I was reading through an article on human rationality that I wanted to share because it's relevant to our perceptions of dice. I think all (most?) of us know that you shouldn't count on a hot (or cold) dice stretch to continue, and that we should play for average dice outcomes**. The article is here at Real Clear Science, but here's the money graf:

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Computational rationality is leading to some elegant and surprising explanations of our biases and errors. One early success consistent with this approach was to examine the mathematics of random sequences like coin tosses, but under the assumption that the observer has a limited memory capacity and could only ever see sequences of finite length. A highly counterintuitive mathematical result reveals that, under these conditions, the observer will have to wait longer for some sequences to arise than others – even with a perfectly fair coin.

The upshot is that for a finite set of coin tosses, the sequences we intuitively feel to be less random are precisely the ones that are least likely to occur. Imagine a sliding window that can only “see” four coin tosses at a time (roughly the size of our memory capacity) while going through a series of results – say from 20 coin tosses. The mathematics show that the contents of that window will hold “HHHT” more often than “HHHH” (“H” and “T” stands for for heads and tails). That’s why we think tails will come after three heads in a row when tossing a coin – demonstrating that humans do make sensible use of the information we observe. If we had unlimited memory, however, we would think differently.

For developing players, I think this is a highly useful piece of information - if you can expand the context of how many dice results you're looking at over the course of the game, you can start to deconvolute where you made mistakes from the typical recriminations about dice variance.

** Caveat: unless you lose on average results. Try not to end up in this boat.

@Rytackle, @Mynock Delta - this might be an interesting tidbit to consider if you guys do a Flight Academy on dice probabilities.

The interesting thing is that both the "blame the dice almost always" and "dice almost never matter" groups are equally wrong.

Ideally, yes, you want more than a narrow window of results, but also ideally you do not want to pollute your analysis with results that are irrelevant. To borrow the "sliding window" metaphor, what a player should strive for is the ability to properly position and look at multiple "results windows," spanning the entire game, all at once.

The thing is, variance is often to blame, even when it evens out over the course of the game.

Just a single bad roll can cost an entire game, at the wrong time. And if the dice weight even a little bit in facvour of one player or the other for the first third of the game, it often doesn't matter how much they will swing back the other way eventually, if you lose half your list before your opponent loses a ship, you're still probably screwed.

So, yes, variance evens out, and yes, we see spikes as bias.

But those spikes matter, because the sliding window can't be arbitrarily positioned; dice rolls in this game happen sequentially, and the earlier ones (can and usually do) matter a lot more than the later ones, both from the perspective of single matches and arguably, whole tournaments - you'd far prefer to have positive swing in later matches against better opponents than earlier against an average opponent who is, broadly, exactly that.

It's important to be aware of it, both because it avoids you blaming it where it really wasn't to blame, AND because it allows you to isolate when it was. Because sometimes you do everything right and still get shat on.

3 hours ago, Jeff Wilder said:

The interesting thing is that both the "blame the dice almost always" and "dice almost never matter" groups are equally wrong.

Ideally, yes, you want more than a narrow window of results, but also ideally you do not want to pollute your analysis with results that are irrelevant. To borrow the "sliding window" metaphor, what a player should strive for is the ability to properly position and look at multiple "results windows," spanning the entire game, all at once.

Just to clarify, what kind of player claims dice don’t matter? I haven’t run into this attitude before, and it’s... different.

Also, I think we might be saying the same thing? I think about the arbitrarily broad window where the dice converge to the average (unless I lose the game with average rolls).

Edit: unless you’re referring to the Fickle Green Dice school where you assume your green dice will all be blanks 😂

Great article mate, it goes well with what I usualy say after I finish a game.

"Its statistically proven that I will occasionally lose a game to bad dice...
... but for the most part I kick ****!"

3 hours ago, thespaceinvader said:

The thing is, variance is often to blame, even when it evens out over the course of the game.

Just a single bad roll can cost an entire game, at the wrong time. And if the dice weight even a little bit in facvour of one player or the other for the first third of the game, it often doesn't matter how much they will swing back the other way eventually, if you lose half your list before your opponent loses a ship, you're still probably screwed.

So, yes, variance evens out, and yes, we see spikes as bias.

But those spikes matter, because the sliding window can't be arbitrarily positioned; dice rolls in this game happen sequentially, and the earlier ones (can and usually do) matter a lot more than the later ones, both from the perspective of single matches and arguably, whole tournaments - you'd far prefer to have positive swing in later matches against better opponents than earlier against an average opponent who is, broadly, exactly that.

It's important to be aware of it, both because it avoids you blaming it where it really wasn't to blame, AND because it allows you to isolate when it was. Because sometimes you do everything right and still get shat on.

