
Using loaded dice to cheat at Settlers of Catan, and p-values to avoid suspicion - jackpirate
https://izbicki.me/blog/how-to-cheat-at-settlers-of-catan-by-loading-the-dice-and-prove-it-with-p-values.html
======
FabHK
Very nice hack and analysis thereof.

I do take issue with this statement, though, in particular the part
highlighted (by me):

> "it’s impossible for your opponents to scientifically prove that you’re
> cheating. _This impossibility is due to methodological defects in the
> current state of scientific practice_ , and we’ll highlight some ongoing
> work to fix these defects."

The impossibility is not due to methodological defects in the current state of
scientific practice. It's intrinsic to the world. If you cheat only a bit, and
only rarely, it might be impossible to detect it statistically.

Of course, there are problems with the current state of scientific practice
and the mindless application of statistics (particularly the 5% significance
level). The replicability crisis in psychology comes to mind.

However, I think statistics is unfairly maligned here. Statistics mostly
delivers on its promise, it just promises less than what some people think.

The world is messy, and statistics can help us to make sense of it, but it can
only do so much. In particular, it's not magic. You need many observations to
make statistical statements, and more observations to make more precise
statements (or statements at a higher significance level). That's intrinsic to
how the world is. It's not a problem with "methodological defects in the
current state of scientific practice". And there aren't simple solutions "to
fix these defects".

(Having said that, the article is great. Many proposals (some of them
mentioned in the article), such as a standard 0.5% significance level, more
Bayes, awareness of power in addition to significance, avoiding p-hacking,
etc., are important and useful steps. But they still won't allow your opponent
to prove that you were cheating :-)

~~~
godelski
> The impossibility is not due to methodological defects in the current state
> of scientific practice. It's intrinsic to the world. If you cheat only a
> bit, and only rarely, it might be impossible to detect it statistically.

Not if you take enough samples. Just have to do it enough to reduce the noise.

But to your main point, yeah, in a game of Catan you can just get unlucky. And
I think a fair amount of people claim that Catan dice aren't fair in the first
place.

Also, did they test the unweighted dice? To understand if there was an
inherent and unknown bias in their dice to start with? If they did, I missed
that part, and they just assumed their dice were fair to begin with.

~~~
cortesoft
I don’t think you are disagreeing with the person you are responding to. They
are trying to say the problem isn’t necessarily the ‘scientific process’, it
is that if you cheat rarely enough you might not give your opponent enough
samples to prove it. Your point that if you had enough samples you could prove
it is actually agreeing with them.

~~~
godelski
Oh, I'm definitely agreeing with the person. I was just being a little snarky
at that one statement.

I am also trying to ask a question that is extremely relevant to the author's
"study".

"Are the original dice fair?"

That was never asked, it was assumed. It is a pretty hefty assumption too. I
think everyone that has played a game of Catan has, at some point, questioned
the fairness of the dice. I'm not saying they are unfair, but because of
manufacturing methods, it is quite possible to get a pair of dice with an ever
so slight bias.

ALWAYS ALWAYS ALWAYS find the bias in the instruments you are using to perform
an experiment. THEY ALL HAVE THEM. NEVER NEVER NEVER assume your instruments
are accurate without first verifying.

~~~
LeifCarrotson
> I think everyone that has played a game of Catan has, at some point,
> questioned the fairness of the dice.

This is why a lot of people play with a set of 36 cards (one 2, two 3's, three
4's...one 12). There really aren't that many rolls in a game, and there's a
lot of variation that may result in a 5 never coming up, which adds more
chance to the game than I like. Cards mean that while the order is random, you
get each number in the deck eventually.

~~~
godelski
I've never heard of anyone playing like that before actually. And it seems to
completely mess with the statistical nature of the game. Even if you shuffled
them each time, shuffling methods are bad and you probably wouldn't get a nice
normal distribution. Which fair dice produce.

