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| '''Liar's dice''' is a class of [[dice games]] for two or more players requiring the ability to deceive and detect an opponent's deception. The genre has its roots in [[South America]], with games there being known as [[Dudo]], Cachito, Perudo or Dadinho; other names include "pirate's dice," "deception dice" and "diception."
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| In "common hand" liar's dice games, each player has a set of dice, all players roll once, and the bids relate to the dice each player can see (their hand) plus all the concealed dice (the other players' hands). In "individual hand" games, there is one set of dice which is passed from player to player. The bids relate to the dice as they are in front of the bidder after selected dice have been re-rolled. The [[drinking game]] version is sometimes called ''Mexicali'' or ''Mexican'' in the United States; the latter term may be a corruption of ''Mäxchen'' ("Little Max"), the name by which a similar game, [[Mia (game)|Mia]], is known in Germany, while Liar's dice is known in Germany as ''Bluff''. It is known by various names in Asia.
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| ==Common hand==
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| [[File:Perudo.jpg|right|thumb|Five six-sided dice are used per player, with dice cups used for concealment.]]
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| Five six-sided dice are used per player, with dice cups used for concealment.
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| Each round, each player rolls their dice under their cups and looks at their new hand while keeping it concealed from the other players. The first player begins bidding, picking a face and a quantity. The bid represents how many of the chosen face value the player believes are showing under all the cups, not just their own. Ones (aces) are often wild, and count as the face of the current bid (unless "ones" are the currently-bid face value).
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| Each player has two choices during his turn: make a higher bid, or challenge the previous bid (typically with a call of "Liar"). Raising the bid means either increasing the quantity, or the face value, or both, according to the specific bidding rules used. There are many variants of allowed and disallowed bids; common bidding variants, given a previous bid of an arbitrary quantity and face value, include:
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| * the player may bid a higher quantity of any face, or the same quantity of a higher face (allowing a player to "re-assert" a face value he believes prevalent if another player increased the face value on his bid);
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| * the player may bid a higher quantity of the same face, or any quantity of a higher face (allowing a player to "reset" the quantity);
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| * the player may bid a higher quantity of the same face or the same quantity of a higher face (the most restrictive; a reduction in either face value or quantity is never allowed).
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| If the current player challenges the previous bid, all dice are revealed. If the bid is valid (at least as many of the face value and any wild aces are showing as were bid), the bidder wins. Otherwise, the challenger wins.
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| ;Variants
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| * Instead of the current player being the only one who can challenge (or "call up") the previously-made bid, any player may challenge a bid at any time. The first challenge made ends the round, and the challenger closest to the current bidder in the direction of play has priority if multiple players challenge at the same time.
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| * If played with the above variant, the player who made the last bid may count aloud from 1 to 10. If he reaches 10 with no one challenging or increasing the bid, the round ends with that player earning back a die. A player may have more than 5 dice that way, and any player who reaches 10 dice that way wins the game.
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| * With the above mentioned variants, some players may stay quiet and win easily. To avoid that, the following rule may be added: Each time a player loses a challenge, he loses a die normally, but the two players sitting to his left and right lose a die as well (unless one of them was the player to win the challenge).
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| * With some bidding systems, a player may elect to choose one or more dice of matching value from under his cup, place them outside the cup in view of the other players, re-roll the remaining dice, and make a new bid of any quantity of that face value.
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| * When a player has no two dice with the same face, he may choose to ''pass'' once in a game round. If he does so, the bid will not be raised. The next player can raise the bid using standard rules, or call the bluff. By doing so, he challenges the claim of the passing player having no two dice with the same face. This is commonly used in multi-round games where dice are removed from the game, as it helps players with few dice left to gain more information about the other dice without risk.
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| * Instead of raising or challenging, the player can claim that the current bid is exactly correct ("Spot On"). If the number is higher or lower, the player loses to the previous bidder, but if they are correct, they win. A "spot-on" claim typically has a lower chance of being correct than a challenge, so a correct "spot on" call sometimes has a greater reward, such as the player regaining a previously lost die.
