I show how a temporary reward suspension (“dilution”) is equivalent to an immediate slash.

We define slashing as removing capital from a validator and burning it. A validator is slashed instantaneously and for simplicity is immedately is able to continue validating the chain after the slashing event.

Dilution is a temporary penalty where a validator earns no rewards for $n$ periods, missing out on compounding gains.

Note, I don’t discuss redistribution explicitly but I believe the results still hold if you assume foregone rewards are distributed to those harmed.

Setup

Assume the total supply of the token at time $t$ is $S_t$. A validator holds some amount of stake $s_t$. The reward rate for performing their duty is $r$. The starting percentage of the network for the validator is $\frac{s_t}{S_t}$. The validator commits some fault at time $t$. The total supply of the network if they hadn’t committed a fault is:

\[S_{t+n}=S_t(1+r)^n\]

Dilution model

Under dilution, the validator is excluded from participating in the network, and thus not earning rewards, for $n$ periods after which they are allowed to participate again.

Trivially, the stakers stake at time $s_{t+n}$ is $s_t$.

The total supply $S_{t+n}$ is:

\[S_{t+n}=(S_t-s_t)(1+r)^n + s_t\]

The validator’s percentage of the total supply becomes:

\[p_{t+n}=\frac{s_t}{(S_t-s_t)(1+r)^n + s_t}\]

Slashing model

The stake at time $t+n$ equals:

\[s_{t+n}=(1-\alpha)s_t(1+r)^n\]

where $\alpha$ is the slashing fraction.

The total supply becomes:

\[S_{t+n}=(S_t-\alpha s_t)(1+r)^n\]

After slashing their percentage is:

\[p_{t+n}=\frac{(1-\alpha)s_t}{S_t-\alpha s_t}\]

Equivalence

For a given $n$, the equivalent slash fraction is equal to:

\[(1-\alpha)s_t(1+r)^n=s_t\] \[\alpha=1-\frac{1}{(1+r)^{n}}\]

For example, assume $r$ is 1% and $n$ is 10 periods. If the validator is deprived of 10 periods worth of rewards that is equivalent to slashing the validator by 9.47%.

For a given $\alpha$, the $n$ that generates an equivalent loss is equal to:

\[n = \frac{\ln(\frac{1}{1-\alpha})}{\ln(1+r)}\]

For example, if the slashing fraction is 20% and $r$ is 1%, then an equivalent punishment would be to dilute the validator for approximately 22.43 periods.

We can also express it in terms of cost to attack.

The cost to attack in the diluted model is:

\[c_{\text{dilution}}=s_t(1+r)^n-s_t=s_t[(1+r)^n-1]\]

whereas for the slashed scenario:

\[c_{\text{slashing}}=s_t(1+r)^n-(1-\alpha)s_t(1+r)^n=\alpha s_t(1+r)^n\]