There are greedy (G) and non-greedy (NG) proposers, as well as censoring and non-censoring builders. Suppose there is a fraction $g$ of G-proposers, and $1-g$ NG-proposers. A fraction $c$ of G-proposers is also censoring (GC-proposer), while $1-c$ doesn’t censor (GNC-proposers). All NG-proposers are non-censoring.

So there is a fraction $gc$ of censoring proposers, and $1-gc$ of non-censoring proposers.

The best builder is censoring (”censoring-best”), so NG-proposers choose a second-best, non-censoring builder, while G-proposers choose the censoring-best builder. The censoring-best builder pays $A$ while the non-censoring builders pay $B$, $A > B$.

Whenever a censoring proposer is faced with an IL they do not wish to satisfy, they either not propose a block and receive $0$, or propose an artificially full block of value $A-\delta$, where $\delta$ is the cost to fill the block, whichever is higher.

Whenever a greedy non-censoring proposer is faced with an IL, they either pick the non-censoring builder block for value $B$ or the artificially full block $A-\delta$, whichever is higher value.

• When NG-proposers make the IL for themselves, they receive $B$ while G-proposers receive $A$

• When NC-proposers make the IL for “the next proposer” (or some other proposer), that proposer is drawn from the distribution of NG-, GC- and GNC-proposers The possible outcomes are:

  • C-proposer making the list for NG-proposer ⇒ $B$, with probability $gc \cdot (1-g)$
  • C-proposer making the list for GNC-proposer ⇒ $A$, with probability $gc \cdot g(1-c)$
  • C-proposer making the list for GC-proposer ⇒ $A$, with probability $gc \cdot gc$
  • NC-proposer making the list for NG-proposer ⇒ $B$, with probability $(1-gc) \cdot (1-g)$
  • NC-proposer making the list for GNC-proposer ⇒ $\max(B, A-\delta)$, with probability $(1-gc) \cdot g(1-c)$
  • NC-proposer making the list for GC-proposer ⇒ $\max(0, A-\delta)$, with probability $(1-gc) \cdot gc$

  The expected payoffs to the IL-bound proposer are:

  • Payoff for NG-proposer ⇒ $B$
  • Payoff for GNC-proposer ⇒ $gc \cdot A+(1-gc) \cdot \max(B, A-\delta) < A$
  • Payoff for GC ⇒ $gc \cdot A + (1-gc) \cdot \max(0, A-\delta) < A$