Spades

The whole deck is dealt out, thirteen cards each. Before a card is played, every seat names a number: how many tricks it thinks it will take. Your bid and your partner's are added into one partnership contract — bid 3 while your partner bids 4 and your side has to win at least seven of the thirteen. Then the hand is played like any trick-taking game: follow the suit that was led if you hold it, and the highest card of that suit wins, except that any spade beats any card of any other suit.

There is one restriction on leading. Spades may not be led until they have been broken — until somebody has played one on a trick they could not follow. The exception is the player with nothing but spades left, who obviously has to be allowed to lead one. Nearly every implementation gets the main rule right and that exception wrong, which leaves a hand unfinishable on trick ten.

You are set, and you lose ten points for every trick you bid — a five-bid that comes home with four tricks is not +40, it is −50. The tricks you did win are worth nothing at all. That asymmetry is why spades is a game about honest bidding rather than a game about winning tricks. Over two thousand played-out hands here a partnership is set on 12% of them, and a set costs 70 points on average against the 57 a made hand is worth — so the swing between bidding right and bidding one too many is more than two hands of ordinary scoring.

Bidding low to stay safe does not work either, because of bags. Every trick you take beyond your bid is worth one point, and one point is fine until the tenth one — at ten accumulated bags your side loses a hundred and the counter drops by ten. So ten overtricks cost more than two made four-bids are worth. The computer seats here finish a hand 0.7 tricks over their bid on average, which is a table that bids close to the bone and still hands back a hundred points to the bag rule about every twelve hands.

A nil is a promise to take no tricks at all. Survive thirteen tricks and your side gets a hundred; take a single one and it costs a hundred. It is not a low bid, it is a different bid: your contract contributes nothing to the partnership total, so your partner is carrying the contract alone while you try to duck everything.

The hands that are safe have no card that can be forced to win. A high spade is the killer, because a trump you are made to win has nothing you can throw under it. A bare ace or king in a side suit is nearly as bad — it gets led at and there is nowhere for it to go. A long run of small cards is the opposite: an eight in a five-card suit will never win anything. The computer seats bid nil on 5.5% of the deals they are dealt and bring it home 64% of the time.

Blind nil is the same bid declared before you look at your cards, worth two hundred either way. This board offers it only when your partnership is a hundred or more behind, which is the usual table rule and the only situation in which it is the right bid rather than a way of losing faster. Your hand stays face down until you answer.

Exactly 3.25 on average — thirteen cards from a deck that is a quarter spades. Knowing the shape of that distribution is most of bidding, because a fourth spade is worth about half a trick and a fifth is worth close to a whole one. The figures below are exact, not simulated: they are binomial coefficients over a 52-card deck.

So the hand you are usually holding has three spades and no trump length worth counting, and the hand that wins tricks by ruffing — five or more — turns up on about one deal in six. A hand with no spade at all turns up roughly once in every 78 deals, and it is the hand where nil is most tempting — though a bare ace in a side suit will still break it.

They count the hand rather than guess at it, and they count trumps and side suits differently, which is the thing a naive bidder gets wrong. A side-suit king is worth well under a trick because somebody is going to ruff it; a fifth spade is worth close to a whole one because by the time the suit is exhausted everywhere else it wins by itself. Short suits only count when there are trumps left to ruff with.

A bare honour counts for half of what a guarded one does, in either column: a king of spades with no small spade behind it, and a singleton king in a side suit, are both cards you will be made to play before you want to. The same logic is why a queen outside the trump suit needs three cards in its suit to count at all.

Across five thousand seeded deals that count puts the four seats' bids at 11.6 of the thirteen tricks available — a table that bids honestly and leaves a little slack, which is what a table of competent players does. A table that bid all thirteen would be set constantly; a table that bid eight would drown in bags. The partner of a nil bidder bids about a trick more than the raw count says, because covering a nil means taking tricks you would otherwise duck.