~~~
philipov
If you don't shuffle until you run out of cards, you mess with the
instantaneous probability of any particular draw, but you hardcode the
distribution. You are guaranteed to get the distribution you put in the deck,
and I think that's a better measure of fairness than having a constant
instantaneous probability.

As the deck becomes smaller, your ability to predict the next draw scales with
your skill at counting cards. I think that's a good feature for a random
number generator to have, as it's what allows games like poker and blackjack
to be about more than simply getting lucky.

~~~
godelski
Well, I agree that a game needs to have more than luck. But something like
blackjack is a horrible example because there's a lot of skill that can be
applied even without counting cards. Not only that, but blackjack still has a
large amount of luck involved with it even when you are highly skilled.

Personally, I find that the best games are ones that have a good balance of
luck and skill. Too much luck and there is no skill, you might as well play a
slot machine (some people enjoy games candy land). Too much skill, and winning
distributions become too low. A highly skilled player will ALWAYS win, and
creates too high of a barrier to entry (connect 4 or dots and squares are an
example here). There are (a few) exceptions to the latter like Go, which has
so many possible moves that you might as well have an element of randomness,
but there is still a steep learning curve.

Catan is one of my favorite games introduction to Euro Games, because the
learning curve is low, and there is enough luck that an intelligent novice can
win. _I don 't actually believe the dice are unfair_, but the low number of
rolls makes each game different. This means the skilled player needs to be
highly adaptive to the changing environment.

Dice create a nice normal distribution that are independent. While over a
large number of games, 6 and 8 are great choices, there will be games where
you just don't do well (they are rare). By not shuffling the cards, you are
creating a flat distribution and really removing the vast majority of luck in
the game (you still have luck in the order of the cards, order of placement,
and order of turn). You now have a dependent probability function, and I think
you could make great arguments that you remove all the things that (I believe,
and laid out above) make the game great. I think you could also make arguments
that the setup is the most important part of the game (when your cards are
dependent events). But it is a game, and these are just opinions.

------
empath75
> To measure this bias, my wife and I spent the next 7 days rolling dice while
> eating dinner.

Some people have very different relationships with their spouses than I do.

~~~
colemannugent
I thought that was weird as well. They only had to do like 100 rolls to have
enough data to draw conclusions from.

~~~
cdancette
100 rolls is not enough. If you keep the same ratio, you can compute the
p-value and I'm sure it will be way over 0.05.

------
TimMurnaghan
The other clue players can use to detect cheating is your behaviour. They can
notice that you are favouring the higher numbers. This is also what can catch
people in gambling games. The betting patterns of card counters are noticeable
- and the eudaemonic folks who found how to predict tilted roulette tables bet
on sectors of the wheel - which again is a distinctive pattern.

So it's pretty much always those behavioural cues that other people will use
to suspect cheating rather than the p-value.

~~~
toblender
If I'm playing with someone that is making AK47's into utensils, I would think
twice about raising my concerns regarding the loaded dice.

Hmm the dice are behaving funny, but that's fine, everything is just fine...

[https://izbicki.me/blog/turning-an-ak-47-into-a-serving-
ladl...](https://izbicki.me/blog/turning-an-ak-47-into-a-serving-ladle.html)

~~~
logfromblammo
Ah yes, the "let the Wookiee win" strategy. You lose all the games, but you do
get to keep playing with both your arms.

I tried strategically throwing games a long time ago. It backfired after I
conceded defeat, and rather than bask in victory, the "winning" player gave me
a narrowed-eyes suspicious look and flipped over my hidden cards. They made it
obvious that I would have won already if I had made the optimal play on my
previous turn. She was _pissed_ , because even though she won the game, she
didn't _beat me_ , because I wasn't even playing the same game as everyone
else. We have some _seriously competitive_ people in my family, and we had a
good argument about whether one was _required_ to make a guaranteed winning
move if it was possible to do so.

------
dep_b
The dice should have been tested before changing them. They might haven't been
perfect dice to begin with. Also the lack of automatization is disturbing. I
find it shocking that no robot arm or pattern recognition was used to do the
tests. Poor wife indeed! "If only my husband was a bigger geek" she must've
thought.

