# Strategic Contract Negotiation in Financial Networks

Akhil Jalan
UT Austin
<EMAIL_ADDRESS>Deepayan Chakrabarti
UT Austin
<EMAIL_ADDRESS>
###### Abstract

How can firms optimally negotiate bilateral contracts with each other in a
financial network? Every firm seeks to maximize the utility it gains from its
portfolio of contracts. We focus on mean-variance utilities, where each firm
has its own beliefs about the expected returns of the contracts and the
covariances between them [Mar52]. Instead of revealing these beliefs, a firm
may adopt a different negotiating position, seeking better contract terms. We
formulate a contract negotiation process by which such strategic behavior
leads to a network of contracts. In our formulation, any subset of firms can
be strategic. The negotiating positions of these firms can form Nash
equilibria, where each firm’s position is optimal given the others’ positions.

We give a polynomial-time algorithm to find the Nash equilibria, if they
exist, and certify their nonexistence otherwise. We explore the implications
of such equilibria on several model networks. These illustrate that firms’
utilities can be sensitive to their negotiating position. We then propose
trade deadlines as a mechanism to reduce the need for strategic behavior. At
the deadline, each firm can unilaterally cancel some or all of its contracts,
for a penalty. In our model networks, we show that trade deadlines can reduce
the loss of utility from being honest. We empirically verify our insights
using data on international trade between $46$ large economies.

###### Contents

  1. 1 Introduction
    1. 1.1 Our Contributions
    2. 1.2 Paper Organization
  2. 2 Background
  3. 3 Strategic Negotiations
  4. 4 Insights and Analysis
    1. 4.1 Two Agents That Can Self-Invest
    2. 4.2 One Investor and Two Hedge Funds
  5. 5 Trade Deadline
    1. 5.1 Trade Deadline Formulation
    2. 5.2 Strategic Negotiation for a Single Agent under the Trade Deadline
  6. 6 Experimental Results
  7. 7 Related Work
  8. 8 Conclusion
  9. 9 Proofs
    1. 9.1 Proof of Theorem 3.4
    2. 9.2 Proof of Theorem 5.3.
    3. 9.3 Proof of Theorem 5.4
  10. A Additional Proofs
    1. A.1 Proof of Lemma 3.7
    2. A.2 Non-Convexity of Constraints
    3. A.3 Proof of Theorem 4.3
    4. A.4 Exact Formulas for Theorem 5.8

##  1 Introduction

We consider negotiations between agents to form a network of bilateral
contracts. An example network is the US over-the-counter (OTC) market for
derivatives. Contracts in the OTC network had a notional value of over $600$
trillion USD in 2022 [ISD23]. Yet, such markets are less well-understood than
public markets [GY16, Eis+23]. This is because the contracts are not
standardized, and deal with products not listed on exchanges. Each contract is
negotiated between two agents, such as banks and hedge funds, and tailored to
their needs. Another example is that of bilateral trade networks between
countries. Countries may trade in unique products (e.g., arms, technology) for
which contracts must be negotiated on a case-by-case basis. Agents with
stronger negotiating positions can extract better contract terms in such
networks. We aim to model and analyze the effect of strategic behaviors in
this setting.

We model the agents as utility maximizers who have heterogeneous beliefs. Let
Alice and Bob denote two agents in a larger network. Then, Alice’s utility
depends on her private beliefs about the risk/reward profile of her portfolio
of contracts with all other agents. But Alice cannot unilaterally set the
terms of her contract with Bob. Instead, Alice and Bob must negotiate a
contract agreeable to both. The final contract depends on Alice’s and Bob’s
claims about the contract’s perceived benefits, which we call their
negotiating positions. Now, Alice’s negotiating position with Bob also affects
Alice’s other contracts. The reason lies in the utility-maximizing behavior of
agents. Since Bob seeks an optimal portfolio of contracts, any changes in the
contract with Alice affects Bob’s contracts with others (e.g. Eve). In turn,
Eve’s contract with Alice is also affected. Hence, Alice must consider the
network effects of her negotiating position. This motivates the following
problem:

What is the optimal negotiating position of each agent in a network?

Algorithm: We present an efficient algorithm to find the optimal negotiating
positions for any chosen subset of agents, or to report when no optimal
solution exists. Our algorithm assumes that the agents use heterogeneous mean-
variance utilities [Mar52, LM79, Pul83, EW09, Sim14]. Under the mean-variance
utility, there is a stable network of contracts that satisfies all agents
[JCS24]. This network depends on the agents’ negotiating positions. However,
[JCS24] assume that agents reveal their private beliefs during contract
negotiations. In contrast, we study agents that deceive others to gain an
advantage. The resulting network can increase an agent’s utility, but only up
to a point. We show that deceptions only provide bounded utility, even if the
other agents’ negotiating positions are known. In practice, an agent will only
know the others’ positions after choosing her position. So, she can choose the
optimal negotiating position assuming that others also make optimal choices.
Hence, the optimal set of negotiating positions is a Nash equilibrium among
deceptions. Our algorithm finds this Nash equilibrium if it exists.

Insights: The Nash equilibria reveal some surprising insights even in simple
networks. For instance, consider a two-agent network. We show that the agents
both have lower utility when they are both strategic than when they are both
honest. This is because if one agent is honest, the other agent can benefit at
her expense by being strategic. The Nash equilibrium has lower utility for
both agents, but neither has an incentive to deviate.

As another example, consider the three-agent network of two hedge funds
trading with an investor. We show that utilities are very sensitive to which
agents act strategically. Suppose the investor is honest and the hedge funds
are strategic. Then, the funds can extract large utility from the investor,
who is left with nearly zero utility. But if the investor is also strategic,
she can force the hedge funds to compete and gains utility at their expense.
Indeed, if all three agents are strategic, the investor is better off than if
all are honest. The opposite is true for the hedge funds.

The above result shows why naive investors may choose to stay away from
bilateral markets. Even sophisticated investors may struggle, since the Nash
solution can depend on the private beliefs of other agents. Two hedge funds in
the same market may well have such knowledge about each other, but an investor
may not. To encourage more participants to join the market, we need to reduce
the sensitivity of outcomes to strategic behavior. This motivates our second
question:

Can we design a mechanism that reduces the need for strategic behavior?

Mechanism: The usual Vickrey-Clarke-Grove mechanism for bilateral trade needs
a market owner to subsidize honesty [Nis+07]. In financial networks such as
those for OTC derivatives or international trade, there is no market owner.
Hence, the applicability of such mechanisms is unclear. Instead, we present a
different mechanism: namely, trade deadlines.

The mechanism operates as follows. First, all agents choose their negotiating
positions, as before. These positions determine the network of contracts.
Second, each agent observes the network and independently decides which
contracts to veto. Third, all veto decisions are simultaneously revealed at a
global trade deadline. A contract is canceled if either party vetoes it.

