Why are mixed sets worth the most?

[ABSOLUTE_MASTER]

Once again, I don't think this applies here because it does not appear the cards are drawn from a finite set but rather randomly generated. As I understand, the only safeguard in place is that the same card is not held by two people at once.

That said, if it was a finite set, it would tip the odds even further towards mixed sets.

Now, when they taught you statistics, did they show you the example of drawing balls of different colors from a box? Did they show the difference between what happens when the balls are replaced vs not replaced? And what about if the balls are replaced but not immediately?

[detlef]

Once again, I don't think this applies here because it does not appear the cards are drawn from a finite set but rather randomly generated. As I understand, the only safeguard in place is that the same card is not held by two people at once.

That said, if it was a finite set, it would tip the odds even further towards mixed sets.

Now, when they taught you statistics, did they show you the example of drawing balls of different colors from a box? Did they show the difference between what happens when the balls are replaced vs not replaced? And what about if the balls are replaced but not immediately?

To answer both questions:

First off, yes, as I mentioned, somebody had played Kamchatka...me, that very turn. Now, even if they're simply automatically shuffling that card back into the mix, the territory is not important in this case, only the color. The fact that it changed colors proves that there are not 42 individual cards out there, each assigned a never-changing color. Rather, it would appear that each time you grab a card, the computer randomly assigns it a color. Thus, it seems that each time, you have the same chance of drawing each color regardless of what is already being held.

As for the second: If, in fact, there is a finite number of cards, the probabilities only change slightly from the basic odds mentioned earlier. However, that slight change affects the chances of drawing 3 of the same color adversely and the chances of drawing a mixed set positively. If you have a red and a blue there are more greens out there than reds and blues so you have a slightly better chance of grabbing one. Combining that with the fact that there are more possible combinations of drawing a mixed set...

[detlef]

Here's a question, since I do not have a physical copy of the game on hand, how is Risk itself set up?

With 42 countries, are there even numbers of red, green, and blue cards (well, soldier, army, and cannon equivalents) in the Risk deck? Or are there more soldiers than cannons, lending a valid reason as to why a set of cannons are worth more than a set of soldiers?

Risk is set up to be escalating.

[dcowboys055]

Here's a question, since I do not have a physical copy of the game on hand, how is Risk itself set up?

With 42 countries, are there even numbers of red, green, and blue cards (well, soldier, army, and cannon equivalents) in the Risk deck? Or are there more soldiers than cannons, lending a valid reason as to why a set of cannons are worth more than a set of soldiers?

There's an even amount of them all (except the "wildcards") in the real life version. And on CC, there is a 33.3% chance of getting a red, 33.3% for a green, and 33.3% of getting a blue every time.

[ABSOLUTE_MASTER]

I think lack should let us know how the script goes before we make more assumptions.

[ABSOLUTE_MASTER]

There's an even amount of them all (except the "wildcards") in the real life version. And on CC, there is a 33.3% chance of getting a red, 33.3% for a green, and 33.3% of getting a blue every time.

One more...

[dcowboys055]

One more what?

[kev82]

Just to throw a spanner in the works, people only seem to be considering the case where there are 3 cards. If you have 4 unplayable cards, for example two of the same type twice (RRGG, RRBB, or GGBB), then the probability of getting a 3 of a kind set with your next card is twice as likely as getting a 1 of each kind set.

Personally, I don't think there is much point discussing this until someone knows for definite whether the card's colours are assigned at game start, and come from a finite deck or whether the colours are generated randomly at each card draw.

Assuming for a second that the probability of each colour is 1/3 at all times, and you count the highest value set you have as soon as you have it, then a simple recursive function can calculate the probabilities. If I've written it correctly (I can post source if anyone wants to check) then I think the probabilities are.

Mixed set 42/99

Any particular single coloured set 19/99

Which makes a mixed set a little more than twice as likely as a particular single colour set, however the probability of any single colour set is 57/99 so you are more likely to get a single colour set that a mixed set.

This however, only stands if the card probabilities are always equally likely and independent.

Edited for bad English and typos.

