Liars Dice Game

This is a game of Liars Dice, which can be played with up to 12 players. This version currently supports 1 user player, with the remaining players filled by computer players that will act with of mix of randomness and probabilistic reasoning.

Theory

The GameAgent (in progress) with m dice and a bet of k of dice d and n total dice remaining, will make bet/call decisions based on

Letting $$X_i = a + (\#\{Z_i = d : i \in [0, i]\})#$ , where a is the number of dice, d, in the hand of the players whose turn it is.

If k - a <= 0; we know that the bet is true so the agent will never call that bet, similarly if k -a > n - m, the agent should always call since this is impossible. Otherwise we define the probability of the bet being true as:

$\mathbb{P}[at\ least\ k\ dice\ of\ face\ d\ |\ oppoenents\ bet] = \mathbb{P}(X_{i, d} \geq k - a | bet) = \sum\limits_{i = 1}^{s} \mathbb{P} [X_{n - m, d} \geq k - a - i | X_{s, d} = i]$

$\mathbb{P} \left[ \# \{ Z_i = d : i \in [1, n] \} \geq k \big| \{ Z_i = d : i \in [1, m] \} = j \right] = \mathbb{P} \left[ \# \{ Z_i = d : i \in [m + 1, n] \} \geq k - j \right]$

Letting $X_{i, d} = \# \{Z_t == d : t \in (1, i) \}$

Further, we estimate the probability that a player is lying or will call our bluff by:

$= \frac{\sum\limits_{i=0}^{min(s, k-a)} \mathbb{P}[X_{n-m-s,d} \geq k-a-i] \cdot \mathbb{P}[X_{s,d} = i]}{\mathbb{P}[X_{n-m} \geq k]} \\ \\ $$ = \frac{\left ( \frac{5}{6} \right )^{s} + \sum\limits_{j = 1}^{min(k, s-a)} \left [1 - \sum\limits_{i = 0}^{j - 1} \binom{n - m -s}{i} \left ( \frac{1}{6} \right )^i \left ( \frac{5}{6} \right )^{n- i} \right ] \cdot \binom{s}{s-j} \left ( \frac{1}{6} \right )^{j} \left ( \frac{5}{6} \right )^{s-j} + \sum\limits_{j = min(k, s-a)}^{k} \left [\binom{s}{s-j} \left ( \frac{1}{6} \right )^{j} \left ( \frac{5}{6} \right )^{s-j} \right ]}{1 - \sum\limits_{l = 0}^{k-a-1} \binom{n}{l} \left ( \frac{1}{6} \right )^{l} \left ( \frac{5}{6} \right )^{n-l}}$$ \\ $$$

mruby7188 / Liars_Dice_Agent

Liars Dice Game

Theory

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