Under the trade deadline, each agent worries about the effect of contract
cancellations on her portfolio. For instance, an agent $i$ may want a
portfolio with two contracts with large but offsetting risks. But if one
counterparty cancels, agent $i$ is left with the other contract. The single
contract incurs a large risk. To prevent such an outcome, agent $i$ will
likely veto both contracts. Knowing this, the counterparties of agent $i$ will
offer her better terms to avoid cancellations. In other words, strategic
agents have to share utility with their counterparties, even if they are
honest. Thus, the mechanism reduces the need for strategic behavior.

###  1.1 Our Contributions

Our work makes several contributions to the literature on network games and
mechanism design. We summarize these below.

Formalizing strategic behavior in contract formation: We characterize an
agent’s negotiating position as her claims about the benefits of a contract.
These claims feed into a mean-variance utility for each agent. The result is a
stable network of bilateral contracts among agents. The optimal negotiating
positions of all agents are Nash equilibria in this setting.

Efficient algorithm for finding Nash equilibria: Our algorithm finds Nash
equilibria among strategic agents, if they exist. We can also analyze
different combinations of honest and strategic agents. This contrasts with
prior work, which mainly considers one strategic agent in a network [GKT20].

Trade deadlines to limit the need for strategic behavior: Without a trade
deadline, strategic agents can induce an honest agent to accept hedging
contracts. Each contract has a large negative utility, but together, they give
a small positive utility. With the deadline, the honest agent judges each
contract on its merits. She will cancel contracts that have significant
negative utility. The strategic agents have to offer better terms to avoid
contract cancellations. Thus, the trade deadline mechanism helps honest agents
and reduces the need for strategic behavior.

Empirical validation on the trade network between OECD countries. Absent the
trade deadline mechanism, we show that a moderately sized economy can gain
significant utility by using their optimal negotiation position. However, if
all nations are strategic, then the same economy is worse-off. With the
mechanism, a strategic nation has limited ability to gain utility. As the
penalty for contract vetoes decreases, the nation is forced to offer better
terms to its counterparties to avoid losing their contracts.

###  1.2 Paper Organization

After covering background material in Section 2, we present our model
formulation and main results in Section 3. Section 4 presents insights based
on analyzing some simple networks. We discuss the trade deadline mechanism in
Section 5, followed by experiments in Section 6. We survey related work in
Section 7 and conclude in Section 8. Section 9 shows some essential proofs,
with the rest deferred to the appendix.

##  2 Background

In this section, we briefly review the model of financial networks with honest
negotiations due to [JCS24].

Notation. We will use lowercase letters $a,b,c,\gamma_{i}$ to denote scalars,
boldface letters $\bm{\mu}_{i},\bm{w}$ to denote vectors, and uppercase
letters $A,B,\Sigma$ to denote matrices. The vectors
$\bm{e}_{1},\ldots,\bm{e}_{n}$ denote the standard basis in $\RR^{n}$, and
$I_{n}$ is the $n\times n$ identity matrix. We use $\bm{v}_{i;j}$ to refer to
the $j^{th}$ component of the vector $\bm{v}_{i}$. We denote the inner product
$\la\bm{v},\bm{w}\ra:=\bm{v}^{T}\bm{w}$. We use $A\succ 0$ to denote that $A$
is positive definite. If $A\in\RR^{m\times n},B\in\RR^{p\times q}$ then
$A\otimes B\in\RR^{mp\times nq}$ denotes their tensor product: $(A\otimes
B)_{ij,k\ell}=A_{ik}B_{j\ell}$, and if $m=p,n=q$ then $A\odot B\in\RR^{m\times
n}$ denotes their Hadamard product: $(A\odot B)_{ij}=A_{ij}B_{ij}$. For an
appropriate matrix $M$, $\textrm{tr}(M)$ calculates its trace and
$\textrm{vec}(M)$ vectorizes $M$ by stacking its columns into a single vector.
For an integer $r\geq 1$, we use $[r]$ to denote $[r]\defeq\\{1,2,\dots,r\\}$.

The network model. Let $W=W^{T}\in\RR^{n\times n}$ denote an undirected
weighted network of contracts between $n$ agents, with $W_{ij}$ being the size
of the contract between $i$ and $j$ and $W_{ii}$ represents $i$’s investment
in its own business. A negative contract $W_{ij}<0$ is valid and represents a
reversed version of a positive contract; for example, in a derivative
contract, $W_{ij}<0$ swaps the roles of the long and short position holders.
During contract negotiations, agent $i$ can pay $P_{ji}$ per unit contract to
agent $j$ to get $j$ to agree to the contract size. Since payments are zero-
sum, $P^{T}=-P$. The contracts size and payments $(W,P)$ together give the
network. At $(W,P)$, agent $i$ has contracts $\bm{w}_{i}:=W\bm{e}_{i}$. Agent
$i$ wants to optimize the utility of their contracts and believes that
contracts have mean return ${\bm{\mu}}_{i}\in\RR^{n}$ and covariance
$\Sigma\succ 0$. Moreover, they have a risk-aversion parameter $\gamma_{i}>0$.
Their utility is then:

$\displaystyle\text{agent $i$'s utility }g_{i}(W,P)$
$\displaystyle:=\bm{w}_{i}^{T}(\bm{\mu}_{i}-P\bm{e}_{i})-\gamma_{i}\cdot\bm{w}_{i}^{T}\Sigma\bm{w}_{i}$
(1)

Note that beliefs do not have to accurate or follow a particular distribution.
The set of all beliefs of all agents is represented in the network setting.

######  Definition  (Network Setting).

A network setting $(M,\Gamma,\Sigma)\in\RR^{n\times n}\times\RR^{n\times
n}\times\RR^{n\times n}$ is a tuple consisting of an arbitrary $M$, a diagonal
$\Gamma\succ 0$, and a symmetric $\Sigma\succ 0$. Agent $i$ has mean belief
$\bm{\mu}_{i}:=M\bm{e}_{i}$, risk-aversion $\gamma_{i}:=\Gamma_{ii}$, and
covariance belief $\Sigma$. Note that $M_{ij}$ and $M_{ji}$ need not be
identical.

######  Definition  (Stable point).

A feasible $(W,P)$ is stable if each agent achieves its maximum possible
utility given prices $P$:

$\displaystyle
g_{i}(W,P)=\max_{(W,P):W=W^{T},P^{T}=-P}g_{i}(W^{\prime},P)\quad\forall
i\in[n].$

######  Theorem 2.1 ([JCS24]).

Given a network setting $(M,\Gamma,\Sigma)$, if $\Gamma\succ 0$ and
$\Sigma\succ 0$, there exists a unique stable point $(W,P)$ such that

$\displaystyle\Sigma W\Gamma+\Gamma W\Sigma$ $\displaystyle=0.5(M+M^{T}),$
$\displaystyle\Sigma^{-1}P\Gamma^{-1}+\Gamma^{-1}P\Sigma^{-1}$
$\displaystyle=\Sigma^{-1}M\Gamma^{-1}-\Gamma^{-1}M^{T}\Sigma^{-1}.$

Equivalently,

$\displaystyle\textrm{vec}(W)$
$\displaystyle=\frac{1}{2}(\Gamma\otimes\Sigma+\Sigma\otimes\Gamma)^{-1}\textrm{vec}(M+M^{T}),$
$\displaystyle\textrm{vec}(P)$
$\displaystyle=(\Gamma^{-1}\otimes\Sigma^{-1}+\Sigma^{-1}\otimes\Gamma^{-1})^{-1}\textrm{vec}(\Sigma^{-1}M\Gamma^{-1}-\Gamma^{-1}M^{T}\Sigma^{-1}).$

Furthermore, agents can efficiently find the stable point through pairwise
negotiations.