[Aimless]

Even assuming that the cards are drawn from a fixed deck, the odds of getting a mixed set are substantially higher than those of an unmixed set. (Better even than in the independent case, as has been said.)

Taking the Classic board (42 cards, 14 of each color) as a starting point, the odds of a pure set after 3 cards is :
14/42*13/41*12/40=0.0317

The odds of a mixed set are :
42/42*28/41*14/40 = 0.239

Further, considering that people will preferentially choose a mixed set over a pure set will tend to ensure that those who don't have a mixed set after 3 cards will wait until they get one or are forced to turn in after 5 cards, the odds of having 5 cards and not getting a mixed set are :
3*28/42*27/41*26/40*25/39*24/38=0.347

Or, with 5 cards, you have a 65.3% chance of having a mixed set.

That said, I think the evidence is rather strong that (at least for now) card generation is independent of previous cards dealt, and thus the odds are as has been posted before. BTW, with independent cards, the odds of not having a mixed set after 5 cards is 3*(2/3)^5 = 0.395. In otherwords, there is a 60.5% chance that someone will redeem a mixed set.

To take things further, it is trivially obvious that, with 5 cards, the chances of having any given pure set is 0.395/3 = 0.132, so the expectation value of any given set of cards (assuming optimum play by the player) is 10*0.605+(8+6+4)*0.132 = 8.426 armies.

Thus, in flat rate games, it can be safely assumed that any time someone turns in cards, it will be a mixed set. IMHO, this makes flat rate games somewhat less interesting, but as the point of a Flat Rate game is to remove the overpowering effect of cards in the late game while maintaining the incentive to attack, it is still effective.

[tahitiwahini]

Thank you Aimless for the very clearly stated and fascinating post.

I've been thinking of how to improve the strength calculation in the greasemonkey script by taking into consideration the number of cards held by a player. Your post has been very helpful in that regard.

[Aimless]

Well, with some work, it would be possible (assuming that the cards are independent - if they aren't, the probabilities are much more involved and vary based on which map you use) to calculate expectation values for 3, 4, and 5 card hands, which might be of use to you. It's too much work for me to do tonight, though - 3 card and 5 card expectation values are straight forward, but the 4 card case is non-trivial. I might try to tackle the problem tomorrow.

On the other hand, I'm not sure it's useful, since game theory says that a rational actor will tend to maximize his outcome by waiting until he has a mixed set or has no possibility of gaining one. About the only situation I can think of in which it would be beneficial to turn in early (barring strategic imperatives in the game itself) is if someone made a 3-card set of blue, in which case the opportunity cost of turning in early is low, and the odds of bettering the set by waiting are equally low.

Hmmm. Actually, now that I think of it, given a 3-card pure set, the odds of bettering that set will always be low; maybe it is beneficial to turn in as soon as you make a set. I'll have to consider it more carefully; it's an interesting problem. And, if the opportunity cost of turning in early is low, then an expectation value for each hand size is useful after all.

Ah well, I'll come back to the issue tomorrow when I have more time.

[tahitiwahini]

Very interesting stuff (at least it is to me).

One factor that's problematical to account for in the decision as to whether to turn in a monochromatic set or wait for a polychromatic set is the identity of the specific countries on the cards. The two army bonus per country owned by the player can be quite attractive. Granted you don't get any say in the deployment of those bonus armies but they can still make a player consider turning in a red set if he were to get bonuses on two or more of the countries. This is particularly true if the player is in a generally defensive rather than an offenisve posture, wherein they are perhaps more concerned with fortification than deployment. Unfortunately, the country of the card held by a player is not public information (that is, it is information that is absent from the game log up until the point at which the card is cashed in).

[IronE.GLE]

Why do people feel the need to post for the sole purpose of telling other people that they shouldn't have posted something?

Trust me, as irritating as the original post may have been (and it this case I don't think it was at all), it's as nothing compared to reading the posts from people saying how the original question shouldn't have been asked.

You don't have to read the post if you don't want to, and you certainly don't have to respond to it.

