Liar's Dice

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.

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 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." 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, is known in Germany, while Liar's dice is known in Germany as Bluff. It is known by various names in Asia.

Five dice are used per player with dice cups used for concealment.

Each round, each player rolls a "hand" of dice under their cup and looks at their hand while keeping it concealed from the other players. The first player begins bidding, announcing any face value and the number of dice that the player believes are showing that value, under all of the cups in the game. Ones are often wild, always counting as the face of the current bid.

Turns rotate among the players in a clockwise order. Each player has two choices during their turn: to 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:

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. The player who loses a round loses one of their dice. The last player to still retain a die (or dice) is the winner. The loser of the last round starts the bidding on the next round. If the loser of the last round was eliminated, the next player starts the new round.

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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