# Insider trading in discrete time Kyle games

Christoph Kühn⋆ Christopher Lorenz⋆

(⋆Institute of Mathematics, Goethe University Frankfurt,
D-60054 Frankfurt am Main, Germany
<EMAIL_ADDRESS>
)

###### Abstract

We present a discrete time version of Kyle’s (1985) classic model of insider
trading. The model has three kinds of traders: an insider, random noise
traders, and a market maker. The insider aims to exploit her informational
advantage and maximise expected profits while the market maker observes the
total order flow and sets prices accordingly.

First, we show how the multi-period model with finitely many pure strategies
can be reduced to a (static) social system in the sense of Debreu (1952) and
prove the existence of a sequential Kyle equilibrium, following Kreps and
Wilson (1982). This requires no probabilistic restrictions on the true value,
the insider’s dynamic information, and the noise trader’s actions.

In the single-period model we establish bounds for the insider’s strategy in
equilibrium. Finally, we prove the existence of an equilibrium for the game
with a continuum of actions, by considering an approximating sequence of games
with finitely many actions. Because of the lack of compactness of the set of
measurable price functions, standard infinite-dimensional fixed point theorems
are not applicable.

Keywords: Information asymmetry, Kyle model, extensive form game, sequential
equilibrium, Komlós’ theorem
2020 MSC Classification: 91A18, 91A27, 91A11, 28A33
JEL Classification: C73, D53, D82, G14

00footnotetext: _Acknowledgments._ We would like to thank Umut Çetin, Ralph
Neininger, Emanuel Rapsch, Marius Schmidt, and participants at the Workshop on
Stochastic Dynamic Games, University of Kiel (2023), for useful comments.

##  1 Introduction

In this paper, we study the discrete time version of Kyle’s (1985) classic
model of a specialist market with asymmetrically informed agents. The model
features three (kinds of) traders: an informed trader, called the insider,
random noise traders, and a market maker.

While the insider acts strategically in order to exploit her informational
advantage about the true value of an asset, the uninformed noise trader
submits orders stochastically independent. The market maker cannot distinguish
between informed and uninformed trades, and thus he sets prices depending only
on the total order flow, from which he learns conditional distributions of the
true value. In a _Kyle equilibrium_ , the insider trades optimally given the
price function of the market maker, and given the insider’s strategy the
market maker’s prices are rational (in the case of risk-neutrality, they
fulfill a zero-profit condition).

Kyle’s model has been a workhorse model for understanding the role of
asymmetric information in the formation of prices. For practitioners it can
serve as a conceptual tool to explain the incentives of market makers to
provide different prices to different market participants based on their level
of informedness. This can be one reason why brokers offer better terms to
retail customers, a phenomenon widely discussed in connection with _payment
for order flow_ (we refer to Lynch (2022) for a recent empirical study).

In his seminal paper, Kyle (1985) proposes a discrete time model assuming
normal distributions and a risk-neutral insider and market maker. The
continuous time model was first comprehensively studied in Back (1992) under
the distributional assumption that noise trades follow a Brownian motion. The
Kyle-Back model has been extended in various directions, and we can only cite
a selection here.

A general principle to establish equilibria follows the _inconspicuous trade_
ansatz. The resulting equilibrium insider strategy is inconspicuous in the
sense that the accumulated total demand process has the same law as the
accumulated demand of the noise trader alone. Furthermore, the true value,
which is independent of the noise trader’s demand, has to be a nondecreasing
function of the accumulated total demand at maturity. The market maker’s price
function of the total demand and time is chosen such that any insider demand
process that leads to the terminal accumulated total demand described above is
optimal. In Çetin and Xing (2013) the existence of an inconspicuous
equilibrium is shown if noise trades follow a Poisson process instead of
Brownian motion. Further extensions include trading with default risk in Campi
and Çetin (2007) and a random trading horizon in Çetin (2018). The case of
dynamic private information where the insider’s information evolves over time
is considered in Back and Pedersen (1998) and Danilova (2010). For a thorough
overview we refer to the monograph Çetin and Danilova (2018).

The inconspicuous trade ansatz does not work for risk-averse market makers. In
the continuous time model, Çetin and Danilova (2016) prove the existence of a
non-inconspicuous equilibrium that is derived from a fixed point (see the
discussion below). In Back et al. (2021) there are several assets, and the
distribution of the true value and the accumulated total demand at maturity
are coupled through an optimal transport map.

In the single-period model, under normality assumptions, Kyle (1985) shows
that there exists a unique equilibrium in the class of affine-linear
equilibria (if the insider’s demand in equilibrium is an affine function in
the true value, then the price function is automatically affine in the total
demand). McLennan et al. (2017) extend uniqueness to a broader class of
equilibria. Çetin and Larsen (2023) study whether Kyle’s affine-linear
equilibrium is stable for different trading times. Rochet and Vila (1994)
study a related single-period model where the insider can observe the trade of
the noise trader and show the existence of a unique equilibrium while relaxing
the assumption of normality. Kramkov and Xu (2022) further relax
distributional assumptions by connecting the problem to the dual problem of a
certain class of optimal martingale transport problems.

In the present paper, we specify the discrete time Kyle model with a risk-
neutral insider and market maker as a (generalised) extensive form game. The
purely discrete approach is original and has the advantage that the model
becomes mathematically tractable without further probabilistic restrictions on
the true value, the noise trader’s orders and in particular regarding the flow
of private information of the insider. The game tree, the so-called extensive
form, gets to the heart of the decision theoretical nature of the Kyle model:
nodes in the tree signify the realised history of order and information flows,
based on which the insider makes a trading decision. Even though the market
maker “does not explicitly maximize any particular objective” (Kyle (1985)),
rational pricing can be considered as an action of the market maker that is
restricted by the assumed behaviour strategy of the insider (we refer to
Remark 2.8 for a detailed discussion). Our purely discrete approach makes the
assumptions of the Kyle model very explicit: The trades of the insider and the
noise trader occur truly simultaneously, and the insider is not able to
observe the current trade of the noise trader. After the orders are submitted
the market maker sets the price.

In continuous time models the chronological sequence is a bit less explicit.
In most models, accumulated trades of the noise trader are given by a Brownian
motion or another diffusion, and admissible insider strategies are of finite
variation. In this constellation, trading volumes are of different orders, and
the chronological sequence of trades becomes irrelevant in the continuous time
limit. On the other hand, there are continuous time generalizations of classic
game trees, but with serious and in some cases even insoluble conceptual
problems caused by the lack of immediate successors of nodes. We refer to
Alós-Ferrer and Ritzberger (2016), especially Subsection 5.7, for a detailed
discussion. It is beyond the scope of the paper to relate this to continuous
time Kyle models. In continuous time timing games (in which each player has
only one move) Riedel and Steg (2017) show the existence of a subgame-perfect
Nash equilibrium.

Our first main result (Theorem 2.16) shows the existence of a _sequential Kyle
equilibrium_ in the sense of Kreps and Wilson (1982) and thus leads an idea
from Kyle’s original work to maturity. Since the market maker does not have
perfect information, a sequential equilibrium is based on his beliefs
regarding the true value that generalize conditional probabilities derived
from Bayes’ rule. The concept of a sequential equilibrium is a refinement of a
subgame-perfect Nash equilibrium. Subgame-perfectness rules out non-credible
strategic plans that are not realised in equilibrium (cf. Subsection 2.3).

The proof of Theorem 2.16 relies on a fixed point theorem for a self-
correspondence acting on the insider’s dynamic strategies and the market
maker’s pricing functions. This is conceptually different to the fixed point
theorem in the continuous time model of Çetin and Danilova (2016) in which the
functional acts on the probability distributions of the accumulated total
demand at maturity, and the dynamic quantities are constructed from a fixed
point distribution.

In Section 3, we establish basic properties of equilibria in single-period
Kyle games. The insider’s demand is nondecreasing in the true value, but there
need not exist a nondecreasing price function of the market maker (Example
3.6). Since the insider uses the noise trades as camouflage to remain
unobserved by the market maker, the following obvious question arises: Are
insider’s demands in equilibrium uniformly bounded for a family of models such
that the (exogenous) noise trader’s demands lie in a fixed bounded interval?
The answer turns out to be positive if the probabilities of the insider’s
demand at the boundaries of the interval are bounded away from zero (see
Theorem 3.3 that also provides the range of insider’s demands explicitly).

In Section 4, we prove the existence of an equilibrium in the continuous state
game (Theorem 4.4) by considering an approximating sequence of discrete state
games. In view of Example 3.6, admissible price functions only need to be
measurable but not continuous in the total demand. This leads to the problem
that the set of price functions (equipped with pointwise convergence almost
everywhere) is not sequentially compact, and standard infinite-dimensional
fixed point theorems of Schauder-Tychonoff’s or Kakutani-Fan’s type (see,
e.g., Theorem 10.1 and Theorem 13.1 in Pata (2019), respectively) cannot be
applied directly to the continuous game.

##  2 The Kyle model as extensive form game

We specify the discrete time, discrete state Kyle model as a generalised
extensive form game. Extensive form games have a reputation of being
notationally burdensome, but possess great interpretive power that is rooted
in its tree structure. In general, we follow the notation of (González-Díaz et
al., 2010, Chapter 3) in the spirit of Selten (1975). In line with the
literature, we label quantities related to the insider by $X$, related to the
noise trader by $Z$, and related to the market maker by $Y$.

###  2.1 Extensive form

Trading takes place over the course of $T\in\mathbb{N}$ trading rounds at
times $t=1,2,\ldots,T$. The insider possesses private information about the
true value of the asset, not known to the noise trader or market maker. There
are $N\in\mathbb{N}$ fundamental information states $1,\ldots,N$, and the true
value is a mapping

$v:I\to\mathbb{R},\quad i\mapsto v^{i},\qquad\mbox{where}\
I:=\\{1,2,\ldots,N\\},$

such that $v^{1}\geq v^{2}\geq\ldots\geq v^{N}$. The flow of information is
specified exogenously by a refining sequence of partitions of $I$:

$\mathcal{I}_{t}:=\\{I_{t,1},\ldots,I_{t,N_{t}}\\}\subseteq 2^{I},\qquad
t\in\\{1,2,\ldots,T\\},\ N_{t}\in\\{1,2,\ldots,N\\}.$

Refining means that for every element $I_{t,i}\in\mathcal{I}_{t}$ with $t\geq
2$, there is $I_{t-1,j}\in\mathcal{I}_{t-1}$, called a predecessor, such that
$I_{t-1,j}\supseteq I_{t,i}$. A subset $I_{t,i}\subseteq I$ represents the
information states in $I$ which are still attainable at time $t$. There can be
multiple states $i$ leading to the same true value. This allows to model the
change of conditional probabilities of the true value over time in a flexible
way. By contrast, for some injective function $i\mapsto v^{i}$, the
conditional probability of a state can only increase or drop to zero over
time. For notational convenience we define

$\mathcal{I}_{T+1}:=\big{\\{}\\{1\\},\\{2\\},\ldots,\\{N\\}\big{\\}}.$

The full revealing of the true value is not a restriction of generality since
at the future fictive date $T+1$, trading has already concluded. Even the
extreme case that the insider has no information at all until the end is not
excluded: the sequence of partitions would be given by
$\mathcal{I}_{1}=\mathcal{I}_{2}=\ldots=\mathcal{I}_{T}=\\{I\\}$.

Each round of trading can be broken down into two steps. Firstly, new
information about the true value of the asset is revealed to the insider.
Secondly, both insider and noise trader _simultaneously_ trade a discrete
quantity of shares from the set

$E_{X}:=\left\\{x^{1},\ldots,x^{K}\right\\}\subseteq\mathbb{R}\quad\mbox{and}\quad
E_{Z}:=\left\\{z^{1},\ldots,z^{L}\right\\}\subseteq\mathbb{R},\quad\mbox{resp.,
where}\ K,L\in\mathbb{N}.$

The set $E_{X}$ could be, e.g., multiples of the minimal (fractional) order
size. The market maker only observes the sum and sets a price. Since the noise
trader does not trade strategically, her action can be placed after that of
the insider in the tree in Definition 2.1 below (see also Figure 2.1). This
sequence of information disclosures, insider and noise trades spans the game
tree, formally given by:

######  Definition 2.1 (Game tree).

The _game tree_ $(\mathbb{T},E)$ is given by the set of nodes

$\displaystyle\mathbb{T}$
$\displaystyle:=\\{r,I_{1,i_{1}},(I_{1,i_{1}},x_{1}),(I_{1,i_{1}},x_{1},z_{1}),(I_{1,i_{1}},x_{1},z_{1},I_{2,i_{2}}),(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}},x_{t}),$
$\displaystyle\qquad(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}},x_{t},z_{t}),(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}},x_{t},z_{t},I_{t+1,i_{t+1}}):$
$\displaystyle\qquad
t\in\\{2,\ldots,T\\},\;(x_{1},\ldots,x_{T})\in(E_{X})^{T},\;(z_{1},\ldots,z_{T})\in(E_{Z})^{T},$
$\displaystyle\qquad(I_{1,i_{1}},\ldots,I_{T+1,i})\in\mathcal{I}_{1}\times\cdots\times\mathcal{I}_{T+1}\;\text{
such that }I_{1,i_{1}}\supseteq\ldots\supseteq I_{T+1,i}\\},$

where $r$ denotes the root node, and by the set of canonical edges
$E\subseteq\mathbb{T}\times\mathbb{T}$, meaning that, for example,
$(r,I_{1,i_{1}})\in E$ and $(I_{1,i_{1}},(I_{1,i_{1}},x_{1}))\in E$.

