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@NickArnosti @giorgiomartini The mechanism of Marek and Peter is equivalent to RSD: for each agent-house pair, the probability that this agent receives this object is the same. Their paper makes an important point that this equivalence cannot be replaced by equality when stating the conjecture

@giorgiomartini Yes, that's right! Equal treatment of equals seems too weak to me, as it imposes no constraints on, e.g. how I can treat two people whose lists differ only in the final two positions.

Random Serial Dictatorship is (i) ex post Pareto efficient (ii) symmetric (as in Pycia and Troyan), (iii) truthful (outcome from truth-telling FOSD outcome from lying). Is RSD is the only mechanism satisfying these properties? As far as I know, this is an open question. (!!!)

@NickArnosti What you get by taking quotient space, as Yannai suggested, is known as the Lebesgue space L_1 en.wikipedia.org/wiki/Lp_space#… For compactness, see math.stackexchange.com/questions/3511…

Analysis Question. Let X be space of increasing functions [0,1]->[0,1], and define d(f,g) =Int|f(x)-g(x)|dx. I want to say that (X,d) is a compact metric space, but d is not a metric, because f(x)=1(x<1) =/= g(x)=1(x<=1). What is the cleanest way to "fix" this?

@NickArnosti Another observation: For n=4, truthfulness can be relaxed to upper-invariance & lower-invariance. In other words, you can drop swap-monotonicity in the decomposition of truthfulness by arxiv.org/pdf/1401.3675.…

@NickArnosti Two papers about NP-hardness, maybe there are follow-ups: arxiv.org/pdf/1409.6076.… arxiv.org/pdf/1304.3169.…

@NickArnosti Absolute continuity is implied by ex-post efficiency. For n=4, RSD happens to be the unique mechanism even if the efficiency requirement is weakened to absolute continuity. I wonder if this generalizes beyond n=4.

@NickArnosti A workaround could be to replace ex-post efficiency with an alternative notion tailored for allocation-matrix representation. One candidate is absolute continuity w.r.t. RSD: an agent cannot get an object with positive probability if she gets it with zero probability for RSD.

@AramGrigoryan_ Now I see, thanks! The basic conjecture is for the number of agents equal to that of objects. More importantly, each agent must receive the same number of objects. Without this assumption, picking an agent at random and allocating all the objects to her satisfies the axioms.

@fsandomi Rule 2 (and therefore the mechanism) is different from RSD. Consider three agents 1,2,3 and two objects A,B. 1 and 2 rank A over B. 3 ranks B over A. Rule 2 will assign B to 3. RSD may not do that. Unless I miss something.

@NickArnosti I once wrote a program that generates a human-readable proof for n=4 agents and objects. Still struggle to publish it as people say it is hard to follow because of the length. Just 70000 pages. dropbox.com/s/snazaq36aett…

@AramGrigoryan_ @NickArnosti Aram, could you clarify how rule 2 works? Maybe it coincides with RSD by itself? By the way, for 3 agents and objects, it is known that RSD is the unique mechanism satisfying the axioms (Bogomolnaia, Moulin 2001). For 4, this also seems to be true (see below)

@NickArnosti ex-post pareto efficiency and symmetry are immediate. to see FOSD, note that the only way an agent can benefit from lying is by misreporting her first choice when rule 2 is used, however, the agent is very likely to miss her first choice if rule 1 is used (with 1-epsilon)