I guess it would be the same answer for those who respond saying, "this has been asked before, use the search you dumb noob." It's because they are self absorbed pricks who think because they found this site before you did, it entitles them to talk down to you like you are a 5 year old kid who stole some bubblegum.

Honestly, who wants to sift through 200 topics that have the keyword 'set' in them? I know I don't want to do that.

[Aimless]

I've thought about the problem some more, and here are the numbers for 3 and 4 card sets assuming the cards are independent events.

For 3 cards, the math is straight forward. There are 27 possible combinations of cards, 6 ways to make a mixed set, and one way each for the pure sets, so the expectation value of a three card hand is 10*6/27+(8+6+4)*1/27 = 2.888 armies.

For a 4 card set, the math is more difficult. There are 3 possible hand types which make a set : 4 cards of one color, 3 cards of one color and an unmatched card, and 2 cards of one color and one each of the others. Obviously, there are 3^4 = 81 possible combinations of cards.

Thus, for any given pure set, there are 1+4C3*2 = 9 possible hands. (One hand with 4 cards of the same color, 8 hands with three of one color and one odd card.) For a mixed set, there are 3*4C2*2 = 36 possible hands. Thus, the odds for any given pure set is 9/81 = 1/9, and for a mixed set is 36/81 = 4/9.

The expectation value of a 4 card hand is thus 10*4/9+(8+6+4)*1/9 = 6.444 armies.

[Edit:]
In light of the below, the case of a player having four cards of the same color is unlikely, since he would have had a three-card set prior to receiving his fourth card, and therefore should have cashed in. Likewise, had he made a mixed set, he would have cashed in as well. Thus, this changes the expectation value of surviving 4-card hands slightly, as the probability of a mixed set is now 3*3C2*2/54 = 1/3, and the probability of each pure set is 8/54 = 4/27.

Therefore, the new expectation value is 10*1/3+(8+6+4)*4/27 = 6 armies.
[/Edit]

On to the other problem to consider - the benefit of waiting on a mixed set versus cashing a pure set early. I'm not going to consider all the possibilities here, as that would be cumbersome. Instead, I'm just going to look at two worst-cases; a three card hand of red, and a four card hand with three red and an odd card.

Given a three card hand of red, the odds of making a mixed set with the remaining two cards is 2/3*1/3 = 2/9. Thus, there is a 7/9 possibility that you will not improve your hand. Thus, the expectation value (after 5 cards) of a hand given the first 3 cards are red is 4*7/9+10*2/9 = 5.333 armies. Thus, the opportunity cost of cashing in early is 1.333 armies. Not that much, considering the two turn wait.

Given a four card hand with three red cards, the chance of making a mixed set on the fifth card is 1/3, so the expectation value is 4*2/3+10*1/3 = 6 armies, so the opportunity cost of cashing early is 2 armies. This one is more questionable, especially since you only need to wait one extra turn.

Obviously, the opportunity costs for early cashing of green and blue sets will be lower.

However, there is a further consideration. If you do wait, and you get lucky and make the mixed set, you will be seeding your hand with two matched cards to begin with, making the likelyhood that your next set is also mixed lower. Conversely, if you do not get lucky, you will be seeding your hand at random (here, the two left-over cards are independent of condition of making a set), so you will not have improved your odds that the next set is mixed.

This event is difficult to factor into the expectation values, however, since it depends on the discount rate of the value of armies in future turns. In any case, the effect is purely negative, thus ultimately decreasing the opportunity cost of cashing in a pure set early.

[tahitiwahini]

Very illuminating analysis, Aimless. Thank you.

Now if you could just explain why when someone loses all his armies in a 16 versus 1 attack, it doesn't necessarily follow that there is any problem with the randomness of the dice.

You would be doing a service to humanity, not to mention eliminating in one fell swoop fully one-third of the posts in this forum.

We're lucky to have you posting here. I hope you're able to continue to find the time.

[Aimless]

Well, considering the number of games play here, and the number of dice rolls per game, I would be highly surprised if there hadn't been at least one case of a player losing a 100v1 fight and coming to the boards to complain about the "obviously" rigged dice.

But that's too much work. I have a midterm in Stat. Mech. due tomorrow.

