Social preferences, accountability, and wage bargaining

We assess the extent of preferences for employment in a collective wage bargaining situation with heterogeneous workers. We vary the size of the union and introduce a treatment mechanism transforming the voting game into an individual allocation task. Our results show that highly productive workers do not take employment of low productive workers into account when making wage proposals, regardless of whether insiders determine the wage or all workers. The level of pro-social preferences is small in the voting game, while it increases as the game is transformed into an individual allocation task. We interpret this as an accountability effect

Partial permutohedra

Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker. For positive integers m and n, the partial permutohedron P(m,n) is the convex hull of all vectors in {0,1,...,n}^m whose nonzero entries are distinct. We study the face lattice, volume and Ehrhart polynomial of P(m,n), and our methods and results include the following. For any m and n, we obtain a bijection between the nonempty faces of P(m,n) and certain chains of subsets of {1,...,m}, thereby confirming a conjecture of Heuer and Striker, and we then use this characterization of faces to obtain a closed expression for the h-polynomial of P(m,n). For any m and n with n ≥ m-1, we use a pyramidal subdivision of P(m,n) to establish a recursive formula for the normalized volume of P(m,n), from which we then obtain closed expressions for this volume. We also use a sculpting process (in which P(m,n) is reached by successively removing certain pieces from a simplex or hypercube) to obtain closed expressions for the Ehrhart polynomial of P(m,n) with arbitrary m and fixed n ≤ 3, the volume of P(m,4) with arbitrary m, and the Ehrhart polynomial of P(m,n) with fixed m ≤ 4 and arbitrary n ≥ m-1

Partial permutohedra

Partial permutohedra are lattice polytopes which were recently introduced and studied by Heuer and Striker [arXiv:2012.09901]. For positive integers m and n, the partial permutohedron P(m,n) is the convex hull of all vectors in {0,1,...,n}^m whose nonzero entries are distinct. We study the face lattice, volume and Ehrhart polynomial of P(m,n), and our methods and results include the following. For any m and n, we obtain a bijection between the nonempty faces of P(m,n) and certain chains of subsets of {1,...,m}, thereby confirming a conjecture of Heuer and Striker. We use this characterization of faces to obtain a closed expression for the h-polynomial of P(m,n). For any m and n with n ≥ m−1, we use a pyramidal subdivision of P(m,n) to establish a recursive formula for the normalized volume of P(m,n), from which we then obtain closed expressions for this volume. We also use a sculpting process (in which P(m,n) is reached by successively removing certain pieces from a simplex or hypercube) to obtain closed expressions for the Ehrhart polynomial of P(m,n) with arbitrary m and fixed n ≤ 3, the volume of P(m,4) with arbitrary m, and the Ehrhart polynomial of P(m,n) with fixed m ≤ 4 and arbitrary n ≥ m−1