# Nash equilibria for relative investors with (non)linear price impact

Nicole Bäuerle  Department of Mathematics, Karlsruhe Institute of Technology
(KIT), D-76128 Karlsruhe, Germany, ORCID 0000-0003-0077-3444
<EMAIL_ADDRESS>and  Tamara Göll  Department of Mathematics, Karlsruhe
Institute of Technology (KIT), D-76128 Karlsruhe, Germany, ORCID
0009-0007-8341-723X<EMAIL_ADDRESS>
###### Abstract.

We consider the strategic interaction of $n$ investors who are able to
influence a stock price process and at the same time measure their utilities
relative to the other investors. Our main aim is to find Nash equilibrium
investment strategies in this setting in a financial market driven by a
Brownian motion and investigate the influence the price impact has on the
equilibrium. We consider both CRRA and CARA utility functions. Our findings
show that the problem is well-posed as long as the price impact is at most
linear. Moreover, numerical results reveal that the investors behave very
aggressively when the price impact is beyond a critical parameter.

Key words:

Portfolio optimization; Price Impact; Nash equilibrium; Relative investor

JEL subject classifications:

C61, C73, G11

##  1\. Introduction

In this paper, we determine the optimal investment strategies of $n$ investors
in a common financial market who interact strategically. The strategic
interaction is caused by two different factors: a relative component inside
the objective function of each investor and by the fact that the stock price
dynamic is affected by the arithmetic mean of the $n$ agents’ investments.

We contribute to two strands of literature. The first one is the literature on
strategic interaction between agents. Strategic interaction in portfolio
optimization problems has been motivated for example by [8] and [25] through
competition between agents. Since then, portfolio choice problems including
strategic interaction between investors have been widely studied. [3] consider
two agents in a continuous-time model which includes stocks following
geometric Brownian motions. They use power utility functions and maximize the
ratio of the two investors’ wealth. [17] also consider stocks driven by
geometric Brownian motions and $n$ agents maximizing a weighted difference of
their own wealth and the arithmetic mean of the other agents’ wealth.
Structurally similar objective functions including the arithmetic mean have
been used by [4]. There, the unique Nash equilibrium for $n$ agents is derived
in a very general financial market using the unique solution to some auxiliary
classical portfolio optimization problem. [29] consider the case of asset
specialization for $n$ agents. They derive the unique constant Nash
equilibrium using both the arithmetic mean under CARA utility and the
geometric mean under CRRA utility. Later, their work has been extended by [28]
to consumption-investment problems including relative concerns. [14, 15] use
forward utilities of both CARA and CRRA type with and without consumption.
More general financial markets (including e.g. stochastic volatility and
incomplete information) were, for example, used in [27], [19] and [21].

The second strand of literature focuses on (large) investors whose trades
affect the price processes of certain assets. [7] gives an overview of
different reasons and methods to incorporate price impact. [22, 23] consider a
discrete time market model in which a single large trader affects the price of
the risky asset. He finds conditions under which there are no arbitrage
opportunities for small traders while the large trader is able to achieve
riskless profit using some market manipulation strategy. [1] introduce a
discrete-time financial market in which the price process of the risky stock
is affected by the investment of a large investor. The impact is divided into
_temporary_ and _permanent_ price impact. They minimize risk and transaction
costs arising from the price impact simultaneously. In [5], the problem of
minimizing the expected cost of liquidating a block of shares over a fixed
time interval is solved in a discrete time financial market. Here, the number
of shares held by a large trader impacts the stock price process linearly.

[13] assume that the investment of a single large investor affects the
interest rate of a riskless asset and the drift and volatility of stock price
processes, which are modeled by Itô-diffusions, simultaneously. They allow for
general square integrable strategies and extend classical results of hedging
contingent claims to their setting. A similar model including stocks paying
dividends was used by [10]. In their setting, the volatility of the stock
prices does not depend on the large investors portfolio and they determine the
optimal consumption strategy of the large investor. [2] use a more general
continuous-time model for the stock prices, but only allow for constant
portfolio processes. They prove necessary and sufficient conditions for the
absence of arbitrage for both small and large investors. [30] consider a
Black-Scholes-type stock price dynamic where the investor’s impact is modeled
by a general price impact function integrated with respect to an Itô process
which models the investment of the large agent. After introducing their market
model they show how to price European options defined therein. [16] also
consider a Black-Scholes-type price process in which the drift is (possibly
nonlinearly) affected by the large investors’ trades and also contains a
stochastic component which depends on the current market state. They maximize
expected utility of the large investor under both complete and incomplete
information. A problem of optimal liquidation in another Black-Scholes-type
market is treated in [20]. Here, the stock price depends linearly on the
dynamics of the large investors selling process. [26] maximize expected
utility in a financial market similar to the one treated in this paper. They
model the price process as a geometric Brownian motion by adding a multiple of
the large traders investment to the constant drift.

The majority of literature considers the case of a single large trader. [33],
however, consider a continuous time financial market where the price impact -
both temporary and permanent - results from the investment of $n+1$ ’strategic
players’. Moreover, [12] considers two agents who interact strategically
through their linear impact on the return of the risk free asset. Maximizing
their terminal wealth under CRRA utility, he derives the unique constant pure-
strategy Nash equilibrium.

In the following, we solve an $n$-agent portfolio problem with relative
performance concerns where we allow that the agents are jointly able to
influence the asset dynamics which is reasonable if $n$ is large and which has
not been done before.

This paper is organized as follows. In the next section we introduce the
linear price impact financial market. In Section 3 we explicitly solve the
problem of maximizing expected utility of exponential type which results in
the unique constant Nash equilibrium. The argument of the utility function
consists of the difference of some agents’ wealth and a weighted arithmetic
mean of the other agents’ wealth. We also examine the influence of the price
impact parameter $\alpha$ to the Nash equilibrium and the stock price attained
by inserting the arithmetic mean of the components of the Nash equilibrium. In
Section 4 we substitute the linear impact of the agents arithmetic mean on the
stock price process by a nonlinear one. We prove that the problem of
maximizing CARA utility is well-posed as long the influence is sublinear and
does not have an optimal solution if the influence is superlinear. In Section
5 we assume that agents use CRRA utility functions (power and logarithmic
utility) and insert the product of some agents wealth and a weighted geometric
mean of the other agents’ wealth into the expected utility criterion. Similar
to the CARA case we are able to explicitly determine the unique constant Nash
equilibrium.

##  2\. Price impact market

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$ be a
filtered probability space and $T>0$ a finite time horizon. Moreover, let $W$
be a standard Brownian motion therein.

The underlying financial market consists of one riskless bond which will for
simplicity be assumed to be identical to 1, and one risky asset (a stock).
Note that it is straightforward to extend the results below to the case of
$d>1$ stocks instead of just one. However, to keep calculations simple, we
only consider one stock.

The price process of the stock, denoted by $(S_{t})_{t\in[0,T]}$, is the
solution to the SDE

$\mathrm{d}S_{t}=S_{t}\left(\left(\mu+\alpha\bar{\pi}_{t}\right)\mathrm{d}t+\sigma\mathrm{d}W_{t}\right),\,S(0)=1.$
(2.1)

Here, the drift $\mu>0$ and volatility $\sigma>0$ are assumed to be
deterministic and constant in time. Our model describes a special case of the
models considered by [13], [10] and [26]. Note that, instead of just one large
investor, we consider the case of $n$ agents who collectively act like one
large investor.

The expression $\bar{\pi}_{t}$ will describe the arithmetic mean of the
investment of $n$ investors into the stock at time $t\in[0,T]$, i.e.

$\bar{\pi}_{t}\coloneqq\frac{1}{n}\sum_{i=1}^{n}\pi^{i}_{t},$ (2.2)

where $\pi^{i}_{t}$ describes either the amount or the fraction of wealth
agent $i$ invests into the stock at some time $t\in[0,T]$. The strategies
$\pi^{i}$ of the $n$ investors are assumed to belong to the set $\mathcal{A}$
of $(\mathcal{F}_{t})_{t\in[0,T]}$-progressively measurable, square-integrable
processes, i.e.

$\mathcal{A}\coloneqq\left\\{\pi:\Omega\times[0,T]\to\mathbb{R}:\,\pi\text{ is
}(\mathcal{F}_{t})_{t\in[0,T]}\text{-progressively measurable,
}\int_{0}^{T}\pi_{t}^{2}\mathrm{d}t<\infty\,\mathbb{P}\text{-a.s.}\right\\}.$
(2.3)

This assumption ensures that the SDE (2.1) has a unique solution (see e.g.
[24]). Let the initial capital of agent $i$ be given by $x_{0}^{i}$.

Finally, $\alpha\in\mathbb{R}$ is some constant that describes the impact of
the investment of the $n$ investors into the stock.

######  Remark 2.1.

  * a)

Some authors argue that $\alpha$ should take both positive and negative values
due to the fact that (large) investors may have both positive and negative
impact on stock returns (see e.g. [9], [11]). On the other hand, [2] prove in
a more general setting that stock prices need to be increasing in terms of
some large investor’s investment. Otherwise it would be possible to construct
some ’In & Out’ arbitrage strategy. Since the optimization problems considered
in this paper have finite optimal solutions, our model appears to be free of
arbitrage. Hence, we allow for both positive and negative values for $\alpha$.

  * b)

Assuming that the drift of the risky stock depends linearly on the agents’
investment makes the model mathematically tractable. However, empirical data
suggests that price impact is concave in order size (see [31] and references
therein). Thus, we also consider the case of nonlinear price impact if
investors use exponential utility functions (see Section 4).

