# Money as a mechanism to intersubjectively compare entirely subjective preferences

Johannes Reich, Martin Härterich

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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 understand under which conditions money allows to interpersonally compare the entirely subjective material preferences of different economic subjects.

**Methodology:** We proceed in several steps. First, we provide a graph-based model for the infinitely scaling network of economic relevant interactions, which resolves the issue that the nodes (economic subjects) and the edges (economic interactions) of this graph cannot be modelled independently. 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 different 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 defined 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 subjective 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 significantly 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 Introduction

Can we meaningfully compare preferences between economic subjects, despite the fact that they are entirely subjective? This question is still fiercely debated in modern economics. The contribution of this article is to show that money can – under well defined 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 ‘unscientific’. 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 ‘scientific’] could only be a ‘mere juxtaposition’ [32, p.132]. Kenneth Arrow [1] still said that ‘interpersonal comparison of utilities has no meaning.’<fnref n="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 different ways of making interpersonal comparisons and their far-reaching consequences on what is permissible in welfare economics.’, culminating in his book ‘Collective Choice and Social Welfare’ in 1970 [34]. But even Amartya Sen stated in 1999 [35, p.68] “If different persons have different preferences (reflected in, say, different 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 defined circumstances, could even be ‘transferred’ 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 utility. 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 interpersonal comparison of preferences, we go through a certain program, which we sketch in the following. The first 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 different interactions.

In other words, we have to find a way to adequately model the infinitely scaling network of economic relevant interactions, a section of which we show

<footnote n="1">¹Economically utility is understood as one form to represent preferences, we come to this issue later.</footnote>
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 different, 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 fill 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 ubiquitous 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 fulfill 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 different subjects, it suffices 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 8 we discuss other approaches from the literature: the concept
of Pareto-efficiency to avoid intersubjective utility comparison alltogether, the
treatment of money as a commodity, and how welfare economics misinterpreted
the utility concept.

In section 7 we 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.
# 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 infinitely scaling network of economically 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><img src="image_2.png" /><figcaption>Figure 1: A section of the infinitely scaling network of economic relevant interactions.</figcaption></figure>

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 modelled 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'<fnref n="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 different ways we name 'externally' and 'internally' [30], as we illustrate in Fig. 2.<fnref n="3" />

Roles of different 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].

<footnote n="2">²This role concept was actually very much inspired by a sociology seminar on Erving Goffman [14] one of the authors took part, which was sympathetically led by Prof. Uta Gerhards at the University of Gießen in the 1990s</footnote>

<footnote n="3">³It 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]</footnote>
We will describe this external composition of roles of different subjects in
detail in section 3.1 for a buyer and a seller role in a trade.

<figure><img src="image_3.png" /><figcaption>Figure 2: The two different ways roles can compose correspond to two different perspectives. They either compose externally, creating an interaction (part A) or internally, creating a coordinating subject (part B).</figcaption></figure>

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 effort 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)<fnref n="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 *q*₀, a
transition relation Δ ⊆ *I*⁶ × *O*⁶ × *Q* × *Q*, where for any alphabet *A*, *A*⁶ = *A* ∪ {*ε*}
holds with *ε* symbolising the empty character. In case of finite interactions, the
success criterion is the set of final states (or *'outcomes'*). In case of infinite inter-
actions, there are different choices for the success criterion [11], one possibility is
the set of all set of states that are occupied infinitely often (Muller acceptance).
Then an outcome would be a set of states.

Instead of $(i,o,p,q) \in \Delta$, we also write $p \xrightarrow{i/o} q$. 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

<footnote n="4">⁴In the informatics literature, such an I/O-A is also called 'transducer', or, in the deterministic case, 'Mealy'-automaton.</footnote>
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 nondeterministic, 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 different 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 deterministic, the decision automaton determines a transition function<fnref n="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 different subjects, i.e. a protocol with explicit decisions, a 'game in interactive form (GIF)' [29], emphasizing the tight relation between the informational protocol and the economic game concept.

## 2.4 Equivalence class construction

In the following we want to describe more complex roles efficiently. 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. $parse_{IO}: I \cup O \rightarrow DocCls \times Param_{IO}$ assigns each character $a \in I \cup O$ a document class $docCls(a) \in DocCls$ and a value of a parameter set $param_{IO}(a) \in Param_{IO}$. We can thereby regard each I/O-character of a decision system as an element of some document class that represents certain parameter values.

<footnote n="5">⁵In the sense of [29], the decision concept becomes a mean to identify an interacting human as a system by a fictitiously assumed system function.</footnote>
2. parse<sub>Q</sub>: Q → Q<sub>mode</sub> × Q<sub>rest</sub> assigns each state value q ∈ Q a mode component mode(q) ∈ Q<sub>mode</sub> and a rest component rest(q) ∈ Q<sub>rest</sub>.

