Liar's poker

:"For the book by Michael Lewis see Liar's Poker."

Liar's poker is a bar game that combines statistical reasoning with bluffing, and is played with the eight-digit serial number on a dollar bill. Normally the game is played with a stack of random bills obtained from the cash register. The object is to make the highest bid of a number that does not exceed the combined total held by all the players. The numbers are usually ranked in the following order: 2,3,4,5,6,7,8,9,0 (10) and 1 (Ace). If the first player bids three 6's, he is predicting there are at least three 6's among all the players, including himself. The next player can bid a higher number at that level (three 7's), any number at a higher level (four 5's) or challenge. The end of the game is reached when a player makes a bid that is challenged all around. If the bid is successful, he wins a dollar from each of the other players, but if the bid is unsuccessful, he loses a dollar to each of the other players.

Liar's dice is a similar game played with dice, often as a drinking game.

Liar's Poker probabilities

The chances that the other players have at least the amount of a number you need to be able to call your bid when challenged, can be determined by the following two formulae:

Formula 1. P(at least X times C) = 1 - binomcdf (Y , 0.1 , X-1)
With:
X = amount of the needed number
C = the needed number, which has a probability of 1/10 = 0.1
Y = the amount of unknown numbers, which is equal to 8 x amount of extra players

"Example 1:" you are playing a 2-player game and you want to determine whether the other player has at least 2 more sixes.
P(at least 2 times six) = 1 - binomcdf (8 , 0.1 , 1) = 0.18670...
So you have a chance of 18.69% that the other player has at least 2 sixes

"Example 2:" you are playing a 5-player game and you want to determine whether the other players have at least 4 more sevens.
P(at least 4 times seven) = 1 - binomcdf (32 , 0.1 , 3) = 0.3997...
So you have a chance of 39.97% that the other 3 players have at least 4 sevens.

Formula 2. In order to calculate the probability of "at least" X times C, you have to subtract each probability from X=1 till X=X-1 from 1.

P(X times C) = Y ⁿC_r X x 0.1^X x 0.9^Y-X
With:
X = amount of the needed number
C = the needed number, which has a probability of 1/10 = 0.1
Y = the amount of unknown numbers, which is equal to 8 x amount of extra players

"Example:" you are playing a 2-player game and you want to determine whether the other player has at least 2 more sixes.
P(at least 2 times six) = 1 - P(no six) - P(1 six)
P(no six) = 8ⁿC_r0 x 0.1⁰ x 0.9⁸ = 0.4305
P(1 six) = 8ⁿC_r1 x 0.1¹ x 0.9⁷ = 0.3826

P(at least 2 times six) = 1 - 0.4305 - 0.3826 = 0.18670...
So you have a chance of 18.69% that the other player has at least 2 sixes

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Overview probabilities of the at least needed amount of a specific number for a 2-player game to a 6 player game.So for example if you need 3 more of a specific number, the chances in a 2 player game are 4%, in a 3 player game 21%, in a 4 player game 44%, et cetera.

Liar's Poker tactic - 'Damned if I do, damned if I don't' - situation

In Liar's Poker it's all about bluffing, just as with regular poker. There are some tactics which are mathematically based and should be fully understood by the players in order to make it a full bluffing game.

Above are the odds that the other players have at least a specific amount of a needed number. It is possible that a player comes in a so-called 'damned if I do, damned if I don't' situation. Assuming that by challenging you will definitely lose, and by raising you will definitely be challenged while not being able to call your bid, you should always raise in a 2-player game, raise in a 3-player game if your odds are above 25%, raise in a 4-player game if your odds are above 33.33% or, in other words, raise in a n-player game if you odds are above (n-2)/(2n-2).

"Example: " You're in a 4-player game. Your serial number is 53653158. The last bid was 7 threes, which you deem is highly possible, since you already hold 2 threes. You can outbid by bidding 7 fives. You need 4 more fives to be able to call your bid, which is a chance of 40%. The tactic above states that you should raise if your odds (40%) are above (n-2)/(2n-2), with n being the amount of players. (4-2) / (2x4 -2) = 0.3333. x100% is 33.33%, which does not surpass your odds of 40%, so statistically you should raise.