##  3\. Optimization under CARA utility with linear price impact

At first, we assume that investors use exponential utility (CARA) functions to
measure their preferences. Hence, define

$U_{i}:\mathbb{R}\to\mathbb{R},\,x\mapsto-\exp\Big{(}-\frac{1}{\delta_{i}}x\Big{)}$
(3.1)

for some parameters $\delta_{i}>0$, $i=1,\ldots,n.$ While using CARA utility
functions, it is more convenient to consider the amount invested into the
risky stock instead of the fraction of wealth or number of shares. Hence, we
interpret $\pi^{i}_{t}$ as the amount of money agent $i$ invests into the
risky stock at some $t\in[0,T],\,i=1,\ldots,n.$ Thus, the wealth process of
agent $i$ is given by

$X_{t}^{i,\pi^{i}}=x_{0}^{i}+\int_{0}^{t}\pi^{i}_{s}\left(\left(\mu+\alpha\bar{\pi}_{s}\right)\mathrm{d}s+\sigma\mathrm{d}W_{s}\right),\
t\in[0,T].$ (3.2)

In this paper, we want to examine the strategic interaction created by the
price impact introduced earlier and a modification of the classical objective
function used in expected utility maximization. Hence, we substitute the
terminal wealth of a single investor inside the expected utility criterion by
a relative quantity which captures the fact that agent $i$ wants to maximize
her terminal wealth while also considering her performance with respect to the
other agents. Similar to Section 2 in [29], we use the difference of agent
$i$’s terminal wealth and a weighted arithmetic mean of the other agents’
terminal wealth. Hence, we insert

$X_{T}^{i,\pi}-\frac{\theta_{i}}{n}\sum_{j\neq i}X_{T}^{j,\pi^{j}}$

into the argument of the utility function of investor $i$. The parameter
$\theta_{i}\in[0,1]$ measures how much agent $i$ cares about her performance
with respect to the other agents.

Our goal will therefore be to find all Nash equilibria to the multi-objective
optimization problem

$\begin{cases}&\sup_{\pi^{i}\in\mathcal{A}}\mathbb{E}\left[-\exp\left(-\frac{1}{\delta_{i}}\left(X_{T}^{i,\pi^{i}}-\frac{\theta_{i}}{n}\sum_{j\neq
i}X_{T}^{j,\pi^{j}}\right)\right)\right],\\\ \text{s.t.
}&X^{i,\pi^{i}}_{T}=x_{0}^{i}+\displaystyle\int_{0}^{T}\pi^{i}_{t}\left(\left(\mu+\alpha\bar{\pi}_{t}\right)\mathrm{d}t+\sigma\mathrm{d}W_{t}\right),\end{cases}$
(3.3)

$i=1,\ldots,n$. A Nash equilibrium for general objective functions $J_{i}$,
$i=1,\ldots,n,$ is defined as follows.

######  Definition 3.1.

Let $J_{i}:\mathcal{A}^{n}\rightarrow\mathbb{R}$ be the objective function of
agent $i$. A vector $\left(\pi^{1,*},\ldots,\pi^{n,*}\right)$ of strategies is
called a _Nash equilibrium_ , if, for all admissible $\pi^{i}\in\mathcal{A}$
and $i=1,\ldots,n$,

$J_{i}(\pi^{1,*},\ldots,\pi^{i,*},\ldots,\pi^{n,*})\geq
J_{i}(\pi^{1,*},\ldots,\pi^{i-1,*},\pi^{i},\pi^{i+1,*},\ldots,\pi^{n,*}).$
(3.4)

I.e. deviating from $\pi^{i,*}$ does not increase agent $i$’s objective
function.

###  3.1. Solution

In order to solve the best response problem (3.3), we fix some investor $i$
and assume that the strategies $\pi^{j}$, $j\neq i$, of the other agents are
given. Under these conditions we can rewrite the optimization problem (3.3)
into a classical portfolio optimization problem in a similar (but not
identical) price impact market. Afterwards, the Nash equilibria can be
determined using the solution to the classical problem.

Define the process $\left(Y_{t}^{i,\varphi^{i}}\right)_{t\in[0,T]}$ by

$Y_{t}^{i,\varphi^{i}}\coloneqq
X^{i,\pi^{i}}_{t}-\frac{\theta_{i}}{n}\sum_{j\neq
i}X^{j,\pi^{j}}_{t},\,t\in[0,T],\,i=1,\ldots,n,$ (3.5)

where we further define the strategy $\varphi^{i}$ by

$\varphi^{i}_{t}\coloneqq\pi^{i}_{t}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}_{t},\,t\in[0,T],\,i=1,\ldots,n,$ (3.6)

which is still square integrable and progressively measurable
($\varphi^{i}\in\mathcal{A}$).

Then we can write $Y_{T}^{i,\varphi^{i}}$ as

$\displaystyle Y_{T}^{i,\varphi^{i}}$
$\displaystyle=X^{i,\pi^{i}}_{T}-\frac{\theta_{i}}{n}\sum_{j\neq
i}X^{j,\pi^{j}}_{T}$ $\displaystyle\eqqcolon
y_{0}^{i}+\int_{0}^{T}\varphi^{i}_{t}\left(\left(\tilde{\mu}^{-i}_{t}+\frac{\alpha}{n}\varphi^{i}_{t}\right)\mathrm{d}t+\sigma\mathrm{d}W_{t}\right),$

where we introduced $y_{0}^{i}\coloneqq
x_{0}^{i}-\frac{\theta_{i}}{n}\sum_{j\neq i}x_{0}^{j}$,
$\bar{\pi}^{-i}_{t}\coloneqq\frac{1}{n}\sum_{j\neq i}\pi^{j}_{t}$ and
$\tilde{\mu}^{-i}_{t}\coloneqq\mu+\alpha\frac{n+\theta_{i}}{n}\bar{\pi}^{-i}_{t}$.

Hence, in order to solve the best response problem associated to (3.3), we can
equivalently solve the following single investor portfolio optimization
problem due to the one-to-one relation between $\pi^{i}$ and $\varphi^{i}$

$\begin{cases}&\sup_{\varphi^{i}\in\mathcal{A}}\mathbb{E}\left[-\exp\left(-\frac{1}{\delta_{i}}Y_{T}^{i,\varphi^{i}}\right)\right],\\\
\text{s.t.
}&Y_{T}^{i,\varphi^{i}}=y_{0}^{i}+\int_{0}^{T}\varphi^{i}_{t}\left(\left(\tilde{\mu}^{-i}_{t}+\frac{\alpha}{n}\varphi^{i}_{t}\right)\mathrm{d}t+\sigma\mathrm{d}W_{t}\right),\,t\in[0,T],\end{cases}$
(3.7)

in a financial market with corrected price impact.

Now assume that $\varphi^{i,*}=\varphi^{i,*}(\tilde{\mu}^{-i})$ is an optimal
solution to (3.7) depending on the drift process $\tilde{\mu}^{-i}$. Then the
optimal solution to the best response problem with respect to (3.3) is
uniquely determined by

$\pi^{i}=\varphi^{i,*}(\tilde{\mu}^{-i})+\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j},\,i=1,\ldots,n.$ (3.8)

Note that we can find a unique Nash equilibrium if and only if problem (3.7)
and the fixed point problem for $\pi^{i}$, given in terms of the system of
equations (3.8), are uniquely solvable.

Using the described technique, we are able to find the unique constant Nash
equilibrium. Note that the restriction to constant Nash equilibria is
necessary since otherwise we would not be able to solve the auxiliary problem
explicitly.

At first, we solve the auxiliary problem (3.7) for investor $i$ under the
assumption that the strategies of the other investors are constant in time and
deterministic.

######  Lemma 3.2.

Let $\theta_{i}\in[0,1]$ and $\delta_{i}>0$, $i=1,\ldots,n$. Moreover, assume
that $n\sigma^{2}-2\delta_{i}\alpha>0$ for all $i=1,\ldots,n$. If, for some
$i\in\\{1,\ldots,n\\}$, the strategies $\pi^{j}$, $j\neq i$, are constant in
time and deterministic, the unique optimal solution to (3.7) is given by

$\varphi^{i,*}_{t}\equiv\frac{n\delta_{i}\tilde{\mu}^{-i}}{n\sigma^{2}-2\delta_{i}\alpha},\,t\in[0,T].$
(3.9)

###### Proof.

Since $\pi^{j}$, $j\neq i$, are constant, the drift $\widetilde{\mu}^{-i}$ is
also constant. The dynamics of the wealth process $Y^{i,\varphi^{i}}$ are
therefore given by

$\displaystyle\mathrm{d}Y_{t}^{i,\varphi^{i}}=\varphi^{i}_{t}\left(\left(\widetilde{\mu}^{-i}+\frac{\alpha}{n}\varphi^{i}_{t}\right)\mathrm{d}t+\sigma\mathrm{d}W_{t}\right),\,t\in[0,T].$

To derive the associated HJB equation used to solve the portfolio optimization
problem, we define the value function

$v(t,y):=\sup_{\varphi^{i}\in\mathcal{A}}\mathbb{E}\Big{[}-\exp\Big{(}-\frac{1}{\delta_{i}}Y_{T}^{i,\varphi^{i}}\Big{)}\Big{|}Y_{t}^{i,\varphi^{i}}=y\Big{]},\
t\in[0,T],\,y\in\mathbb{R}.$ (3.10)

The maximum value in (3.7) is thus given by $v(0,y_{0}^{i}).$ The Hamilton
Jacobi Bellman (HJB) equation for this problem reads

