Aces
Aces is a brand new virtual game and a dual of the game of Spades.
In mathematics we often use duality to simplify the solving of a problem. I will attempt to give you a brief overview of the concept. We have a problem that has proven difficult to solve in a particular branch (lets call it b1). We map (a type of mathematical transfer) the problem over into branch b2. We choose b2 because it’s a rather simple problem in b2. Once we solve it in b2 we then map the solution back into b1. Some of the great unsolved problems (at that time) have been conquered in this manner. All the examples that come to mind are rather arcane and would likely not be clear or helpful to most of you. So please excuse my hand waving here and focus on the concept, not the details.
My purpose here is twofold. Initially I was looking for a dual game to more clearly bring home the point of card distribution and the many traps that are easy to fall into. The more I thought about this game and the more analysis I did about its combinatorial underpinnings, the more I realized that the beautiful symmetry between the 2 games might actual make it a fun game to play!
Aces has all the same rules as Spades! You play to 500 (or whatever) or -200 (or whatever). You collect bags and bagout at multiples of 10 (or whatever). You have to follow suit if you can and if you cant then you may sluff or ruff (cut, trump, etc). You must be curious how a game can have all the same rules and yet be different. There are many ways of course but this particular one is a dual and therefore has many interesting properties. I constructed this game so that you could play it at home with a standard card deck. Both Spades and Aces have the identical 52 cards.
The difference is this: Aces has 13 suits and 4 ranks. I will assume I don’t have to tell you what those suits are. The trump suit is aces! The ranks, in descending order are: SHDC, or to express that mathematically: S>H>D>C. This is the same ranking that suits have in bridge. This was done for consistency. But don’t be misled here. In Aces this is not the rank of the suits! This is the rank of the cards in the same way that in Spades we have J>8>5>4>2. Because Aces is a dual of Spades some interesting artifacts are: the total # of hands is the same in both (obviously). What may not be so obvious is that the total # of possible distributions is also the same (39)! For clarity and as an example I will list the top 3 (most common) patterns in both games.
Spades:
4432 21.6%
5332 15.5%
5431 12.9%
Aces:
2221111111000 19.1%
2222111110000 18.8%
3221111110000 16.7%
Another duality is between your aces and your spades.
In Aces your aces are like spades (in Spades) and your spades are like aces (in Spades)!
Just to make sure you are still with me, I will give a few examples. Remember that your trump suit is aces! We have 4 trumps: A of spades, A of hearts, A of diamonds and the A of clubs. Lets say I have the opening lead and I decide to lead the spade 7 (in Spades we typically write this 7S, in Aces I will write it as S7). This way we are consistently putting the rank 1st and the suit 2nd. If you have a 7 you must follow suit. If you don’t you may ruff if you have an A or you may choose to sluff a side suit. Note that the S7 is the boss in the 7 suit. So unless someone ruffs, it will win the trick. This is similar for the S2 (the S2 is the boss card in the ‘2 suit’). So now you should see why spades in Aces are like aces in Spades!
I will now pick a few hands to demonstrate the key point. You are dealt the following hand:
S7 H4 H9 CJ SK H3 D8 C7 DQ DK SQ S9 S8
If I display this hand in a Spades room at GP it will be shown thusly: S:KQ987 H:943 D:KQ9 C:J7 or something similar where all the spades are grouped together, as are the hearts, diamonds and clubs.
If I display it in the Aces room it will appear as: A:- K:SD Q:SD J:C T:- 9:SHD 8:S 7:SC 6:- 5:- 4:H 3:H 2:-
Let us now look at the distribution patterns from both perspectives.
Spades: 5332 15.5%
Aces: 3222111100000 12.6%
Not much difference.
Let’s consider another:
Spades: KT7 Q963 A542 J8
A pretty typical looking hand, 4432 (21.6%) from a Spades pattern perspective. But this same hand picked up over in our Aces room would look like this:
Aces: A:D K:S Q:H J:C T:S 9:H 8:C 7:S 6:H 5:D 4:D 3:H 2:D
A pretty rare hand (.011%) and not typical at all! 1111111111111 distribution!
And for symmetry’s sake let’s reverse it. And look at one more hand:
Spades: KJT7542 AQJ652
7600 distribution (.0056%)
Aces: H S H SH S - - S H SH S - SH
2221111111000 distribution (19.1%), our most common pattern
Now we see an extremely rare hand from a Spades perspective is the most common from an Aces viewpoint. And yet, we are talking about the exact same hand!
How about this doozie: S: AKQJT98765432
From a pattern perspective: Spades (13 0 0 0), Aces (1111111111111) If we let B stand for 52C13 (the total # of possible hands) then p(of that hand) = 1/B no matter whether your playing Spades or Aces. But p(that pattern) from a Spades viewpoint = 4/B (essentially zero) but from an Aces perspective = .011%. And while that is also a pretty small # if we divide the 2 to get a proportional weighting we will get 2E15 (2 followed by 15 0s). Human beings are pattern recognition machines. We see faces on Mars, in tortillas, clouds, rocks ‘ wherever we look.
All this brings me to my main point (finally, they say).
In case you’re a bit woozy from all this parallel universe mumbo jumbo, let’s go back to Spades.
Looking at hands after the fact is fraught with danger.
If a priori (before the fact) I ask, ‘What’s the chance I’ll get a 5332 type hand’. The answer is 15.5%. If I had asked, ‘What’s the chance I will get the specific hand listed above (S:KQ987 H:943 D:KQ9 C:J7)’. The answer is 1/B. The same chance I will get S: AKQJT98765432. Stop and think about this and let it soak in. When we see the 5332 hand, we are glad to get such a nice hand but we don’t write home about it. But if we saw a 13 spades hand when we viewed our cards, we wouldn’t shut up for years. Why is this? The answer probably lies in the discipline known as Evolutionary Psychology. We have been selected for, based on our ability to recognize patterns. A corollary to this is our desire to categorize. But I digress.