3. parse<sub>D</sub>: D → DCls × Param<sub>D</sub> assigns each decision value d ∈ D a decision class component dCls(d) ∈ DCls and a value of a parameter set param<sub>D</sub>(d) ∈ Param<sub>D</sub>. We can thereby regard each decision as an element of a given decision class that represents certain parameter values.

And we introduce a set of conditions cond as functions cond<sub>dCls(d),mode(p),docCls(i)</sub> : Param<sub>D</sub> × Q<sub>rest</sub> × Param<sub>IO</sub> → {true, false}, 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 define the desired equivalence relation:

**Definition 1.** Two transitions t, t' ∈ Δ are equivalent, symbolically t ~<sub>Δ</sub> t', if their values for the decision and document classes, the mode, and the condition are equal, that is iff dCls(d) = dCls(d'), docCls(i) = docCls(i'), docCls(o) = docCls(o'), mode(p) = mode(p'), mode(q) = mode(q'), and cond<sub>dCls(d),mode(p),docCls(i)</sub> = cond<sub>dCls(d'),mode(p'),docCls(i')</sub>.

Thus with dCls(d), mode(p), docCls(i), and cond<sub>dCls(d),mode(p),docCls(i)</sub> 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 Δ<sub>l</sub> we also write:

$$mode(p) \xrightarrow{\frac{dCls(d),docCls(i),cond_{dCls(d),mode(p),docCls(i)}}{docCls(o)}} mode(q) \quad (1)$$

To help recognize decisions, mode-values and document classes, we prefix 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 receives a purchase order document *Order* from a customer, decides @Accept to accept the order and passes accordingly from mode value LISTENING to ORDERED, provided that the customer is assessed as *trustworthy*, and, accordingly, returns a confirmation document *Confirmation* and notate this as:

$$LISTENING \xrightarrow{\frac{@Accept,Order,\text{isTrustworthy}}{Confirmation}} ORDERED$$

It is important to keep in mind, that unlike the $p \xrightarrow{i/o} q$ notation, which denotes a single transition, Eq. (1) denotes a whole class of equivalent transitions.
# 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 price<sub>a</sub>, together with an increment in the quantity $n_a^B$ of commodity a for B and a decrement in the quantity $n_a^S$ of commodity a for S in the handover step.

$$ \text{External coupling: } \begin{array}{l} (i) \quad price_a := cost_a^B = revenue_a^S; \\ (ii) \quad n_a^B \rightarrow n_a^B + 1; \quad n_a^S \rightarrow n_a^S - 1 \end{array} \qquad (2) $$

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 $mod^B$ being her mode state componente, $n^B$ being the number of desired commodities a, $b^B$ 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, $b_0$, val<sub>a</sub>, 'undefined'). B proceeds in the following steps:

1. Being in mode WISHFUL B decides @Order<sub>p</sub> to buy a commodity a for some amount of money that will be her cost cost with $b_0 \ge$ 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 notification 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 commodity. If it arrives, $n_B$ is increase by one and B transits to mode DELIVERED.

4. Being in mode DELIVERED, B has no alternative then to decide @Pay to pay, i.e. to hand over the amount of money cost to S that was determined in step 1 beforehand. Thus, her budget $b^B$ shrinks by this amount while transiting into mode SETTLED.

5. Being in mode SETTLED, B waits for the settlement notification settled. In case seller S is of the opinion that the trade had not been settled, she transits back to DELIVERED.
A bit simplified, *B* has two make two decisions<fnref n="6" />: First, she has to decide @Order_p to buy a commodity *a* for given costs *cost*, and secondly, she has to decide @Pay to pay. The first decision @Order_p is free with respect to this interaction as is not determined by the success criterion of the interaction, because WISHFUL is an allowed final state. Instead, @Pay is enforced by the success criterion, as DELIVERED is not an allowed final state and to decide @Pay is the only way to proceed.

Now we turn to Seller *S*. He has the state (*mod*, *n*, *b*, *cost*, *revenue*)⁵ with *mod*⁵, *n*⁵, *b*⁵ with an equivalent meaning as for *B* and *cost*⁵ being his cost to somehow acquire, for example by production, the commodity *a* and *revenue*⁵ 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 final state SETTLED. Receiving too little money, he has to do a little bookkeeping and sends back a *NotSettled*-Notification 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 first decision is free with respect to the interaction as it is not determined by *S*' success 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<fnref n="7" /> *comp*<sup>external</sup>-operator such that it composes a *B*- and an *S*-role to a *trade* = *comp*<sup>external</sup>(*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:

<footnote n="6">⁶Effectively, @Order_p is a class of mutual exclusive decisions that determine her cost.</footnote>

<footnote n="7">⁷We 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.</footnote>
<figure><img src="image_4.png" /><figcaption>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 coupling 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 final states are indicated by an extra frame</figcaption></figure>

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 commodity 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 interaction rules that *S* delivers the ordered commodity and *B* pays the correct amount of money.

