Top Secret Alliances

03-11-24

I was awake for a little while in the night recently. I'd gone to bed early, so I woke up quite fully at some point, and couldn't get back to sleep. After some time trying a few things I felt ready to give lying with my eyes shut a go.

The downside of that approach of course is that it's rather boring. So I decided to supplement it with a little bit of thinking, despite the alleged dangerousness of that particular pastime.

What I found myself thinking about was alliance card games. You have probably spotted the problem here already — that is far too stimulating a topic to be any use in getting to sleep! Fortunately for you, the rest of this post is about these types of games, and not the details of my somnetic struggles (it turns out, after some time, I slept once more).

I cooked up a couple of games. Or to be more accurate, somewhere between games, and ideas for games. Maybe one is around 80%, and the other 40%-ish. So about 1.2 games total, at an estimate.

The first is not super interesting, at least currently. I'll write it up at some point, but will spare you for now. The second, though, has something interesting, I think. If I follow it through to a conclusion, the final game (or games) will probably be fairly different to the sketch (or even the concept) here. But I think it is an interesting starting point.

Afterwards, this second game also span off into a couple of other somewhat related games, which I've also sketched out a little here. Again, they are just rough ideas for now, but might be nice kernels upon which to build.

But it started out with thinking about the different type of alliances that occur in different games, which led to a rough go at a type of classification.

Some of the more popular ways to decide an alliance.

Alliance games

Alliance card games are the category of games where overall the game is played as individuals, but each hand has players put together in teams (or 'alliances') of some sort. I'm mainly thinking of trick-takers, though I'm sure there are other classes of games with this mechanism.

There are a few different types of alliance games, in terms of what players know about the alliance at the start of play. I'm sure people more knowledgeable than I have categorised games in this way previously, but as I am not currently aware of such a categorisation, I will divvy them up myself in the way I have been thinking about them. The names aren't great, but hopefully illustrative enough.

Many games fit into more than one category, depending on the type of contract played. So perhaps best to think of these as 'game types' for individual hands, rather than the overall game itself.

Full-knowledge alliances

In alliance games of this type, the teams are fully determined before the hand begins in some public way, generally an auction of some sort. These tend to be one player (a declarer) against the others acting as a team, but need not be (for example Prop and Cop in Solo).

Games of this type include (very much non-exhaustively):

Alliances decided in play

A special category of these type of fully-public alliances are those that are determined during the course of the hand. These are not alliances that are fixed, but unknown, but ones that haven't yet been determined by an auction, and are instead determined during play. Once determined, they are immediately known to everyone.

Games in this category include:

There are probably more of these that I am unaware of / can't bring to mind for now.

One-public alliances

These are alliance games where the team of a single player is known publicly, while the team of the remaining players is not known publicly at the start of play, and only becoming apparent at a later stage. Players know their own team status.

Almost all (if not all) of these games are of the format of one player winning an auction, and specifying a card, the holder of which will be their partner. In many instances the holder of this card will (alone) know the team alliance of all players.

Games of this type include:

Non-duo declarer alliances

Not really a separate category, but an interesting subset of these games are those where the player whose team is known publicly ('declarer') is not a team of two. When they are a team of two, their partner knows everyone's teams. But in other cases they do not - no player knows the full team status of everyone.

For this type, all the versions I am aware of are a single player playing against the others, under the pretense of (potentially) playing two-against-the-rest (calling a card they hold):

In such cases the other players will not realise they are playing a game of this subset until later in the hand.

The other semi-plausible possibility would be a game of six (or possibly five) where a player calls for two partners. However I am not aware of such a game.

Self-knowledge alliances

In alliance games of this class, no information is publicly known about any teams, but each player knows their own team.

All of the games I am currently aware of in this class have the holders of two specific cards being partners against the rest, and are closely related:

Top-secret alliances

In games of this type, not only are no alliance affiliations publicly known, but players themselves do not know which team they are on.

I am not certain if these types of games are truly meaningfully different from self-knowledge alliances, but I think they might be.

I don't actually know of any games in this class. The observant reader may well see where this is headed — indeed this is why I have gathered you all here today. I have a very rough sketch of such a game, and then a semi-related idea.

Mostly-secret alliances

In games of this type, one player's team is publicly known, but the other players do not know which team they are on. So declarer's partner is not initially aware of that fact.

Again, I'm not aware of any such games, but I have a sketch for one.

