How to balance Dixit?

Question

The rules of Dixit that the game ends when the deck runs out (leaving players with full hands at the end). Every time I have played, the players have commented that this means that some players have been the storyteller more than others, and it doesn't seem fair. The players have wanted to continue playing after the deck has run out, until everyone has been the storyteller the same number of times, either by playing with fewer cards to choose from their hands, or by reshuffling the used cards.

What is the reason for not continuing until all players have been the storyteller? Do these methods of balancing the game have any unintended side effects?

Answer 1

That particular rule isn't particularly balanced, and is included for convenience. However, playing with fewer cards in hand also imbalances the game, as storytellers have fewer cards to choose from to come up with a good hint, and other players have fewer options to play in.

Turns as the storyteller aren't the same value as other turns either; the storyteller is probably more likely to get their 3 points, but other players stand to gain more points by misleading other players and guessing the card correctly.

One way to fix this is, as you mentioned, to reshuffle cards until everyone has had an equal number of turns. When I play, we agree to finish at the end of however many more rounds.

Additionally, if you have extra Dixit card packs, you probably don't want to keep playing until you run out of cards.

Answer 2

Definitions

• A turn consists of one player being the story teller and all other players guessing.
• A round consists of as many turns as there are players, so that every player has been the story teller once.
• A game is completely fair if only full rounds have been played, and players have the same number of cards to choose from in each turn.

A fair game

Since the only way to achieve a fair game is by playing only full rounds, we need to either reuse discarded cards by shuffling them back in, or calculate how many rounds can be played by using the cards only once. This may mean playing fewer turns than the number of cards allows for.
I would advice discarding the excess cards before playing, to avoid accidentally starting a round you can't finish.

Calculation

When playing with 4–6 players, each turn every player discards one card, so for n players, n cards are discarded every turn. After a full round, n² cards will have been discarded and after r rounds, r × n² cards will have been discarded.
Also, every player needs to have 6 cards to choose from for the final turn, so after that final turn, every player will have 5 cards left in their hand for a total of 5 × n cards.
So for 4 and more players, the formula is c = r × n² + n × 5 for a fair game.

When playing with 3 players, players play with 7 cards in their hand of which they play two, except for the story teller who only plays one. This means that 5 cards are discarded every turn (the story teller 1, the other players 2 each). After a full round 15 cards will have been discarded.
After the final turn they should have 5 + 5 + 6 cards left in their combined hands, for a total of 16 cards.
This makes for the slightly different formula c = 15 × r + 16

From this, we can calculate the number of cards c in the deck that yields a completely fair game of r rounds. Excess cards can be discarded after shuffling.

3 players

c = 15 × r + 16

You can play 4 rounds for a total of 12 turns. You need to discard 8 cards from the deck and play with 76 cards. If more cards are available, add 15 for each additional round.

4 players

c = 16 × r + 20

You can play 4 rounds for a total of 16 turns. You can play with the full deck of 84 cards. If more cards are available, add 16 for each additional round.

5 players

c = 25 × r + 25

You can play 2 rounds for a total of 10 turns. You need to discard 9 cards from the deck to play with 75 cards. If more cards are available, add 25 for each additional round.

6 players

c = 36 × r + 30

You can play 1 round of 6 turns. Discard 18 cards from the deck to play with 66 cards. If more cards are available, add 36 for each additional round.

7 players

c = 49 × r + 35

While the rest of the game doesn't provide for 7 players (neither the scoring board nor the voting tokens), you could play 1 round with 7 players using the full deck of 84 cards. If more cards are available, you would need to add 49 for each additional round.