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Otto Matic, the referee

When the rest of a hand can't go any other way, the referee says so. You can lay them down, or play it out and watch.

The rule

A hand is a lay down when one side takes every remaining trick whatever either player does. Not "probably", not "with the right line": there is no card the losing side can play, and no card the winning side can play, that changes a single trick. It can happen anywhere from the first lead down to two cards left. The last trick is always played.

How the referee decides

Otto Matic sees both hands, which Marta's own play never does. After every trick, with the winner about to lead, it asks one question: can the other side take a trick from here, no matter what anyone does? It answers by trying every legal lead and every legal reply, all the way down. Two shapes come up:

• The follower can never win. They hold no spades, and in every suit the leader still has, the follower either has none of it or only cards under the leader's lowest.
• The follower trumps everything. They hold nothing but spades and the leader has none, so every lead gets trumped and the spades run the rest.

A looser test was tried first, "does a winning line exist". It called a hand where the winner could still throw a trick away by leading a low spade, and playing the hand out proved it wrong. So the rule is the strict one, and the reasons the referee gives are the ones you can check against your own cards.

How Marta thinks as she climbs

Marta's strength is one number, 0 to 100, and the ratchet moves it after each of your games. It buys three different things as it climbs. Up to 60 it turns her old habits up: she bids closer to her hand, leads high while she owes tricks, plays to set a nil, remembers more of the cards that have gone by. From 60 to 80 a different Marta takes over, and the share of hands she thinks all the way through rises from none to every one of them. From 80 to 100 she thinks every hand and the card ladder below decides how much she is shown while doing it.

Up to 80 she never sees your cards. She knows what anyone at the table knows: her hand, both discards, every trick played, the suits you have failed to follow, your bid, and, if you have a record here, how your bids compare with what you take. From that she deals you a few hundred hands you could be holding, throws out the ones you would not have bid the way you did, and solves each of the rest to the end with both sides playing perfectly. Then she plays the card that does best across all of them. She bids the same way: the count she makes in most of those worlds, less one trick, because a hand solved as if she will know your cards runs about a trick optimistic. Once her bid is safe the arithmetic turns on you, and the lines that leave you short of your bid, or hand you bags, are the ones that score.

Above 80 that changes, and the ladder is the only thing that changes it. At 81 she is dealt three of your ten cards face up, one more every few points, all ten at 100. A card she has been shown is not a guess: it is pinned into every world she deals, and the rest of the hand is sampled around it as before. At ten there is nothing left to sample — one world, and it is your actual hand. That is the advantage you are climbing into, which is why it is written here rather than left for you to notice.

What it is worth, on 200 identical deals against the practice-table bot: the old ruthless Marta won 65.5% of them, thinking in the play alone 78.0%, thinking in both the bid and the play 80.0%. Handed your actual cards she would win 95.0%, which is the most perfect play is worth in this game and the reason there is still room above her.

The technique is called determinization, or Perfect Information Monte Carlo. It is how Matthew Ginsberg's GIB played bridge at expert level, GIB: Imperfect Information in a Computationally Challenging Game (2001), and how later programs play Skat and Bridge, Perfect Information Monte Carlo with Postponing Reasoning (2024). The solver under it is the lay-down referee's search carried to a whole hand: every legal line, scored the way the hand will be scored, with bags, sets and nil. Her thinking clock is 150 milliseconds a card at 100 (less lower down), so early in a hand she solves a handful of worlds and by the last few cards, hundreds. The code is utilities/marta_mind.py.

The proof

Otto played Marta 10,000 seeded games twice: once laying hands down when the referee called them, once playing every card to the end. Every hand's bids, tricks, scores and bags had to match between the two runs.

Called with this many cards left: 2 23,5873 14,7004 7,8085 3,8886 1,6437 6918 2659 14310 3

The card ladder, above 80

Thinking alone takes her to about 80% against the practice-table bot, and everything up to that point she does on public information. Above 80 the ratchet starts handing her something she has not earned at the table: at 82 she is shown one of your ten cards, at 84 two, and so on to all ten at 100. The cards are drawn at random when the hand is dealt and stay the same for that whole hand, her bid included, so what she knows never jumps mid-hand.

It rides the same machinery as her thinking rather than sitting beside it. A card she has been shown is simply evidence, so it is pinned into every hand she deals you and the rest is sampled as before. That is why the top of the dial is not a different Marta but the same one with less to guess about, and why all ten cards means exactly one possible hand, which is perfect play.

Your game-over screen says how many cards she is being shown, and the number is on this page rather than hidden in the code, because a disclosed advantage is a challenge and a hidden one is a cheat.

What each card is worth was measured before the rungs were placed, on 200 identical deals at each count. One or two cards barely narrow what she is already working out for herself, so the ladder starts at three and the climb is in the back half.

Cards she is shown	Marta won	Otto won	Marta win rate
0 of 10	154	46	77.0%
2 of 10	156	44	78.0%
4 of 10	162	38	81.0%
6 of 10	174	26	87.0%
8 of 10	181	19	90.5%
10 of 10	192	8	96.0%

Marta's dial, measured

Marta's strength (0 to 100) is the number the ratchet moves after each of your games. This is the proof that a higher number is a harder Marta: easy Otto played her 1,000 seeded games at each setting, the same deals every time, so only Marta changed.

Marta at	Marta won	Otto won	Marta win rate
0	506	492	50.6%
10	530	467	53.0%
20	578	417	57.8%
30	588	410	58.8%
40	611	386	61.1%
50	617	377	61.7%
60	649	349	64.9%
65	81	39	67.5%
70	91	28	75.8%
75	92	28	76.7%
80	88	32	73.3%
85	94	26	78.3%
90	105	15	87.5%
95	107	13	89.2%
100	113	7	94.2%

The rows up to 60 are 1,000 deals each. The thinking rows above it are 120 deals, which carries about four points either way, so a single step that dips is the sample and not a weaker Marta. The climb across the band is far bigger than that: she takes about two games in three at 60 and nineteen in twenty at 100.

Forty calls, played out

Chosen for variety: every size from ten cards down to two, both winners, the special cards in play. Otto sits in the player's seat. Under each call is the played-out result from the second run.

Regenerated by utilities/referee_proof.py, last run 2026-09-08.