How Monks Really Work

Article written by MrFixitOnline Staff with additions from Neilkaz and AndyHRE (Hosted with permission from MrFixitOnline)
Published on 09-18-2008; updated on 08-17-2014
Tags: Statistics, Hidden Mechanics

Introduction

With the release of two expert Castle Age strategies featuring Monks (the Sheriff’s SMUSH and Crexis and Methos_ST’s C&M Express), these preacher/healers have suddenly become all the rage. But how exactly do they work? What little is known of how the work comes from the kind posting of Ensemble Studios’ ES_Sandyman on AOKH. This post, combined with players’ anecdotal evidence, has come to form the generally accepted monk theory, and has provided the backbone to most strategic thinking of how monks work.

However, MFO's empirical testing has revealed that Monks operate very differently than expected. Monks are generally slower and less likely to convert than has been revealed to date, and units with conversion bonuses are extremely difficult to convert in anything close to a reasonable timeframe.

Some Quick Conclusions

  • No monk will convert before about 5 to 5.25 seconds.
  • Using more than one monk will improve your chances a lot, once you pass the 5 second threshold.
  • When facing scouts, do not use monks. And conversely, when facing monks, crank out the Scouts.
In the rest of this article, the MFO staff will share Neilkaz’s conclusions about how monks really work, give you the details on converting regular and resistant units, and share some strategic conclusions. After that, we provide some hopes for further research and then let you get down and dirty with the data in our appendices.

How Monks Really Work

  • All unit conversions take between approximately 5 and 15 seconds. Buildings take much longer.
  • It takes between 0.18 and 0.3 seconds for a Monk to start up the conversion process.
  • The conversion process is measured in 1.22 second intervals, plus or minus 0.04 seconds.
  • Four Intervals (5 to 5.25 seconds) must pass before the monk can convert a unit. Contrary to previously published reports, the monk makes only one conversion attempt (not four) during this five second period.
  • There are no partial-interval conversions.
  • After twelve intervals (i.e., about 15 seconds) unconverted units are always converted.
  • Units without any conversion resistance have about a 28.4% chance of being converted on each attempt. Standard rules of probability apply to adding additional monks.
  • Scouts and Light Cavalry, which have a single level of conversion resistance, have only a 9.55% percent chance of being converted in each interval, making it very difficult to convert a scout/light cav in any reasonable time.

The Details: Normal Units

Conv. Interval Time (Seconds) Conv. % (1 Monk) Conv. % (2 Monks) Conv. % (3 Monks)
Start Up .18 0 0 0
1 1.40 0 0 0
2 2.62 0 0 0
3 3.84 0 0 0
4 5.06 26.9 48.7 63.3
5 6.28 48.1 73.7 86.5
6 7.50 63.1 86.5 95.1
7 8.72 74.2 93.1 98.2
8 9.94 81.4 96.5 99.3
9 11.16 86.1 98.2 99.8
10 12.38 90.8 99.1 99.9
11 13.60 94.0 99.5 100.0
12 14.82 100.0 100.0 100.0

Note: The 2 and 3 Monk Columns are theoretical results subject to further testing

The Details: Scouts/Light Cavalry

Scouts and Light Cavalry are more resistant to conversion than normal units. We ran over a 1000 trials and discovered that after a .30 second start up, scouts are converted in the same intervals as normal units, however they do take longer to convert. But despite the scout’s conversion resistance, scouts are always converted after approximately fifteen seconds, just like regular units.

