# Hat-game strategies in Hanabi

This past week or two, I’ve been once again playing around with my framework for Hanabi-playing bots. In particular I’d like to talk about a very bot-friendly strategy due to Jeff Wu.

Hanabi (BoardGameGeek page) is a card game that’s been around since about 2010. It is a pure cooperative game, in which all the players win or lose together as a team; there is no competitive aspect. Mechanically, it’s sort of like Klondike meets Indian poker: the players are trying to build up foundation piles with the values 1 to 5 in each of the five colors (suits) in the deck, but with the twist that the players’ hands are held facing outward, so each player can see all the cards available to play… except their own. On your turn, you may expend one of the group’s eight “clock tokens” in order to give a hint to another player about the contents of their hand; or (blindly) play a card from your hand to the piles; or (blindly) discard a card in order to regain one clock token for the team.

The only kind of hint allowed is to choose either a color or a value visible in the target player’s hand, and indicate to them exactly which of their cards are of that color or value — and, by process of elimination, which cards must not have that color, or that value. For example, a valid hint might be to say, “Bob, your leftmost and rightmost cards are green.”

Typically, a group of Hanabi players will establish certain conventions in order to complete their foundation piles and declare victory. Some conventions are almost self-evident: Don’t discard if you could safely play a card instead. If you must discard, discard your oldest unknown card (because if nobody’s spending hints on it, it’s likely worthless). Some conventions are non-trivial but still highly intuitive for human beings: If I’m giving you a hint about some specific card, it’s probably because I want you to play it right now.

(This reminds me of a thought-experiment I read a long time ago — perhaps by Douglas Hofstadter — about something akin to the categorical imperative but a bit less philosophical. In this experiment, you and a friend are both arriving in, let’s say, San Francisco; and you need to meet up together; but neither of you have any means of communicating with each other. Where do you go, hoping that your friend will also go there?

(Okay, that was a hard one. Suppose you had already agreed to meet at City Hall; that’s still a pretty big area that you could wander around for quite a while. How would you maximize your chances of finding your friend? If you’re like me, you’d head straight for the main entrance and hang out right there. Why the main entrance in particular, as opposed to the side entrance or around back or the second-floor restroom? This convention is non-trivial but still highly intuitive for human beings.)

Immanuel Kant on Hanabi:

Act only according to that maxim whereby you can, at the same time, will that it should become a universal law.

## Hat-guessing and Hanabi

Jeff Wu wrote a Hanabi bot that uses a clever strategy to achieve perfect scores about 90% of the time in a four-player game. This BGG thread calls the strategy “hat-guessing,” because of its similarity to a number of logic puzzles involving colored hats.

There are 10 prisoners. Each prisoner is assigned a hat with either one stripe, or no stripe. The prisoners are lined up single-file such that each can see the hats in front of him but not behind. Starting with the prisoner in the back of the line and moving forward, they must each, in turn, say “one” or “zero”. If the number they say matches their own hat’s number of stripes, they are released; if not, they are killed on the spot. A friendly guard warns them of this test beforehand and tells them that they can formulate a plan whereby 9 of the 10 prisoners will definitely survive. What is the plan?

The prisoner at the back of the line has no information on his own hat at all. So, he’s going to be taking one for the team here. He counts up all the stripes he sees, and loudly declares that sum (mod 2). So if he sees 3 stripes total, he’ll say “One!” (Followed, fifty percent of the time, by a gunshot.) The ninth prisoner looks forward and counts 3 stripes, which is an odd number; so he knows that the prisoner behind him (who shouted “One!”) must have seen those same 3 stripes, and no fourth stripe on his own hat. Therefore he confidently announces “Zero!” and is saved. The next prisoner looks and sees only 2 stripes; he deduces that his own hat must have one of the 3 stripes the previous two prisoners saw, confidently announces “One!”, and is saved. And so on down the line until 9.5 prisoners have been saved.

How does this translate into a bot-friendly Hanabi strategy?

Let’s say that each player wants to know whether their leftmost card is playable or not. The player whose turn it is has no information on his own hand at all. So, he’s going to be taking one for the team here. He counts up all the playable leftmost cards he sees, and loudly declares that sum (mod 2) in the form of a hint — let’s say, if the sum (mod 2) is 0, he’ll give a color hint, and if it’s 1, he’ll give a value hint.

Alice's hand: 2r 3g 2y 1r
Bob's hand:   4r 5w 1g 3y
Carol's hand: 3y 1w 1b 5r
Dan's hand:   2g 3r 3r 1y

Current top cards: 1r 2y 1g 1b 1w


Alice sees the leftmost cards “4r, 3y, 2g.” That’s two playable cards. She gives a color hint — doesn’t matter to whom — let’s say, “Dan, your middle two cards are red.”

Bob sees the leftmost cards “3y, 2g” (and Alice’s 2r, which doesn’t count). That’s two playable cards, which agrees with the hint Alice gave. Bob realizes that his leftmost card is not playable.

Carol sees the leftmost cards “2g, 4r” (and Alice’s 2r, which doesn’t count). That’s only one playable card, which disagrees with the hint Alice gave. Carol realizes that her leftmost card is playable.

