A few changes to paper

casper4/papers/discouragement.tex

@@ -26,9 +26,13 @@ We model an economic consensus mechanism as being a game where there is an infin
 
 Note that if revenues to participants are dominated by transaction fees, then $p=1$ will hold.
 
-An attacker pays $\frac{1-\frac{h}{r}}{x^p}$ where $r$ is the \textit{proportional loss ratio} - the penalty incurred by victims divided by the penalty divided by the attackers where both are expressed as a fraction of the sizes of the two groups. The relationship between $r$ and the \textit{griefing factor} (ratio of victim losses over attacker losses expressed in absolute terms) is simple: $$g = r * \frac{\alpha}{1-\alpha},$$ where $\alpha$ is the portion of participants controlled by the attacker.
+Each participant controlled by the attacker pays $\frac{1-\frac{h}{r}}{x^p}$ where $r$ is the \textit{proportional loss ratio}. The proportional loss ratio is the ratio between the loss the victims suffer and the loss the attacker suffers, where both losses are expressed in percentage terms. For example, if an attack that causes the attacker to lose $1\%$ of deposits of all participants that they control causes everyone else to lose $2\%$, then the proportional loss ratio is 2.
 
-We now rephrase the problem into the language of supply and demand: there exist a set of players, each of which has some \textit{reserve interest rate} at which they are willing to become participants in the consensus mechanism. This is the demand curve, where the interest rate is the price. The protocol, which offers interest rates for participation in the consensus mechanism, sets the supply curve. If $p=0$, the supply curve is horizontal - the protocol offers that interest rate to an unlimited number of participants. If $p=\infty$, the supply curve is vertical. For any other $p$, the supply curve is declining with a constant elasticity of $\frac{1}{p}$. We model the attacker as having unilateral power to set $d$ (by attacking), and this pushes down the supply curve.
+The reason behing the above formulas is as follows. We assume that there is some ``base interest rate'' paid to all participants, which is proportional to some inverse power of the total size of the participant set. There is not necessary a principled in-protocol notion of the ``extent'' to which an attacker is attacking, so we define our own: the extent of an attack is $h$ if the victims' return decreases to $\frac{1-h}{x^p}$. We assume the proportional loss ratio $r$ is fixed, hence the attacking participants' return must be $\frac{1-\frac{h}{r}}{x^p}$.
+
+In contracts the \textit{griefing factor}, another way of comparing attacker and victim losses, is defined in absolute terms: for example, if in such a scenario the attacker controls $\frac{1}{3}$ of the total participant set, then the set of victims is twice as large as the attacker, and so altogether the victims lose four times more than the attacker, and so the griefing factor would be 4. The relationship between the proportional loss ratio $r$ and griefing factor is simple: $$g = r * \frac{1-\alpha}{\alpha},$$ where $\alpha$ is the portion of participants controlled by the attacker. In our above example, $\alpha = \frac{1}{3}$, so $g = 2 * \frac{\frac{2}{3}}{\frac{1}{3}} = 4$.
+
+We now rephrase the problem into the language of supply and demand: there exist a set of players, each of which has some \textit{reserve interest rate} at which they are willing to become participants in the consensus mechanism. This is the demand curve, where the interest rate is the price. The protocol, which offers interest rates for participation in the consensus mechanism, sets the supply curve. If $p=0$, the supply curve is horizontal - the protocol offers that interest rate to an unlimited number of participants. If $p=\infty$, the supply curve is vertical. For any other $p$, the supply curve is declining with a constant elasticity of $\frac{1}{p}$. We model the attacker as having unilateral power to set $h$ (by attacking), and this pushes down the supply curve.
 
 We model the demand curve as also being a simple exponential function, $x^d$. In general, we expect there to be wide disparities between the reserve interest rates of different players, as they have different levels of wealth, technical capability to operate a node in the consensus mechanism, and willingness to lock up their capital; additionally, we expect many players will be readily willing to lock up $50\%$ of their capital, somewhat willing to lock up $80\%$, hard pressed to lock up $95\%$, and not willing at all to lock up $100\%$. Hence, $d > 1$ seems likely, though we will consider the problem abstractly and give results for various values of $d$.