Conv. Interval Time (Seconds) Conv. % (1 Monk) Conv. % (2 Monks) Conv. % (3 Monks)
Start Up .30 0 0 0
1 1.52 0 0 0
2.74 0 0 0
3 3.96 0 0 0
4 5.18 9.8 18.2 26.0
5 6.40 18.5 33.1 45.2
6 7.62 25.0 45.2 59.5
7 8.84 33.7 55.2 70.0
8 10.06 40.4 63.3 77.8
9 11.28 45.3 70.0 83.6
10 12.50 49.6 75.5 87.8
11 13.72 54.3 79.9 91.0
12 14.94 100.0 100.0 100.0

Note: The 2 and 3 Monk Columns are theoretical results subject to further testing

Strategic Consequences

Assuming the theoretical assumptions for multiple monks hold true, then for regular units, doubling up your monks gives you a pretty big boost. For example, after 5.06 seconds a single monk has a 28.4% chance to convert a unit, while a two-monk effort has a 48.7% chance. You also get to the 90% probability threshold much faster, in 8.72 seconds instead of 12.38 seconds. A third monk boosts the first conversion opportunity to 63% and reaches 90% in 7.50 seconds. However, adding further monks quickly yields diminishing returns.

For Scouts, the results are less clear. Consider a single monk. For a regular unit, the monk needs 6.28 seconds to get about a 50% chance of conversion. Against a scout, however, a single Monk needs 12.5 seconds to reach a 50% chance of conversion. How often will a Monk get 12.5 seconds of uninterrupted chanting? So adding a second monk clearly helps (it takes two monks 8.84 seconds to pass 50%).

On the other hand, adding a second monk does nothing to speed your time to a 90% chance of conversion. In either case, you will need the full fifteen seconds before you get over 90% (via the automatic conversion feature).

Using up to four monks seems necessary to get any sort of high probability in a reasonable time, and even five monks seem to take forever to reach a 90% chance (13.72 seconds). So if you need certainty against a scout, you really need to amass some monastic muscle. However, if your monks are protected for the full fifteen seconds, you wouldn’t need to double your monks, because they will automatic convert after 15 seconds.

The real lesson might be: When facing scouts, do not use monks. And conversely, when facing monks, crank out the Scouts.

More Research to Follow

Is this the counter to the Saracen Monk Rush? MFO hopes to tell you soon. We’ll be sending Neilkaz back into his windowless room for a new series of tests on Monks vs. Scouts in realistic settings to see if the lessons given here hold true in the heat of battle.

Currently our conclusions for multiple monks are still based on some untested theory, so Neilkaz will have to turn to whether doubling up is really as powerful as it looks.

And, if we can pry the resignation letter out of his hands and toss it into the trash, we’ll see if he dares test how a Scout with Faith (i.e. two bonuses) behaves. No telling what new Myths will be shattered in the future.

Appendices

Monk Conversions according to Ensemble Studios

ES illuminated the theoretical workings of Monks and conversion in a post made on Age of Kings Heaven in early January:
Q) Can you supply us with the exact formula concerning a monk’s single conversion chance?
A) He has a 33% chance per second. However, he MUST chant for at least 4 seconds (15 seconds vs. a building) before conversion. In effect, this means he frequently gets the guy in 4 seconds, because the chance to succeed is being checked during the first few seconds and if it works, after the 4-second time limit is up, he gets the conversion immediately.

Q) Also how does faith and any other civ bonuses affect conversions as well as how much harder are specific units with a built in resistance to convert?
A) Faith, being a Teuton, and being a light cav/scout all affect your chances of conversion by halving them.

Q) What is the formula for determining conversion chances when using multiple monks?
A) Each monk checks for success independently, but there is still a 4-second time limit.
Based on ES’s post, it is possible to compute theoretical conversion times. In fact, there is a fairly easy formula for determining the probability of a conversion when "m" monks have "s" seconds to get the job done, and each monk has "p" chance of conversion each second:
Prob = 1 - [(1-p)^(m*s)]
Everything in the square brackets can be described as the probability that all m monks fail to convert in each of their s tries given a p chance per try. Therefore, the complement (i.e., 1 minus that number) is the chance that this does not happen, i.e. the chance that at least one of the m monks succeeds in at least one of the s tries, and that’s what really matters when you’re trying to convert.
For example, with m=1 monk with a p= 0.33 chance of converting with s=6 seconds of chanting, you will have a little less than a 91% chance of converting a unit.
Of course, since the term to the right of the minus sign will always be less than 1, there is always a chance that all the monks will all fail. However, as the number of monks and the time dedicated increases, the chances of conversion grow.

