Editing Random ballot (section)

===Properties===
[[Allan Gibbard]] proved the '''random dictatorship theorem'''.<ref>{{Cite journal |last=Gibbard |first=Allan |date=1977 |title=Manipulation of Schemes that Mix Voting with Chance |url=https://www.jstor.org/stable/1911681 |journal=Econometrica |volume=45 |issue=3 |pages=665–681 |doi=10.2307/1911681 |issn=0012-9682 |jstor=1911681 |hdl-access=free |hdl=10419/220534}}</ref> It says that RD is the only rule that satisfies the following three properties:

* [[Anonymity (social choice)|'''Anonymity''']]: the lottery does not discriminate in advance between different voters.
* '''[[Strategyproofness]]''': any false report by an agent results in an outcome that is weakly [[Stochastic dominance|stochastically dominated]].
* [[Pareto-efficiency|'''Ex post Pareto-efficiency''']]: the outcome is Pareto-efficient.
** In fact, with strict preferences, RD satisfies a stronger efficiency property called ''SD-efficiency'': the resulting lottery is not stochastically dominated. With weak preferences, RSD satisfies ex-post efficiency, but violates SD-efficiency.
** Even with strict preferences, RD violates the stronger property called PC-efficiency: the resulting lottery might be dominated in the sense of pairwise-comparisons (for each agent, the probability that another lottery yields a better alternative than the RD lottery is larger than the other way around).

RD also satisfies a property called agenda consistency. It is the only rule satisfying the following properties:<ref>{{Cite journal |last1=Pattanaik |first1=Prasanta K. |last2=Peleg |first2=Bezalel |date=1986 |title=Distribution of Power under Stochastic Social Choice Rules |url=https://www.jstor.org/stable/1912843 |journal=Econometrica |volume=54 |issue=4 |pages=909–921 |doi=10.2307/1912843 |issn=0012-9682 |jstor=1912843|url-access=subscription }}</ref>

* Strong contraction consistency ("regularity"): probabilities cannot decrease when removing arbitrary alternatives.
* Ex-post efficiency.
* A probabilistic version of [[independence of irrelevant alternatives]].

Subsequent research have provided alternative proofs, as well as various extensions.<ref name=":0" />{{Rp|15}} One impossibility result relates to extending the theorem to weak preferences. It says that, with weak preferences, the properties of anonymity, SD-efficiency and SD-strategyproofness are incompatible when there are at least 4 agents and 4 alternatives.<ref>{{Cite journal |last1=Brandl |first1=Florian |last2=Brandt |first2=Felix |last3=Eberl |first3=Manuel |last4=Geist |first4=Christian |date=2018-01-31 |title=Proving the Incompatibility of Efficiency and Strategyproofness via SMT Solving |url=https://doi.org/10.1145/3125642 |journal=Journal of the ACM |volume=65 |issue=2 |pages=6:1–6:28 |arxiv=1604.05692 |doi=10.1145/3125642 |issn=0004-5411 |s2cid=1135734}}</ref>

RD satisfies an axiom called ''population consistency'', and an axiom called ''cloning-consistency'', but violates ''composition consistency''.{{Clarify|reason=What is composition consistency?|date=July 2024}}