On the Equivalence of Weak- and Cyclic-Monotonicity

When is a monotone function cyclically monotone?

“…In revealed preference theory, Chambers and Echenique (2018) use it to establish that a demand function is strongly rationalizable with a quasilinear utility if and only if it satisfies a continuity condition and the law of demand (i.e., the negative of the demand function is monotone) (see also Amir et al 2017). In producer choice theory, the working paper version (Kushnir and Lokutsievskiy 2019) shows that any weakly rationalizable supply functions with a finite range that is positive homogeneous of degree zero is characterized by the law of supply (i.e., the supply function is monotone). We also explain how the reduction of cyclic monotonicity to the requirement of an allocation being monotone is helpful in solving spatial allocation problems.…”

“…In the working paper version(Kushnir and Lokutsievskiy 2019), we also show how the result for singlepeaked preferences byMishra et al (2014) follows from Theorem 1.…”

“…Nerve is then a special weighted hypergraph 1 See Lavi and Swamy (2009), Mishra and Roy (2013), and Carbajal and Mu'alem (2020). The cyclic monotonicity condition also has applications in revealed preference theory, producer theory, and spatial allocation (see Chambers andEchenique 2018, Kushnir andLokutsievskiy 2019). 2 The result with a similar flavor for environments without transfers also appeared in Pycia (2012).…”

“…11 See Lavi and Swamy (2009), Mishra and Roy (2013), and Carbajal and Mu'alem (2020). The cyclic monotonicity condition also has applications in revealed preference theory, producer theory, and spatial allocation (see Chambers and Echenique 2018, Kushnir and Lokutsievskiy 2019). …”