Liar's poker

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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 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.


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[edit] 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 substract each probability from X=1 till X=X-1 from 1.

P(X times C) = Y nCr X x 0.1X x 0.9Y-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) = 8nCr0 x 0.10 x 0.98 = 0.4305
P(1 six) = 8nCr1 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

Overview probabilities of the at least needed amount of a specific number for a 2-player game to a 6 player game.

At least needed amount / 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.

[edit] Liar's Poker tactic - 'Damn if I do. Damn if I don't'-situation

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

Above is described what the odds are 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 'damn if I do, damn 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 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 probabilities which need to be surpassed to raise in a "Damn if I do. Damn 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 only hold on on these statics and tactics.

[edit] 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 player's decisions in your benifit, while keeping these statistics in the back of your mind.

[edit] The Most Famous Game Never Played

One hand, one million dollars, no tears. This now famous quote was John Gutfreund's challenge to John Meriwether, two titans of 1980s Wall Street.

Quoted in Michael Lewis' book Liar's Poker. John Gutfreund then CEO of Salomon Brothers - an investment bank that gained notoriety in the 1980s. As CEO, Gutfreund became the icon for the excess that defined the 1980s culture in America. John Meriwether founder of Long-Term Capital Management, an initially enormously successful hedge fund with annualized returns of over 40% in its first years, in 1998 it lost $4.6 billion in less than four months and became a prominent example of the risk potential in the hedge fund industry.

What Gutfreund said has become a legend at Salomon Brothers, and a visceral part of its corporate identity. He said: 'One hand, one million dollars, no tears.' Meriwether grabbed this meaning instantly. The King of Wall Street, as Business Week had dubbed Gutfreund, wanted to play a single hand of a game called Liar's Poker for a million dollars. He played the game most afternoons with Meriwether and the six young bond arbitrage traders who worked for Meriwether, and was usually skinned alive. Some traders said Gutfreund was heavily outmatched. Others who couldn't imagine John Gutfreund as anything but omnipotent - and there were many - said that losing suited his purpose, though exactly what that might be was a mystery.

The peculiar feature of Gutfreund's challenge this time was the size of the stake. Normally his bets didn't exceeded a few hundred dollars. A million was unheard of. The final two words of his challenge, 'no tears', meant that the loser was expected to suffer a great deal of pain, but wasn't entitled to whine, bitch or moan about it. He'd just have to hunker down and keep his poverty to himself. But why? you might ask if you were anyone other that the King of Wall Street. Why do it in the first place? Why, in particular, challenge Meriwether instead of some lesser managing director? It seemed an act of sheer lunacy. Meriwether was the King of the Game, the Liar's Poker champion of the Salomon Brothers' trading floor.

On the other hand, one thing you learn on a trading floor is that winners like Gutfreund always have some reason for what they do; it might not be the best of reasons, but at least they have a concept in mind. I was not privy to Gutfreund's innermost thoughts, but I do know that all the boys on the trading floor gambled, and that he wanted badly to be one of the boys. What I think Gutfreund had in mind in this instance was a desire to show his courage, like the boy who leaps from the high dive. Who better than Meriwether for the purpose? Besides, Meriwether was probably the only trader with both the cash and the nerve to play.

(...)

The code of the Liar's Poker player was something like the code of the gunslinger. It required a trader to accept all challenges. because of the code - which was his code - John Meriwether felt obliged to play. But he knew it was stupid. For him, there was no upside. If he won, he upset Gutfreund. No good came of this. But if he lost, he was out of pocket a million bucks. This was worse than upsetting the boss. Although Meriwether was by far the better player of the game, in a single hand anything could happen. Luck could very well determine the outcome. Mertiwether spent his entire day avoiding dumb bets, and he wasn't about to accept this one. 'No, John,' he said, 'if we're going to play for those kind of numbers, I'd rather play for real money. Ten Million dollars. No tears.'

Ten million dollars. It was a moment for all players to savour. Meriwether was playing Liar's Poker before the game even started. He was bluffing. Gutfreund considered the counter proposal. It would have been just like him to accept. Merely to entertain the thought was a luxury that must have pleased him well. (It was good to be rich.)

On the other hand, 10 million dollars was, and is, a lot of money. If Gutfreund lost, he'd have only 30 million or so left. (...) So Gutfreund declined. In fact, he smiled his own brand of forced smile and said, 'You're crazy.' No, thought Meriwether, just very, very good.

[edit] In popular culture

  • 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.