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Money as a mechanism to intersubjectively
compare entirely subjective preferences
Johannes Reich, johannes.reich@sap.com
Martin Härterich, martin.haerterich@sap.com
July 27, 2024
Abstract
Purpose: This article contributes a money model that let us under-
stand under which conditions money allows to interpersonally compare
the entirely subjective material preferences of dierent economic subjects.
Methodology: We proceed in several steps. First, we provide a graph-
based model for the innitely scaling network of economic relevant interac-
tions, which resolves the issue that the nodes (economic subjects) and the
edges (economic interactions) of this graph cannot be modelled indepen-
dently. It rests on the role concept, where roles can compose externally to
an interaction (an edge) and internally to a coordinating subject (a node).
Second, we choose the preference concept and its possible representation
by a utility function to model how an economic subject takes its decisions.
Then we introduce two complementary roles of buyer and seller together
with two dierent ways to compose both, namely
'externally'
to a trade
and
'internally
to a trader. We provide a simple utility representation
that is
'compositionally consistent'
as we dened it for both compositions.
In fact, we prove that this utility representation is the only one with this
property.
Findings: In this money model, interpersonal comparison of subjec-
tive material preferences is possible under the following conditions: (1)
The decisions of an economic subject in a trade can be modelled by a
preference model where preferences are expressable as utility. And (2) all
buyers have the same total budget and the same access to the same set of
alternatives.
Value: Since the (now disproved) conviction that one cannot compare
subjective preferences interpersonally had signicantly shaped economic
theory in the past, the authors assume that the (now proven) fact that
money allows exactly such a comparison under certain conditions will
shape economic theory and practice in the future as well.
1
1 Introduction
Can we meaningfully compare preferences between economic subjects, despite
the fact that they are entirely subjective? This question is still ercely debated
in modern economics. The contribution of this article is to show that money
can under well dened circumstances provide exactly that kind of 'magic':
a mechanism to interpersonally compare entirely subjective preferences.
In fact, in the middle of the 20th century, it was mainstream to think that
interpersonal comparison of preferences was 'unscientic'. To quote some well
known scholars of that time: Lionel Robbins [31], said
'every mind is inscrutable
to every other mind and no common denominator of feeling is possible'
. This
conviction seemed to be the base of his belief that the relation between ethics
[viewed as
'speculative'
] and economics [viewed as
'scientic'
] could only be a
'mere juxtaposition'
[32, p.132]. Kenneth Arrow [1] still said that
'interpersonal
comparison of utilities has no meaning.'
1
. In his historical perspective on the
problem of interpersonal comparisons of utility, Stavros Drakopoulos [9] states in
1989 that
the majority of economists are prepared to declare that interpersonal
comparisons [with respect to utility] are not possible.
According to his own account, Amartya Sen [36] was
'much concerned with
incorporating dierent ways of making interpersonal comparisons and their far-
reaching consequences on what is permissible in welfare economics.'
, culminat-
ing in his book 'Collective Choice and Social Welfare' in 1970 [34]. But even
Amartya Sen stated in 1999 [35, p.68]
If dierent persons have dierent prefer-
ences (reected in, say, dierent demand functions), there is obviously no way
of getting interpersonal comparisons from these diverse preferences
If this were true, economics would face a severe dilemma: How should it say
anything sensible about the distribution of scarce but exchangeable goods based
on the preferences of the people, without being able to relate these preferences
to one another?
In sharp contrast to this dismissive attitude, it is well known since many
decades that utility, under well dened circumstances, could even be 'trans-
ferred' from one person to another. This property is the base for game theorists
like John C. Harsanyi [15] (pp. 111) or Roger B. Myerson [24] (pp. 384) to
classify cooperative games into those with and those without transferable util-
ity. The base for this transferability is the representation of utility in a special,
namely a linear function.
To convincingly show that money can be a mechanism that allows for in-
terpersonal comparison of preferences, we go through a certain program, which
we sketch in the following. The rst step is to provide a model of the economic
subject that accounts for both, its relevant decision-driven interactive behavior
and its ability to sensibly coordinate its dierent interactions.
In other words, we have to nd a way to adequately model the innitely
scaling network of economic relevant interactions, a section of which we show
1
Economically utility is understood as one form to represent preferences, we come to this
issue later.
2
in Fig. 1. Mathematically, such a network can be viewed as a graph with the
economic subjects as nodes and their interactions as edges.
But, unfortunately, neither the economic subjects (the nodes) nor the inter-
actions (the edges) can be modelled independently. The key idea here is to use
one common building block for modeling both, the subjects as well as their in-
teractions, we call 'roles', and couple these roles in two dierent, complementary
ways, namely externally and internally [30]. Their external coupling results in
an interaction and their internal coupling, which we name 'coordination', results
in a (more complete) economic subject.
Next, we ll the leeway of the nondeterministic interactions with a model of
the human mind and action: the decision and preference concept. We assume
that we decide along our preferences for outcomes, an ubiquitious approach in
economics.
As is well known, preferences can, under certain restrictive circumstances, be
represented by a utility function. The freedom we have to express our preference
relation with a utility function paves the way to our money model as we can
impose on it several desirable restrictions and can still be sure to talk about our
preferences while instead refering to utility values.
In section 4.3 we introduce the money model in an intuitive way by using
the traditional micro economic concept of the utility of a buyer and a seller role.
But beyond traditional economics, we take both, the interaction perspective as
a trade as well as the coordination perspective as a trader into account.
In the following section we additionally prove that the ad hocly chosen simple
linear utility functions of buyer and seller in the former section are indeed the
only ones that fulll our intuition of money as a utility transfer mechanism,
assuming certain requirements which we name
'compositional consistent'
.
In fact, for the remaining step to intersubjectively compare the preferences
of dierent subjects, it suces that there exists at least one such utility repre-
sentation. For the preferences of competing buyers to become comparable, all
buyers are required to have the same total budget and the same access to the
same set of alternative outcomes.
In section 8we discuss other approaches from the literature: the concept
of Pareto-eciency to avoid intersubjective utility comparison alltogether, the
treatment of money as a commodity, and how welfare economics misinterpreted
the utility concept.
In section 7we extend our model to labor trade with a certain caveat.
In the concluding discussion of section 9, we explore the normative aspects
of our money model in two respects. First, we, the authors, argue that the mon-
etary mechanism as a society should be designed in such a way that it achieves
its design goal of making our material preferences at least approximately com-
parable. And secondly, taking the repercussions of the money mechanism on our
ability to freely decide seriously, we advocate to shape the money mechanism as
a society in a way that it strongly support us in making truly free rather then
enforced economic decisions.
3
2 A model of the economically acting subject
2.1 Interactions and coordination of the economic subject
We model the economic world as a virtually innitely scaling network of eco-
nomically interacting subjects, a section of which we show in Fig. 1. Thus, we
propose to model this network as a graph, where the nodes are the subjects and
the edges are the interactions.
Figure 1: A section of the innitely scaling network of economic relevant inter-
actions.
Beyond traditional network theory in economics (e.g. [3]) or organizational
psychology (e.g. [5]), our model takes into account that the nodes (economic
subjects) and the edges (economic interactions) of this graph cannot be mod-
elled independently: we cannot describe the interactions without referring to
the participating subjects, nor can we describe the subjects and what they do
without refering to their interactions.
The key idea, taken from computer science [29,30], is to use a projection
of the subjects, which we call a 'role'
2
, as the common building block for both,
the edges as well as the nodes of this graph.
These roles can compose, or couple, in two dierent ways we name
'exter-
nally'
and
'internally'
[30], as we illustrate in Fig. 2.
3
Roles of dierent subjects couple externally by the exchange of information
or goods resulting in an
'interaction'
, the edges of the graph. In informatics,
another term is
'protocol'
[18,7].
2
This role concept was actually very much inspired by a sociology seminar on Erving
Goman [14] one of the authors took part, which was sympathetically led by Prof. Uta
Gerhards at the University of Gieÿen in the 1990s
3
It was mainly Arend Rensink's talk Compositionality huh? at the Dagstuhl Workshop
Divide and Conquer: the Quest for Compositional Design and Analysis in December 2012,
who drew the attention of one of the authors to the prominent position of the composition
concept with regard to the issues of interaction and interoperability [28]
4
We will describe this external composition of roles of dierent subjects in
detail in section 3.1 for a buyer and a seller role in a trade.
A) B)
Figure 2: The two dierent ways roles can compose correspond to two dierent
perspectives. They either compose externally, creating an interaction (part A)
or internally, creating a coordinating subject (part B).
Roles of a single subject couple internally by what we call a
'coordination'
,
resulting in a (more complete) economic subject, a node of the graph. We will
describe this internal composition of roles of a single subject in detail in section
3.2 for a buyer and a seller role composing to a trader.
2.2 The role model in mathematical terms
To comprehend this role model and its consequences in detail, we describe it
mathematically. The somewhat higher conceptual eort associated with this
precision is more than compensated for by the possibility of a much better
understanding of the decision-making concept in economic interactions and the
relevance of the preference (and hence, utility) concept for modelling our free
decisions only.