$0=v_{t}+\max_{a\in\mathbb{R}}\left\\{v_{y}\widetilde{\mu}^{-i}a+\left(\frac{\alpha}{n}v_{y}+\frac{\sigma^{2}}{2}v_{yy}\right)a^{2}\right\\}$
(3.11)

for $y\in\mathbb{R}$, $t\in[0,T]$, with terminal condition
$v(T,y)=-\exp(-\frac{1}{\delta_{i}}y).$ Note that we omitted the arguments of
$v$ to keep notation simple. The maximum in (3.11) is attained at

$a^{*}=-\frac{n\widetilde{\mu}^{-i}v_{y}}{n\sigma^{2}v_{yy}+2\alpha v_{y}}.$
(3.12)

Inserting the maximum point into (3.11) yields

$0=v_{t}-\frac{1}{2}\frac{n(\widetilde{\mu}^{-i})^{2}v_{y}^{2}}{n\sigma^{2}v_{yy}+2\alpha
v_{y}}.$ (3.13)

We use the ansatz $v(t,y)=-f(t)\exp(-\frac{1}{\delta_{i}}y)$ for some
continuously differentiable function $f:[0,T]\to\mathbb{R}$ satisfying
$f(T)=1$. Then (3.13) simplifies to the ODE

$f^{\prime}(t)+\rho f(t)=0,\,f(T)=1,$ (3.14)

where
$\rho=\frac{1}{2}\frac{n(\widetilde{\mu}^{-i})^{2}}{n\sigma^{2}-2\alpha\delta_{i}}$.
The unique solution to this ODE is given by $f(t)=e^{\rho(T-t)},\,t\in[0,T]$.
Finally, $v(t,y)=-\exp(\rho(T-t)-\frac{1}{\delta_{i}}y)$,
$t\in[0,T],\,y\in\mathbb{R},$ solves the HJB equation. Inserting $v$ into
(3.12) yields

$\varphi^{i,*}\equiv\frac{n\delta_{i}\widetilde{\mu}^{-i}}{n\sigma^{2}-2\delta_{i}\alpha}.$
(3.15)

A standard verification theorem (see for example [6], pp.280-282, [32], [18]
for similar versions) concludes our proof. ∎

Lemma 3.2 together with (3.8) introduces a system of linear equations whose
solutions constitute Nash equilibrium strategies. The next theorem displays
the unique solution to this system and thus, the unique constant Nash
equilibrium.

######  Theorem 3.3.

Assume that $n\sigma^{2}-2\delta_{j}\alpha>0$ for all $j=1,\ldots,n.$ If
$1-\hat{\theta}\neq\sum_{j=1}^{n}\frac{n\alpha\delta_{j}}{(n+\theta_{j})\left(n\sigma^{2}-\delta_{j}\alpha\right)},$
the unique constant Nash equilibrium to (3.3) is given by

$\displaystyle\pi^{i,*}$
$\displaystyle=\frac{n}{n+\theta_{i}}\frac{n\delta_{i}\mu}{n\sigma^{2}-\delta_{i}\alpha}+\left(\frac{\theta_{i}}{n+\theta_{i}}+\frac{n\alpha\delta_{i}}{(n+\theta_{i})(n\sigma^{2}-\delta_{i}\alpha)}\right)\cdot\frac{\sum_{j=1}^{n}\frac{n}{n+\theta_{j}}\frac{n\delta_{j}}{\left(n\sigma^{2}-\delta_{j}\alpha\right)}\cdot\mu}{1-\hat{\theta}-\sum_{j=1}^{n}\frac{n\alpha\delta_{j}}{(n+\theta_{j})\left(n\sigma^{2}-\delta_{j}\alpha\right)}},$

$i=1,\ldots,n,$ where
$\hat{\theta}=\sum_{j=1}^{n}\frac{\theta_{j}}{n+\theta_{j}}.$ If
$1-\hat{\theta}=\sum_{j=1}^{n}\frac{n\alpha\delta_{j}}{(n+\theta_{j})\left(n\sigma^{2}-\delta_{j}\alpha\right)}$,
there is no constant Nash equilibrium.

###### Proof.

Using Lemma 3.2, the unique optimal solution to the auxiliary problem (3.7) is
given by

$\varphi^{i,*}=\frac{n\delta_{i}\widetilde{\mu}^{-i}}{n\sigma^{2}-2\delta_{i}\alpha}.$
(3.16)

Note that this is obviously a constant and deterministic strategy. Moreover,
we defined $\varphi^{i,*}=\pi^{i}-\frac{\theta_{i}}{n}\sum_{j\neq i}\pi^{j}$
and $\widetilde{\mu}^{-i}=\mu+\frac{n+\theta_{i}}{n^{2}}\alpha\sum_{j\neq
i}\pi^{j}.$ Hence, we need to solve the following system of linear equations
to determine the unique constant Nash equilibrium

$\pi^{i}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}=\frac{n\delta_{i}}{n\sigma^{2}-2\delta_{i}\alpha}\mu+\frac{\delta_{i}\alpha}{n\sigma^{2}-2\delta_{i}\alpha}\frac{n+\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}.$ (3.17)

Rearranging (3.17) and adding $\pi^{i}$ in the sum yields

$\pi^{i}=\frac{n}{n+\theta_{i}}\frac{n\delta_{i}}{n\sigma^{2}-\delta_{i}\alpha}\mu+\left(\frac{\theta_{i}}{n+\theta_{i}}+\frac{n\delta_{i}\alpha}{(n+\theta_{i})(n\sigma^{2}-\delta_{i}\alpha)}\right)\sum_{j=1}^{n}\pi^{j}.$
(3.18)

Summing over all $i\in\\{1,\ldots,n\\}$ on both sides then yields

$\sum_{j=1}^{n}\pi^{j}=\sum_{j=1}^{n}\frac{n}{n+\theta_{j}}\frac{n\delta_{j}}{n\sigma^{2}-\delta_{j}\alpha}\mu+\left(\hat{\theta}+\sum_{j=1}^{n}\frac{n}{n+\theta_{j}}\frac{\delta_{j}\alpha}{n\sigma^{2}-\delta_{j}\alpha}\right)\sum_{j=1}^{n}\pi^{j}.$
(3.19)

Solving for $\sum_{j=1}^{n}\pi^{j}$ (which is possible if and only if
$\sum_{j=1}^{n}\frac{n\alpha\delta_{j}}{(n+\theta_{j})(n\sigma^{2}-\delta_{j}\alpha)}\neq
1-\hat{\theta}$) yields

$\sum_{j=1}^{n}\pi^{j}=\frac{\sum_{j=1}^{n}\frac{n}{n+\theta_{j}}\frac{n\delta_{j}}{\left(n\sigma^{2}-\delta_{j}\alpha\right)}\cdot\mu}{1-\hat{\theta}-\sum_{j=1}^{n}\frac{n\alpha\delta_{j}}{(n+\theta_{j})\left(n\sigma^{2}-\delta_{j}\alpha\right)}}.$
(3.20)

Finally, we can insert (3.20) into (3.18) to obtain the claimed representation
of $\pi^{i,*}$ which concludes our proof. ∎

######  Remark 3.4.

Theorem 3.3 contains the two special cases $\alpha=0$ (no price impact) and
$\theta_{i}=0$ for all $i=1,\ldots,n$ (no relative concerns in the objective
function). For $\alpha=0$ we obtain

$\pi^{i,*}=\left(\frac{n\delta_{i}}{n+\theta_{i}}+\frac{\theta_{i}}{(1-\hat{\theta})(n+\theta_{i})}\sum_{j=1}^{n}\frac{n\delta_{j}}{n+\theta_{j}}\right)\cdot\frac{\mu}{\sigma^{2}}>0$
(3.21)

for $i=1,\ldots,n$ which coincides (as expected) with the Nash equilibrium in
[4] (Remark 4.1). If $\theta_{i}=0$ for all $i=1,\ldots,n,$ we deduce

$\pi^{i,*}=\frac{n\delta_{i}\mu}{n\sigma^{2}-\delta_{i}\alpha}+\frac{\alpha\delta_{i}}{n\sigma^{2}-\delta_{i}\alpha}\cdot\frac{n\sum_{j=1}^{n}\frac{\delta_{j}}{n\sigma^{2}-\delta_{j}\alpha}}{1-\alpha\sum_{j=1}^{n}\frac{\delta_{j}}{n\sigma^{2}-\delta_{j}\alpha}}\cdot\mu,$
(3.22)

$i=1,\ldots,n.$

###  3.2. Influence of the parameter $\alpha$

We consider two different features of our solution that are affected by the
choice of $\alpha$ which regulates the price impact. Throughout this
subsection, we assume that $\alpha$ satisfies the conditions of Theorem 3.3,
i.e.
$\alpha<\frac{n\sigma^{2}}{2\delta_{\text{max}}}\eqqcolon\alpha_{\text{max}}$,
where $\delta_{\text{max}}\coloneqq\max\\{\delta_{1},\ldots,\delta_{n}\\}$ and

$\hat{s}(\alpha)\coloneqq\sum_{j=1}^{n}\frac{n\alpha\delta_{j}}{(n+\theta_{j})(n\sigma^{2}-\delta_{j}\alpha)}+\hat{\theta}\neq
1.$

Indeed, it is possible to show that there exists a unique
$\alpha_{0}\in(0,\alpha_{\text{max}})$ such that $\hat{s}(\alpha_{0})=1.$ This
can be seen as follows: First $\alpha\mapsto\hat{s}(\alpha)$ is strictly
increasing and continuous on $(-\infty,\alpha_{\text{max}}].$ Further, we have
$\hat{s}(0)=\hat{\theta}<1$ and $\hat{s}(\alpha_{\text{max}})>1.$ Thus, the
intermediate value theorem implies the statement. We have to exclude this
$\alpha_{0}$ from our considerations. The specific value of $\alpha_{0}$ does
not depend on the type of the agent. It is the same for all investors.

