How to Ruin Board Games: Perudo

Introduction

There’s a moment in Pirates of the Caribbean: Dead Man’s Chest (in our opinion the most underrated of the Pirates trilogy, although our editor keeps trying to tell us there were more films than 3 for some reason) where Orlando Bloom’s character Will Turner engages Davy Jones in a dice game. The stakes? His father’s soul against 100 years of servitude on Davy Jones’s ship, the Flying Dutchman. His father joins in to protect Will, and a 3-way game proceeds. In the end, Will’s father takes the fall for his son, agreeing to serve an eternity on the ship. Masterful cinema, even if the first one was better.

Now, a couple of noteworthy observations from that game:

1. I don’t know if the life-changing stakes are what were responsible for how exciting the game looked, or if it was the game design mechanics, but I assume you’re all interested in finding out.
2. In that game, all three players had different intentions. Davy Jones earnestly wanted to win, Will Turner only played so he could goad Davy Jones into revealing the location of a key he was after (seriously, watch the movie), and Will’s father Bill Turner just wanted to make sure his son didn’t lose. This made them all play in different ways.

That game, known as Liar’s Dice to most, but now rebranded as Perudo, is the subject of this article. It’s a game of probability, giving an advantage to those who pride themselves on their mathematical cunning. It’s also a game of bluffing… in my experience, almost entirely a game of bluffing (chorus of boos from the maths fans).

How to play

If you’re not familiar with Perudo, here is the game’s structure, lovingly condensed to its essentials:

1. The Setup: Each player gets five dice and a cup. The cup hides your dice, but also hides your tears (it’s a multipurpose cup). The game continues until only one player has any dice left.
2. The Bid: Every player rolls their dice, takes a peek, and pretends they’re totally unimpressed by what they see. The first player begins with a bold announcement, like “There are four threes on the table.” If you think that sounds absurd, just wait—this is only the beginning.
3. The Escalation: The next player has two options. They can either:
   1. Raise the Bid: Claim there are even more dice showing a specific value (“Seven twos!”) or,
   2. Challenge It: Call out the previous player by yelling “Dudo!” which means “I think you’re full of it!” (courtesy of Google Translate).
4. The Reveal: If “Dudo!” is called, all dice are revealed. If the bid was correct (or even too low—rare, but deliciously humiliating), the challenger loses a die. If the bid was nonsense, the liar loses a die. And so it goes, with players steadily whittled down until only one dice-wielding liar remains.
5. Wild Ones: Ones are special in Perudo; they’re like wild cards, counting toward any number a player might claim. This little rule creates a whole new layer of mayhem, perfect for sowing the seeds of doubt among even the most trustworthy friends. To illustrate this, let’s say you roll your five dice, and the results are 1, 1, 3, 3, 4. You technically have 4 threes in your hand, or 2 fives (even though you actually don’t have any fives).

There are some intricacies to follow on from this. For example,

• If the bidding is getting too high for you, you can half the current bid (number of dice, and round up) and call ones (e.g. “10 fives, eh? I can’t go higher. I bid 5 ones”). Once the bidding gets into ones, all bidders must stay in the ones, and if they wish to exit and bid for another number, they need to double and add 1 to the number of dice (e.g. “6 ones? I bid 13 fives.”)
• The first time any player gets down to one die, a lightning round is played in which they call a number, and bidding may not leave that number (just for this round, ones count only as ones).

The model

What we were most interested in was how to approach a game of Perudo. During APR’s Board Game Nights (we’re very cool like that), we get a range of playstyles, going from what we like to call “rabbit in headlights”, a strategy where the player makes as few big moves as possible and tries to float under the radar, to “consummate bluffer”, a strategy where the player relies on blind luck and accumulated chaotic energy to throw everyone else off their rhythm. Through experience, I’ve lost all my friends doing this the player has lost all their friends doing this.  We wanted to see if a game-theoretic approach, where we specify types of players and interact them in a simulated game of Perudo, would yield any helpful insights.