Also worth noting that variance on red dice has a much larger impact than variance on green dice.

Rolling the low probability 4 hits with no mods has a far greater impact on the game than the 4 evades with no mods. Simply because extra hits are never wasted, where as extra evades are often wasted.

14 minutes ago, Roundy1161 said:

Also worth noting that variance on red dice has a much larger impact than variance on green dice.

Rolling the low probability 4 hits with no mods has a far greater impact on the game than the 4 evades with no mods. Simply because extra hits are never wasted, where as extra evades are often wasted.

Never say never. 4 hits on an A-Wing with no shields that rolls **** wastes a couple of hit results.

This is frankly, why more cards like Heroic need to be in the game.

Like a lot of players I play on odds. I should get x hits etc. but I don’t do something banking on the odds being average.

I have lost Scum Fenn to a range 3 shot from Escape Shuttle Lando, he rolled hit, crit. I rolled 4 blanks. Crit was direct hit. You wouldn’t bank on that happening, but it did, I laughed at the BS odds, but that’s X Wing.

Dice are dice. They hate you no matter who you are 🙂

When some players make the comparison that this is chess with dice, I say NO because the very nature of chess is there is no dice. Not to say you can't have a strategic game with random elements such as cards or dice or a spinner wheel. But yeah for strategy games they seem to get thrown into to categories, chess or other.

3 hours ago, PaulRuddSays said:

Just to clarify, what kind of player claims dice don’t matter? I haven’t run into this attitude before, and it’s... different.

Its generally the idea that critiquing your dice will not help you get better at the game, as ideally you will never be in a position for your green dice to fail you, and you will have enough shots that red die variance won't matter. Personally, it doesn't manifest during the game as much as it does after it. When it does manifest, its more in shutting down any sort of "oh my dice were hot" with "no, they were average" The challenge is to be able to have that as a tool to get better while still "blaming" dice for when something happens during the game so you don't get into a spiral of negativity (or send your opponent into one, for that matter). It also isn't as vocal all the time because it not something you talk about directly when you win, because its not fun to be around.

4 hours ago, PaulRuddSays said:

Just to clarify, what kind of player claims dice don’t matter? I haven’t run into this attitude before, and it’s... different.

You haven't run into players telling you that your dice don't matter, because if you were a good enough player, you wouldn't be depending on them anyway? Lucky you. (No, seriously ... lucky you.)

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Also, I think we might be saying the same thing? I think about the arbitrarily broad window where the dice converge to the average (unless I lose the game with average rolls).

It doesn't actually sound like we're saying quite the same thing. I'm saying that while it's true you shouldn't be looking at arbitrary "windows" of results and drawing variance conclusions from them, it is also not useful to look at the dice for an entire game and draw variance conclusions from that. What a skilled player should be able to do is look at the many different windows of results within the game that actually had an effect on the outcome.

In other words, just for example, you can't get unlucky when you're rolling two attack dice against a five-green-dice target, with focus and evade, so that's not a result that matters. But on the other side, you can certainly get lucky in the same situation -- your opponent rolls five blanks -- and that's a data point to factor in when you consider the overall variance of a game.

Only once a player can skillfully recognize the real effects of variance on a game can that player "filter out" the variance and build a meaningfully accurate perception of how skillfully he or she -- and his or her opponent -- played the game.

Or just say screw green dice and fly the Ghost like I do! Haha

2 hours ago, Jeff Wilder said:

What a skilled player should be able to do is look at the many different windows of results within the game that actually had an effect on the outcome.

I think a useful window is an engagement phase, and whether it changed the game with unexpected results.

To give an example, I had a game recently where we both agreed that I should have lost a ship and his should still be alive but heavily damaged. But due to bad/good luck, mine just survived and his went down.

That had obviously a large effect on the rest of the game.

In another example it took me 3 turns to push a single damage onto a quadjumper. Again, obviously a large effect on the game. Here the window was even larger than a single engagement phase, from my point of view.

All that being said, it is in my opinion and experience revealing to see what kind of player talks about decisions and talks about dice after a game.

6 hours ago, Jeff Wilder said:

You haven't run into players telling you that your dice don't matter, because if you were a good enough player, you wouldn't be depending on them anyway? Lucky you. (No, seriously ... lucky you.)

It doesn't actually sound like we're saying quite the same thing. I'm saying that while it's true you shouldn't be looking at arbitrary "windows" of results and drawing variance conclusions from them, it is also not useful to look at the dice for an entire game and draw variance conclusions from that. What a skilled player should be able to do is look at the many different windows of results within the game that actually had an effect on the outcome.