][c]0.25 . . . . . . . .
$\\{I\\}$$I_{1,1}$$\quad\cdots$$I_{1,2}$$\quad\cdots$$I_{1,i_{1}}$$I_{1,N_{1}}$$\quad\cdots$$\dots$$\dots$$I_{2,j_{1}}$$\quad\cdots$$I_{2,j_{2}}$$\quad\cdots$$I_{2,j_{i}}$$\quad\cdots$$\dots$

(a) The insider’s private information.

][c].7 $X$ $Z$ $X$ $Z$ . ⋮. ⋮⋮. . . . . . . . . . . . . . .
⋮⋮⋮$r$$I_{1,1}$$\quad\cdots$$I_{1,2}$$\quad\cdots$$I_{1,i_{1}}$$I_{1,N_{1}}$$\quad\cdots$$\dots$$\dots$$x^{1}$$\quad\cdots$$x^{2}$$x^{K}$$\quad\cdots$$\dots$$z^{1}$$\quad\cdots$$z^{2}$$\quad\cdots$$z^{L}$$\dots$$I_{2,j_{1}}$$\quad\cdots$$I_{2,j_{2}}$$I_{2,j_{i}}$$\quad\cdots$$\dots$$x^{1}$$\quad\cdots$$x^{2}$$\quad\cdots$$x^{K}$$\quad\cdots$$\dots$

(b) The game tree.

Figure 2.1: Diagram of the game tree from Definition 2.1. The dotted boxes
denote the set of nodes where the insider (marked $X$) and the noise trader
(marked $Z$) take a decision.

The set of nodes where new information is revealed, and where the insider and
the noise trader make a move are given by

$\displaystyle\mathbb{T}_{V}:=\\{\tau\in\mathbb{T}:\;\tau=r\text{ or
}\tau=(I_{1,i_{1}},x_{1},z_{1},\ldots,z_{t})\text{ for
}t\in\\{1,\ldots,T\\}\\},$
$\displaystyle\mathbb{T}_{X}:=\\{\tau\in\mathbb{T}:\;\tau=I_{1,i_{1}}\text{ or
}\tau=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}})\text{ for
}t\in\\{2,\ldots,T\\}\\},\ \text{and}$
$\displaystyle\mathbb{T}_{Z}:=\\{\tau\in\mathbb{T}:\;\tau=(I_{1,i_{1}},x_{1})\text{
or }\tau=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}},x_{t})\text{ for
}t\in\\{2,\ldots,T\\}\\},$

respectively. Next, we furnish these sets of nodes with transition
probabilities, starting with the _nature player_. For a finite set $A$, the
simplex $\Delta A:=\\{p\in[0,1]^{A}:\ \sum_{a\in A}p(a)=1\\}$ is identified
with the set of probability distributions over the points in $A$ by
$p(\\{a\\}):=p(a)$.

######  Definition 2.2 (Probability assignments).

We fix $\nu\in\Delta I$ with $\nu>0$ and $\zeta\in\Delta E_{Z}$ with
$\zeta>0$. The _probability assignment_ $p_{V}$ assigns to every
$\tau\in\mathbb{T}_{V}$ a probability distribution over the direct successors
of $\tau$ with probabilities $p_{V}(\tau,I_{1,i_{1}}):=\sum_{i\in
I_{1,i_{1}}}\nu(\\{i\\})$ for $\tau=r$, and

$\begin{aligned} p_{V}(\tau,I_{t+1,i_{t+1}}):=\frac{\sum_{i\in
I_{t+1,i_{t+1}}}\nu(\\{i\\})}{\sum_{j\in
I_{t,i_{t}}}\nu(\\{j\\})}\end{aligned}\qquad\begin{aligned} &\text{for
}t\in\\{1,\ldots,T\\},\ I_{t+1,i_{t+1}}\subseteq I_{t,i_{t}},\\\
&\tau=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}},x_{t},z_{t}).\end{aligned}$

The _probability assignment_ $p_{Z}$ is given by

$p_{Z}:\mathbb{T}_{Z}\to\Delta E_{Z},\qquad p_{Z}(\tau,\,\cdot\,):=\zeta.$
(2.1)

In (2.1), with slight abuse of notation, we identify each direct successor of
$\tau\in\mathbb{T}_{Z}$ with an order $z\in E_{Z}$. The insider, on the other
hand, observes besides the fundamental information about the true value also
the noise trader’s past trades before making a trading decision (the latter is
a standard assumption in the Kyle model motivated by the observability of the
past prices that depend on the total demand):

######  Definition 2.3 (Behaviour strategy).

A _behaviour strategy_ $\xi$ is a mapping from the insider’s nodes
$\mathbb{T}_{X}$ to the set of probability distributions over trades $E_{X}$,

$\xi:\mathbb{T}_{X}\to\Delta E_{X},\qquad\tau\mapsto\xi(\tau,\;\cdot\;).$

With $\Xi$ we denote the set of behaviour strategies.

For a behaviour strategy $\xi\in\Xi$ and a node $\tau\in\mathbb{T}_{X}$, one
interprets $\xi(\tau,\\{x\\})$ with $x>0$ ($x<0$) as the probability of buying
(selling) $\lvert x\rvert$ shares at node $\tau$. We define the set of
terminal nodes by

$\Omega:=\left\\{\tau\in\mathbb{T}:\
\tau=(I_{1,i_{1}},x_{1},z_{1},\ldots,\allowbreak I_{T+1,i})\right\\}.$

The realisation probability $p$ is a mapping from $\Xi$ to $\Delta\Omega$,
assigning to every strategy a probability distribution over outcomes according
to

$\begin{split}p^{\xi}(\omega):=\nu(\\{i\\})\prod_{s=1}^{T}\xi((I_{1,i_{1}},x_{1},z_{1},\ldots,I_{s,i_{s}}),\\{x_{s}\\})\;\zeta(\\{z_{s}\\})\qquad\\\
\qquad\text{for
}\omega=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{T+1,i}).\end{split}$ (2.2)

###  2.2 Kyle equilibrium

It is a defining feature of Kyle-type models that the market maker can not
distinguish the orders of the insider and the noise trader. In each trading
round, he only observes the total order flow $y\in E_{Y}:=\\{x+z:\;x\in
E_{X},z\in E_{Z}\\}$. Following (Kyle, 1985, Equation (3.3)), a price $S_{t}$
at time $t=1,\ldots,T$ is a function of the total order flow
$(x_{1}+z_{1},\ldots,x_{t}+z_{t})$.

######  Definition 2.4 (Pricing system).

A _pricing system_ $S:=(S_{t})_{t\in\\{1,\ldots,T\\}}$ of the market maker is
a family of functions

$S_{t}:(E_{Y})^{t}\to[v^{N},v^{1}],\qquad(y_{1},\ldots,y_{t})\mapsto
S_{t}(y_{1},\ldots,y_{t}).$

With $\mathcal{S}$ we denote the set of pricing systems.

Given a pricing system $S\in\mathcal{S}$, the payoff of the insider is defined
by

$U(\omega,S):=\sum_{t=1}^{T}\left[v^{i}-S_{t}(x_{1}+z_{1},\ldots,x_{t}+z_{t})\right]\,x_{t}\quad\text{for
}\omega=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{T+1,i}),$

and her objective is to maximise the expected utility

$u(\xi,S):=\sum_{\omega\in\Omega}p^{\xi}(\omega)\;U(\omega,S)\longrightarrow\max_{\xi\in\Xi}!$
(2.3)

Assuming that the insider plays $\xi\in\Xi$, the joint probability of
fundamental information $i$ and order flow $(y_{1},\ldots,y_{t})$ is given by

$p^{\xi}_{Y}(i,y_{1},\ldots,y_{t}):=\quad\sum_{\mathclap{\begin{subarray}{c}(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{T+1,i})\in\Omega\\\
\text{s.t. }x_{s}+z_{s}=y_{s}\text{ for all }s=1,\ldots,t\end{subarray}}}\quad
p^{\xi}(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}},x_{t},z_{t},\ldots,I_{T+1,i}),$
(2.4)

where $i$ uniquely determines $(I_{1,i_{1}},I_{2,i_{2}},\ldots,I_{T+1,i})$.
Using Bayes’ rule, the market maker can infer the conditional probability of
$i$ given $(y_{1},\ldots,y_{t})$ assuming the insider plays $\xi$. If
$p^{\xi}_{Y}(y_{1},\ldots,y_{t}):=\sum_{i=1}^{N}p^{\xi}_{Y}(i,y_{1},\ldots,y_{t})>0$,
it reads

$p^{\xi}_{Y}(i\mid
y_{1},\ldots,y_{t}):=\frac{p^{\xi}_{Y}(i,y_{1},\ldots,y_{t})}{p^{\xi}_{Y}(y_{1},\ldots,y_{t})}.$
(2.5)

######  Definition 2.5 (Rational pricing).

A pricing system $S\in\mathcal{S}$ for the market maker is _rational_ assuming
the insider plays $\xi\in\Xi$ if

$S_{t}(y_{1},\ldots,y_{t})=\sum_{i=1}^{N}p^{\xi}_{Y}(i\mid
y_{1},\ldots,y_{t})\;v^{i},$ (2.6)

for all $(y_{1},\ldots,y_{t})\in(E_{Y})^{t}$, $t=1,\ldots,T$, with
$p_{Y}^{\xi}(y_{1},\ldots,y_{t})>0$, i.e, the price equals the expectation of
the true value of the asset conditional on the market maker’s information.

In equilibrium, the assumption of the market maker coincides with the strategy
played by the insider. In addition, the insider’s strategy is optimal given
the prices quoted by the market maker.

######  Definition 2.6 (Kyle equilibrium).

A _Kyle equilibrium_ is a pair
$(\xi^{\star},S^{\star})\in\Xi\times\mathcal{S}$ satisfying

  1. (i)

Profit maximisation: Given $S^{*}$, the strategy $\xi^{\star}$ is optimal
according to (2.3),

  2. (ii)

Rational pricing: Given $\xi^{\star}$, the pricing system $S^{\star}$ is
rational according to (2.6).

The existence of an equilibrium can be shown without probabilistic
restrictions on the true value, the insider’s dynamic information, and the
noise trader’s actions.

######  Theorem 2.7.

Every Kyle game has a Kyle equilibrium.

###### Proof.

Follows from Theorem 2.16 below. ∎

######  Remark 2.8 (Kyle model as a dynamic social system in the sense of
Debreu (1952)).

We can now place the interaction of the insider and the market maker in a game
theoretic context, even though the latter does not maximise any particular
objective. Rational pricing in the sense of Definition 2.5 can be seen as a
constraint on market maker’s allowed actions (at each observed order flow
$(y_{1},\ldots,y_{t})$) that depends on the behaviour strategy he assumes the
insider chooses. If $p_{Y}^{\xi}(y_{1},\ldots,y_{t})>0$, the market maker has
only one choice at $(y_{1},\ldots,y_{t})$. In a (standard) extensive form
game, an action of a player can only depend on past actions of other players
but not on their behaviour strategies, i.e., not on the probabilities of all
possible actions.

Debreu (1952) extends strategic games by allowing that the choice of an action
by a player can be restricted by the (simultaneous) actions of the other
players. This idea is applied to dynamic games by Butler (2017), having
constraints that depend on behaviour strategies of other players. We refer to
(Butler, 2017, Chapter 2) for a detailed discussion about conceptual issues,
but note that the Kyle game does not fit into this framework, since there are
infinitely many possible asset prices the market maker can set.

###  2.3 Sequential Kyle equilibrium

In extensive form games, Nash equilibria can be based on irrational plans of
some players that need not be realised since the plans refer to nodes that are
not reached in the equilibrium that they produce (“non-credible threat”). To
rule out such equilibria, Selten introduced the stronger criterion of a
subgame perfect equilibrium (we refer to (Selten, 1975, Section 5)): if the
game can be restarted in a node, the equilibrium strategies have to be optimal
also for the induced subgame. There are several refinements of this concept
for games with imperfect information. The most popular one is that of a
sequential equilibrium introduced by Kreps and Wilson (1982) (for a detailed
discussion we refer to (González-Díaz et al., 2010, Section 3.5)).

In the following, we adapt this concept to Kyle games. While the insider has
perfect information the market maker has not. Up to time $t$, he only observes
the total order flow $(y_{1},\ldots,y_{t})$. The union of nodes at time $t$
such that $x_{1}+z_{1}=y_{1}$, …, $x_{t}+z_{t}=y_{t}$ is called an information
set of the market maker. In information sets $(y_{1},\ldots,y_{t})$ with
$p^{\xi}_{Y}(y_{1},\ldots,y_{t})=0$ the conditional probability (2.5) is not
defined. Consequently, following (Kreps and Wilson, 1982, Section 4), the
market maker has to form more subtle _beliefs_ regarding the fundamental
information states.

