Liar's Dice


Overview:

Liar's Dice is a simple, fun, strategic dice game. There are many variations. Here is the one. The object of the game is to be the last player with remaining dice. If a player loses all of his dice, he is out of the game. During each round, each player rolls all of his dice without revealing them to any other player. The player who begins the round announces that there are at least x number of dice showing y dots (pips). The next player (clockwise) may challenge this statement if he thinks that there are not at least x number of dice with the stated dots (eg. 7 6’s) present among all of the players' dice or he may continue the game by making his own statement of there being at least x number of dice with y dots. This goes on until a player challenges. For example: player B challenges player A. All of the every player’s dice are revealed and if there are at least x number of dice showing y dots, say z number of dice, then player A was correct and he does not lose any dice. In this case, player B loses z - x dice. However, if B was correct and there were not at least x number of dice showing y dots on the table, then player B is correct and loses no dice. Player A loses x - z dice. For example, if A said there were 4 dice showing 6’s, but there were actually 7, then B, who challenged, would lose 7 - 4 = 3 dice. But if there were only 2 dice showing sixes, A would lose 4 - 2 = 2 dice. After this, the round is over and the winner of the previous round is the player who starts the new round. This is repeated until only one player is left with any dice.


Game Requirements:

  1. 2 or more players. 
  2. Dice. Each player gets the same number. 8 dice/player is suggested. If there are 5 players and each get 8 dice, then 40 dice are required. 
  3. Non-transparent cups (it is important to not be able to see the dice through the cup). Each player gets one cup.


Detailed Gameplay:

[This step happens only at the very beginning of the game:]

  1. To decide who starts the game: Everybody rolls one dice.  The player with the highest roll begins (see step 2 below).  In the event of a tie, re-roll until one player wins.


[The following steps happen during each round:]

  1. All players put all of their dice in their cup and roll them, then flip the cup upside down on the table.  Each player may look at their dice under the cup while practicing caution to not allow the other players to see their dice.

  2. If beginning the (very first round of the) game, the highest rolling player (see step A above) begins.  If beginning any round after the first, the winner of the previous round begins.  The beginning player calls out two numbers: x and y; these represent his (perhaps false) belief that there are at least x number of dice showing y dots.  For example, "4 5's" means that this player is claiming that there are at least 4 dice showing 5's on the table.  

    1. 1's are special.  They are known as stars.  1's count as any other number.  For example, if there are 2 2's and 3 1's on the table, then there are actually 5 2's on the table.  There are also 3 1's (stars) on the table as no other number counts as a star.

  3. The next player (going in a counterclockwise fashion), player B, may continue playing or challenge the previous player, player A.   

    1. If player B decides to continue playing, then he must call out two numbers representing his belief that there are at least x number of dice showing y dots as before.  But now, at least one of the numbers x and y must be higher than before (see i. below).  After B has made his play, then repeat step 3 (without rolling any additional dice).  

      1. For example, if the player A had said 4 5's, then valid plays would include 4 6's (x is the same and y is higher), 5 2's, 5 3's, 5 4's, 5 5's, 5 6's, 6 2's, 6 3's, etc. (x is higher).  

      2. Stars (1's) are special.  They count as 2 of another number in the context of when they are able to be played.  For example, in this case, a valid play would be 2 stars (1’s) since 2 stars counts as 4 of any other number.  Suppose player C plays 3 stars.  Then valid plays for player D are 4 stars or 7 (or higher) of any other number.  Note that you only need to match the number x with the stars (eg. 2 stars is a valid play after 4 of any other number), but you need to go up one number to be a valid play for stars (eg. 6 of any number is not a valid play for 3 stars; 7 is).  

      3. An option that is available is called “Show and Re-roll.”  This requires the player to leave at least one dice on the table and then re-roll all remaining dice.  The player still needs to make a valid play of x number of y dots.  The dice that the player chooses to display can be anything (but it is probably advantageous to display the dice that correspond to y).  Note that the player must have more than 1 dice in order to use Show and Re-roll.

    2. If player B decides to challenge, all players must reveal all of their dice.  The round ends after at least one player loses dice - see below for the three possible situations.  When a player(s) loses dice, place the lost dice in the center of the table.  After the challenge has been made, if more than one player has dice, start a new round (step 1) with the winning player going first.  Otherwise, if only one player has remaining dice, this player wins.

      1. If there are more y dice than what player A said, then the player B loses the difference between the actual number of dice, z, and the number of dice the first player said.  For example, if player A said there were 4 5's, and there were actually 6 5's (6 is the sum of the number of 1's and the number of 5's), then player B would lose 6 - 4 = 2 dice.  Player A is said to have won this round and begins the next round.  

