On uniqueness of equilibrium in the Kyle model

Although it is linguistically natural and convenient to treat the market maker as a single individual, one should really imagine a continuum of market makers competing with each other for order flow.

We should emphasize that the problem of uniqueness of equilibrium in the dynamic versions of the Kyle model is different in an essential way from the question of uniqueness in the one-period model. [1] shows that in the continuous time setting the derived equilibrium is unique within a nice class of functions. In the dynamic setting the uniqueness of the Kyle equilibrium was studied by [12] for the class of tame equilibria, where theoretical results and computational experiments suggest the presence of multiple equilibria!

We feel that we should clearly state our understanding of the relationship between our work and BKL. The statement of our Theorem 2.1 improves the statement of their Corollary 1 by replacing an unspecified positive constant a with 1. The proof in BKL actually proves the result with \(a = 1\), but under BKL’s maintained assumption of “admissibility,” which requires that the variance of the informed trader’s order is finite. Our proof is much simpler and more direct, and does not require this assumption. Our Theorem 2.2 has the same statement as the first part of BKL’s Result 2, but again their result has the maintained assumption of admissibility, and our proof is simpler and more direct. BKL has no analogue of Proposition 3.3, which implies Theorem 1.2. Our Lemma 4.1 is Corollary 2 of BKL, and the respective proofs are similar. BKL has no analogues of our Corollary 4.2 and Propositions 4.3 and 4.4. The assertions of the first two of these would follow immediately, at least under the assumption of admissibility, if the overall argument of BKL was correct, but it has not yet been accepted by the profession, and it remains to be seen whether it can be brought to a satisfactory condition.