######  Definition 2.9 (System of beliefs).

A _system of beliefs_ $\mu=(\mu_{t})_{t=1,\ldots,T}$ of the market maker is a
collection of probability distributions over $I$, indexed by the total order
flow:

$\mu_{t}:(E_{Y})^{t}\to\Delta
I,\qquad(y_{1},\ldots,y_{t})\mapsto\mu_{t}(\;\cdot\,\mid y_{1},\ldots,y_{t}).$

With $\mathcal{M}$ we denote the set of all systems of beliefs.

######  Definition 2.10 (Pricing with beliefs).

A system of beliefs $\mu\in\mathcal{M}$ induces a pricing system
$S^{\mu}\in\mathcal{S}$ by

$S^{\mu}_{t}(y_{1},\ldots,y_{t}):=\sum_{i=1}^{N}\mu_{t}(i\mid
y_{1},\ldots,y_{t})\;v^{i}\quad\mbox{for all}\
(y_{1},\ldots,y_{t})\in(E_{Y})^{t},\ t=1,\ldots,T.$

######  Definition 2.11 (Rational beliefs).

A system of beliefs $\mu\in\mathcal{M}$ is rational assuming $\xi\in\Xi$ if

$\mu_{t}(i\mid y_{1},\ldots,y_{t})=p^{\xi}_{Y}(i\mid
y_{1},\ldots,y_{t})\qquad\text{for all }i\in I$ (2.7)

and for all $(y_{1},\ldots,y_{t})\in(E_{Y})^{t}$, $t=1,\ldots,T$, with
$p_{Y}^{\xi}(y_{1},\ldots,y_{t})>0$.

For information sets $(y_{1},\ldots,y_{t})$ with
$p_{Y}^{\xi}(y_{1},\ldots,y_{t})>0$ the beliefs $\mu_{t}(i\mid
y_{1},\ldots,y_{t})$ are uniquely determined by $\xi$ according to (2.7). In
information sets that cannot be realised, (Kreps and Wilson, 1982, Section 5)
argue that rational beliefs should be the limit of conditional probabilities
that result from an approximating sequence of strategies where every node is
reached with positive probability. For the Kyle game, this leads to the
following definitions.

######  Definition 2.12 (Completely mixed behaviour strategies).

The set $\Xi^{0}$ of completely mixed behaviour strategies is given by

$\Xi^{0}:=\\{\xi\in\Xi:\;\xi(\tau,\\{x\\})>0\text{ for all
}\tau\in\mathbb{T}_{X},x\in E_{X}\\}.$

######  Definition 2.13 (Consistent beliefs).

A system of beliefs $\mu\in\mathcal{M}$ is _consistent_ with $\xi\in\Xi$ if
there is an approximating sequence
$(\xi^{n},\mu^{n})_{n\in\mathbb{N}}\subseteq\Xi\times\mathcal{M}$ with
$(\xi^{n},\mu^{n})\to(\xi,\mu)$ as $n\to\infty$ such that

  1. (i)

$\xi^{n}$ is completely mixed for all $n\in\mathbb{N}$ (see Definition 2.12),

  2. (ii)

$\mu^{n}$ is rational assuming $\xi^{n}$ for all $n\in\mathbb{N}$ (see
Definition 2.11).

(Elements of $\Xi\times\mathcal{M}$ can be identified with elements of
$\mathbb{R}^{d}$ for some $d$, and convergence is understood accordingly.)

The insider is the only player who appears in the game tree. Since she has
perfect information, she need not have beliefs, and at any node a subgame can
be started. For notational convenience, we only start subgames at
$\tau\in\mathbb{T}_{V}$ and do this in the following way:

For $\tau=r$, the subgame is the game itself. For
$\tau=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}},x_{t},z_{t})\in\mathbb{T}_{V}$,
$t\in\\{1,\ldots,T\\}$, we restrict the game tree to the subtree with root
node $\tau$ (which leads to a restriction of terminal nodes to those that come
after $\tau$), and realisation probabilities $p_{\tau}^{\xi}$ conditional on
starting in $\tau$, analogously to (2.2), are given by

$\begin{split}p_{\tau}^{\xi}(\omega):=\frac{\nu(\\{i\\})}{\sum_{j\in
I_{t,i_{t}}}\nu(\\{j\\})}\prod_{s={t+1}}^{T}\xi((I_{1,i_{1}},x_{1},z_{1},\ldots,I_{s,i_{s}}),\\{x_{s}\\})\;\zeta(\\{z_{s}\\})\qquad\\\
\qquad\text{for
}\omega=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}},x_{t},z_{t},I_{t+1,i_{t+1}},x_{t+1},z_{t+1},\ldots,I_{T+1,i}).\end{split}$
(2.8)

The expected utility for the subgame starting in $\tau\in\mathbb{T}_{V}$
becomes, analogously to (2.3),

$u_{\tau}(\xi,S):=\sum_{\begin{subarray}{c}\omega\in\Omega\\\ \omega\text{
after }\tau\end{subarray}}p_{\tau}^{\xi}(\omega)\;U(\omega,S).$

######  Definition 2.14 (Subgame optimality).

A behaviour strategy $\xi^{\star}\in\Xi$ is _subgame optimal_ given
$S\in\mathcal{S}$ if at every node $\tau\in\mathbb{T}_{V}$,

$\xi^{\star}\in\operatorname*{arg\,max}_{\xi\in\Xi}u_{\tau}(\xi,S).$ (2.9)

Subgame optimality corresponds to _sequential rationality_ in (Kreps and
Wilson, 1982, Section 4) (not to be confused with rational pricing). Because
the insider has perfect information, subgame optimality is equivalent to the
notion of _subgame perfection_ in (Selten, 1975, Section 5).

######  Definition 2.15 (Sequential Kyle equilibrium).

A _sequential Kyle equilibrium_ is a pair
$(\xi^{\star},\mu^{\star})\in\Xi\times\mathcal{M}$ satisfying:

  1. (i)

Profit maximisation: Given $S^{\mu^{\star}}$, the strategy $\xi^{\star}$ is
subgame optimal (see Definition 2.14),

  2. (ii)

Rational pricing: The system of beliefs $\mu^{\star}$ is consistent with
$\xi^{\star}$ (see Definition 2.13).

Whereas (Kyle, 1985, Equations (3.5), (3.7)) already envisions a form of
subgame optimality for the insider, the novel aspect of a sequential Kyle
equilibrium is the consistency condition regarding the beliefs of the market
maker. The main result of this section is the following.

######  Theorem 2.16.

Every Kyle game has a sequential Kyle equilibrium.

###  2.4 Proof of Theorem 2.16

The proof of Theorem 2.16 is based on the classical ideas that certain
extensive form games have a _(trembling hand) perfect equilibrium_ , see
(Selten, 1975, Section 11), and every perfect equilibrium is a sequential
equilibrium, see (Kreps and Wilson, 1982, Proposition 1). However, we have to
adapt this approach to accommodate the structure of a Kyle game. The proof we
present is self-contained, without formally introducing Selten’s _agent normal
form_ , only requiring the reader to be familiar with Kakutani’s fixed point
theorem (see, e.g., (González-Díaz et al., 2010, Theorem 2.2.1)).

For $\epsilon>0$ small enough such that $\epsilon\lvert E_{X}\rvert<1$, the
_$\epsilon$ -perturbed game_ is the Kyle game with insider strategies in

$\Xi^{\epsilon}:=\\{\xi\in\Xi:\;\xi(\tau,\cdot)\in\Delta
E_{X}^{\epsilon}\text{ for all }\tau\in\mathbb{T}_{X}\\},$

where $\Delta E_{X}^{\epsilon}:=\\{p\in\Delta
E_{X}:p(\\{x\\})\geq\epsilon\text{ for all }x\in E_{X}\\}$. We write
$\xi_{\tau}:=\xi(\tau,\,\cdot\;)$ and follow the usual notation that

$\xi=(\xi_{-\tau},\xi_{\tau})\qquad\text{where}\qquad\xi_{-\tau}:\mathbb{T}_{X}\setminus\\{\tau\\}\to\Delta
E_{X},\quad\xi_{-\tau}(\tau^{\prime})=\xi_{\tau^{\prime}}.$

For a node
$\tau:=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}})\in\mathbb{T}_{X}$ we
define, analogously to (2.8),

$\begin{split}p_{\tau}^{\xi}(\omega):=\frac{\nu(\\{i\\})}{\sum_{j\in
I_{t,i_{t}}}\nu(\\{j\\})}\prod_{s=t}^{T}\xi((I_{1,i_{1}},x_{1},z_{1},\ldots,I_{s,i_{s}}),\\{x_{s}\\})\;\zeta(\\{z_{s}\\})\qquad\\\
\qquad\text{for
}\omega=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}},x_{t},z_{t},I_{t+1,i_{t+1}},x_{t+1},z_{t+1},\ldots,I_{T+1,i})\end{split}$

and

$u_{\tau}(\xi,S):=\sum_{\begin{subarray}{c}\omega\in\Omega\\\ \omega\text{
after }\tau\end{subarray}}p_{\tau}^{\xi}(\omega)\;U(\omega,S).$

Next, we define the self-correspondence

$\begin{gathered}\begin{aligned}
F^{\epsilon}:\Xi^{\epsilon}\times\mathcal{M}&\rightrightarrows\Xi^{\epsilon}\times\mathcal{M},\\\
(\xi,\mu)&\mapsto\prod_{\tau\in\mathbb{T}_{X}}f^{\epsilon}_{\tau}(\xi,\mu)\times
f_{0}^{\epsilon}(\xi),\qquad\text{where}\end{aligned}\\\
f_{\tau}^{\epsilon}(\xi,\mu):=\operatorname*{arg\,max}_{p\in\Delta
E_{X}^{\epsilon}}\;u_{\tau}((\xi_{-\tau},p),S^{\mu})\ \text{ and }\
f_{0}^{\epsilon}(\xi):=\\{\mu^{\prime}\in\mathcal{M}:\mu^{\prime}\text{
consistent with }\xi\\}.\end{gathered}$

The first component of the correspondence $F^{\epsilon}$ maps a pair
$(\xi,\mu)\in\Xi^{\epsilon}\times\mathcal{M}$ to the set of “locally optimal”
strategies in $\Xi^{\epsilon}$ given the pricing system $S^{\mu}$ and the
insider’s strategy $\xi$ in all other nodes. The second component maps
$(\xi,\mu)$ to the set of beliefs consistent with $\xi$. Working with locally
optimal strategies makes sense because of the dynamic programming principle.

######  Lemma 2.17.

For every $\epsilon\in(0,1/|E_{X}|)$, $F^{\epsilon}$ has a fixed point.

Such a fixed point can be seen as an equilibrium of the $\epsilon$-perturbed
Kyle game when the insider maximises her utility by choosing trades separately
at each node and considering the choices at other nodes as given.

###### Proof of Lemma 2.17.

We fix $\epsilon\in(0,1/|E_{X}|)$. One can identify $\Xi$ and $\mathcal{M}$
with subsets of $\mathbb{R}^{d}$ for some $d$, and topological properties are
to be understood accordingly. In order to apply Kakutani’s fixed point
theorem, we need to verify that $F^{\epsilon}$ is upper hemicontinuous, non-
empty-, closed-, and convex-valued (see, e.g., (González-Díaz et al., 2010,
Theorem 2.2.1)). It is sufficient to prove these properties for the components
$f_{\tau}^{\epsilon}$, $f_{0}^{\epsilon}$.

_Step 1._ Fix $\tau\in\mathbb{T}_{X}$ and consider
$f_{\tau}^{\epsilon}:\Xi^{\epsilon}\times\mathcal{M}\rightrightarrows\Delta
E_{X}^{\epsilon}$. Any element of $\Delta E_{X}^{\epsilon}$ is a convex
combination of the _pure strategies_ $\delta_{x}$, $x\in E_{X}$. Since (2.2)
is linear in the transition distribution at a single node and by (2.3), the
set $f_{\tau}^{\epsilon}(\xi,\mu)$ consists of those convex combinations where
suboptimal pure strategies receive only the minimal weight $\epsilon$, i.e.,

$\begin{split}f_{\tau}^{\epsilon}(\xi,\mu)=\\{p\in\Delta
E_{X}^{\epsilon}:\;p(\\{x\\})=\epsilon\ \mbox{for all }x\in
E_{X}\setminus\widehat{E}_{X}(\xi,\mu,\tau)\\},\end{split}$ (2.10)

where

$\widehat{E}_{X}(\xi,\mu,\tau):=\\{x\in
E_{X}:u_{\tau}((\xi_{-\tau},\delta_{x}),S^{\mu})\geq
u_{\tau}((\xi_{-\tau},\delta_{x^{\prime}}),S^{\mu})\ \mbox{for all
}x^{\prime}\in E_{X}\\}.$

Consequently, the set $f_{\tau}^{\epsilon}(\xi,\mu)$ is obviously closed,
convex, and non-empty. It remains to show upper hemicontinuity (see, e.g.,
(González-Díaz et al., 2010, page 21) for a definition). The conditional
probabilities $p^{\xi}_{\tau}$ in (2.8) are continuous in $\xi$. As the
concatenation of continuous functions, $(\xi,\mu)\mapsto
u_{\tau}((\xi_{-\tau},\delta_{x}),S^{\mu})$ is continuous for all $x\in
E_{X}$. Thus,
$\widehat{E}_{X}(\xi^{\prime},\mu^{\prime},\tau)\subseteq\widehat{E}_{X}(\xi,\mu,\tau)$
for all $(\xi^{\prime},\mu^{\prime})$ in a neighborhood of $(\xi,\mu)$ since a
suboptimal pure strategy remains suboptimal under a slight perturbation of
$(\xi,\mu)$ and there are only finitely many of them. In conjunction with
(2.10), we obtain $f_{\tau}^{\epsilon}(\xi^{\prime},\mu^{\prime})\subseteq
f_{\tau}^{\epsilon}(\xi,\mu)$, which implies upper hemicontinuity.