      2. If there are less y dice than what player A said, then player A loses the difference between z and x.  For example, if player A said there were 4 5's, and there were actually 2 5's, then player A loses 4 - 2 dice. Player B is said to have won this round and begins the next round.  

      3. If there are exactly y dice on the table, then player A is said to have won the round (because he claimed that there are at least y dice) and every player other than player A loses 1 dice.  The exception is that if any player other than player B has only 1 dice remaining, he does not lose this dice.  If player B has only 1 dice remaining, he loses this dice and is out of the game.  


Strategy:

Liar’s Dice is a relatively straightforward game.  The main strategy involves the decision of whether to continue playing or to challenge the previous player.  Here are some things to consider:

  1. During any given round, the total number of dice in the game is known.  Simply take the total number of dice and subtract the number of dice in the middle (that players have lost).  For example, if 5 players begun the game and they each started with 8 dice, and if 4 dice had been lost in previous rounds, then there are now 36 dice in play.  

  2. The expected number q of z dice can be easily calculated.  Assume that all dice are fair.  Continuing with the above example where that are 36 dice and assuming that nobody has utilized a Show and Re-roll, the expected number of 2’s is 6/36 = 6.  The expected number of 3’s, 4’s, 5’s, 6’s, and 1’s is also 6/36 = 6.  For any one dice, each number has a 1 in 6 chance of being selected.  So If there are 36 total dice, then each number would be expected to occur 6 times.  This is very valuable to consider.  Without taking into consideration anybody else’s plays and recalling that stars (1’s) count for any other number, then the expected number of stars is 6 and the expected number of 2’s, 3’s, 4’s, 5’s, and 6’s is actually 12.  

  3. We do have to consider what other people are playing, however.  It would be unusual for there to be the exactly the expected number of dice.  If a lot of players have been going with 5’s, for example, then perhaps there are a lot of 5’s (including stars) on the table.  It could be, of course, that some or all of the players are bluffing and that there are actually not many 5’s.  This is something to take into consideration.

  4. It is better to calculate expectations differently than described in step 2.  You obviously know how many eg. 5’s (and stars) you have.  You also know how many 5’s (and stars) have been displayed in Show and Re-roll.  You also know how many non-5’s and non-stars you have and that have been displayed in Show and Re-roll.  This should all be factored into the expected number of dice.  For example, if there are 2 1’s, 1 4, 1 5, and 1 6 showing from Show and Re-roll and if, in your hand you have 1 1, 3 5’s, and 3 of the other numbers, then you know that there are certainly 7 5’s (3 1’s and 4 5’s) in the game.  There are also 5 non-5’s in the game.  This leaves 24 dice that you don’t know about.  Among these 24 dice, you expect there to be 8 5’s (4 stars and 4 5’s).  But you already know that there are 7 5’s.  So the new, working expected number of 5’s is 15.

  5. Use the Show and Re-roll to your advantage.  If the bidding is getting really high (maybe higher than your calculated expected number from 4.), then challenging may be dangerous.  It stops the game and either you (and maybe others) lose dice or the previous player loses dice.  But if you continue playing, then the player after you may or may not challenge.  If they don’t challenge, this is most likely good for you.  Anyway, a strategy that will probably make it less likely for the player after you to challenge is to reveal the dice corresponding to your play (eg. if you are saying there are a certain number of 5’s, you would most likely want to reveal all of your 1’s and 5’s).  Doing this gives the other players the information that some of the previously unknown dice are certainly 5’s.  Another use of Show and Re-roll is to try and get more of a certain kind of dice with the re-roll.  Even if you don’t have, eg. a 5, and you are playing 5’s, it might be a good strategy to show another number and then re-roll your other dice in an attempt to get 5’s.  Everything else equal, it is probably better to use Show and Re-roll when the bidding gets higher.

  6. Bluffing can be useful on occasion.  One potential occasion is when there are only two players left.  Another occasion could be early on when you have lots of, eg. 5’s, but not any 4’s.  You could say a fairly high number of 4’s to try and get the other people to think that there are lots of 4’s.  This can artificially inflate people’s perception of how many 4’s there are and possibly cause a player to lose a lot of dice.

  7. Betting a high, but not too high, number of dice may make is less likely that you will go again later in the round.  For example, if there are 6 players and the expected number of 6’s (including stars) is 12, then starting with 13 6’s as your play might be smart because it is unlikely that the next player will challenge you and it is also likely that there will be a challenge before it gets around to you again.

  8. Stars can be very useful.  If there is a situation where the bidding has gotten high, eg. 14 6’s and you are scared that you will get called if you say 15 6’s, then consider playing 7 stars.  The expected losses decrease a lot.  Say there are actually 5 6’s and 5 stars on the table.  Then if you had said 15 6’s, you would lose 5.  If you had said 7 stars, you would only lose 2.  


Author: Todd Jones