Cheat (BS) Card Game & Bluff Calculator

The rank climbs 2, 3, 4 … K, A and round again whoever is playing, so you cannot choose what to claim — only how many cards to put down and whether they really are that rank. Click cards to select them, then Play. When somebody else plays you may challenge: a wrong challenge costs you the pile, a caught lie costs them. Everything runs in your browser.

A lying claim of four is always catchable

There are 4 cards of every rank. If you hold h of them and somebody claims to be playing k, then k + h > 4 makes the claim impossible — not unlikely, impossible. That one inequality carries most of the game, and it makes big claims the dangerous ones rather than the bold ones:

The bottom row is not a rounded ninety-nine-point-something. If you claim all four of a rank and do not hold all four, the ones you are missing are sitting in somebody's hand, and that somebody knows for a fact that you are lying. There is no deal on which it can be got away with — while a lying claim of one gets away with it 97.4% of the time.

A lying claim of three from somebody holding none of the rank is certain too, by the same kind of argument rather than by probability: all four cards are spread among the three opponents, and four cards in three hands forces one hand to hold two. Pigeonhole, not odds — which is why both of these come out at exactly one rather than nearly one.

Note the gap between the two columns. Against a three-card claim a single opponent is sure only 25.7% of the time, but somebody at the table is sure 73.7% of the time. Most of the certainty at a Cheat table is distributed — it exists, but not in your hand.

What the deal gives you

The four cards of a rank land in four random positions of 52 and you own thirteen of them, so the split is exact hypergeometric arithmetic. You hold exactly one of each rank on average — four cards shared four ways — but the average is the least likely single outcome after one:

Nearly a third of the time you hold none of a rank at all, which is what makes lying compulsory rather than optional in this game — the rank comes round whether you can meet it or not. And the rule works on a total: a card you hold and a card you have watched being revealed rule out a claim equally, because both say the same thing about how many are left.

And when you cannot be certain, there is no "challenge frequency"

The natural next question is what rate to challenge at against somebody who lies at some fixed rate q. It has no answer, because it presumes a mixed strategy that is never called for. Challenging wins the pile when they were lying and loses it when they were not, so at pile size S the expected swing is p · S · (2q − 1) — linear in p. Against a fixed opponent the best reply is always pure.

The threshold is exactly one half, and the pile size cancels completely — a big pile raises the stakes in both directions and moves no decision. That is worth knowing at the table, because a large pile feels like a reason to hold off and is not one.

Mixed strategies belong to the equilibrium against an opponent who adapts to you. Against a liar with a fixed rate, no intermediate rate can beat both pure ones — it can only tie them, at the threshold itself.