How effective are monks and how good is Faith?

These questions currently have no good answer for groups of more than 1 monk so I decided to write a program to test it. Apparently monks have a 25% conversion chance, a 1-1.2s conversion interval period, and have independent chances except at the minimum and maximum intervals of 3 intervals and 12 intervals. Beyond those intervals the chance is 0% and 100% respectively (except for scouts/eagles with faith where its like 85% at the max). Other people testing in game have yielded values like 28.4% chance but if you run a binomial distribution simulation with 500 events at a known 25% chance you can get +/-3.4% fairly regularly. In 100 simulations of 500 events each +/-3.4% occurred 6 times. +/- 3% occurred 15 times. So I will assume the hard-coded value of 25% is accurate. Faith cuts this down to 1/3rd the value or 8.33%

The specific question to analyze faith is: "Given a set number of monks, what do the conversion numbers look like over time relative to not having faith?" Turns out it does a lot. To demonstrate here are 2 distributions using 5 monks and 5 targets. These were generated via simulation and conditional binomial distributions (i.e. the distribution at time t is conditional on the distribution at time t-1). Results for both methods were very similar. With no faith and an assumed conversion interval of 1.2s we get:

With Faith this changes to:

For context, it takes ~8 seconds for a husbandry-upgraded paladin from 12 range to first hit a monk. So that is about 6-7 conversion intervals depending on whether the monks click first or not.

The last two columns are the metrics by which cost efficiency can be judged. Either you save some units from being converted or you force your opponent to use more monks. One other benefit is negating all the gold spent on monk improvements should your opponent switch away from using monks. However you also prevent the enemy from obtaining free units.

For example in terms of resources relative to your opponent a converted paladin puts you at a resource swing of -120f and -150g. Lets say you have some paladins in your army and the enemy has 5 monks. You kill the monks in 12 game seconds or 10 conversion intervals.

Without Faith you're expected to lose 4.33 paladins for a resource swing of -520f and -150g.

With Faith you're expected to lose 1.76 for a resource swing of -211.2 and +236g. Total savings relative to baseline: +308f and +386g.

For reference, with Heresy you are expected to lose 4.33 but they die for a swing of -260f and +175g. Total savings from Heresy: +260f and +325g.

With both Faith and Heresy 1.76 paladins die for a swing of -105.6f and +368g. Total savings relative to baseline: +414.4f and +518g

Also these are the minimum savings. Since the net resource trade from a battle is non-linear (see Lanchester's laws), by maintaining your units you save resources in the ensuing battle as well.

In terms of the expected conversion time for a unit: Without Faith the mean time to convert time is 5.66 intervals, rounded up to 6. With Faith this becomes 11.77 intervals, rounded up to 12. So Faith doubles the expected time before a conversion occurs, which gives you double the time to kill the monks or move their target. The relevant median times are 6 and 13 which is even better.

For discussion: In terms of strategies, have people been underestimating Faith? It seems to have a return better than Heresy both in terms of the conversions and in terms of the ensuing battle (because you have a higher ratio) but unfortunately comes much later. Allowing twice as much time before conversion makes this even better because the losses happen later.

Anyway just thought I would put this information here since I found it interesting.