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We consider two models of fair division with indivisible items: one for goodsand one for bads. For goods, we study two generalized envy freeness proxies(EF1 and EFX for goods) and three common welfare (utilitarian, egalitarian andNash) efficiency notions. For bads, we study two generalized envy freenessproxies (1EF and XEF for goods) and two less common diswelfare (egalitarian andNash) efficiency notions. Some existing algorithms for goods do not work forbads. We thus propose several new algorithms for the model with bads. Our newalgorithms exhibit many nice properties. For example, with additive identicalvaluations, an allocation that maximizes the egalitarian diswelfare or Nashdiswelfare is XEF and PE. Finally, we also give simple and tractable cases whenthese envy freeness proxies and welfare efficiency are attainable incombination (e.g. binary valuations, house allocations).

Almost Envy Freeness and Welfare Efficiency in Fair Division with Goods or Bads