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Speculative Physics

Groups Redux

by Mendel Schmiedekamp
Sep 23,2005

 

Groups Redux

Last month I talked about designing two 24 hour RPGs. One of those relied on a mathematical group as its core mechanic. Abstract algebra is an incredibly rich place to find structures to serve as the inspiration for a holistic design. More about groups and their related algebraic structures can be found here.

Breaking Out of Trouble

Lets consider an infinite non-commutative group, generated by three elements A, B, and C. In practice the members of this group will consist of strings of A, B, and C, as well as their inverses. But we can make things more interesting by making these generators follow a more complex rule. For example, if A is its own inverse (meaning that AA = identity, so any two consecutive A's cancel out), if B's inverse is BB (meaning BBB cancels out), and C's inverse is CCC (meaning CCCC cancels out as well). Under these rules we only need to keep track of the three generators, since they produce their own inverses.

The final result is a series of strings containing no consecutive A's, no triples of B's, and no quartets of C's. For example:

  • ABBABBABBABBABBCCC
  • CBCCBBCCCBAB
  • ACCCACCBCCA

But all the following strings reduce:

  • ABBABBCBBBABBCCC -> ABBABBCABBCCC
  • CBCCBBCCCBBAABCBB -> CBCCBBCCCBBBCBB -> CBCCBBCCCCBB -> CBCCBBBB -> CBCCB
  • ACCCCACBCCA -> AACBCCA -> CBCCA

This reduction can be useful, as suddenly inserting a symbol can cause a sequence to rapidly decrease in size, but the degree of reduction varies dramatically. For practical purposes this sort of sequence can be represented by a chain of poker chips, each element being assigned a color of chip. To make a contest out of it we could allow one player to place one to two chips somewhere in the length of the string (perhaps costing more the further back the chips go from the right end) and then letting the other player add only to the right end of the string, but able to add one or more chips at a time (again with some associated cost).

This sort of mechanic can make a very potent dueling mechanic, where each player has a string of chips, and seeks to collapse their opponent's string to penetrate their defenses. Lets decide that the attack cost is just the number of chips you skip to place your chip times the number of chips, while at the same time you may place up to three chips in defense, A costing one, B costing two, and C costing three, perhaps making A your positioning, B your parrying, and C your feints and mind games. The players start with a default defense costing their full initiative (which is the total cost of attack and defense they can manage).

So lets consider two combatants, both with initiative eight. They start with the following strings:

  1. CBC
  2. BACB

The combatants take turn. The first chooses to add AB between his opponent's A and C, costing 4 points, he also chooses to put AC on the end of his string. This gives the strings:

  1. CBCAC
  2. BAABCB

The other combatant then chooses to add an AC after the C, costing 2, he also chooses to place a CAB at the end of his string. This gives the results:

  1. CBCAACC
  2. BAABCBCAB

Then these reduce to the following:

  1. CBCCC
  2. BBCBCAB

After that the rounds continue until someone's string is reduced below a minimum length (based on the severity of the duel).

A Group of Thrones

Now lets consider the exact same group as above, but making it commutative. In this case we end up with only a finite number of unique elements in the group (24 to be exact). This is because we can group together the common generators, so the reduced form never has more than one A, two B's, and three C's.

Now taking this group we can use the combination of elements as a way to determine the story relationship between two elements which can then be associated with a faction in the setting. While this will result in a symmetric result if all the elements are chosen, we can select a small subset and later add factions as needed using the same approach. To determine the nature of the relationships, we specify what each generator means in this case. Lets say that A indicates antagonism, B determine the level of military interaction, and C indicates the level of cultural and social interaction.

First we take two random selections:

  1. BBCC
  2. AC

The result of combining these two factions is ABBCCC, which indicated a highly antagonistic and past between the two factions, both in court and in the battlefield. We can then add a new faction:

  1. BBCC
  2. AC
  3. ABBCCC

This third faction has a relationship with both of the other factions, with the first faction, is has AC, which indicated a negative relationship, but with only a small amount of interaction, and wholly in the social arena. On the other hand the third faction's relationship to the second is BB, which indicates allies who have fought on the same side often, although it appears never against the first faction.

Now as it stands, our interpretation generates only two broad factions, split between having A and not in the original string. We could instead choose to have the presence of no B's as being neutral, one as being allied, and two as being opposed. This allows for a much more complex arrangement, for example, making factions 1 and 3 opposed to faction 2, but indifferent to each other.

Structure

Mathematical structures provide a rich field of ideas and tools to piece together a design. By careful application, they can provide both flexibility and freedom, and the overall constraints required to retain an RPG's identity. Next week I will be talking about another game design I've been working on, in this case a cyberpunk game about speed and debt.

Next Month: Need for Speed

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