The object allocation problem with favoring upper ranks

We introduce (strict) favoring upper ranks, which is an extension of favoring higher ranks for random assignments. We demonstrate that ex post favoring ranks implies (strict) favoring upper ranks, and envy-freeness implies favoring upper ranks. Moreover, for at least four agents, no mechanism satisfies strict favoring upper ranks and either equal division lower bound or equal treatment of equals and lower invariance. Finally, we proved that the (modified) eating algorithm provides a random assignment that is (strict) favoring upper ranks.