I wasn't going to jump on any kind of Valentine's Day gaming bandwagon, but being as there isn't one, maybe I'll balance things out by posting on the theme anyway, and put forward a idea or two on showing in-game relationships mechanically.
The first is just an adaption of a simple system I posted last year, for identifying acquaintances.
The basic mechanism is a roll using a die with any number of faces. When creating a new character, each player rolls this die. The result represents their nature, interests or needs in a given area, whether they are fully aware of these or not, in interpersonal relationships for the sake of the post. The higher the number, the more specific they are.
You could modify the result by a number based on a relevant characteristic, or even fudge it to suit a particular character concept, or hidden purposes, and in less tactical games a character could have multiple rolls, use a larger die or be given a range.
If you roll a maximum, roll again and add the second result to the first. Keep adding for each extra maximum. To reduce linearity, you could even set up a matrix.
When a character or group meets another person, roll for that person too. If two individuals have matching numbers there's a connection of some kind. If it's not an exact match, the closer the two numbers are, the better the two individuals bond, or the more intimately they understand one another, or are interested one in the other.
A test of some kind could be made to check how reciprocal this is. If the minimum number represents the broadest general nature, and rising numbers ever more specific, it could be a need to roll equal or above, modified by the difference in the numbers of the two individuals. A failed test would be a lack of connection on the part of that individual.
What practical effect this might have on the game can be left to GM and players to decide. As a minimum it could be a non-binding prompt, or add colour to encounters, but it could also manifest mechanically in positive or negative modifiers to rolls, for insight or persuasion maybe, or just when the individuals in question are together.
This won't be everyone's cup of tea, and some of us may even find it uncomfortable, but if we can abstract other physical, mental and spiritual interactions, why not this kind?
The second idea is a bit of a plug. If you have a copy of Triffles: Abandoned Space, you might have seen the suggestion each Strand could be interpreted as an appropriate response in a given context. If you haven't tried this yet, it's no surprise I recommend it.
The complete Fabric in Abandoned Space would work especially well for interactions in this kind of location, but also more generally between world-weary individuals, or those under stress, and the blank Fabric can be used to get new webs of potential behaviour.
You could get a similar effect by mapping out areas of mood or patterns of behaviour in lesser or greater detail for a given character and moving between these in some way in response to events, whether more or less randomly or according to a set of criteria.
All thoughts are welcome, but don't hang around if you have other places be creative.
_

2 responding:
An interesting way to "gamify" something people usually leave to role-playing. I think it could lead to some interesting results.
I agree, and I'd hope so. There can be a tension in gaming as a whole between the understanding players have of a situation and the rules governing it, as well as scope for and interest in the introduction of material to a game that isn't covered by the overall ruleset. This is something that offers a starting point somewhere in the middle of all of that, whether prompts for more playing of roles, or a foundation for fuller expression in mechanisms.
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