We model a role as a nondeterministic, stateful, discrete input/output tran-
sition system together with a success-criterion as an input/output-automaton
(I/O-A)
4
. It consists of three sets of characters,
I
,
O
and (non empty)
Q
, so
called
'alphabets'
, for input, output and internal state values, a start state
q0
, a
transition relation
∆⊆Iϵ×Oϵ×Q×Q
, where for any alphabet
A
,
Aϵ=A∪{ϵ}
holds with
ϵ
symbolising the empty character. In case of nite interactions, the
success criterion is the set of nal states (or
'outcomes'
. In case of innite inter-
actions, there are dierent choices for the success criterion [11], one possibility is
the set of all set of states that are occupied innitely often (Muller acceptance).
Then an outcome would be a set of states.
Instead of
(i, o, p, q)∈∆
, we also write
pi/o
→q
. In this model, a character
of an alphabet might not carry any other semantics than being distinguishable
from all other characters of its alphabet. Then this would correspond to the
4
In the informatics literature, such an I/O-A is also called 'transducer', or, in the deter-
ministic case, 'Mealy'-automaton.
5
notion of information in the sense of Claude E. Shannon [37]. But it could also
denote some 'real'-world entity like apples.
2.3 Decisions of the economic subject
In general, the interactions will be nondeterminstic, stateful and asynchronous.
Nondeterministic and asynchronous because anything else would not scale. And
stateful, because otherwise nondeterminism would imply complete randomness.
In a nondeterministic transition relation, for a given input
i
and internal
state
p
, there could be several possible transitions leading to dierent output
characters
o
and new internal states
q
.
The transition relation can be determined with an additional alphabet
D
which is disjoint to
I∪O∪Q
in a way that each nondeterministic transition
is complemented by a unique character
d∈D
. We call these extra characters
d∈D
'decisions' and the complemented I/O-A a
decision automaton
.
As the transition relation of a decision automaton is per construction de-
terministic, the decision automaton determines a transition function
5
. In this
model we can say that the nondeterminism of our interactions leaves us the
latitude to determine our actions with our decisions.
We call a decision
'free'
if it is not determined by the success criterion of the
interaction and
'enforced (by the success criterion)'
otherwise.
We call the external coupling of these decision automata of dierent sub-
jects, i.e. a protocol with explicit decisions, a
'game in interactive form (GIF)'
[29], emphasizing the tight relation between the informatical protocol and the
economic game concept.
2.4 Equivalence class construction
In the following we want to describe more complex roles eciently. We therefore
introduce a certain abstraction by summarizing the elements of the transition
relation in equivalence classes similar to [27].
The key idea is to create equivalence classes that preserve the time structure
of the transitions by identifying a combination of document classes, the most
relevant state value components as well as additional conditions in a way that
we can still attribute one set of entities to time
t
(like
i
and
p
) and the other
entities to
t′=successor(t)
(like
o
and
q
).
We introduce three injective functions:
1.
parse
IO :I∪O→DocCls ×
Param
IO
assigns each character
a∈I∪O
a document class
docCls(a)∈DocCls
and a value of a parameter set
paramIO(a)∈
Param
IO
. We can thereby regard each I/O-character of
a decision system as an element of some document class that represents
certain parameter values.
5
In the sense of [29], the decision concept becomes a mean to identify an interacting human
as a system by a ctitiously assumed system function.
6
2.
parse
Q:Q→Qmode ×Qrest
assigns each state value
q∈Q
a mode com-
ponent
mode(q)∈Qmode
and a rest component
rest(q)∈Qrest
.
3.
parse
D:D→DCls ×P aramD
assigns each decision value
d∈D
a de-
cision class component
dCls(d)∈DCls
and a value of a parameter set
paramD(d)∈P aramD
. We can thereby regard each decision as an ele-
ment of a given decision class that represents certain parameter values.
And we introduce a set of conditions
cond
as functions
conddCls(d),mode(p),docC ls(i):P aramD×Qrest ×P ar amIO → {true, f alse}
,
which, for each decision class, mode and document class, evaluates the values of
the remaining state and the parameters of the decisions and input documents
to either true or false.
With these tools we now dene the desired equivalence relation:
Denition 1.
Two transitions
t, t′∈∆
are equivalent, symbolically
t∼∆t′
, if
their values for the decision and document classes, the modi, and the condition
are equal, that is i
dCls(d) = dCls(d′)
,
docCls(i) = docCls(i′)
,
docCls(o) =
docCls(o′)
,
mode(p) = mode(p′)
,
mode(q) = mode(q′)
, and
conddCls(d),mode(p),docC ls(i)=conddCls(d′),mode(p′),docC ls(i′)
.
Thus with
dCls(d)
,
mode(p)
,
docCls(i)
, and
conddCls(d),mode(p),docC ls(i)
we
relate the transition equivalence explicitly to all possible values of
(p, d, i)
at
time
t
and with
mode(q)
and
docCls(o)
we ignore
rest(q)
and
param(o)
for all
times
t′=succ(t)
.
Instead of the set of all transitions of a single equivalence class
∆l
we also
write:
mode(p)
(dCls(d),docC ls(i),conddCls(d),mode(p),docC ls(i))/docCls(o)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ mode(q)
(1)
To help recognize decisions, mode-values and document classes, we prex
all decision names with an '@', put all mode values in small caps and write all
document class with an initial large letter.
We now can say in the following article that, for example, a seller who re-
ceives a purchase order document
Order
from a customer, decides
@Accept
to ac-
cept the order and passes accordingly from mode value
listening
to
ordered
,
provided that the customer is assessed as
trustworthy
, and, accordingly, returns
a conrmation document
Conrmation
and notate this as:
listening
(
@Accept
,
Order
,isT rustworthy )/
Conrmation
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
ordered
It is important to keep in mind, that unlike the
pi/o
→q
notation, which de-
notes a single transition, Eq. (1) denotes a whole class of equivalent transitions.
7
3 Interaction and coordination: the composition
of buyer and seller roles to trade and trader
3.1 Trade as interaction: external composition
In our trade interaction, the
external
coupling of two subjects in their buyer
(
B
) and seller (
S
) roles is created by the exchange of money and commodity
a
. The coupling condition is therefore the identity between
B
's cost and
S
's
revenue which we name
pricea
, together with an increment in the quantity
nB
a
of commodity
a
for
B
and a decrement in the quantity
nS
a
of commodity
a
for
S
in the handover step.
External coupling:
(i)pricea:=
cost
B
a=
revenue
S
a;
(ii)nB
a→nB
a+ 1; nS
a→nS
q−1
(2)
In Fig. 3, we illustrate the external composition of a seller role
S
and a buyer
role
B
to a simple trade interaction for a commodity
a
.
Buyer
B
has the state
(mod, n, b,
val
,
cost
)B
with
modB
being her mode
state componente,
nB
being the number of desired commodities
a
,
bB
being her
budget,
val
B
being her valuation or maximal amount she is willing to pay for
her desired commodity
a
, and
cost
being her cost she will have to pay to get an
a
. We assume the initial state as
(
wishful
,0, b0,
val
a,
'undened'
)
.
B
proceeds
in the following steps:
1. Being in mode
wishful
B
decides
@Orderp
to buy a commodity
a
for
some amount of money that will be her cost
cost
with
b0≥
cost
and
indicates her intention by documenting her transition to her new mode
ordered
for
S
with sending
S
an
Order
document with the necessary
information of commodity
a
and amount
cost
.
2. Being in mode
ordered
,
B
awaits the delivery of the notication whether
S
accepts the order or not. In case she receives a
Reject
, she transits back
to mode
wishful
. In case she receives an
Accept
she transits to mode
confirmed
, waiting for the commodity.
3. Being in mode
confirmed
,
B
awaits the delivery of the ordered commod-
ity. If it arrives,
nB
is increase by one and
B
transits to mode
delivered
.
4. Being in mode
delivered
,
B
has no alternative then to decide
@P ay
to
pay, i.e. to hand over the amount of money
cost
to
S
that was determined
in step 1 beforehand. Thus, her budget
bB
shrinks by this amount while
transiting into mode
settled
.
5. Being in mode
settled
,
B
waits for the settlement notifcation
settled
.
In case seller
S
is of the opinion that the trade had not been settled, she
transits back to
delivered
.
8
A bit simplied,
B
has two make two decisions
6
: First, she has to decide
@Orderp
to buy a commodity
a
for given costs
cost
, and secondly, she has
to decide
@P ay
to pay. The rst decision
@Orderp
is free with respect to
this interaction as is not determined by the success criterion of the interaction,
because
wishful
is an allowed nal state. Instead,
@P ay
is enforced by the
success criterion, as
delivered
is not an allowed nal state and to decide
@P ay
is the only way to proceed.
Now we turn to Seller
S
. He has the state
(mod, n, b, cost, revenue)S
with
modS
,
nS
,
bS
with an equivalent meaning as for
B
and
cost
S
being his cost to
somehow acquire, for example by production, the commodity
a
and
revenue
S
the amount of money he gets in exchange for the commodity.
