Any number of people may play Liars Poker, but the optimal number of players seems to be three to six. The exact statement of some of these rules depends upon the number of players. Where necessary, we will use n to refer to the number of players.

To begin the game, each player obtains a random eight digit number. The most common method is to have each player choose a bill (of US currency, generally a one dollar bill (A more primitive version of Liars Poker is sometimes called "dollar poker.")). A player's number for that round is the serial number on the bill selected. Play begins as one player makes the opening bid. A typical bid might be "5 sevens." This means that the player estimates that the total number of sevens in all players’ numbers, including his own, is at least 5. The turn then passes clockwise to the next player on the left. For his turn, each player must either make a stronger bid or challenge the previous bid. A bid is stronger if it calls for at least the same number of occurrences of a higher rank (e.g., "5 nines") or a greater number of occurrences (e.g., "6 threes"). The zero is considered the highest rank (usually referred to as "ten" as in "7 tens").

Eventually, a bid will be challenged by all the other n-1 players. At this time, the bidder may ask for the count, and then each player reveals how many of the selected rank he has. If the total number equals or exceeds the number bid, the bidder wins one unit from each of the other players. If the bid is not made, then the bidder loses one unit to each of the other players. Whether or not the bid is made, the final bidder is the first bidder in the next round.

The most important enhancement of the basic game is the right of rebidding. If a player's bid is challenged by all of the other players, he has the option of playing this bid or of making a new, stronger bid. However, only one rebid is allowed. If this new bid is also challenged by all the players, the bidding then stops and this new bid is the final bid. If this new bid is not challenged by all of the remaining bidders and one of them makes a stronger bid, the bidding continues with each player, including the bidder who just made a rebid, having the right to rebid if challenged all the way around.

The allowance of rebids greatly extends the strategic scope of Liars Poker. The necessity to bluff and determine if others are bluffing are major features of the game.

The greater the number of occurrences bid for, the more difficult it is to make the bid. To encourage higher bidding, the odds are tilted in favor of the bidder for high bids. To give a greater incentive to bid, and to generally spice things up, if a bid of n+3 of a kind is made, the successful bidder wins two units from each player instead of one unit. Thus, in a four-person game, a bid of '7 fours" is worth double. However, if the bidder is unsuccessful at this level, he only loses one unit to each of the other players. Similarly, a successful bidder of n+5 of a kind wins three units from each player while only risking one unit to each player should he not make the bid. A bid of n+7 of a kind gets four to one odds; n+9 gets five to one odds, etc.

Due to fervently asserted but statistically unverified beliefs that sixes are more difficult to make than other ranks, any bid of sixes is given preferential odds of twice what the bid would otherwise get. For example, in a five player game, a bid of "7 sixes" will net the bidder two units from each player if successful, but only cost him one unit if unsuccessful. A bid ·of "10 sixes" in a five-person game is especially attractive: the bidder can only lose one unit to each player but he can win six units from each player (a multiplier of three because of the n+5 rule and a multiplier of two due to the sixes rule for a total multiplier of two times three, i.e., six).

Occasionally one can make a successful bid without having any of the number bid. Such a "heroic" bid entitles the bidder to a bump in the multiplier-the bid is worth one more than it would otherwise be. For example, in a five-person game, if a person bids and makes "6 sixes" without having any sixes himself, he wins three units from each player (the multiplier of two due to the sixes rule is bumped to three because of the hero rule).

If no player has any of the number bid for (including the bidder), instead of losing, the bidder wins. The multiplier is 2n-6, independent of how many are bid or whether or not the bid is for sixes. Thus, in a five person game, if the skunk rule applies, the bidder wins four from each player, whether the bid is "6 threes", "7 sixes", or "8 twos." In a three person game, the skunk rule multiplier is zero-if there are none and the game is a push. In a two person game, the skunk rule does not apply.

The normal stake for each round is one unit, though it is commonly agreed by all players involved to increase the stakes at times; a usual agreement is that the last hand of the night is for double stakes. Progressive stakes is a formal way of varying the basic stake throughout the session. The basic stake for each round is determined by the prior hand. It is the unit multiplier the bidder won or would have won. For example, in a five-person game, suppose the hand's final bid is "8 sixes." By the n+3 rule and the sixes rule, this bid merits a multiplier of four. Whether or not this bid is actually made, the basic stake for the next hand is four units. Suppose in this next hand the final bid is "8 threes." This bid is a double by the n+3 rule. Since the basic stake is four units for this hand, if the bid is made, the bidder wins eight units from each player. If the bidder is unsuccessful, he loses four units to each player. For the following hand, the stake is again increased, this time to two units (even though a successful bid would have won eight units per player, the multiplier due to the bid was only two units). If the final bid in this hand is "6 twos." the stake for the following hand reverts back to one unit. An exception is generally made for the skunk- rule: after a player "goes for the none," the next hand is just a double, independent of the number of players. If a player has none, he is presumed to be going for the none and not the hero for purposes of determining the stake for the next hand.

When using US currency for playing Liars Poker, there are certain conventions the adherence to which demonstrates to others that a player is truly cultured. First, the selection of new bills for the next round must be done with proper decorum. When starting a new session, the choice of the first bill goes to the player whose fortunes as of late have been most disappointing; Thereafter, the final bidder from the previous round has the honor of choosing the first bill. The choice then rotates clockwise to the next player and continues around until all players have chosen their bills. In the first hand, the opening bidder is determined by looking at the two letters which flank the serial number. The lowest pair by alphabetical order is given the first bid. Thereafter, the final bidder of the previous hand starts off the bidding in the next round.

