What the helper does
Enter how many dice each player still has, your own hand, the current bid, and whether the round is palifico (wilds disabled). The engine computes, for every legal action — raise, Dudo (liar), or Calza (exact) — the exact probability that the underlying count of the bid face is satisfied by the dice you cannot see.
Hidden dice are modelled as independent uniform on {1..6}. Per the standard 'common hand' wild rule, a non-ace face has per-die probability 2/6 (the face itself plus the wild ace), and aces or palifico rounds have 1/6. The count of matching unseen dice is therefore Binomial(U, p), and we evaluate P(count ≥ k) and P(count = k) from the exact PMF.
The recommended action is the one with the highest survival probability — i.e. the lowest chance of losing a die this turn. For raises that is P(bid succeeds); for Dudo it is P(bid fails); for Calza it is P(bid is exact).
Why Perudo cannot be 'solved' like chess
Perudo is an extensive-form game of imperfect information with chance: each player's dice are private and the action set branches with the table size. Even the simplified 1-vs-1 Dudo with two dice per side has ≈10⁴ information sets, and the full five-dice four-player game blows past 10¹². Backward-induction style 'solving' is therefore out of reach.
The right solution concept is a Nash equilibrium of the extensive-form game. For two-player zero-sum variants, Koller and Pfeffer (1997) showed that the sequence form turns the equilibrium problem into a linear program of size polynomial in the game tree, which solves small Liar's Dice variants exactly.
For the full game, the modern recipe is Counterfactual Regret Minimization (CFR), introduced by Zinkevich et al. (2008). CFR is an iterative no-regret algorithm whose average strategy provably converges to a Nash equilibrium in two-player zero-sum games; sampled variants (MCCFR, Lanctot et al. 2009) extend it to much larger trees. Neller and Hnath (2011) applied a fixed-strategy iteration variant of CFR specifically to 2-player 1-die vs 1-die and 2-die vs 2-die Dudo, producing the first published near-optimal Dudo strategies. Their tutorial 'An Introduction to Counterfactual Regret Minimization' (Neller & Lanctot, 2013) walks through Kuhn poker and Dudo in full and is the de facto reference.
What strong CFR policies actually do
When you inspect a converged CFR policy for Dudo, three patterns dominate. First, the player almost always raises along the 'expected count' frontier — a bid is comfortable when its quantity is ≤ the expected matching count given the player's dice, and uncomfortable above it. Second, the bluff rate on a face the player does not hold is small but non-zero — exactly the mixed-strategy ingredient that makes Dudo unsolvable by pure heuristics. Third, Dudo is called once the cumulative bid jumps far above the expected count, with a probability that grows smoothly rather than as a hard threshold.
BoardSolve's helper exposes the deterministic part of that policy: the exact bid probabilities and the survival-maximising action. Adding the mixed-strategy bluff layer requires opponent modelling and history, which is exactly what the CFR papers below provide for those who want to go further.