To do this, we modelled the game of Perudo in R, using R6 to play around with object-oriented programming (OOP) principles. We made a simplification by not modelling the lightning round, as we felt it would have a relatively low impact on the eventual winner’s identity (which is what we were interested in).

We added a few different player “playstyles”:

• Safe: A player that tends to play conservatively. They would calculate the expected number of dice of each bid and pick the lowest such bid. They would not risk going higher than necessary and would call Dudo instead. Perfect for the most boring member of the group.
• Smart: A player based on the Safe archetype, but who uses the previous bid to help inform their guesswork, rather than only what’s in their hand. They have a similar attitude to risk to the Safe player.
• Bluffer: A player that knows the rough expected number of dice for each bid, but is willing to go up to two higher than expected. Lives for chaos. Bids like their dice have been blessed by the gods.
• Always One: A fundamentally risk-averse player who will halve the bid to ones whenever it’s plausible. Yes, we really made this one.
• Trust Nobody: A Cold War era spy whose experiences in espionage have led to fundamental trust issues no therapist has yet been able to solve. Will doubt even the most sensible bids and call Dudo on anyone who bids up to two levels below the expected number of dice (for example: they expect to see 10 fives, and call Dudo on 8 fives or above).

We wrapped these playstyles in code to simulate full games of Perudo, and then ran a load of simulations to make the results more reliable. The model can be customised by:

• Number of players: it will accommodate as many players round a table as told – note that real Perudo comes pre-packaged for 2-6 players (30 dice are included).
• Number of dice: the starting dice count of each player, defaulted to 5 as per the real game.
• Types of players: the playstyle of each player.

To quickly illustrate the idea of “expected number of dice”, which our champions hinge their strategies on, let’s use an example:

It’s the start of a 6-player game, and Davy Jones has rolled 1,1,1,3,5. Including ones, he has 4 threes on the table, or 4 fives. There are 25 more dice on the table. He would reasonably think (without any other information) that 1/6^th of those dice have rolled a 5, and 1/6^th have rolled a 1. So, counting up the fives:

4 (mine) + (1/6)*25 (the 5s) + (1/6)*25 (the 1s) = 12.33. Therefore, we expect there to be 12 fives on the table.

The playstyles are built around this expected dice count.

Does going first in Perudo actually matter?

A timeless question for board game enthusiasts: does kicking things off give you an edge? In games like chess, the first-move advantage is legendary. In Go, there’s a whole handicap to your score designed around who holds the black pieces. For others, like Monopoly, the advantage probably exists but we’re too busy asking Uncle Jimmy to pay up on our hotel to actually figure it out.

In Perudo, turn order is shuffled with each round since the loser of the previous round starts the next. This dynamic should, in theory, level the playing field. But we don’t live in “theory,” do we? So, we tested it. To isolate the effects of turn order, we used five players, all following the Safe playstyle (because they’re boring, predictable, and won’t rock the boat).

What did we find?

As expected, there was some minor variation in win rates, but nothing to suggest starting the game first makes a dramatic difference. Switching out the five Safe players for any five players of the same style didn’t produce a pattern either. Intuitively this makes sense: each round in Perudo resets the stakes as another die is lost, giving the person who kicked off the game a fleeting few minutes in the limelight before they’re just like everyone else.

So, let’s call it—going first in Perudo doesn’t matter. Move on, chess fans, nothing to see here.

Which playstyle reigns supreme?

Every Perudo player has that one friend who insists their way is best. As regular readers will know, we exclusively hang around with our fellow actuaries and haven’t made a new friend since 2014, so it’s fair to say OUR friends are all number-crunchers. You’d have to travel a long way out of our circle to find someone who plays Perudo because they “trust their gut”. Let’s be honest, though—not every approach to Perudo is created equal. For every cool, calm and collected player ripping out their inscribed abacus at Board Game Night, there’s someone out there confidently bidding 10 fives when there are only eight dice around the table. (We all know that person. We all love that person).

