A matching game
Players: i,j (think: male, female)
Preferences: U(IO, II),
IO: Interest overlap.
II: Interest Intensity.
(i,j) have (n,m) interests (they don’t necessarily have equally many), (ni,mj). Let (ki) be the subset of individual i’s interests from the total interest set (ni) which is non-overlapping with the interests set (mj) (non-shared interests), and let (li) be the subset of interests from (ni) which do overlap with (mj) (shared interests). Assume that individual i’s total (negative) utility contribution from the interest set (ki) is equal to [-ki*(aiNO*qiNO)] – where II here enters the model as a scaling vector aiNO with 0 < aiNO < 1, where 0 denotes no interest and 1 denotes high interest, where the NO-part denotes ‘Non-Overlapping’ interests and where q is a relevance factor – some interests are intense but we don’t care if the partner shares them. To get a model one can always solve you probably need to assume q is bounded, but in the real world it often isn’t (‘dealbreakers’). Similarly, the interest set (li) which enter both utility functions Ui and Uj contributes individual i with a utility of [li*(aiO*qiO)] to total utility from entering the relationship, where Oi denotes the interests of individual i which ‘Overlaps’ with interests from the interest set (mj). Let the reservation utility be zero and total utility from entering the relationsship for individual i be li*(aiO*qiO) – ki*(aiNO*qiNO). Do note that the problem is not perfectly symmetric as the scaling parameter qi is in general not equal to qj, even if (li) = (lj). There’s also the problem that the common interest factor might enter (at least in part) the utility function as a share of total interest space – 2 common interests out of 4 might be better than 2 common interests out of 30. Though you might in some cases be able to let this effect enter the model via q.
Utility matters but we need a matching likelihood (ML) as well. Let the likelihood that (i,j) meet be a function of l*(aC), where dML/dl and dML/daC are both positive – so people are more likely to meet if they have many common interests and they are more likely to meet the more intense the interests are (the latter is more dubious than the former, ie. compare internet chess with ballet). Arguably one might include qC in the ML, because some people’s interests choices are ‘potential partner-relevant’, but it’s easier if we leave that out for now. Assume further that…
The model I was beginning to outline above had zero dynamics, no risk, no ‘family preferences’, ‘income/status’-variables, ‘age/looks’ -ll-, geography, beliefs… You might want to remember this model outline next time you hear a social scientist talk about this or that. A very simple model like the one above with few variables and simple relations between the variables can still be quite difficult to solve because you have to think very hard about what’s going on, what you’re assuming along the way and how to implement decision rules in the model that make the resulting equilibrium(/a) appear plausible (and how to get rid of implausible equilibria). Social behaviour is difficult to model and it’s hard to get good results in micro setups like these because there are too many variables at play and way too much interaction going on.
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