The rules form a ladder. The bottom rung never bids unless the dealer is forced to; each rung above adds one rule. Every rung was played against a team that bids like the bot, on the same deals, each deal twice with the seats swapped so that card luck cancels. The cards were played the same way at all four seats, so bidding was the only difference.
Two columns, because a hand is worth more to a better card player: basic card play is a simple rule-based player (lead your highest, win the trick cheaply if you can); expert card play is the bot's own search. Most people fall between the two.
What each rule buys
| # | Rule | Basic card play | Expert card play | ||
|---|---|---|---|---|---|
| buys | vs the bot | buys | vs the bot | ||
| 0 | Never bid unless you are the dealer and everyone else has passed. | – | −1.14 (0%) | – | −2.03 (0%) |
| 1 | If nobody has bid, bid 4 when you hold the ace of your suit. | +0.27 ± 0.03 | −0.87 (24%) | +0.67 ± 0.06 | −1.35 (33%) |
| 2 | If an opponent has bid 4, bid 5 when you hold the ace of your suit. | +0.53 ± 0.03 | −0.34 (70%) | +0.94 ± 0.06 | −0.41 (80%) |
| 3 | If an opponent has bid 5, bid 6 when you hold the ace of your suit and another top card. | +0.18 ± 0.02 | −0.16 (86%) | +0.30 ± 0.04 | −0.12 (94%) |
| 4 | For average card players: do not bid when the ace is your only top card and you also hold the 3 of your suit. | +0.05 ± 0.01 | −0.12 (90%) | −0.07 ± 0.02 | −0.18 (91%) |
| 5 | If an opponent has bid 4 and you do not hold the ace of your suit, still bid 5 with three top cards. | +0.04 ± 0.01 | −0.08 (93%) | +0.09 ± 0.03 | −0.09 (96%) |
| 6 | If your table competes for the bid: unless you are the dealer, open at 5 instead of 4 when you hold the ace of your suit and another top card. | +0.12 ± 0.02 | +0.05 (104%) | +0.29 ± 0.05 | +0.20 (110%) |
| For comparison: the baseline bot's own rules, which bid by counting trump | – | −1.04 (8%) | – | −1.32 (35%) | |
"Buys" is points per hand gained over the rung below, with its 95% interval. "Vs the bot" is points per hand against a team bidding like the bot, where 0 means bidding as well as it does; the percentage is how far that is from never bidding (−1.14 and −2.03) to the bot (0).
Reading it
- The ace is the whole first rule. One card to look for decides the opening bid.
- Rule 2 is worth more than rule 1. Letting an opponent keep the bid at 4 when you hold an ace is the most expensive habit measured.
- Rules 3 and 6 are one idea. The ace plus another top card is a strong hand: bid 6 over an opponent's 5 with it, and open at 5 so an opponent cannot take the bid cheaply.
- Three rules do most of the work. They close 86% of the gap with basic card play and 94% with expert card play.
- Counting trump does little. The baseline bot's rules close 8% with basic card play.
Two rules with conditions
Rule 4 (pass on the ace alone when you also hold the 3) helps with basic card play (+0.05 ± 0.01) and costs with expert card play (−0.07 ± 0.02). The 3 is worth three points to whoever captures it, and a lone ace cannot protect it; a strong card player can.
Rule 6 (open at 5 with the ace and another top card) is the one rule whose value depends on who is across the table. Against opponents who compete for the bid it stops them taking it cheaply at 5. Against opponents who rarely bid it costs: the table below shows both.
Against different tables
The same ladder played against other kinds of opponents, basic card play, 50,000 deals each. Points per hand for a team using the rules down to that row; in brackets, what that row's rule buys against that opponent.
| Rules used | Never bid | Count-your-trump bidders | A team using rules 1 and 2 | A team using rules 1 to 3 | A team using rules 1 to 5 | A team using all six rules |
|---|---|---|---|---|---|---|
| 0. Never bid | +0.00 | −0.16 | −1.00 | −1.00 | −1.07 | −0.85 |
| 1. Open at 4 with the ace | +1.00 (+1.00) | +0.63 (+0.79) | −0.75 (+0.26) | −0.75 (+0.26) | −0.83 (+0.24) | −0.65 (+0.20) |
| 2. Bid 5 over a 4 with the ace | +1.00 (+0.00) | +0.88 (+0.25) | +0.00 (+0.75) | −0.04 (+0.70) | −0.13 (+0.70) | −0.53 (+0.12) |
| 3. Bid 6 over a 5 with ace + 1 | +1.00 (+0.00) | +0.88 (−0.00) | +0.04 (+0.04) | +0.00 (+0.04) | −0.10 (+0.03) | −0.13 (+0.40) |
| 4. Lone ace with the 3: pass | +1.07 (+0.07) | +0.95 (+0.07) | +0.10 (+0.06) | +0.05 (+0.05) | −0.06 (+0.04) | −0.09 (+0.04) |
| 5. No ace, 3 top cards: bid 5 over a 4 | +1.07 (+0.00) | +0.95 (+0.01) | +0.13 (+0.03) | +0.10 (+0.04) | +0.00 (+0.06) | −0.08 (+0.01) |
| 6. Open at 5 with ace + 1 | +0.85 (−0.22) | +0.82 (−0.14) | +0.53 (+0.40) | +0.13 (+0.03) | +0.08 (+0.08) | +0.00 (+0.08) |
With expert card play, against count-your-trump bidders (10,000 deals): 1. Open at 4 with the ace +0.38; 2. Bid 5 over a 4 with the ace +1.05; 3. Bid 6 over a 5 with ace + 1 +1.10; 4. Lone ace with the 3: pass +1.01; 5. No ace, 3 top cards: bid 5 over a 4 +1.10; 6. Open at 5 with ace + 1 +1.02.
- Rules 1, 2 and 5 never cost against any opponent tried; rule 3 does nothing against passive opponents and gains against competitive ones.
- Against a typical trump-counting table the first three rules are worth about a point a hand.
- The rules do not beat a team using the same rules, as they should not.