_Step 2._ Now we turn to
$f_{0}^{\epsilon}:\Xi^{\epsilon}\rightrightarrows\mathcal{M}$. Any strategy in
$\xi\in\Xi^{\epsilon}$ is completely mixed, and thus has exactly one system of
beliefs that is consistent with $\xi$, namely the conditional probabilities
$\mu:=p^{\xi}_{Y}$ from (2.5), where the denominator is strictly positive for
all $(y_{1},\ldots,y_{t})$. The singleton $\\{\mu\\}$ is obviously non-empty,
closed and convex. From (2.5) and strict positivity it follows that $\mu$
considered as a function of $\xi$ is continuous, and thus $f_{0}^{\epsilon}$
is upper hemicontinuous.

Applying Kakutani’s fixed point theorem to $F^{\epsilon}$ one obtains a fixed
point. ∎

######  Lemma 2.18.

For $\epsilon>0$ let $(\xi^{\epsilon},\mu^{\epsilon})$ be a fixed point of
$F^{\epsilon}$. Then the strategy $\xi^{\epsilon}$ is subgame optimal in
$\Xi^{\epsilon}$ given $S^{\mu^{\epsilon}}$.

###### Proof.

Let $(\xi^{\epsilon},\mu^{\epsilon})$ be a fixed point of $F^{\epsilon}$
according to Lemma 2.17. As such, $\xi^{\epsilon}$ is locally optimal given
$S:=S^{\mu^{\epsilon}}$, i.e.,

$\xi^{\epsilon}_{\tau}\in\operatorname*{arg\,max}_{p\in\Delta
E_{X}^{\epsilon}}\;u_{\tau}\big{(}(\xi_{-\tau}^{\epsilon},p),S\big{)}\quad\text{for
all }\tau\in\mathbb{T}_{X}.$ (2.11)

We prove (2.9) by backward induction over the period $t$ in which the node
$\tau\in\mathbb{T}_{V}$ lies. This means, we have to show that

$u_{\tau}(\xi^{\epsilon},S)\geq u_{\tau}(\xi,S)\qquad\text{for all
}\tau=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t-1,i_{t-1}},x_{t-1},z_{t-1}),\
\xi\in\Xi^{\epsilon},$

using that the assertion holds for $t$ instead of $t-1$ (the base case $t-1=T$
follows by the same arguments as below). Making use of the special structure
of the Kyle game, the proof is shorter than the original one by Selten for
general extensive form games (see (Selten, 1975, Lemma 6)).

Step 1. Throughout the proof, we fix a competing strategy
$\xi\in\Xi^{\epsilon}$. In the first step, we consider nodes of the form
$\tau^{\prime}:=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}})\in\mathbb{T}_{X}$,
i.e., nodes in which new insider information is already revealed, and define
the strategy
$\xi^{\prime}:=(\xi^{\epsilon}_{-\tau^{\prime}},\xi_{\tau^{\prime}})$. From
(2.11) it follows that

$u_{\tau^{\prime}}(\xi^{\epsilon},S)\geq u_{\tau^{\prime}}(\xi^{\prime},S).$

On the other hand, by definition of $u_{\tau^{\prime}}$ and
$u_{\tau^{\prime}_{k,l}}$, one has

$u_{\tau^{\prime}}(\xi^{\prime},S)=\sum_{k=1}^{K}\sum_{l=1}^{L}\xi((I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}}),\\{x^{k}\\})\;\zeta(\\{z^{l}\\})\;u_{\tau^{\prime}_{k,l}}(\xi^{\epsilon},S)$
(2.12)

where
$\tau^{\prime}_{k,l}:=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}},x^{k},z^{l})\in\mathbb{T}_{V}$.
From the induction hypothesis it follows that the RHS of (2.12) dominates the
LHS of

$\sum_{k=1}^{K}\sum_{l=1}^{L}\xi((I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t,i_{t}}),\\{x^{k}\\})\;\zeta(\\{z^{l}\\})\;u_{\tau^{\prime}_{k,l}}(\xi,S)=u_{\tau^{\prime}}(\xi,S).$

Put together, we arrive at $u_{\tau^{\prime}}(\xi^{\epsilon},S)\geq
u_{\tau^{\prime}}(\xi,S)$.

Step 2. For nodes
$\tau:=(I_{1,i_{1}},x_{1},z_{1},\ldots,I_{t-1,i_{t-1}},x_{t-1},z_{t-1})\in\mathbb{T}_{V}$,
the estimate $u_{\tau}(\xi^{\epsilon},S)\geq u_{\tau}(\xi,S)$ follows from
Step 1 by weighting $u_{\tau^{\prime}}(\xi^{\epsilon},S)$ and
$u_{\tau^{\prime}}(\xi,S)$ by the transition probabilities in the insider’s
private information tree, which are exogenous. ∎

The proof of Theorem 2.16 is completed by the following lemma.

######  Lemma 2.19.

There exists a sequence
$(\epsilon_{n})_{n\in\mathbb{N}}\subseteq\mathbb{R}_{+}$ with
$\epsilon_{n}\downarrow 0$, a sequence
$(\xi^{\epsilon_{n}},\mu^{\epsilon_{n}})_{n\in\mathbb{N}}$ with
$(\xi^{\epsilon_{n}},\mu^{\epsilon_{n}})\in\Xi^{\epsilon_{n}}\times\mathcal{M}$,
and $(\xi^{*},\mu^{*})\in\Xi\times\mathcal{M}$ such that
$(\xi^{\epsilon_{n}},\mu^{\epsilon_{n}})$ is a fixed point of
$F^{\epsilon_{n}}$ for all $n\in\mathbb{N}$ and
$(\xi^{\epsilon_{n}},\mu^{\epsilon_{n}})\to(\xi^{*},\mu^{*})$ as $n\to\infty$.
In addition, $(\xi^{*},\mu^{*})$ is a sequential Kyle equilibrium. (Elements
of $\Xi\times\mathcal{M}$ can be identified with elements of $\mathbb{R}^{d}$
for some $d$, and convergence is understood accordingly.)

###### Proof.

For every $n\in\mathbb{N}$, let
$(\xi^{\nicefrac{{1}}{{n}}},\mu^{\nicefrac{{1}}{{n}}})\in\Xi^{\nicefrac{{1}}{{n}}}\times\mathcal{M}$
be a fixed point of $F^{\nicefrac{{1}}{{n}}}$ provided by Lemma 2.17. Since
$\xi^{\nicefrac{{1}}{{n}}}(\tau,\\{x\\})\in[0,1]$ for all
$\tau\in\mathbb{T}_{X}$ and $x\in E_{X}$, there exists a subsequence
$(n_{k})_{k\in\mathbb{N}}$ such that
$\xi^{\nicefrac{{1}}{{n_{k}}}}\to\xi^{\star}$ as $k\to\infty$ for some
$\xi^{\star}$ with $\xi^{\star}(\tau,\\{x^{i}\\})\geq 0$, $i=1,\ldots,K$ and
$\sum_{i=1}^{K}\xi^{\star}(\tau,\\{x^{i}\\})=1$. Because $1/{n_{k}}>0$ for all
$k\in\mathbb{N}$, every $\mu^{\nicefrac{{1}}{{n_{k}}}}$ is the unique system
of beliefs that is rational assuming $\xi^{\nicefrac{{1}}{{n_{k}}}}$. There
exists a further subsequence $(n_{k_{j}})_{j\in\mathbb{N}}$ and
$\mu^{\star}\in\mathcal{M}$ such that
$\smash{\mu^{1/n_{k_{j}}}}\to\mu^{\star}$ as $j\to\infty$. In conclusion,
$\smash{(\xi^{\nicefrac{{1}}{{n_{k_{j}}}}},\mu^{\nicefrac{{1}}{{n_{k_{j}}}}})}\to(\xi^{\star},\mu^{\star})$
as $j\to\infty$, $\smash{\xi^{\nicefrac{{1}}{{n_{k_{j}}}}}}$ is completely
mixed, and $\smash{\mu^{\nicefrac{{1}}{{n_{k_{j}}}}}}$ is rational assuming
$\xi^{n_{k_{j}}}$, so $\mu^{\star}$ is consistent with $\xi^{\star}$ (see
Definition 2.13).

It remains to show that $\xi^{\star}$ is subgame optimal given
$S^{\mu^{\star}}$. W.l.o.g. $\smash{n_{k_{j}}}=n$ for all $j\in\mathbb{N}$. We
fix a node $\tau\in\mathbb{T}_{V}$ and a competing strategy
$\widetilde{\xi}\in\Xi$. We construct an approximating sequence
$(\widetilde{\xi}^{n})_{n\in\mathbb{N}}$ with
$\widetilde{\xi}^{n}\in\Xi^{\nicefrac{{1}}{{n}}}$ by
$\widetilde{\xi}^{n}(\tau,\\{x\\}):={\nicefrac{{1}}{{n}}}+\widetilde{\xi}(\tau,\\{x\\})(1-\lvert
E_{X}\rvert/n)$ for all $n>\lvert E_{X}\rvert$. Following Lemma 2.18 the
strategy $\xi^{\nicefrac{{1}}{{n}}}$ is subgame optimal in
$\Xi^{\nicefrac{{1}}{{n}}}$ given $S^{\mu^{\nicefrac{{1}}{{n}}}}$ which
implies

$u_{\tau}\big{(}\xi^{\nicefrac{{1}}{{n}}},S^{\mu^{\nicefrac{{1}}{{n}}}}\big{)}\geq
u_{\tau}\big{(}\widetilde{\xi}^{n},S^{\mu^{\nicefrac{{1}}{{n}}}}\big{)}\quad\mbox{for
all }n>|E_{X}|.$

Because $(\xi,\mu)\mapsto u_{\tau}(\xi,S^{\mu})$ is jointly continuous,
$(\xi^{\nicefrac{{1}}{{n}}},\mu^{\nicefrac{{1}}{{n}}})\to(\xi^{\star},\mu^{\star})$,
and
$(\widetilde{\xi}^{n},\mu^{\nicefrac{{1}}{{n}}})\to(\widetilde{\xi},\mu^{\star})$
as $n\to\infty$, we can pass to the limit $n\to\infty$ on both sides to get

$u_{\tau}\big{(}\xi^{\star},S^{\mu^{\star}}\big{)}\geq
u_{\tau}\big{(}\widetilde{\xi},S^{\mu^{\star}}\big{)}.$

∎

##  3 Structure of equilibria in single-period models

In this section, we establish basic properties of equilibria in single-period
Kyle games, i.e., $T=1$ and
$\mathcal{I}_{1}=\\{\\{1\\},\\{2\\},\ldots,\\{N\\}\\}$, and show that the
insider’s demands are uniformly bounded if the noise trader’s demands lie in
$[-1,1]$ and the probabilities of $\\{-1\\},\\{1\\}$ are bounded away from
zero. It also becomes apparent that in discrete state Kyle games the selection
of $E_{Z}$ and $E_{X}$ is a crucial issue. We denote
$E_{V}:=\\{v^{N},v^{N-1},\ldots,v^{1}\\}$ and write an insider strategy as
$\xi(v,\\{x\\})$, $v\in E_{V}$, $x\in E_{X}$. Throughout the section, we
assume that

$\\{-1,1\\}\subseteq E_{Z}\subseteq[-1,1],\qquad\\{0\\}\in E_{X},$

and use the notation

$E^{\xi}_{X}:=\\{x\in E_{X}:\exists v\in E_{V}\ \mbox{with}\
\xi(v,\\{x\\})>0\\}.$

######  Assumption 3.1.

One has $((E_{X}+E_{Z})-E_{X})\cap{\rm conv}(E_{Z})\subseteq E_{Z}$. In other
words, for every $x_{1},x_{2}\in E_{X}$, $z_{1}\in E_{Z}$,

$x_{2}\in[x_{1}+z_{1}-1,x_{1}+z_{1}+1]\quad\implies\quad\exists z_{2}\in
E_{Z}\ \mbox{such that}\ \ x_{2}+z_{2}=x_{1}+z_{1}.$

The assumption is satisfied, for instance, if $E_{Z}$ is an equidistant grid
and distances between elements of $E_{X}$ are multiples of the size of this
grid. It rules out the effect that by the precise knowledge of $x+z$ the
market maker can infer $x$, although $x+z$ is no extreme point of
$E_{X}+E_{Z}$.