In reality, such a verification 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 specified.
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 different for both roles and we need it mainly to agree on a
meta-level on the design of trade-interactions.

As a final 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 different trade interactions
1 and 2 in Fig. 4.<fnref n="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 different, coordinated interactions need not.

T has the state `((mode<sup>S</sup>, mode<sup>B</sup>), n, b, revenue, cost)<sup>T</sup>` and the coordination condition is:

1. Coordinating function of money: there is only one state for the budget $b^T = b^B = b^S$.

2. Identity of traded commodity: there is only one state for the possession of the commodity $n^T = n^B = n^S$.

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 *Order*₁ for a commodity *a* and price *p*₁ in her trade interaction 1 as *S*₁ which would incur a revenue *revenue* = *p*₁ 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.

<footnote n="8">⁸There 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.</footnote>
3. Being in mode (ORDERED, ORDERED) $T$ in her role $B_2$ receives either a *Reject$_2$*, requiring a backward transition to (ORDERED, WISHFUL), or an *Accept$_2$*, initiating the transition to (ORDERED, CONFIRMED)

4. Being in mode (ORDERED, CONFIRMED) $T$ must decide @Accept, sends an Accept$_1$ and transits to (CONFIRMED, CONFIRMED).

5. Being in mode (CONFIRMED, CONFIRMED) $T$ awaits the reception of commodity *Product$_2$*, 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 *Product$_1$*, transiting to (DELIVERED, DELIVERED) and decrease its commodity counter $n$ again by 1.

7. Being in mode (DELIVERED, DELIVERED) she waits for the *Money$_1$*. If the received money suffices, i.e. it was at least price $p_2$ then she transits to (SETTLED, DELIVERED), sends *Settled$_1$* and increases her budget $b$ by this amount.

8. Being in mode (SETTLED, DELIVERED), she has to decide @Pay to pay the *Money$_2$* to transit to (SETTLED, SETTLED) and reduce her budget $b$ by this amount.

9. Being in mode (SETTLED, SETTLED), $T$ awaits the last *Settled$_2$-notification*.

All other transitions of the unrestricted product automaton are eliminated by this coordination.

Again, more formally we define a within subject composition operator *comp*<sup>internal</sup> for a homogeneous composition such that we combine the two roles *B* and *S* of the same subject to a trader role *T* = *comp*<sup>internal</sup>(*B*, *S*). Part of *comp*<sup>internal</sup> are two projection functions<sup>9</sup> *proj*<sub>*B*,*S*</sub>: *Q*<sup>*T*</sup> ∪ *I*<sup>*T*</sup> ∪ *O*<sup>*T*</sup> ∪ *D*<sup>*T*</sup> → *Q*<sup>*T*</sup> ∪ *I*<sup>*T*</sup> ∪ *O*<sup>*T*</sup> ∪ *D*<sup>*T*</sup> with *proj*<sub>*B*</sub>(*Q*<sup>*T*</sup> ∪ *I*<sup>*T*</sup> ∪ *O*<sup>*T*</sup> ∪ *D*<sub>*T*</sub>) ⊆ *Q*<sup>*B*</sup> ∪ *I*<sup>*B*</sup> ∪ *O*<sup>*B*</sup> ∪ *D*<sup>*B*</sup> and *proj*<sub>*S*</sub>(*Q*<sup>*T*</sup> ∪ *I*<sup>*T*</sup> ∪ *O*<sup>*T*</sup> ∪ *D*<sup>*T*</sup>) ⊆ *Q*<sup>*S*</sup> ∪ *I*<sup>*S*</sup> ∪ *O*<sup>*S*</sup> ∪ *D*<sup>*S*</sup>. Δ<sup>*T*</sup> fulfills the requirements, that the two projections of the transition relation Δ<sup>*T*</sup> of *T* result in the transition relations Δ<sup>*B*</sup> and Δ<sup>*S*</sup> and the success criterion of *T* is that both success criteria of *B* and *S* can be fulfilled on the projected sets.

In the proposed coordination, $T$ has to take 4 decisions, but only the first of them is still free. Namely she freely either decides @Reject to reject the received order *Order*<sub>1</sub> in interaction 1 or @Order<sub>*p*<sub>2</sub></sub> to order the commodity herself in interaction 2 for price *p*<sub>2</sub>. All the other three decisions, @Accept to accept the order in interaction 1, @Deliver to deliver the commodity in interaction 1 and @Pay to pay the price *price*<sub>2</sub> in interaction 2, are enforced 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,

<footnote n="9">⁹for a projection function *h* it holds that sequential application does not change the mapping, i.e. *h* = *h* ◦ *h*</footnote>
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 @Order<sub>*p*</sub> to order the commodity herself in interaction 2 for price *p*<sub>2</sub>. But this is invisible in interaction 1. We see that coordination changes the structure of the decision making.

