Dave Computes Opening Betting in Poker with an Ante Structure

In a blind structure, a player should generally open with a raise or fold. Once a player folds, it's as if they were never in the hand to any players who have left to act.

Things are more complicated in an ante structure, where a player can open with a bet or check. Later players must consider that they may be beat by an earlier player who checked.

2 Players

P1 (the first player to act) assumes P2 (the second to act) is likely to have a 50th-percentile hand. Thus, P1 expects a top-50% hand is likely to win.

If P1 bets any top-50% hand, P2 figures that P1 is likely to have a hand in the middle of that range: a 25th-percentile hand. Thus, P2 needs a top-25% hand for a likely win. P2 might therefore raise with a top-25% hand and fold the rest.

Alternatively, if P1 checks a bottom-50% hand, then P2 figures that P1 is likely to have a hand in the middle of that range: a 75th-percentile hand. Thus, P2 only needs a top-75% hand for a likely win. P2 might therefore open by betting a top-75% hand and checking the rest.

We can summarize this play as:

P1 bet top 50% -> P2 raise top 25%

P1 check bottom 50% -> P2 bets top 75%

(Of course, we're simplifying things considerably by ignoring possibilities of check-raising or bluffing.)

3 Players

Suppose P1 has a top-10% hand. There's a 90% chance they've got P2 beat and a 90% chance they've got P3 beat. That's a 0.9 * 0.9 = 81% chance they've got both players beat. Suppose P1 holds a top-x hand For an even chance of holding the best hand, (1 - x)(1 - x) = 0.5, which is satisfied when x = 29%. Therefore, P1 might open by betting a top 29% hand and checking the rest.

Suppose P1 bets. P2 can expect to beat P1 with a top-14.5% hand (29/2), but things are worse when we consider that P3 might still beat such a hand. P2 needs a top-12% hand for a likely win over both players.

Alternatively, if P1 checks, P2 expects P1 to have a bottom-71% hand and P3 to have any hand. P2 needs a top-40% hand for a likely win over both now.

Now considering all situations that P3 might face, we can summarize this play as:

P1 bets top 29% -> P2 raises top 12% -> P3 reraises top 4%

P1 bets top 29% -> P2 folds bottom 88% -> P3 raises top 14%

P1 checks bottom 71% -> P2 bets top 40% -> P3 raises top 19%

P1 checks bottom 71% -> P2 checks bottom 60% -> P3 bets top 47%

4 Players

P1 bets top 20% -> P2 raises top 8% -> P3 reraises top 3% -> P4 caps top 1%

P1 bets top 20% -> P2 raises top 8% -> P3 folds -> P4 reraises top 3%

P1 bets top 20% -> P2 folds bottom 92% -> P3 raises top 9% -> P4 reraises top 3%

P1 bets top 20% -> P2 folds bottom 92% -> P3 folds bottom 91%- > P4 raises top 9%

P1 checks bottom 80% -> P2 bets top 26% -> P3 raises top 11% -> P4 reraises top 4%

P1 checks bottom 80% -> P2 bets top 26% -> P3 folds -> P4 raises top 12%

P1 checks bottom 80% -> P2 checks bottom 74% -> P3 bets top 33% -> P4 raises top 16%

P1 checks bottom 80% -> P2 checks bottom 74% -> P3 checks bottom 67% -> P4 bets 41%

5 Players

P1 bets top 15% -> P2/P3/P4/P5 raises top 5 - 7%

P1 checks bottom 85% -> P2 bets top 19% -> P3/P4/P5 raises top 7 - 9%

P1 checks bottom 85% -> P2 checks bottom 81% -> P3 bets top 23%

-> P4/P5 raises top 10 - 11%

P1 checks bottom 85% -> P2 checks bottom 81% -> P3 checks bottom 77%

-> P4 bets top 28% -> P5 raises top 13%

P1 checks bottom 85% -> P2 checks bottom 81% -> P3 checks bottom 77%

-> P4 checks bottom 72% -> P5 bets top 33%