S
proceeds in the
following steps:
1. Being in mode
listening
S
has produced a commodity
a
with costs
cost
and waits for orders. If an order arrives,
S
transits to mode
ordered
.
2. Being in mode
ordered
,
S
has to decide whether he wants to accept the
order or not. If he decides to reject, he transits back into mode
listening
,
indicating this by sending a
Reject
-document. If he decides to accept,
S
transits into the new mode
confirmed
, indicating by sending an
Accept
.
3. Being in mode
confirmed
,
S
has no choice then to decide to deliver
commodity
a
to
B
and to transit to
delivered
.
4. Being in mode
delivered
,
S
waits for the reception of the right amount of
money. Receiving it,
S
increases his budget
b
by this amount and transits
in its nal state
settled
. Receiving too little money, he has to do a little
bookkeeping and sends back a
NotSettled
-Notication and keeps waiting
for the rest.
To sum up the decisions of
S
: First,
S
has alternatively to decide either
@Accept
or
@Reject
and second, in case he decided
@Accept
, he has to decide
@Deliver
. Again, the rst decision is free with respect to the interaction as it
is not determined by
S
' succcess criterion, while the second decision is enforced
by the success criterion.
To put it more formally, we model the resulting trade interaction
trade
as
the result of the 'between subject', inhomogenous
7
compexternal
-operator such
that it composes a
B
- and an
S
-role to a
trade =compexternal (B, S )
: the state
of
trade
is the product state of
B
and
S
, the transition relation of
trade
is the
restriction of the product transition relation, restricted by the character and
commodity exchange and a trade is successful, if both roles are successful.
Observe that the success of the interaction rests on the truthfulness of the
trading partners:
6
Eectively,
@Orderp
is a class of mutual exclusive decisions that determine her cost.
7
We name an operator 'homogeneous' if its domain is the same as its codomain or any
codomain product (example
op :A×A→A
) and 'inhomogeneous' otherwise.
9
Figure 3: A seller and a buyer engaged in a trade interaction as described
in the text. All transitions of the buyer are lined up vertically and all
transitions of the seller are lined up horizontally. Each dot represents a
combind state of both, buyer and seller. Each transition is labelled with
(
decision
,
condition
,
incoming document
)/
outgoing document
. The external cou-
pling results in a restricted reachable state space of the product automaton,
illustrated by the 90
°
-arrows. A possible path of a complete trade is shown.
Possible nal states are indicated by an extra frame
1. The exchanged items are truly identical for both roles. That is, what we
denote for both roles with the same character like
′a′
for the commod-
ity is truly identical for both roles. The same holds for the characters
representing the amount of money.
2. Both take their enforced decisions, that is, both stick to the interac-
tion rules that
S
delivers the ordered commodity and
B
pays the correct
amount of money.
In reality, such a verication can be very complex and therefore trust is an
essential part of these kind of interactions.
We intentionally refrained from using the term 'price' in the description of
the interaction to illustrate that there is no necessity for both roles to use or
even agree on this term during a trade. They only have to stick to the rules of
the interaction to use the exchange-related characters as specied.
10
However, introducing the term 'price' makes it much easier for us to under-
stand what is going on during a trade, as it underlines the complementary role
of the exchanged money: for seller, the price is the money it receives, for buyer,
the price is the money it has to provide. So the meaning of the term 'price' is
complementarily dierent for both roles and we need it mainly to agree on a
meta-level on the design of trade-interactions.
As a nal remark in this section we want to point that already John Hicks
in [17] emphasized the stepwise model of trading.
3.2 Trader as coordination: internal composition
As said before, there is a second,
'internal'
way, to compose the roles of a single
subject, namely by
'coordination'
. We illustrate our example of a trader
T
internally coordinating her roles of
B2
and
S1
of two dierent trade interactions
1 and 2 in Fig. 4.
8
Coordinating roles means to restrict the product transition space by coordi-
nation rules. First, the state is
not
the product state of the roles, but we have
to declare which parts of the state of both roles are identical. Secondly, we have
to state which transitions become unavailable, and thirdly, which decisions may
become dependent. Coordination requires all decisions to be named uniquely.
The I/O-characters of the dierent, coordinated interactions need not.
T
has the state
((modeS, modeB), n, b, revenue, cost)T
and the coordination
condition is:
1. Coordinating function of money: there is only one state for the budget
bT=bB=bS
.
2. Identity of traded commodity: there is only one state for the possession
of the commodity
nT=nB=nS
.
3. Before
T
signals to agree selling
a
as seller
S
in interaction 1, she has to
assure to acquire
a
in interaction 2 as
B
.
In detail, the transitions of
T
are:
1. While in her mode
(
listen
,
wishful
)
she receives an
Order1
for a com-
modity
a
and price
p1
in her trade interaction 1 as
S1
which would incur
a revenue
revenue =p1
and transits to
(
ordered
,
wishful
)
.
2. Being in mode
(
ordered
,
wishful
)
she has to decide
@Order
to order
as
B2
the same commodity for a price
p2
with
p1> p2
which would incur
costs
cost =p2
and transit to
(
ordered
,
ordered
)
while sending the
respective
Order2
.
If she instead decides
@Reject
to reject the order,
T
has to transit back
to
(
listen
,
wishful
)
while sending a
Reject1
in her trade interaction 1.
8
There is actually another way to coordinate the production and consuming function of
buyer and seller cancelling the interaction part, namely as a self-supporter that consumes
what she produces by herself.
11
3. Being in mode
(
ordered
,
ordered
)T
in her role
B2
receives either a
Reject2
, requiring a backward transition to
(
ordered
,
wishful
)
, or an
Accept2
, initiating the transition to
(
ordered
,
confirmed
)
4. Being in mode
(
ordered
,
confirmed
)T
must decide
@Accept
, sends an
Accept1
and transits to
(
confirmed
,
confirmed
)
.
5. Being in mode
(
confirmed
,
confirmed
)T
awaits the reception of com-
modity
P roduct2
, which initiates her transition to
(
ordered
,
delivered
)
and increases her commodity counter state to
n+ 1
.
6. Being in mode
(
ordered
,
delivered
)
, she has to decide
@Deliver
to
deliver the received commodity as
P roduct1
, transiting to
(
delivered
,
delivered
)
and decrease its commodity counter
n
again by 1.
7. Being in mode
(
delivered
,
delivered
)
she waits for the
Money1
. If the
received money suces, i.e. it was at least price
p2
then she transits to
(
settled
,
delivered
)
, sends
Settled1
and increases her budget
b
by this
amount.
8. Being in mode
(
settled
,
delivered
)
, she has to decide
@P ay
to pay
the
Money2
to transit to
(
settled
,
settled
)
and reduce her budget
b
by this amount.
9. Being in mode
(
settled
,
settled
)
,
T
awaits the last
Settled2
-notication.
All other transitions of the unrestricted product automaton are eliminated
by this coordination.
Again, more formally we dene a within subject composition operator
compinternal
for a homogeneous composition such that we combine the two roles
B
and
S
of the same subject to a trader role
T=compinternal(B, S )
. Part of
compinternal
are two projection functions
9
projT
B,S
:
QT∪IT∪OT∪DT→
QT∪IT∪OT∪DT
with
projT
B(QT∪IT∪OT∪DT)⊆QB∪IB∪OB∪DB
and
projT
S(QT∪IT∪OT∪DT)⊆QS∪IS∪OS∪DS
.
∆T
fullls the requirements,
that the two projections of the transition relation
∆T
of
T
result in the transition
relations
∆B
and
∆S
and the success criterion of
T
is that both success criteria
of
B
and
S
can be fullled on the projected sets.
In the proposed coordination,
T
has to take 4 decisions, but only the rst of
them is still free. Namely she freely either decides
@Reject
to reject the received
order
Order1
in interaction 1 or
@Orderp2
to order the commodity herself in
interaction 2 for price
p2
. All the other three decisions,
@Accept
to accept
the order in interaction 1,
@Deliver
to deliver the commodity in interaction 1
and
@P ay
to pay the price
price2
in interaction 2, are enforded by the success
criteria of both interactions and the coordination condition.
Interestingly, although viewed in isolation, both,
B
and
S
each had a free
decision, one of these free decisions is eliminated by this special coordination,
9
for a projection function
h
it holds that sequential application does not change the map-
ping, i.e.
h=h◦h
12
namely
S
's
@Accept
. From an intuitive position, we would say that
T
has
already decided to accept the order for internal processing with her decision
@Orderp
to order the commodity herself in interaction 2 for price
p2
. But this
is invisible in interaction 1. We see that coordination changes the structure of
the decision making.
Figure 4: A trader coordinates its two roles of buyer and seller. Like in-
teraction, coordination also restricts the possible reachable state space of the
product automaton of the involved roles by transition elimination and decision-
dependencies.
Thus, we have demonstrated that what seems to be a free decision in one
interaction may just be the consequence of the subject being involved in some
other interaction: Coordinating several roles can reduce (up to complete elimi-
nation) as well as increase the number of free decisions.
Other coordinations for
T
are possible, each with dierent chances and risks.
