Start with the odds.
Use your own dice and the chances for unseen dice to judge a bid. All seven heads-up leaders avoided deliberate random bluffs. That does not make every bid true or prove bluffing always loses.
Liar’s Dice simulation
Strategy results & counters
The recurring heads-up pattern
Bid from probability.
Skip deliberate bluffs.
All seven two-player leaders used probability-based bidding with deliberate bluffing set to zero. A useful starting point in this tested field; the best choice still depends on the rules and opponents.
7 / 7
heads-up leaders
had zero deliberate bluffing
3,377,152 confirmation games7 rule profiles2–6 players · 5 starting dice
What to take to the table
Call selectively.
The heads-up leaders’ base call cutoffs ranged from 12.5% to 20% estimated bid truth. They tolerated plausible bids and challenged when the estimate fell below their cutoff.
Learn the responses.
The supported Perudo counters learned from revealed rounds: how often opponents called or raised at different bid probabilities. This helped against a known style, with a cost against varied opponents.
Best observed heads-up choices
Each leader’s average against its frozen challenger population. Opponents receive equal weight, including a 50% self-play reference.
| Rules | Strategy | Win rate | 95% interval |
|---|---|---|---|
| Classic | Probability-led biddingrefined-classic-32 | 66.1% | 64.8–67.5% |
| Calza | Probability-led biddingrefined-calza-40 | 68.5% | 67.1–69.9% |
| No-wild | Probability-led biddingresearch-no-wild-32 | 73.2% | 71.7–74.6% |
| Perudo | Probability-led biddingtruthful | 65.7% | 64.3–67.1% |
| Perudo-Calza | Disciplined probability playrefined-perudo-calza-25 | 64.7% | 63.3–66.0% |
| Single-loser-agree | Probability-led biddingrefined-single-loser-agree-40 | 74.5% | 73.1–75.9% |
| Spot-on | Probability-led biddingrefined-spot-on-40 | 74.2% | 72.8–75.6% |
Calza is unavailable when a game starts heads-up, so Classic/Calza and Perudo/Perudo-Calza have identical rules here. Their searches, challenger populations, and seeds differ.
The clearest separation was in No-wild. Its leader beat the closest pure finalist by 2.37 percentage points (simultaneous interval: 0.27–4.46) under equal challenger weighting. Giving each strategy family equal total weight kept the same observed leader but left that comparison unresolved. Most other leading comparisons also remain unresolved.
The heads-up leaders’ bidding parameters
“Support” is the estimated chance that a bid is true. Opening and raise targets are preferences, not guarantees. Challenge cutoffs are base thresholds for estimated bid truth, not the chance a challenge succeeds. These describe the tested bots, not universal rules for human play.
| Rules | Opening target | Raise target | Call below |
|---|---|---|---|
| Classic | 99.0% | 85.4% | 12.5% |
| Calza | 65.1% | 89.0% | 13.9% |
| No-wild | 67.6% | 68.3% | 12.9% |
| Perudo | 84.0% | 78.0% | 20.0% |
| Perudo-Calza | 86.5% | 98.0% | 16.4% |
| Single-loser-agree | 98.8% | 89.7% | 16.1% |
| Spot-on | 99.0% | 88.5% | 17.7% |
With three to six players
Table size changes the choice. These are the highest observed averages across four equally weighted opponent mixes: broad, defensive, finalists, and pressure.
For Classic at 4–6 players, disciplined probability play led. The preset is named calza-disciplined, but its exact-call feature is inactive under Classic rules. None of the 35 mixed-field settings, including the two-player panels in the full data, proves its leader beats every tested finalist.
Classic · 3–6 player results & rules
Ones are wild; bids name faces 2–6. Raise the quantity, or keep the quantity and raise the face. There is no exact-count call.
| Players | Highest observed strategy | Win rate | 95% interval |
|---|---|---|---|
| 3 | Probability-led biddingresearch-classic-32 | 44.8% | 39.4–50.3% |
| 4 | Disciplined probability playcalza-disciplined | 36.6% | 31.2–42.1% |
| 5 | Disciplined probability playcalza-disciplined | 30.8% | 25.3–36.3% |
| 6 | Disciplined probability playcalza-disciplined | 27.7% | 22.2–33.1% |
research-classic-32: The base challenge threshold is 16.9% estimated bid truth. Preferred opening and raise support targets are 75.7% and 72.4%; these are preferences, not guarantees. The configured deliberate-bluff rate is 0.0%.calza-disciplined: The base challenge threshold is 20.0% estimated bid truth. Preferred opening and raise support targets are 84.0% and 78.0%; these are preferences, not guarantees. The configured deliberate-bluff rate is 0.0%.