I assure you; the probabilities of the 2 hands are the same. Try to consider which pair of glasses you have on.
Let’s look at that other hand above: S:KT7 H:Q963 D:A542 C:J8
This is another ho-hum hand from a Spade’s perspective. But when we put on our Ace’s glasses, it looks like this: A:D K:S Q:H J:C T:S 9:H 8:C 7:S 6:H 5:D 4:D 3:H 2:D a hand with a distribution pattern of 1111111111111. This is definitely a hand to write home about if you are an Aces player. This pattern will occur ~ 1 in 10,000 hands’wow! This is the only distribution pattern in Aces that has no void. So while we are talking about the exact same hand, how typical or unusual you perceive it to be has much to do with your frame of reference. What bias affects the patterns you look for and see? Matt correctly touched on this on a distant post. I will restate it for emphasis now.
S=> r =>C
The Service (the Mersenne - twister ) produces a random vector of size 52. This vector is passed to any particular Client that will produce an outcome relevant to whatever our Client is supposed to do. Here at GP that Client groups the 52 elements into 4 hands and displays them in a form that’s relevant to the game of spades. To allow for any errors that might have occurred between S and C we need to gather our data at the last link in the chain. This is what has been done to run the tests to check for the validity of the final data (the actual cards that show up in your hands at the table).
Mathematicians that specialize in Probability, Decision theory and/or Utility theory have long been aware how unexpected are the patterns that occur and how counter intuitive they can be. A very significant point is this: based on what the function of our Client is, the output will produce predictable patterns. In the ‘real world’ most data points are disjoint, overlapping and intertwined in many ways. But here in our very specialized corner of the world, we have a situation where the results are totally connected. If a flaw exists anywhere in the chain leading from S to C then it will be reflected in all views of the output. We can put on our Spades goggles or maybe our Aces goggles. Any flaws will produce consistently good or consistently bad results. It’s logically impossible for some views to be valid while others are faulty. In the messy real world, Inferential Statistics is our tool of choice. We don’t have idealized pseudo-randomness. We are dealing with entities (usually people). We have all kinds of intertwined relationships.
To argue over the size of samples you need to take, misses the point. In the real world trying to make predictions and forecasts necessitates the need to take samples from the pool you are interested in. If there are 100,000,000 people who can vote in an election but only 20% will vote then our population of interest is just 20,000,000. And based on the confidence level we desire we can calculate the sample size we need to achieve that degree of confidence. This type of statistics is routinely taught at the middle school level now in America and abroad. But let’s talk about the actual situation we have here at GP. Our potential population is our B as defined above (~6E11). But our actual population is the hands played here! For any of you to test it you need to determine just how big a sample you need gather to feel confident at whatever level you choose. And to that extent the discussion between galt and 000000 is relevant. But for me to be concerned about just how many of the several million hands I need to sample is ridiculous since I can examine them all in a matter of seconds. There is absolutely no need to be concerned about how many I should choose! I choose them all.
Don’t think for a second that I don’t understand your argument galt. But it’s a straw man holding a red herring. You can pose examples like: look at this 4432 hand ( AKQJ AKQJ AKQ AK ). Sure its got a typical 4432 distribution but look at the imbalance of strength! My response is: while that hand has a typical distribution pattern its probability is 12/B. It’s essentially 0 for all practical purposes. If anybody ever saw this type of hand then you have a solid argument that something’s wrong. But nobody has and nobody will. The same is true for 100 coin flips that go HTHTHT’. For 100 flips. While that gives us a 50-50 split, that event has 1 chance in E30 (1 followed by 30 zeros). If you think we would say that it is a normal occurrence just because its 50% heads then you misjudge our analysis. We could ask all 200 players in the clubhouse to each flip a coin 10 times and record their results. We would see things like HHTHTTTHTH. We can write that as 2113111. It has the pattern 3211111. We could analyze all 200 patterns and see how they correspond to what we would expect. As a mathematician I don’t need to list out all 1024 possible results to see what the patterns are. Just as in spades I don’t need to list all the B possible hands to know what they are and what patterns occur, what percentage of the time. I use combinatorial analysis to calculate them. If someone said I got HTHTHTHTHT (1111111111) I would say wow! That sequence is 1 in 1024. That pattern is 1 in 512. But with 200 people it’s not that amazing, although not expected.
My main point is we don’t need to worry about how big a sample to take when we have access to both all the actual hands played here at GP, and also we know the exact nature of all B (635 billion) possible hands. There is no need to poll. There is no need to wonder what the cards are thinking! We aren’t dealing with people here (with respect to the hands we hold). We aren’t even dealing with cards. We are merely dealing with a randomized vector of size 52. The fact that v(7) has a symbol in it such as 9c or Qd or 3h is totally beside the point. It could be seeded with 52 types of fruit (4 genera, 13 species) and the Client might be playing ‘Fruit basket bingo’ ( I love that game ).
The cards we get at the table here at GP are totally dependent on the randomness of our dealer (S). The ACM has given this algorithm its stamp of approval as the best random # generator on the planet.
Think hard about the difference between a priori (before the fact) reasoning and post priori (after the fact) reasoning.
Whether we look at these hands as Spades hands or Aces hands or Heart hands or No Trump Bridge hands, we get the same result! They are as randomly distributed as we could possible hope for. If the deals are determining who wins most of the time, what does that tell you? It says you are playing pairs that are close to your ability. Whether your pair is expert, average or rank beginner, the less differential between you and your opponents, the more likely the outcome will be determined by the cards. This is assuming you’re playing ‘rubber spades’ not RepSpades or RepAces.