No matter how crazy the bidding may get, it eventually comes to an end. At this time, the players commence the count. The historical method, still widely observed, is for the remaining players, starting with the player to the bidder's left and then continuing clockwise, to hold up their hand with as many fingers raised as they have digits with the correct rank. A defiant fist indicates the utter absence of the desired number. "Getting fisted" is finding all the players around you waving their fists at you - certainly a cause for dejection, unless you can turn the tables on them and skunk them!

Like most truly valuable pieces of our culture, Liars Poker isn't immune to the onrush of new technology. Even given rather aggressive assumptions about future inflation and money supply growth, it was becoming increasingly obvious that the available supply of bills would be exhausted long before the appetites of Liars Poker players could be satisfied. The solution-- strips of computer generated random numbers known as Salomon's Liars Poker Strips (SLIPS™)-- arrived not a moment too soon. The use of these sheets of fifteen random numbers has necessitated several additional conventions to preserve the integrity of the game.

Play begins with each player choosing a SLIP™ of fifteen numbers, using the same conventions described above for choosing bills. In the first round, each player uses the first number at the top of his SLIP™. To determine which number is used for the second hand, each player gives the parity (odd/even) of the last digit of the number just played. For each odd parity, one un-played number is skipped. If the bottom of the SLIP is reached, the skipping wraps around back to the top. Only ten numbers are played on each SLIP™. The idea of skipping is so that with progressive stakes, players cannot set up higher stakes when they know the next number they’ll be playing will be a juicy one, like a number containing four 6’s. The tenth number played is for double the stakes it otherwise would have been played for.

LP goes back at least to 1972, when a game of liars poker featured at about the 12th minute of the movie ‘The Long Goodbye” with Elliot Gould. Also, the game has been played with different rules at different places e.g. the story of Aaron Brown at Kidder told in his book, the Poker Face of Wall St, and the Quants. Other variants are ‘I doubt it’ and Perudo. Wiki entry on the game here: https://en.wikipedia.org/wiki/Liar%27s_poker

The basic math is pretty straightforward, and we all had a good sense of the probability distribution of digits in a random selection of serial numbers. Let’s say you’re playing in a 6 person game, so there are 5 other players besides yourself. Say that you have 3 8s. How many 8s are there likely to be among the other 5 players? Well, there are 5 times 8, or 40 digits in total, and so we’d expect there would be 4 of each digit among those players. In fact, the probability that there are 4 or more occurrences of a given digit is a bit higher than 50%, as you can see from the full calculation. Most players probably did these calculations, but what’s really important is having a sense for the distribution of occurrences conditional on being challenged.

Unconditional probability of x occurrences in n total number of digits = (9/10)n-x(1/10)x n!/(n-x)!/x!

The winner’s curse in LP results from the tendency to get challenged when the other players don’t have the number you’re bidding. For example, in a five man field, with 5 times 8, or 40 digits out there, you’d expect 4 of each digit, but conditional on people only challenging if they have 0 or 1 of the bid digit, the expected number of each digit is only 2 1/2.

Brad: 15101952 Leo: 39540096 Mike: 93004455

Brad starts the bidding with 3 5s, Leo challenges even though he has one 5, but figures Mike will go up, which he does, bidding 3 8s, a decoy. Brad has no 8s so challenges, and Leo has no 8s so challenges too. Mike now switches to 4 fives, hoping that Brad’s opening bid meant he had a few. Brad suspects Mike’s bluffing, and anyway, chances are Leo has none as he challenged first time around, and the chance of him having none is higher than him having 1, so Brad challenges as he thinks he has a high chance of losing 2 units if he bids 5 5s, vs a high chance of losing just one unit if he challenges 4 5s. Leo figures that Mike probably has two 5s but he’s not sure about Brad as all he did was start with 3 fives, so he challenges too. It’s 2nd time around, so Mike has to count, and there are five 5s, and so Mike wins, and Brad and Leo both wish they had gone up. Brad and Leo have to pay Mike one unit each.

LP also probably sharpened our trading skills. It even had some similarities to the calculus of bidding for US treasury bonds at auction, although that’s quite another story altogether. It was a zero sum game, and so you had to keep asking yourself what were the mistakes these other smart players might be making, and often one worried that the best one could hope for was to break even.

If you played enough, you could see a number of decision-making biases of the Kahneman-Tversky variety in action (we never heard of those guys at the time), including confirmation bias, anchoring, herding (we called it ‘glomming’), and the fallacy of sunk costs, driving some players to double down and bid more aggressively to try to dig themselves out of hole, and bidding that sometimes seemed aimed at minimizing regret rather than maximizing gains. Above all, people tend to be over-confident, much as 90% of drivers believe themselves to be above average. Most players didn’t like to see themselves as passive, and so didn’t like to follow a strategy heavy on challenging, which made the more passive approach a good one, and had the added benefit of allowing you to be a pretty dangerous snake in the grass on occasion. Sometimes a challenging strategy would come into vogue, and then it would be time to become a more active bidder (which may provide some answer to the topical question about what happens as index investing becomes bigger and bigger).

Other variations not mentioned in 1986 rules document: PYB (Pick your best), around the horn, elimination (usually for determining who would pay for a group dinner), roll ‘em.

A score sheet kept by the most meticulous of the group. It was an honor to be trusted with the sheet. Generally the sheet was settled monthly. It was not expected that the sheet keeper would keep or share statistics with players beyond the past month’s tally. On one occasion, the sheet keeper gave a life-to-date tally to one of the players, who then dropped out of the game forever more, much to the vexation of the rest of the players.

There are no apps we’d recommend, or good computerized players to practice against, and as far as we know, no solution to the game, but it does seems solvable with machine learning algorithms if not with other approaches too.

More statistics and analysis: http://wizardofodds.com/games/liars-poker/

Thanks to Yung Lim (an honoured former keeper of the sheet) for providing a copy of the 1986 Salomon rules.