Structural Holes are 'a gap between two individuals who have complementary sources of information.' Topological Dictators- situations where the Pareto rule is homotopic, i.e. continuously deformable, to Dictatorship- arise where there are 'holes' in the domain of the decision problem- i.e. it is not perfectly contractible. If preferences inform ethos- i.e. constitute some essential aspect, or ipseity, of the agent- then all information about an alterity's preference profiles is complementary- i.e. not univocal even if indiscernibly identical. But this means preference profiles are not collapsible. Thus, Social decision problems must have structural holes in their domain such that preferring a Pareto improvement is homotopic to Dictatorship. This does not however mean anything very much except that people who think it does must also accept Godel's proof of God.
If we think of Structural Holes as arbitrage opportunities between co-ordination and dis-coordination games, then
1) either there is a homotopy between preference profiles such that they can be deformed into each other across Lewis-Stalnacker 'closest possible worlds'. In this case, 'ethos' or 'ipseity' is 'contractible' such that agents share a potentially univocal identity.
2) no such homotopy exists. In this case, there can be no univocal shared 'Social' identity. Alterity is ubiquitous and irreducible.
If (1) then for any domain featuring ipseity and alterity with three or more agents- or just one agent with non-collapsible meta-preferences or meta-meta preferences and so on- a Teritius Gaudens must exist if Pareto improvements are univocally desirable- in which case the Social Welfare Function is collapsible. Since all contractible spaces are collapsible but not vice versa, the Tertius Gaudens' grinning 'free face', under (2), gains by making the opposite appear the case.
This is the third, Deliberative Reason always has with it- a Jesting Pilate with a vengeance.
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