<figure><img src="image_5.png" /><figcaption>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.</figcaption></figure>

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 different chances and risks.
For example, with enough budget it would be possible to order first, not knowing
whether she will be successfully able to sell the produced commodity at a higher
price than the purchase price. Or *T* could even order first without budget,
hoping to be able to pay for the purchased commodity 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-
notification. We see that spontaneous transitions of roles are central for their
flexible 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 fits 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 finite interactions, we have a set of final state values to
characterize success and we take a certain free decision because we prefer one
of these final state values. In case of infinite interactions, an outcome could be,
as mentioned, a set of state values, where each of them is taken infinitely often.

Given two distinguishable outcomes *a* and *b*, we can then say that being
in state *p* ∈ *Q* and receiving some input character *i* ∈ *I*<sup>*e*</sup>, with two possible
transitions *p* <sub>(*d*<sub>1</sub>, *i*)/*o*<sub>1</sub></sub>→ *q*<sub>1</sub>, finally leading to outcome *a* and *p* <sub>(*d*<sub>2</sub>, *i*)/*o*<sub>2</sub></sub>→ *q*<sub>2</sub>
finally leading to outcome *b*, we prefer outcome *a* over *b*, notated as *a* ≻<sub>*p*,i*</sub> *b* if
we would take the decision *d*<sub>1</sub> that leads to *a* instead of *d*<sub>2</sub> leading to *b*. If we
are indifferent between outcome *a* and *b* we could chose randomly and notate
*a* ~<sub>*p*,i*</sub> *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 $\tilde{\sim} \subseteq (C \times M)^2$ relates pairs of numbers of commodities and amounts of money. To keep our notation simple we refer only to a single class of commodities.
For example, let us assume that we prefer to have 10€ and no cinema ticket over having only 5€ and such a ticket. In symbols: $(0_{\text{cinema ticket}}, 10€) \succ (1_{\text{cinema ticket}}, 5€)$. In this case we would not pay more than 5€ for the ticket.

Why should we do such a seemingly artificial extension of our preference relation? Because we intuitively, that is without reflection, 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 finally 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 $\succsim$, if for every two outcomes *a* and *b* $util(a) \ge util(b)$ is equivalent to $a \succsim b$. 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. Mathematicians 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 appropriate 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 fixed 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 infinite items on each level.

As is well known, a utility function over-specifies its corresponding preference relation. If one such utility function $f$ exists, then any concatenation $u \circ f$ with a strongly monotonic function $u: \mathbb{R} \to \mathbb{R}$ is also a corresponding utility function to the same preference relation.
But beside the issue of over-specification, 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 defined as having exclusively 'selfish' preference that are directed only towards oneself [12].

To account for uncertain outcomes Johann von Neumann and Oskar Morgenstern [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 $(mod, n, b, val_a, cost_a)^B$ to B with the initial values of $(WISHFUL, 0, b_0, val_a, 'undefined')$.

B has the free decision @Order to buy or not to buy her preferred commodity a at costs $cost_a^B$. With its initial budget $b_0^B$, B would pay at most $val_a^B < b_0^B$ in exchange for a.

We now assume B's preference relation to relate to pairs $(b, n)$. B is indifferent per definition towards $(b_0^B, 0) \sim (b_0^B - val_a^B, 1)$ and $-$ assuming 1 as the unit of money $-$ has the following preferences:

$$ (b_0^B, 0) \sim (b_0^B - val^B, 1) \prec (b_0^B - val^B + 1, 1) \prec \dots \quad (3) $$

Thus, we can define a utility function for the decision @Order, where B gains in a trade interaction the difference between her valuation $val_a^B$ and her $cost_a^B$ to acquire a.

$$ \text{Buyer:} \quad \text{util}_a^B = \text{val}_a^B - \text{cost}_a^B \quad (4) $$

We see immediately that for any $cost_a^B \le val_a^B$ this function is indeed a utility function for B, as for a given valuation $val_a^B$ the equivalence $util_a^B(cost_1) > util_a^B(cost_2) \Leftrightarrow cost_1 < cost_2 \Leftrightarrow (1, b_0^B - cost_1) > (1, b_0^B - cost_2)$ 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 offer for a commodity a for a price p or to reject it.