Overview of probabilities which need to be surpassed to raise in a "Damned if I do, damned if I don't" - situation:

As is stated before, Liar's Poker is all about bluffing, so you shouldn't stick closely to these statistics and tactics.

Example game

If every player follows the exact mathematical formulae, a possible game is the following. Keep in mind that the order of least to most valuable number is 2-3-4-5-6-7-8-9-0-1.

Player 1: 21068274
Player 2: 44789800
Player 3: 27706500
Player 4: 63523655

Player 1 begins

Player 1: 3 twos (has 2 two's - 92% chance others have another two)
Player 2: 4 fours (has 2 fours - 71% chance others have another two fours)
Player 3: 4 zeros (has 3 zeros - 92% chance others have another zero)
Player 4: 5 fives (has 3 fives - 71% chance others have another two fives)
Player 1: Challenge (can only outbid if others have at least 4 more of two, six, seven or eight, which is a chance of 21%, and 21%<33%)
Player 2: 5 zeros (has 2 zeros - 44% chance others have another three zeros)
Player 3: 6 zeros (has 3 zeros - 44% chance others have another three zeros)
Player 4: Challenge (can only outbid if others have at least 4 more fives, which is a chance of 21%, and 21%<33%)
Player 1: Challenge (can only outbid if others have at least 5 more twos, which is a chance of 9%, and 9%<33%)
Player 2: Challenge (can only outbid if others have at least 7 more fours, eights or zeros, which is a chance of 1%, and 1%<33%)

Player 3 has been challenged by all the other players. Each player tells his amounts of zeros. For Player 3 to win, together they have to have at least 6 zeros. They have exactly 6, so Player 3 wins and the other Players have to pay him the agreed amount.

This game was played with four players who fully understood and applied the mathematical formulae, but in Liar's Poker it's about bluffing and trying to influence other players' decisions to your benefit, while keeping these statistics in the back of your mind.

In popular culture

*In his 1989 book "Liar's Poker", Michael Lewis details how Salomon Brothers traders would play liar's poker.
*In the January 22nd edition (week-long) of Poker After Dark, Phil Hellmuth claims to have taught Daniel Negreanu how to play liar's poker.
*A version of liar's dice is played in Pirates of the Caribbean: Dead Man's Chest between Will Turner, Bootstrap Bill, and Davy Jones.
*A game of liar's poker was played in an episode of the TV series "Hustle" (Season 3, Episode 3) where one of the main characters plays and loses against two merchant bankers, the result of which is that he has to bend over and get the same number of strokes from the cane as the level the bidding reached (it got to nine 6's).
*Councilmen Tommy Carcetti and Tony Gray play a game of liar's poker interrupted by a visit from a Baltimore Sun reporter in an episode of the HBO TV series "The Wire" (Season 3, Episode 3, "Dead Soldiers")
*Liar's poker was played in an episode of the TV series "WKRP in Cincinnati" (Season 2, Episode 15 "Herb's Dad") where Herb Tarlek Sr. bested Venus Flytrap and Dr. Johnny Fever with the long-shot bid of nine 6's. Herb Tarlek Jr. later beat Venus and Johnny with the same bid. It was then revealed that he used the same dollar as his father.
*Characters on the show Quincy M.E. were often seen playing Liar's poker.
* Anne O Faulk's "Holding Out" uses the game as a plot point.
* In the 1977 movie "Semi-Tough", Burt Reynolds' and Jill Clayburg's characters play an ongoing game of liar's poker periodically throughout the movie.
* In the 1973 movie " The Long Goodbye" Elliot Gould and Jim Bouton play in an early scene.
* in the 1965 movie "Cat Ballou", the game is mentioned at about minute 21.