We review the theory of fairness as it pertains to concretely specified problems of resource allocations. We present punctual notions designed to evaluate how well individuals, or groups, are treated in relation to one another: no-envy, egalitarian-equivalence, individual and collective lower or upper bounds on welfare, notions of equal or equivalent opportunities, as well as various families extending these notions. We also introduce relational notions specifying how allocation rules should respond to changes in resources (resource monotonicity), technologies (technology monotonicity), preferences (welfare domination under preference replacement), and population (population monotonicity, consistency, converse consistency). We investigate the implications of these properties, in various combinations, in the context of various models: the “classical” problem of dividing an unproduced bundle, economies with production, economies with public goods, economies with single-peaked preferences, economies with indivisible goods, and economies in which the dividend is a non-homogeneous continuum. We consider economies in which resources are owned collectively, economies in which they are owned privately, and economies in which ownership is mixed. We offer a number of characterizations of particular allocation rules.