~~~
Simon_says
He got his wife to do it. That's the original automation.

------
kd5bjo
By this analysis, you can average between 5 and 15 more resources over the
course of the game vs unloaded dice, and then goes on to show that there
aren't enough dice rolls in a game to prove the dice are loaded.

The problem here is that you don't get the expected number of resources in
every game, and there's no analysis of the variance. I suspect that the result
of insignificance is correct, in that this "cheat" provides such a slight
advantage that it won't materially affect a single game. By the time you've
played enough games to reliably use the advantage, your opponents will have
seen enough die rolls to show their bias.

~~~
pc86
> _By the time you 've played enough games to reliably use the advantage, your
> opponents will have seen enough die rolls to show their bias._

I have a hard time believing that after 100 games someone would say "you know
what, it appears to me that sixes have been rolled slightly more over the
previous 100 games than I would expect" and an even harder time believing the
next logical would be "you must have altered the dice!"

~~~
adrianN
Looks like you only play boardgames with casuals.

~~~
logfromblammo
If you're OP, you keep a running mental tally of other player's resource
cards, so it is immediately obvious which rolls have been coming up more often
than chance, and which ones have been coming up less.

There are 21 different ways to load 2 dice with one weighted face each. Here's
a table showing which rolls are advantaged or disadvantaged, according to
which faces are weighted.

    
    
           02 03 04 05 06 07 08 09 10 11 12
      1,1   <  <  <  <  <  <  >  >  >  >  >
      1,2   <  <  <  <  <  =  >  >  >  >  >
      1,3   <  <  <  <  <  =  >  >  >  >  >
      1,4   <  <  <  <  <  =  >  >  <  >  >
      1,5   <  <  =  =  <  =  >  =  =  <  >
      1,6   <  =  =  =  =  >  =  =  =  =  <
      2,2   =  <  <  <  =  <  =  >  >  >  =
      2,3   =  <  <  <  <  =  <  >  >  >  =  <<< GOOD
      2,4   =  <  =  <  >  =  >  >  =  >  =
      2,5   =  =  <  =  =  >  =  =  <  =  =
      3,3   =  =  <  =  >  <  >  =  >  =  =
      3,4   =  =  =  =  <  >  <  =  =  =  =  <<< BEST
      4,4   =  =  >  =  >  <  >  =  <  =  =
      5,2*  =  =  <  =  =  >  =  =  <  =  =
      5,3   =  >  =  >  >  =  >  <  =  <  =
      5,4   =  >  >  >  <  =  <  <  <  <  =  <<< GOOD
      5,5   =  >  >  >  =  <  =  <  <  <  <
      6,1*  <  =  =  =  =  >  =  =  =  =  <
      6,2   >  <  =  =  >  =  <  =  =  <  <
      6,3   >  >  <  >  >  =  <  <  <  <  <
      6,4   >  >  >  >  >  =  <  <  <  <  <
      6,5   >  >  >  >  >  =  <  <  <  <  <
      6,6   >  >  >  >  >  <  <  <  <  <  <
    

* duplicated to show symmetry

So for this game, I'd probably weight 3 and 4, to advantage 7s over 6s and 8s,
and build more on the 5s and 9s. Most players that have no knowledge of the
load of the dice will prefer to have at least one settlement on a 6 or 8. The
2,3 and 5,4 pairs also disadvantage 6 and 8, with bias for higher or lower
numbers.

------
smallnamespace
Note that if your _only_ goal was to test whether the dice was loaded, using a
test against only one outcome (in this case, only 6s) is not the best way.

Use either a Kolmogorov-Smirnov test[1] or the Anderson-Darling test[2].

The intuition is that these tests are more powerful because they take the
difference between the entire empirical distribution minus the expected
probability mass distribution. You're using 'all the numbers' simultaneously
to check for cheating.

Funnily enough, I first learned about these tests a nearly a decade ago,
precisely because I wanted to know whether Settlers dice were loaded.

[1]
[https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_tes...](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test)

[2]
[https://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test](https://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test)

~~~
FabHK
I think for this case (with 6 discrete outcomes) you could just use Pearson's
Chi-Square test. What you describe is general enough for continuous
distributions.

