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We fully solve an assignment problem with heterogeneous firms and multiple heterogeneous workers whose skills are imperfect substitutes, that is, when production is submodular. We show that sorting is neither positive nor negative and is characterized sufficiently by two regions. In the first region, mediocre firms sort with mediocre workers and coworkers such that output losses are equal across all these pairings (complete mixing). In the second region, high skill workers sort with a low skill coworker and a high productivity firm, while high productivity firms employ a low skill worker and a high skill coworker (pairwise countermonotonicity). The equilibrium assignment is also necessarily characterized by product countermonotonicity, meaning that sorting is negative for each dimension of heterogeneity with the product of heterogeneity in the other dimensions. The equilibrium assignment as well as wages and firm values are completely characterized in closed form. We illustrate our theory with an application to show that our model is consistent with the observed dispersion of earnings within and across U.S. firms. Our counterfactual analysis gives evidence that the change in the firm project distribution between 1981 and 2013 has a larger effect on the observed change in earnings dispersion than the change in the worker skill distribution.

The multistochastic Monge–Kantorovich problem on the product X = ∏ i = 1 n X i of n spaces is a generalization of the multimarginal Monge–Kantorovich problem. For a given integer number 1 ≤ k n we consider the minimization problem ∫ c d π → inf on the space of measures with fixed projections onto every X i 1 × … × X i k for arbitrary set of k indices { i 1 , … , i k } ⊂ { 1 , … , n } . In this paper we study basic properties of the multistochastic problem, including well-posedness, existence of a dual solution, boundedness and continuity of a dual solution.

The multistochastic (n, k)-Monge–Kantorovich problem on a product space $$\prod _{i=1}^n X_i$$ is an extension of the classical Monge–Kantorovich problem. This problem is considered on the space of measures with fixed projections onto $$X_{i_1} \times \cdots \times X_{i_k}$$ for all k-tuples $$\{i_1, \ldots , i_k\} \subset \{1, \ldots , n\}$$ for a given $$1 \le k < n$$ . In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution $$\pi $$ to the following important model case: $$n=3, k=2, X_i = [0,1]$$ , the cost function $$c(x,y,z) = xyz$$ , and the corresponding two-dimensional projections are Lebesgue measures on $$[0,1]^2$$ . We prove, in particular, that the mapping $$(x,y) \rightarrow x \oplus y$$ , where $$\oplus $$ is the bitwise addition (xor- or Nim-addition) on $$[0,1] \cong {\mathbb {Z}}_2^{\infty }$$ , is the corresponding optimal transportation. In particular, the support of $$\pi $$ is the Sierpinski tetrahedron. In addition, we describe a solution to the corresponding dual problem.