1 / 25

How to Add up Uncountably Many Numbers? (Hint: Not by Integration)

How to Add up Uncountably Many Numbers? (Hint: Not by Integration) Peter P. Wakker, Econ., UvA & Horst Zank, Econ., Univ. Manchester. We consider binary relations  on sets X n , where X is connected topological space, and homomorfisms of ( X n ,  ) in (  , ). 2.

mleech
Download Presentation

How to Add up Uncountably Many Numbers? (Hint: Not by Integration)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. How to Add up Uncountably Many Numbers? (Hint: Not by Integration) Peter P. Wakker, Econ., UvA& Horst Zank, Econ., Univ. Manchester We consider binary relations  on sets Xn, where X is connected topological space, and homomorfisms of (Xn, ) in (, ).

  2. 2 Homomorfisms in (, ) are functions V : Xn  that represent: (f1,…,fn)  (g1,…,gn)  V(f1,…,fn)  V(g1,…,gn). Often V is of a special form, e.g. V(f1,…,fn) = V1(f1)+ … + Vn(fn) (additive homomorfism), or V(f1,…,fn) = p1U(f1) + … + pnU(fn). V(f1,…,fn) = p1U1(f1) + … + pnUn(fn). Later,  on sets XS where S is infinite.

  3. 3 Outline: 1. Economic applications: - allocation of prizes over agents; - decision under uncertainty. 2. Classical results for finite sets (Theorem of Debreu, 1960). 3. Extension to infinite sets: the basic research question. 4. Basic result for infinite sets; - simple functions; - bounded functions. Not: unbounded functions, applications.

  4. 4 Applications: 1. Allocation of prizes over agents. {1,…,n} is set of agents, X is a set of prizes. E.g. prizes are monetary amounts, X= ;  X is a set of houses;  X is a set of health states. As said, X is a connected topological space.

  5. 5 f = (f1,…,fn)  Xn: allocation, assigning fj to agent j, j = 1,…,n. f is a function from the agent set to the prize set. An arbitrator must choose between several available allocations. (f1,…,fn)  (g1,…,gn): Arbitrator prefers (f1,…,fn) to (g1,…,gn). Question: What are sensible kinds of preference relations?

  6. 6 Utilitarianism: Determine the subjective value Vj(fj) of prize fj for agent j. Evaluate allocation (f1,…,fn) by V(f1,…,fn) = V1(f1) + … + Vn(fn). Choose from available allocations the one valued highest. Is utilitarianism a wise method? It does, in a way, ignore social interactions.

  7. 7 Or: V(f1,…,fn) = p1U(f1) + … + pnU(fn). Or: V(f1,…,fn) = p1U1(f1) + … + pnUn(fn).

  8. 8 2. Decision under uncertainty Elections in a country. {1,…,n}: set of participating candidates. Exactly one of them will win, and it is unknown which one. (f1,…,fn): investment, yielding fj if candidate j wins. So, investments map candidates to prizes. (f1,…,fn)  (g1,…,gn): you prefer the left investment.

  9. 9 Expected utility: Determine (subjective) utility U(fj) of prize fj. Determine (subjective) probability pj that candidate j will win. Evaluate investment (f1,…,fn) by V(f1,…,fn) = p1U(f1) + … + pnU(fn), its expected utility. Choose from available investments the one with highest expected utility.

  10. 2 10 Alternative homomorfisms: V(f1,…,fn) = p1U1(f1) + … + pnUn(fn) (state-dependent expected utility). Or: V(f1,…,fn) = V1(f1) + … + Vn(fn). Are expected utility, or one of the mentioned alternative homomorfisms, wise methods? These theories ignore specific kinds of risk attitudes (certainty effect, …).

  11. 11 Which conditions on  are necessary/sufficient for homomorfisms as described? 1. is a weak order:  is complete:  f,g  Xn: f  g or g  f.  is transitive: [f  g & g  h]  f  h. 2.  is continuous:  fXn:  {g  Xn: g  f} is closed;  {g  Xn: f  g} is closed.

  12. 12 Notation: X is “identified with” the constant function (,…, ).  on X is derived from  on Xnthrough  (,…,)(,…,). U:X  is monotonic if  U()U().