An unknown alliance condition

The various types of alliances all come about due to how the alliance conditions are known. For full-knowledge alliances, these are settled concretely in the auction. For one-public alliances, declarer announces a card which determines the condition. In self-knowledge games the conditions are fixed before play, by certain cards being 'paired'.

For a top-secret alliance then, one would need a condition that is not known by any player at the start, but which is nevertheless determined in some fashion (and similarly for mostly-secret alliances, but more on that later.).

A game of missing cards

Here's my first idea. It isn't a full game, just a sketch of a mechanism. Whether it is worth fleshing out into a proper game remains to be seen. To use a light espionage theme, for now let's call it Cambridge Circus.

For four players, remove an entire suit from the deck (let's say spades), leaving a 39-card deck. Deal out nine cards to each player. Of the remaining three, put two face up in the middle of the table, and the third face down beneath them. This face-down card is called the alliance card.

Card-play rules are Whist-like (f,tr). If both face-up cards are the same suit, that is trumps. If they are different suits, the missing suit is trumps.

Play is 'usually' two against two, with the two players holding the cards of the same rank as the alliance card being partners with one another. If instead one player holds both of these, or one is one of the face-up cards, they play alone against the three. If both the face-up cards are the same rank (i.e. the three last cards were all the same rank), then each player plays for themselves.

I wrote 'usually' in quotes referring to two-against-two, as I have not checked the probabilities to determine how often this is the case. However, a quick over-the-thumb calculation suggests this probably happens about three-quarters of the time.

How would players figure the situation out? Well suppose amongst other cards I hold 8♣. I have not seen any other 8 be played. I have a few diamonds as well. I lead one, and discover that all other players are out. As I do not have 8♦, and it has not been played, I know it must be the alliance card. No other player knows this for now, as they do not know what diamonds I still hold. I thus know the holder of the 8♥ is my partner (I would know concretely who this is had it been played earlier). I'd probably want to try and get this card out now, so I know who I'm with, and so I lead hearts.

There are a few variations on this, but that's probably the rough shape of how it goes. Of course, ahead of time you don't know it's an 8, so you'd have to be doing this kind of reasoning for all ranks. You might also be able to do this deduction, but not hold the card yourself — i.e. you've seen both other 8's come out, and deduced the third is missing. Then you would also know who your partner was.

This level of processing might necessitate cards being left face-up in front of players after having been played, stacked vertically/horizontally depending on who won, as is sometimes done (albeit not face-up) in games where reconstructing play matters.

The aim

Well, that's about as far as I've got. I think a point-trick game would be more interesting for this, as when a player has got some information about partnerships, smearing allows the alliances to be exploited in a more interesting way than simply letting partner win tricks when you could top them. It allows for more opportunities for players to figure out partnerships, once some of them begin to be known.

I think the main fun of this game would be in trying to determine who's with who. So it would be useful if there was a fairly concrete advantage to having done so. I also worry that this would actually be a bit tedious/formulaic — any player with length in the suit of the alliance card's suit is likely to determine the missing card sooner, so maybe players don't have enough agency with this. But then I suppose the fun is trying to give minimal information to other players — playing cards that you hold two of a rank of, say. It might also be nice to contrast this with wishing to play cards of certain values, so it would be good to think about the points distribution here.

I think as well it would be useful if there was some slight asymmetry between the two teams, so that it 'matters' more which team you are on. The holders of the cards have the advantage of being able to discover their team more easily (as it's easier to prove a positive than a negative), but it feels like there should be something a bit more. Maybe they get the cards on the table at the end? Or they could exchange such cards (in exchange for revealing their status)? Or a different 'target' score?

Like I say, this is just a sketch of an idea, really just a proof-of-principle at the moment. But I might explore developing it into something a bit more concrete, in which case dear reader I'm sure you shall be the first to know.

Proving a positive

When fiddling around with this 'partners of the missing rank' idea, I started to become a bit more focussed on the 'deducing the missing card' angle, rather than the actual secret partnerships themselves. It got me wondering — might the opposite situation be interesting? In other words, having an additional card in the deck that players try to deduce?

I scratched my head on the logistics of this for a bit. If you added a single spades card, say, then there wouldn't be very much mystery. I wondered about having all cards the same suit, but then the game you play while you are trying to figure things out wouldn't be very interesting.

But then I came up with a procedure that I think might work. It is a little bit fiddly, and quite possibly too fiddly to do every hand, but maybe I can figure out a way to make the game so bloody entertaining that players won't mind a bit of extra faff before play.

A sketch with an extra

Let's call this Overflow. Here's how it works.