##  3 Strategic Negotiations

We formalize the contract negotiation process as follows.

######  Definition 3.1 (Strategic Contract Negotiation $(M,\Gamma,\Sigma,S)$).

Suppose there is a network setting $(M,\Gamma,\Sigma)$. We also have a public
set $S\subseteq[n]$ of strategic agents; the remaining agents are honest. The
contract negotiation is a three-stage process:

  1. 1.

Strategy Phase: Each strategic agent $k\in S$ chooses a
$\bm{\delta}_{k}\in\RR^{n}$. The honest agents $j\not\in S$ choose
$\bm{\delta}_{j}=0$. The strategic agents cannot see the choices of other
strategic agents before making their choice.

  2. 2.

Contract Formation Phase: Define a matrix $\Delta$ whose $i$’th column is
$\bm{\delta}_{i}$, and let $M^{\prime}=M+\Delta$. The contract network is
formed based on the network setting $(M^{\prime},\Sigma,\Gamma)$.
Specifically, we use Algorithm 1 of [JCS24] to find the stable point $(W,P)$.

  3. 3.

Reward Collection Phase: Each agent $i\in[n]$ receives a reward $g_{i}(W,P)$
with respect to the true network setting $(M,\Gamma,\Sigma)$.

Since all agents make strategic choices independently, they cannot adapt their
strategy to the others’ choices. Instead, they can choose a strategy based on
a Nash equilibrium, defined below.

######  Definition 3.2 (Nash Equilibrium).

A Nash Equilibrium for a Strategic Contract Negotiation $(M,\Gamma,\Sigma,S)$
is a matrix $\Delta$, as defined in Definition 3.1, with the following
property. For each $k\in S$, if all other strategic agents follow $\Delta$,
then agent $k$ gains the highest utility by also following $\Delta$ in the
Strategy Phase.

Input: Network setting $(M,\Gamma,\Sigma)$ with $n$ agents, strategic agent
set $S\subseteq[n]$.

Output: Nash equilibria set $\mathcal{E}$.

$K\leftarrow\Sigma\otimes\Gamma+\Gamma\otimes\Sigma$;

$\textrm{vec}(W)\leftarrow\frac{1}{2}K^{-1}\textrm{vec}(M+M^{T})$;

$\Pi\leftarrow\text{the $n^{2}\times n^{2}$ matrix such that
$\Pi\textrm{vec}(M)=\textrm{vec}(M^{T})$}$ ;

/* commutation matrix */

$L\leftarrow\frac{1}{2}(K^{-1}+K^{-1}\Pi)$;

$T\leftarrow 0,\bm{y}\leftarrow\bm{0}\text{ where }T\in\RR^{n\abs{S}\times
n\abs{S}},\bm{y}\in\RR^{n\abs{S}}$;

for  _$k$ in $S$_ do

Define $L^{(p,q)}\leftarrow\text{the $p^{th}$ row and $q^{th}$ column block of
$L$; each block has size $n\times n$}$;

$T^{(k,k)}\leftarrow
L^{(k,k)}+(L^{(k,k)})^{T}-2\gamma_{k}(L^{(k,k)})^{T}\Sigma L^{(k,k)}$;

for  _$j$ in $S\setminus\\{k\\}$_ do

$T^{(k,j)}\leftarrow\big{(}I-2\gamma_{k}(L^{(k,k)})^{T}\Sigma\big{)}L^{(k,j)}$;

end for

$\bm{y}^{(k)}\leftarrow(2\gamma_{k}(L^{(k,k)})^{T}\Sigma-I)W\bm{e}_{k}$;

end for

$\mathcal{F}\leftarrow\\{X\in\RR^{n\times|S|}:T\textrm{vec}(X)=\bm{y}\\}$;

$\mathcal{E}\leftarrow\bigg{\\{}\Delta\in\RR^{n\times n}:\exists
X\in\mathcal{F}\quad\text{s.t.}\quad\forall
i\in[n],\quad\Delta\bm{e}_{i}=\begin{cases}X\bm{e}_{i},&\text{ if }i\in S\\\
\bm{0},&\text{ otherwise}\end{cases}\bigg{\\}}\bigg{\\}}$;

return _$\mathcal{E}$ if $|\mathcal{E}|>0$ else report ”No Nash Equilibrium”_

ALGORITHM 1 Nash Equilibria Computation

Our main result is an algorithm that provably finds all Nash equilibria or
reports if no such equilibria exist (Algorithm 1).

######  Theorem 3.3.

Algorithm 1 returns the set of Nash Equilibria for the strategic contract
negotiation process $(M,\Gamma,\Sigma,S)$ if it is nonempty, and otherwise
returns “No Nash Equilibrium.”

To prove Theorem 3.3, we first prove that no agent can gain unbounded utility
by being strategic.

######  Theorem 3.4 (Concave utility given others’ choices).

Given any network setting $(M,\Gamma,\Sigma)$ and any set of choices
$\\{\bm{\delta}_{i};i\neq k\\}$ for all agents except $k$, the utility of $k$
is a quadratic function of $\bm{\delta}_{k}$ with a negative definite Hessian.

######  Corollary 3.5 (Strategy yields bounded utility).

No choice of negotiating position lets agent $k$ achieve unbounded utility,
even if agent $k$ has full information about other agents’ beliefs and
choices.

######  Proof 3.6.

By Theorem 3.4, there exists a symmetric matrix $A\succ 0$, a vector $\bm{b}$,
and a scalar $c$ such that if other agents use negotiating positions
$(\bm{\delta}_{j})_{j\neq k}$, then
$g_{k}(\bm{\delta}_{k})=-\bm{\delta}_{k}^{T}A\bm{\delta}_{k}+\bm{\delta}_{k}^{T}\bm{b}+c$.
The optimal $\bm{\delta}_{k}$ is at the unique critical point
$\bm{\delta}_{k}=\frac{1}{2}A^{-1}\bm{b}$. Therefore the utility gain of $k$
for any choice of $\bm{\delta}_{k}$ is bounded by
$g_{k}(\frac{1}{2}A^{-1}\bm{b})-g_{k}(\bm{0})=\frac{1}{4}\bm{b}^{T}A^{-1}\bm{b}$.

We also need the following Lemmas.