######  Lemma 3.2.

Let $(\xi,S)$ be an equilibrium in a discrete single-period Kyle game. We have
that

  * (i)

$\xi(v^{i},\\{x\\})>0$, $v^{i}<v^{j}$ $\implies$
$\xi(v^{j},E_{X}\cap[x,\infty))=1$ for all $v^{i},v^{j}\in E_{V},x\in E_{X}$
$\xi(v^{i},\\{x\\})>0$, $v^{i}>v^{j}$ $\implies$
$\xi(v^{j},E_{X}\cap(-\infty,x])=1$ for all $v^{i},v^{j}\in E_{V},x\in E_{X}$

  * (ii)

$x>0$, $\xi(v,E_{X}\cap[x-2,x))=0$ for all $v\in E_{V}$ $\implies$ $\int
S(x^{\prime}+z)\zeta(dz)=v^{\prime}$
for all $x\in E_{X}$, $x^{\prime}\in E_{X}\cap[x,\infty)$, $v^{\prime}\in
E_{V}$ with $\xi(v^{\prime},\\{x^{\prime}\\})>0$
(the inverse assertion for $x<0$ holds as well)

  * (iii)

For $y_{1},y_{2}\in E_{Y}$ with $p^{\xi}_{Y}(y_{1}),p^{\xi}_{Y}(y_{2})>0$ and
$y_{2}\geq y_{1}+2$, we have $S(y_{1})\leq S(y_{2})$.

  * (iv)

For $y_{1},y_{2}\in E_{Y}$ with $p^{\xi}_{Y}(y_{1}),p^{\xi}_{Y}(y_{2})>0$,
$y_{2}\geq y_{1}$, and $S(y_{1})=v^{1}$, we have $S(y_{2})=v^{1}$.
For $y_{1},y_{2}\in E_{Y}$ with $p^{\xi}_{Y}(y_{1}),p^{\xi}_{Y}(y_{2})>0$,
$y_{2}\geq y_{1}$ and $S(y_{2})=v^{N}$, we have $S(y_{1})=v^{N}$.

Property (i) says that the insider’s demand is nondecreasing in the true value
she observes. Property (ii) states that the gaps between the insider’s order
sizes (that depend on the true value) should not be larger than the range of
the noise trader’s order sizes. Otherwise, the market maker could infer the
true value and profits vanish. Maybe surprisingly, the price function $S$ is
in general not nondecreasing, see (Counter-)Example 3.6, but we have the
weaker properties (iii) and (iv).

###### Proof of Lemma 3.2.

Ad (i). By symmetry, we only have to prove the first implication. Consider the
gain function

$x\mapsto x(v-\int S(x+z)\zeta(dz)).$ (3.1)

If $x$ is a maximiser (not necessarily unique) for $v=v^{i}$, it strictly
dominates $x^{\prime}<x$ for $v=v^{j}$.

Ad (ii). Let $x_{1}\in E_{X}\cap(0,\infty)$ with
$\xi(v,E_{X}\cap[x_{1}-2,x_{1}))=0$ for all $v\in E_{V}$. Define
$x_{2}:=\inf\\{x\in E^{\xi}_{X}\cap[x_{1},\infty)\\}$, $v^{\star}:=\inf\\{v\in
E_{V}:\xi(v,\\{x_{2}\\})>0\\}$, and $x_{3}:=\sup\\{x\in
E_{X}:\xi(v^{\star},\\{x\\})>0\\}$. Here, the case $x_{2}=\infty$ is trivial
and thus excluded. By optimality of $\xi$, $0\in E_{X}$, and part (i), we have
that

$\int S(x+z)\zeta(dz)\leq v^{\star}\leq v\ \mbox{for all}\ x\in
E_{X}\cap[x_{1},x_{3}],v\in E_{V}\ \mbox{with}\ \xi(v,\\{x\\})>0.$ (3.2)

On the other hand, for each $x\in E^{\xi}_{X}\cap[x_{1},x_{3}]$, $z\in E_{Z}$,
the price $S(x+z)$ lies in the convex hull of $\\{v\in E_{V}:\exists
x^{\prime}\in E_{X}\ \mbox{with}\ \xi(v,\\{x^{\prime}\\})>0,z^{\prime}\in
E_{Z}\ \mbox{such that}\ x^{\prime}+z^{\prime}=x+z\\}$ by rational pricing
(cf. Definition 2.5 and observe that the set is nonempty by $x\in
E^{\xi}_{X}$). Consequently, for each $x\in E^{\xi}_{X}\cap[x_{1},x_{3}]$, the
average price $\int S(x+z)\zeta(dz)$ lies in the convex hull of
$M_{x}:=\\{v\in E_{V}:\exists x^{\prime}\in E_{X}\ \mbox{with}\
\xi(v,\\{x^{\prime}\\})>0\ \exists z,z^{\prime}\in E_{Z}\ \mbox{such that}\
x^{\prime}+z^{\prime}=x+z\\}$. But for $v\in M_{x}$ we must have that $v\geq
v^{\star}$ by $x^{\prime}\geq x-2\geq x_{1}-2$ and again part (i). Together
with (3.2), we obtain that $v=v^{\star}$ for all $x\in
E^{\xi}_{X}\cap[x_{1},x_{3}]$ and $v\in M_{x}$. This implies equality in (3.2)
and even more that

$S(x+z)=v^{\star}=v\ \mbox{for all}\ x\in E_{X}\cap[x_{1},x_{3}],v\in E_{V}\
\mbox{with}\ \xi(v,\\{x\\})>0,z\in E_{Z}.$ (3.3)

Now, define $x_{4}:=\inf\\{x\in E_{X}:\xi(v,\\{x\\})>0\ \mbox{for some\ }v\in
E_{V}\cap(v^{\star},\infty)\\}$. We want to show that $x_{4}>x_{3}+2$ and
assume by contradition that this does not hold. Then, by $x_{4}\geq x_{3}$ and
Assumption 3.1, there would exist a $z^{\prime}\in E_{Z}$ such that
$x_{4}+z^{\prime}=x_{3}+1$ which would imply that $S(x_{3}+1)>v^{\star}$, a
contradiction to (3.3). By definition of $x_{3}$ and $x_{4}$, the estimate
$x_{4}>x_{3}+2$ yields $\xi(v,E_{X}\cap[x_{4}-2,x_{4}))=0$ for all $v\in
E_{V}$. This means that $x_{4}>x_{1}$ satisfies the properties that we
required for $x_{1}$. By proceeding analogously, we obtain the assertion
because of (3.3).

Ad (iii). Let $x_{1}:=\sup\\{x\in E^{\xi}_{X}:\exists z\in E_{Z}\ \mbox{such
that}\ x+z=y_{1}\\}\leq y_{1}+1\leq y_{2}-1$. Part (i) and basic properties of
the conditional expectation yield that

$S(y_{1})\leq\frac{\sum_{i}\nu(v^{i})\xi(v^{i},\\{x_{1}\\})v^{i}}{\sum_{i}\nu(v^{i})\xi(v^{i},\\{x_{1}\\})}\leq
S(y_{2}).$

Ad (iv). By symmetry, we only have to prove the first implication. We assume
by contradiction that there exist $x_{2}\in E^{\xi}_{X}$ with
$\xi(v^{1},\\{x_{2}\\})<1$ and $z_{2}\in E_{Z}$ such that $x_{2}+z_{2}=y_{2}$.
By $p^{\xi}_{Y}(y_{1})>0$, there must exist $x_{1}\in E^{\xi}_{X}$ and
$z_{1}\in E_{Z}$ such that $x_{1}+z_{1}=y_{1}$.

Case $x_{1}\leq x_{2}$. It follows from part (i) that
$\xi(v^{1},\\{x_{1}\\})<1$, a contradiction to $S(y_{1})=v^{1}$.
Case $x_{1}>x_{2}$. We have that $x_{2}\in[y_{1}-1,y_{1}+1]$. By Assumption
3.1, there exists a $z^{\prime}\in E_{Z}$ with $x_{2}+z^{\prime}=y_{1}$ that
is a contradiction to $S(y_{1})=v^{1}$. ∎

######  Theorem 3.3.

In any equilibrium $(\xi,S)$ of a discrete single-period Kyle game, either
insider’s buy orders cannot be executed at a price below the maximal true
value, i.e., $S(x+z)=v^{1}$ for all $x\in E_{X}\cap(0,\infty)$, $z\in E_{Z}$,
or their sizes are bounded by $6+6/\zeta(\\{1\\})$, i.e.,
$E^{\xi}_{X}\cap(0,\infty)\subseteq(0,6+6/\zeta(\\{1\\}))$. Analogously,
either insider’s sell orders cannot be executed at a price above the minimal
true value or their sizes are bounded by $6+6/\zeta(\\{-1\\})$.

######  Example 3.4.

Both the true value and the demand of the noise trader are $\pm 1$ with
probability $1/2$. In addition, $E_{X}=\\{-2n,-2(n-1),\ldots,0,2,\ldots,2n\\}$
for some fixed $n\in\mathbb{N}$. Then, one equilibrium is given by
$\xi(v,\cdot)=\delta_{2n}1_{(v=1)}+\delta_{-2n}1_{(v=-1)}$ and
$S(y)=-1_{(y<0)}+1_{(y>0)}$ for all $y\in E_{Y}$.

###### Proof of Theorem 3.3.

We fix a Kyle game and an equilibrium $(\xi,S)$ of this game. By symmetry, it
is sufficient to show the assertion regarding the insider’s buy orders. Let
$x_{1}:=\sup\\{x\in E_{X}\cap(0,\infty):\xi(v^{1},\\{x\\})>0\ \mbox{and}\ \int
S(x+z)\zeta(dz)<v^{1}\\}$.

Case $x_{1}=-\infty$. Let $x\in E^{\xi}_{X}\cap(0,\infty)$ (if no such $x$
exists we are done). By Lemma 3.2(i), we have that
$\xi(v^{1},E_{X}\cap[x,\infty))>0$. This means that there is an order size
$x^{\prime}\in E^{\xi}_{X}\cap[x,\infty)$ with $\xi(v^{1},\\{x^{\prime}\\})>0$
that is an optimizer of (3.1) when the true value takes the maximal value
$v^{1}$. But, by $x_{1}=-\infty$, the gain $v^{1}-\int
S(x^{\prime}+z)\zeta(dz)$ cannot be positive. Consequently, we must have that
$\int S(x^{\prime\prime}+z)\zeta(dz)=v^{1}$ for all $x^{\prime\prime}\in
E_{X}\cap(0,\infty)$ since otherwise $x^{\prime}$ could not be optimal given
$v^{1}$. It follows that $S(x^{\prime\prime}+z)=v^{1}$ for all $z\in E_{Z}$.
We conclude that the insider’s buy orders are never executed at a price below
$v^{1}$.

Case $x_{1}>0$. Now, we turn to the “main” case. W.l.o.g. $x_{1}\geq 6$ since
otherwise $x_{1}<6+6/\zeta(\\{1\\})$ and the upper bound could be verified
similar to the previous case. Let $v^{\star}:=\inf\\{v\in
E_{V}:\xi(v,x_{1})>0\\}$. By definition of $x_{1}$ and $v^{\star}$, we have
that $\int S(x_{1}+z)\zeta(dz)<v^{1}$ and $\int S(x_{1}+z)\zeta(dz)\leq
v^{\star}$. Let us find an $x_{2}\in E_{X}$ with

$S(x_{2}+1)\leq\int S(x_{1}+z)\zeta(dz)\quad\mbox{and}\quad
S(x_{2}+1)<v^{\star}.$ (3.4)

First, consider the case that $v^{\star}=v^{1}$. The first inequality in (3.4)
is satisfied by any $x_{2}\in E^{\xi}_{X}\cap(-\infty,x_{1}-4]$ because of
Lemma 3.2(iii). Lemma 3.2(ii) guarantees that there exists an $x_{2}\in
E^{\xi}_{X}\cap[x_{1}-6,x_{1}-4)$ since otherwise the insider could not make a
profit on average with $x_{1}$ when $v^{1}$ occurs. Then, the second
inequality in (3.4) follows from $\int S(x_{1}+z)\zeta(dz)<v^{1}$.

