From ee9a621f0317492ea44ab1475da9af3298e147ab Mon Sep 17 00:00:00 2001 From: Vitalik Buterin Date: Sun, 13 Aug 2017 21:37:53 -0400 Subject: [PATCH] Fixed discouragement paper --- papers/casper/discouragement.pdf | Bin 264719 -> 264569 bytes papers/casper/discouragement.tex | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/papers/casper/discouragement.pdf b/papers/casper/discouragement.pdf index bb371019d5a6127ed39a55de30ba264acafc7afe..9d0d388a27b7824d775bd66151597e4ccb4db643 100644 GIT binary patch delta 29112 zcmV($K;yrUlo0un5RfGSIhUcO0Vt797zy@PWYHdy92iTHZ#93(KIOl4(=MDxzi<0? z*r+m(o2@QX{*Kns@9_I=+I)HV1N6B}8lg?8P@mPJb|%|uD@?6e-Ctew$L<^G0I~Py z*pA(Rve_04u@!JjCu*IzKg2^jw&$MN?>Cjq!+B_Y+{nBOEC~Eofw&V@RkGN-^ZqDYftB{Nxwx~L_++6o7PTVq6F>!jf86i2^{iN9F#Z>2J#G>R-%vzz|j zR61`rTcuRqem@<@wo2eb9^TdaJ zOG1Y&J5H8Dm*_YxtYvHb&`td$Hd|vIL~5(G%U>DcD3E_ZzZ!x2iZ1J>IPqzCWU3!- zxE*kVNINF#E_TD7IeUmNTUM}5COLK2h4XZL0WMkP9bD41y!QNAyb`zbE9ot>5P!qi z(9N)n0lzLE+vdDW`U0YcmromOb1=Vy 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both strategies are unprofitable. For values $r > 2$, the proof would need to be more conditional on specific values of $p$. We can make the claim that, if the griefing factor is bounded by $GF$, i.e. $r \le GF * \frac{\alpha}{1-\alpha}$, then a discouragement attack cannot be profitable if and only if $p \le \frac{1}{GF}$. -We can check this at the boundary $h = 1$ as follows. We want to show that $\frac{1 - hp * \frac{1-\alpha}{\alpha}}{(\alpha + (1-\alpha)(1-h)^{\frac{d}{d+p}})^p} \le 1$, so we show that the numerator is less than or equal to the denominator. At $h = 1$, the numerator simplifies to $1 - \frac{p}{\alpha} + p$ and the denominator to $\alpha^p$. At $\alpha=1$, the two are equal. To show that the numerator is strictly less for $\alpha<1$, we can take the derivative of both with respect to $\alpha$; the numerator becomes $\frac{p}{\alpha^2}$ and the denominator becomes $p * \alpha^{p-1}$, and since $\alpha < 1$ the derivative of the numerator is clearly greater, so for $\alpha < 1$ the original fraction will be less than one. Checking for $0 < h < 1$ is much harder, but analytically it can be verified that it holds. +We can check this at the boundary $h = 1$ as follows. We want to show that $\frac{1 - hp * \frac{1-\alpha}{\alpha}}{(\alpha + (1-\alpha)(1-h)^{\frac{1}{d+p}})^p} \le 1$, so we show that the numerator is less than or equal to the denominator. At $h = 1$, the numerator simplifies to $1 - \frac{p}{\alpha} + p$ and the denominator to $\alpha^p$. At $\alpha=1$, the two are equal. To show that the numerator is strictly less for $\alpha<1$, we can take the derivative of both with respect to $\alpha$; the numerator becomes $\frac{p}{\alpha^2}$ and the denominator becomes $p * \alpha^{p-1}$, and since $\alpha < 1$ the derivative of the numerator is clearly greater, so for $\alpha < 1$ the original fraction will be less than one. Checking for $0 < h < 1$ is much harder, but analytically it can be verified that it holds. Hence, if the griefing factor is bounded by 2, we want $p \le \frac{1}{2}$, and similarly for other griefing factors.