For example, with enough budget it would be possible to order rst, not knowing
whether she will be successfully able to sell the produced comodity at a higher
price than the purchase price. Or
T
could even order rst without budget,
hoping to be able to pay for the purchased comodity exclusively by the revenue
it hopes to generate selling it. Or, in a mass case,
T
could use a mixed approach.
However, the possibilities to coordinate her interactions depends on the ex-
tend that the protocol roles allow it. Our chosen coordination depends on the
separation of the reception of the order from the sending of the accept-/reject-
13
notication. We see that spontaneous transitions of roles are central for their
exible coordination.
4 How we decide
Until now we have just introduced little more than what traditional informat-
ics provides to model interactions: I/O-automata, supplemented by a decision
concept together with the mathematical concept of composition.
Now comes the genuine economic part as we enrich our interaction/coordina-
tion model with a model of our mind on how we decide.
4.1 A simple model of mind: preferences
If we want to hold up our model of free decisions, but nevertheless want to be
able to know 'why' we act like we do, that is, we want to explain our decisions,
we have to come up with a model of our mind. A quite simple and still expressive
model of our mind, which is used extensively in contemporary economics, is the
preference concept. It ts very nicely in our interaction/coordination model.
With the preference concept, we postulate that within an interaction we
act as we want in the sense that we take a certain free decision because we
prefer some 'outcome' over some alternative 'outcome'. What is an 'outcome'?
Intuitively we might say that the decision related outcome were the target state
value of the transition the decision is involved in. We think that it is more
appropriate to relate the outcome to the interaction as a whole and identify
the outcome with the success criterion of the interaction, compatible to game
theory. That is, for nite interactions, we have a set of nal state values to
characterize success and we take a certain free decision because we prefer one
of these nal state values. In case of innite interactions, an outcome could be,
as mentioned, a set of state values, where each of them is taken innitely often.
Given two distinguishable outcomes
a
and
b
, we can then say that being
in state
p∈Q
and receiving some input character
i∈Iϵ
, with two possible
transitions
p(d1, i)/o1
−−−−−−→ q1
, nally leading to outcome
a
and
p(d2, i)/o2
−−−−−−→ q2
nally leading to outcome
b
, we prefer outcome
a
over
b
, notated as
a≻p,i b
if
we would take the decision
d1
that leads to
a
instead of
d2
leading to
b
. If we
are indierent between outcome
a
and
b
we could chose randomly and notate
a∼p,i b
. The subscript can be dropped, if it is clear from its context.
The essential idea of the money mechanism is to extend our preference rela-
tion over those commodities, that are exchangeable to a certain degree, to pairs
of such commodities and
'money'
. As our preference relation refers to part of
our state, this extension is easily achievable. We call these preferences
'material
preferences'
.
Mathematically speaking, being
M
the set of money values and
C
the set of
indexed commodities, then the preference relation
≿⊆(C×M)2
relates pairs
of numbers of commodities and amounts of money. To keep our notation simple
we refer only to a single class of commodities.
14
For example, let us assume that we prefer to have 10
e
and no cinema ticket
over having only 5
e
and such a ticket. In symbols:
(0
cinema ticket
,10e)≻
(1
cinema ticket
,5e)
. In this case we would not pay more then 5
e
for the ticket.
Why should we do such a seemingly articial extension of our preference
relation? Because we intuitively, that is without reection, grasp the usefulness
of this concept in supporting us in our challenge to coordinate our many social
interactions. Part of the money we gain in one interaction we can put on the
table in another interaction. We can now start to realize a lot more actions,
according to our original preference relation, because of this money mechanism
why shouldn't we prefer something that makes this possible?
So, we can state: some real world entity that can be exchanged and can
be represented by some state gets a pivotal role in our preference relation with
regard to all commodities that we view as exchangable to a certain degree. And
to be able to talk about it easily, we name it
'money'
. Money in this sense works,
because of its capability to support us in coordinating our diverse economically
relevant interactions and nally because we prefer it to do so as whole subjects.
4.2 Utility
As is explained in virtually every textbook on economics (for example [22]),
the key idea of the utility concept is to simplify our model of preferences even
further by introducing a single valued function
util
, mapping all outcomes to
real numbers such that we can replace our preference relation by a comparison
of these 'utility' values.
We say that a function
util
, mapping outcomes to real numbers, represents
our preference relation
≿
, if for every two outcomes
a
and
b
util
(a)≥util(b)
is
equivalent to
a≿b
. We thereby reduce a property of a binary relation between
two outcomes to a property, the utility, that can be attribute to a single outcome.
There are a number of propositions describing under which conditions it is
possible to represent preference relations with utility functions. Above all, the
preference relation necessarily has to be transitive and complete. Mathemati-
cians call such a relation a
'total order'
. In economics, such a preference relation
is often termed
'rational'
. We think from an economic perspective a more appro-
priate naming for such a total order would be
'consistent'
. Because on the one
hand, being rational in colloquial terms implies much more than having a (in
the above sense) consistent preference relation and in real life having consistent
preferences is probably rather the exception then the rule even for decisions,
most of us would describe as rational in colloquial terms.
Then, in the countable case with a xed preference relation, there always
exists a representing utility function. Also for continuous preference relations
over convex domains [6]. However, no such representation exist in the important
case of hierarchical preference relations with innite items on each level.
As is well known, a utility function over-species its corresponding preference
relation. If one such utility function
f
exists, then any concatenation
u◦f
with
a strongly monotonic function
u:R→R
is also a corresponding utility function
to the same preference relation.
15
But beside the issue of over-specication, talking about utility (where this is
allowed per assumption) is really the same as talking about preferences. Utility
maximization in our model just means to realize what we prefer most. Thus, the
popular equation of utility maximization with egoism is false in our model, as the
utility concept in our model does not determine the content of the preferences
it is supposed to express. Egoism is much better dened as having exclusively
'selsh'
preference that are directed only towards oneself [12].
To account for uncertain outcomes Johann von Neumann and Oskar Mor-
genstern [41] looked at probability distributions over outcomes, named them
'lotteries'
, and viewed these lotteries as the to-be-preferred alternatives.
So, in summary, for us utility is nothing we look for in the brains of people
but is a mathematical concept we use to simplify our considerations on human
preferences where mathematics allows us to do so.
4.3 How buyer and seller decide in a trade
We now apply the introduced concepts of preferences and utility to explain how
buyer
B
and seller
S
decide in a trade interaction according to our model.
4.3.1 Buyer
As described in section 3.1 we attributed the state
(mod, n, b, vala,
cost
a)B
to
B
with the initial values of
(
wishful
,0, b0, vala,
'undened'
)
.
B
has the free decision
@Order
to buy or not to buy her preferred commodity
a
at costs
cost
B
a
. With its initial budget
bB
0
,
B
would pay at most
valB
a< bB
0
in exchange for
a
.
We now assume
B
's preference relation to relate to pairs
(b, n)
.
B
is indif-
ferent per denition towards
(bB
0,0) ∼(bB
0−valB
a,1)
and assuming 1 as the
unit of money has the following preferences:
(bB
0,0) ∼(bB
0−valB,1) ≺(bB
0−valB+ 1,1) ≺. . . .
(3)
Thus, we can dene a utility function for the decision
@Order
, where
B
gains in a trade interaction the dierence between her valuation
valB
a
and her
cost
B
a
to acquire
a
.
Buyer:
util
B
a=valB
a−
cost
B
a
(4)
We see immediately that for any
cost
B
a≤valB
a
this function is indeed a util-
ity function for
B
, as for a given valuation
valB
a
the equivalence
util
B
a(
cost
1)>
util
B
a(
cost
2)⇔
cost
1<
cost
2⇔(1, bB
0−
cost
1)≻(1, bB
0−
cost
2)
holds.
4.3.2 Seller
Let's turn to the seller role '
S
'. According to our trade interaction,
S
has the
free decision to accept an oer for a commodity
a
for a price
p
or to reject it.
Initially,
S
has a certain amount of money
bS
0
before the production of com-
modity
a
. For production,
S
had to spend a part of his money, his
cost
S
a
. With
16
selling he wants at least to get compensated for his expenses. Hence, he is indif-
ferent towards the following possessions
(bS
0−
cost
S
a,1) ∼(bS
0,0)
and the more
money he additionally gets in the trade, the better:
(bS
total −
cost
S
a,1) ∼(bS
0,0) ≺(bS
0+ 1,0) ≺. . .
(5)
Thus, we can dene the utility of
S
, he gains from a trade interaction as the
dierence between his
revenue
S
a
and
cost
S
a
to produce
a
.
Seller:
util
S
a=
revenue
S
a−
cost
S
a
(6)
Again, it's easy to verify, that this is indeed a utility function. In economics,
there have been many other terms to name these quantities.
util
S
a
is also named
'prot'
or
'supplier rent'
, while
util
B
a
of the last section is also named
'consumer
rent'
.
4.3.3 The trade
According to our assumption of freedom of choice, both parties will only agree
on this trade if both take at least some utility out of it. Hence,
B
's valuation as
well as
S
's cost determine the possible price range in so far as
S
's cost determine
the lowest and
B
's valuation the highest possible price. Within this price range,
this trade constitutes a 'win-win'-situation for both.
