From 4b8248b7c35ffe1c57782b8e3ae317f0d918e904 Mon Sep 17 00:00:00 2001 From: Vitalik Buterin Date: Sun, 27 Aug 2017 09:55:46 -0400 Subject: [PATCH] Started on economics paper --- papers/casper-basics/casper_basics.tex | 5 +- papers/casper-economics/appendix.tex | 2 +- .../casper_economics_basic.pdf | Bin 127459 -> 253659 bytes .../casper_economics_basic.tex | 273 ++++++++---------- 4 files changed, 120 insertions(+), 160 deletions(-) diff --git a/papers/casper-basics/casper_basics.tex b/papers/casper-basics/casper_basics.tex index 4dc3c12..2621881 100644 --- a/papers/casper-basics/casper_basics.tex +++ b/papers/casper-basics/casper_basics.tex @@ -3,20 +3,17 @@ Vitalik Buterin \\ Ethereum Foundation} - \documentclass[12pt, final]{article} %\input{eth_header.tex} - - - \usepackage{color} \newcommand*{\todo}[1]{\color{red} #1} \newcommand{\eqref}[1]{eq.~\ref{#1}} \newcommand{\figref}[1]{Figure~\ref{#1}} \usepackage{amsthm} \newtheorem{theorem}{Theorem} +\newtheorem{definition}{Definition} \usepackage{graphicx} \graphicspath{{figs/}{figures/}{images/}{./}} diff --git a/papers/casper-economics/appendix.tex b/papers/casper-economics/appendix.tex index 700a114..f7ee939 100644 --- a/papers/casper-economics/appendix.tex +++ b/papers/casper-economics/appendix.tex @@ -4,7 +4,7 @@ \section{Safety Failure} \label{app:safetyfailure} -\TODO{Put the full description/definition of conflicting blocks here.} +\todo{Put the full description/definition of conflicting blocks here.} diff --git a/papers/casper-economics/casper_economics_basic.pdf b/papers/casper-economics/casper_economics_basic.pdf index 0996c790ba6eb747e15f2cfc11133965a9e953f4..60a991bee1a37eab3908f465b3abc127b0693617 100644 GIT binary patch delta 226122 zcma(2V|QR-w{Qu^wry8z+pgHC*fw@-+qP}nPAX2NVzXlOeR}lMZ@=T5(NFJRus^Le z$6WI|IDp>jMFuF9#U&Y77&+i57gk2s;n+x+NgRGz!}0UOG0B_RTew=1uyAm)lKh_^ z9FwG#t*e^Lh=?TLhnue!lQ~;*{=~~ zztV8U+NTS4C~y9@i`B}ZGqWhFbV#5nbJ z*T)~jhO+M+k!o0L__#>EoJ7_J+xAWT1X*3xn-)w~T}^h9O^-dVp3gQUv1pNV!P2-t 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Prepare or Commit to the same hash. Causing a safety failure requires violating one of the Casper Commandments (Section \ref{sect:casperprotocol}) and thus ensures immense loss in deposits. In the second case, in a chain that has not been finalized for $\epoch - \epochLF$ epochs, the penalty to attackers is at least, \begin{equation} -\begin{split} -\min \left[ \NPP \left(\frac{1}{3}\right) + \NPCP\left(\frac{1}{3}\right), \NCP \left(\frac{1}{3}\right) + \NCCP\left(\frac{1}{3}\right) \right] &* \BP(\totaldeposit, \epoch - \epochLF) \\ -\left(\frac{1}{3}\right) \min \left[ \NPP + \NPCP, \NCP + \NCCP \right] &* \BP(\totaldeposit, \epoch - \epochLF) \\ -\end{split} +\min \left[ \NPP \left(\frac{1}{3}\right) + \NPCP\left(\frac{1}{3}\right), \NCP \left(\frac{1}{3}\right) + \NCCP\left(\frac{1}{3}\right) \right] * \BP(\totaldeposit, \epoch - \epochLF) \\ +\left(\frac{1}{3}\right) \min \left[ \NPP + \NPCP, \NCP + \NCCP \right] * \BP(\totaldeposit, \epoch - \epochLF) \\ \end{equation} To enforce a ratio between validator losses and loss to protocol utility, we set, @@ -321,7 +287,7 @@ To enforce a ratio between validator losses and loss to protocol utility, we set \begin{equation} \BP(\totaldeposit, \epoch - \epochLF) \equiv \frac{k_1}{\totaldeposit^p} + k_2 * \lfloor \log_2(\epoch - \epochLF) \rfloor\; . \end{equation} -\TODO{what is $p$ in the in the above equation?} +\todo{what is $p$ in the in the above equation?