We proceed with the case that $v^{\star}<v^{1}$. Since $\int
S(x_{1}+z)\zeta(dz)\leq v^{\star}$, $\xi(v^{1},\\{x_{1}\\})>0$ and by rational
pricing (cf. Definition 2.5), there must exist $x\in E^{\xi}_{X}$, $z_{1},z\in
E_{Z}$, and $v\in E_{V}\cap(-\infty,v^{\star})$ with $\xi(v,\\{x\\})>0$ and
$x+z=x_{1}+z_{1}$. One has that $x\in E^{\xi}_{X}\cap[x_{1}-2,\infty)$.
Consequently, for all $x^{\prime}\in E^{\xi}_{X}\cap(-\infty,x_{1}-4)$ we get
$S(x^{\prime}+1)\leq v<v^{\star}$ by Lemma 3.2(i) and rational pricing. Thus,
$x_{2}$ from above can also be taken in the case $v^{\star}<v^{1}$ and (3.4)
is shown. In addition, since the total demand $x_{1}+1$ can only occur if
$v\geq v^{\star}$, rational pricing yields

$S(x_{1}+1)\geq v^{\star}.$ (3.5)

Define $s:=S(x_{2}+1)$ and $\Delta:=x_{1}(v^{\star}-\int
S(x_{1}+z)\zeta(dz))-x_{2}(v^{\star}-\int S(x_{2}+z)\zeta(dz))$. We have that

$\displaystyle\Delta$ $\displaystyle=(x_{1}-x_{2})(v^{\star}-\int
S(x_{1}+z)\zeta(dz))+x_{2}(\int S(x_{2}+z)\zeta(dz)-\int S(x_{1}+z)\zeta(dz))$
$\displaystyle\leq(x_{1}-x_{2})(v^{\star}-s)+x_{2}(s-v^{\star})\zeta(\\{1\\})$
$\displaystyle\leq(v^{\star}-s)(6-x_{2}\zeta(\\{1\\})),$

where we use (3.5) to estimate $S(x_{2}+z)-S(x_{1}+z)$ for $z=1$. Since
$x_{1}$ maximises (3.1) given $v^{\star}$, we have $\Delta\geq 0$. By
$v^{\star}>s$, we obtain that $x_{2}\leq 6/\zeta(\\{1\\})$ and arrive at
$x_{1}\leq 6+6/\zeta(\\{1\\})$. ∎

######  Remark 3.5.

We note that (3.5) need not hold for $S(x_{1}+z)$ with $z<1$ instead of
$S(x_{1}+1)$. This is the reason why the upper bound $6+6/\zeta(\\{1\\})$
depends on the probability of the noise trader’s demand at the boundary. For a
sequence of discrete models, the bound could tend to infinity if
$\zeta(\\{1\\})\to 0$.

######  Example 3.6 ($\nexists$ nondecreasing equilibrium price function).

We provide a minimalist example of a single-period Kyle game in which the
equilibrium is unique and the equilibrium price function is strictly
decreasing for some demands. The true value takes a high value $v^{1}:=1$, a
medium value $v^{2}:=\nicefrac{{1}}{{2}}$, or a low value $v^{3}:=0$, with
uniform distribution
$\nu:=\nicefrac{{1}}{{3}}\delta_{0}+\nicefrac{{1}}{{3}}\delta_{\nicefrac{{1}}{{2}}}+\nicefrac{{1}}{{3}}\delta_{1}$.
Both insider and noise trader can buy or sell one share of the asset or not
trade at all, i.e., $E_{X}:=E_{Z}:=\\{-1,0,1\\}$, and so the market maker
observes a total order flow in $E_{Y}=\\{-2,-1,0,1,2\\}$. The crux in our
example is that the noise trader shows bearish sentiment and is more likely to
sell, namely

$\zeta:=\frac{6}{8}\delta_{-1}+\frac{1}{8}\delta_{0}+\frac{1}{8}\delta_{1}.$

This leads to the effect that when the market maker observes $y=0$ he is more
likely to believe that the insider and noise trader traded $x=1$ and $z=-1$,
respectively, than he would be, ceteris paribus, under uniformly distributed
noise trades. The conditional probability of $x=1$ can be higher under the
condition $y=0$ than under $y=1$. This is the reason why the example can work
although the insider’s demand is nondecreasing in the true value (in the sense
of Lemma 3.2(i)) and the market maker’s price is the conditional expectation
of the true value given $y$.

In the following, we want to show that the Kyle equilibrium is unique, and its
price function $S$ satisfies $S(0)>S(1)$. For this, let $(\xi,S)$ be an
arbitrary equilibrium.

_Step 1:_ Let us first show that $\xi(1,\,\cdot\,)=\delta_{1}$ and
$\xi(0,\,\cdot\,)=\delta_{-1}$, i.e., for the extreme true values
buying/selling is the unique optimal action for the insider. Assume by
contradiction that $\xi(1,\,\cdot\,)\not=\delta_{1}$, i.e., buying is at least
not the only optimal action when $v=1$. Because $S\leq 1$ and
$E_{X}=\\{-1,0,1\\}$, this can only be the case if the profits are zero, i.e.,
the average price $\int S(1+z)\zeta(dz)$ equals $1$, and this is equivalent to
$S(2)=S(1)=S(0)=1$. But regardless of $\xi(\nicefrac{{1}}{{2}},\,\cdot\,)$ and
$\xi(0,\,\cdot\,)$, the noise trader always ensures that $y=0$ is reached with
positive probability when $v=\nicefrac{{1}}{{2}}$ or $v=0$, and so $S(0)<1$ by
rational pricing, a contradiction. Hence, we must have that
$\xi(1,\,\cdot\,)=\delta_{1}$. By the same arguments,
$\xi(0,\,\cdot\,)=\delta_{-1}$ is the unique optimal action when $v=0$.

_Step 2:_ Now we turn to the optimal strategy when $v=\nicefrac{{1}}{{2}}$.
Using the notation
$\xi(\nicefrac{{1}}{{2}},\,\cdot\,)=\alpha_{1}\delta_{1}+\alpha_{0}\delta_{0}+\alpha_{-1}\delta_{-1}$,
with $\alpha_{1},\alpha_{0},\alpha_{-1}\geq 0$ and
$\alpha_{1}+\alpha_{0}+\alpha_{-1}=1$, we can compute the rational prices
given $\xi$:

$\displaystyle S(2)$ $\displaystyle=\frac{1\cdot
1+\alpha_{1}\cdot\frac{1}{2}}{1+\alpha_{1}}=\frac{1+\frac{1}{2}\alpha_{1}}{1+\alpha_{1}},$
$\displaystyle S(1)$ $\displaystyle=\frac{1\cdot
1+(\alpha_{1}+\alpha_{0})\cdot\frac{1}{2}}{1+\alpha_{1}+\alpha_{0}}=\frac{1+\frac{1}{2}(\alpha_{1}+\alpha_{0})}{1+\alpha_{1}+\alpha_{0}},$
$\displaystyle S(0)$ $\displaystyle=\frac{6\cdot
1+(6\alpha_{1}+\alpha_{0}+\alpha_{-1})\cdot\frac{1}{2}+1\cdot
0}{6+6\alpha_{1}+\alpha_{0}+\alpha_{-1}+1}=\frac{6+\frac{1}{2}(6\alpha_{1}+\alpha_{0}+\alpha_{-1})}{7+6\alpha_{1}+\alpha_{0}+\alpha_{-1}},$
$\displaystyle S(-1)$
$\displaystyle=\frac{(6\alpha_{0}+\alpha_{-1})\cdot\frac{1}{2}+1\cdot
0}{6\alpha_{0}+\alpha_{-1}+1}=\frac{\frac{1}{2}(6\alpha_{0}+\alpha_{-1})}{1+6\alpha_{0}+\alpha_{-1}},$
$\displaystyle S(-2)$ $\displaystyle=\frac{6\alpha_{-1}\cdot\frac{1}{2}+6\cdot
0}{6\alpha_{-1}+6}=\frac{\alpha_{-1}}{2+2\alpha_{-1}}.$

We observe that $S(2),S(1),S(0)>\nicefrac{{1}}{{2}}$ and so the average price

$\int S(1+z)\zeta(dz)>\frac{1}{2},$

which makes buying in $v=\nicefrac{{1}}{{2}}$ suboptimal. As a result
$\alpha_{1}=0$. In addition, the above formulas show immediately that
$S(0)<1$, $S(-1)<\nicefrac{{1}}{{2}}$, and $S(-2)\leq\nicefrac{{1}}{{4}}$.
Thus, we get the rough estimate

$\int
S(-1+z)\zeta(dz)=\frac{1}{8}S(0)+\frac{1}{8}S(-1)+\frac{6}{8}S(-2)<\frac{6}{16}<\frac{1}{2}.$

Consequently, selling in $v=\nicefrac{{1}}{{2}}$ is not profitable either, so
$\alpha_{-1}=0$, and $\alpha_{0}=1$. In other words, the optimal strategy is
$\xi(\nicefrac{{1}}{{2}},\,\cdot\,)=\delta_{0}$. In conclusion, the unique
Kyle equilibrium $(\xi,S)$ is given by

$\xi(v,\,\cdot\,):=\delta_{2v-1},\qquad S(-2)=0,\ S(-1)=\frac{3}{7},\
S(0)=\frac{13}{16},\ S(1)=\frac{3}{4},\ S(2)=1,$

also depicted in Figure 3.1. The price function $S$ is decreasing between $0$
and $1$.

$v=0$$-1$$-2$$-1$$-1$$0$$0$$1$$v=\frac{1}{2}$$0$$1$$-1$$0$$0$$1$$1$$v=1$$1$$0$$-1$$1$$0$$2$$1$$X$$Z$$y$
(a) The optimal (non-randomising) strategy of the insider in the game tree.

$y$$S(y)$$-2$$-1$$0$$1$$2$$0$$\frac{1}{2}$$1$ (b) The rational price function
$S$ is decreasing between $y=0$ and $y=1$.

Figure 3.1: The unique equilibrium from Example 3.6.

##  4 Continuous state game

In this section, we prove the existence of an equilibrium in the single-period
Kyle game when the true value and the noise trader’s demand are probability
measures over the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ with
compact but not necessarily finite support. The main challenge is that the
bounded price function of the market maker in the continuous game can only be
expected to be measurable but in general not a continuous function of the
total demand. Consequently, the insider’s profit need not be continuous in her
demand. The set of $[-1,1]$-valued Borel-measurable functions equipped with
pointwise convergence almost everywhere is not sequentially compact. The
standard example is a sequence of Rademacher functions (see, e.g., Example 3.2
in (Balder, 2021)) that cannot have a convergent subsequence. Because of the
lack of compactness of the set of price functions, we cannot apply standard
infinite-dimensional fixed point theorems of Schauder-Tychonoff’s or Kakutani-
Fan’s type (see, e.g., Theorem 10.1 and Theorem 13.1 in Pata (2019),
respectively) directly to the continuous game.

Instead, we consider equilibria of a sequence of finite Kyle games that are
derived from the continuous game by discretising the true value and the noise
trader’s demand. We assume that the noise trader’s demand is absolutely
continuous w.r.t. the Lebesgue measure $\lambda$. This tames the expected
gains of the insider, and a weak limit of her discrete equilibrium strategies
(that exists along a subsequence by compactness of probability distributions
on $[-1,1]$) is part of an equilibrium in the continuous model. On the other
hand, after passing to forward convex combinations, the discrete equilibrium
price functions of the market maker possess a pointwise limit $\lambda$-almost
everywhere. Together with the insider strategy, it is an equilibrium price
function in the continuous model.

Let us introduce the continuous state model. Let $E_{V}:=[0,1]$,
$\nu\in\mathcal{P}(E_{V},\mathcal{B}(E_{V}))$, $E_{Z}:=[-1,1]$, and
$\zeta\in\mathcal{P}(E_{Z},\mathcal{B}(E_{Z}))$. We assume that

$\zeta\ \text{is absolutely continuous w.r.t. the Lebesgue measure
}\lambda,\text{ i.e.,\ }\zeta\ll\lambda.$

######  Definition 4.1 (Insider strategy).

Let $E_{X}=[\underline{x},\overline{x}]$ for some fixed
$\underline{x},\overline{x}\in\mathbb{Z}$ with $\underline{x}<0<\overline{x}$.
Based on $(E_{V},\mathcal{B}(E_{V}),\nu)$, an insider strategy is a Young
measure $\xi:E_{V}\times\mathcal{B}(E_{X})\to[0,1]$, i.e.,

  1. (i)

$\xi(\;\cdot\;,B)$ is Borel-measurable for every $B\in\mathcal{B}(E_{X})$,

  2. (ii)

$\xi(v,\;\cdot\;)$ is a probability measure for every $v\in E_{V}$.

With $\Xi:=\Xi(E_{V},\nu;E_{X})$ we denote the set of insider strategies.

The interpretation is analogous to that of discrete behaviour strategies (see
Definition 2.3). We refer to Balder (2021) and the references therein for an
overview of the theory of Young measures and their applications in optimal
control theory. The market maker observes the total order flow $y:=x+z\in
E_{Y}:=E_{X}+E_{Z}=[\underline{x}-1,\overline{x}+1]$ and sets a price.

######  Definition 4.2 (Price function of market maker).

A price function is a Borel-measurable function

$S:E_{Y}\to E_{V},\qquad y\mapsto S(y).$

With $\mathcal{S}$ we denote the set of price functions.