[https://en.wikipedia.org/wiki/Pearson%27s_chi-
squared_test](https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test)

------
hinkley
My old boss was into Settlers, and they stayed late to play some games a few
times. I got invited once and weird stuff happened.

Somehow I ended up next to both 6's on the map.

As near as I can tell, one of the dice we used would not land on 4. Whatever
was going on, the number of 6's and 9's rolled that night were abnormally
high. By the time people figured out we were rolling as many 6's as we had 7's
and 8's combined, I already had a town on the coast and pretty much
steamrolled everybody with sheep and wheat.

Really the game wasn't that fun at that point, so I just tried to end it as
fast as possible.

When you're playing an RPG, everybody seems to gravitate toward their 'lucky
dice' which are most likely defective in the right way. But if you're playing
a game with others you probably want the fairest dice you can find.

I remember years ago seeing a sales video from some retired aerospace engineer
that was making geometrically perfect dice. He'd worked out the resins so they
cured uniformly. He'd stack his dice next to some random set and point out how
the other guy's stack curved to one side while his was perfectly straight.

Probably didn't sell a lot of those to DND players, except perhaps GMs.

~~~
vanderZwan
> Probably didn't sell a lot of those to DND players, except perhaps GMs.

GM's don't have to show their dice rolls and can overrule the outcome if it
fits their storyline better, so the rolling is mostly ceremonial anyway.

------
logfromblammo
I'd be interested to see the results by loading the 1-2 edge of the die, or
even the 1-2-3 corner. The former should result in more 5s and 6s, and the
latter more 4s, 5s, and 6s. And since the weight would be not centered on a
face, would make the die more likely to tumble over to a preferred face if the
die had to shed its sideways momentum. The unbalanced die would prefer to
tumble or skip when a non-preferred face shows, and slide or rock when a
preferred face shows.

The problem, of course, is that there is no way to prevent your opponent from
accidentally benefiting from your cheating. If you have biased the dice to
favor 8s over 6s, they might just plop down on an 8 before you can place your
settlements. You would have to use two sets of loaded dice, one biased high,
and the other biased low, and swap them out after first settlements are
placed.

Also, in my experience, the other players will still accuse you of cheating
whether they can prove it scientifically or not, because they prefer to
believe that you are better than they are at being sneaky and underhanded than
better at honest strategy.

For instance, Clue (aka Cluedo) is a game where one skilled player can
repeatedly curbstomp lesser skilled players. If you do, they then grab your
player sheet to look at all the strange and indecipherable symbols you have on
it that are not simple Xs, and they accuse you of cheating. You can either
explain the mathematical advantage of the additional information you record,
lose all future advantage against those players, and get accused of "breaking
the game", or you can remain silent and still not be able to play it again
because you're a "cheater".

Besides that, what self-respecting board gamer doesn't just leave the crappy
wooden dice in the box, and use the good dice from their dice bag?

I like this analysis, though. It reminds me of the guy who designed a one-
sided die.

------
cdancette
This post explains in a very simple way the scientific method (p-values) to
prove that a result is significant or not, and the problems with it.

It's crazy that by using a p-value of 0.05, it means that 5% of all scientific
results might be false.

~~~
mehrdadn
> It's crazy that by using a p-value of 0.05, it means that 5% of all
> scientific results might be false.