######  Lemma 3.7 (Utility from Strategy).

If agent $k$ reports $\bm{\delta}_{k}^{\prime}$ resulting in
$(W^{\prime},P^{\prime})$, then $k$’s utility is

$\displaystyle g_{k}(W^{\prime},P^{\prime})$
$\displaystyle=-\la\bm{\delta}_{k},\bm{w}_{k}^{\prime}\ra+\gamma_{k}\la\Sigma\bm{w}_{k}^{\prime},\bm{w}_{k}^{\prime}\ra,\quad\text{(where
$\bm{w}_{k}^{\prime}=W^{\prime}\bm{e}_{k}$)}.$

######  Lemma 3.8 (Commutation Matrix [HJ08]).

There exists a permutation matrix $\Pi:\RR^{n^{2}}\to\RR^{n^{2}}$ such that
for $X\in\RR^{n\times n}$, $\Pi\textrm{vec}(X)=\textrm{vec}(X^{T})$.

We are ready to prove Theorem 3.3.

######  Proof 3.9 (Proof of Theorem 3.3.).

Let $(\bm{\delta}_{j})_{j\in S}$ be the tuple of negotiating positions for the
strategic agents. Let $\Delta_{M}\in\RR^{n\times n}$ have $i^{th}$ column
$\bm{\delta}_{i}^{\star}$ if $i\in S$ and zero otherwise. Let
$(W^{\prime},P^{\prime})$ be the stable point resulting from a choice of
$M^{\prime}:=M+\Delta_{M}$ as the agents’ negotiating positions in the
Strategy Phase. From Theorem 2.1, we have

$\displaystyle\textrm{vec}(W^{\prime})$
$\displaystyle=\textrm{vec}(W)+0.5(\Sigma\otimes\Gamma+\Gamma\otimes\Sigma)^{-1}\textrm{vec}(\Delta_{M}+\Delta_{M}^{T})$
$\displaystyle\Rightarrow\textrm{vec}(W^{\prime}-W)$
$\displaystyle=L\textrm{vec}(\Delta_{M})$
$\displaystyle\Rightarrow\bm{w}_{k}^{\prime}-\bm{w}_{k}$
$\displaystyle=L^{(k,k)}\bm{\delta}_{k}+\sum\limits_{j\in S:j\neq
k}L^{(k,j)}\bm{\delta}_{j},$

where the second line follows from Lemma 3.8, and the matrix $L$ is defined as
in Algorithm 1.

Let $A_{k}:=L^{(k,k)}$ and $\bm{b}_{k}:=\sum\limits_{j\in S:j\neq
k}L^{(k,j)}\bm{\delta}_{j}^{*}$. By Lemma 3.7, we have:

$\displaystyle g_{k}$
$\displaystyle=-\la\bm{\delta}_{k},\bm{w}_{k}^{\prime}\ra+\gamma_{k}\la\bm{w}_{k}^{\prime},\Sigma\bm{w}_{k}^{\prime}\ra$
$\displaystyle=-\la\bm{\delta}_{k},(\bm{w}_{k}+A_{k}\bm{\delta}_{k}+\bm{b}_{k})\ra+\gamma_{k}\la(\bm{w}_{k}+A_{k}\bm{\delta}_{k}+\bm{b}_{k}),\Sigma(\bm{w}_{k}+A_{k}\bm{\delta}_{k}+\bm{b}_{k})\ra$
$\displaystyle=-\la\bm{\delta}_{k},A_{k}\bm{\delta}_{k}\ra-\la\bm{\delta}_{k},\bm{b}_{k}\ra-\la\bm{\delta}_{k},\bm{w}_{k}\ra+2\gamma_{k}\la\Sigma\bm{w}_{k},A_{k}\bm{\delta}_{k}+\bm{b}_{k}\ra$
$\displaystyle+\gamma_{k}\la
A_{k}\bm{\delta}_{k}+\bm{b}_{k},\Sigma(A_{k}\bm{\delta}_{k}+\bm{b}_{k})\ra+\gamma_{k}\la\bm{w}_{k},\Sigma\bm{w}_{k}\ra$
$\displaystyle\Rightarrow\nabla_{\bm{\delta}_{k}}g_{k}$
$\displaystyle=-(A_{k}+A_{k}^{T})\bm{\delta}_{k}-\bm{b}_{k}+2\gamma_{k}A_{k}^{T}\Sigma\bm{w}_{k}+2\gamma_{k}A_{k}^{T}\Sigma
A_{k}\bm{\delta}_{k}+2\gamma_{k}A_{k}^{T}\Sigma\bm{b}_{k}-\bm{w}_{k}$

By Theorem 3.4, the optimal negotiating position $\bm{\delta}^{\star}$ is the
critical point of $g_{k}$ with respect to $\bm{\delta}_{k}$. Setting
$\nabla_{\delta_{k}}g_{k}=0$, we obtain

$\displaystyle(A_{k}+A_{k}^{T}-2\gamma_{k}A_{k}^{T}\Sigma
A_{k})\bm{\delta}_{k}^{\star}$
$\displaystyle=(2\gamma_{k}A_{k}^{T}\Sigma-I)\bm{w}_{k}+(2\gamma_{k}A_{k}^{T}\Sigma-I)\bm{b}_{k}$

Rearranging terms, we obtain precisely the linear system of Algorithm 1:

$\displaystyle(2\gamma_{k}(L^{(k,k)})^{T}\Sigma-I)\bm{w}_{k}$
$\displaystyle=\bigg{[}(L^{(k,k)}+(L^{(k,k)})^{T}-2\gamma_{k}(L^{(k,k)})^{T}\Sigma
L^{(k,k)})\bm{\delta}_{k}^{\star}$
$\displaystyle+\big{(}I-2\gamma_{k}(L^{(k,k)})^{T}\Sigma\big{)}\sum\limits_{j\in
S:j\neq k}L^{(k,j)}\bm{\delta}_{j}^{\star}\bigg{]}$
$\displaystyle\Rightarrow\bm{y}^{(k)}$
$\displaystyle=T^{(k,k)}\bm{\delta}_{k}^{\star}+\sum\limits_{j\in S:j\neq
k}T^{(k,j)}\bm{\delta}_{j}^{\star},$ (2)

where $T^{(p,q)}$ and $\bm{y}^{(k)}$ are defined as in Algorithm 1. Therefore
if a Nash equilibrium exists, Algorithm 1 finds it. Conversely, a tuple
$(\bm{\delta}_{i}^{\star})_{i\in S}$ that solves Eq. (2) for all $k$ is such
that $\bm{\delta}_{i}^{\star}$ is the optimal $\bm{\delta}_{i}$ for all $i\in
S$ given that other agents report $(\bm{\delta}_{j}^{\star})_{j\in
S\setminus\\{i\\}}$. Therefore, if no Nash equilibrium exists, Eq. (2) cannot
be simultaneously satisfied for all $k$, so Algorithm 1 returns “No Nash
Equilibrium.”