Initially, S has a certain amount of money $b_0^S$ before the production of commodity a. For production, S had to spend a part of his money, his $cost_a^S$. With
selling he wants at least to get compensated for his expenses. Hence, he is indif-
ferent towards the following possessions ($b_0^S - cost_a^S, 1) \sim (b_0^S, 0)$ and the more
money he additionally gets in the trade, the better:

$$
(b_{total}^{S} - cost_{a}^{S}, 1) \sim (b_{0}^{S}, 0) \prec (b_{0}^{S} + 1, 0) \prec \dots \quad (5)
$$

Thus, we can define the utility of S, he gains from a trade interaction as the
difference between his revenue<sup>S</sup><sub>a</sub> and cost<sup>S</sup><sub>a</sub> to produce a.

Seller: \quad $util_a^S = revenue_a^S - cost_a^S \qquad (6)$

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_a^S is also named
‘profit’ or ‘supplier rent’, while util_a^B 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 *price*<sub>*a*</sub> is just the sum of the
individual utilities and does not depend on the price.

$$
\text{Total utility (trade): } \text{util}_{a,\text{total}} = \text{util}_a^B + \text{util}_a^S = \text{val}_a^B - \text{cost}_a^S \quad (7)
$$

Note that this form of total utility depends on our assumptions of the simple
form of Eqs. 4 and 6 for the utility of *B* and *S*. If we had chosen a monotonically
transformed version of both, then the total utility of trade would look different.
For example, we could have defined *util*<sub>*a*</sub><sup>*B*</sup> = log(*val*<sub>*a*</sub><sup>*B*</sup> - *cost*<sub>*a*</sub><sup>*B*</sup>) and *util*<sub>*a*</sub><sup>*S*</sup> =
log(*revenue*<sub>*a*</sub><sup>*S*</sup> - *cost*<sub>*a*</sub><sup>*S*</sup>). Then total utility would have been

$$
\begin{align*}
\text{util}_{a,\text{total}} &= \log(\text{val}_a^B - \text{cost}_a^B) + \log(\text{revenue}_a^S - \text{cost}_a^S) \\
&= \log((\text{val}_a^B - \text{cost}_a^B)(\text{revenue}_a^S - \text{cost}_a^S)) \\
&= \log((\text{val}_a^B - \text{price}_a)(\text{price}_a - \text{cost}_a^S)),
\end{align*}
$$

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.
We give an example: A book producer and a book enthusiast have an initial budget of 100€ each, making a total asset of 200€. The book producer now produces a book which costs him 30€, resulting in a remaining budget of 70€ plus the new book. The book enthusiast would give away 90€ of his 100€ to get this book. Both agree on a price of 50€. Subsequently, the book producer has 120€ and the book enthusiast has 50€ and a book he values 90€, making an asset of 140€. Together, after the trade, both have a total asset of 260€. The asset grew by the total utility created by the trade of 20€ for the book producer and 40€ for the book enthusiast. So both did benefit from the trade, though not equally.

## 4.4 How trader decides

Lets first look at our trader T as a black box, ignoring any possible inner structure. As T resells any commodity she buys, keeping the number of commodities $n_a$ finally constant, her preference relation – relating outcomes – actually relates only to her budget. The more budget she gets from trading, the better.

$$ (b_0^T, n) \prec (b_0^T + 1, n) \prec (b_0^T + 2, n) \prec \dots \qquad (8) $$

It's easy to show that the difference between her revenue revenue as sales price $p_1$ she receives in her selling interaction 1, and her cost as purchase price $p_2$ she has to pay in her buying interaction 2, is a utility-representation of her preference relation:

$$ \text{Utility (trader): } \text{util}^T = \text{revenue} - \text{cost} = p_1 - p_2 \qquad (9) $$

Obviously, the utility of T depends only on the price(s): T will sell a commodity for $p_1$ only if she succeeds in buying it for a lower price $p_2$, that is, if $p_1 > p_2$.

Now, lets turn to the inner structure of T in case we composed her as $T = comp^{internal}(B, S)$ with a coordination condition as we described in section 3.2. Then for B, its successful interaction 2 changes $n_a$ by $\Delta n_a = 1$ and the budget $b$ by $\Delta b = -p_2$ as long as $p_2 < val_a^B$ otherwise both remain constant. For S, its successful interaction 1 has $\Delta n_a = -1$ and at least $\Delta b = p_1$ as long as $p_2 > cost_a^S$.

Hence, as successful trading implies successful buying and selling, $n_a$ remains constant and her budget will either remain constant of increase by $\Delta b = p_1 - p_2$, which corresponds to eq. 9.

