Monday, September 14, 2020

Game length scaling by player count

That is, how to tell if the game box is lying by looking at the rules (it almost always is, but still...).

My basic framework here is that for many games, the length is simply the product of the number of turns and the amount of time taken per turn. This obviously doesn't work for games without turns, such as real-time games like Space Cadets Dice Duel.  There also may be overhead that doesn't scale the same way,  e.g. once per day activities in Robinson Crusoe, or once per round activities in Agricola.

Many games are the same length to first order, independent of player count: Battlestar Galactica, Carcassonne, Pandemic, Kingdomino. These games have a roughly fixed number of total turns. Some of these may have strong second-order terms to affect game length, for example Pandemic, which tends to last more turns with more players as it's harder to get cards in a single hand. Other games could take more time per turn with more players due to group discussion, which may be play group dependent.  These games can be identified by the fact that the there's some shared resources (e.g. cards or tiles) that advances the players towards the end (Battlestar Galactica, Pandemic), or ends the game when it runs out (Carcassonne, Kingdomino*).

Let me expand my thoughts on Kingdomino. First, for 2-player I'm focused on the "Mighty Duel" 7x7 variant (which I've almost always played). This alone makes the 2 player and 4 player games take the same amount of time, at least to first order. If instead we looked at 2 player 5x5 games, we'd expect them to be roughly half the length of a 4 player game.  Second, we should probably break up the two aspects of the game: selecting tiles and placing them. In Mighty Duel and 3-4 player games, all tiles/dominos are available to choose at some point. Also, each round three tiles are chosen, and the fourth is forced to whoever is left. Thus, this part of the game is the same whether there are 2, 3, or 4 players. The question is whether placing takes a different amount of time. Perhaps it's slightly different in a 3 player game, but I often think of it as incidental to your choice of tile in the next round (as it happens at the same time). I can see how 4 player games we may expect to take more time, but I don't think it'd be linear.

Many games are proportional to the number of players to first order: Ticket to Ride, Splendor, Agricola. These games have a roughly fixed number of turns per player. Some of these games (e.g. Agricola) are honest and state that the play time is a certain number of minutes per player. Others, like Ticket to Ride, do not.  These games can be identified by an explicit number of rounds (Agricola), the end of the game triggered by a single player's resources depleting (Ticket to Ride), or games which are a race to a victory condition (Splendor).

Sometimes, though, a game doesn't fit neatly into one of the above categories, as some elements are adjusted by player count.  First, let's consider Bohnanza.  There's a shared deck of cards and the game ends as soon as the players exhaust it three times (see more below).  At first, we'd think that this would be a fixed length game, however things get more complicated when we consider the variant rules below, which allow the full 7 players.

Bohnanza, 45 minutes

104 normal cards

expansion: +50 cards: 154

3 player: -4 cards: 150 cards

4-5 players: -24 cards: 126 cards

6-7 players: -4 -6 cards: 144 cards

3 players: only two times through the deck. 3 bean fields

4-7 players: 3 times, 2 bean fields

new rules: flip 2 cards, draw 1 card per player per turn (original: draw three cards)

There are three things coming into play here.  First, the number of cards in the deck varies with player count.  Second, with 3 players you only go through the deck twice and each player has three bean fields instead of two.  Third, the number of cards drawn per turn is a function of the number of players.

Let's calculate the number of turns it takes to get through the deck the first time.

3: 150 / (2+3) = 30

4: 126 / (2+4) = 21

5: 126 / (2+5) = 18

6: 144 / (2+6) = 18

7: 144 / (2+7) = 16

If subsequent times through deck has no card removal (almost certainly not true), then we can calculate the following number of total turns.

3 players: 60 turns

4 players: 63 turns

5 players: 54 turns

6 players: 54 turns

7 players: 48 turns

Practically, each time through the deck actually takes much less time than the first.  Some of this may just be as players get into the flow of the game, but there are two structural reasons as well.  First, cards are essentially removed from play as they are sold and converted to coins.  Second, a number of bean cards are still in players' fields as they go through the deck the second and third time.  Let's look at the total number of bean fields in each player count.

3 players: 9 fields

4 players: 8 fields

5 players: 10 fields

6 players: 12 fields

7 players: 14 fields

This effect would further skew decrease the number of turns in large player count games relative to smaller player count games.  However, with more players there are more possible negotiation considerations.  It's more likely that there are multiple other players who have something you want, and similarly more likely that there are multiple players who want something you have.  Three player games can go very quickly, as players get committed to what they are going for and you have more need to manage the large number of plantings you'll do.  But in a 7 player game, your turn is much more rare, so trading is more important.  While I do not have data to quantify it, I suspect that this at least equalizes the length of the game and probably even pushes larger player length games to be longer.  If there's any data on this, it'd be interesting to see.

As another example, let's look at Century: Spice Road.  Here what's interesting is that the game ends when one player gets a certain number of point cards, but that number depends on the number of players in the game.

Century: Spice Road, 30-45 minutes

2-3 players: 6 cards

4-5 players: 5 cards

Thus, if each card takes "x" turns to obtain, then the relative game length is as follows.

2 players: 12x

3 players: 18x

4 players: 20x

5 players: 25x

As we can see, we'd expect a 5 player game to take more than twice as long as a 2 player game.  This ignores the fact that obtaining the 6th point card likely takes fewer turns than the first, as you've already built up an engine.  This clearly indicates that the 30-45 minute ranges is incorrect, as the max range should be closer to twice the minimum.

Some games get shorter with more players: Sushi Go Party, at least to first order, given negligible overhead for scoring more players, actually has fewer cards per player in higher player count games.

Sushi Go Party!

2-3: 10 cards each

4-5: 9 cards each

6-7: 8 cards

8: 7 cards

There are a host of other considerations in game length that I haven't touched on here, many of which have clear implications as to whether they increase or decrease game length, but how to quantify that effect is less clear.

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