So, to settle a debate which has raged on since we first invited our colleagues to think about an article like this one, we pit five contenders against each other in a cage match. We put one of each playstyle above into the mixer for fairness, unconvinced as we were that reality would ever find an Always One to show us it was a good modelling choice. Let’s cut short the preamble. Who dominated? Who flopped? Who needs to rethink their life choices?

The Results

Pretty shameful, right? To not even APPEAR on the leaderboard? Shocker – our two “joke” strategies (Trust No One and Always One) went about as successfully as you’d expect. Unsurprisingly, Smart players cleaned up, winning over half the games. It seems paying attention to probabilities is, in fact, a winning strategy. Meanwhile, Bluffers held their own, securing a respectable third of the wins by presumably staring their opponents dead in the eye and raising the bid to something absurd. Safe players? They’re the second-tier insurance policy no one wants—statistically sound but just not exciting enough to close the deal.

But why does Safe lose to Bluffer? It’s a clash of predictability vs noise. Bluffers thrive on chaos and risk, and in a game with uncertainty baked in, their antics outperform the conservative approach of Safe players.

What happens when we double the player count?

Motivated by a question of whether the Bluffer, in a larger player count game, might take advantage of the added uncertainty and chaos to mount a comeback, we slapped another five of each player on the table. Here’s what happened:

As we can see, the impact is the other way round—the Smart playstyles win by an even greater margin, with the Safe playstyle essentially becoming inefficient. The Bluffers combined for 123 of the 500 wins (c.25% of the total simulations). This is a slight decline on the original run, implying that increasing the player count actually makes the noise inherent in a Bluffer strategy less valuable!

We see that the win rate of the Smart player is now 72%, much more reliable (and at the expense of Safe players) than in the five-person game. It’s not obvious how to explain this, but with more players the games become longer and more arduous. Perhaps the divergence from the “probabilistic” choice therefore costs more in the long run. The decline of the Safe player is interesting, but perhaps they are directly eaten by the Smart player, who are essentially them but with additional information and therefore better predictions.

What happens when Smart and Bluffer go head-to-head?

One of the real takeaways here is that the Smart and Bluffer players are the only ones who can consistently approximate actual competition (as opposed to whatever bloodbath the other three contestants get themselves into). We tried to cut through all the noise, especially given the fact Safe players seemed to be donating wins to the Smart players, and we ran a four-player simulation with only Bluffers and Smart players. Here’s what we got:

So, in a turn up for the books, the Bluffer starts to win more often when they only need to contend with Smart players. Indeed, aggregating the results from the same playstyle together, we get:

It’s practically a dead heat! This suggests that Bluffers are the Smart player’s kryptonite, forcing them into a psychological chess match where chaos sometimes triumphs over probability. So, while Smart may be king against other playstyles, Bluffers level the playing field in a straight duel.

It’s easy to intuitively explain this – the smart player probably has a better strategy, but the Bluffers don’t bid realistically. Since Smart players base their guess on information that includes the previous bid, a Smart player next to a Bluffer will likely overestimate and go too far. Therefore, we see that the more Bluffers around the table, the more “scrambled” the Smart player algorithm.

We can see this demonstrated when we run a four-player game with three Bluffers and one Smart player. The win percentage for Smart players drops down to c.16% (which, if we had 4 Smart players instead, would be c.25%). Similarly, if we replace the Smart player with a Safe player, suddenly the Bluffers’ chaos doesn’t have an impact and the Safe player wins 38% of all encounters (again, compared to an expectation of c.25%). So we end up with an interesting little triangle:

• Bluffers’ chaos messes up the rhythm of Smart players.
• Smart players use more information and the same strategy to beat Safe players.
• Safe players are immune to the stupidity of Bluffers.

Limitations

Of course, we wouldn’t be doing our due diligence as actuaries without making sure the key limitations of this model are noted!