} The first term serves to take profits for non-committers away; the second term creates a penalty which is proportional to the loss in protocol utility. @@ -338,7 +304,7 @@ We define the degree that malicious validators can create penalties for honest v \begin{equation} \GF{ \gamesymbol,C } \equiv \max_{S \in strategies(T \setminus C)} \frac{loss(C) }{\min[ 0, loss(Players \setminus C) ] } \; . \end{equation} -\TODO{I need to work on this equation more. I don't like it yet.} +\todo{I need to work on this equation more. I don't like it yet.} @@ -359,9 +325,9 @@ A strategy that imposes a loss to outsiders either at no cost to a coalition, or \label{fig:GF} \end{figure} -Then to define the griefing factor over the entire game, we sum the area under the curve in \Figref{fig:GF} leading to, +Then to define the griefing factor over the entire game, we sum the area under the curve in \figref{fig:GF} leading to, \begin{equation} -\GF{ \gamesymbol } \equiv \int_{0}^{1} \mathscr{GF}( \gamesymbol, \alpha ) \; d\alpha \; . +\GF{ \gamesymbol } \equiv \int_{0}^{1} GF( \gamesymbol, \alpha ) \; d\alpha \; . \end{equation} @@ -384,14 +350,11 @@ Let us now analyze the attack types: \centering \renewcommand{\arraystretch}{2} \begin{tabular}{l l l } - \toprule - \thead{Attack} & \thead{Amount lost by \\ malicious validators} & \thead{Amount lost by \\ honest validators} \\ - \midrule + \textbf{Attack} & \textbf{Amount lost by malicious validators} & \textbf{Amount lost by honest validators} \\ Minority of size $\alpha < \frac{1}{2}$ non-Prepares & $\NPP * \alpha + \NPCP(\alpha) * \alpha$ & $\NPCP(\alpha) * (1-\alpha)$ \\ Majority censors $\alpha < \frac{1}{2}$ Prepares & $\NPCP(\alpha) * (1-\alpha)$ & $\NPP * \alpha + \NPCP(\alpha) * \alpha$ \\ Minority of size $\alpha < \frac{1}{2}$ non-Commits & $\NCP * \alpha + \NCCP(\alpha) * \alpha$ & $\NCCP(\alpha) * (1-\alpha)$ \\ Majority censors $\alpha < \frac{1}{2}$ Commits & $\NCCP(\alpha) * (1-\alpha)$ & $\NCP * \alpha + \NCCP(\alpha) * \alpha$ \\ - \bottomrule \end{tabular} \caption{Attacks on the protocols and their costs to malicious validators and honest validators.} \end{table} @@ -404,8 +367,8 @@ There is a symmetry between the non-Prepare case and the non-Commit case, so we \begin{figure}[h!bt] \centering - \subfloat[Utility with function of $p$]{ \includegraphics[width=3in]{goodness-with-p.pdf} \label{fig:utility} } - \subfloat[\NCCP and \NPCP as a function of $\alpha$]{ \includegraphics[width=3in]{cs.pdf} \label{fig:collectivepenalties} } + Utility with function of $p${ \includegraphics[width=3in]{goodness-with-p.pdf} \label{fig:utility} } + \NCCP and \NPCP as a function of $\alpha${ \includegraphics[width=3in]{cs.pdf} \label{fig:collectivepenalties} } \caption{Plotting the griefing factor as a function of the proportion of players coordinating to grief.} \label{fig:GF} @@ -417,7 +380,7 @@ In the normal case, anything less than $\frac{1}{3}$ Commits provides no economi Now, let us analyze the griefing factors, to try to determine an optimal shape for $\NCCP$. The griefing factor for non-Committing is, \begin{equation} -\mathscr{GF} = \frac{(1-\alpha) * \NCCP(\alpha)}{\alpha * (1 + \NCCP(\alpha))} \; . +GF = \frac{(1-\alpha) * \NCCP(\alpha)}{\alpha * (1 + \NCCP(\alpha))} \; . \end{equation} The griefing factor for censoring is the inverse of this. If we want the griefing factor for non-Committing to equal one, then we could compute: @@ -464,4 +427,4 @@ The above analysis gives a parametrized scheme for incentivizing in Casper, and \input{appendix.tex} \end{document} - \ No newline at end of file +