Week 9: Multi-way pots

Which of this course's two-player results survive a third player at the table, tested against a real Monte Carlo simulation rather than assumed

19 April 2027

An audit, not a new formula#

Every result so far — pot odds, Bayesian updating, MDF — was built for exactly one opponent. Adding a second doesn’t invalidate all of it; it invalidates the parts that quietly assumed “one opponent” without saying so. This week is that audit.

What survives unchanged#

The pot-odds formula itself — required equity = call / (pot + call) — never assumed a player count. It only needs hero’s true equity to beat everyone still in the hand. Bayes’ rule is the same: nothing about it requires exactly one hidden variable, only that updating now happens over a joint distribution of multiple opponents’ likely holdings instead of one. Both formulas survive; what changes is how expensive the equity or the update is to actually compute.

What doesn’t survive cleanly: MDF#

Week 7’s minimum defense frequency came from a single indifference equation between exactly one bettor and one caller. With a second player still to act behind hero, hero’s own calling frequency now interacts with what that player does too — a call correct against one opponent can invite a squeeze from the other. A clean single-equation multi-way MDF is not something this course derives; flagging that gap here rather than quietly reusing the two-player formula as if it still applied.

The one folk shortcut this week actually tests#

A natural guess for multi-way equity: if a hand has 50.4% equity against one random hand (week 1’s own pocket-2s number), a tempting shortcut is squaring that figure for two opponents, cubing it for three — treating each opponent as an independent 50.4%-equity coin flip.

What the simulation actually says#

Running the same seeded Monte Carlo harness from week 1 (300,000 trials, seed 42), extended to deal multiple opponents from the same board:

The independence shortcut underestimates multi-way equity, and by a growing margin — the same “the shortcut isn’t wrong, it’s the wrong shape” pattern week 3 found in the rule of 4 and 2.

Why the shortcut misses#

Opponents’ hands aren’t independent trials once they share a board: when the board doesn’t help hero, it often doesn’t help every opponent equally either. Treating each opponent as an independent coin flip throws that correlation away, and it happens to throw away equity hero actually has.

Before next week#

Multi-way equity is still a single-hand snapshot — one pot, decided by one showdown. Week 10 moves to a setting where winning a hand isn’t the same as winning value: a tournament, where chips are worth less than their face count once elimination is on the table.