At first, we consider the impact of the choice of $\alpha$ on the optimal
strategy of agent $i$, i.e. the $i$-th entry $\pi^{i,*}$ of the Nash
equilibrium. At first, it can be easily shown that $\pi^{i,*}>0$,
$i=1,\ldots,n,$ if and only if $\alpha<\alpha_{0}$. Moreover, we can compute
the derivative of $\pi^{i,*}$ with respect to $\alpha$ and deduce that it is
strictly positive on $(-\infty,\alpha_{\text{max}})\setminus\\{\alpha_{0}\\}.$
Note however that $\pi^{i,*}$ is only piecewise increasing on
$(-\infty,\alpha_{0})$ and $(\alpha_{0},\alpha_{\text{max}})$ due to the
discontinuity located at $\alpha_{0}$.

The second property of $\alpha$ we want to consider is the influence on the
equilibrium stock price $(S_{t}^{*})_{t\in[0,T]}$ that is obtained by
inserting the Nash equilibrium from Theorem 3.3 into the stock price dynamic.
At first, it is not clear whether $S_{t}^{*}$ is smaller or larger than the
stock price with drift $\mu$ and volatility $\sigma$ without the $n$
investors’ impact. It obviously suffices to consider the drift of $S_{t}^{*}$
compared to $\mu$ since the volatility does not depend on the $n$ agents’
investments.

From the proof of Theorem 3.3, we know that the arithmetic mean of the
components of the Nash equilibrium is given by

$\frac{1}{n}\sum_{j=1}^{n}\pi^{j,*}=\frac{(\hat{s}(\alpha)-\hat{\theta})\cdot\mu/\alpha}{1-\hat{s}(\alpha)}.$
(3.23)

Therefore, the drift of $S_{t}^{*}$ is equal to

$\displaystyle\mu+\frac{\alpha}{n}\sum_{j=1}^{n}\pi^{j,*}=\mu\cdot\frac{\hat{s}(\alpha)-\hat{\theta}}{1-\hat{s}(\alpha)}.$

Since the constant $\frac{\hat{s}(\alpha)-\hat{\theta}}{1-\hat{s}(\alpha)}$ is
strictly positive if and only if $\alpha\in(0,\alpha_{0})$, we deduce that the
drift of $S_{t}^{*}$ is larger (smaller) than $\mu$ if and only if
$\alpha\in(0,\alpha_{0})$
($\alpha\in(-\infty,0)\cup(\alpha_{0},\alpha_{\text{max}})$). Moreover, since
we already saw that $\sum_{j=1}^{n}\pi^{j,*}$ is piecewise increasing in terms
of $\alpha$, we infer that $S_{t}^{*}$ is also piecewise increasing in terms
of $\alpha$ on $(-\infty,\alpha_{0})$ and $(\alpha_{0},\alpha_{\text{max}})$.

Figure 1 shows the behavior of $\pi^{1,*}$ (cf. Theorem 3.3) in terms of
$\alpha$ for the two different risk aversion parameters $\delta=1$ and
$\delta=4$. The vertical lines (dotted) show the discontinuity $\alpha_{0}$
for the different parameter choices. The gray horizontal line (dashed) marks
the value zero while the orange and blue horizontal lines (dashed) display the
optimal solution to the classical problem of maximizing expected terminal
wealth under CARA utility without price impact and relative concerns given by
$\delta\mu\sigma^{-2}$ (Merton ratio). There are two ways the agents may try
to influence the stock price to their advantage. By buying the stock they may
jointly increase the stock value and thus raise their utility or by jointly
short-selling the stock and thus decrease its value. Our analysis shows that
in case of a small price impact the agents go for the first option and in case
of a larger price impact they go for the latter option. Of course, this is
only true under the exponential utility where short-selling is no problem.
Under an increasingly negative price impact, the investors engage less in the
financial market.

Figure 1. Illustration of $\pi^{1,*}$ from Theorem 3.3 in terms of
$\alpha\in(-0.04,\alpha_{\text{max}})$ for $n=12$, $\mu=0.03$, $\sigma=0.2$
and $\alpha_{\text{max}}=n\sigma^{2}/8$.
$\theta_{1}=0.3,\,\delta_{1}\in\\{1,4\\}$ and the parameters $\theta_{j}$ and
$\delta_{j}$, $j\geq 2$ are increasing from $0$ to $1$ with step size $0.1$
and from $0.5$ to $2.7$ by step size $0.2$, respectively.

##  4\. Optimization under CARA utility with nonlinear price impact

At the beginning of Section 2, we assumed that the price impact of the $n$
investors in our financial market is given as a linear function in terms of
the arithmetic mean of the $n$ investors’ strategies. While the use of the
arithmetic mean seems intuitive and reasonable since we assumed that investors
are ’small’, one could ask whether using a different function than a linear
one would lead to a different optimization problem and hence also a different
optimal investment.

In Theorem 3.3 we were able to find an explicit solution to the associated
multi-objective portfolio optimization problem using exponential utility (if
the parameters are chosen accordingly). The proof highly relies on the
linearity of the price impact, so we will not be able to give an explicit
solution to the resulting optimization problem in general. However, we will
discuss that using a function $g$ that grows superlinearly yields a problem
that does not have a finite optimal solution while a function $g$ that grows
sublinearly yields a finite optimal solution. If $g$ is a linear function, it
depends on the parameter choices whether or not there exists a finite optimal
solution (cf. Theorem 3.3). Since, in the linear case, the optimally invested
amount is close to zero for decreasing price impact (i.e. if $\alpha<0$, see
Theorem 3.3 and Figure 1) we only consider price impact which is increasing in
order size.

More explicitly, the price impact will now be modeled by some strictly
increasing and continuous function  $g:\mathbb{R}\to\mathbb{R}$ with $g(0)=0$.
Therefore, the stock price process will be given as the solution to the SDE

$\mathrm{d}S_{t}=S_{t}\left(\left(\mu+g\left(\bar{\pi}_{t}\right)\right)\mathrm{d}t+\sigma\mathrm{d}W_{t}\right),\,S_{0}=1,$
(4.1)

which is, of course, still just a stochastic exponential.

As before, we have to restrict ourselves to constant Nash equilibria.
Therefore, from the view of investor $i$, we can rewrite the expression
$g(\bar{\pi}_{t})$ in the previous SDE as follows

$g(\bar{\pi}_{t})=g\left(\frac{1}{n}\sum_{j=1}^{n}\pi^{j}_{t}\right)=g\left(\frac{1}{n}\pi^{i}_{t}+\frac{1}{n}\sum_{j\neq
i}\pi^{j}\right)\eqqcolon\widetilde{g}(\pi^{i}_{t}),$ (4.2)

where $\widetilde{g}(p)\coloneqq g\left(\frac{p}{n}+\frac{1}{n}\sum_{j\neq
i}\pi^{j}\right),\,p\in\mathbb{R}$. Of course, we assumed that the strategies
$\pi^{j}$, $j\neq i$, of the other investors are fixed, constant and
deterministic. It also follows that $\widetilde{g}$ is still strictly
increasing and satisfies $\widetilde{g}\left(-\sum_{j\neq i}\pi^{j}\right)=0$.

Again, strategies $\pi^{i}$ are restricted to the set $\mathcal{A}$ of
admissible strategies. In the following, we will prove that

$\begin{cases}&\sup_{\pi^{i}\in\mathcal{A}}\mathbb{E}\left[-\exp\left(-\frac{1}{\delta_{i}}\left(X_{T}^{i,\pi^{i}}-\frac{\theta_{i}}{n}\sum_{j\neq
i}X_{T}^{j,\pi^{j}}\right)\right)\right],\\\ \text{s.t.
}&X^{i,\pi^{i}}_{T}=x_{0}^{i}+\displaystyle\int_{0}^{T}\pi^{i}_{t}\left(\left(\mu+g(\bar{\pi}_{t})\right)\mathrm{d}t+\sigma\mathrm{d}W_{t}\right),\end{cases}$
(4.3)

has an optimal solution if $g$ grows sublinearly and there exists no optimal
strategy if $g$ grows superlinearly. Here, $\pi^{j}$, $j\neq i$, are assumed
to be constant.

The following theorem summarizes the first assertion of this section, which
treats the case that $g$ grows superlinearly.

######  Proposition 4.1.

If $\lim_{x\to\pm\infty}\frac{g(x)}{x}=\infty$, (4.3) does not have an optimal
solution.