Calza · 3–6 player results & rules
Classic bidding with an exact-count claim after a bid. A correct Calza restores one lost die, up to the starting cap; a wrong call costs a die. Available only with at least three active players.
| Players | Highest observed strategy | Win rate | 95% interval |
|---|---|---|---|
| 3 | Disciplined probability playresearch-calza-25 | 49.9% | 44.4–55.3% |
| 4 | Disciplined probability playcalza-disciplined | 40.1% | 34.6–45.5% |
| 5 | Disciplined probability playcalza-disciplined | 35.0% | 29.6–40.5% |
| 6 | Disciplined probability playresearch-calza-25 | 31.6% | 26.2–37.1% |
research-calza-25: The base challenge threshold is 21.1% estimated bid truth. Preferred opening and raise support targets are 75.9% and 84.7%; these are preferences, not guarantees. The configured deliberate-bluff rate is 0.0%. Calza is declined at the die cap and otherwise requires positive estimated recovery value.calza-disciplined: The base challenge threshold is 20.0% estimated bid truth. Preferred opening and raise support targets are 84.0% and 78.0%; these are preferences, not guarantees. The configured deliberate-bluff rate is 0.0%. Calza is declined at the die cap and otherwise requires positive estimated recovery value.
No-wild · 3–6 player results & rules
Ones are ordinary dice. Bids can name any face, 1–6. Raise the quantity, or keep the quantity and raise the face. There is no exact-count call.
| Players | Highest observed strategy | Win rate | 95% interval |
|---|---|---|---|
| 3 | Disciplined probability playresearch-no-wild-25 | 49.9% | 44.4–55.4% |
| 4 | Disciplined probability playresearch-no-wild-25 | 40.9% | 35.4–46.4% |
| 5 | Disciplined probability playresearch-no-wild-25 | 34.2% | 28.8–39.7% |
| 6 | Search informed by public bidsrefined-no-wild-28 | 29.6% | 24.1–35.0% |
research-no-wild-25: The base challenge threshold is 15.9% estimated bid truth. Preferred opening and raise support targets are 97.2% and 81.4%; these are preferences, not guarantees. The configured deliberate-bluff rate is 0.0%.refined-no-wild-28: Uses bounded sampled next-response search with 16 rollouts per candidate and a configured raise-candidate limit of 8. It weights possible hidden states using recent public bids. This is not a full-game equilibrium solver.
Perudo · 3–6 player results & rules
Ones are wild, with special bids on ones: switch to ones with a quantity of at least half the previous quantity, rounded up; switch away with at least twice the ones quantity plus one. Ordinary rounds cannot open on ones. A player’s first drop to one die triggers Palifico while more than two players remain: ones stop being wild and the opening face stays fixed. There is no exact-count call.
| Players | Highest observed strategy | Win rate | 95% interval |
|---|---|---|---|
| 3 | Disciplined probability playcalza-disciplined | 49.1% | 43.7–54.6% |
| 4 | Search over likely next responsesresearch-perudo-45 | 43.2% | 37.7–48.6% |
| 5 | Search over likely next responsesresearch-perudo-37 | 35.7% | 30.2–41.1% |
| 6 | Search over likely next responsesresearch-perudo-45 | 35.1% | 29.7–40.6% |
calza-disciplined: The base challenge threshold is 20.0% estimated bid truth. Preferred opening and raise support targets are 84.0% and 78.0%; these are preferences, not guarantees. The configured deliberate-bluff rate is 0.0%.research-perudo-45: Uses bounded sampled next-response search with 16 rollouts per candidate and a configured raise-candidate limit of 8. This is not a full-game equilibrium solver.research-perudo-37: Uses bounded sampled next-response search with 16 rollouts per candidate and a configured raise-candidate limit of 8. This is not a full-game equilibrium solver.
Perudo-Calza · 3–6 player results & rules
Perudo bidding and Palifico, plus Calza die recovery. Calza is unavailable during Palifico or with two active players.
| Players | Highest observed strategy | Win rate | 95% interval |
|---|---|---|---|
| 3 | Disciplined probability playcalza-disciplined | 49.2% | 43.7–54.6% |
| 4 | Disciplined probability playcalza-disciplined | 40.5% | 35.1–46.0% |
| 5 | Search over likely next responsesresearch-perudo-calza-29 | 34.5% | 29.0–39.9% |
| 6 | Search over likely next responsesrefined-perudo-calza-29 | 31.1% | 25.7–36.6% |
calza-disciplined: The base challenge threshold is 20.0% estimated bid truth. Preferred opening and raise support targets are 84.0% and 78.0%; these are preferences, not guarantees. The configured deliberate-bluff rate is 0.0%. Calza is declined at the die cap and otherwise requires positive estimated recovery value.research-perudo-calza-29: Uses bounded sampled next-response search with 16 rollouts per candidate and a configured raise-candidate limit of 4. This is not a full-game equilibrium solver.refined-perudo-calza-29: Uses bounded sampled next-response search with 8 rollouts per candidate and a configured raise-candidate limit of 8. This is not a full-game equilibrium solver.
