definition.tex

%!TEX root = egalitarianism.tex

\section{Defining egalitarianism}\label{sec:definition}


Having established the basics of consensus mechanisms, we now propose the first
definition of an economic measure of \emph{egalitarianism} in cryptocurrencies.
Before we present our definition, let us first state the \emph{desiderata} of such a
definition. First of all, we want to allow concrete measurements to be
performed on cryptocurrencies and data to be extracted in a manner that is
quantitative and not vague. Thus far, the claims for egalitarianism in various
cryptocurrencies have been rather informal, using a rhetoric which fails to include
exact data~\cite{van2013cryptonote,mcmillan2013}. As such, different cryptocurrencies claim egalitarianism over the
others, without demonstrating the claims or provide conclusive arguments.
Secondly, a definition of egalitarianism must measure the protocol maintenance 
returns of a ``rich dollar'' compared to that of a ``poor dollar.''
We thus desire a
measure which, for a particular cryptocurrency, extracts a smaller value
to indicate a \emph{lack of egalitarianism} (\eg a
case where large wealth generates blocks disproportionately faster than
small wealth) and a larger value to indicate \emph{perfect egalitarianism} (where
every invested dollar has exactly equal power in terms of cryptocurrency
generation).

As a means towards establishing our egalitarianism definition, we define the
\emph{egalitarian curve} $f$ of a cryptocurrency. The horizontal axis of this
curve plots the financial capital which is available for investment denominated
in a fiat currency, USD.\footnote{Given that we explore
a small investment duration, it makes little difference whether these are
nominal USD or real USD, as long as they are the same when applying comparisons.}  The
vertical axis plots the Return On Investment (ROI), which measures the
cryptocurrency amount that is freshly generated in the investment period and
remains unspent at the end of the investment period,
given an optimal allocation of the initial capital. We require
the Return On Investment is necessarily \emph{freshly generated}
cryptocurrency; thus, it must be newly mined or minted, and not part of the
initial capital. Of course, purchasing
cryptocurrency which has already been generated is an investment option, but it
is immaterial to our egalitarianism definition, which focuses on measuring the
egalitarianism of freshly generated cryptocurrency.  Finally, the curve is
plotted with a fixed investment duration in mind --- in this paper, we use a
duration of 1 year.  Naturally, curves of different cryptocurrencies can be
compared only if they use the same duration.

\begin{definition}[Egalitarian curve]
    Given a cryptocurrency $c$, an investment period interval $d$, the set of
    all possible investment strategies $\mathcal{B}$, we define the \emph{egalitarian curve}
    $f_{c,d}: \mathbb{R}^+ \longrightarrow \mathbb{R}^+$ of $c$ for
    investment period $d$ as:

    \[
        f_{c,d}(v) = \frac{\underset{B \in \mathcal{B}}{\max}{\mathbb{E}[B(v)]} - v}{v}
    \]
\end{definition}

The value $\underset{B \in \mathcal{B}}{\max}{\mathbb{E}[B(v)]}$ identifies the maximum expectation of
returns across all investment strategies $\mathcal{B}$, \ie the amount of
returns which the \emph{optimal} strategy ensures for a given initial capital $v$.
The expectation is taken with the blockchain execution as a random variable,
since returns vary by execution (the randomness of the execution can affect the
returns of the strategy, as the same strategy can bring larger returns if the
participant is ``lucky'' e.g., it happens to produce many blocks~\cite{equitability}).

We remark first that we do allow strategies to reinvest capital. For instance,
returns earned from mining rewards can be reinvested in electricity costs for
future mining. Furthermore, for unit consistency, we assume the strategy
$B(v)$ returns the freshly generated coins denominated in the same units as the capital $v$ was given in, such that $f$ represents a ROI; thus, we denominate the
generated cryptocurrencies in USD using the market exchange rate.
Second, we assume participants act independently and follow the protocol
according to its specifications. 

Based on the above, it
is now straightforward to define the \emph{ideal egalitarian curve}. In this
case, the ROI is stable regardless of capital invested. Under these ideal
conditions, the amount of freshly generated cryptocurrency is exactly
proportional to the money invested. Thus, the ideal curve is any constant
curve.

% , as depicted in Figure~\ref{fig:ideal}.
%
% \begin{figure}
%     \centering
%     \includegraphics[width=0.4 \columnwidth,keepaspectratio]{figures/ideal.pdf}
%     \caption{The ideal egalitarian curve of an ideal cryptocurrency.}
%     \label{fig:ideal}
% \end{figure}

As an interesting thought experiment, consider the egalitarian curve which is
decreasing. In this case, the poor would receive proportionally more newly
created cryptocurrencies for every dollar they invest, \ie it would be a
redistribution of wealth from the rich to the poor. However, one
can quickly see that, in decentralized cryptocurrencies where the identities of
the participants are unknown, it is impossible to
hope for something better than the constant curve. Indeed, the fact that
decentralized cryptocurrencies allow anonymous generation of new
identities~\cite{douceur2002sybil}
allows a rich investor to split their investment into smaller ones.  Thus, if
the curve were ever to have a negative slope, the sum of the smaller splits of
the rich investment would achieve a higher gain. By the definition of the
curve, which mandates that it depicts the ROI of an \emph{optimal} investment,
this would be a contradiction. The following lemma makes the above intuition
more precise:

\begin{restatable}[Sybil strategies]{lemma}{restateLemSybilStrategies}
\label{lem:sybil}
    Fix a cryptocurrency $c$ and an investment period interval $d$. Given capital $v$,
    for every natural number $i \in \mathbb{N}^\star$, it
    holds that $f_{c,d}(v) \leq f_{c,d}(i \cdot v)$.
\end{restatable}

The proof of this Lemma is available in Appendix~\ref{sec:proofs}.

Using our definition of the egalitarian curve, we now define egalitarianism as
a concrete number. We begin by considering the initial capital $v$ as a random
variable following a certain distribution $\mathcal{D}$. Egalitarianism is
defined as the variance of the expected ROI when the capital is chosen from the
given distribution.

\begin{definition}[Egalitarianism]
  Given a cryptocurrency $c$, an investment period duration $d$ and an initial
  capital distribution $\mathcal{D}$, we define the \emph{egalitarianism} $e$ of $c$
  for investment duration $d$ under initial capital distribution $\mathcal{D}$
  as follows:

  \[
    e_{c,d,\mathcal{D}} = -\textsf{Var}_{v \gets \mathcal{D}}[f_{c,d}(v)]
  \]

  where $f$ is the egalitarian curve of $c$.
\end{definition}

The intuition behind this definition is that, to have egalitarianism, the ROI
must remain the same across different capital investments. As such, any
deviation from the mean is non-egalitarian. Naturally, if the
egalitarianism of a certain cryptocurrency is \emph{higher} than another's, we
say that the former is \emph{more egalitarian} than the latter. Of course, to be
accurate, such comparisons must only be made after fixing the parameters $c$
and $d$ as well as the initial capital distribution $\mathcal{D}$. We will now
fix the distribution $\mathcal{D}$ to be the uniform distribution between a
minimum and a maximum capital. This choice corresponds to the intuition that the
returns are the same for all initial capitals alike. Clearly a cryptocurrency
with an ideal egalitarian curve is perfectly egalitarian, as we now define.

\begin{definition}[Perfect egalitarianism]
  A cryptocurrency $c$ is \emph{perfectly egalitarian} for investment duration
  $d$ and initial capital distribution $\mathcal{D}$ if
  $e_{c,d,\mathcal{D}} = 0$.
\end{definition}