Endogamy is computed at four scenarios that cross the two snapshots' structures (minority share, m) and behaviours (in-group preference, p):
Each cell uses a closed-form expression for the compositional matching DGP. With α = 1 + p, a group's endogamy from its own perspective is:
endogamy = (own share · α²) / (own share · α² + other share)
The matching shown on the canvases is one stochastic realisation of the same DGP — illustrative, not where the displayed numbers come from.
A symmetric (Shapley) decomposition averages T1-anchored and T2-anchored estimates of each effect so they sum exactly to the observed change:
Δ_struct = ½ · [ (T1·T2 − T1·T1) + (T2·T2 − T2·T1) ] Δ_behav = ½ · [ (T2·T1 − T1·T1) + (T2·T2 − T1·T2) ] Δ_struct + Δ_behav = Δ_obs
Each group's endogamy is the share of its own paired members whose partner is in the same group. Majority and minority can move very differently — minority endogamy is much more sensitive to composition (Blau structural opportunity), while majority's is dominated by preference.
Rising inter-ethnic unions in a diversifying society can look like growing openness — but they can also be entirely structural, driven by who the partner market now contains rather than by changing preferences. In Sweden 1991–2017, observed same-ancestry unions among Swedish-origin women fell roughly 12 percentage points; yet holding the partner market constant, in-group preference moved the other way and tightened by ~5 pp. The decline was structural; symbolic boundaries, if anything, became slightly stronger. The logic flips for minorities, whose endogamy is highly sensitive to group size (Blau's structural opportunity): a larger co-ethnic pool can increase their same-group share even at constant preference. We can't read openness directly off observed unions — we have to construct what they would have looked like under a counterfactual market.