How probable is extreme (un)luck? A mathematical study

[dwarftough]

After a game you often see chatlines like "I was soo unlucky, -20% inflicted" or "he was too lucky, +10% over the whole game" and so on, and so on. Luck in Wesnoth is a never-ending source of emotional comments. And at some point, the question arose in my head: How much luck/unluck is really too much?

In this study I'll consider the most obvious metric of luck in Wesnoth (with Standard RNG): percentages of inflicted/taken strikes in the "Statistics" menu (highlighted by red).

The statistics we discuss highlighted in red

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I will use a simple model to get reasonable estimates on the probabilities and luck ranges: each real game is a lot more complex but these numbers should at least give you some idea of how it can go.

This forum post is a summary of my results. The full article is presented in an attached pdf and will consist of several parts: basic model and results, discussion and some extra info, and detailed calculations that give the results.

Main Result: Probable luck ranges for units on 50% defense
That would be our simplified model: suppose we have only units fighting on 50% defense (for example, Orcish Grunts on hills). We want to estimate which percents in the stats window (for strikes) we'll see with the probability of, let say, 95%. We also suppose, for simplicity of calculation, that the number of strikes n is enough large. Here is the result:

Main result formula

200 and 95% are easy to remember numbers, but there isn't anything magical about 95%: a similar formula can be derived for other probabilities. For 98% the formula's numerator is 233, for 99% it's 257. Below is the table with some precalculated results for different probabilities and numbers of strikes n.

Table of precalculated probable ranges

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Some conclusions

What are some brief conclusions we can derive from these results and what do I have to say about this?

1. The possible percentage range gets narrower when n rises, but at a slower pace

As you can see from the table, ±20% strikes luck range is achieved (with 95% probability) only at n = 100 strikes (which is a long enough game). And ±10% is reached only at n = 400 strikes (that's an excruciatingly long game)

Don't forget that it's the range that occurs with 95% probability, here goes the second conclusion (let's take n = 100 for example)

2. For n = 100 roughly 1 of 20 games will face more than 20% stats, roughly 1 of 50 games will face more than 23.3%, roughly 1 of 100 games will face more than 25.7%

As small as it may seem, 1% isn't that small probability and such games will appear once in a while. And this 25.7% range is calculated for long enough games (n = 100), if the game is shorter, the range may be bigger than that.

[the_kaygan]

[Vilebeggar]

Calculations aside i want to talk how bad rolls affect a competitive game. If you are generally unlucky in a game that will negatively affect your chances of winning since your units wont be able to hit/dodge. But what is worse in my opinion in unluck in KEY moments, like failing to kill a unit on an important chokehold or an important unit of yours dying despite being on favourable terrain.

It is known that a match between a superior player and an inferior one will most likely result in the superior players victory. However due to the rng and faction matchup factor anyone can beat any player at least once in a wesnoth game. Sometime extreme unluck is enough to settle a game. However a game between 2 even players in my opinion will most likely be decided by the rng or the faction matchup which has been a longstanding issue that frankly cannot be fixed since this game has 6 of them and multiple different maps.

Though some matchups like for example a loyalist vs dwarf or undead vs drake have always been problematic and on certain maps have been borderline unplayable despite the rng. Many attempts have been made by some parties (namely the ladder era guys) to fix the balance issues between factions for a spam of over 20 years that somehow are still considered a work "in progress". New patches of updates are still released frequently that only further mish mash the unit pool and widen the inbalance. To me they are only trying to hide the fact that they have failed.

So yes i would say wesnoth is an imperfect game where luck or the matchup type will get the best of you sometimes but you must play it with a grain of salt.