One neat thing about the formula that comes out of ES’s description is that, except for the initial four-second period, having m monks try for s seconds has the same effect as having s monks try for m seconds. In other words, five monks have the same probability of conversion over six seconds as six monks over five seconds. Therefore, it makes some sense to think in terms of the Monk-Sec, equal to the work of one monk in one second. However, because each monk needs four seconds of prep time before any conversion is possible, Monk-Sec analysis requires a caveat that monks and seconds are only interchangeable if the number of seconds exceeds three. So while five monks for six seconds and six monks for five seconds both provide 30 monk-secs of benefit, two monks for two seconds are not the equivalent of one monk for four seconds. In fact, two monks for two seconds are actually ZERO monk-secs, since according to ES the monk’s conversion is on-hold until four seconds is reached.

Resistance according to ES

Based on ES’s description, there are three special categories, each which confers 50% resistance to conversion, each of which is cumulative. These are Light Cavalry (and Scout) units, Units with Faith, or Units in a team game with a Teutonic Ally. Since each by itself confers 50% bonus, the "p" value for converting a unit with one of these bonuses drops from 0.33 to 0.165. If the unit has two such bonuses (for example, a light cavalry unit with faith), the p value is cut in half again, to 0.0825. Finally, if the unit has all three bonuses, the p value is one-eighth the size of a basic unit’s, shrinking to 0.04125.

Debunking the Myths

However, MFO testing has revealed that much of the theory of monks is a myth.

MFO testing led by our crack empiricist, Neilkaz, reveals that, as ES says, Monks never convert before four conversion intervals elapse. Other than that, though, much of the generally accept theory of monks is less than perfectly true.

Myth One: The basic time period per conversion is 1 second.

According to Neilkaz, it takes approximately 0.18 seconds for a Monk to even contemplate beginning the conversion process (0.3 for a unit with conversion resistance). Neilkaz suspects this approx. 0.18 time may represent it takes for converted units to change color after the conversion is successful.

Once this overhead time is incurred, then a conversion attempt general occurs every 1.22 seconds (+/- .04 seconds).

Therefore, it makes sense to think of a Monk-Sec as really being a Monk-Interval, where each interval is approximately 1.22 seconds long and the first interval is slightly longer, at 1.40 seconds, to account for the 0.18 start-up cost. Nevertheless, for the rest of this analysis, we will use the term Monk-Sec, with the understanding that the seconds used in the calculate of Monk-Secs is approximately 1.22 seconds long.

Throughout the rest of this article, the expression "Conversion Interval" will be used to describe this approximately 1.22-second interval.

Myth Two: Monks attempt four conversions during the initial warm-up period, holding their results in reserve.

Based on MFO testing, this appears to be false. The probability of conversion after four monk-intervals vs. the probability of conversion in five or six monk-intervals makes is consistent with the assumption that the first three intervals are a simple warm-up period, and that there is only one check made during the fourth Conversion Interval. Thus, a single monk chanting for four Conversion Intervals makes one conversion attempt, chanting for five CI makes two conversion attempts, etc.

Myth Three: Monks have a basic rate of p = 0.33.

After shattering myths one and two, the model for a conversion looks like this: a short initiation period, three warm-up intervals with no conversion attempt, and then a fourth interval, during which the Monk makes his first conversion attempt. If p=0.33, then over the long run (simulated here with one thousand tests), we’d expect about 330 conversions after four intervals. Similarly, if 100 monks were to try to convert 100 units, we expect about 33 to get converted after four conversion intervals. MFO performed thousands of tests on normal units and, on average, 28.4% of the units were converted after four intervals. Over this many trials, the probability of p being 0.33 and still having only 28% convert is effectively zero.

Based on this, MFO is of the opinion that the correct p value is 0.28.4, not 0.33.

Myth Four: Each additional conversion attempt made by a monk is at the same basic rate.