(a) Both agents believe the mean reward of their contract is $5$.

(b) Results for P1 if she negotiates using $5+\delta$.

Figure 1: One strategic agent: (a) The network setting is as in Example 3.10
with $\bm{\mu}_{1;2}=\bm{\mu}_{2;1}=5$. (b) Suppose agent P1 reports
$\bm{\delta}_{1;2}=\delta<0$. Then P1 gets a smaller contract but a better
price. Even with the better price, the smaller contract means that their
expected return is worse for any $\delta\neq 0$ (right). But their risk
shrinks faster than expected return, so the highest utility is with
$\delta=-3$.

######  Example 3.10 (Nash Equilibrium for $2$-agent setting.).

Consider a network with two strategic agents, and $\Sigma=\Gamma=I$ and
$M=\begin{bmatrix}0&\bm{\mu}_{2;1}\\\ \bm{\mu}_{1;2}&0\end{bmatrix}$. Then, we
have

$\displaystyle T$ $\displaystyle=\left[\begin{matrix}0.5&0&0&0\\\
0&0.375&0.125&0\\\ 0&0.125&0.375&0\\\ 0&0&0&0.5\end{matrix}\right],$
$\displaystyle\bm{y}$ $\displaystyle=\left[\begin{matrix}0\\\
-0.125\bm{\mu}_{12}-0.125\bm{\mu}_{21}\\\
-0.125\bm{\mu}_{12}-0.125\bm{\mu}_{21}\\\ 0\end{matrix}\right].$

Solving the system gives

$\displaystyle\Delta_{M}$
$\displaystyle=-\frac{1}{4}\begin{bmatrix}0&\bm{\mu}_{1;2}+\bm{\mu}_{2;1}\\\
\bm{\mu}_{1;2}+\bm{\mu}_{2;1}&0\end{bmatrix}$

From Theorem 2.1, it is easily shown that prices under
$(M+\Delta_{M},\Sigma,\Gamma)$ are the same as those under
$(M,\Sigma,\Gamma)$, while the contracts are smaller. Therefore, everyone is
worse-off. Figure 1 shows an example with one strategic agent.

##  4 Insights and Analysis

Now, we study the Nash equilibria for several network settings. While these
are small networks, they offer insights that carry over to the larger real-
world network discussed later in Section 6.

###  4.1 Two Agents That Can Self-Invest

Consider a network with two agents (P1 and P2) who can form a contract with
each other, and each can invest in themselves (“self-invest”). The network
setting is as follows.

$\displaystyle M=\begin{bmatrix}M_{11}&M_{12}\\\
M_{21}&M_{22}\end{bmatrix},\quad\Sigma=\begin{bmatrix}1&\rho\\\
\rho&1\end{bmatrix},\quad\Gamma=\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}$ (3)

The parameter $\rho\in(-1,1)$ is the correlation between an agent’s returns
from self-investment versus trading. For $\rho\approx 1$, returns from self-
investing and trading move in lockstep. So, each agent must hedge between
self-investing and trading, hoping to benefit from any differences in their
returns. But as $\rho$ goes to $-1$, the risks from self-investing and trading
offset. If both offer positive returns, an agent can gain nearly risk-free
reward. Hence, negative correlations can lead to higher utility for agents.

######  Theorem 4.1.

Consider the network setting of Eq. 3. Let
$\kappa:=\rho(M_{11}+M_{22})-(M_{12}+M_{21})$, and let $\Delta$ be the
negotiation positions (Definition 3.1) for all agents at the Nash equilibrium.

  1. 1.

If only agent $k$ is strategic, then

$\displaystyle\Delta_{ij}$ $\displaystyle=\begin{cases}\kappa/3&\text{if
$i\neq k,j=k$}\\\ 0&\text{otherwise}\end{cases}$

  2. 2.

If both agents are strategic, then

$\displaystyle\Delta_{ij}$ $\displaystyle=\begin{cases}\kappa/4&\text{ if
$i\neq j$}\\\ 0&\text{otherwise}\end{cases}$

######  Remark 4.2.

We can show that it is strategic for $i$ to report $\Delta_{ii}=0$ even if
$\Sigma_{1;1}\neq\Sigma_{2;2}$ and $\Gamma_{11}\neq\Gamma_{22}$.

Figure 2: Nash equilibria for two agents: The utility for either agent when
both are honest (solid line) is higher than when both are strategic (dashed
line). When only agent P1 is strategic, P1 gains the highest utility (dotted
circles) while P2’s utility is lowest (dotted squares). The network settings
are shown in the bottom row, with an arrow from $i$ to $j$ corresponding to
$M_{ji}$.

Figure 2 shows the agents’ utility for honest versus strategic negotiating
positions over a range of $\rho$. We can make several observations.

Strategic agents report self-investing returns truthfully. Suppose agent $i$
claims that her self-investments have higher returns than in reality (that is,
$M^{\prime}_{ii}>M_{ii}$). If agent $j$ wants to trade with $i$, then $j$ will
have to offer better trading terms via better prices. Thus, high self-
investing returns are a plausible negotiating strategy. However, Theorem 4.1
shows that $\Delta_{ii}=0$ at the Nash equilibrium, so
$M^{\prime}_{ii}=M_{ii}$. This is because if both agents make untrue claims
about self-investing, they get smaller contracts, lowering utility.

Payments to others can increase when the agent becomes strategic. It may
appear that strategic agents can only increase their utility by extracting
higher payments from others. However, this need not be true. Suppose both
agents have a utility of $1$ from self-investing and $0$ from trading. By
symmetry, if both agents are honest, they make no payments. Now, suppose only
agent P1 is strategic and $\rho\approx 1$. By Theorem 4.1, P1 will claim to
have higher returns from trading than her actual returns. This implies that P1
pays P2 during contract formation. But the contract size also changes. With
the new contract size, P1 still gains utility at the expense of P2.

Utility is lower when both agents are strategic. Figure 2 shows several
instances where the agents are worse off when both are strategic versus when
both are honest. This is because the agents face a Prisoner’s Dilemma. If both
are honest, they cooperate, and both gain high utility. However, being
strategic is a dominant strategy. This forces both to be strategic, leading to
lower utility for both.

Negative correlations amplify the effect of negotiating positions. Suppose
self-investing and trading both have positive expected returns. When
correlations are negative, their risks cancel while their returns add. So, an
agent can take large positions and achieve high utility. But, as noted in the
previous paragraph, there is a drop in utility when both agents are honest
versus strategic. We find a larger drop for negative correlations. Hence,
under negative correlations, the effect of strategic behavior is also more
pronounced.

###  4.2 One Investor and Two Hedge Funds

Consider a $3$-agent network where one investor interacts with two hedge funds
under the following network setting.