###### Proof.

In order to prove that (4.3) does not have an optimal solution, we will prove
that, even if we only consider constant strategies for agent $i$, the optimal
value is zero and the associated strategy is infinite. If $\pi^{j}$ is
constant for all $j\in\\{1,\ldots,n\\}$, we obtain

$\displaystyle X_{T}^{i,\pi^{i}}-\frac{\theta_{i}}{n}\sum_{j\neq
i}X_{T}^{j,\pi^{j}}$ $\displaystyle=$ $\displaystyle
x_{0}^{i}-\frac{\theta_{i}}{n}\sum_{j\neq
i}x_{0}^{j}+\Big{(}\pi^{i}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}(\mu+g(\bar{\pi}))T+\Big{(}\pi^{i}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}\sigma W_{T}$ $\displaystyle\eqqcolon$ $\displaystyle
y_{0}^{i}+\mu(\pi^{i})T+\sigma(\pi^{i})W_{T}.$

Hence, for fixed $\pi^{j}$, $j=1,\ldots,n$, the value of the objective
function in (4.3) is given by

$\displaystyle\mathbb{E}\Big{[}-\exp\Big{(}-\frac{1}{\delta_{i}}\Big{(}y_{0}^{i}+\mu(\pi^{i})T+\sigma(\pi^{i})W_{T}\Big{)}\Big{)}\Big{]}$
(4.4) $\displaystyle=$
$\displaystyle-\exp\Big{(}-\frac{1}{\delta_{i}}y_{0}^{i}\Big{)}\cdot\exp\Big{(}-\frac{1}{\delta_{i}}\Big{(}\mu(\pi^{i})-\frac{\sigma(\pi^{i})^{2}}{2\delta_{i}}\Big{)}T\Big{)}.$
(4.5)

Thus, maximizing the objective function of $\eqref{eq:problem_g}$ with respect
to constant strategies $\pi^{i}$ is equivalent to maximizing
$\mu(\pi^{i})-\frac{\sigma(\pi^{i})^{2}}{2\delta_{i}}.$ Reinserting the
definition of $\mu(\pi^{i})$ and $\sigma(\pi^{i})$ yields

$\displaystyle\mu(\pi^{i})-\frac{\sigma(\pi^{i})^{2}}{2\delta_{i}}$
$\displaystyle=$
$\displaystyle\pi^{i}g(\bar{\pi})-\frac{\sigma^{2}}{2\delta_{i}}(\pi^{i})^{2}+\pi^{i}\Big{(}\mu+\frac{\sigma^{2}\theta_{i}}{n\delta_{i}}\sum_{j\neq
i}\pi^{j}\Big{)}-\frac{\theta_{i}}{n}g(\bar{\pi})\sum_{j\neq
i}\pi^{j}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{(}\mu+\frac{\sigma^{2}\theta_{i}}{2n\delta_{i}}\sum_{j\neq
i}\pi^{j}\Big{)}$

which converges to $\infty$ if $\pi^{i}$ converges to $\pm\infty$ using the
assumption that $g$ grows superlinearly. Therefore,

$\displaystyle 0\geq$
$\displaystyle\sup_{\pi^{i}\in\mathcal{A}}\mathbb{E}\Big{[}-\exp\Big{(}-\frac{1}{\delta_{i}}\Big{(}X_{T}^{i,\pi^{i}}-\frac{\theta_{i}}{n}\sum_{j\neq
i}X_{T}^{j,\pi^{j}}\Big{)}\Big{)}\Big{]}$ $\displaystyle\geq$
$\displaystyle\sup_{\begin{subarray}{c}\pi^{i}\in\mathcal{A}\\\ \pi^{i}\text{
constant}\end{subarray}}\mathbb{E}\Big{[}-\exp\Big{(}-\frac{1}{\delta_{i}}\Big{(}X_{T}^{i,\pi^{i}}-\frac{\theta_{i}}{n}\sum_{j\neq
i}X_{T}^{j,\pi^{j}}\Big{)}\Big{)}\Big{]}=0.$

Hence, the optimal value of (4.3) is zero, which implies that the argument
inside the exponential function needs to be infinite. Hence, the problem does
not have an optimal solution. ∎

As a result we cannot hope for a Nash equilibrium in this case. Now we can
consider the case of sublinear growth of $g$. Hence, we assume that

$\lim_{x\to\pm\infty}\frac{g(x)}{x}=0.$

Then we can prove that there exists an optimal policy for (4.3). In order to
do so, let $a^{*}$ be a maximum point of

$a\mapsto\Big{(}a-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}\Big{(}\mu+\widetilde{g}(a)\Big{)}-\frac{\sigma^{2}}{2\delta_{i}}\Big{(}a-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}.$ (4.6)

Due to our assumption on $g$, a maximum point $a^{*}$ exists and is finite.
Then we obtain the following result.

######  Proposition 4.2.

If $\lim_{x\to\pm\infty}\frac{g(x)}{x}=0$, an optimal strategy for (4.3) is
given by $\pi_{t}^{i}\equiv a^{*}$, where $a^{*}$ is the maximum point from
(4.6).

###### Proof.

For the moment we restrict to bounded strategies $(\pi_{t}^{i}),$ i.e. there
exists a constant $K>0$ such that $|\pi_{t}^{i}|\leq K$ for all $t\in[0,T].$
For constants $\pi^{j}$, we obtain

$\displaystyle-\frac{1}{\delta_{i}}\Big{(}X_{T}^{i,\pi^{i}}-\frac{\theta_{i}}{n}\sum_{j\neq
i}X_{T}^{j,\pi^{j}}\Big{)}$ $\displaystyle=$
$\displaystyle-\frac{1}{\delta_{i}}\Big{(}x_{0}^{i}-\frac{\theta_{i}}{n}\sum_{j\neq
i}x_{0}^{j}\Big{)}-\frac{1}{\delta_{i}}\Bigg{(}\int_{0}^{T}\Big{(}\pi^{i}_{t}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}(\mu+g(\bar{\pi}_{t}))\mathrm{d}t+\sigma\int_{0}^{T}\Big{(}\pi^{i}_{t}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}\mathrm{d}W_{t}\Bigg{)}$
$\displaystyle-\frac{\sigma^{2}}{2\delta_{i}^{2}}\int_{0}^{T}\Big{(}\pi^{i}_{t}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}\mathrm{d}t+\frac{\sigma^{2}}{2\delta_{i}^{2}}\int_{0}^{T}\Big{(}\pi^{i}_{t}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}\mathrm{d}t.$

Now define a new probability measure $\mathbb{Q}$ by

$\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\mathbb{P}}=\exp\Bigg{(}-\frac{\sigma^{2}}{2\delta_{i}^{2}}\int_{0}^{T}\Big{(}\pi^{i}_{t}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}\mathrm{d}t-\frac{\sigma}{\delta_{i}}\int_{0}^{T}\Big{(}\pi^{i}_{t}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}\mathrm{d}W_{t}\Bigg{)}.$

Note that this expression is a density since $\pi_{t}^{i}$ is bounded. Thus,
we can write the (negative) target function of (4.3) as

$\displaystyle\mathbb{E}\left[\exp\left(-\frac{1}{\delta_{i}}\left(X_{T}^{i,\pi^{i}}-\frac{\theta_{i}}{n}\sum_{j\neq
i}X_{T}^{j,\pi^{j}}\right)\right)\right]=\exp\Bigg{(}-\frac{1}{\delta_{i}}\Big{(}x_{0}^{i}-\frac{\theta_{i}}{n}\sum_{j\neq
i}x_{0}^{j}\Big{)}\Bigg{)}$
$\displaystyle\cdot\mathbb{E}_{\mathbb{Q}}\left[\exp\Bigg{(}-\frac{1}{\delta_{i}}\Bigg{(}\int_{0}^{T}\Big{(}\pi^{i}_{t}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}(\mu+g(\bar{\pi}_{t}))-\frac{\sigma^{2}}{2\delta_{i}}\Big{(}\pi^{i}_{t}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}\mathrm{d}t\Bigg{)}\Bigg{)}\right].$

But now in order to minimize the expectation we can do this pointwise under
the integral which leads to maximizing (4.6). Since the maximizing point is
not at the boundary, the assumption of bounded policies is no restriction.
Thus, we have solved the problem.

∎

Whether or not a Nash equilibrium exists in this case depends now on the
precise choice of $g.$

######  Remark 4.3.

The structure of the function (4.6) considered in the proof of Proposition 4.2
implies that there exist at least one and at most two global maxima.

##  5\. Optimization under CRRA utility with linear price impact

In this section, we assume that agents use CRRA utility functions (power or
logarithmic) to measure their preferences. Hence, we let

$U_{i}:(0,\infty)\rightarrow\mathbb{R},\,x\mapsto\begin{cases}\left(1-\frac{1}{\delta_{i}}\right)^{-1}x^{1-\frac{1}{\delta_{i}}},\,&\delta_{i}\neq
1,\\\ \ln(x),\,&\delta_{i}=1\end{cases}$ (5.1)

for some preference parameter $\delta_{i}>0$, $i=1,\ldots,n$. By $\ln(\cdot)$
we denote the natural logarithm.

While using CRRA utility functions, it is mathematically more convenient to
optimize the invested fraction of wealth instead of the amount or number of
shares. Thus, throughout this subsection, $\pi^{i}_{t}$, $i=1,\ldots,n,$
denotes the fraction of agent $i$’s wealth invested into the risky stock at
some time $t\in[0,T]$. The wealth process of agent $i$ is therefore given as
the solution to the SDE

$\mathrm{d}X_{t}^{i,\pi^{i}}=X_{t}^{i,\pi^{i}}\pi^{i}_{t}\left(\left(\mu+\alpha\bar{\pi}_{t}\right)\mathrm{d}t+\sigma\mathrm{d}W_{t}\right),\,X_{0}^{i,\pi^{i}}=x_{0}^{i}.$
(5.2)

Similar to Section 3 in [29], we include the strategic interaction component
into our problem by inserting the product of agent $i$’s and a weighted
geometric mean of the other agents’ terminal wealth into the expected utility
criterion of the portfolio optimization problem. Therefore, the portfolio
optimization problem of agent $i$ is given by

$\begin{cases}&\sup_{\pi^{i}\in\mathcal{A}}\,\mathbb{E}\Bigg{[}\frac{\delta_{i}}{\delta_{i}-1}\Bigg{(}X_{T}^{i,\pi^{i}}\Big{(}\prod_{j\neq
i}X_{T}^{j,\pi^{j}}\Big{)}^{-\frac{\theta_{i}}{n}}\Bigg{)}^{\frac{\delta_{i}-1}{\delta_{i}}}\Bigg{]},\\\
\text{s.t.
}&\mathrm{d}X_{t}^{i,\pi^{i}}=X_{t}^{i,\pi^{i}}\pi^{i}_{t}\left(\left(\mu+\alpha\bar{\pi}_{t}\right)\mathrm{d}t+\sigma\mathrm{d}W_{t}\right),\,X_{0}^{i,\pi^{i}}=x_{0}^{i}.\end{cases}$
(5.3)

In order to find an explicit solution for the Nash equilibrium we need to
restrict ourselves to constant strategies. Since the reduction to some
auxiliary problem containing only one instead of all $n$ agents is not
possible in this setting, we need to directly solve the best response problem
in order to determine the Nash equilibrium. Then the unique constant Nash
equilibrium is given in the following theorem.

