Originally posted by DoctorScribbles
Have no such fear. Nothing is too fantastical for the Spirituality forum, after all. And how much more so for a theoretical model of such a place! Let your fantasies run wild. That's how great ideas are born. (Ref. the bar scence depicted in A Beautiful Mind.)
This is going to be one hell of a difficult model to solve.
It must also be a dynamic game in which different strategies are played over time, resulting in equilibrium paths.
Now one (not really so simple) way to model this would be to consider the struggle between the Holier-Than-Thou Alliance (HTT) and the Rationals. These groups compete each period for converts (a fraction from a unit measure continuum of Undecideds). At the end of each period (must be defined), HTT and Rationals convert some fraction of undecideds depending on their strategies. In the next period we have a new pool of Undecideds to convert.
A utility function for a member of the HTT would be a function of the fraction of all players that are HTT, the fraction of HTT that are members of their particular cult, and a personal satisfaction term which gives increased weight to the second fraction in periods where the member plays differentiate. The form must be such that in equilibrium players face a tradeoff between focusing on converting more xtians through cooperating and persuading these converts to their cult (The function must be seperable in these inputs.).
A utility function for a Rational consists of an entertainment term which rises with the comedic value of mocking HTT's as well as component which rises with the fraction players that are enlightened Rationals.
HTT: In any given period each member of the HTT can choose cooperate or differentiate.
----> cooperate: This strategy specifies joining in a unified multidenominational assault on the Rationals and also giving Amen's to each others posts.
----> differentiate: Do not participate in the group assualt and instead focus on turning more of the current period HTT converts to that players particular cult. Perhaps a Protestant could point out that the Catholic Church is really a led by Satan, or a Catholic could remind new converts that Protestants are heretics that left the One True Church.
----> reason: This strategy calls for rationals to band together in patient, reasoned refutation of the HTT's superstitious claims. This increases the fraction of undecideds that choose Rationality in a period.
----> mock: Here Rationals increase their individual utilities at the expense of seeming as ridiculous as the HTT. This decreases the expected return on converts, but increases the utility of current members highlighting their intellectual superioty to the HTT.
The fractions of converts to HTT and the Rationals is determined by the following equation:
HTTCon = 50% + (% of HTT that play cooperate - % of Rationals that play reason)
RationCon= 1 - HTTCon
The fractions of HTTCon that choose Protestant and Catholic are determined by these equations.
HTTCat: (# of Catholics that play differentiate) / (# of all HTT players that play differentiate)
HTTPro: 1 - HTTCat
So the more Catholics and Protestants cooperate the more they increase their fraction of converts. However, if one HTT member (say Protestant) plays differentiate and the rest play cooperate, then all the HTT converts from that period become Protestants! If however most Catholics and Protestants play differentiate, this backbiting causes the Rationals to gain a greater percentage of converts.
In equilibrium HTT members must find a balance between gaining xtian converts and shoring up these converts into their One True Faith.
Rationals must balance certain utility from mocking HTT's and the potentially greater utility from freeing undecideds.
I can already think of many improvements to this, but they could get wildly complicated. For instance, members of HTT's and Rationals could move into the Undecided pool. From here they could go to Rational or HTT or remain Undecided. We would need to incorporate some factor that allowed for decoversion.
Playing Reason repeatedly could cause could have diminishing returns over time. Thus at some point, Rationals will become tired of countering the brainwashing techniques of HTT. Or perhaps playing Mock could have some potential to upset HTT's and drive a wedge into their alliance so that fewer HTT members play cooperate and instead choose differentiate.
We could also make the equilibria Weak Perfect Bayesian Nash Equilibria, where members have beliefs about how all other members will play. This seems particular fitting given the nature of this forum.
Ok, well please check the math. I think though with well-specified preferences we could make this work.