With two people, there is one interaction. If we bring in a third person, there are two more interactions, one with each earlier member. With the fourth member, three new interactions emerge. In other words…
x = 0 + 1 + 2 + 3 + … + n-1 = (n – 1) * (n / 2) (I’m going to be lazy and not bother with the proof.)
How can we put this into use?
Depends on what we want to do. If we want a game where the players themselves produce much of the narrative, we need more of these interactions, whereas if we want to keep the amount of time used on bickering to a minimum, we want less of these interactions.
With three players we have (3 – 1) * (3 / 2) = 3 interactions. With five players, we have (5 – 1) * (5 / 2) = 10 interactions. Therefore by adding two people into the group, they will interact with each other over three times as much as the three player group, thus taking much more time and potentially bogging down the game.
Obviously, its not always this straightforward. Certain players will use more time than others, certain players will only agree with other players and so forth. Also, different genres will work differently. Combat-oriented games with little planning will not work the same way and adding players will not break it in the same manner.
Still, understanding this principle and how it will change the picture, should be taken into account when planning games or designing adventures, although I do doubt anyone in the world does so.