• Rigid playstyles: Real players don’t stick to one playstyle. People adapt, pivot, and occasionally throw everything out the window on their last die.
• The lightning round? What lightning round?: We didn’t model the infamous final lightning round—the showdown when players get down to their last die and everything is on the line. Why? Because we were busy simulating actual fun This omission could leave a glaring hole in our findings, but you didn’t pay for lightning round results, did you?
• The Smart player is unrealistically… well, smart: Our Smart players base all of their logic on probability and the last bid made. But real people aren’t calculators with dice. They’re overconfident, error-strewn, and prone to trusting that one friend who definitely isn’t bluffing.
• The Safe players aren’t human either: Speaking of unrealistic behaviour, the safe players in our model quietly follow hard probability as a rule. In our experience, safe players often abandon their strategy once someone throws “coward” their way.
• We forgot that people are lazy: Our model assumes players vaguely know the probabilities in play and the expected number of dice. Real Perudo is mumbling “well, I have 2 sixes, so maybe there are, like, 10?” before slapping down a very wrong bid. Not everyone wants to calculate at Board Game Night.

What next?

Now that we’ve thoroughly dismantled our own credibility, let’s dream big. There are fascinating questions about Perudo that our model just isn’t ready to tackle, and may never be. What could the next model, probably built by AI, tell us?

1. How does emotional baggage affect gameplay?
   Does losing three rounds in a row turn a Smart player into a Bluffer? Will a player who’s had a terrible day at work suddenly start calling “Dudo!” like they didn’t read this article and see the zero wins that kind of play gets you? Our model doesn’t account for emotions, but we’re betting that frustration makes for some spectacularly bad decisions.
2. What about alliances and betrayals?
   Perudo is rarely played in a social vacuum. Friends gang up on the reigning champion, siblings hold grudges over Christmas dinner, and someone always decides to spite Craig just because. These dynamics can upend even the most carefully calculated strategies, but sadly, our model is blissfully unaware of Craig’s existence.
3. Does trash talk work?
   If you loudly proclaim, “I have 4 sixes and I dare you to doubt me!” does it make people less likely to call your bluff? Our model doesn’t include the psychological impact of banter.
4. How does alcohol change the game?
   Perudo and beverages often go hand in hand, and let’s be honest: the strategy changes dramatically after a drink or two. We didn’t model tipsy Bluffers or wine-fueled Smart players, but maybe the whole movie changes if Davy Jones has had a bottle or two of rum?
5. What’s the real cost of a lightning round meltdown?
   At the final stage of the game, when players are down to one die each, tensions skyrocket, and things get… dramatic. Who folds under the pressure? Who bids 10 sixes with no sixes in their hand just to make everyone gasp?

Conclusion: How to win at Perudo (according to science-ish)

So, what have we learned from this exercise? Well, firstly, if you’ve been reading the Trust No One or Always One playstyles and thinking “that sounds like a bit of me”, stop. Really, you’re not just losing, you’re losing your dignity.

If, however, you’re looking to up your Perudo game, here are a few tips:

1. Be Smart, but don’t be a robot: The Smart playstyle is the gold standard for a reason. Try to calculate where you can, and don’t just look at your own hand, think about what other people are doing too. But remember – if the maths says you’re being ridiculous, you’re probably being ridiculous.
2. Bluff like you mean it: The Bluffer strategy tries to take entropy and construct a ladder out of it. You know what? It works (sometimes)! Confidence is your weapon: go ahead and jump from 2 fives to 8 sixes, and maybe your opponents will frazzle themselves before you have to think again. Just don’t make a habit of doing it ALL the time.
3. Being Safe isn’t that safe: Playing conservatively and never taking any risks might keep you in the game longer than the absolute amateurs, but it’ll also win you about as many trophies as them.
4. Where you start doesn’t matter: Go with the flow at the start. It’s in the middle where the winners get ahead.

If you follow this guide, you too can become as competent as RStudio at Perudo. Remember, if you’ve followed all this, and you’re still losing repeatedly, it’s just a game. And it’s just a model. Have fun, you don’t always need to play to win, and every game needs a wildcard! Just make sure you bring snacks to the next game!

John Nicholls & Cristian Redondo Loures

January 2025