When Linear Equilibria Are Unique in Kyle’s Trading Model
Summary
The document surveys research on whether Kyle’s linear demand and pricing rules are the only equilibria. In the single-period model, the cited results depend on the noise-trader distribution and model assumptions: nonlinear strategies can occur with Bernoulli noise, while work on continuous noise and on Kyle’s original setup finds a unique linear equilibrium. Other cited work describes linear strategies as robust relative to nonlinear alternatives in a single-period setting.
For the multi-period model, the response reports uniqueness of linear strategies for beliefs in the elliptical-distribution class, which includes the Gaussian case used by Kyle. In continuous time, nonlinear insider strategies may exist, but the cited result restricts them to smooth, monotone functions of total order size. These findings qualify the original conjecture across model variants; they do not establish one answer for every possible distribution or set of assumptions.
Key ideas
- Equilibrium uniqueness in Kyle’s model depends on the time structure and assumptions about noise or beliefs.
- Nonlinear strategies can arise in a single-period model with Bernoulli noise.
- For the original single-period setup and specified continuous-noise cases, cited studies find a unique linear equilibrium.
- In multi-period models with elliptical beliefs, linear strategies are reported as unique.
- Continuous-time models may admit nonlinear strategies that are smooth and monotone in total order size.
Tags
Full text
# Equilibrium in the Kyle (1985) model # 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? Thanks in advance. ## Answer by kurtosis (score 2, accepted) https://quant.stackexchange.com/a/57052 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.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.