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| ===Elements of strategy===
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| As with any game of chance, probability is highly important. The key element is the "expected quantity": the quantity of any face value that has the highest probability of being present. For six-sided dice, the expected quantity is one-sixth the number of dice in play, rounded down. When wilds are used, the expected quantity is doubled as players can expect as many aces, on average, as any other value. Because each rolled die is independent of all others, any combination of values is possible, but the "expected quantity" has a greater than 50% chance of being correct, and the highest probability of being exactly correct. For example, when 15 dice are in play and wilds are used, the expected quantity is 5. The chances of a bid of 5 being correct are about 59.5%; in contrast, the chances of a bid of 8 being correct are only about 8.8%.
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| However, a high bid is not necessarily incorrect, because bids incorporate information the player knows. A player who holds several dice of a single value (for instance, four out of the five dice in his hand are threes) may make a bid, with fifteen dice on the table, of "six threes". To an outside observer who sees none of the dice, this has an extremely low probability of being correct (even with wilds), however since the player knows the value of five of those dice, the player is actually betting that there are two additional threes among the ten unknown dice. This is far more likely to be true (about 40%).
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| Each bid gives others at the table information. Players, through subsequent bids, reveal the players' confidence in the quantity of each face value rolled. A player with two or three of a certain face value under his or her own cup may make a bid favoring that face value. Players can thus use these bids to build a mental picture of the unknown values, which either strengthens or weakens their confidence in a bid they are considering. Others may consider a bid as evidence it is true, and if their own dice support the same conclusion, may increase the bid on that face value, or if their dice refute it may bid on a different face, or challenge the previous bid.
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| Conversely, bids can also be bluffs. Bluffs in liar's dice can be split into two main categories: early bluffs and late bluffs.{{cn|date=October 2012}} An early bluff is likely to be correct by simple probability (depending on the number of players), but other players may believe the bidder made that bid because his or her dice supported it. Thus, the bluff is false information that can lead to incorrect higher bids being made on that face value. Players will thus attempt to trick other players into overbidding by use of early bluffs to inflate a particular face value. A late bluff, on the other hand, is usually less voluntary; the player is often unwilling to challenge a bid, but as a higher bid is even more likely to be incorrect it is even less appealing. A late bluff is thus a critical part of the game; convincing bluffs, as well as reliable detection of bluffs, allow the player to avoid being challenged on an incorrect bid.
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| Playing Liar's dice involves interpersonal skills similar to other bluffing games such as [[poker]]. Being able to reliably detect bluffs through giveaways, or "tells", and analyzing a player's bidding history for patterns that can indicate the likelihood of a bluff, are important skills here just as in poker.
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| ===Dice odds===
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| For a given number of unknown dice ''n'', the probability that ''exactly'' a certain quantity ''q'' of any face value are showing, ''P(q)'', is
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| :<math>\ P(q) = C(n,q) \cdot (1/6)^q \cdot (5/6)^{n-q}</math>
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| Where ''C(n,q)'' is the [[binomial coefficient|number of unique subsets]] of ''q'' dice out of the set of ''n'' unknown dice. In other words, the number of dice with any particular face value follows the [[binomial distribution]] <math>B(n,\tfrac{1}{6})</math>.
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| For the same n, the probability ''P'(q)'' that ''at least q'' dice are showing a given face is the sum of ''P(x)'' for all ''x'' such that ''q ≤ x ≤ n'', or
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| :<math>\ P'(q) = \sum_{x=q}^n C(n,x) \cdot (1/6)^x \cdot (5/6)^{n-x}</math>
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| These equations can be used to calculate and chart the probability of exactly ''q'' and at least ''q'' for any or multiple ''n''. For most purposes, it is sufficient to know the following facts of dice probability:
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| * The expected quantity of any face value among a number of unknown dice is one-sixth the total unknown dice.
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| * A bid of the expected quantity (or twice the expected value when playing with wilds), rounded down, has a greater than 50% chance of being correct and the highest chance of being exactly correct.<ref name="Christopher P Ferguson">{{cite web|last2=Ferguson|first2=Thomas S|title=Models for the Game of Liar's Dice|url=http://www.math.ucla.edu/~tom/papers/LiarsDice.pdf|publisher=University of California at Los Angeles|accessdate=16 January 2013|last1=Ferguson|first1=Christopher P|language=English}}</ref>
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| ==Individual hand==
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| {{unsourced-section|date=January 2013}}
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| [[File:Liar Dice.jpg|thumb|A set of [[poker dice]] being rolled behind a screen, played as in the "individual" hand version of liar's dice.]]
