Hanabi strategy and heuristics

A fun cooperative card game of imperfect information

Hanabi is a very interesting co-op card game. It's one of the very few games that manage to combine low complexity of rules with very high skill ceiling (like chess, the holy grail of game design). The goal of Hanabi is to play a number of cards in the right order. There are 5 colors and 5 numbers. They must be played (go on the table) in order for each color, though it doesn't matter which colored card is played when. In other words, you can play White 1, then White 2, then Red 1, OR White 1, Red 1, then White 2. If you played White 2 before White 1, it would be a misplay. See the photo below (4 Yellows have been played in order, and 1 card of Blue, Red, Green). Game design wise, it's an imperfect information game, because you hold your hand of 5 cards backwards, so that other players can see your cards, but you cannot see your own. Thus, if there are 3 players, there are 15 cards in hands, of which you can see 10, thus 2/3 of the currently available information. Furthermore, you draw from a pile of shuffled cards. Thus, you can also count cards to figure out what remains in the pile, and also what must be in your hand. The game ends when you run out of cards to draw, or misplay cards 3 times. Your score is the number of cards you played in the right order. Thus, if you played all numbers from all colors, then you will get 25 points, the maximum.

The game proceeds in turns. On your turn, you must do one of 3 options:

1. Play a card (may be in any position in your hand). If it fits the cards on the board, it is a succesful play and stays there, otherwise, it is a misplay and is discarded (e.g., you cannot play a Red 4 if the Red 3 has not been played).

2. Discard a card (may be in any position), and flip a used hint token.

3. Give a hint (flip a hint token).

The hinting is the bread and butter of the game. You must tell a player all of their cards with a certain color or number. If that player has multiple Red cards, you cannot just tell them that one card is Red. The same is true for the numbers. Thus, sometimes it is not possible to single out specific cards (e.g., you want to draw attention to Red 3, but they have multiple 3's and multiple Reds).

Because the game is on a timer (the pile of cards will run out), the hints must be efficient enough to allow players to play. Since you have 25 cards to play (5 colors * 5 numbers), if you use a single hint for each card, you would need 25 hints. You begin the game with 9 available hints.

In most cases, given the time constraint, it is not possible to hint a specific card (e.g. Blue 3) a player has in their hand because this would require 2 hints per card, first tell them about the color (Blue) and then about the number (3). In total, this would require 50 hints (25*5). There are several copies of most cards (3 of the Ones, 2 of the Twos-Threes-Fours, and 1 of the Fives). Thus there are 10 cards of each color, and thus in total 50 cards. If you are 3 players, you start with 15 cards in hands, and thus have 35 left in the draw pile. Adding this to your initial pool of hint token (9), means that you have 44 turns in the game before the final round.

At the game start, the most important thing to do is tell players which cards in their hands are 1's. Unfortunately, they might be the wrong colors if they have duplicates or some 1's have already been played. So even if you want a player to play a 1, you might instead tell them about their Red cards instead.

But which card to play and which to discard? Given the time constraints, since there isn't enough hints to use to tell people specifically, so you must rely on heuristics. The game is thus in essence a meta-game about coming up with and correctly using a set of agreed upon heuristics that guide behavior. We played this game 100 times or so, and these are our heuristics:

1. When in doubt, discard the most rightward card in your hand. New cards drawn go into the leftmost position of the cards you haven't been hinted anything about. Thus, any card in the discard position has usually sat in your hand for at least a couple of turns, giving other players a chance to hint you about them if they should be played or must be saved for later.

2. Your hand is split in groups of cards based on what hints you were given about them. Thus, to begin with, all of your cards are in the unknown (no hints) group. If someone tells you that some cards are Green, you move those cards to the leftmost position in the hand in a separate group of Green cards. If we label the cards in your hand after their position in reverse order (right-to-left A-to-E) (so that it is in the normal order for the other players looking at your hand), the leftmost card is E, and the rightmost card is A. If you are later hinted that some cards are 3's, which overlaps with the Green block, then you have 4 blocks: Green not-3's, Green 3's, non-Green 3's, and unknowns (may also be non-Green non-3's). Thus, if you see someone has a 5 in their discard position (A), it is important to move that card out of the discard position.

3. When hinting, hint the closest player to yourself. Because each turn costs time in the form of cards in the deck (if they are discarded or played), it is important to play as much as possible. Thus, the heuristic is to hint the player who goes after you if they have anything useful to play.

4. Related to the above, if given a choice between playing and something else, default to playing. For instance, if the next player has an important card in discard position that they don't know about, and there are no hints left to use, you have to discard instead of playing to allow them to avoid discarding that card.

5. There is a presumption that when you are given a hint, the other player intends you to play something. If you know nothing else, it is the leftmost card in the hint you got (if it included multiple cards). This is a defeasible reason because if you have good reason to believe it is dangerous to play, you would not play it. A good reason not to play something is that it was in your discard position. In some cases it is obvious, like a Red 5 (currently unplayable), which if discarded would result in a non-perfect game (max score cannot be attained). But if you are a told about a 2 in the discard position, it would be unclear if it should be saved or played. Looking at the board (cards already played and those discarded) often clears up things. The reason this play leftmost card heuristic works most of the time is that it makes it possible to indicate a specific playable card in a hand for a player using only 1 hint, which is very efficient. Sometimes 2 playable cards can be hinted, e.g. 2 Twos.

Following these heuristics, you will have a reasonable chance at winning with 2 or 3 players (winning meaning getting the maximum score). The game gets progressively harder with more players. I've never won with 4 or more players. I'm curious to hear if others have come up with better or other strategies. There are also the optional higher difficulty Rainbow cards, which can either be treated as a 6th color by itself, or used as joker colors (they count as every color). The latter is particularly hard because every color hint becomes ambiguous, as it is unclear whether the cards you were told were Blue are actually Blue or Rainbow.

The above heuristics are human-friendly, but in mathematics, a number of people have proven that you can adopt algorithm-friendly meta-heuristics that almost always win (and others propose Hanabi as a topic for AI training). These involve using and interpreting hints in ways which allows one to almost always indicate a specific action (e.g. play card in position 3/C, or discard card in position 2/B). So hinting 3's in positions A, D, E might not relate to 3's at all, but might mean play card B.

For those interested in playing, you can play this game online in various places, such as boardgamesarena.