We would like to make a couple of points:

1. We might at first have thought that now, as a commodity is first 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 final state values in this finite interaction. In a sense there is no double utility because the won commodity is forseeable lost again.
2. We might also have thought that the relation between the preference relation $\succcurlyeq^T$ and $\succcurlyeq^B$ as well as $\succcurlyeq^S$ is that of a restriction like $\succcurlyeq^B = \succcurlyeq^T|_{proj_B^T(F)}$. This is not the case. As we have seen, the preference relation of $T$ does not relate to $n_a$ 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 fit 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 monotone transforms, possibly different 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 $comp^{external}(comp^{internal}(B, S))$? 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 definitions for a buyer $B$ and a seller $S$ in a trade interaction 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 suffices 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 specific 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.
First, we define:

**Definition 2.** We name the utility functions, representing the material preferences of a buyer B and a seller S in a trade, 'compositional consistent' if they fulfill the following constraints:

* Trade-interaction constraints:

(a) The utility of B depends only on the difference between her valuation and her costs: $util_a^B = f(val_a^S - cost_a^B)$, and becomes zero if her valuation equals her costs, $f(0) = 0$, and becomes her valuation if the costs are zero, $f(val_a^S) = val_a^S$.

(b) The utility of S depends only on the difference between his revenue and his cost: $util_a^S = g(revenue_a^S - cost_a^S)$, and becomes zero if his revenue equals his cost, $g(0) = 0$, and becomes his revenue if his costs are zero, $g(revenue_a^S) = revenue_a^S$.

(c) The total utility resulting from the trade as the sum of the utility of S and B is the difference between B's valuation and S's cost and is therefore independent of the price: $util_{a,total} = util_a^B + util_a^S = val_a^B - cost_a^S$.

* Trader constraint: Any strongly monotonic transform of the utility functions 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 $util_a^S$ and $util_a^B$ as given by (4) and (6) are the only compositional consistent utility functions.

To prove this theorem, we first have to provide the general utility functions for S and B in a trade interaction. As we have seen, $val_a^B - cost_a^B$ is a possible utility function of Buyer and $revenue_a^S - cost_a^S$ is a possible utility function of Seller. Then, the general utility functions for B and S, which depend only on the required difference, are provided by two additional monotonic functions $f, g: \mathbb{R} \to \mathbb{R}$ such that

$$ 
\begin{aligned}
util_a^B &= f(val_a^B - cost_a^B) \\
util_a^S &= g(revenue_a^S - cost_a^S)
\end{aligned}
 $$

Now, the trader constraint requires both function to be identical: $f = g$. And the external coupling condition (Eq. 2) makes the first constraint to

$$ 
\begin{aligned}
util_{a,total} &= util_a^B + util_a^S \\
&= f(val_a^B - price_a) + f(price_a - cost_a^S)
\end{aligned}
 $$

With the next lemma, we prove that this requires $f$ to be linear with zero intercept.
**Lemma 1.** Let $f$ be a real monotonic function such that whenever $a, b, c \in \mathbb{R}$ with $a \ge b \ge c$ 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 $f(val_a^B) = val_a^B$, $f$ has to be the identity function.

Considering money as a commodity like any other, Léon Walras discovered the linear influence 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 indifferent with respect to two bundles $x, y \in \mathbb{R}^n$, $x \sim y$, then any additional amount of money (component 1) does not change that:

$$ (x + \alpha e_1) \sim (y + \alpha e_1), \forall \alpha \in \mathbb{R}, e_1 = (1, 0, \dots, 0) \quad (10) $$

2. Money (component 1) is desirable in the sense that everything else constant, we prefer more money over less money:

$$ (x + \alpha e_1) \succ (x), \forall \alpha > 0 \quad (11) $$

This is what our intuition about money is about. If money works as intended, then it should not influence 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 transfer utility in the defined 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 difference 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 consumption, 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€ for a single apple we want to eat now or in a few days, we will not pay 1000€ for 1000 apples in a single moment, as we would perhaps even have to pay additional money to dispose most of them afterwards somehow.
# 6 Intersubjective comparison of preferences

Based on the role-based nature of a trade, we can compare the preferences of different trading subjects in several different ways. We can compare the valuations of competing roles and the utility of complementary roles. And, finally, we can compare the total utilities generated in different trades.

## 6.1 Intersubjective comparison of preferences between potential 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 fix 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 $val_a^A$ on a commodity a and another buyer B to spend a maximum of $val_a^B$ allows us to draw a direct conclusion about their preferences. Then for $A (0_a, val_a)^A \sim (1_a, 0)^A$ holds and for $B (0_a, val_a)^B \sim (1_a, 0)^B$ holds. We can thus define a relation 'A prefers a more than B', symbolically $A \succ_a B$, if and only if $val_a^A > val_a^B$

