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Decision Theory Lecture 4
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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
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Individual decision making – under Certainty Choice functions Choice Choice function Weak axiom of revealed preference (WARP)
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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
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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)
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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)
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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)
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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)
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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
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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
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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
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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
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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
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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
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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
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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)
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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)
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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)
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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}
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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
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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)
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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
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23 Prove that the previous definitions are equivalent Homework
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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
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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
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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
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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
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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
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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
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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
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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
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32 Rule #1 – meets’em all: WARP, , , (intuitively – the evaluation does not depend on the menu, will be formalized later) The solution
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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
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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
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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
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