######  Theorem 5.1.

Assume that the following assumptions hold

  1. a)

$(n+\theta_{i})\left(n\sigma^{2}-\delta_{i}\alpha\right)-\theta_{i}\delta_{i}\sigma^{2}>0$
for all $i=1,\ldots,n$,

  2. b)

$n\sigma^{2}-2\delta_{i}\alpha>0$ for all $i=1,\ldots,n$,

  3. c)

$1-\sum_{j=1}^{n}\frac{(n-\theta_{j})\alpha\delta_{j}-n\theta_{j}(\delta_{j}-1)\sigma^{2}}{(n+\theta_{j})(n\sigma^{2}-\alpha\delta_{j})-n\theta_{j}\delta_{j}\sigma^{2}}\neq
0$.

Then the unique (up to modifications) constant Nash equilibrium to (5.3) in
terms of invested fractions is given by

$\displaystyle\pi^{i,*}=$
$\displaystyle\frac{n^{2}\delta_{i}\mu}{(n+\theta_{i})(n\sigma^{2}-\delta_{i}\alpha)-n\theta_{i}\delta_{i}\sigma^{2}}+\frac{(n-\theta_{i})\alpha\delta_{i}-n\theta_{i}(\delta_{i}-1)\sigma^{2}}{(n+\theta_{i})(n\sigma^{2}-\delta_{i}\alpha)-n\theta_{i}\delta_{i}\sigma^{2}}$
(5.4)
$\displaystyle\cdot\Bigg{(}1-\sum_{j=1}^{n}\frac{(n-\theta_{j})\alpha\delta_{j}-n\theta_{j}(\delta_{j}-1)\sigma^{2}}{(n+\theta_{j})(n\sigma^{2}-\alpha\delta_{j})-n\theta_{j}\delta_{j}\sigma^{2}}\Bigg{)}^{-1}\sum_{j=1}^{n}\frac{n^{2}\delta_{j}\mu}{(n+\theta_{j})(n\sigma^{2}-\delta_{j}\alpha)-n\theta_{j}\delta_{j}\sigma^{2}}.$
(5.5)

###### Proof.

Let $i\in\\{1,\ldots,n\\}$ be arbitrary but fixed and assume that the other
agents use constant strategies $\pi^{j}$, $j\neq i$, which will also be
assumed to be arbitrary but fixed. Now define the stochastic process
$(Y_{t}^{-i})_{t\in[0,T]}$ by $Y_{t}^{-i}=\prod_{j\neq i}X_{t}^{j,\pi^{j}}$,
$t\in[0,T]$.

At first, we determine the dynamics of the process
$\Big{(}(Y_{t}^{-i})^{-\frac{\theta_{i}}{n}}\Big{)}_{t\in[0,T]}$. To simplify
our calculations, we first consider the logarithm of this process. We obtain

$\ln\Big{(}(Y_{t}^{-i})^{-\frac{\theta_{i}}{n}}\Big{)}=-\frac{\theta_{i}}{n}\sum_{j\neq
i}\ln\Big{(}X_{t}^{j,\pi^{j}}\Big{)}$ (5.6)

for $t\in[0,T]$. The Itô-Doeblin formula implies

$\displaystyle\mathrm{d}\ln\Big{(}X_{t}^{j,\pi^{j}}\Big{)}$
$\displaystyle=\pi^{j}((\mu+\alpha\bar{\pi}_{t})\mathrm{d}t+\sigma\mathrm{d}W_{t})-\frac{\sigma^{2}}{2}(\pi^{j})^{2}\mathrm{d}t.$

Hence, using (5.6),

$\displaystyle\mathrm{d}\Big{(}\ln\Big{(}(Y_{t}^{-i})^{-\frac{\theta_{i}}{n}}\Big{)}\Big{)}=-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}((\mu+\alpha\bar{\pi}_{t})\mathrm{d}t+\sigma\mathrm{d}W_{t})+\frac{\theta_{i}}{n}\frac{\sigma^{2}}{2}\sum_{j\neq
i}(\pi^{j})^{2}\mathrm{d}t.$

Using the Itô-Doeblin formula a second time then yields

$\displaystyle\mathrm{d}\Big{(}(Y_{t}^{-i})^{-\frac{\theta_{i}}{n}}\Big{)}=\mathrm{d}\Big{(}\exp\Big{(}\ln\Big{(}(Y_{t}^{-i})^{-\frac{\theta_{i}}{n}}\Big{)}\Big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\,(Y_{t}^{-i})^{-\frac{\theta_{i}}{n}}\left(-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}((\mu+\alpha\bar{\pi}_{t})\mathrm{d}t+\sigma\mathrm{d}W_{t})+\frac{\sigma^{2}}{2}\frac{\theta_{i}}{n}\sum_{j\neq
i}(\pi^{j})^{2}\mathrm{d}t+\frac{\sigma^{2}}{2}\Big{(}\frac{\theta_{i}}{n}\Big{)}^{2}\Big{(}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}\mathrm{d}t\right).$

Hence, we can use partial integration to find the dynamics of the process
associated to the argument of the utility function in (5.3):

$\displaystyle\mathrm{d}\Big{(}X_{t}^{i,\pi^{i}}(Y_{t}^{-i})^{-\frac{\theta_{i}}{n}}\Big{)}$
$\displaystyle=$ $\displaystyle
X_{t}^{i,\pi^{i}}\left(Y_{t}^{-i}\right)^{-\frac{\theta_{i}}{n}}\Bigg{(}-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}((\mu+\alpha\bar{\pi}_{t})\mathrm{d}t+\sigma\mathrm{d}W_{t})+\frac{\sigma^{2}}{2}\frac{\theta_{i}}{n}\sum_{j\neq
i}(\pi^{j})^{2}\mathrm{d}t+\frac{\sigma^{2}}{2}\Big{(}\frac{\theta_{i}}{n}\Big{)}^{2}\Big{(}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}\mathrm{d}t$
$\displaystyle+\pi^{i}_{t}((\mu+\alpha\bar{\pi}_{t})\mathrm{d}t+\sigma\mathrm{d}W_{t})-\frac{\theta_{i}}{n}\sigma^{2}\pi^{i}_{t}\sum_{j\neq
i}\pi^{j}\mathrm{d}t\Bigg{)}$ $\displaystyle=$ $\displaystyle
X_{t}^{i,\pi^{i}}\left(Y_{t}^{-i}\right)^{-\frac{\theta_{i}}{n}}\Bigg{(}\pi^{i}_{t}(\mu\mathrm{d}t+\sigma\mathrm{d}W_{t})+\frac{\alpha}{n}(\pi^{i}_{t})^{2}\mathrm{d}t+\Big{(}\frac{\alpha}{n}-\frac{\alpha\theta_{i}}{n^{2}}-\frac{\theta_{i}}{n}\sigma^{2}\Big{)}\pi^{i}_{t}\sum_{j\neq
i}\pi^{j}\mathrm{d}t$ $\displaystyle+\Big{(}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}\frac{\theta_{i}}{n}\Big{(}\frac{\theta_{i}}{2n}\sigma^{2}-\frac{\alpha}{n}\Big{)}\mathrm{d}t-\frac{\theta_{i}}{n}\sum_{j\neq
i}\pi^{j}(\mu\mathrm{d}t+\sigma\mathrm{d}W_{t})+\frac{\theta_{i}}{2n}\sigma^{2}\sum_{j\neq
i}(\pi^{j})^{2}\mathrm{d}t\Bigg{)},$

where we used the last step to separate the summands depending on $\pi^{i}$
from the ones that do not depend on $\pi^{i}$. Now a simple caluculation
yields that we can rewrite

$X_{t}^{i,\pi^{i}}\cdot\left(Y_{t}^{-i}\right)^{-\frac{\theta_{i}}{n}}=\widetilde{X}^{i,\pi^{i}}_{t}\cdot\left(\widetilde{Y}^{-i}_{t}\right)^{-\frac{\theta_{i}}{n}},$
(5.7)

where the process $\widetilde{Y}^{-i}$ does not depend on $\pi^{i}$. More
specifically, the dynamics of $\widetilde{X}^{i,\pi^{i}}$ and
$\widetilde{Y}^{-i}$ are given by