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 first, that already [39] showed that under comparable valuation conditions, any allocation resulting from free trade in our sense becomes efficient 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 complementary 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 define 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 arbitrarily unfair. Although a free trade is, per definition, 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 definition [of fairness]", which we would like
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 first thing we have to realize is, that reasoning over this case, we aggregate utility of different 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 first possibility is to average utility first and then to distribute it evenly by deriving the individual prices thereof. Let us look again at the book producer who now produces 2 books at 30€. There are two book enthusiasts, one with a valuation of 90€, the other, a bit less enthusiastic, with a valuation of 60€. The average total utility then is total valuation minus total cost divided by the number of participants $(90€ + 60€ - 30€ - 30€)/3 = 30€$. If the first book is sold for 60€ and the second book for 30€, every participant gets off individually with the average utility of 30€. In summary, we proceeded in two steps:

$$ \text{Step 1 [Determine average utility]:} \quad \bar{u} = \frac{1}{n+1} \sum_{i} (val_{i} - cost_{i}) = \frac{n}{n+1} (val - cost) $$

$$ \text{Step 2 [Determine } n \text{ prices]:} \quad p_i = val_i - \bar{u} $$

Thus, for increasing $n$, the average utility everyone realizes, converges against the difference 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 first price would determine to 60€ and the second price to 45€. The utility of the first buyer would be 30€, of the second buyer 15€ and for our seller it would be 45€.

$$ \text{Step 1 [Determine } n \text{ prices}] \quad p_i = (val_i - cost_i)/2 $$

$$ \text{Step 2a [average utility of the } n \text{ buyers]} \quad \bar{u}^B = \frac{1}{n} \sum_{i} (val_i - p_i) = val - \bar{p} $$

$$ \text{Step 2b [utility of seller]} \quad u^S = \sum_{i} (p_i - cost_i) = \sum_{i} u_i^S = n\bar{u}^S $$

The average utility for all participants is identical in both cases, but the distributions are different. Now all buyers get on average the difference between their average valuation and the average price, and seller accumulates all his utilities from each trade.

So, in the first case all participants benefit equally, in the second case, the
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 employees.

## 6.3 Aggregation of utilities of different trades: markets

With the fulfillment of the comparable valuation condition we can directly compare all buyers' valuations. Together with the objectively determinable costs of all sellers and our requirement, that the total utility of a single trade is independent 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 $j_{max}$ as this price creates the largest enclosed area and thereby maximizes total utility.

<figure><img src="image_6.png" /><figcaption>Figure 5: Supply and demand of a market mechanism.</figcaption></figure>

We call a market mechanism that unfolds its effects in a social context where the comparable valuation condition is being fulfilled, and which therefore does create a price that is most preferable in the sense of all, a 'valid' market mechanism.

With the comparable valuation condition being violated, we still can technically add up all differences 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 'maximized 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.
The social impact of large budget differences of market participants might be mitigated if the exchangebility is almost perfect and where quite superficial features segementize the market in different 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 fulfill hierarchical needs could be to deliberately incentivize an oversupply.

# 7 Extension to trading labor

## 7.1 The discretionary component of the employee's decision

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 difference 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 influenced 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 first glance, it may seem that this inverse relation enforce an indemnification 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 extent on the price determination mechanism.
## 7.2 The effects of additional hierarchical preferences for fairness of the employee

Beside our preference for some job over some other job, what else could determine the price for work? Here is one important example: what happens, if employees have an additional hierarchically superordinated preference for fairness? It renders the preference relations recursive<fnref n="10" />.

Let us assume that an employer has to offer a job that she valuates at 130.000€. The employee with a superordinated preference for local fairness has an initial internal valuation of 10.000€. Knowing little about each other, employer and employee agree on an initial salary of 20.000€ and the employee starts working.

After a while, the employee discovers, that the employer's valuation is indeed 130.000€ and therefore a fair salary would have (initially) amounted to 70.000€. What sum would this employee have to claim to remain in the company? If he claims 70.000€ as the minimal salary to stay in the company, he in fact relates to his own, older valuation of 10.000€ - which has just changed to 70.000€. So assuming 70.000€ as minimal acceptable salary, a fair salary would amount to 100.000€, and so on and so forth. The preference relation becomes recursive and the fixpoint is the valuation of the employer, in this case, 130.000€.

As a result, the internal valuation of the employee fits 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].

<footnote n="10">¹⁰According 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].</footnote>
## 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 organization 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 efficiency: avoiding interpersonal utility comparison

Vilfredo F. Pareto (1848-1923) was the first 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 cardinal 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-efficient', if no other such allocation exists, that makes some consumer better off without making some other consumer worse off [22, p. 313].

In this sense, a Pareto-efficient 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-efficiency intersubjective comparison of preferences is unnecessary. It suffices if all the participants do or don't prefer to transit from a given into some target state.