The total utility for both after trading
a
at
pricea
is just the sum of the
individual utilities and does not depend on the price.
Total utility (trade):
util
a,total =
util
B
a+
util
S
a=valB
a−
cost
S
a
(7)
Note that this form of total utility depends on our assumptions of the simple
form of Eqs. 4and 6for the utility of
B
and
S
. If we had chosen a monotonically
transformed version of both, then the total utility of trade would look dierent.
For example, we could have dened
util
B
a= log(valB
a−
cost
B
a)
and
util
S
a=
log(
revenue
S
a−
cost
S
a)
. Then total utility would have been
util
a,total = log(valB
a−
cost
B
a) + log(
revenue
S
a−
cost
S
a)
= log (valB
a−
cost
B
a)(
revenue
S
a−
cost
S
a)
= log (valB
a−pricea)(pricea−
cost
S
a),
which would not be independent of the price.
With the assumed utility functions of
B
and
S
, both parties contribute to
the total utility.
B
with a high valuation and
S
with low costs. As in our model
total utility does not depend on the price, we can attribute it to the interaction
as an invariant and say: a free trade creates a utility surplus of x. The price
only distributes the utility between
B
and
S
in the trade hence, the notion of
money as a mechanism to distribute utility.
17
We give an example: A book producer and a book enthusiast have an initial
budget of 100
e
each, making a total asset of 200
e
. The book producer now
produces a book which costs him 30
e
, resulting in a remaining budget of 70
e
plus the new book. The book enthusiast would give away 90
e
of his 100
e
to
get this book. Both agree on a price of 50
e
. Subsequently, the book producer
has 120
e
and the book enthusiast has 50
e
and a book he values 90
e
, making
an asset of 140
e
. Together, after the trade, both have a total asset of 260
e
.
The asset grew by the total utility created by the trade of 20
e
for the book
producer and 40
e
for the book enthusiast. So both did benet from the trade,
though not equally.
4.4 How trader decides
Lets rst look at our trader
T
as a black box, ignoring any possible inner struc-
ture. As
T
resells any commodity she buys, keeping the number of commodities
na
nally constant, her preference relation relating outcomes actually relates
only to her budget. The more budget she gets from trading, the better.
(bT
0, n)≺(bT
0+ 1, n)≺(bT
0+ 2, n)≺. . .
(8)
It's easy to show that the dierence between her revenue
revenue
as sales
price
p1
she receives in her selling interaction 1, and her cost as purchase price
p2
she has to pay in her buying interaction 2, is a utility-representation of her
preference relation:
Utility (trader):
util
T=
revenue
−
cost
=p1−p2
(9)
Obviously, the utility of
T
depends only on the price(s):
T
will sell a com-
modity for
p1
only if she succeeds in buying it for a lower price
p2
, that is, if
p1> p2
.
Now, lets turn to the inner structure of
T
in case we composed her as
T=
compinternal(B, S )
with a coordination condition as we described in section 3.2.
Then for
B
, its successful interaction 2 changes
na
by
∆na= 1
and the budget
b
by
∆b=−p2
as long as
p2< valB
a
otherwise both remain constant. For
S
, its successful interaction 1 has
∆na=−1
and at least
∆b=p1
as long as
p2> costS
a
.
Hence, as successful trading implies successful buying and selling,
na
remains
constant and her budget will either remain constant of increase by
∆b=p1−p2
,
which corresponds to eq. 9.
We would like to make a couple of points:
1. We might at rst have thought that now, as a commodity is rst bought
and sold second,
T
creates utility twice but this is not the case. Instead,
the preferences, and therefore utility, only relates to the outcomes, that
is, the nal state values in this nite interaction. In a sense there is no
double utility because the won commodity is forseeable lost again.
18
2. We might also have thought that the relation between the preference rela-
tion
≿T
and
≿B
as well as
≿S
is that of a restriction like
≿B=≿T|projT
B(F)
.
This is not the case. As we have seen, the preference relation of
T
does not
relate to
na
at all, while that of
B
and
S
do. So, deducing the preference
relations of
B
and
S
from
T
implies an appropriate decision structure of
the single roles as well as separating the give and take of commodity
a
.
3. We achieved the perfect t of the utilities of
B
,
S
, and
T
only because
we ad hocly assumed the simple forms of the utilities and did not apply
additional strongly monotoneous transforms, possibly dierent for each
role. This will be one of the additional requirements in the next section,
where we will generalize our money model: For roles that can be combined
in one subject, any possible such transformation must be uniform for all
roles.
What happens if a trader trades with herself, that is what is the result of
compexternal(compinternal(B , S))
? In this case nothing happens, as it is not
possible to sell a commodity to a higher price than to buy it. This can be
generalized to all closed chains of traders:
Theorem 1.
A closed chain of traders does not perform any trade and therefore
creates no utility.
Such a closed chain is a simple example where free decisions can be enforced
or prevented by external circumstances, whose existence cannot be deduced from
the local interaction itself. It means that any working trade (chain) assumes
that in the end there are at least some real producers and consumers involved.
More general it means that much of contemporary economic theory depends
heavily on the assumption of true independence of its economic subjects.
5 Money as a utility transfer mechanism
In the last two sections we have seen that our assumption of two simple, namely
linear, utility function denitions for a buyer
B
and a seller
S
in a trade inter-
action together with the coupling condition of a price, leads to a simple form
of total utility, which was independent of the price. This function of money to
work as a utility transfer mechanism is well known in economics [15,24].
In fact, for our aim to show that subjective preferences become comparable
with the money mechanism, it suces to have at least one utility representation
with our desired properties. In other words, We can use the freedom of the utility
concept not to insist on any particular form, but to choose exactly that form
with which we achieve our aim to make the interpersonal comparison of utility
meaningful and which thereby allows us to compare preferences interpersonally.
That said, we now show that, with respect to our specic roles
S
and
B
and their external and internal couplings of trade and trader, that the utility
representations we introduced for both are the only ones that are compositional
consistent with respect to our external and internal composition of both roles.
19
First, we dene:
Denition 2.
We name the utility functions, representing the material prefer-
ences of a buyer
B
and a seller
S
in a trade, 'compositional consistent' if they
fulll the following constraints:
Trade-interaction constraints:
(a) The utility of
B
depends only on the dierence between her valuation
and her costs: util
B
a=f(
val
S
a−costB
a)
, and becomes zero if her val-
uation equals her costs,
f(0) = 0
, and becomes her valuation if the
costs are zero,
f(
val
S
a) =
val
S
a
.
(b) The utility of
S
depends only on the dierence between his revenue
and his cost: util
S
a=g(revenueS
a−costS
a)
, and becomes zero if his
revenue equals his cost,
g(0) = 0
, and becomes his revenue if his costs
are zero,
g(revenueS
a) = revenueS
a
.
(c) The total utility resulting from the trade as the sum of the utility of
S
and
B
is the dierence between
B
's valuation and
S
's cost and
is therefore independent of the price: util
a,total =utilB
a+utilS
a=
valB
a−costS
a
.
Trader constraint: Any strongly monotonic transform of the utility func-
tions of
B
and
S
must be identical.
The claim is now,
Theorem 2.
The utility functions of a seller
S
and a buyer
B
in a trade
interaction according to Def. 2, namely util
S
a
and util
B
a
as given by
(4)
and
(6)
are the only compositional consistent utility functions.
To prove this theorem, we rst have to provide the general utility functions
for
S
and
B
in a trade interaction. As we have seen,
valB
a−costB
a
is a possible
utility function of Buyer and
revenueS
a−costS
a
is a possible utility function
of Seller. Then, the general utility functions for
B
and
S
, which depend only
on the required dierence, are provided by two additional monotonic functions
f, g :R→R
such that
utilB
a=f(
val
B
a−
cost
B
a)
utilS
a=g(
revenue
S
a−
cost
S
a)
Now, the trader constraint requires both function to be identical:
f=g
.
And the external coupling condition (Eq. 2) makes the rst constraint to
utila,total =utilB
a+utilS
a
=f(valB
a−pricea) + f(pricea−costS
a)
With the next lemma, we prove that this requires
f
to be linear with zero
intercept.
20
Lemma 1.
Let
f
be a real monotonic function such that whenever
a, b, c ∈R
with
a≥b≥c
then
f(a−b) + f(b−c)
does not depend on
b
. Then
f
is a linear
function.
We give the proof of the lemma in Appendix A
Thus,
f
is a linear function. To complete the proof of our theorem, we note
that together with our boundary conditions
f(0) = 0
, and
f(valB
a) = valB
a
,
f
has to be the identity function.
Considering money as a commodity like any other, Léon Walras discovered
the linear inuence of money on our utility representation of a trade and coined
the term
'numeraire'
[42] for the money term. Economists traditionally say that
our preferences are
'quasilinear'
with respect to the commodity money, which
just means that they are linear in the money term and arbitrary in any other
term.