Given a price function $S\in\mathcal{S}$, the insider’s objective is to
maximise the _expected utility_

$u(\xi,S):=\iiint\left[v-S(x+z)\right]x\;\zeta(dz)\xi(v,dx)\nu(dv)\rightarrow\max_{\xi\in\Xi}!$
(4.1)

Analogous to the discrete Kyle game, a price function $S$ is _rational_
assuming a strategy $\xi\in\Xi$ of the insider if for all
$A\in\mathcal{B}(E_{Y})$:

$\iiint S(x+z){\rm 1\rule{0.43057pt}{6.54443pt}\hskip
0.86108pt}_{{}_{\scriptstyle A}}(x+z)\;\zeta(dz)\xi(v,dx)\nu(dv)=\iiint
v\;{\rm 1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}(x+z)\;\zeta(dz)\xi(v,dx)\nu(dv).$ (4.2)

An equilibrium in the continuous state Kyle game has the familiar structure of
simultaneously requiring optimality for the insider, and rational pricing for
the market maker:

######  Definition 4.3 (Continuous Kyle equilibrium).

A _continuous Kyle equilibrium_ is a pair
$(\xi_{\star},S_{\star})\in\Xi\times\mathcal{S}$ satisfying

  1. (i)

Profit maximisation: Given $S_{\star}$, the strategy $\xi_{\star}$ maximises
(4.1).

  2. (ii)

Rational pricing: Given $\xi_{\star}$, the price system $S_{\star}$ is
rational according to (4.2),

The main result of this section is the following theorem, which we prove in
the remainder of the section.

######  Theorem 4.4.

The single-period, continuous state Kyle game admits a Kyle equilibrium.

###  4.1 Discretisation and embedding

We construct a sequence of discrete state games acting on the refining
sequence of _dyadic, equidistant grids_

$\begin{gathered}E_{V}^{n}:=\left\\{\frac{k}{2^{n}}:k=0,1,\ldots,2^{n}\right\\},\quad
E_{Z}^{n}:=\left\\{\frac{k}{2^{n}}:k=-2^{n},\ldots,0,\ldots,2^{n}\right\\},\\\
\text{and}\quad
E_{X}^{n}:=\left\\{\frac{k}{2^{n}}:k=2^{n}\underline{x},\ldots,0,\ldots,2^{n}\overline{x}\right\\},\quad
n\in\mathbb{N}.\end{gathered}$

We embed the discrete strategies and price functions into the continuous model
through piecewise constant continuation between the points of the $n$th grid:

$\displaystyle\Xi^{n}$
$\displaystyle:=\left\\{\xi\in\Xi:\;\xi(v,\,\cdot\,)=\xi\left(\frac{\left\lfloor
2^{n}v\right\rfloor}{2^{n}},\;\cdot\;\right)\text{ and }\xi(v,E_{X}\setminus
E^{n}_{X})=0\text{ for all }v\in E_{V}\right\\},$ (4.3)
$\displaystyle\mathcal{S}^{n}$
$\displaystyle:=\left\\{S\in\mathcal{S}:\;S(y)=S\left(\frac{\left\lfloor
2^{n}y\right\rfloor}{2^{n}}\right)\text{ for all }y\in E_{Y}\right\\}.$

It is straightforward to check that the insider’s expected utility from (2.3)
in the $n$th discrete game that is created by discretising the measures $\nu$
and $\zeta$, can be expressed in terms of the continuous quantities by

$u^{n}(\xi,S)=\iiint x\left[\frac{\left\lfloor
2^{n}v\right\rfloor}{2^{n}}-S\left(x+z\right)\right]\,\zeta(dz)\xi(v,dx)\nu(dv)\qquad\text{for
}(\xi,S)\in\Xi^{n}\times\mathcal{S}^{n}.$ (4.4)

Analogously, the rational pricing condition from Definition 2.5 for
$S\in\mathcal{S}^{n}$ given $\xi\in\Xi^{n}$ reads

$\begin{split}&\iiint S(x+z){\rm 1\rule{0.43057pt}{6.54443pt}\hskip
0.86108pt}_{{}_{\scriptstyle A}}\left(x+z\right)\,\zeta(dz)\xi(v,dx)\nu(dv)\\\
&\qquad=\iiint\frac{\left\lfloor 2^{n}v\right\rfloor}{2^{n}}{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}\left(x+z\right)\,\zeta(dz)\xi(v,dx)\nu(dv)\end{split}$ (4.5)

for all sets $A$ of the form $A=[k/2^{n},(k+1)/2^{n})$,
$k\in\\{(\underline{x}-1)2^{n},(\underline{x}-1)2^{n}+1,\ldots,(\overline{x}+1)2^{n}-1\\}$.

###  4.2 Approximation and existence of a limit point

For the convenience of the reader, we repeat the following definition.

######  Definition 4.5 (Narrow convergence).

A sequence of Young measures
$(\xi^{n})_{n\in\mathbb{N}}\subseteq\Xi(E_{V},\nu;E_{X})$ converges _narrowly_
to $\xi\in\Xi(E_{V},\nu;E_{X})$ if for all
$A\in\mathcal{B}(E_{V}),f\in\mathcal{C}_{b}(E_{X})$

$\iint{\rm 1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}(v)f(x)\;\xi^{n}(v,dx)\nu(dv)\to\iint{{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}(v)}f(x)\;\xi(v,dx)\nu(dv).$

An immediate consequence of narrow convergence is that

$\displaystyle\nu\otimes\xi^{n}\to\nu\otimes\xi\quad\text{weakly, where}$
(4.6) $\displaystyle\nu\otimes\xi^{n}(A\times
B):=\int_{A}\xi^{n}(v,B)\;\nu(dv)\ \text{for }A\times
B\in\mathcal{B}(E_{V})\times\mathcal{B}(E_{X})$

(to see this, one applies Fubini’s theorem for transition probabilities). For
further information regarding narrow convergence we refer to Balder (2021) and
the references therein.

######  Lemma 4.6 (Convergence of utilities).

Let $(\xi^{n},S^{n})_{n\in\mathbb{N}}\subseteq\Xi^{n}\times\mathcal{S}^{n}$
(that are not necessarily equilibria) such that $\xi^{n}\to\xi$ narrowly for
some $\xi\in\Xi$ and $\sum_{k=0}^{k_{n}}\lambda_{n,k}S^{n+k}\to S$
$\lambda$-a.e. for some $S\in\mathcal{S}$ and forward convex combinations
$(\lambda_{n,k})_{n\in\mathbb{N},k=0,1,\ldots,k_{n}}\subseteq\mathbb{R}_{+}$,
$k_{n}\in\mathbb{N}$ with $\sum_{k=0}^{k_{n}}\lambda_{n,k}=1$. Then

$\sum_{k=0}^{k_{n}}\lambda_{n,k}u^{n+k}(\xi^{n+k},S^{n+k})\xrightarrow{n\to\infty}u(\xi,S).$

###### Proof.

We split the integrals in (4.4) and (4.1) into two parts and handle them
separately. We start with the first parts including the true value. Let
$\epsilon>0$. For every $m\in\mathbb{N}$, we get

$\displaystyle\iint\left\lvert x\frac{\left\lfloor
2^{m}v\right\rfloor}{2^{m}}-xv\right\rvert\;\xi^{m}(v,dx)\nu(dv)$
$\displaystyle\leq\max(|\underline{x}|,|\overline{x}|)\sup_{v\in[0,1]}\left\lvert\frac{\left\lfloor
2^{m}v\right\rfloor}{2^{m}}-v\right\rvert$ (4.7)
$\displaystyle\leq\max(|\underline{x}|,|\overline{x}|)\frac{1}{2^{m}}.$

By (Balder, 1988, Theorem 2.2(b)), $\xi^{n}\to\xi$ narrowly implies that there
exists $n\in\mathbb{N}$ such that

$\left\lvert\iint xv\;\xi^{n+k}(v,dx)\nu(dv)-\iint
xv\;\xi(v,dx)\nu(dv)\right\rvert<\epsilon\quad\mbox{for all}\
k\in\mathbb{N}_{0}.$ (4.8)

Combining the two estimates (4.7) and (4.8) yields that for $n$ large enough

$\displaystyle\left\lvert\sum_{k=0}^{k_{n}}\lambda_{n,k}\iint
x\frac{\left\lfloor
2^{n+k}v\right\rfloor}{2^{n+k}}\;\xi^{n+k}(v,dx)\nu(dv)-\iint
xv\;\xi(v,dx)\nu(dv)\right\rvert$
$\displaystyle\quad\leq\sum_{k=0}^{k_{n}}\lambda_{n,k}\left\lvert\iint
x\frac{\left\lfloor
2^{n+k}v\right\rfloor}{2^{n+k}}\;\xi^{n+k}(v,dx)\nu(dv)-\iint
xv\;\xi^{n+k}(v,dx)\nu(dv)\right\rvert$
$\displaystyle\quad\quad+\sum_{k=0}^{k_{n}}\lambda_{n,k}\left\lvert\iint
xv\;\xi^{n+k}(v,dx)\nu(dv)-\iint xv\;\xi(v,dx)\nu(dv)\right\rvert$
$\displaystyle\quad\leq\sum_{k=0}^{k_{n}}\lambda_{n,k}\left(\max(|\underline{x}|,|\overline{x}|)\frac{1}{2^{n+k}}+\epsilon\right)<2\epsilon.$
(4.9)

Now we turn to the second parts of the integrals in (4.4) and (4.1) including
the price functions. This is more challenging as both integrand and integrator
vary with $n$. The key is that the absolute continuity of the noise trader’s
demand $\zeta$ smooths the realised price $\int S^{n}(x+z)\zeta(dz)$ per share
for an order of size $x$. Define

$f^{n}(x):=x\int S^{n}(x+z)\;\zeta(dz),\qquad n\in\mathbb{N}.$ (4.10)

Since $\zeta\ll\lambda$, there exists a density $g\in L^{1}(\lambda)$ with
$\zeta(A)=\int_{A}g\,d\lambda$ for all $A\in\mathcal{B}(\mathbb{R})$. Due to
the translation invariance of the Lebesgue measure, we can write

$f^{n}(x)=x\int S^{n}(x+z)g(z)\,\lambda(dz)=x\int
S^{n}(y)g(y-x)\,\lambda(dy).$

Since $g\in L^{1}(\lambda)$, we have that $\int\left\lvert
g(z-h)-g(z)\right\rvert\lambda(dz)\to 0$ as $h\to 0$, see, e.g., (Kallenberg,
2021, Lemma 2.7). The reason is that $g$ can be approximated in
$L^{1}(\lambda)$ by (uniformly) continuous functions, and for a uniformly
continuous function $\widetilde{g}$ we can use the estimate
$\int\left\lvert\widetilde{g}(z-h)-\widetilde{g}(z)\right\rvert\lambda(dz)\leq\sup_{z\in
E_{Z}}\left\lvert\widetilde{g}(z-h)-\widetilde{g}(z)\right\rvert\lambda(E_{Z})$
which tends to 0 as $h\to 0$. As a result, we obtain

$\left\lvert f^{n}(x+h)-f^{n}(x)\right\rvert\leq\sup_{x\in E_{X}}\left\lvert
x\right\rvert\sup_{y\in E_{Y}}\left\lvert S^{n}(y)\right\rvert\int\left\lvert
g(y-x-h)-g(y-x)\right\rvert\,\lambda(dy)\xrightarrow{h\to 0}0.$ (4.11)

The uniform estimate $\left\lvert S^{n}\right\rvert\leq 1$ yields that the
family $(f^{n})_{n\in\mathbb{N}}$ is equicontinuous at every fixed $x$.
Consequently, we can apply Lemma A.1 and (4.6) to obtain that

$\sup_{m\in\mathbb{N}}\;\left\lvert\iint f^{m}(x)\xi^{n}(v,dx)\nu(dv)-\iint
f^{m}(x)\xi(v,dx)\nu(dv)\right\rvert\xrightarrow{n\to\infty}0.$

It follows that for $n$ large enough and all $k\in\mathbb{N}_{0}$

$\left|\iint f^{n+k}(x)\xi^{n+k}(v,dx)\nu(dv)-\iint
f^{n+k}(x)\xi(v,dx)\nu(dv)\right|<\epsilon.$ (4.12)

Furthermore, by $\zeta\ll\lambda$, $\sum_{k=0}^{k_{n}}\lambda_{n,k}S^{n+k}\to
S$ $\lambda$-a.e., and dominated convergence, we obtain

$\sum_{k=0}^{k_{n}}\lambda_{n,k}f^{n+k}(x)=x\int\sum_{k=0}^{k_{n}}\lambda_{n,k}S^{n+k}(x+z)\;\zeta(dz)\longrightarrow\;x\int
S(x+z)\;\zeta(dz)=:f(x)$

pointwise. Thus, dominated convergence yields that for $n$ large enough

$\left|\iint\sum_{k=0}^{k_{n}}\lambda_{n,k}f^{n+k}(x)\;\xi(v,dx)\nu(dv)-\iint
f(x)\;\xi(v,dx)\nu(dv)\right|<\epsilon.$ (4.13)