$\displaystyle M=\begin{bmatrix}0&a&a\\\ m&0&0\\\
m&0&0\end{bmatrix},\quad\Sigma=\begin{bmatrix}1&0&0\\\ 0&1&\rho\\\
0&\rho&1\end{bmatrix},\quad\Gamma=I.$ (4)

The first column corresponds to the investor, and the others to the hedge
funds. Under this setting, the hedge funds do not want to trade with each
other, and none of the agents want to self-invest. Also, the hedge funds are
correlated with each other (via $\rho$), and uncorrelated with the investor.

######  Theorem 4.3.

Consider the the network setting of Eq. 4, where strategic agents can only
modify the non-zero entries in their column of $M$. Define

$\displaystyle\nu$
$\displaystyle=\frac{1}{2}\left(\frac{1}{2-\rho}+\frac{1}{2+\rho}\right),$
$\displaystyle\eta$
$\displaystyle=\frac{1}{2}\left(\frac{1}{2-\rho}-\frac{1}{2+\rho}\right),$
$\displaystyle\zeta$ $\displaystyle=\frac{\nu-\eta}{\nu+(\nu-\eta)(1-\nu)}.$

  1. 1.

Honest investor and strategic hedge funds:

$\displaystyle M^{\prime}_{21}=M^{\prime}_{31}$ $\displaystyle=m,$
$\displaystyle M^{\prime}_{12}=M^{\prime}_{13}$
$\displaystyle=\frac{a\nu-m(1-\nu)(\nu-\eta)}{\nu+(1-\nu)(\nu-\eta)}.$

  2. 2.

All agents strategic: If the investor does not distinguish between the hedge
funds, then

$\displaystyle M^{\prime}_{21}=M^{\prime}_{31}$
$\displaystyle=\frac{m-a\zeta}{1+\zeta},$ $\displaystyle
M^{\prime}_{12}=M^{\prime}_{13}$
$\displaystyle=\frac{a\nu-M^{\prime}_{21}(1-\nu)(\nu-\eta)}{\nu+(1-\nu)(\nu-\eta)}.$

(a) Network diagram with entries of entries of $M$ marked on edges.

(b) Utilities of the Investor and a single Fund under Theorem 4.3. Note that
the two Funds achieve the same utility.

Figure 3: Network with three agents: (a) An investor trades with two hedge
funds, with the investor gaining $5$ per unit contract while the hedge funds
gain $1$. (b) We show the utility for the investor (left) and either hedge
fund (right) for various strategic behaviors. The investor has low utility
when she is honest, but is better off than the hedge funds when she is
strategic.

From Theorem 4.3 and Figure 3(b), we observe the following.

The investor’s utility is very sensitive to her negotiating position. Suppose
the investor is honest and both hedge funds are strategic. Then, the investor
will accept worse terms from the funds and achieve less utility. But the
situation is reversed if the investor is also strategic (Figure 3(b)). The
investor now achieves higher utility than either fund. Thus, the investor’s
outcome is very sensitive to her negotiating position.

The sensitivity to negotiating positions increases as $\rho\to-1$. Figure 3(b)
shows that when $\rho$ decreases, the investor loses utility if she is honest
but gains utility if she is strategic. The reason is that as $\rho\to-1$, the
investor wishes to invest almost equally in both funds to reduce her risk. The
hedge funds only form one contract each. Since they cannot hedge their risk,
they prefer much smaller contracts than the investor. The investor can extract
higher payments for this, increasing her utility.

Strategic behavior can reduce utility. Suppose the investor is honest. As
$\rho\approx-1$, the hedge funds are worse off being strategic than if they
were both honest (Figure 3(b)). If both funds are honest, their contract sizes
match their risk preference. However, if both are strategic, each fund worries
about its competitor. So, both funds end up with worse terms.

##  5 Trade Deadline

The previous section showed that an agent’s utility is sensitive to her
negotiating position. For instance, an investor can extract utility from hedge
funds if she is strategic, but loses utility to them if she is honest. The
need for strategic behavior may dissuade new investors. Furthermore, the Nash
equilibrium formula can depend on the private beliefs of other agents (e.g.,
in Theorem 4.3). Established agents may have the relevant historical or
insider knowledge, which new entrants lack. Hence, to grow the network, we
need a mechanism that reduces the need for strategic behavior. We propose such
a mechanism, namely, a trade deadline.

###  5.1 Trade Deadline Formulation

We implement the trade deadline between the Contract Formation and Reward
Collection phases of the contract negotiation process (Definition 3.1). After
Contract Formation, we have a stable point $(W,P)$. We formally define the
trade deadline mechanism as follows.

######  Definition 5.1 (Trade deadline mechanism with penalty $\theta$).

Given $(W,P)$, each agent $i\in[n]$ submits a veto vector
$\bm{v}_{i}\in\\{0,1\\}^{n}$, where $\bm{v}_{i;j}=1$ if $i$ wishes to retain
her contract with $j$, and $\bm{v}_{i;j}=0$ if she wishes to exit the
contract. The agents choose their veto vectors without coordination. All
vectors are revealed simultaneously at the trade deadline, and the appropriate
contracts are canceled. The new contract network is given by
$W^{\prime}_{ij}=W_{ij}\cdot\bm{v}_{i;j}\cdot\bm{v}_{j;i}$. The ensuing Reward
Collection phase is based on $(W^{\prime},P)$. Each agent also pays a penalty
$\theta\geq 0$ for every exit, for a total penalty of
$\theta(n-\bm{v}_{i}^{T}\bm{1})$ for agent $i$.

As the penalty $\theta\to\infty$, every agent will prefer to keep all
contracts. Conversely, as $\theta\to 0$, agents can exit contracts at will. We
assume all agents agree upon a $\theta$ before contract negotiation. Now, an
agent $i$ cannot directly select the $\bm{v}_{i}$ that maximizes her utility
since she does not know the others’ choices. Thus, a natural goal is to seek a
robust veto, as defined next.

######  Definition 5.2 (Robust Veto).

Let $v_{-i}:=(v_{1;i},\dots,v_{i-1;i},v_{i+1;i},\dots,v_{n;i})^{T}$. The
robust veto vector $\bm{v}_{i}$ for agent $i$ is given by:

$\displaystyle\bm{v}_{i}^{\star}$
$\displaystyle=\operatorname*{arg\,max}\limits_{\bm{v}_{i}\in\\{0,1\\}^{n}}\min\limits_{\bm{v}_{-i}\in\\{0,1\\}^{n-1}}g_{i}(W^{\prime}(\bm{v}_{i},\bm{v}_{-i}),P)-\theta(n-\bm{v}_{i}^{T}\bm{1})$

Next, we show two negative results. First, any deadline mechanism leads to
agents having lower utility than if they were all honest. Second, for the
specific case of the robust veto, finding the optimal vector vector is NP-hard
under randomized reductions.

######  Theorem 5.3 (Trade deadlines lower utility compared to honesty).

Consider a network $(W,P)$ formed by honest agents that is modified to
$(W^{\prime},P)$ by a trade deadline mechanism based on veto vectors.
Irrespective of how agents choose their veto vectors, we have for all
$i\in[n]$,

$\displaystyle g_{i}(W^{\prime},P)$ $\displaystyle\leq g_{i}(W,P).$

The inequality is strict unless none of $i$’s contracts are vetoed.

