Liar's Dice

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Rules

• Each player roll dice at the beginning of the game and does not reveal the result.
• Starting with the first player, each player can call liar/spot-on (except the first player) or raise the bid.
• If one player call liar/spot-on, everyone reveals their rollouts.
• If the call is not successful, the caller loses one die and becomes the first player for the next round unless the loser lost all dice, under which the next player becomes the first player.
• If the call is successful, and the player called
  • liar: the previous player will lose one die and becomes the first player for the new round.
  • spot-on: everyone loses one die except the player and the player starts the next round.
• The game terminates when there is only one player left, and this player is the winner.
• "1" is wild, means it will be counted as any denomination.

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Order of bets:

We use a pair of integers to represent bets. For example, $[3,5]$ stands for the bet "three fives". We say $[a,b]>[c,d]$ if

a (2*a if b==1) > c (2*c if d==1) or a (2*a if b==1) == c (2*c if d==1) & b>d

For instance, $[3,1]>[5,6], [4,5]>[4,3], [6,2]>[3,1]$