Dan sees “4r, 3y”, which disagrees with Alice’s hint, and realizes that his leftmost card is playable as well.

Thus, with just one hint, Alice has managed to convey one bit of useful information to each of the three other players! And of course she also told Dan quite a bit about Dan’s hand.

This convention is not practically useful for human beings, as far as I know — we aren’t that good at mental bit-twiddling — but it is great for computer players. Jeff’s contribution (besides a really clean and pleasant Rust implementation) was the observation that we can squeeze a lot more bits of information into Alice’s hint, by formalizing and generalizing this idea!

## The generalized InfoBot strategy

All of the following steps use only the players’ common knowledge, unless otherwise stated.

• For each player, we devise a HintStrategy, which is an integer k along with a reversible mapping from the integers [0..k) to a set of hints that would (according to common knowledge) be legal for that player. In the simple strategy above, k was 2, and the mapping was “0: give a color hint. 1: give a value hint.”

• Alice (whose turn it is) now has a set of sum(k) possible hints she could give, to unambiguously communicate any integer in [0..sum(k)). That is, if she wants to announce the number “0”, she gives Bob a color hint. For “1”, she gives Bob a value hint. For “2”, she gives Carol a color hint. For “3”, she gives Carol a value hint. For “4” and “5”, she gives Dan hints. She can express any integer in [0..6).

• For each player, using only common knowledge, we devise a prioritized list of Questions that that player would like to have answered. In the simple strategy above, each player only had one question: “Is my leftmost card playable? 0 for no, 1 for yes.” But let’s say that they’d also like to know whether their rightmost card was playable, too. That becomes the second Question in their list. Each player can have many questions, as long as the product of all the possible answers remains less than Alice’s sum(k) — which in our example is 6. (Remember, this is all based on common knowledge! Bob can compute his own HintStrategy and therefore his own k without any recourse to eyesight. He can also compute his own Questions, and also the Questions of anyone else at the table.)

• Alice (whose turn it is) considers each player’s Questions and multiplies out the answers into one combined Answer per player. For example:

  Alice's hand: 2r 3g 2y 1r
Bob's hand:   2r 5w 1g 3y  # this has changed from above
Carol's hand: 3y 1w 1b 5r
Dan's hand:   2g 3r 3r 1y

Current top cards: 1r 2y 1g 1b 1w


Alice’s sum(k) is 6. Bob’s leftmost and rightmost cards are both playable, so his Answer is 112 = 3. Only Carol’s leftmost card is playable, so her Answer is 102 = 2. Dan’s Answer is also 2.

• Alice announces the sum of the answers (mod k). In this case, 3+2+2 is 7, which is 1 (mod 6), so Alice announces “1” by giving a value hint to Bob.

• Bob works out Carol’s and Dan’s Answers the same way Alice did and deduces that his own Answer must have been 3 = 112, and thus that his leftmost and rightmost cards must be playable. He also makes note of Carol’s and Dan’s Answers, contributing to the store of common knowledge for future turns. Meanwhile, Carol and Dan are going through the same process.

This concludes the description of a generic hat-guessing hint system for Hanabi!

I call this “generic” because it is parameterized on HintStrategy and Questions. We can improve our bot’s performance by finding improvements to those two parameters.

In his Rust implementation, Jeff Wu observed that we can always increase k to 3 per player, because with only three copies of each card in the deck and four cards in hand, we are guaranteed to be able to give

• a color hint that touches the leftmost card,

• a value hint that touches the leftmost card, or

• a hint that does not touch the leftmost card.

Furthermore, if it is common knowledge that a particular player’s hand has at least one color besides the leftmost card’s color, and at least one value besides the leftmost card’s value, then we can increase k to 4 for that player! Jeff’s bot implements this improvement to HintStrategy, which means that his equivalent of Alice is able to announce numbers ranging as high as sum(k)=9 (worst case) or sum(k)=12 (common case) instead of our example’s paltry sum(k)=6.

Jeff also improves on the selection of Questions for each player.

• The first kind of question he asks is, for each card, “is this card playable?” This consumes one bit of the Answer space for each card that isn’t already known-unplayable. Also, we consider this kind of question only if this player has no already-known-playable card.

• The other kind of question he asks is, for each card, “what is its exact color and value?” But since there aren’t usually enough bits of Answer space to get a clean answer, he buckets the possibilities into just as many buckets as there are possible Answers (plus one special bucket for any card that is “worthless” and can safely be discarded). So for example, if sum(k) is known to be 9, we might say that answer “0” indicates “1w or 2r or 3y”, and answer “1” indicates “2g or 2y”, and answer “2” indicates “5w or 5r”, and so on.

I’ve thought of other possible Questions that one could ask:

• “What is the index of my leftmost playable card?”

• “What is the index of my leftmost worthless card?”

And it ought to be possible to increase k even more, in common cases. But in each case, either I haven’t managed to implement my new idea correctly yet, or I’ve implemented it only to find that it is outperformed by Jeff’s original HintStrategy and Question selections. His bot is really quite good!

The complete code in Rust can be seen here, and my C++ port here.