######  Theorem 5.4 (Hardness of Finding $\bm{v}_{i}^{\star}$ in Definition
5.2).

If RP (Randomized Polynomial time) does not equal NP, then there is no
deterministic polynomial-time algorithm to find $\bm{v}_{i}^{\star}$.

Our proof is by a careful analysis of the original family of hard lattices in
the seminal result of [Ajt98], who proved hardness of the $\ell_{2}$-Shortest
Vector Problem assuming $\textbf{RP}\neq\textbf{NP}$. Given the hardness of
finding $\bm{v}_{i}^{\star}$, we focus on a special case of the robust veto.

######  Definition 5.5 (Simplified Veto).

Let $g_{i;j}(W,P):=(M_{ji}-P_{ji})W_{ji}-\gamma_{i}\Sigma_{jj}W_{ji}^{2}$ be
the utility that agent $i$ receives solely from her contract with agent $j$.
Agent $i$ selects her veto vector as

$\displaystyle\bm{v}_{i;j}=\begin{cases}1,&\text{if
$g_{i;j}(W,P)\geq-\theta$}\\\ 0,&\text{otherwise.}\end{cases}.$

In other words, $i$ retains her contract with $j$ if $i$’s utility given
$(W,P)$ and $\bm{v}_{i}=\bm{e}_{j}$ is not worse than paying the penalty. Note
that under the Robust Veto (Definition 5.2), $g_{i;j}(W,P)\geq-\theta$ is a
necessary condition for $\bm{v}_{i;j}^{*}=1$, but not sufficient.

###  5.2 Strategic Negotiation for a Single Agent under the Trade Deadline

Under the trade deadline mechanism, an agent $i$ must choose both her
negotiating position $\bm{\delta}_{i}$ and her veto vector $\bm{v}_{i}$. We
consider the following form of this problem.

######  Problem 5.6 (Single Agent under Trade Deadline).

Suppose agent $i$ knows that other agents will use
$\bm{\delta}_{-i}:=(\bm{\delta}_{j})_{j\neq i}$ as their negotiating
positions, and simplified veto vectors. How should $i$ select the optimal
$\bm{\delta}_{i}$ so that no counterparty will cancel their contract with $i$?
Formally, $i$ seeks a negotiating position $\bm{\delta}_{i}^{\star}$ such that

$\displaystyle\bm{\delta}_{i}^{\star}=\arg\sup\limits_{\bm{\delta}_{i}}g_{i}(W(\bm{\delta}_{i},\bm{\delta}_{-i}),P(\bm{\delta}_{i},\bm{\delta}_{-i}))$
(5) s.t. $\displaystyle
g_{j;i}(W(\bm{\delta}_{i},\bm{\delta}_{-i}),P(\bm{\delta}_{i},\bm{\delta}_{-i}))\geq-\theta\quad\text{for
all $j\neq i,j\notin\mathcal{S}$,}$ (6)

where $\mathcal{S}\subseteq[n]$ is a subset of “safe partners” for whom agent
$i$ makes an exception. Agent $i$ chooses
$\bm{\delta}_{i}=\bm{\delta}_{i}^{\star}$ and $\bm{v}_{i}=\bm{1}$ if Eq. 6 has
a feasible solution, and $\bm{\delta}_{i}=\bm{0},\bm{v}_{i}=\bm{0}$ otherwise.

Unfortunately, it can be shown that the constraints need not be convex sets
(Appendix A.2). Hence, Problem 5.6 is not convex. However, we can find local
minima using standard solvers.

######  Remark 5.7.

The safe partners can include agents with whom agent $i$ has outside
agreements, such as contracts signed before the deadline.

Example: One investor trading with two hedge funds. When an honest investor
trades with two strategic hedge funds, the funds can extract utility from the
investor (Section 4.2). But under the trade deadline mechanism, each fund must
ensure that the investor also gains at least $-\theta$ from each trade. This
constrains the hedge funds, as we show next.

######  Theorem 5.8.

Suppose the investor reports her negotiating position as
$M^{\prime}_{21}=M^{\prime}_{31}=\mu$. Let $\tilde{\mu}$ be the hedge funds’
optimal negotiating position when there is no deadline (Theorem 4.3). Consider
the trade deadline mechanism with penalty $\theta$ and no safe partners. Then
there exist constants $\ell_{\theta}^{\dagger}<h_{\theta}^{\dagger}$,
depending on the network setting and $\mu$, such that the Nash-optimal
negotiating position for each hedge fund under the deadline is given by

$M^{\prime}_{12}=M^{\prime}_{13}=\min(h_{\theta}^{\dagger},\max(\tilde{\mu},\ell_{\theta}^{\dagger})).$

We give exact formulas for $\ell_{\theta}^{\dagger}$ and
$h_{\theta}^{\dagger}$ in Appendix A.4.

Thus, the mechanism forces each hedge fund to choose a negotiating position in
the interval $[\ell_{\theta}^{\dagger},h_{\theta}^{\dagger}]$. When the Nash
equilibrium value $\tilde{\mu}$ is within this interval, the hedge funds are
unaffected by the trade deadline.

Figure 4: Effect of trade deadlines: The setting is the same as in Figure
3(b). When $\theta$ is small (left), the utilities of the investor and hedge
funds are all close. As $\theta$ grows, strategic behavior begins to affect
outcomes, especially at negative $\rho$. Setting $\theta\to\infty$ (right) is
equivalent to having no deadline, as in Figure 3(b).

Figure 4 shows results for various choices of $\theta$. The mechanism has a
strong effect for negative correlations. Without the trade deadline, an honest
investor accepts large contracts with each fund since the negative correlation
reduces the risk. However, each individual contract has a large negative
utility. Under the trade deadline mechanism, the investor would cancel these
contracts. Knowing this, the hedge funds must choose negotiating positions
that give the investor at least $-\theta$ utility for each contract. This
improves the investor’s utility, even when she is honest.

##  6 Experimental Results

We study the international trade network of $n=46$ large economies from
2010-2020 [OEC22]. Nodes represent nations, and we observe the trading network
for every fiscal quarter. The weight of an edge $W_{ij}^{t}$ between countries
$i$ and $j$ at quarter $t$ is the sum of the quarter’s trade flows from $i$ to
$j$ and vice versa. There are no self-loops ($W_{ii}^{t}=0$). From the
$W^{t}$, we construct the network setting $(M^{t},\Sigma^{t},\Gamma^{t})$ for
each quarter $t$. Following [JCS24], we set $\Gamma^{t}=I$ for all $t$, and
$\Sigma^{t}=\Sigma$, where $\Sigma$ is obtained from a Semidefinite Program
(SDP). Given $\Sigma^{t}$ and $W^{t}$, we solve for $M^{t}$ using Theorem 2.1.

