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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.

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

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  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.

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