For the book by Michael Lewis, see Liar's Poker

Liar's Poker

Liar's Poker is a non-fiction, semi-autobiographical book by Michael Lewis describing the author's experiences as a bond salesman on Wall Street during the late 1980s...

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For the card game sometimes known as Liar's Poker, see Commune (card game)

Commune (card game)

Commune is a bluffing-based card game that requires knowledge of the poker hands and much strategic thinking. It is best played in a large group of people, and does not require use of a table or playing surface.- Object :...

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Liar's poker is a bar game that combines statistical reasoning with bluffing, and is played with the eight-digit serial number on a U.S. dollar bill

United States one-dollar bill

The United States one-dollar bill is the most common denomination of US currency. The first president, George Washington, painted by Gilbert Stuart, is currently featured on the obverse, while the Great Seal of the United States is featured on the reverse. The one-dollar bill has the oldest...

. Normally the game is played with a stack of random bills obtained from the cash register. The objective 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 6s, he is predicting there are at least three 6s among all the players, including himself. The next player can bid a higher number at that level (three 7s), any number at a higher level (four 5s) 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

Liar's dice

Liar's dice, or Liar dice, with roots originating in South America and popularized in early Spanish History, was brought to Spain by the Spanish conqueror Francisco Pizarro during the 16th century. It became extremely popular in Hong Kong and consequencially China...

 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 4 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.

Number Needed	Extra players
	1 player	2 players	3 players	4 players	5 players
1	0.56	0.81	0.92	0.97	0.99
2	0.19	0.49	0.71	0.84	0.92
3	0.04	0.21	0.44	0.63	0.78
4	0.01	0.07	0.21	0.40	0.58
5	0.00	0.05	0.09	0.21	0.37
6	0.00	0.00	0.03	0.09	0.21
7	0.00	0.00	0.01	0.04	0.10
8	0.00	0.00	0.00	0.01	0.04

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

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 5-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 number of players. (5-2) / (2x5 -2) =0.375x100% = 37.5%<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:

	2-player game	3-player game	4-player game	5-player game	6-player game
(n-2)/(2n-2)	always raise	0.25	0.33	0.38	0.40
max. needed numbers	always raise	2 or less needed	3 or less needed	4 or less needed	4 or less needed

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 twos - 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 5 more fours, eights or zeros, which is a chance of 9%, and 9%<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
  Liar's Poker
  Liar's Poker is a non-fiction, semi-autobiographical book by Michael Lewis describing the author's experiences as a bond salesman on Wall Street during the late 1980s...
  
  , Michael Lewis
  Michael Lewis (author)
  Michael Lewis is an American non-fiction author and financial journalist. His bestselling books include The Big Short: Inside the Doomsday Machine, Liar's Poker, The New New Thing, Moneyball: The Art of Winning an Unfair Game, The Blind Side: Evolution of a Game, Panic and Home Game: An...
  
   details how Salomon Brothers
  Salomon Brothers
  Salomon Brothers was a bulge bracket, Wall Street investment bank. Founded in 1910 by three brothers along with a clerk named Ben Levy, it remained a partnership until the early 1980s, when it was acquired by the commodity trading firm Phibro Corporation and then became Salomon Inc. Eventually...
  
   traders would 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
  Hustle (TV series)
  Hustle is a British television drama series made by Kudos Film and Television for BBC One in the United Kingdom. Created by Tony Jordan and first broadcast in 2004, the series follows a group of con artists who specialise in "long cons" – extended deceptions which require greater commitment, but...
  
  (Season 3, Episode 3) where one of the main characters plays and loses against two merchant bankers.
• 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
  Semi-Tough
  Semi-Tough is a 1977 film directed by Michael Ritchie and starring Burt Reynolds, Kris Kristofferson, Jill Clayburgh, Lotte Lenya, Bert Convy, and Brian Dennehy. The plot involves a love triangle between the characters portrayed by Reynolds, Kristofferson and Clayburgh...
  
  , Burt Reynolds
  Burt Reynolds
  Burton Leon "Burt" Reynolds, Jr. is an American actor. Some of his memorable roles include Bo 'Bandit' Darville in Smokey and the Bandit, Lewis Medlock in Deliverance, Bobby "Gator" McCluskey in White Lightning and sequel Gator, Paul Crewe and Coach Nate Scarborough in The Longest Yard and its...
  
  ' and Jill Clayburg's characters play an ongoing game of liar's poker periodically throughout the movie.