Combining the two estimates (4.12) and (4.13), we get that for $n$ large
enough

$\displaystyle\Bigg{|}\sum_{k=0}^{k_{n}}\lambda_{n,k}\iiint
xS^{n+k}(x+z)\;\zeta(dz)\xi^{n+k}(v,dx)\nu(dv)$ $\displaystyle\hskip
160.00024pt-\iiint xS(x+z)\;\zeta(dz)\xi(v,dx)\nu(dv)\Bigg{|}$
$\displaystyle\quad=\left\lvert\sum_{k=0}^{k_{n}}\lambda_{n,k}\iint
f^{n+k}(x)\;\xi^{n+k}(v,dx)\nu(dv)-\iint f(x)\;\xi(v,dx)\nu(dv)\right\rvert$
$\displaystyle\quad\leq\sum_{k=0}^{k_{n}}\lambda_{n,k}\left\lvert\iint
f^{n+k}(x)\;\xi^{n+k}(v,dx)\nu(dv)-\iint
f^{n+k}(x)\;\xi(v,dx)\nu(dv)\right\rvert$
$\displaystyle\qquad+\left\lvert\iint\sum_{k=0}^{k_{n}}\lambda_{n,k}f^{n+k}(x)\;\xi(v,dx)\nu(dv)-\iint
f(x)\;\xi(v,dx)\nu(dv)\right\rvert$ $\displaystyle\quad<2\epsilon.$

Together with (4.2), the assertion follows. ∎

###### Proof of Theorem 4.4.

Step 1: For each $n\in\mathbb{N}$, we choose a Kyle equilibrium
$(\xi^{n}_{\star},S^{n}_{\star})\in\Xi^{n}\times\mathcal{S}^{n}\subseteq\Xi\times\mathcal{S}$
from the embedded discrete model (4.3)/(4.4)/(4.5) that exists by Theorem 2.7.
The sequence $(\xi^{n}_{\star})_{n\in\mathbb{N}}$ of Young measures is
obviously tight in the sense of (Balder, 2021, Definition 3.3) since their
support $[\underline{x},\overline{x}]$ is bounded. From Prohorov’s theorem for
Young measures (Balder, 2021, Theorem 4.10), it follows that
$(\xi^{n}_{\star})_{n\in\mathbb{N}}$ is relatively sequentially compact in the
narrow topology. That is, there exists a subsequence
$(n_{j})_{j\in\mathbb{N}}$ and a limit point
$\xi_{\star}\in\Xi(E_{V},\nu;[\underline{x},\overline{x}])$ such that
$\xi^{n_{j}}_{\star}\to\xi_{\star}\quad\text{narrowly as }j\to\infty$.
W.l.o.g. $n_{j}=j$ for all $j\in\mathbb{N}$.

Next, we apply a version of Komlós’ theorem to obtain a limiting price
function. Since $|S^{n}|\leq 1$ for all $n\in\mathbb{N}$, (Delbaen and
Schachermayer, 1994, Lemma A1.1) guarantees the existence of a measurable,
$E_{V}$-valued function $S$ and forward convex combinations
$(\lambda_{n,k})_{n\in\mathbb{N},k=0,1,\ldots,k_{n}}\subseteq\mathbb{R}_{+}$,
$k_{n}\in\mathbb{N}$ with $\sum_{k=0}^{k_{n}}\lambda_{n,k}=1$ such that

$\sum_{k=0}^{k_{n}}\lambda_{n,k}S^{n+k}_{\star}\to
S_{\star}\quad\lambda-\text{a.e. as }n\to\infty.$

It remains to show that $(\xi_{\star},S_{\star})$ is a Kyle equilibrium in the
continuous model.

Step 2 (Optimality of $\xi_{\star}$ given $S_{\star}$): Let $\xi_{0}\in\Xi$.
We have to show that

$u(\xi_{0},S_{\star})\leq u(\xi_{\star},S_{\star}).$ (4.14)

Let us discretise $\xi_{0}$ according to Lemma A.2, such that
$\xi_{0}^{n}\in\Xi^{n}$ is a trading strategy of the $n$th discrete game and
$\xi_{0}^{n}\to\xi_{0}$ narrowly. But $\xi^{n}_{\star}$ from Step 1 is an
optimal strategy (in the $n$th discrete game) given $S^{n}_{\star}$ and hence
$u^{n}(\xi_{0}^{n},S^{n}_{\star})\leq u^{n}(\xi^{n}_{\star},S^{n}_{\star})$
for all $n$, and so

$\sum_{k=0}^{k_{n}}\lambda_{n,k}u^{n+k}(\xi_{0}^{n+k},S^{n+k}_{\star})\leq\sum_{k=0}^{k_{n}}\lambda_{n,k}u^{n+k}(\xi^{n+k}_{\star},S^{n+k}_{\star})$
(4.15)

for the forward convex combinations
$(\lambda_{n,k})_{n\in\mathbb{N},k=0,1,\ldots,k_{n}}$ from Step 1. Using that
$\sum_{k=0}^{k_{n}}\lambda_{n,k}\allowbreak S^{n+k}_{\star}\to S_{\star}$
$\lambda$-a.e., we can apply Lemma 4.6 to both sides of (4.15) to conclude
(4.14).

Step 3 (Rationality of $S_{\star}$ given $\xi_{\star}$): Our goal is to show
that from the rational pricing condition (4.5) of the discrete games for all
$n\in\mathbb{N}$, we can deduce the rational pricing condition (4.2) of the
continuous game. Sets of the form

$A=[k/2^{m},(k+1)/2^{m}),\ m\in\mathbb{N},\
k\in\\{(\underline{x}-1)2^{m},(\underline{x}-1)2^{m}+1,\ldots,(\overline{x}+1)2^{m}-1\\}$
(4.16)

constitute a $\cap$-stable generator of the $\sigma$-algebra
$\mathcal{B}(E_{Y})$. Thus, by a Dynkin-argument (see, e.g., (Kallenberg,
2021, Theorem 1.1)), it is sufficient to verify (4.2) for $A$ from (4.16). We
fix a set $A$ of this form with $m\in\mathbb{N}$. For $n\geq m$, the rational
pricing condition (4.5) for the $n$th discrete game implies that

$\begin{split}\iiint S^{n}_{\star}(x+z){\rm 1\rule{0.43057pt}{6.54443pt}\hskip
0.86108pt}_{{}_{\scriptstyle
A}}\left(x+z\right)\,\zeta(dz)\xi^{n}_{\star}(v,dx)\nu(dv)\qquad\\\
\qquad=\iiint\frac{\left\lfloor 2^{n}v\right\rfloor}{2^{n}}{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}\left(x+z\right)\,\zeta(dz)\xi^{n}_{\star}(v,dx)\nu(dv).\end{split}$ (4.17)

We note that (4.17) only needs to hold for $n\geq m$ since $\zeta$ is already
the limiting measure and integrating the function ${\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}(x+\cdot\,)$ for a strict subset $A$ of $[l/2^{n},(l+1)/2^{n})$ could
produce a bias.

We show the convergence of (4.17) to (4.2) for the left- and right-hand sides
separately, starting with the right-hand side. The function $(v,x)\mapsto
v\int{\rm 1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}\left(x+z\right)\,\zeta(dz)$ is continuous in $x$ by the same reasons which
lead to (4.11). Since it is furthermore measurable in $v$, it is a suitable
integrand for narrow convergence according to (Balder, 1988, Theorem 2.2(b)).
From the narrow convergence of $\xi^{n}_{\star}\to\xi_{\star}$ it follows that

$\iiint v{\rm 1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}\left(x+z\right)\,\zeta(dz)\xi^{n}(v,dx)\nu(dv)\to\iiint v{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}\left(x+z\right)\,\zeta(dz)\xi(v,dx)\nu(dv).$

Then, the convergence follows from (4.7) as in the proof of Lemma 4.6.

Now we turn to the left-hand side in (4.17). The convergence of the right-hand
side already implies the convergence of the left-hand side (without passing to
convex combinations), but it remains to verify that $\iiint S(x+z){\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}(x+z)\;\zeta(dz)\xi(v,dx)\nu(dv)$ is the limit. This follows as in the
proof of Lemma 4.6 by passing to convex combinations. The only difference is
that instead of (4.10) we consider the functions $\widetilde{f}^{n}(x):=\int
S^{n}(x+z){\rm 1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}(x+z)\,\zeta(dz)$, $n\in\mathbb{N}$. ∎

##  Appendix A Appendix

In the proof of Theorem 4.4 we need the following lemma, for a condensed proof
see, e.g., (Bogachev, 2018, Theorem 2.2.8).

######  Lemma A.1.

(Rao, 1962, Theorem 3.2) Suppose that a sequence of probability measures
$(\mu^{n})_{n\in\mathbb{N}}$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$
converges weakly to a probability measure $\mu$, and let $F$ be a family of
uniformly bounded, pointwise equicontinuous functions from $\mathbb{R}$ to
$\mathbb{R}$. Then

$\sup_{f\in F}\;\left\lvert\int f(y)\mu^{n}(dy)-\int
f(y)\mu(dy)\right\rvert\to 0\quad\mbox{as}\ n\to\infty.$

Next we show how an insider strategy in the continuous Kyle game can be
approximated in the narrow topology through a sequence of strategies in the
discretised games (cf. Subsection 4.1).

######  Lemma A.2.

Let $\xi\in\Xi(E_{V},\nu;E_{X})$ be an insider strategy in the continuous Kyle
game. There exists a sequence of strategies $\xi^{n}\in\Xi^{n}$,
$n\in\mathbb{N}$, in the discrete games such that $(\xi^{n})_{n\in\mathbb{N}}$
converges narrowly to $\xi$.

###### Proof.

Let $\xi\in\Xi(E_{V},\nu;E_{X})$. To shorten formulae, we assume w.l.o.g. that
$\nu(\\{1\\})=\xi(v,\\{\overline{x}\\})=0$ for all $v\in E_{V}$. We define
approximating Young measures lying in $\Xi^{n}$ by

$\begin{aligned} \xi^{n}(v,\,\cdot\,):=\sum_{\begin{subarray}{c}k=0\\\
\nu(D^{n}_{k})>0\end{subarray}}^{2^{n}-1}{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
D^{n}_{k}}}(v)\sum_{l=2^{n}\underline{x}}^{2^{n}\overline{x}-1}\delta_{l2^{-n}}\frac{\nu\otimes\xi(D^{n}_{k}\times
D^{n}_{l})}{\nu(D_{k}^{n})}+\sum_{\begin{subarray}{c}k=0\\\
\nu(D^{n}_{k})=0\end{subarray}}^{2^{n}-1}{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
D^{n}_{k}}}(v)\,\delta_{0},\end{aligned}\quad\begin{aligned} v\in E_{V},\\\
n\in\mathbb{N},\end{aligned}$

where $D^{n}_{j}:=[j2^{-n},(j+1)2^{-n})$. By (Balder, 1988, Theorem 2.2(d)),
it is sufficient to show

$\iint{\rm 1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}(v)f(x)\,\xi^{n}(v,dx)\nu(dv)\to\iint{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}(v)f(x)\,\xi(v,dx)\nu(dv)$ (A.1)

for all sets $A=[k/2^{m},(k+1)/2^{m}),\ m\in\mathbb{N},\
k\in\\{(\underline{x}-1)2^{m},(\underline{x}-1)2^{m}+1,\ldots,(\overline{x}+1)2^{m}-1\\}$
and $f\in\mathcal{C}_{b}(E_{X})$. For every $n\geq m$, the endpoints of $A$
fall onto the $n$th grid, and we get

$\displaystyle\iint{\rm 1\rule{0.43057pt}{6.54443pt}\hskip
0.86108pt}_{{}_{\scriptstyle A}}(v)f(x)\xi^{n}(v,dx)\nu(dv)$
$\displaystyle\quad=\iint f(x)\sum_{\begin{subarray}{c}k=0\\\
\nu(D^{n}_{k})>0,\ D^{n}_{k}\subseteq
A\end{subarray}}^{2^{n}-1}\sum_{l=2^{n}\underline{x}}^{2^{n}\overline{x}-1}{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
D^{n}_{k}}}(v)\frac{\nu\otimes\xi(D^{n}_{k}\times
D^{n}_{l})}{\nu(D_{k}^{n})}\;\delta_{l2^{-n}}(dx)\nu(dv)$
$\displaystyle\quad=\sum_{\begin{subarray}{c}k=0\\\ D^{n}_{k}\subseteq
A\end{subarray}}^{2^{n}-1}\sum_{l=2^{n}\underline{x}}^{2^{n}\overline{x}-1}f(l2^{-n})\int_{D^{n}_{k}}\xi(v,D^{n}_{l})\nu(dv)$
$\displaystyle\quad=\iint{\rm 1\rule{0.43057pt}{6.54443pt}\hskip
0.86108pt}_{{}_{\scriptstyle A}}(v)f\left(\frac{\lfloor
2^{n}x\rfloor}{2^{n}}\right)\xi(v,dx)\nu(dv).$

By pointwise convergence of ${\rm 1\rule{0.43057pt}{6.54443pt}\hskip
0.86108pt}_{{}_{\scriptstyle A}}(v)f(2^{-n}\lfloor 2^{n}x\rfloor)\to{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle A}}(v)f(x)$ as
$n\to\infty$, bounded convergence leads to

$\iint{\rm 1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}(v)f\left(\frac{\lfloor
2^{n}x\rfloor}{2^{n}}\right)\,\xi(v,dx)\nu(dv)\to\iint{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}_{{}_{\scriptstyle
A}}(v)f(x)\,\xi(v,dx)\nu(dv),$

which shows that $\xi^{n}\to\xi$ narrowly. ∎

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