Online Appendix Decentralized Exchange
نویسندگان
چکیده
Proof of Lemmas A.2 and A.1. {Λi}i is an equilibrium if and only if {Λi}i = F ({Λi}i). By direct calculation, defining Xi = (Λi + αiVN(i)), we get {Λi}i = F ({Λi}i) if, and only if, {Xi}i = G({Xi}i). That F and G are monotone increasing follows because matrix inversion Y → Y −1 is monotone decreasing (e.g., Horn and Johnson (2013)), and projection onto N(i) Y → YN(i) is monotone increasing in the positive semidefinite order. Proof of Proposition 6. Pick an arbitrary starting tuple {X0 i }i such that {X0 i }i ≤ G({X0 i }i). By direct calculation, the corresponding price impacts Λi = (X 0 i ) −αiVN(i) satisfy {Λi }i ≥ F ({Λi }i). Since map F is continuous and monotone with respect to the defined partial order, recursively applying F to inequality {Λi }i ≥ F ({Λi }i), we can see that F({Λi }i) is monotone decreasing and hence converges to a fixed point of F . For the price impact tuple satisfying {Λi }i ≤ F ({Λi }i), the sequence F({Λi }i) is monotone increasing. Therefore, to prove convergence to a fixed point, we need to show that it is bounded from above. To this end, pick α > 0 sufficiently large so that {Λ̃i}i defined by Λ̃i = αdiag((I(n)− 2))n∈N(i) (42)
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تاریخ انتشار 2017