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Stream: practice: industry

Topic: Analyze this Model of a Market

Julius Hamilton (Aug 31 2024 at 17:31):

Let $C$ denote a universal set of commodities.

Let $Q$ denote a set of quantities.

Let $V$ denote a function $V : C \times C \to Q$.

Let $P$ denote a set of people.

Let $V_p$ denote a $V$-function associated with person $p$.

For all $a, b, c \in C, p \in P$:

• $V_p(a, b) = \frac{1}{V_p(b, a)}$
• $V_p(a, c) = V_p(a,b) * V_p(b,c)$
• $V_p(a, a) = 1$

What kind of algebraic object is this?

John Baez (Aug 31 2024 at 17:40):

You seem to be assuming $Q$ is a group (since you're multiplying its elements, dividing them and talking about an element $1$), but you're not telling us that it's a group - you're calling it a mere 'set'. This will make mathematicians say 'tut-tut'. I'm warning you because I've seen some of them around here.

Julius Hamilton (Aug 31 2024 at 17:41):

Right.

• $Q$ needs inverses, an identity, and a closed operation $*$. We might want more properties, like associativity, commutativity and total order.

Julius Hamilton (Aug 31 2024 at 17:44):

• $V_p$ maps elements of $C \times C$ into $Q$, but $C$ and $P$ are bare sets with no structure. $V$ is an index. We can say that $Q$ is an algebraic structure imposed on an indexed family of elements $\{ V_{p, a, b}\}$.

John Baez (Aug 31 2024 at 17:48):

You have to make up your mind on these issues before I can say anything interesting. Some choices are 'mathematically better' than others, in the sense that mathematicians can tell you more interesting things about them. Thus, I would want $Q$ to be a group, and probably an abelian group. But order structure on it would be just distracting at this stage, because it throws mud into what is otherwise a rather beautiful question.

Julius Hamilton (Aug 31 2024 at 17:49):

Thanks!
I’ll keep adding details :)

John Baez (Aug 31 2024 at 17:50):

That is: it's beautiful if your set of people consists of one person. If there's more than one, you can separately analyze the situation one person at a time, since you've put in no interactions between people.

(This would be like talking about a set that is a group in two different ways, with no interaction between those ways: mathematically boring.)

Julius Hamilton (Aug 31 2024 at 17:50):

RIGHT! And that is what I will do :)

Julius Hamilton (Aug 31 2024 at 19:50):

$Q$ can be a commutative ring.

Each person $p$ is associated with a function $C \to Q$.

The family of functions $\{C \to Q\}_p$ is called an economy. It is the disjoint union of this family.

Denote the economy $E$, and the range of possible states of an economy $\mathcal{E}$. A transfer is an update function $t : P \times P \times C \times Q \times \mathcal{E} \to \mathcal{E}$, which takes a giver, a getter, a commodity, and a quantity for an economy and updates it:

$t(p_{\text{give}}, p_{\text{get}}, c, q, \mathcal{E})$ is $\mathcal{E}$ except where $q$ in $(c,q)_{p_{\text{give}}}$ is now $q_{\text{previous}} - q$, and where $q$ in $(c,q)_{p_{\text{get}}}$ is now $q_{\text{previous}} + q$.

I would like help improving the formulation before moving on. There will be a set of states $S$ to index the economy, to define $t : \mathcal{E}_{s \in S} \to \mathcal{E}_{s + 1 \in S}$.

Julius Hamilton (Aug 31 2024 at 20:17):

This is a different formulation I am liking.

Let there be:

1. A group of states, $(S, +)$, represented by the integers.
2. A set of owners, $O$.
3. A set of commodities, $C$.
4. A commutative ring of quantities, $Q$.

There is a function $\text{has}: S \to O \to C \to Q$.
There is a function $\text{willing-to-trade} : S \to O \to C \to C \to Q$.

The economy $E$ at state $S$ is identified with the set of values of the 'has' function.

There is a function
$\text{give}: S \to O \to O \to C \to Q \to \text{has}$
which returns a new version of the 'has' function (in which owner 1 has q less quantity of C, and owner 2 has q more).

Julius Hamilton (Sep 04 2024 at 21:14):

Here is an updated work-in-progress model I am quite happy about. I’m open to any comments, questions or criticisms. Thanks.

We extend ZFC with this signature:

$\Sigma :=$
$\{\{\text{Thing}, \text{Owner}, \text{Time}, \text{Amount}, \text{Economy}\},$
$\{\text{give}, \text{own}, \text{offer}\},$
$\{\}\}$

and these axioms:

• $\text{own} : \text{Time} \to \text{Owner} \times \text{Thing} \to \text{Amount}$
• $\text{offer} : \text{Time} \to \text{Owner} \times \text{Thing} \times \text{Thing} \to \text{Amount} \times \text{Amount}$
• $\text{give} : \text{Time} \to \text{Owner}^2 \times \text{Thing} \times \text{Amount}$
• $\text{Economy} \in \text{Amount}^{\text{Owner} \times \text{Thing} \times \text{Time}}$

Some observations:

• The ownership function tells us who owned what when. It describes the evolution of an economy over a continuum of states. But there are different possible such economic histories. That’s why an economy is an element in the set of all possible economies, which is the set of all possible ways of mapping owner-thing-time triples to amounts.
• $\text{give}$ is a function which describes who gave what to whom when. It can also be considered an update function which takes an economy at time $t$ and returns the ownership function of an economy at time $t + 1$. In future versions of this draft, it may be crucial to differentiate between an economy at time $t_k$ vs. the set of states of an economy for all times $t_i$.
• A constraint of the $\text{give}$ relation is that at time $t_i$, the amount $a$ of a thing $c$ given by an owner $o_1$ to an owner $o_2$ must satisfy an identity between $\text{own}_{t}$ and $\text{own}_{t + 1}$, which is that $\text{own}(t, o_1, c) - a = \text{own}(t + 1, o_1, c)$ and $\text{own}(t, o_2, c) + a = \text{own}(t + 1, o_2, c)$.