$\displaystyle\mathrm{d}\widetilde{X}^{i,\pi^{i}}_{t}$
$\displaystyle=\widetilde{X}^{i,\pi^{i}}_{t}\pi^{i}_{t}\Bigg{(}\Big{(}\mu+\frac{\alpha}{n}\pi^{i}_{t}+\frac{\alpha}{n}\Big{(}1-\frac{\theta_{i}}{n}\Big{)}\sum_{j\neq
i}\pi^{j}\Big{)}\mathrm{d}t+\sigma\mathrm{d}W_{t}\Bigg{)},$
$\displaystyle\mathrm{d}\widetilde{Y}^{-i}_{t}$
$\displaystyle=\widetilde{Y}^{-i}_{t}\Bigg{(}\sum_{j\neq
i}\pi^{j}\Big{(}\Big{(}\mu+\frac{\alpha}{n}\sum_{j\neq
i}\pi^{j}\Big{)}\mathrm{d}t+\sigma\mathrm{d}W_{t}\Big{)}+\frac{\sigma^{2}}{2}\Big{(}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}\mathrm{d}t-\frac{\sigma^{2}}{2}\sum_{j\neq
i}(\pi^{j})^{2}\mathrm{d}t\Bigg{)}$

with
$\widetilde{X}_{0}^{i,\pi^{i}}=x_{0}^{i},\,\widetilde{Y}_{0}^{-i}=\prod_{j\neq
i}x_{0}^{j}.$

The previously introduced processes $\widetilde{X}^{i,\pi^{i}}$ and
$\widetilde{Y}^{-i}$ simplify the derivation of the HJB-equation in this
setting. In order to derive a HJB-equation we define the following value
function ($t\in[0,T],\,x,y\in(0,\infty)$)

$\displaystyle v(t,x,y)$
$\displaystyle\coloneqq\sup_{\pi^{i}\in\mathcal{A}}\mathbb{E}\left[\frac{\delta_{i}}{\delta_{i}-1}\left(\widetilde{X}^{i,\pi^{i}}_{T}\left(\widetilde{Y}^{-i}_{T}\right)^{-\frac{\theta_{i}}{n}}\right)^{\frac{\delta_{i}-1}{\delta_{i}}}\Bigg{|}\widetilde{X}^{i,\pi^{i}}_{t}=x,\,\widetilde{Y}^{-i}_{t}=y\right].$

We can derive a HJB equation using classical arguments (see e.g. [32], [6],
[18]) and obtain

$\displaystyle 0=$ $\displaystyle v_{t}+yv_{y}\Bigg{\\{}\sum_{j\neq
i}\pi^{j}\Big{(}\mu+\frac{\alpha}{n}\sum_{j\neq
i}\pi^{j}\Big{)}+\frac{\sigma^{2}}{2}\Big{(}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}-\frac{\sigma^{2}}{2}\sum_{j=1}^{n}(\pi^{j})^{2}\Bigg{\\}}+\frac{\sigma^{2}}{2}y^{2}v_{yy}\Big{(}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}$
$\displaystyle+\sup_{\pi^{i}\in\mathbb{R}}\Bigg{\\{}xv_{x}\pi^{i}\mu+\left(\frac{\alpha}{n}\left(1-\frac{\theta_{i}}{n}\right)xv_{x}+\sigma^{2}xyv_{xy}\right)\pi^{i}\sum_{j\neq
i}\pi^{j}+\Big{(}\frac{\alpha}{n}xv_{x}+\frac{\sigma^{2}}{2}x^{2}v_{xx}\Big{)}(\pi^{i})^{2}\Bigg{\\}},$

where we omitted the arguments of $v$ and its derivatives for notational
convenience. The supremum is attained at

$\pi^{i,*}=-\frac{xv_{x}\mu+\Big{(}\frac{\alpha}{n}\Big{(}1-\frac{\theta_{i}}{n}\Big{)}xv_{x}+\sigma^{2}xyv_{xy}\Big{)}\sum_{j\neq
i}\pi^{j}}{2\Big{(}\frac{\alpha}{n}xv_{x}+\frac{\sigma^{2}}{2}x^{2}v_{xx}\Big{)}},$
(5.8)

which reduces the HJB equation to the PDE

$\displaystyle 0=$ $\displaystyle v_{t}+yv_{y}\Bigg{\\{}\sum_{j\neq
i}\pi^{j}\Big{(}\mu+\frac{\alpha}{n}\sum_{j\neq
i}\pi^{j}\Big{)}+\frac{\sigma^{2}}{2}\Big{(}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}-\frac{\sigma^{2}}{2}\sum_{j=1}^{n}(\pi^{j})^{2}\Bigg{\\}}+\frac{\sigma^{2}}{2}y^{2}v_{yy}\Big{(}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}$
$\displaystyle-\frac{\Big{(}xv_{x}\mu+\Big{(}\frac{\alpha}{n}\Big{(}1-\frac{\theta_{i}}{n}\Big{)}xv_{x}+\sigma^{2}xyv_{xy}\Big{)}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}}{4\Big{(}\frac{\alpha}{n}xv_{x}+\frac{\sigma^{2}}{2}x^{2}v_{xx}\Big{)}}$

with terminal condition

$v(T,x,y)=\frac{\delta_{i}}{\delta_{i}-1}\Big{(}xy^{-\frac{\theta_{i}}{n}}\Big{)}^{\frac{\delta_{i}-1}{\delta_{i}}},\,x,y>0.$
(5.9)

For the solution, we make the following ansatz for $v$

$v(t,x,y)=f(t)\frac{\delta_{i}}{\delta_{i}-1}\Big{(}xy^{-\frac{\theta_{i}}{n}}\Big{)}^{\frac{\delta_{i}-1}{\delta_{i}}}$
(5.10)

for some continuously differentiable function $f:[0,T]\to(0,\infty)$ with
$f(T)=1$. Hence, inserting the ansatz for $v$ reduces the HJB equation to the
ODE

$\displaystyle 0=f^{\prime}(t)+\rho f(t)$ (5.11)

with terminal condition $f(T)=1$, where we defined the constant

$\displaystyle\rho=$
$\displaystyle-\frac{\theta_{i}}{n}\frac{\delta_{i}-1}{\delta_{i}}\Bigg{(}\sum_{j\neq
i}\pi^{j}\Big{(}\mu+\frac{\alpha}{n}\sum_{j\neq
i}\pi^{j}\Big{)}+\frac{\sigma^{2}}{2}\Big{(}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}-\frac{\sigma^{2}}{2}\sum_{j=1}^{n}(\pi^{j})^{2}\Bigg{)}$
$\displaystyle+\frac{\sigma^{2}}{2}\frac{\theta_{i}}{n}\frac{\delta_{i}-1}{\delta_{i}}\Big{(}1+\frac{\theta_{i}}{n}\frac{\delta_{i}-1}{\delta_{i}}\Big{)}\Big{(}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}-\frac{\Big{(}n\delta_{i}\mu+\Big{(}\alpha\delta_{i}\Big{(}1-\frac{\theta_{i}}{n}\Big{)}-\sigma^{2}\theta_{i}(\delta_{i}-1)\Big{)}\sum_{j\neq
i}\pi^{j}\Big{)}^{2}}{4\Big{(}n\sigma^{2}-2\alpha\delta_{i}\Big{)}^{2}}.$

The unique solution to (5.11) is given by

$f(t)=e^{\rho(T-t)},\,t\in[0,T].$ (5.12)

Inserting the solution $v$ of the HJB equation into the maximizer $\pi^{i,*}$
yields

$\pi^{i,*}=\frac{n\delta_{i}\mu+\Big{(}\alpha\delta_{i}\Big{(}1-\frac{\theta_{i}}{n}\Big{)}-\sigma^{2}\theta_{i}(\delta_{i}-1)\Big{)}\sum_{j\neq
i}\pi^{j}}{n\sigma^{2}-2\alpha\delta_{i}}.$ (5.13)

Application of a standard verification theorem (see for example [32], [18],
[6] for similar arguments) implies that $\pi^{i,*}$ is the unique solution to
the best response problem. Moreover, since $\pi^{j}$ were assumed to be
constant, $\pi^{i,*}$ is constant, as well. To conclude the proof, we need to
solve the system of linear equations defined by (5.13) for $i=1,\ldots,n$. By
adding an appropriate multiple of $\pi^{i}$ on both sides and simplifying the
equation, we obtain

$\pi^{i}=\frac{n\delta_{i}\mu}{(n+\theta_{i})\Big{(}\sigma^{2}-\frac{\delta_{i}\alpha}{n}\Big{)}-\sigma^{2}\theta_{i}\delta_{i}}+\frac{\alpha\delta_{i}\Big{(}1-\frac{\theta_{i}}{n}\Big{)}-\sigma^{2}\theta_{i}(\delta_{i}-1)}{(n+\theta_{i})\Big{(}\sigma^{2}-\frac{\delta_{i}\alpha}{n}\Big{)}-\sigma^{2}\theta_{i}\delta_{i}}\sum_{j=1}^{n}\pi^{j}.$
(5.14)

Summing over all $i\in\\{1,\ldots,n\\}$ on both sides and solving for
$\sum_{j=1}^{n}\pi^{j}$ then yields

$\sum_{j=1}^{n}\pi^{j}=\Bigg{(}1-\sum_{j=1}^{n}\frac{\alpha\delta_{j}\Big{(}1-\frac{\theta_{j}}{n}\Big{)}-\sigma^{2}\theta_{j}(\delta_{j}-1)}{(n+\theta_{j})\Big{(}\sigma^{2}-\frac{\delta_{j}\alpha}{n}\Big{)}-\sigma^{2}\theta_{j}\delta_{j}}\Bigg{)}^{-1}\sum_{j=1}^{n}\frac{n\delta_{j}\mu}{(n+\theta_{j})\Big{(}\sigma^{2}-\frac{\delta_{j}\alpha}{n}\Big{)}-\sigma^{2}\theta_{j}\delta_{j}}$
(5.15)

Finally, inserting (5.15) into (5.14) yields the unique constant Nash
equilibrium given by ($i=1,\ldots,n$)