Single-loser-agree · 3–6 player results & rules
Classic bidding plus an exact-count call on your turn. If correct, only the bidder loses one die; if wrong, the caller loses one.
| Players | Highest observed strategy | Win rate | 95% interval |
|---|---|---|---|
| 3 | Search informed by public bidsresearch-single-loser-agree-36 | 50.4% | 44.9–55.8% |
| 4 | Probability-led biddingrefined-single-loser-agree-40 | 42.7% | 37.3–48.2% |
| 5 | Search informed by public bidsresearch-single-loser-agree-36 | 35.0% | 29.5–40.4% |
| 6 | Search informed by public bidsresearch-single-loser-agree-36 | 29.7% | 24.3–35.2% |
research-single-loser-agree-36: Uses bounded sampled next-response search with 16 rollouts per candidate and a configured raise-candidate limit of 6. It weights possible hidden states using recent public bids. This is not a full-game equilibrium solver.refined-single-loser-agree-40: The base challenge threshold is 16.1% estimated bid truth. Preferred opening and raise support targets are 98.8% and 89.7%; these are preferences, not guarantees. The configured deliberate-bluff rate is 0.0%.
Spot-on · 3–6 player results & rules
Classic bidding plus an exact-count call on your turn. If correct, every other active player loses one die; if wrong, the caller loses one.
| Players | Highest observed strategy | Win rate | 95% interval |
|---|---|---|---|
| 3 | Search informed by public bidsrefined-spot-on-28 | 49.8% | 44.3–55.2% |
| 4 | Search informed by public bidsrefined-spot-on-28 | 39.8% | 34.3–45.3% |
| 5 | Search informed by public bidsrefined-spot-on-04 | 31.2% | 25.7–36.7% |
| 6 | Adapt to table size and dice leftresearch-spot-on-10 | 27.8% | 22.4–33.3% |
refined-spot-on-28: Uses bounded sampled next-response search with 16 rollouts per candidate and a configured raise-candidate limit of 6. It weights possible hidden states using recent public bids. This is not a full-game equilibrium solver.refined-spot-on-04: Uses bounded sampled next-response search with 16 rollouts per candidate and a configured raise-candidate limit of 8. It weights possible hidden states using recent public bids. This is not a full-game equilibrium solver.research-spot-on-10: The base challenge threshold is 21.1% estimated bid truth. Preferred opening and raise support targets are 66.3% and 91.3%; these are preferences, not guarantees. The configured deliberate-bluff rate is 0.0%. Challenge thresholds increase by 10 percentage points heads-up and decrease by four in multiplayer; one remaining die subtracts a further six. Raise targets decrease by 12 points heads-up and increase by three in multiplayer, subject to the policy's bounds.
The heads-up table above uses a broader challenger population. Its averages and leaders can differ from the two-player mixed panels included in the downloadable data.
Counters with supported advantages
Every opponent at these tables uses the named target strategy. Each listed counter exceeded the symmetric reference of 1 ÷ player count under the simultaneous confidence bounds.
Perudo
Response learning found substantial advantages against these selected search-based opponents. After each reveal, the counter updates its estimates of how opponents react to plausible and implausible bids.
| Players | Counter & target | Win rate | 95% interval | Reference |
|---|---|---|---|---|
| 3 | Learn opponents’ responsesrefined-perudo-27Against: research-perudo-45 | 61.3%1,536 games | 50.4–72.3% | 33.3% |
| 4 | Learn opponents’ responsesrefined-perudo-27Against: research-perudo-45 | 47.6%2,048 games | 36.7–58.5% | 25.0% |
| 5 | Learn opponents’ responsesrefined-perudo-43Against: research-perudo-37 | 37.3%2,560 games | 26.4–48.3% | 20.0% |
| 6 | Learn opponents’ responsesrefined-perudo-03Against: research-perudo-45 | 42.9%3,072 games | 32.0–53.9% | 16.7% |
Perudo-Calza
The four-player counter searches likely next responses. The five-player counter learns opponents’ responses from revealed rounds. These are two distinct targeted results.
| Players | Counter & target | Win rate | 95% interval | Reference |
|---|---|---|---|---|
| 4 | Search over likely next responsesrefined-perudo-calza-29Against: research-perudo-calza-33 | 36.6%2,048 games | 25.6–47.5% | 25.0% |
| 5 | Learn opponents’ responsesrefined-perudo-calza-27Against: research-perudo-calza-29 | 38.8%2,560 games | 27.9–49.7% | 20.0% |
Single-loser-agree
The supported heads-up counter uses probability-led bidding: a 98.8% opening support target, an 89.7% raise target, a 16.1% base call cutoff, and zero deliberate bluffing.
| Players | Counter & target | Win rate | 95% interval | Reference |
|---|---|---|---|---|
| 2 | Probability-led biddingrefined-single-loser-agree-40Against: research-single-loser-agree-28 | 61.9%1,024 games | 51.0–72.8% | 50.0% |
A specialist is not always the best default. All four Perudo counters above lost to the corresponding mixed-field leader in supported paired comparisons against varied opponents. These tests support a counter to a specific target, not superiority to every other candidate.
Seven settings are shown, choosing the observed leader in each. Nine candidate/table combinations passed the criterion in total. The other targeted settings remain unresolved; this does not establish that no counter exists.