An example deck ready for a hand of Overflow. Note that only one rank is in all four suits, each other rank only appearing in three of them.

Take a full 52-card deck, and divide it into thirteen piles of four cards each by rank, so that you have one pile with all the Aces, one with all the twos, and so on. Shuffle each pile, and then take one card from each, without exposing any of these cards. This gives you a pile of cards, one of each rank, with mixed suits. Shuffle these, and pick one at random. This will be the extra rank. Set aside the remaining twelve (that take no further part in the hand), and put the chosen one card together with the remaing thirteen piles of three, and mix the whole lot together to make your 40-card deck for the hand.

Four players get ten cards each. The primary aim is to determine which rank has cards of all four suits, rather than just three. Each suit will have a random mixture of ranks present in it, which will vary from hand to hand.

And that's it! I haven't figured out anything beyond that — what the shape of the game is like, what the 'reward' is for deducing the extra card. This also has an issue of advantage, where a player holding more than one of the extra rank probably has quite an advantage in being able to deduce its presence, but it feels like it may not be insurmountable. The procedure also allows a simple modification so that the rank is any possible rank, but could be restricted to, for example, Ten, Jack, Queen, King, Ace. Having it be a 'high-value' card like these (potentially) are might mean that players are less keen on hoarding these for their deduction power as they will want to play them to win useful cards.

Not-quite-so-secret

In the course of writing up these ramblings, and specifically trying to clarify the alliance categorisations, it occured to me that there is another possible category, related to the final one. In these games, one player has a team publicly known, but the other players don't know who is allied with them (what I refer to as 'mostly-secret alliances').

Again, I have a little sketch for how this works, which I'll outline here, and maybe flesh out in the future. It is like Cambridge Circus with an auction and card-calling. For a working name let's stick with the theme and call it Manchurian (referring to the candidate, not the region).

For four players, deal 12 cards each from a 52-card deck, with the remaining four constituting a talon. There is an auction - for now let's say players in turn can either 'play' or 'pass'. If the first three pass, dealer must play. Whoever bids 'play' becomes the declarer.

Declarer takes up the four cards of the talon into their hand. They discard three cards face-up on the table. The fourth is placed face-down, as the alliance card. Declarer's partner will be whoever has the card of the same rank and colour as the alliance card. So for instance if the alliance card is 7♣, their partner is the player holding 7♠. The partner will not become aware of this fact until some point in the course of play. Declarer can lay an alliance card that pairs with a card they hold, in which case they play alone against the other three, again without the other players realising this initially.

Declarer chooses trumps freely before play begins. Card-play is Whisty rules, as are rankings. Game is point-trick, with Aces 11, face cards 5 each, Tens 10, 9-5 are 1 each, and 4,3,2 each count their face value, for 200 points in the deck. Declarer aims to win over half the points of the deck (last trick winner gets the three face-up cards, the alliance card always going to declarer).

A successful declarer and teammate are each paid by the defenders 50 points, plus the number of points taken above 100. So a declarer taking 130 card-points would score 50 + 30 = 80 points (as would their partner). An unsuccessful declarer pays 100 points plus the amount falling below 100.

I'm not sure why this particular game I've fleshed out a bit more than the other two — I can't claim that I have put very much thought into it. I guess it just feels a little simpler, and so maybe a fairly straightforward game structure might 'just work'.

Notation

When I was just a wee one I was always taught that, when performing, it is always best to leave the audience on some notation.

As an adult, I have come to the belief that the same probably applies to blog posts.

And so, to slightly expand on my alliance categorisation above, let's introduce some notation. We can classify (in one way) alliance games by two numbers:

So we can write the classification as two (comma-separated) numbers. We will use \(n\) to refer to 'all players', so that we don't need to necessarily refer to the overall player number of a game.

I'll finish then with a summary table, showing how this notation works, with reference to labels mentioned above, and a few examples.

Alliance category Label Example games
\((n,n)\) Full-knowledge alliance Skat, Solo Whist, 4p French Tarot
\((1,n)\) One-public alliance Schafkopf, Koenigrufen (rufer games), Hungarian Tarokk, Nomination Whist
\((0,n)\) Self-knowledge alliance Doppelkopf (normal game), Kop
\((0,0)\) Top-secret alliance Cambridge Circus
\((1,1)\) Mostly-secret alliance Manchurian

I'm sure with this tool at your hand, future communications on this matter will be magnificently succinct. Dear reader, you are welcome.