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| The "individual hand" version is for two players. The first caller is determined at random. Both players then roll their dice at the same time, and examine their hands. [[List of poker hands|Hands are called in style]] similar to poker, and the game may be played with [[poker dice]]:
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| *Five of a kind: e.g., 44444
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| *Four of a kind: e.g., 22225
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| *Full house: e.g., 66111
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| *High straight: 23456
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| *Low straight: 12345
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| *Three of a kind: e.g., 44432
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| *Two pair: e.g., 22551
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| *Pair: e.g., 66532
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| *Runt: e.g., 13456
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| One player calls their hand. The other player may either call a higher-ranking hand, call the bluff, or re-roll some or all of their dice.{{huh?|date=January 2013}} When a bluff is called, the accused bluffer reveals their dice and the winner is determined.<ref>Hoyle's Rules of Games, Third Revised and Updated Edition. Albert H. Morehead and Georffrey Mott-Smith - Revised and Updated by Philip D. Morehead</ref>
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| ==Drinking game version==
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| The first player rolls two dice under a cup and claims a roll. Most claims are scored by reading the higher die as the 10s place and the lower as the 1s, e.g., a roll of 1 and 4 is read as "41". Doubles are higher than "65", and what would be the lowest roll 2-1, is a "Mexican" and higher than 6-6.
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| Special rolls:
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| *3-1 Social (everyone drinks, cancel all previous rolls, roll again to open)
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| *3-2 Reverse (change direction and previous player drinks one sip (maybe two if he/she is thirsty), cancel all previous rolls, roll again to open)
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| *2-1 Mexican (if the cup is lifted revealing a Mexican, the incorrect challenger drinks twice, if the player does not challenge, the player must still drink, since nothing is higher than Mexican)
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| The next player may do one of two things. If he believes the roller, he simply takes the dice (without looking at the result), rolls, and claims a higher scoring roll. If he does not believe the roller, the cup is lifted, revealing the roller's hand. Either the bluffer or incorrect challenger must drink.
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| ==Commercial versions==
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| {{Expand list|date=October 2012}}
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| *1974 ''Liars Dice'', published by E.S. Lowe
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| *1984 ''Liars Dice'', Milton Bradley, designed by Richard Borg.
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| *1993 ''Call My Bluff'', by FX Schmid, designer [[Richard Borg]], won the 1993 [[Spiel des Jahres]] and [[Deutscher Spiele Preis]] awards.<ref>[http://www.funagain.com/control/productaward/~award_type=SDJ/~award_year=1993 1993 Spiel des Jahres]</ref>
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| *1994 ''[[Perudo]]'', published by University Games, designed by Cosmo Fry.
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| *2001 ''Bluff'', from [[Ravensburger]] (after acquiring FX Schmid), reissue of ''Call My Bluff'', won the 2006 [[Årets Spel]] adult game of the year award.
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| *2002 ''Liars Dice'', by Endless Games
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| *2010 ''Deception Dice'', by Tumblin Dice, LLC
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| *2011 ''Diception'' by Four Clowns Game and Toy Co
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| *2011 ''Liar's Dice Live'' by FrontDev
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| *2012 ''Roll Call'', by DiceCasters.
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| ==Depictions in media==
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| Liar's dice is shown being played in the film ''[[Pirates of the Caribbean: Dead Man's Chest]]'' with the stakes being years of undead service aboard the [[Flying Dutchman (Pirates of the Caribbean)|Flying Dutchman]] under [[Davy Jones (Pirates of the Caribbean)|Davy Jones]].
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| Liar's dice is also a playable subgame in the videogame ''[[Red Dead Redemption]]'' and the browser game ''[[TirNua]]''.
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| == See also ==
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| *[[Cheat (game)|Cheat]], a card game with a similar emphasis on bluffing and detecting bluffs
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| *[[Liar's poker]], a structurally similar game using the digits of the serial numbers on dollar bills
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| ==References==
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| {{reflist}}
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| ==External links==
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| * {{bgg title|45|Liar's Dice}}
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| {{Spiel des Jahres}}
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| {{Drinking games}}
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| {{DEFAULTSORT:Liar's Dice}}
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| [[Category:Dice games]]
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| [[Category:Drinking games]]
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| [[Category:Spiel des Jahres winners]]
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| [[Category:Richard Borg games]]
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| [[Category:Ravensburger games]]
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| [[Category:Endless Games games]]
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| [[de:Liar Dice]]
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