We use the abbreviation $DTP$ to refer to a “Detection True Positive” event. This refers to the event in which a guilty operator is accused. Similarly, $DFP$ refers to a “Detection False Positive” event, which refers to event where an innocent operator is accused. We further use $CTP, CFP, OTP, OFP$ to refer to “Court True (False) Positive” events and “Overall True (False) Positive events”. The first ($CTP, CFP$) refer to the scenario where a guilty (innocent) operator that is being processed in a court is convicted. The latter ($OTP, OFP$) to the event that, in overall, a guilty (innocent) operator is convicted.
For $i\in \{DTP, DFP, CTP, CFP, OTP, OFP\}$ we let $p_i$ be the probability that $i$ occurs. One has:
$$ \begin{equation}\begin{aligned} p_{DTP}&=P(accused|guilty)\\ p_{CTP}&=P(convicted|accused \wedge guilty)\\ p_{OTP}&=P(convicted|guilty)\\ p_{DFP}&=P(accused|innocent)\\ p_{CFP}&=P(convicted|accused \wedge innocent)\\ p_{OFP}&=P(convicted|innocent) \end{aligned} \end{equation} $$
In previous notes [Analysis of operators’ economic incentives, Bonding requirements for operating the courts] we encountered the need to estimate some of these probabilities, namely $p(t){DTP}, p{CTP}, p_{CFP}$. In this note, we discuss how to find all the probabilities in (1). First, we deal with $p_{OTP}$ and $p_{OFP}$ by observing that
$$ \begin{align*}&p_{OTP}=P(convicted|guilty)= p_{CTP}\cdot p_{DTP},\\ &p_{OFP}=P(convicted|innocent)=p_{CFP}\cdot p_{DFP}\end{align*} $$
These equalities follow from observing that:
$$ \begin{aligned}&P(convicted|guilty)=\\ & \quad \quad P(convicted|accused)P(accused|guilty) + P(convicted|not.accused)P(not.accused|guilty)=\\ &\quad \quad P(convicted|accused) P(accused|guilty)+ 0\cdot P(not.accused|guilty) =\\ &\quad \quad p_{CTP}p_{DTP} \end{aligned} $$
The second equality follows similarly.
$p_{DTP}$ and $p_{DFP}$ depend on how long the time period in question is. So it makes more sense to look at a decay model. Let $\tau_{DTP}$ be the mean lifetime for how long a guilty party will be undetected. Hence the probability that a guilty party will be accused over a time period $t$ is
$$ p(t){DTP}=1-e^{-t/\tau{DTP}}. $$
Similarly, we obtain
$$ p(t){DFP}=1-e^{t/\tau{DFP}}. $$
It is unclear how the probabilities in (1) and the parameters $\tau_{DTP}$ (and $\tau_{DFP}$) can be estimated from theoretical considerations alone. This mean lifetime will depend on the ability of other parties to produce evidence that could incriminate a white-labeling operator (c. ref. White-labeling evidence types)
To deal with this situation, we can set up an experiment where a known set of white-label nodes is set up on the Ethereum network. By offering one-time rewards (such as via a grant or a competition) we can attract external parties with the appropriate expertise to monitor the entire data set and look for these white labels. The experiment can also include a phase where jurors evaluate some cases. An example is given later.
We will initially estimate the probabilities (1) and the time averages $\tau_{DTP}, \tau_{DFP}$ assuming the data produced in the experiment corresponds to “real-world” data.
The true ground truth parameters from “real life” and those observed in the experiment will likely differ non-negligibly (The higher the competition incentives the better we should expect the approximation to be). However, a rough approximation seems to be enough for our purposes.