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:
The expected payoffs to the IL-bound proposer are: