# A Mathematical Model of Copyright

May 26, 2023 — What is copyright, from first principles? This essay introduces a mathematical model of a world with ideas, then adds a copyright system to that model, and finally analyzes the predicted effects of that system.

# Part 1: The world

### The world

W

The world W contains observers who can make point-in-time observations O_t1 of W.

### Ideas

I: f(O_{t_1}, t_\Delta) → O_{{t_1}+{t_\Delta}}

An idea I is a function that given input observations at time t_1 can generate expected observations at time {t_1}+{t_\Delta}.

### Thinkers

T

A thinker2 T can store ideas I in T.

### Skillsets

S

A skillset S is the set of \set{I_i, \mathellipsis, I_n} embedded in T.

### Idea Creation

\alpha: f(S, O, t) → I_{new}

A thinker can generate a new idea I_{new} from its current skillset S and new observations O in time t.

### Value

V: f(I, O_{predictions}, O_{actual}) → \sum{(O_{actual} - O_{prediction})}^2

An idea I can be valued by a function V which measures the accuracy of all of the predictions produced by the idea O_{{t_1}+{t_2}} against the actual observations of the world W at time {}_{{t_1}+{t_2}}. Idea I_i is more valuable than idea I_j if it produces more accurate observations holding the size of |I| constant.

### Fictions

F

A fiction3 F is an I that does not accurately model W.

### Messages

M

Thinkers can communicate I to other thinkers by encoding I into messages M_I.

### Signal

\Omega: \frac{\sum{V(I)}}{|M|}

The Signal \Omega of a message is the value of its ideas divided by the size of the message.

### Fashions

Z

A fashion Z_{M_I} is a different encoding of the same idea I.

### Teachers

\tau

A teacher is a T who communicates messages M to other T. A thinker T has access to a supply of teachers \tau within geographic radius r so that \tau = \set{T|T < r}.

### Learning

L: f(M_I, T) → T^\prime

The learning function L applies M_I to T to produce T^\prime containing some memorization of the message M_I and some learning of the idea I.

### Objectives

B

A thinker T has a set of objectives B_T that they can maximize using their skillset S_T.

### Technologies

X

T can use their skillset S to modify the world to contain technologies X.

### Technology Creation

\Pi: f(\set{T},\set{X}, t) → X_{new}

Technology creation is a function that takes a set of thinkers and a set of existing technologies as input to produce a new technology X_{new}.

### Artifacts

A

With X M_I can be encoded to a kind of X called an artifact A.

### Creators

\chi

A creator \chi is a T who produces A.

### Outliers

\sigma

An outlier \sigma is a \chi who produces exceptionally high quality A.

### Copies

K

A copy K_A is an artifact that contains the same M as A.

### Derivatives

A^{\prime}

A derivative A^{\prime} is an artifact updated by a \chi to better serve the objectives B of \chi.

### Libraries

J

A library J is a collection of A.

### Attention

N

Thinkers T have a finite amount of attention N to process messages M.

### Distribution

D: f(A_o, T_o) → A_{T_o}

Distribution is a function that takes artifact A at location o and moves it to the thinker's location T_o.

### Publishers

Q

A publisher is a set of T specializing in production of A.

### Censors

U: U(D)

A censor is a function that wraps the distribution function and may prevent an A from being distributed.

### Masters

\Psi

A master \Psi is now legally assigned to each artifact for duration d so A becomes A^{\Psi}.

### Royalties

R

A royalty R is a payment from T to \Psi for a permission on A^\Psi.

### Permission Function

P: f(A^\Psi, T) → \{-1, 0, R\} * (\theta = Pr(\Psi, A^\Psi)) \text{ in } t < d

For every A^\Psi used in \Pi a permission function P must be called and resolve to >-1 and royalties of \sum{R_{A^\Psi}} must be paid. If any call to P returns -1 the creation function \Pi fails. If a P has not resolved for A^{\Psi} in time d it resolves to 0.4 P always resolves with an amount of uncertainty \theta that the \Psi is actually the legally correct A^\Psi.

### Classes

T = \begin{cases} T_{R+} &\text{if } R_{in} - R_{out} > 0 \\ T_{R-} &\text{if } R_{in} - R_{out} \leq 0 \end{cases}

The Royal Class T_{R+} is the set of T who receive more R than they spend. Each member of the Non-Royal Class T_{R-} pays more in R than they receive.

### Public Domain

A^0

A public domain artifact A^0 is an artifact claimed to have no \Psi or an expired d. The P function still must be applied to all A^0 and the uncertainty term \theta still exists for all A^0.

