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 (, ).
Schijnt dat homomorfisme maar een-kant-op implicatie is. Embedding moet injectief zijn.
Wiskundigen moeten measure-theory er meteen bij hebben, ze zijn er tezeer aan gewend en het niet doen gaf veel verwarring.
This lecture will introduce a new functional, generalizing integrals, that can be interpreted as an addition of uncountably many numbers. It will be ordinally equivalent, in a way specified below.
At X say: X is “quite general.”
Term (weak) preference laten vallen.

2. Homomorfisms in (, ) are functions V : Xn   that represent :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. 1. Economic applications: - allocation of prizes over agents; 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. 1. Allocation of prizes over agents. {1,…,n} is set of agents, 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.
Bij connected topological spaces: Mathematicians say “Ah I know what that is” and then it is fine. With economists I would be spending at least five minutes here discussing whether connectedness is empirically realistic. Well, it is never perfectly satisfied, especially with finite sets, so then you discuss how approximate it is, and how serious its restrictions are for the particular economic applications that you should have made clear at the outset.

5. f = (f1,…,fn)  Xn: allocation, assigning fj to agent j, j = 1,…,n. 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. Determine the subjective value Vj(fj) of prize fj for agent j. 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.
Bij additive representation zeggen:
Evaluation is done in two stages:
(1) Evaluate each coordinate in isolation.
(2) Aggregate them, simply by addition.

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

8. 2. Decision under uncertainty8
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.
At investment, say that uncertainty about candidates means uncertainty about profit resulting from investment.

9. Determine (subjective) utility U(fj) of prize fj. 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.
At end of page, discuss a bit subjective probs. Possibly discuss Bayesian statistics.

10. Alternative homomorfisms: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, …). 2

11.  {g  Xn: g  f} is closed;  {g  Xn: f  g} is closed.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.
Weak order: ranking with ties allowed.

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

13. Notation: if is (f with fi replaced by )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. Lemma. Joint independence is necessary for additive homomorphisms.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.
Say that really easy, but central, so I give the proof.
Aan eind zeggen: If one common coordinate doesn’t matter then, by repeated application, any number of common coordinates doesn’t matter.

15. Statement (ii) is necessary for Statement (i):
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.
and
sufficient
Uniqueness results: ...

16. Statement (ii) is necessary for Statement (i):
If n  3, then 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.
and sufficient
Say that only green things are different than preceding slide.

17. We characterized homomorfisms through17
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!
After the state-dependent formula, talk some, first relating it to the other two. Explain unidentifiability in words. Will not be very clear to public. That is exists iff additive homomorfism, that each is more general than the other, can easily be understood, and from that it can already be concluded that state-dependence is overdone, inefficient parameters.

18. 18
Now we turn to the extensions of functionals from S = {1,…,n} to infinite (general) S.
f : {1,…,n}  X;
homomorfism:
f : S  X;
homomorfism:
 pjU(fj)
j=1 n
S U(f(s))dP(s)
 pjUj(fj)
j=1 n
S Us(f(s))dP(s)
Of course, before question mark, let them spend some time in suspense.
Vj(fj)
j=1 n ?

19. PART 2. Theorems for Infinite S19
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. Notation. For f : S  X, g : S  X, A  S, the function fAg : S  X20
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. A  S is null if fAg ~ g for all f,g.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.
Explain ~. Weak ordering is ranking with ties, this is the tie-relation, the symmetric part of the binary relation.
If I know what a function is outside of A, I need not know what is in A to already determine in what equivalence class, what level, of the ranking system it is.

22.  Each VA is continuous;  Each VA is monotonic;   is represented by 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.
Show immediately all of Statement (i) (done this way for consistency with later theorem where this layout is really desirable).

23. How about nonsimple functions?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).
Pointwise monotonicity: automatically satisfied in finite case. Our choice if we want it in general, let’s do it it’s so natural!
Warn them that some technical, boring, conditions are coming.

24. Simple-function denseness of :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 V is defined similarly.
Existence-of-certainty-equivalents:
 f:SX   X s.t. f ~*
where *: SX is the constant- function.

25.  Each VA is simple-continuous;  Each VA is monotonic; 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.