Grafting linkers with open ends of complementary single-stranded DNA makes a flexible tool to tune interactions between colloids, which facilitates the design of complex self-assembly structures. Recently, it has been proposed to coat colloids with mobile DNA linkers, which alleviates kinetic barriers without high-density grafting, and also allows the design of valency without patches. However, the self-assembly mechanism of this novel system is poorly understood. Using a combination of theory and simulation, we obtain phase diagrams for the system in both two and three dimensional spaces, and find stable floppy square and CsCl crystals when the binding strength is strong, even in the infinite binding strength limit. We demonstrate that these floppy phases are stabilized by vibrational entropy, and "floppy" modes play an important role in stabilizing the floppy phases for the infinite binding strength limit. This special entropic effect in the self-assembly of mobile DNA-coated colloids is very different from conventional molecular self-assembly, and it offers new axis to help design novel functional materials using mobile DNA-coated colloids.Nucleic acids are ubiquitous in nature because of their capability of encoding large amounts of information via canonical Watson-Crick base-paring interactions [1]. With the help of chemical methods to make synthetic oligonucleotides of arbitrary sequences, one can use specific binding interactions between single-stranded DNA (ssDNA) chains to program selective interactions between different colloidal particles. For example, one can graft DNA linkers to the surface of colloidal particles with open ends of ssDNAs. These ssDNA tails serve as "sticky ends" that bind specifically to other colloids coated with ssDNA tails of complementary sequence, which offers a novel way of manipulating the self-assembly of colloidal particles [2,3]. By using DNA-coated colloids (DNACCs), a number of ordered crystals [4-10] and selfassembled "colloidal molecules" [11] have been obtained in experiments, while the self-assembly mechanism of DNACCs is still not well understood [9,12,13]. For example, the diffusionless transformation from a floppy crystal to the other compact crystal has been observed in experimental systems while the underlying mechanism remains not fully resolved [13].Recently, a novel system of mobile DNA-coated colloids (mDNACCs) was introduced that displays qualitatively new properties [14,15]. Compared with immobile DNA-coated colloidal systems, mDNACCs have a broader temperature window for self-assembly, and therefore allow the better control over the assembly process [14]. Mobility of DNA linkers also allows particles to more easily roll around each other and rearrange [14,16], without grafting of very high density. Moreover, unlike colloids with patches in specific locations [11,16], the interaction in mDNACCs is intrinsically a many-body potential, which could be employed to control the "valency" of particles without patches by tuning nonspecific repulsions between the p...