\varLambda: f(A_i, A_j^\Psi) → A_{ij}

Advertising is a function \varLambda that takes an A and combines it with an orthogonal artifact A_j^\Psi that serves B_\Psi.

# Part 3: Predicted effects of copyright

## Effect on Z

\Uparrow {Z \over I}

We should expect the ratio of Fashions Z to Ideas I to significantly increase since there are countless M that can encode I and each unique M can be encoded into an A^\Psi that can generate R for \Psi.

## Effect on F

\Uparrow F

We should expect the number of Fictions F to increase since R are required regardless if the M encoded by A accurately predicts the world or not. \Psi are incentivized to create A encoding F that convince T to overvalue A^\Psi.

## Effect on \varLambda

\Uparrow \varLambda

We should expect a significant increase in the amount of advertising \varLambda as \chi are prevented from generating A^{\prime} with ads removed.

## Effect on |M|

\Uparrow |M|

We should expected the average message size |M| to increase because doing so increases R by decreasing \theta and increasing A^\Psi.

## Effect on \Omega

\Downarrow \Omega

We should expect the average signal \overline{\Omega} of messages to decrease.

## Effect on K

\Uparrow {K \over I_{new}}

We should expect the ratio of number of copies K to new ideas I_{new} to increase since the cost of creating a new idea α is greater than the cost of creating a copy K and royalties are earned from A not I.

## Effect on \Pi

\Downarrow \Pi

We should expect the speed of new artifact creation to slow because of the introduction of Permission Functions P.

## Effect on J

\Uparrow {{Z + F + K} \over I}

We should expect libraries to contain an increasing amount of fashions Z, fictions F, and copies K relative to distinct ideas I.

## Effect on S

\Downarrow S

We should expect a decrease in the average thinker's skillset \overline{S} as more of a thinker's N is used up by Z, F, K and less goes to learning distinct I.

## Effect on I^\prime

\Downarrow I^\prime

We should expect the rate of increase in new ideas to be lower due to the decrease in \overline{S}.

## Effect on Classes

f^{\prime}(R, T_{R+}) > 0
f^{\prime}(R, T_{R-}) < 0

We should expect the Royal Class T_{R+} to receive an increasing share of all royalties R as surplus R is used to obtain more R streams.

## Effect on \sigma

\sigma → T_{R+} > 0

We should expect a small number of outlier creators to move from T_{R-} to T_{R+}.

## Effect on A^0

\Downarrow {{A^0} \over A^\Psi}

We should expect a decrease in the amount of A^0 relative to A^\Psi as T_{R+} will be incentivized to eradicate A^0 that serve as substitutes for A^\Psi. In addition, the cost to T of using any A^0 goes up relative to before because of the uncertainty term \theta.

## Effect on A^{\prime}

\Downarrow A^{\prime}

We should expect the number of A^{\prime} to fall sharply due to the addition of the Permission Functions P.

## Effect on B

\Uparrow {{B_\Psi} \over {B_T}}

We should expect A to increasingly serve the objective functions of \Psi over the objective functions B_T.

## Effect on Q

\Downarrow Q

We should expect the number of Publishers Q to decrease due to the increasing costs of the permission functions and economies of scale to the winners.

## Effect on U

\Uparrow U

We should expect censorship to go up to enforce copyright laws.

We should expect the number of A promoting © to increase to train T to support a © system.

## Effect on T_{R-}

\Downarrow T_{R-}

We should expect the Non-Royal Class T_{R-} to pay an increasing amount of R, deal with an increasing amount of noise from {Z + F + K}, and have increasingly lower skillsets \overline{S}.

# Conclusions

New technologies X_{new} and specifically A_{new} can help T maximize their B_T and discover I_{new} to better model W.

A copyright system would have no effect on I_{new} but instead increase the noise from {Z + F + K} and shift the \overline{A} from serving the objectives B_T to serving the objectives B_\Psi.

A copyright system should also increasingly consolidate power in a small Royal Class T_{R+}.

### Notes

1 The terms in this model could be vectors, matrices, tensors, graphs, or trees without changing the analysis.

2 We will exclude thinkers who cannot communicate from this analysis.

3 The use of "fictions" here is in the sense of "lies" rather than stories. Fictional stories can sometimes contain true I, and sometimes that may be the only way when dealing with censors ("artists use lies to tell the truth").

4 If copyright duration is 100 years then that is the max time it may take P to resolve. Also worth noting is that even a duration of 1 year introduces the permission function which significantly complicates the creation function \Pi.