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

(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), orV(f1,…,fn) = p1U(f1) + … + pnU(fn).V(f1,…,fn) = p1U1(f1) + … + pnUn(fn).

Later, on sets XS where S is infinite.

Homomorfisms in (, ) are functions V : Xn that represent :

3

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.

Outline:

4

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.

Applications:

5

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?

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.

6

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.

Utilitarianism:

7

V(f1,…,fn) = p1U(f1) + … + pnU(fn).

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

Or:

8

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.

2. Decision under uncertainty

9

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.

Expected utility:

10

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, …). 2

Alternative homomorfisms:

11

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.

Which conditions on are necessary/sufficient for homomorfisms

as described?

12

X is “identified with”the constant function (,…, ).

on X is derived from on Xn

through (,…,) (,…,).

U : X is monotonic if U() U().

Notation:

13

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

Notation: if is (f with fi replaced by )

14

Proof.

if ig

Vi() +jiVj(fj) Vi() + jiVj(gj)

Vi() +jiVj(fj) Vi() + jiVj(gj)

if ig.

Lemma. Joint independence is necessary for additive homomorphisms.

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.

15

Theorem (Debreu 1960).If n 3, then

and sufficient

Uniqueness results: ...

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.

16

Theorem (Wakker 1989).If n 3, then

and sufficient

17

V1(f1) + … + Vn(fn) (additive)

andp1U(f1) + … + pnU(fn).

What about

p1U1(f1) + … + pnUn(fn)?

Decomposition of Vj = pjUj is unidentifiable!

We characterized homomorfisms through

18

Now we turn to the extensions of functionals from S = {1,…,n} to infinite (general) S.f : {1,…,n} X;

homomorfism:

pjU(fj)j=1

n

pjUj(fj)j=1

n

Vj(fj)j=1

n

f : S X;

homomorfism:

S U(f(s))dP(s)

S Us(f(s))dP(s)

?

19

Let {A1,…,An} be a finite partition of S.

(A1:f1, …, An:fn) is the function assigning fj to all sAj.

Such functions are simple.

P.s., measure-theory: soit!

PART 2. Theorems for Infinite S

20

f : S X, g : S X, A S, the function fAg : S X

agrees with f on Aand with g on Ac.

Notation. For

21

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.

A S is null if fAg ~ g for all f,g.

(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.

22

Theorem. If partition of S with three or more nonnull sets, then the following two statements are equivalent for simple functions:

23

Let’s only do “bounded” ones.

f : S X is bounded if , X s.t. f(s) for all sS.

Pointwise monotonicity of : sS: f(s) g(s) f g.

Pointwise monotonicity of V: S: sS: f(s) g(s) V(f) V(g).

How about nonsimple functions?

24

f g, simple f', g' s.t.f f' g' g, and sS: f(s) f'(s) and g'(s) g(s).

Simple-function denseness of Vis defined similarly.

Existence-of-certainty-equivalents: f:SX X s.t. f ~*where *: SX is the constant- function.

Simple-function denseness of :

(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.

25

Theorem. If partition of S with three or more nonnull sets, then following statements are equivalent for bounded functions: