On the "group non-bossiness" property

We extend the concept of non-bossiness to groups of agents and say that a mechanism is group non-bossy if no group of agents can change the assignment of someone else while theirs being unaffected by misreporting their preferences. First, we show that they are not equivalent properties. We, then, prove that group strategy-proofness is sufficient for group non-bossiness. While this result implies that the top trading cycles mechanism is group non-bossy, it also provides a characterization of the market structures in which the deferred acceptance algorithm is group non-bossy

Dihedral Group Frames with the Haar Property

We consider a unitary representation of the Dihedral group $D_{2n}=\mathbb{Z}_{n}\rtimes\mathbb{Z}_{2}$ obtained by inducing the trivial character from the co-normal subgroup $\left\{0\right\}\rtimes\mathbb{Z}_{2}.$ This representation is naturally realized as acting on the vector space $\mathbb{C}^{n}.$ We prove that the orbit of almost every vector in $\mathbb{C}^{n}$ with respect to the Lebesgue measure has the Haar property (every subset of cardinality $n$ of the orbit is a basis for $\mathbb{C}^{n}$) if $n$ is an odd integer. Moreover, we provide explicit sufficient conditions for vectors in $\mathbb{C}^{n}$ whose orbits have the Haar property. Finally, we derive that the orbit of almost every vector in $\mathbb{C}^{n}$ under the action of the representation has the Haar property if and only if $n$ is odd. This completely settles a problem which was only partially answered in \cite{Oussa}