$\displaystyle\pi^{i,*}=$
$\displaystyle\frac{n\delta_{i}\mu}{(n+\theta_{i})\Big{(}\sigma^{2}-\frac{\delta_{i}\alpha}{n}\Big{)}-\sigma^{2}\theta_{i}\delta_{i}}+\frac{\alpha\delta_{i}\Big{(}1-\frac{\theta_{i}}{n}\Big{)}-\sigma^{2}\theta_{i}(\delta_{i}-1)}{(n+\theta_{i})\Big{(}\sigma^{2}-\frac{\delta_{i}\alpha}{n}\Big{)}-\sigma^{2}\theta_{i}\delta_{i}}$
$\displaystyle\cdot\Bigg{(}1-\sum_{j=1}^{n}\frac{\alpha\delta_{j}\Big{(}1-\frac{\theta_{j}}{n}\Big{)}-\sigma^{2}\theta_{j}(\delta_{j}-1)}{(n+\theta_{j})\Big{(}\sigma^{2}-\frac{\delta_{j}\alpha}{n}\Big{)}-\sigma^{2}\theta_{j}\delta_{j}}\Bigg{)}^{-1}\sum_{j=1}^{n}\frac{n\delta_{j}\mu}{(n+\theta_{j})\Big{(}\sigma^{2}-\frac{\delta_{j}\alpha}{n}\Big{)}-\sigma^{2}\theta_{j}\delta_{j}}.$

∎

######  Remark 5.2.

Similar to Remark 3.4, Theorem 5.1 contains the special cases $\alpha=0$ and
$\theta_{i}=0$, $i=1,\ldots,n$. For $\alpha=0$ (no price impact), we deduce

$\pi^{i,*}=\Bigg{(}\frac{n\delta_{i}}{n+\theta_{i}(1-\delta_{i})}+\frac{\theta_{i}(1-\delta_{i})}{n+\theta_{i}(1-\delta_{i})}\cdot\frac{\sum_{j=1}^{n}\frac{n\delta_{j}}{n+\theta_{j}(1-\delta_{j})}}{1-\sum_{j=1}^{n}\frac{\theta_{j}(1-\delta_{j})}{n+\theta_{j}(1-\delta_{j})}}\Bigg{)}\cdot\frac{\mu}{\sigma^{2}},$
(5.16)

$i=1,\ldots,n$. In the special case without relative concerns inside the
objective function, we obtain

$\pi^{i,*}=\frac{n\delta_{i}\mu}{n\sigma^{2}-\delta_{i}\alpha}+\frac{\alpha\delta_{i}}{n\sigma^{2}-\delta_{i}\alpha}\cdot\frac{\sum_{j=1}^{n}\frac{n\delta_{j}\mu}{n\sigma^{2}-\alpha\delta_{j}}}{1-\sum_{j=1}^{n}\frac{\alpha\delta_{j}}{n\sigma^{2}-\alpha\delta_{j}}},$
(5.17)

$i=1,\ldots,n.$ A comparison with Remark 3.4 shows that the Nash equilibria in
the special case of $\theta_{i}=0$ for all $i=1,\ldots,n$ are actually the
same, although $\pi^{i,*}$ represents the invested amount for exponential and
the invested fraction for power utility.

##  6\. Conclusion

In this paper we derive Nash equilibria for agents with relative performance
measures in financial markets with price impact. We show that as long as the
price impact is not more than linear, the individual optimization problems are
well-defined. Whereas without price impact, the agents would always invest a
positive amount in the stock in our model, the situation changes dramatically
when the price impact is beyond a certain threshold. Then the agents massively
short-sell the stock and try to take advantage of falling prices. Thus, a
larger price impact leads to a more aggressive behavior of the agents in our
model.

Acknoledgement: The authors would like to thank Dirk Becherer and Johannes
Muhle-Karbe for helpful discussions and hints to literature.

Statements and Declarations: The authors have no relevant financial or non-
financial interests to disclose.

## References

  * [1] Robert Almgren and Neil Chriss, _Optimal execution of portfolio transactions_ , Journal of Risk 3 (2001), 5–40.
  * [2] Peter Bank and Dietmar Baum, _Hedging and portfolio optimization in financial markets with a large trader_ , Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics 14 (2004), no. 1, 1–18.
  * [3] Suleyman Basak and Dmitry Makarov, _Competition among portfolio managers and asset specialization_ , Paris December 2014, Finance Meeting EUROFIDAI-AFFI Paper (2015).
  * [4] Nicole Bäuerle and Tamara Göll, _Nash equilibria for relative investors via no-arbitrage arguments_ , Mathematical Methods of Operations Research (2022), 1–23.
  * [5] Dimitris Bertsimas and Andrew W Lo, _Optimal control of execution costs_ , Journal of Financial Markets 1 (1998), no. 1, 1–50.
  * [6] Tomas Björk, _Arbitrage theory in continuous time_ , 2. ed., Oxford University Press, 2004.
  * [7] Jean-Philippe Bouchaud, _Price impact_ , arXiv:0903.2428 (2009).
  * [8] Stephen J. Brown, William N. Goetzmann, and James Park, _Careers and survival: Competition and risk in the hedge fund and CTA industry_ , The Journal of Finance 56 (2001), no. 5, 1869–1886.
  * [9] Henrik Cronqvist and Rüdiger Fahlenbrach, _Large shareholders and corporate policies_ , The Review of Financial Studies 22 (2008), no. 10, 3941–3976.
  * [10] Domenico Cuoco and Jakša Cvitanić, _Optimal consumption choices for a ‘large’investor_ , Journal of Economic Dynamics and Control 22 (1998), no. 3, 401–436.
  * [11] Giuliano Curatola, _Portfolio choice of large investors who interact strategically_ , Available at SSRN 3404491 (2019).
  * [12] by same author, _Price impact, strategic interaction and portfolio choice_ , The North American Journal of Economics and Finance 59 (2022), 101594.
  * [13] Jakša Cvitanić and Jin Ma, _Hedging options for a large investor and forward-backward sde’s_ , The Annals of Applied Probability 6 (1996), no. 2, 370–398.
  * [14] Goncalo Dos Reis and Vadim Platonov, _Forward utilities and mean-field games under relative performance concerns_ , From Particle Systems to Partial Differential Equations, Springer, 2019, pp. 227–251.
  * [15] by same author, _Forward utility and market adjustments in relative investment-consumption games of many players_ , arXiv:2012.01235 (2020).
  * [16] Zehra Eksi and Hyejin Ku, _Portfolio optimization for a large investor under partial information and price impact_ , Mathematical Methods of Operations Research 86 (2017), 601–623.
  * [17] Gilles-Edouard Espinosa and Nizar Touzi, _Optimal investment under relative performance concerns_ , Mathematical Finance 25 (2015), no. 2, 221–257.
  * [18] Wendell H Fleming and Halil Mete Soner, _Controlled markov processes and viscosity solutions_ , vol. 25, Springer Science & Business Media, 2006.
  * [19] Guanxing Fu, Xizhi Su, and Chao Zhou, _Mean field exponential utility game: A probabilistic approach_ , arXiv:2006.07684 (2020).
  * [20] Hua He and Harry Mamaysky, _Dynamic trading policies with price impact_ , Journal of Economic Dynamics and Control 29 (2005), no. 5, 891–930.
  * [21] Ruimeng Hu and Thaleia Zariphopoulou, _$n$ -player and mean-field games in It$\hat{o}$-diffusion markets with competitive or homophilous interaction_, arXiv:2106.00581 (2021).
  * [22] Robert A Jarrow, _Market manipulation, bubbles, corners, and short squeezes_ , Journal of Financial and Quantitative Analysis 27 (1992), no. 3, 311–336.
  * [23] by same author, _Derivative security markets, market manipulation, and option pricing theory_ , Journal of Financial and Quantitative Analysis 29 (1994), no. 2, 241–261.
  * [24] Ioannis Karatzas and Steven E. Shreve, _Methods of mathematical finance_ , vol. 39, Springer, 1998.
  * [25] Alexander Kempf and Stefan Ruenzi, _Tournaments in mutual-fund families_ , The Review of Financial Studies 21 (2008), no. 2, 1013–1036.
  * [26] Holger Kraft and Christoph Kühn, _Large traders and illiquid options: Hedging vs. manipulation_ , Journal of Economic Dynamics and Control 35 (2011), no. 11, 1898–1915.
  * [27] Holger Kraft, André Meyer-Wehmann, and Frank Thomas Seifried, _Dynamic asset allocation with relative wealth concerns in incomplete markets_ , Journal of Economic Dynamics and Control 113 (2020), 103857.
  * [28] Daniel Lacker and Agathe Soret, _Many-player games of optimal consumption and investment under relative performance criteria_ , Mathematics and Financial Economics 14 (2020), no. 2, 263–281.
  * [29] Daniel Lacker and Thaleia Zariphopoulou, _Mean field and $n$-agent games for optimal investment under relative performance criteria_, Mathematical Finance 29 (2019), no. 4, 1003–1038.
  * [30] Hong Liu and Jiongmin Yong, _Option pricing with an illiquid underlying asset market_ , Journal of Economic Dynamics and Control 29 (2005), no. 12, 2125–2156.
  * [31] Johannes Muhle-Karbe, Zexin Wang, and Kevin Webster, _Stochastic liquidity as a proxy for nonlinear price impact_ , Available at SSRN 4286108 (2022).
  * [32] Huyên Pham, _Continuous-time stochastic control and optimization with financial applications_ , vol. 61, Springer Science & Business Media, 2009.
  * [33] Torsten Schöneborn and Alexander Schied, _Liquidation in the face of adversity: stealth vs. sunshine trading_ , EFA 2008 Athens Meetings Paper, 2009\.