property (Chernoff property)

22. 22 If something not picked from menu B’, shan’t be picked from a bigger one: If we add to B 1 some new alternatives B 2, then the choice will either not change, or something out of new alternatives should be picked  property differently

23. 23 Prove that the previous definitions are equivalent Homework

24. 24 BC 1 (B)C 2 (B)C 3 (B)C 4 (B)C 5 (B) {a} {b} {c} {a,b}{a} {b}{a,b} {a,c}{a} {c}{a} {b,c} {b} {c}{b} {a,b,c}{a} {a,b}{b}{a} WARPyes no  yes no yes An exercise – check the  property for C 1 -C 5

25. 25 Conclusion for the previous exercise –  and WARP differ (let’s look for other properties) Definition (  property): Take B =2 X. C() meets  property, if the following holds: if form some B’ in B we have x,y  C(B’), than for each B, B’  B, we have x  C(B)  y  C(B). Intuitively: if x and y are picked in a menu B’, then their status is equal in every greater menu B.  property

26. 26 BC 1 (B)C 2 (B)C 3 (B)C 4 (B)C 5 (B) {a} {b} {c} {a,b}{a} {b}{a,b} {a,c}{a} {c}{a} {b,c} {b} {c}{b} {a,b,c}{a} {a,b}{b}{a} WARPyes no  yes no yes  no An exercise – check  property for C 1 -C 5

27. 27 Definition (  property): Assume B =2 X. C() meets , if the following holds: if for every menu B i out of a family of menus we have x  C(B i ), then for B=  B i we have x  C(B). Intuitively: if x is picked in every menu (in a family of menus), than it is also picked in a joint menu  property

28. 28 BC 1 (B)C 2 (B)C 3 (B)C 4 (B)C 5 (B) {a} {b} {c} {a,b}{a} {b}{a,b} {a,c}{a} {c}{a} {b,c} {b} {c}{b} {a,b,c}{a} {a,b}{b}{a} WARPyes no  yes no yes  no  yes no An exercise – check  property for C 1 -C 5

29. 29 BC 1 (B)C 2 (B)C 3 (B)C 4 (B)C 5 (B) {a} {b} {c} {a,b}{a} {b}{a,b} {a,c}{a} {c}{a} {b,c} {b} {c}{b} {a,b,c}{a} {a,b}{b}{a} WARPyes no  yes no yes  no  yes no The complete solution

30. 30 Assume C1-C5 can be used in a public tender (a,b,c denote offers) Take C 3 ({a,b})={a}, C 3 ({b,c})={b}, C 3 ({a,b,c})={a,b} – different choice for a complete problem (b may be selected), – different when short listing – … pairise comparisons also change the outcome – b „better than” c, a „better than” b, hence a – putting c on the table impacts the chocie (favours b – possible alliance) Properties and manipulation

31. 31 Public tender Alternatives – offers described by: price and time to deliver (quality is constant) Rule #1: – minimize the expression  price i +   time i (for some weights ,  determined irrespectively of set of offers) Rule #2: – calculated the minimal price (MP) and minimal time (MT) for all offers (assume MP>0 and MT>0) – minimize the expression price i /MP + time i /MT Which rule do you like? An exercise

32. 32 Rule #1 – meets’em all: WARP, , ,  (intuitively – the evaluation does not depend on the menu, will be formalized later) The solution

33. 33 Rule #2 – doesn’t meet a single one Take B={x,y,z}, x=(4,4), y=(1,9), z=(16,1) – what will be selected? Try to find some modifications in order to show how , ,  are broken The solution

34. 34 Different views on decision making – choice and choice functions – preferences – utility function We can judge not only alternatives, but also choice rules – not meeting some properties yields a risk of being manipulated – different properties, not all of them equivalent Summing up

35. 35 Materials Compulsory: – A. MasColell, M. Whinston, J. Green Microeconomic Theory, Oxford University Press, 1995, rozdz. 1 Supplementary: – A. Sen, Choice Functions and Revealed Preference, The Review of Economic Studies, 1971, 38(3), s. 307-317