Two consequences from the linear utility representation of money are well
known (see e.g. [22, p. 45]):
1. if someone is indierent with respect to two bundles
x, y ∈Rn
,
x∼y
, then
any additional amount of money (component 1) does not change that:
(x+αe1)∼(y+αe1),∀α∈R, e1= (1,0, ..., 0)
(10)
2. Money (component 1) is desirable in the sense that everything else con-
stant, we prefer more money over less money:
(x+αe1)≻(x),∀α > 0
(11)
This is what our intuition about money is about. If money works as intended,
then it should not inuence our preference relation of material goods and it
should always be better to have a bit more than a bit less money.
However, allthough in the presented model the money mechanism can trans-
fer utility in the dened sense, it does not represent or measure utility in the
sense that the more money we have, the more utility we got. So, the statement
Money is transferable utility is false in this model. True is that money is a
mechanism of utility transfer where individual utility manifests itself on both
sides of a trade only as a dierence between some sort of revenue/valuation and
some sort of cost.
Also the sentence, The marginal utility of money decreases with increasing
amount of money is false in this model. First, we do not attribute utility to
money. And second, within this model utility is a strict linear function of any
money term.
But true is, that our valuation per item of all commodities we buy for con-
sumption, like apples, dramatically decreases with increasing numbers, that is,
the valuations for consumption commodities are not additive. While we may be
willing to pay 1
e
for a single apple we want to eat now or in a few days, we will
not pay 1000
e
for 1000 apples in a single moment, as we would perhaps even
have to pay additional money to dispose most of them afterwards somehow.
21
6 Intersubjective comparison of preferences
Based on the role-based nature of a trade, we can compare the preferences
of dierent trading subjects in several dierent ways. We can compare the
valuations of competing roles and the utility of complementary roles. And,
nally, we can compare the total utilities generated in dierent trades.
6.1 Intersubjective comparison of preferences between po-
tential buyers
How can we compare the subjective preferences of potential buyers, for example,
who prefers a given commodity most? Posed in this abstract manner, this
question actually does not aim at the utility competing buyers gain in some
interaction like a trade. As it ignores any cost (or, more precisely, assumes a x
cost for all potential buyers) it aims only at their valuation.
If two buyers have the same budget and access to the same alternatives, then
the willingness of one buyer
A
to spend a maximum of
valA
a
on a commodity
a
and another buyer
B
to spend a maximum of
valB
a
allows us to draw a direct
conclusion about their preferences. Then for
A(0a, vala)A∼(1a,0e)A
holds
and for
B(0a, vala)B∼(1a,0e)B
holds. We can thus dene a relation '
A
prefers
a
more than
B
', symbolically
A≻aB
, if and only if
valA
a> valB
a
We therefore name the condition under which the valuations of two buyers
become comparable, namely to have the same budget as well as access to the
same alternatives, the
'comparable valuation condition'
.
We would like to note rst, that already [39] showed that under compara-
ble valuation conditions, any allocation resulting from free trade in our sense
becomes ecient and envy-free (see also [23] for a recent discussion).
And secondly, we note that subjective valuations are indeed measurable with
price-determination mechanisms like Vickrey second-price auctions [40].
6.2 Intersubjective comparison of utility between comple-
mentary roles of
B
and
S
Our assumption to add a buyer's and a seller's utility to a total utility in a trade
implies the comparability of the utility of both roles. It allows us to dene a
'fair' price as the price which provides both the same utility in their trade.
It is immediately apparent, that if a price can be fair, it can also be arbi-
trarily unfair. Although a free trade is, per denition, a win-win-situation, it
nevertheless can be arbitrarily unfair in the sense that the advantageous party
somehow sucks almost all utility from the wronged party. We thereby get a
glimpse of the power of money as a mechanism to distribute utility: we might
be able to design it such that it distributes utility by and large fairly or we
might design it willingly or unwillingly to become a mechanism to distribute
utility systematically in an extremely unfair manner.
However, fairness is a rather complex concept and, to quote Hervé Moulin
[23],
There is no single compelling denition [of fairness]
, which we would like
22
to underline by what we distinguish as 'local' versus 'global' fairness in trades.
With local fairness, we refer to a single trade. With global fairness, we refer to
many trades.
6.2.1 Fairness in 1:n relations
What happens if a seller sells his commodities not to a single but to
n
buyers?
The rst thing we have to realize is, that reasoning over this case, we aggregate
utility of dierent subjects. The validity of our reasoning therefore depends on
the validity of our assumptions, that everyone has the same total budget and
has the same access to all relevant alternatives. In this 1:n case, the order of
utility partition and price determination plays a decisive role for fairness.
The rst possibility is to average utility rst and then to distribute it evenly
by deriving the individual prices thereof. Let us look again at the book pro-
ducer who now produces 2 books at 30
e
. There are two book enthusiasts, one
with a valuation of 90
e
, the other, a bit less enthusiastic, with a valuation of
60
e
. The average total utility then is total valuation minus total cost divided
by the number of participants (90
e
+ 60
e
- 30
e
- 30
e
)/3 = 30
e
. If the rst
book is sold for 60
e
and the second book for 30
e
, every participant gets o indi-
vidually with the average utility of 30
e
. In summary, we proceeded in two steps:
Step 1 [Determine average utility]:
¯u=1
n+1 Pi(vali−
cost
i)
=n
n+1 val −cost
Step 2 [Determine
n
prices]:
pi=vali−¯u
Thus, for increasing
n
, the average utility everyone realizes, converges against
the dierence between the average valuation and the average cost.
The second possibility is to determine the local 'fair' price in each trade
directly and afterwards look at the utilities of every participant. As
S
takes
part in every trade interaction, he now gets his share
n
-times. The rst price
would determine to 60
e
and the second price to 45
e
. The utility of the rst
buyer would be 30
e
, of the second buyer 15
e
and for our seller it would be 45
e
.
Step 1 [Determine
n
prices]
pi= (vali−
cost
i)/2
Step 2a [average utility of the
n
buyers]
¯uB=1
nPi(vali−pi)
=val −¯p
Step 2b [utility of seller]
uS=Pi(pi−
cost
i)
=PiuS
i=n¯uS
The average utility for all participants is identical in both cases, but the
distributions are dierent. Now all buyers get on average the dierence between
their average valuation and the average price, and seller accumulates all his
utilities from each trade.
So, in the rst case all participants benet equally, in the second case, the
23
seller's total utility increases with the number of buyers (or sold items) and
thereby shows the same scaling behaviour as with a constant price for all trades.
This distinction becomes of great relevance at a company employing
n
em-
ployees.
6.3 Aggregation of utilities of dierent trades: markets
With the fulllment of the comparable valuation condition we can directly com-
pare all buyers' valuations. Together with the objectively determinable costs of
all sellers and our requirement, that the total utility of a single trade is inde-
pendent of the price, this makes a market mechanism maximize total utility of
all possible trades.
To prove that, we just have to order all sellers according to their increasing
cost and all buyers according to their decreasing valuation, and, as Fig. 5
shows, the intersection of both curves determines the market price
p∗
together
with the index
jmax
as this price creates the largest enclosed area and thereby
maximizes total utility.
Figure 5: Supply and demand of a market mechanism.
We call a market mechanism that unfolds its eects in a social context where
the comparable valuation condition is being fullled, and which therefore does
create a price that is most preferable in the sense of all, a
'valid'
market mech-
anism.
With the comparable valuation condition being violated, we still can techni-
cally add up all dierences between the individual valuations and costs to some
value and the market mechanism will still generate some price that maximizes
this value, but this price will not relate in any simple sense to the preferences
of the participants. So we are not allowed to speak about this value as a 'max-
imized total utility'.
It could be, that the violation of the comparable valuation condition is one
of the most common reasons for severe market failure nowadays.
24
The social impact of large budget dierences of market participants might
be mitigated if the exchangebility is almost perfect and where quite supercial
features segementize the market in dierent luxury levels, like for watches or
pictures/paintings which are sold in a price range from single Euros to several
(hundred) millions. But it could become devastating if hierarchical needs are
touched, such as with food, health, education or housing and these essential
ressources become scarce. Here, for a society, a pragmatic means to keep prices
in a sensible range for basic commodities that fulll hierarchical needs could be
to deliberately incentivize an oversupply.
7 Extension to trading labor
7.1 The discretionary component of the employee's deci-
sion
How do we trade our labor? In a free world, we should ad. Like sugar, as
Karl Marx has suggested [20, p. 399]? Then, as a seller of its labor, that is as
an employee, the salary would be the subject's revenue, its cost would be its
expenses for reproduction, and the dierence would be its utility.
Does this makes sense? Compared to selling an apple, which takes place in
a virtual second, our labor time is an essential part of our life and thus it is to
be expected, that in a free world, we would like to express our preference for
some work over others.
Thus, for trading labor in a free world, we have to assume also a discretionary
component on the employee's side, symmetric to that of the buyer, that is the
employer, in the original model. Then the cost term of the employee is no longer
determined exclusively by objective facts, but represents the
'internal valuation'
of the work to be done in the sense of some minimal acceptable amount of money
for which the employee would still opt for doing the job. The more the employee
likes the job, the lower the minimum salary the employer will at least have to
pay for it.
