Update!

While responding to a comment to my previous post, I found an error in my work: the cumulative damage against a fighter with Evade ought to be 1.51, but was listed as 0.92. Fixing this error also changes the interpretation of the data–basically, it explains some things that were previously ambiguous. Most of the changes are in the “The Test”, “Conclusions”, and “Last Words” sections.

I’m embarrassed that I’m issuing such an extensive correction after such a mistake, but it seems intellectually honest to leave the prior post up so people can see where I went wrong. I plead my week-long bout with the flu and my increasingly busy schedule for the oversight… and it is correct this time, I swear.

Introduction

When last we spoke, I was making a mathematical point about the use of Focus over Evade. Conversation has continued, both here at at the BoardGameGeek thread that inspired my original post.

In response to my assertion that Focus is a better choice if you expect to be fired on by two or more opponents, user Theorist made the following point (among others):

I also agree that over 2 defensive rolls, a Focus token provides more eye results than an Evade token. But where I disagree is in what that means in gameplay — an Evade token is used intelligently to knock off 1 damage, but it means less to turn 3 eyes into 3 evades when the attacking ship only rolled 1 strike. Focus is unpredictable, often rolling high at the wrong times or not being usable (42.2%) when needed.

The conclusion is similar. Focus if you have a shot you’ll get to take. Evade if you don’t because it provides better defense.

My response was to offer to run a simulation of multiple X-wings attacking a single TIE fighter, and see which action choice for the TIE resulted in less damage. Read on for the results!

A Brief Note On Method

Here’s what I did using the software package SAS, one million times:

  • Simulate three attack rolls, each using three Attack dice and rerolling blanks and eyeball results, as if three X-wings were attacking with Target Lock at Range 2.
  • Simulate one defense roll using three Agility dice, as if a TIE fighter were defending against the X-wings’ attacks at Range 2.
  • Assign and use tokens:
  • In the “Evade” run, assign an Evade token to the TIE and use it the first time the TIE would take uncanceled damage results.
  • In the “Focus” run, assign a Focus token to the TIE and use it the first time there are uncanceled damage results and there is at least one eyeball result. (That is, it’s not saved for some “intelligent” time–it’s just used the first time it would do any good.)
  • I used the same set of random seeds for both runs, meaning that the dice rolls are identical each time. The only difference between the two simulations is which defensive token the TIE has at the start of the attack.
  • I didn’t stop the simulation when the “TIE” accrued enough damage to be destroyed. Think of this as a situation where all three X-wings have only this TIE in their firing arcs, so any damage they would have dealt beyond 3 is wasted.

Finally, after some prompting by Theorist on the initial version of this post, I verified the simulation results the old-fashioned way–by working out joint distributions and using Bayesian logic. It was a pain, but allowed me to correct an error that was working against my hypothesis….

Expected Results

Based on my prior analysis, Focus should be more effective than Evade at preventing damage from multiple attacks. The difference shouldn’t be huge, but it should be noticeable; I calculated that with three attacks, a Focus token should be worth about 1.2 Evade tokens.

Judging from his prior comments–and he’s welcome to defend himself or correct me if I’m wrong–but Theorist expected Focus to perform worse than Evade because it provides a more consistently effective defense.

The Test

Here are the tables that illustrate the results after each of the three X-wing shots. These are cumulative values, so each row represents the state of the TIE fighter after it’s been shot 1, 2, and 3 times. Mean Damage is the expected number of damage cards dealt to the TIE fighter, Spent Token is the fraction of simulated fighters that have spent their Evade or Focus, and Survival is the number of simulated fighters that are still alive.

First, the ship with Focus:

Table of attack data.

3 Attack + Target Lock, firing three times at 3 Agility + Focus.

And the same ship with Evade instead:

Corrected table of data.

3 Attack + Target Lock, firing three times at 3 Agility + Evade.


And here’s the result of all three shots, in graph form. Each chart represents the distribution of damage for the corresponding simulation run.

Damage distribution

Distribution of damage vs. Focus. Click to enlarge.

Damage distribution.

Distribution of damage vs. Evade. Click to enlarge.

Conclusions

After the second attack, Evade and Focus are essentially equivalent defensive choices with respect to the amount of damage taken. Evade is better by four one-hundredths of a point of damage. That difference is less than 3%, and is so small that it would take hundreds of matches for it to observe a countable difference in play, so I think it’s fair to call them even. Additionally, the table points out another, possibly more powerful piece of evidence: the effect of choosing one token over the other has essentially no impact on the long-term survival of your ship.

Here’s why. On the first attack, Focus is (as predicted) more effective than Evade in some cases–but those cases, where there are both multiple uncanceled damage results and multiple eyeballs on the defensive dice–aren’t very common. (Reanalyzing the data the long way says that they come up about 10% of the time.) In about another 37% of cases, Focus is exactly as effective as Evade, preventing only one damage. Finally, 53% of the time you either don’t need Focus or there aren’t any eyeballs on the defensive dice.

However, when the second attack comes, there’s about a 50% chance that the ship with Focus still has a token to spend–versus about 30% for the ship with Evade. The second attack provides another set of chances for Focus to shine, and if it still doesn’t help you out, don’t worry. There’s a third attack coming, and it’s very unlikely that your wingman with Evade still has his or her token.

I was expecting Focus to pull ahead much more quickly than it did, and I (still) suspect Theorist is right about the reason why: those situations where Focus represents an improvement over Evade are simply too rare. But while I was wrong about the magnitude of the effect, I’m still encouraged by the result. BGG user Piqsid, whose post kicked off this whole idea, posted a smart followup earlier this week:

I don’t think our goal should be to prove that Focus is better than evade. It might be, but it doesn’t need to be. It only needs to be at least the same…

All I want to do is show that Focus is likely to be just as good as evade for defense. I am much more concerned about offense.

I like his point here, and agree with it. There are a number of tactical reasons to use Focus rather than Evade, most of which revolve around the value of offense over defense in X-wing (and I think Theorist would agree…) Given that fact, the question we’re really interested in is whether it’s substantially worse than Evade on defense, and you can consult the simulation results for the answer.

However, I want to guard against over-generalizing from these results. This scenario (3 Attack + action, against 3 Agility + action) is an important test case for Imperial squads, but keep in mind that if your attacker isn’t as effective–fewer dice, or no token to buff the attack–Focus is probably less valuable than it appears here. Similarly, if your ship has better Agility than I’ve assumed–you’re at Range 3 or have a Stealth Device–Focus is better than it looks here.

Last Words

So the bottom line from the previous post needs to be amended. (And, you know, now I’m amending it again.) Evade provides a drastically better defense against the first attack, and it’s a better choice if you’re feeling defensively minded and your TIE or A-wing expects only one incoming attack.

But in any case where your fighter is in multiple firing arcs, you should strongly consider Focus regardless of whether you expect to attack. Focus is always stronger on offense, but after the first attack it has a very similar impact on your defense. Using Focus for defense, rather than Evade, is no less likely to save your life if you’re facing multiple attacks.