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In the multiple-partners job market, introduced in (Sotomayor, 1992), each firm can hire several workers and each worker can be hired by several firms, up to a given quota. We show that, in contrast to what happens in the simple assignment game, in this extension, the firms-optimal stable rules are neither valuation monotonic nor pairwise monotonic. However, we show that the firms-optimal stable rules satisfy a weaker property, what we call firm-covariance, and that this property characterizes these rules among all stable rules. This property allows us to shed some light on how firms can (and cannot) manipulate the firms-optimal stable rules. In particular, we show that firms cannot manipulate them by constantly over-reporting their valuations. Analogous results hold when focusing on the workers. Finally, we extend to the multiple-partners market a known characterization of the fair-division rules on the domain of simple assignment games. |
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