In other words, just for example, you can't get unlucky when you're rolling two attack dice against a five-green-dice target, with focus and evade, so that's not a result that matters. But on the other side, you can certainly get lucky in the same situation -- your opponent rolls five blanks -- and that's a data point to factor in when you consider the overall variance of a game.

I haven’t run into those players, but I’ve only gotten limited tournament time since I started playing. I’m expecting to get more in 2019, so we’ll see if my luck holds up.

And we may not have been saying quite the same thing, but that may be because I’m bad at writing. I agree with @GreenDragoon that there are typically a few key engagements where it matters much more. A lot of the rest of the game is perception bias where we remember ‘negative variance’, even though it didn’t materially change the game state.

As a personal example - I played a guy in our league night last week that mostly plays Legion, and he brought (Rebel) AAXX, with a smattering of upgrades. I brought three trick shot SFs and Midnight. He jousted in our initial engagement and ended up behind a rock, so I got off three focused trick shots versus one A-Wing... who took no damage at all. In return, one of my SFs had their shields wiped out at R3 by an X-Wing. In the late mid-game, his I3 X-Wing managed to solo Midnight by dropping 3 attacks with natural 3 hits (versus average green dice). In the entire game, my opponent never took a single reposition with any ships. (Reminder: A-WINGS!)

The guy I played is super nice, but he thought it was a really close game. It was not - in a few key windows, he had dice results that put him in the sub-10% probability range. If he thinks about the whole game, he acknowledges hot dice but can’t deconvolute to see where he should have flown better. If he considers only the individual windows that moved the game state materially, then it doesn’t seem as unlikely (a 1/20 chance will probably happen a few times each game when each each player fields 4 ships). A skilled player should be capable of blending the two views and pick engagements where they know they (should) come ahead on average and also don’t get screwed by standard deviation (which I think is almost uniformly ignored when many people talk about averages).

16 hours ago, Captain Lackwit said:

This is frankly, why more cards like Heroic need to be in the game.

I think I've had 5-6 games using Heroic on A and X-wings and incredibly it hasn't triggered once!

It's come up frequently on Vassal when players look at the results in black and white and assume they weren't 'lucky', but if your 3 evades come when they only had 1 hit and you then blank out and die they miss the point of bad results at bad times.

I always try to look for what I could have done better though and those who dwell on dice tend to stagnate in terms of player skill in my opinion.

59 minutes ago, Just Paulos said:

I think I've had 5-6 games using Heroic on A and X-wings and incredibly it hasn't triggered once!

I can't tell if you're being sarcastic or finally having that "Ah hah!" moment when you realize Heroic is actually a terrible card.

14 minutes ago, bydand said:

I can't tell if you're being sarcastic or finally having that "Ah hah!" moment when you realize Heroic is actually a terrible card.

It’s not terrible, it’s situational. Like trick shot, if a bit more extreme an example, considering you can’t really try to roll blanks on purpose. It’s one point because of the situationality of it, but when it comes up and works, it’s well worth that one point.

1 hour ago, Just Paulos said:

I think I've had 5-6 games using Heroic on A and X-wings and incredibly it hasn't triggered once!

That happens. But you'll be glad when it does.

19 hours ago, Roundy1161 said:

Also worth noting that variance on red dice has a much larger impact than variance on green dice.

Rolling the low probability 4 hits with no mods has a far greater impact on the game than the 4 evades with no mods. Simply because extra hits are never wasted, where as extra evades are often wasted.

I thought this was a nice observation. There is of course the exception when you "overkill" a ship with extra hits. In that case the extra hits are wasted.

15 minutes ago, radon86 said:

I thought this was a nice observation. There is of course the exception when you "overkill" a ship with extra hits. In that case the extra hits are wasted.

It’s actually not true that overkill is wasted. It actually benefits your opponent a slight amount, counterintuitively. It’s a variation on the Monty Hall problem, which is legitimately interesting on its own. Highly recommend that you look into it and think really hard about it.

Edit: the corollary to the above is that, if your opponent kills your ship and you could spend a focus/evade/force to reduce the number of cards dealt... don’t.

Filtering the deck is definitely beneficial. It always annoys me when my opponent goes "yeah it's dead" and don't deal the cards.

I want as much info as I can get!

4 hours ago, Roundy1161 said:

Filtering the deck is definitely beneficial. It always annoys me when my opponent goes "yeah it's dead" and don't deal the cards.

I want as much info as I can get!

Not that useful, given that you haven't any control over what is drawn from it and when*

*obviously, in some cases you do, and in those cases, it's potentially useful but the benefit is very rarely worth hassling over.