In truth, from Neilkaz’s testing shown above, it appears that for a basic unit, each additional conversion attempt after the first has another 28.4% chance of succeeding, until the ninth attempt. While the standard assumption would imply that 12 Conversion Intervals should result in a 95% chance of success, in one thousand tests, every unit was converted by the 12th Conversion Interval.

Therefore, at least for basic units, it appears that all unconverted units will be automatically converted in the 12th CI. This would have to be a special condition written in the code, which ES should be able to confirm. We welcome any input from our friends at Ensemble!

Myth Five: Units with conversion bonuses decrease the probability of an individual conversion by 50%

ES said that having any one conversion bonus would halve the probability of conversion. Based on the MFO revisions to other commonly held myths, we would expect the first conversion to occur after 4 CI, and to happen with approximately 14.2% probability (i.e. half of 28.4%). We’d then expect to see a steady increase consistent with the 14.2% base probability.

In truth, we do not. Neilkaz got out his Scenario Editor and tried twenty iterations of 50 monks versus 50 scouts. The recent notoriety of The_Sheriff’s Saracen Monk Rush increased our interest in these tests. The results were very revealing.

We see a pattern much more in line with an approximately 9.55% probability. This means that Scouts are just about three times as resistant to conversion as a regular unit. In fact, over the course of 8 CI (about fourteen seconds), a single Monk rates only about a 55.2% chance of converting a Scout. That’s a long time to chant for slightly better than even odds.

One small note: For some reason, with Scouts, the initial interval, previously measured as about 0.18 seconds, came in at a larger 0.30 seconds. MFO has no strong suspicions for what causes this.

Mathematical Formulas

For the mathematically inclined, the impact of correcting all of these misstatements is that the original formula needs to be written as:
For s<9,
Prob = 1 - [(1-p)^(m*s)], where s = Maximum [# of conversion intervals – 3, 0];
For s>=9,
Prob =1.
For regular units, p=28.4% (approx.)
For scouts, p=9.55% (approx.)
If we define the "Sec" in Monk Sec to mean the number of CI minus three (or zero if the number is negative), then the formula remains the same. In fact, this makes the Monk-Sec concept even easier, since if it takes one monk six attempts (i.e. 9 CI, including three that do no count as conversion attempts), it takes two monks three attempts or three monks two attempts. If we define the "sec" in Monk-Sec to mean all conversion intervals after the first three, then you can really divide Monk-Secs by the number of monks and get the correct number of Secs (i.e. those after the first three) to reach 90% certainty.

Based on all of the above, the correct 90% Conversion Rate is:

Number of Conversion Bonuses   90% Conv Rate (seconds)
Zero   7
One   24*
Two   ?**
Three   ?**

* Note that any number of monks will always convert after 12 CI, so the 24 Monk-Sec number only applies with three or more monks.

** To be completed in a follow-up article.

Testing Methodology

Here is Neilkaz’s testing methodology, direct from the Master Empiricist himself:

"I spent most of the afternoon running 20 trials of a scenario where I have 50 monks converting 50 targets. I pause immediately and assign each monk its respective victim and then hit HC to start a villager before un-pausing. This allows me to time very accurately in quarter-second intervals based upon how far along the villager is to completion. I am also able to pause on slow speed a couple times during each quarter second interval to help me get a more accurate feel for what is happening.

In 20 trials of 50 monks it was very clear that any monk selected at the same time as other clergymen would convert at the same time. In all my trials there were only nine points in time at which conversions occurred. Thus it's clear that all monks selected at the same time will have the same intervals of chance for a conversion to occur or not. In other words, there were never any partial-interval conversions.

This is interesting because I am certain that the intervals between conversion attempts varied slightly across my tests, perhaps due to the long chanting rituals. I am certain of this because in a couple trials I achieved my first group of conversions prior to five seconds passing. In most trials it was a bit after five seconds and in one it took almost 5.25 sec.

After nine intervals have passed any unconverted units were always converted. This end time varied from a couple trials of about 14.6 sec to a few of almost 15.0 sec."