  13. 13 Notation: if is (f with fi replaced by )  on Xn is monotonic if if if . For additive homomorfisms (V1(f1) + … + Vn(fn)), the following condition is necessary. Joint independence: if ig if ig

  14. 14 Lemma. Joint independence is necessary for additive homomorphisms. Proof. if ig  Vi()+jiVj(fj)  Vi()+jiVj(gj)  Vi() +jiVj(fj)  Vi() +jiVj(gj)  if ig.

  15. and sufficient If n 3, then 15 Theorem (Debreu 1960). Statement (ii) is necessary for Statement (i): (i) Vj : X , j=1,…,n, s.t. represent  additively through V(f1,...,fn) = V1(f1)+…+Vn(fn);  are continuous;  are monotonic. (ii) satisfies:  weak ordering;  monotonicity;  continuity;  joint independence. Uniqueness results: ...

  16. If n 3, then and sufficient 16 Theorem (Wakker 1989). Statement (ii) is necessary for Statement (i): (i) U : X , pj>0, j=1,…,n, s.t.  U is continuous;  U is monotonic;  is represented through V(f1,...,fn) = p1U(f1) + … + pnU(fn). (ii) satisfies:  weak ordering;  monotonicity;  continuity;  joint independence&tradeoff consistency.

  17. 17 We characterized homomorfisms through V1(f1) + … + Vn(fn) (additive) and p1U(f1) + … + pnU(fn). What about p1U1(f1) + … + pnUn(fn)? Decomposition of Vj = pjUj is unidentifiable!

  18. n pjU(fj) j=1 n pjUj(fj) n j=1 Vj(fj) j=1 18 Now we turn to the extensions of functionals from S = {1,…,n} to infinite (general) S. f : S  X; homomorfism: f : {1,…,n}  X; homomorfism: SU(f(s))dP(s) SUs(f(s))dP(s) ?

  19. 19 PART 2.Theorems for Infinite S Let {A1,…,An} be a finite partition of S. (A1:f1, …, An:fn) is the function assigning fj to all sAj. Such functions are simple. P.s., measure-theory: soit!

  20. 20 Notation. For f : S  X, g : S  X, A  S, the function fAg : S  X agrees with f on A and with g on Ac.

  21. 21 A  S is null if fAg ~ g for all f,g. Monotonicity: For all nonnull A1, (A1:f1,A2:f2,…,An:fn)  (A1:f1’,A2:f2,…,An:fn)  f1 f1’. Joint independence: cAf  cAg cA’f  cA’g.

  22. 22 Theorem. If partition of S with three or more nonnull sets, then the following two statements are equivalent for simple functions: (i) A  S VA : X  s.t.  Each VA is continuous;  Each VA is monotonic;  is represented by V(A1:f1,..., An:fn) = VA1(f1) + … + VAn(fn). (ii) satisfies:  weak ordering;  monotonicity;  continuity;  joint independence.

  23. 23 How about nonsimple functions? Let’s only do “bounded” ones. f : S  X is bounded if ,  X s.t.  f(s)  for all sS. Pointwise monotonicity of : sS: f(s)  g(s)  f  g. Pointwise monotonicity of V: S: sS: f(s)  g(s)  V(f)  V(g).

  24. 24 Simple-function denseness of :  f  g,  simple f', g' s.t. f  f'  g'  g, and  sS: f(s)  f'(s) and g'(s)  g(s). Simple-function denseness of Vis defined similarly. Existence-of-certainty-equivalents:  f:SX  X s.t. f ~* where *: SX is the constant- function.

  25. 25 Theorem. If partition of S with three or more nonnull sets, then following statements are equivalent for bounded functions: (i) A  S VA : {fA}  s.t.  Each VA is simple-continuous;  Each VA is monotonic;  is represented by V satisfying pointw.mon., simple-fion-densensess, and: V(f) = VA1(fA1) + … + VAn(fAn) for each partition A1,…,An of S. (ii) satisfies:  weak ordering, monotonicity, simple- continuity, joint independence;  pointw. mon., existence-of-certainty- eq.s, simple-function denseness.

More Related