[silvanthalas]

Risk is set up to be escalating.

That's not what I asked.

Besides, with the variant Risk games out there, such as LotR Risk, I'm guessing the 4/6/8/10 values came from one of them.

There's an even amount of them all (except the "wildcards") in the real life version.

Thanks for the straight-forward answer.

[AAFitz]

they are chosen independently...so the odds of getting three green are the same as a mixed set...the amount of armies just changes...and i was on a site with different values for the sets...very depressed to get 4 armies for a blue set

[Aimless]

One last change to my above calculations. Taking only "surviving hands" into consideration - that is, assuming that players cash in immediately after making a set - it occurs to me that my previously stated expectation value for a 5 card hand is wrong, since a large number of combinations involve making a set prior to the fifth card.

It is trivial that the only way to have 4 cards without having a set is with two pair. Thus, upon drawing the fifth card, there is a 1/3 chance of making a mixed set, and a 2/3 chance of making a pure set. As the odds for each pure set are identical, then any given pure set has a 2/3*1/3 = 2/9 chance. Thus, the new expectation value of a five card hand is 10*1/3 + (8+6+4)*2/9 = 7.333 armies, substantially lower than the value listed above.

[detlef]

they are chosen independently...so the odds of getting three green are the same as a mixed set...the amount of armies just changes...and i was on a site with different values for the sets...very depressed to get 4 armies for a blue set

They may be chosen independently, but that does not mean that there's just as much chance of getting a pure set as a mixed. That much has been very much established by about a dozen or so posts in this thread.

[AK_iceman]

The colors are chosen randomly, there isn't a real deck. The odds of getting a green/red/blue card are the same regardless of how many are already out.
So therefore, the odds of getting a mixed set are the same as the regular sets.

[Aimless]

The colors are chosen randomly, there isn't a real deck. The odds of getting a green/red/blue card are the same regardless of how many are already out.

True.

So therefore, the odds of getting a mixed set are the same as the regular sets.

False.

[detlef]

The colors are chosen randomly, there isn't a real deck. The odds of getting a green/red/blue card are the same regardless of how many are already out.
So therefore, the odds of getting a mixed set are the same as the regular sets.

So then it is some special kind of "random" that defies all laws of probability?

Once again, if it is, in fact, random, then you are twice as likely to draw a mixed set than all the one-color sets combined after 3 cards.

Can we please just understand this simple fact as being established and proven beyond any doubts and move on from this point?

[tahitiwahini]

The colors are chosen randomly, there isn't a real deck. The odds of getting a green/red/blue card are the same regardless of how many are already out.
So therefore, the odds of getting a mixed set are the same as the regular sets.

I will accept the first part of your statement as correct: namely, the odds of getting a green/red/blue cared are the same. That is, the cards are selected independently (there being no real deck).

But, the second part of your statement is not correct, as several previous posters have demonstrated. Assuming you have an equal chance of getting a green/red/blue card, the result when you consider sets of three cards is that you are much more likely to get a mixed set as you are to get a pure (monochromatic) set. This has to do purely with combinatorics and the assumption that the first part (about the independent nature of the card selection) is true.

I think if you go back and read the earlier posts, that they will make this argument quite convincingly.

[AAFitz]

they are chosen independently...so the odds of getting three green are the same as a mixed set...the amount of armies just changes...and i was on a site with different values for the sets...very depressed to get 4 armies for a blue set

They may be chosen independently, but that does not mean that there's just as much chance of getting a pure set as a mixed. That much has been very much established by about a dozen or so posts in this thread.

actually there is a lot of conjecture...about whether they are picked out of a deck or not....they arent..which is the point of my thread...there is a 1/3 chance of getting each one regardless of what you have....this has been posted by lack...im not guessing

but if you want a different answer....the mixed set is worth more because it just makes sense conceptually to have it that way because the odds are different than those of getting any individual colored set...so if the odds of getting one are more...then the answer is its a bonus...if you think the odds are the same...then its just because it makes sense...conceptually

either way...a mixed being worth the most makes the most sense...odds or not...and as AK says the its how the actual risk game is played...