Actually, this model entails some interesting consequences. As for selling
ordinary goods, the total utility can be inuenced by both participants. The
employer by creating a productive environment where it makes sense to pay an
employee a lot of money that sounds ok and the employee by preferring this
work so much, that he is willing to do the job for almost nothing, even negative
amounts are conceivable that sounds strange. At rst glance, it may seem
that this inverse relation enforce an indemnication semantics on every salary
where high salaries can only be achieved if they compensate for all the scourges
of the job.
But in reality, bad jobs are usually badly paid. And on a second glance we
see that the valuation of the employee does only represent the lower boundary
of the possible salary range and not the salary itself. The actual salary will
depend to a large extend on the price determination mechanism.
25
7.2 The eects of additional hierarchical preferences for
fairness of the employee
Beside our preference for some job over some other job, what else could de-
termine the price for work? Here is one important example: what happens, if
employees have an additional hierarchically superordinated preference for fair-
ness? It renders the preference relations recursive
10
.
Let us assume that an employer has to oer a job that she valuates at
130.000
e
. The employee with a superordinated preference for local fairness
has an initial internal valuation of 10.000
e
. Knowing little about each other,
employer and employee agree on an initial salary of 20.000
e
and the employee
starts working.
After a while, the employee discovers, that the employer's valuation is indeed
130.000
e
and therefore a fair salary would have (initially) amounted to 70.000
e
.
What sum would this employee have to claim to remain in the company? If he
claims 70.000
e
as the minimal salary to stay in the company, he in fact relates
to his own, older valuation of 10.000
e
which has just changed to 70.000
e
. So
assuming 70.000
e
as minimal acceptable salary, a fair salary would amount to
100.000
e
, and so on and so forth. The preference relation becomes recursive
and the xpoint is the valuation of the employer, in this case, 130.000
e
.
As a result, the internal valuation of the employee ts exactly the external
valuation of the company. The utility of both, employer and employee shrink to
zero. But this just means that both the employee and the employer achieve what
they prefer most: for the employee that he is treated fairly and for the employer
that she employs its employees to reach her business goals the usual case for
organizations that are not owned by someone, like associations, foundations,
etc.Another recursive relation between the preferences of employer and employee
might be created by a dependency of the external valuation of the employer on
the internal valuation of the employee. The employer might expect that an
employee who highly valuates doing a task will on average perform this task
better than another one with lower internal valuation. In this case it would make
sense for the employer if she payed employees with higher internal valuation
more than employees with lower internal valuation - although the latter would
possibly start working only for higher salaries.
One consequence is that labor markets don't work the same way as markets
for ordinary goods, which is already common sense in behavioural economics
[8].
10
According to our knowledge, recursive preferences were introduced to economics by David
M. Kreps and Evan L. Porteus [19] and Larry Selden [33] to distinguish the separate roles of
time and risk. See also the work of Larry G. Epstein and Stanley E. Zin [10] and Philippe
Weil [43].
26
7.3 Employees as part of an organization
Up to this point, our model to trade labor views the employee and the employer
still as two otherwise non-related but interacting systems.
However, it might be more appropriate to view an organization as a system
that comes into existence because of all of its interacting employees. Then, the
employees become parts of the organization and the organization becomes the
supersystem. Now, from a logical point of view, a supersystem cannot interact
in any form with its parts, the employees [28] like a family does not interact
with the mother or father or child. In this latter model, the employees do not
sell their labor any longer, but cooperate within the organization and get their
money not in direct exchange for their work but the money distribution within
the organization becomes part of its internal coordination mechanism. Why
should such an orgnization not distribute all utility it gained to its employees?
And it would be natural to determine the valuation of the jobs collectively.
8 Other approaches
8.1 Pareto eciency: avoiding interpersonal utility com-
parision
Vilfredo F. Pareto (1848-1923) was the rst to make the distinction between
'ordinal'
and
'cardinal'
utility and introduced the idea to handle the analysis
of economic equilibria with ordinal utility [2]. An ordinal interpretation of the
utility-numbers in this sense allows only to use their ranking while a cardi-
nal interpretation also allows to use their distance for calculation purposes like
equilibrium determination.
Honouring his achievements, a feasible allocation of resources is nowadays
said to be
'Pareto-ecient'
, if no other such allocation exists, that makes some
consumer better o without making some other consumer worse o [22, p. 313].
In this sense, a Pareto-ecient state is a state that cannot be changed further
on a strict voluntary base, as any change must be preferred by at least someone
and must not be objected by at least someone else.
Obviously, for determining Pareto-eciency intersubjective comparison of
preferences is unnecessary. It suces if all the participants do or don't prefer
to transit from a given into some target state.
However, for Pareto-eciency to make any sense from a societal point of
view, it is essential that already the given state was reached according to this
same criterion, that is on a completely voluntary base. Otherwise Pareto-
eciency degenerates into a justication of the status quo.
But this is almost never the case, as it is well known from both, a theoretical
as well as an empirical perspective, that Pareto-eciency is a bad model for
human decision making which can easily be demonstrated with the example of
the Prisoners' Dilemma [24, p. 98].
Why should we not take away the money a thieve has stolen? Why shouldn't
we redistribute money distributed by mechanisms realizing any fairness model
27
only to a very limiting extend, perhaps even invaliding our desired money se-
mantics to make our material preferences intersubjectively comparable? Why
should we not be able to correct the eects of erroneous preferences, even against
the beneciaries of the errors?
In line with Amartya Sen [36], we think that the wide acceptance of the
Pareto criterium in economics as a model for
'rational behavior'
was a direct
consequence of the denial of the possibility to interpersonally compare prefer-
ences. Thereby, the denial of the possibility to interpersonally compare utility
made economics a doctrine to justify the economic status quo. In its nal con-
sequence, it even removed the issue of social justice from the economic agenda,
culminating in Friedrich Hayek's strange view of social justice being
'strictly
empty and meaningless'
[16, p. 68].
It is interesting to note that from a purely logical point of view, the con-
clusion from the impossibility to intersubjectively compare entirely subjective
preferences to the non-existence of social justice as a sensible concept would
entail the conclusion from the existence of social justice as a sensible concept
to the possibility to intersubjectively compare entirely subjective preference, as
logically for two propositions
A
and
B
,
A→B⇔ ¬B→ ¬A
holds.
Thus, accepting Friedrich Hayek's conclusion and also accepting the obvi-
ous relevance of social justice as a concept, we therefore must conclude that
interpersonal comparison of subjective preference is (somehow) possible.
8.2 Treating money as a commodity
Traditionally, motivated by handling money as any other commodity (see for
example [22,26]), the utility representation of a buyer's preferences is modelled
a bit dierently from our approach.
First, it is necessary to set a 'price' for money, which is usually chosen ad
hoc as 1. Then,
B
's utility is not given as the utility exclusively arising from
the trade, but before the trade as its budget
bB
total
and after the trade as its
reduced budget
bB
total
′=bB
total −
cost
B
a
, supplemented by her valuation
valB
a
:
Buyer:
util
′B
a=bB
total
′+valB
a.
(12)
This is indeed a utility representation of
B
, as again the equivalence
util′B
a(cost1)>
util′B
a(cost2)⇔cost1< cost2⇔(1, bB
total −cost1)≻(1, bB
total −cost2)
holds. We
could have concluded this immediately by recognizing that this form of utiltiy
is the same as eq. 4increased by the initial total budget:
util′B
a=
util
B
a+bB
total
.
Lets see what the consequences are for the external and internal coupling of
B
and
S
.
For the external coupling of
B
and
S
in a trade, we get the total utility as
before, but with an additional dependency on
B
's initial budget:
util′
a,total =util′B
a+
util
S
a
=bB
total +valB
a−costS
a
(13)
28
However, with this utility representation, the internal composition of
B
and
S
to a trader does not work, as the trader constraint allowing only identical
utility transforms for
B
and
S
is violated.
Thus, a model that treats money as an ordinary commodity, except its (quite
extraordinary) linear utility representation, where utility is somehow attributed
to all commodities and therefore also to money, is not compatible with a holistic
view on economic subjects. Such a holistic view, which takes into account all of
the subject's interactions in its roles and the coordination of its roles, prohibits
a utility scale in an absolute sense.
As we have just proven that our utility representation for compositional
consistent utility is uniquely dened by denition 2, this is actually no surprise.
A trade is not a symmetric, but an asymmetric situation with respect to the
exchanged items. The symmetry breaks down because it's the money we use to
coordinate our dierent interactions and not the apples.
8.3 Welfare economics
Welfare economics is an area of economics that suered particular from the
avoidance to intersubjectively compare material preferences. It is simply not
possible to do welfare economics without weighing the welfare of dierent sub-
jects against each other. The only question is, what kind of individual welfare
concept to use?
In the tradition of utilitarism, some economists model the individual good
with a unidimensional utility-value [15,34], where every subject
i
draws a well
dened, real-valued 'utility'
Ui(x)
from each state
x
of the world (or as a von
Neumann-Morgenstern lottery). However, as Marc Fleurbaey and Peter J. Ham-
mond [13] indicate, the interpretation of this value is quite heterogeneous. It
could be the extend of pleasure or pain in the tradition of Jeremy Bentham,
happiness, desire satisfaction [4], or, following most economists, preference sat-
isfaction.