However, for Pareto-efficiency 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-efficiency degenerates into a justification 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-efficiency 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
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 effects of erroneous preferences, even against
the beneficiaries 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 final 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 differently 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 $b_{total}^B$ and after the trade as its reduced budget $b_{total}^{B'} = b_{total}^B - cost_a^B$, supplemented by her valuation $val_a^B$:

$$
\text{Buyer: } \text{util}_a^{B'} = b_{\text{total}}^{B'} + \text{val}_a^{B}. \qquad (12)
$$

This is indeed a utility representation of B, as again the equivalence $util_a'^B(cost_1) > util_a'^B(cost_2) \Leftrightarrow cost_1 < cost_2 \Leftrightarrow (1, b_{total}^B - cost_1) > (1, b_{total}^B - cost_2)$ holds. We could have concluded this immediately by recognizing that this form of utility is the same as eq. 4 increased by the initial total budget: $util_a'^B = util_a^B + b_{total}^B$. 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:

$$
\begin{align}
\text{util}'_{a, \text{total}} &= \text{util}'_{a}^{B} + \text{util}_{a}^{S} \tag{13} \\
&= b_{\text{total}}^{B} + \text{val}_{a}^{B} - \text{cost}_{a}^{S} \nonumber
\end{align}
$$
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 defined by definition 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 different interactions and not the apples.

## 8.3 Welfare economics

Welfare economics is an area of economics that suffered particular from the avoidance to intersubjectively compare material preferences. It is simply not possible to do welfare economics without weighing the welfare of different subjects 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 defined, real-valued 'utility' $U_i(x)$ from each state *x* of the world (or as a von Neumann-Morgenstern lottery). However, as Marc Fleurbaey and Peter J. Hammond [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 satisfaction.

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 enthusiast 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 influence 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
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 fluctuate enormously, depending on changing circumstances. Perhaps our book enthusiast, who was just willing to pay 90€ for the book, reads the first 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€ as she now possess only 50€ 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 € 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€ – why should we pay much more?

In summary, the material wealth of a society, as the sum of all its valuations, is quite dynamic and is governed by sources and sinks. It essentially is a flow quantity and thereby depends on the relation between its creation and annihilation and the stability of its measures. Our presumption is, that depending 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 extent. 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 different 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.
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 fulfill this function at least approximately or
not? We, the authors, think clearly yes. Otherwise the societal consequences of
efficient price determination mechanisms like markets seem to be quite unclear.

Since the conviction that one cannot compare subjective preferences inter-
personally had significantly 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 suffering from addiction knows intuitively the difference 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 flexibility in adapting to
variations of our social interactions, taking over new roles, abandoning old ones
or modify existing roles virtually on the fly. If some of our interactions do start to
determine our actions, the first 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 flexibility 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 influence 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
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 Christiane Schwieren for their encouragement to put these ideas on paper. And finally, 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 emphasizing 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 $a, b, c \in \mathbb{R}$ with $a \ge b \ge c$ 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

$$
f(a-b) + f(b-c) = 2f((a-c)/2). \tag{14}
$$

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

$$
\frac{f(x) + f(y)}{2} = f\left(\frac{x+y}{2}\right) \quad \text{for all } x, y \ge 0. \tag{15}
$$

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 fulfilled by every linear function, but conversely it is also true that every monotonic function fulfilling this condition is linear.

We first show the linearity for rational coefficients and in a second step extend this result to general real numbers. W.l.o.g. let $f(0) = 0$ (The general case with $f(0) \neq 0$ 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 $z \ge 0$ and $n = 0, 1, 2, \dots$ follows by induction: The cases $n = 0$ and $n = 1$ are trivial, by our assumption $f(0) = 0$. For the inductive step, let $n \ge 1$ and set $x = 2nz$ and $y = 2z$ in equation (15). Then the right hand side simply is $f(\frac{2nz+2z}{2}) = f((n+1)z)$, whereas the left hand side becomes $\frac{f(2nz)+f(2z)}{2} = \frac{2nf(z)+2f(z)}{2} = nf(z) + f(z) = (n+1)f(z)$ (using the inductive hypothesis).

By substituting $z$ with $\frac{z}{n}$ in $f(nz) = nf(z)$, we get $f(\frac{z}{n}) = \frac{f(z)}{n}$ for all $z \ge 0$ and $n = 1, 2, \dots$. Combining this with the above we obtain $f(qz) = qf(z)$ for any rational coefficient $q = \frac{m}{n}$ (with $m, n = 1, 2, 3, \dots$). In particular, we have $f(q) = qf(1)$ for any rational $q$.

To extend the result to general, real coefficients $r$ we use the monotonicity of $f$: Namely we have $f(r) \ge f(q) = qf(1)$ for any rational $q \le r$ which shows that $f(r)$ cannot be smaller than $rf(1)$ and, likewise, we find that $f(r)$ cannot be larger than $rf(1)$ either, hence $f(r) = rf(1)$. $\square$