That would only be the case if scientists were robots who immediately
published anything with a p-value up to 0.05. They're not, though. If they get
clearly nonsensical results, they will obviously re-evaluate it. In other
words, the p-value doesn't incorporate the fact that the experiment passed
sanity checks in your own head (and the reviewers') before it was published.
(And yes, there are bad actors in every field who game the system, but my
point still stands.)

~~~
adrianN
From what I've heard on HN, scientists are actually robots who massage their
data until they get a p-value <0.05 and then immediately publish.

~~~
nonbel
No, the publishing process takes a long time. Sometimes it could be years.

~~~
adrianN
Yeah, but in Russia they use nine women to produce a baby in only one month.

~~~
nonbel
This sounds like some kind of comment about divisibility of the work to
publish something. I don't get it though.

After the bulk of the paper is written, I can easily proofread, typeset, etc
everything myself in less than a week. Now get someone else to double check
that. Lets say that is another week.

After that the only thing is to get someone worthwhile to spend some time on
your paper and point out anything confusing or erroneous. Granted, this could
take a month or so of study. However, I never really saw that happen in
practice. In reality you would be lucky to get people to glance over it one
evening.

So what is taking so long?

~~~
adrianN
In my experience a significant fraction of the time it takes to publish a
paper is spent waiting for the journal. During that time you can do other
useful research. The long delay between submitting, getting through the
reviewers and the actual publication is one of the reasons why for example in
CS a lot of the interesting stuff happens in conference publications with fast
turnarounds and the journal versions of the same paper appear a year or two
later.

~~~
nonbel
>"waiting for the journal"

Yes, what are they doing?

------
jimrandomh
This is a neat project! The statistical test used could be a lot more
sensitive, by taking advantage of information about the physics of biased
dice. They test whether 6 is rolled more often than it should be relative to
the other 5 numbers, but they should instead look at whether it's rolled more
often relative to 1. Since these are on opposite sides of the die, and
unfairness in 6-sided dice comes from uneven distribution of weight, these go
together; you get to combine the information from too-many 6s and too-few 1s
and get a test that needs only about half as many rolls to reach a given
confidence level.

~~~
jackpirate
I'm the author, and I totally agree!

Originally, I was going to point out other null hypotheses that we could try
to reject. This has the advantage that some of them (like the one you propose)
model the physics. And it also has the disadvantage that if we propose too
many hypotheses to test, then we will be more likely to get false positives.
But the article was already too long, and this is a HUGE can of worms to open.

------
rando444
I really enjoy these fun thought experiments.

The only thing I didn't see clarified though is whether the die was loaded
using the water only once, or whether they re-loaded the die every night
during the week they were testing.

~~~
jackpirate
Hi, author here. I loaded the dice only once. The weighed 0.65 ounces before
hand, and 0.75 ounces afterward. They kept that weight the whole week.

~~~
nonbel
So a suspicious player could detect the loaded dice simply by weighing them?
That actually makes a lot of sense. If you look into actual dice/coin
manufacturing they do not run tests like "roll a sample of dice a million
times each, then check p-value". Instead they have a clever manufacturing
process with stringent tolerances. If they run such tests at all it would be
to find the upper bound on deviation from fairness over the lifetime of the
dice (eg 100k rolls).

------
tzs
I wonder if there are any good, simple procedures that could be applied during
a game to mitigate against biased dice (without giving up on physical dice)?

For games that only use one die, perhaps something like generating a random
number in 1 <= r <= 6n[1], using some procedure that all the players have
input to so they can all agree to trust the number, and then shifting all die
roll results by that number? So if the number were 3 and the die came up 2, it
would count as 5.

That would not remove the bias from the die, but it would move it to a
different number. For games where some numbers are consistently good for you
and some are consistently bad for you that would be enough to make it so that
biasing the die does not work in the long run--some games it would end up in
your favor and some it would end up against you.

For games with two dice, that might not work as well. If a single random
number were picked and used to shift both die results, it would shift the bias
just like with the single die game.

However, the dice will still be biased to come up matching, and in some games
matching numbers on the two is significant.

One might try to address this by generating two random numbers at the start of
the game, one for each die, for the shifts. That would have bias against
getting a match on the two dice, so would provide an advantage in games where
a match is bad.

[1] Generalizing to n-sided dice is left as an exercise for the reader.

