Efficiency and fairness are two desiderata in market design. Fairness requires randomization in many environments. As one of the few successful matching mechanisms, Top Trading Cycle is best known for being efficient to solve deterministic allocation problems , but it is inadequate to incorporate randomization efficiently and fairly. We propose a class of Fractional Top Trading Cycle mechanisms to solve allocation problems in which agents have fractional endowments or are ranked by coarse priorities . Dropping the graph-based definition of Top Trading Cycle , we use parameterized linear equations to describe how agents trade endowments or priorities  . Our mechanisms are ex-ante efficient. They satisfy various fairness axioms when parameter values in the equations are properly chosen. We apply our mechanisms to a couple of market design problems and obtain efficient and fair assignments in all of them.
Efficiency and fairness are two desiderata in market design. Fairness requires randomization in many environments. As one of the few successful matching mechanisms, Top Trading Cycle (TTC) is well known for being efficient to solve deterministic allocation problems . But it is inadequate to incorporate randomization efficiently and fairly. We propose a class of Fractional TTC mechanisms to solve random allocation problems efficiently and fairly . Dropping the graph-based definition of TTC , we use parameterized linear equations to describe how agents trade endowments or priorities at each step . Our mechanisms are ex-ante efficient. They satisfy various fairness axioms when parameters are properly chosen. We apply them to a couple of market design problems and obtain efficient and fair assignments in all of them.
R
is best
(TTC) is well
R
, but
. But
R
Top Trading Cycle
TTC
R
allocation problems in which agents have fractional endowments or are ranked by coarse priorities
random allocation problems efficiently and fairly
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Top Trading Cycle
TTC
A
at each step
R
parameter values in the equations
parameters
R
our mechanisms
them