The Rookery

Madeline's thoughts on social deduction games, forum/community meta, and any other philosophical musings
 Thumb up

Polynomial Proportionality

msg tools
No mountains, no valleys
Never argue with idiots; they'll drag you down to their level and then beat you on experience.
Microbadge: The Resistance fanMicrobadge: Werewolf fanMicrobadge: I was here for BGG's Twentieth Anniversary!Microbadge: I write marathon postsMicrobadge: The Resistance fan
There are lots of games in which the amount of some resource determined in setup depends on the number of players. For instance, in Suburbia, you start by setting up three stacks of location tiles. Each stack has size 15 for two players, 18 for three players, or 21 for four players. Then you add in several more tiles to the last pile to randomize the end of the game: 6, 9, or 12 for two, three, or four players respectively.

Helpfully, the game board has some iconography to remind you of this, so you don't have to memorize it. But if you did, you could note that the number of tiles in a stack is 9+3n, where n represents the number of players, and the number of extra tiles is just 3n. In math, this is an example of "linear proportionality;" the output value depends on the number of players, but it increases in a consistent way. The jump from two to three is the same size as the jump from three to four.

I've mentioned before that I suspect one reason why some players find the mechanics of Spirit Island appealing is the way it uses scale. Setup requires 4 Fear Markers and one island board per player. Sometimes you have to add X amount of Blight per player. Sometimes there are bonus powers that trigger on each board, so its effect is linearly proportional to the number of players. So if someone is looking for a co-op that has a similar balance or rhythm with four players as it does as a solo game, this might be of interest. (In comparison to games like Forbidden Island where the "game" takes "turns" between each player's turn, and that can create some swinginess based on how long an individual player goes between actions that they can synergize.)

There are some games where the printed "estimated time" on the box depends on the number of players: like, "30 minutes plus 15 minutes per player." I can't think of any off the top of my head, but I feel like I've seen them around, maybe just on BGG discussions. All of these are "linear" in the sense that the jump from two to three is about the same as the jump from three to four. You don't need to have any scary n^2 values in your computations. But what about social games like werewolf?

Compare a game with 9 players to a game with 19. The latter will probably have more wolves, and this increase might be sort of linear--2 maxes in 9 players is about proportional to 4 maxes in 19. The number of players also influences the number of game days: because there are more wolves, it will take more kills to either eliminate all the max evils or have the wolves reach parity. So we can guess that the number of days in a werewolf game, on average, is linearly proportional to player count. Mathematically, we can write D ~ aN, where D is the number of days, N is the number of players, a is some constant, and ~ means "about proportional to."

What about the total number of posts in the game? Well, a longer game will have more days with players alive and arguing compared to a shorter one. But also, say we only compare D1 to D1: if there are more players in a bigger game, there will be more people posting. So the total post count is probably something like P ~ b*N*D, where b is some other constant. Substituting in our formula from above, we have P ~ b*N*a*N, or P ~ (ab)*N^2. This is a nonlinear relationship!

Again, consider a D1 scenario. If we imagine that every player makes, on average, 20 posts per day (this is an arbitrary figure), then that would give a total of 180 posts on D1 of a 9-player game versus 380 in a 19-player game. Once people start to die, these numbers will decrease, but linearly, so the overall total will still be a quadratic function.

But is this even a realistic assumption to make? Even looking at one player, the chances are higher that they will have more to comment on in the larger game, because there has been more content--or because your buddy posted a cute gif you want to quote, or someone said something rude you want to get vengeance for, etc. Posts beget posts. Interactions beget interactions. The total possibility for social interaction, for better or for worse, is probably nonlinear itself.

The number of pairs of players is quadratic. I suspect that matters when monitoring for conflict: even if player A is normally fine on their own, they might cause problems by antagonizing (or egging on) player B. So the number of potential problems is proportional to N^2. (Alternatively, the probability that nothing goes wrong is proportional to 1/N^2.)

Hypothetically, if there were only Nice People and Jerks, and only the Jerks created problems, and Nice People never caused problems among themselves, then a mod could just be like "I need N Nice People to fill this game, and the time it will take to fill is proportional to N." Of course, that would require that the mod could reliably distinguish between Nice People and Jerks, and that the Jerks wouldn't react poorly to being called Jerks, both of which are preposterous. But I think it's worth remembering both the good news: that in fact, the vast majority of people are not going to be jerks for the sake of it. And the bad news: that even if the vast majority of people have good intent, there are many people who can still create conflict in specific contexts.
Twitter Facebook
1 Comment
Subscribe sub options Thu Sep 2, 2021 12:21 am
Post Rolls
  • [+] Dice rolls
Loading... | Locked Hide Show Unlock Lock Comment     View Previous {{limitCount(numprevitems_calculated,commentParams.showcount)}} 1 « Pg. {{commentParams.pageid}} » {{data.config.endpage}}
    View More Comments {{limitCount(numnextitems_calculated,commentParams.showcount)}} / {{numnextitems_calculated}} 1 « Pg. {{commentParams.pageid}} » {{data.config.endpage}}