~~~
jknz
Unbiasing a coin is easy (the Von Neuman technique): toss it twice, if the
results are different then output the first toss. If the results are the same,
forget both and start again.

You can use this technique to unbias a 4-die or 8-die by throwing a coin
multiple times (apply the above technique for each digit in base 2).

To produce a fair 6-die from a biased coin, the following should work.

\- Using the Von-Neumann technique, creates 3 independent unbiased coin X, Y,
Z. Then X= 1 + X + 2Y + 2Z is uniform in {1,...,8}, we basically randomize
each digit in base 2.

\- Finally, throw the unbiased 8-die until you obtain a number different than
7,8. The result is unbiased in {1,...6}.

    
    
        import numpy as np
        
        def biased_coin():
            return np.random.choice([0,1], p=[0.1,0.9])
    
        def unbiased_coin():
            a = biased_coin()
            b = biased_coin()
            if a != b:
                return a
            else:
                return unbiased_coin()
    
        def unbiased_eight():
            return 1 + unbiased_coin() + 2*unbiased_coin() +4*unbiased_coin()
    
        def unbiased_six():
            d = unbiased_eight()
            if d < 7:
                return d
            else:
                return unbiased_six()
    
        np.bincount([unbiased_six() for i in range(6000)])
    
    

Edit: there was a mistake in the previous version of the comment because the
digits in base 2 were not be independent. The last version is correct, I
believe :)

------
punnerud
Could it be that your dice was biased to begin with? Or have you already
rolled them ~5000 times before you placed them in water?

~~~
askvictor
I was wondering this; is a control unnecessary here as we assume an equal
distribution, or is that assumption false?

~~~
punnerud
I think it’s false because every mechanical device have some variation, but is
it so small it can don’t “affect” the result?

------
nmg
Einstein: "God does not play dice."

Hawking: "God definitely plays dice, but He sometimes confuses us by throwing
them where they can't be seen"

Izbicki: "And if the dice are loaded, we can't prove it."

There is some plain & profound truth in this article underlying some pretty
cool math.

------
simonebrunozzi
I really like Settlers of Catan, but I have never found a nicely designed
online site to play it. Does any of you know of anything?

Also surprised that there's no "free" \+ "open source" version of it, like
FreeCiv or FreeCol or FreeOrion.

~~~
ovrdrv3
Tabletop Simulator is good, once you get used to it and learn a few keyboard
shortcuts! Only downside is it's usually $30 and I am not 100 percent sure if
the catan ports are safe to forever remain? If anyone has insight on the
legalities of Tabletop Simulator I am curious.

~~~
bllguo
In the US it's $20, with discounts to $10 (look for the upcoming Steam sale).
I am not qualified to speak on legality but a cursory search suggests that the
catan ports are technically not safe... still, there are many versions in the
workshop as catan is such a popular game. It's hard to imagine that they all
get pulled without replacement.

~~~
ovrdrv3
Thanks for the reply. I hope they continue on there!

>In the US it's $20, with discounts to $10 (look for the upcoming Steam sale).

Thanks, looks like it will finally be time to gift it to a friend!

------
IncRnd
A hypothesis was presented: using higher numbers will give 5-15 more resource
cards over the course of a game. Then, conclusions were drawn that standard
statistical techniques are deficient for not being able to detect the bias.

Where are the results of the actual experiment, say of 10 games played using
the biased strategy, against a control opponent? Perhaps, standard statistical
techniques are correct.

Without empirical tests of the hypothesis, where is the science? Perhaps, the
dice are only loaded for a short time, in which time all the skew occurred.
Without testing of the hypothesis, it is impossible to know.

------
maherbeg
A nice mitigation technique is for everyone to bring their own dice, put them
into a hat and require random selection of two of them per turn.

Of course if you suspect your friends are cheating you have bigger problems.

