Equilibrium in the Kyle (1985) model

In his 1985 paper, Kyle presents 3 versions of the same model: a single period model, a multiple period model and the continuous time limit of the multiple period model.

When he formalizes the equilbrium problem for the discrete time multiple period model, he restricts himself to recursive linear pricing (P) and demand (X) rules. He writes on page 1322: " We suspect, but have not been able to prove, that equilibria with nonlinear X and P do not exist."

Has anyone ever proved or disproved this conjecture? Is it a redundant restriction as he suspected, or is it something binding and there are equilibria that would yield greater profit to the insider, albeit with nonlinear demand schedules?

If this appears to remain an open question, is there any relatively recent reference that says we're still unclear about this?

There are a few works examining nonlinear strategies $$X$$ and the uniqueness of linear strategies in Kyle (1985).

Single-Period Kyle Model

Cho and El Karoui (2000) find a nonlinear strategy for the single-period Kyle model if they use a Bernoulli distribution for the noise term. For continuous noise (i.e. non-atomic distributions), they also characterize the existence of a unique (linear) equilibrium.

Boulatov, Kyle, and Livdan (2012) show the linear strategy is unique for the original single-period Kyle model setup. Boulatov and Bernhardt (2015) also examine a single-period case and show that the linear strategy is unique and robust while nonlinear strategies are not robust. Thus the linear strategy is the equilibrium.

Multi-Period Kyle Model

Foster and Viswanathan (1993) show that for multi-period Kyle models, the linear strategy is a unique equilibrium for beliefs in the class of elliptical distributions (e.g. the Gaussian distribution used by Kyle).

Continuous-time Kyle Model

Back (1992) shows that in the continuous-time Kyle model, there may be nonlinear strategies. The strategies $$X$$ are, however, smooth and monotone in the total order size.

As an interesting aside, Back and Baruch (2004) study conditions where the continuous-time Kyle model converges to the same equilibrium as the Glosten and Milgrom (1985) model.

• That was very helpful. Thanks for taking the time to pull out references on the subject! It's really appreciated. Aug 3 '20 at 15:19
• I just looked at FV(1993) and I am not sure how their proof excludes nonlinear equilbria. Their proof conjectures a quadratic objective and a linear recursion for the evolution of prices. Am I missing something or did they just establish that the equilibrium they identified in Prop. (7) is unique WITHIN the class of linear recursions -- i.e., there might be nonlinear equilibria. Otherwise, the result would also be valid for I=1 insider and T=1 period... so, BKL(2012) would have wasted their time proving a less general result. Or am I missing something? Aug 5 '20 at 23:10
• (cont.) I do see how BLK(2012) manage to prove that nonlinear equilibria cannot exist in the single-period Kyle model, as long as trading strategies are suitably differentiable -- otherwise, a first order condition is violated. Aug 5 '20 at 23:13
• the BKL(2012) link is broke. Would you please write out the full reference so that even if the link is broke people can still search for the paper?
– Hans
Jan 21 '21 at 0:56
• @Hans Putting "Boulatov, Kyle, and Livdan (2012)" into a search engine gives you multiple pages with the paper. Take your pick from whichever is available to you in your locale (which varies by user). Jan 22 '21 at 1:14