We will not go into a detailed critique of these approaches because we think
that there is a fundamental misconception right at the beginning: in our model
total material wealth is given by the sum of all our valuations (+all budgets)
and not by some trade-realizable utility. But it makes sense to add up our
valuations only if we all have the same total budget and the same access to the
same alternatives with respect to ourselves.
The example of the trade between our book producer and our book enthusi-
ast has shown, that a trade between economic subjects increases total wealth by
shifting goods from subjects with lower to subjects with higher valuation and
utility measures (in our model) the extend of the wealth gain and shift between
the trading parties.
But trade is only one out of many mechanisms that inuence our wealth.
As said before, consumption destroys valuation quite naturally. We do not
materially valuate an apple that we have eaten any longer, it's gone. We can
also valuate something we have created by ourselves in the sense that we come
29
to the conviction that we will not sell it below some minimum price for whatever
reason. And, there is also bequeath and donation, etc.
And our valuation for one commodity can uctuate enormously, depending
on changing circumstances. Perhaps our book enthusiast, who was just willing
to pay 90
e
for the book, reads the rst few pages and changes her mind, now
thinking:
Oh what a rubbish! Never ever again should I spend any cent on this
kind of trash!
With this re-valuation she has cut her wealth in one swoop by
90
e
as she now possess only 50
e
together with a book where she would not
even pay a single cent for, which even may create costs for disposal. Phillip
Nelson [25] introduced the term
'experience good'
for goods whose value can
be accurately estimated only after they have been experienced. Or, we have
invested hundreds of Billions
e
in highly valued military equipment because
we thought we were encircled by foes, but one day we realize that all of them
became friends. As a result our material wealth diminishes dramatically, despite
our much better overall situation. Or, for some commodity we just know that
everybody else only pays 2
e
why should we pay much more?
In summary, the material wealth of a society, as the sum of all its valua-
tions, is quite dynamic and is governed by sources and sinks. It essentially is
a ow quantity and thereby depends on the relation between its creation and
annihilation and the stability of its measures. Our presumption is, that depend-
ing on these factors, every society has a certain, characteristic maximal wealth
accumulation capacity. A throwaway society can produce very much without
accumulating valuable goods, whereas a stable society which heavily invests in
long-valuable goods can accumulate a lot of material wealth even with much
less production capabilities.
But, very important, because of the hierarchical nature of our preferences
and also because there are a lot of things we prefer that we cannot buy directly,
like health, education, freedom, democracy, peace, etc. the sum of all valuations
for our material goods represent societal wealth in a holistic sense only to a
rather limited extend. Actually this is well known, and as a result, there is a
thorough discussion about further indicators to more validly estimate societal
wealth in a more holistic sense [38].
9 Discussion
Yes, the money mechanism can achieve the miracle of sensibly relating totally
subjective material preferences of dierent individuals. But, only if the valued
items are somehow nonhierarchically interchangable and if everybody competing
on these items has to have the same total budget and the same access to all
relevant alternatives.
Our key thesis is that we can understand the money mechanism only if we
take both perspectives, the interaction as well as the coordination perspective
of our economic subjects into account. Money gets its special role only because
it becomes part of our preference relations in
all
our economic relevant exchange
interactions that we have to coordinate.
30
As money can but not necessarily does make our material preferences com-
parable, depending on the societal context, one of the remaining key question
is: is it desireable? Should we as a society work in the direction to establish cir-
cumstances that enable money to fulll this function at least approximately or
not? We, the authors, think clearly yes. Otherwise the societal consequences of
ecient price determination mechanisms like markets seem to be quite unclear.
Since the conviction that one cannot compare subjective preferences inter-
personally had signicantly shaped economic theory in the past, we assume that
the now proven fact that money allows exactly such a comparison under certain
conditions will shape economic theory and practice in the future as well.
Another key question adressed by our model is: Do truly free decisions really
exist?
One corner stone of our model of an economically acting subject was the
assumption of free decisions, governed by preferences. But, our model also
shows that what appears to be a free decision in one interaction may actually be
enforced by within-subject coordination with another interaction or by between-
subject circular interactions.
A good indication for truly free decisions is the obvious evidence that our
physiological apparatus to maintain our autonomy is itself plastic and can be
severly damaged by addictive substances. Anybody who had to deal with peo-
ple suering from addiction knows intuitively the dierence between free and
enforced decision.
In our opinion the best indication for truly free decisions in our sense is
indeed our social robustness in the sense of an extreme exibility in adapting to
variations of our social interactions, taking over new roles, abandoning old ones
or modify existing roles virtually on the y. If some of our interactions do start to
determine our actions, the rst thing we loose is our social coordination ability.
Poverty is a good example, as Karl Marx accurately pointed out, writing:
The
realm of freedom begins, in fact, only when the work determined by necessity and
external expediency comes to an end.
[21] (p. 828).
It is this exibility that allows money to become a coordinating mean. Thus,
according to our money model, the coordinating function of money is in itself
a good indication for our capability for truly free decisions. Actually, any rel-
evant deviation from the two consequences of the linear utility representation
of money, that money does not inuence our preference relation over material
goods and we always prefer a bit more over a bit less money, indicates a devi-
ation from our money model and a possible tainting of our preferences which
in turn indicates an impairment of our physiological apparatus to maintain our
decision freedom.
In this respect, money can support our social coordination ability very much
and thereby increase our autonomy: we can use what we earn in one role in
many other roles. But, by the very same function, money can also be a mean
to destroy our freedom of decision and drive us into desolate slavery.
Additionally, despite the fact that a single free trade is always a win-win
situation, with its unlimited utility transfer functionality, we can distribute total
utility with money arbitrarily unfair not only on a local, but on a gigantic global
31
scale.
For us, because of the repercussions of money on our ability to make free
decisions, a theory of money cannot be ethical neutral, but always contains
a strong normative component. That is, we could design the societal money
handling mechanisms in a way that they best support our economic freedom
or, alternatively, in a way that they drive most of us into misery and abyss.
And we, the authors, prefer the former over the latter and therefore view it as
essential to understand which handling of money fosters the humanly preferable
over the rather unpreferable.
Acknowledgement
We would like to thank Marvin Deversi and Leoni
Bossemeyer who gave very valuable feedback. Also Martin Kocher and Chris-
tiane Schwieren for their encouragement to put these ideas on paper. And
nally, it was Christine Reich, who had the idea to assign the term 'valuation'
also to the cost-term of the employee with respect to its work and thereby em-
phasizing the symmetric relation between the external valuation of the work by
the employer and its inner valuation by the employee.
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A Proof of Lemma 1
Here, we supplement the proof of our main technical lemma which we skipped
in the main text body.
Lemma 1.
Let
f
be a real monotonic function such that whenever
a, b, c ∈R
with
a≥b≥c
then
f(a−b) + f(b−c)
does not depend on
b
. Then
f
is a linear
function.
Proof.
If we chose
b= (a+c)/2
we see that
f(a−b) + f(b−c) = 2f((a−c)/2) .
(14)
This value even depends only on
a−c
and not on
a
and
c
individually. As
f(a−b) + f(b−c)
is supposed not to depend on
b
at all, equation (14) holds
for every
b
.
Substituting
x=a−b
and
y=b−c
we can rewrite this as
f(x) + f(y)
2=fx+y
2
for all
x, y ≥0.
(15)
That is, the mean of the function values of
x
and
y
equals the function value
of the mean of
x
and
y
. This is certainly fullled by every linear function, but
conversely it is also true that every monotonic function fullling this condition
is linear.
We rst show the linearity for rational coecients and in a second step
extend this result to general real numbers. W.l.o.g. let
f(0) = 0
(The general
case with
f(0) = 0
can be reduced to this one by taking
g(x) = f(x)−f(0)
and
looking at
g
instead of
f
.)
f(nz) = nf (z)
for all
z≥0
and
n= 0,1,2, . . .
follows by induction: The
cases
n= 0
and
n= 1
are trivial, by our assumtion
f(0) = 0
. For the inductive
step, let
n≥1
and set
x= 2nz
and
y= 2z
in equation (15). Then the
right hand side simply is
f2nz+2z
2=f((n+ 1)z)
, whereas the left hand side
becomes
f(2nz)+f(2z)
2=2nf(z)+2f(z)
2=nf(z) + f(z)=(n+ 1)f(z)
(using the
inductive hypothesis).
By substituting
z
with
z
n
in
f(nz) = nf (z)
, we get
fz
n=f(z)
n
for all
z≥0
and
n= 1,2, . . .
. Combining this with the above we obtain
f(qz) = qf (z)
for
any rational coecient
q=m
n
(with
m, n = 1,2,3, . . .
). In particular, we have
f(q) = qf (1)
for any rational
q
.
To extend the result to general, real coecients
r
we use the monotonicity
of
f
: Namely we have
f(r)≥f(q) = qf (1)
for any rational
q≤r
which shows
that
f(r)
cannot be smaller than
rf (1)
and, likewise, we nd that
f(r)
cannot
be larger than
rf (1)
either, hence
f(r) = rf (1)
.
36