~~~
maksimum
This solves the problem by reducing it to a trusted random number generator
(the hat). At that point you may as well skip a step and draw individual
outcomes from the hat.

------
andrepd
I don't get this.

>It’s impossible for your opponents to scientifically prove that you’re
cheating.

Then two paragraphs later you scientifically prove you were cheating (simply
roll dice thousands of times).

~~~
burkaman
> while playing a game.

Your opponents can't prove cheating based on the game alone.

~~~
godelski
Which is kind of a dumb thing to say. They even note that a typical Catan game
only has 60 rolls. You could have many suspicious looking rolls that would
result from fair dice. The fact is that 60 rolls isn't going to be enough even
if you were cheating a lot more than what was shown in this post.

------
9mit3t2m9h9a
What does the calculation of advantage mean there? The advantage per one nice
and one naughty roll (a pair of rolls) is multiplied by the total number of
single rolls in a 4-player game.

And I guess an honest player would pick the nice locations as often as naughty
ones (there is no reason to actually prefer them). I don't know the rules, so
I don't know if there are enough naughty locations to roll only on them
without _that_ being suspicious.

------
jps359
Cool analysis. Doesn't every player use the same die for a game like Catan?

~~~
chalence
Yes, but presumably only you would know the bias of the dice and would exploit
it by preferentially building next to higher numbered spaces.

------
zejn
> posted on 2018-12-14

OP cheats at time too, posting this a full year ahead of schedule. :)

------
darepublic
I can finally beat my in-laws at this game this holiday season!

~~~
logfromblammo
Skilled players can beat unskilled players regardless of the bias of the dice.
It doesn't matter if you get 5-15 more brick cards if it's an ore game. And
how do you keep the other players from building on the bias-favored numbers?

If you're going to cheat, you're better off sleeving your most critical
resource cards, to protect them from 7-rolls, soldiers, and monopoly cards
taking them from your hand. You unsleeve and swap when you get a favorable
resource roll, and hope nobody is OP enough to keep a running total of all
opponent resources in their head.

~~~
rockostrich
The problem is, how do you know those numbers are bias-favored if your sample
size is a few dice rolls? And if there is a bias in those few dice rolls,
there's no guarantee it will continue. Sure, a more skilled player will always
beat a much less skilled player, but the randomness of the dice certainly
closes the skill gap.

It's much different than a game like Agricola or Dominion where you're playing
100% against what the other people do and not relying on the outcomes of 4
dice rolls to determine what you want to do on your turn.

~~~
logfromblammo
The randomness of the board set up partially cancels the bias you introduce
into the dice. There are 20 different possible numeric token layouts, and each
of those has trillions of different possible tile combinations.

You really have no practical way of knowing in advance which resource tiles
will be under the number tokens you have biased for by loading the dice. So
cheating in this fashion doesn't really provide that much of an advantage over
the other players. Since the intentional cheating is scientifically
indistinguishable from randomness, all good players will already be able to
compensate for unfavorable variations in die rolls.

~~~
logfromblammo
It's actually ( 12864852000 * 20 ) possible board combinations. Not _quite_
trillions; more like a quarter of a trillion.

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SurrealSoul
Now I have something else to blame besides luck when I lose

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pougetj
Great article! One nitpick however in the informal definition of a p-value:
the p-value is the probability of getting results similar _or more extreme_ to
the results we observed if the dice are not biased.

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tzahola
Our set came with plastic dice. Crap!

~~~
andresgottlieb
FYI: [https://www.wikihow.com/Load-Dice](https://www.wikihow.com/Load-Dice)