Negotiating with honest agents. We run our experiment for a fixed node $i=44$,
corresponding to the United Kingdom (other countries have qualitatively
similar results). For each time $t$, we solve Problem 5.6 for node $i$ for
various choices of $\theta$, assuming all other nodes are honest
($\bm{\delta}_{j}=\bm{0}$ for all $j\neq i$). This yields $i$’s negotiating
position, from which we calculate $i$’s utility.

To choose the proper scale for $\theta$, we calculate the average utility
$\beta$ of all individual contracts under honest reporting across all time
periods. We set $\theta=\tilde{\theta}\cdot\beta$ for
$\tilde{\theta}\in\\{10^{-6},10^{-4},10^{-2},1\\}$. As $\tilde{\theta}$
increases, the penalty for exiting a contract increases, and the effect of the
trade deadline mechanism weakens. We designate as safe partners all nodes $j$
for which $g_{j;i}<-\theta$ when all nodes (including $i$) report honestly.
With these safe partners, Problem 5.6 has a feasible solution in all cases. We
compare these results to the setting with no trade deadline
($\tilde{\theta}=\infty$) and the baseline case where $i$ is honest
($\bm{\delta}_{i}^{t}=\bm{0}$).

Figure 5 (left plot) shows the results for all quarters. There is significant
variability across quarters due to changes in $M_{t}$. We find that $i$’s
utility is the smallest when $i$ is honest (baseline setting) and largest when
there is no trade deadline mechanism. The utility under the trade mechanism is
always in between these two. Smaller values of $\tilde{\theta}$ impose more
constraints on $i$’s negotiating position, leading to lower values for $i$’s
utility.

Negotiating with strategic agents. All agents other than $i$ set their
negotiating positions as the Nash equilibrium values (Algorithm 1). For the
baseline, we set $i$ too at its Nash equilibrium value. The safe partners at
time $t$ are all $j\neq i$ such that $g_{j;i}<-\theta$ at the baseline. Figure
5 (right plot) shows the results for all quarters. We see that $i$’s utility
is lower than when all other agents were honest. Thus, the presence of other
strategic agents lowers $i$’s utility.

Figure 5: Trade deadlines for OECD network: The utility gain from strategic
behavior decreases under the trade deadline mechanism. As $\theta\to 0$, the
penalty for canceling contracts with the strategic agent goes to $0$. Hence,
the strategic agent must negotiate such that their contracts are not canceled,
and is therefore forced into a worse negotiating position.

##  7 Related Work

We review prior on network games, trade and contract negotiations, and trade
deadlines.

Network games. Network games involve $n$ agents whose payoffs depend on their
neighbors’ actions in a network [Tar04]. These games are widely studied in
economics [GY16, EG22] and computer science [Len+20, Ros+22].

Our work closely relates to endogenous network formation models for economic
networks [AA20, SG21, JCS24]. We study stable points of a network formation
process in which each agent wants to form edges to maximize its utility. Prior
works have assumed that the formation process is based on the actual
preferences of the agents. Our focus is on strategic manipulation of the
network formation process.

Another stream of work considers the strategic manipulation of network games.
[GGG20] prove that a planner can maximize welfare in a game with strategic
complements/substitutes by modifying marginal returns. [GKT20] and [CR21]
study adversarial perturbations in opinion dynamics. Both works only consider
a single strategic actor. In contrast, any subset of agents can be strategic
in our setting.

Bespoke contract negotiations. In a financial network, each pair of agents
negotiates a contract with custom terms. Ideal mechanisms for bilateral
contract negotiation under information asymmetry do not exist in general
[MS83]. In the special case of principal-agent problems with information
acquisition, [PW22] provide efficient algorithms to solve the principal’s
minimum payment problem with limited liability. However, there are tradeoffs
between the limited liability condition and other desirable aspects of the
contract negotiation mechanism, such as individual rationality [Alo+23].

Trade negotiations. Two-player matrix games are used to model aspects of
bilateral trade negotiations, such as one-off trade wars [Hug07, SP16] or
repeated Prisoner’s Dilemma games [Khu22]. In the $n=2$ case of our model, we
obtain a Prisoner Dilemma-type payoff matrix in which each agent can negotiate
honestly or strategically. However, we go beyond the matrix game setting by
(i) using a network model with arbitrarily many agents, and (ii) considering
real-valued negotiating positions that give agents more flexibility to act
strategically. Models with $n>2$ are better suited to model multilateral
settings such as strategic trade negotiations between the European Union (EU)
and the Association for Southeast Asian Nations (ASEAN) [Lac12]. Moreover,
trade negotiations can involve a continuum of strategic positions, such as
import tariffs for nations seeking to join the WTO [Rog20].

Trade deadline mechanisms. We design a trade deadline mechanism to limit the
sensitivity of outcomes to strategic behavior. While incentive-compatible
mechanisms such as the Vickrey-Clarke-Groves (VCG) mechanism are known for
bilateral trade, the VCG mechanism requires a subsidy if the buyer values the
good more than the seller [Nis+07]. If there is neither a subsidy nor a tax,
the Gibbard-Satterthwaite theorem [Gib73, Sat75] forbids the existence of non-
trivial decision rules in dominant strategies for many settings [Jac00]. In
general, there is tension between the twin goals of (i) a dominant strategy
incentive-compatible mechanism, and (ii) subsidies and taxes adding up to zero
[Jac00].

Our trade deadline mechanism has no subsidy, and is most impactful when the
penalty for vetoing contracts is zero. Trade deadlines for player trades have
been studied in sports economics. Unlike our setting, the purpose of sports
trade deadlines is not to limit strategic behavior. Sports trade deadlines
create new opportunities for strategic behavior via over or under-bidding on
certain players [Zim02] and creating inefficiencies in secondary betting
markets [DFÖ18].

##  8 Conclusion

Contract negotiations in the real world need a judicious mix of cooperation
and competition. One who is too honest and cooperative leaves himself open to
manipulation. However, unrestricted competitive behavior leads to a breakdown
of trust. Without trust, the economy falters [Ho22].

Our paper is an attempt to model this phenomenon in a tractable manner. In our
model, all agents trade off reward against risk via personal utility
functions. The agents cooperate by accommodating each other’s utility
functions and paying a price when necessary to make the contract agreeable to
others. They compete by deceiving others about their utility functions. But,
like in the real world, deception has limited utility. The larger the
deception, the less the other agents are willing to engage. This leads to
smaller contracts for the deceiver, reducing his utility. Thus, our model
leads to Nash equilibria among strategic agents. Furthermore, we present an
efficient algorithm to find these equilibria.

The above algorithm needs to know both the agents’ public deceptions and
private beliefs. Hence, those with better information can be more strategic,
again echoing the real world. Being strategic confers a significant jump in
utility compared to being honest. To reduce the need for strategy, we
introduce the trade deadline mechanism. The mechanism allows agents to veto
contracts at the deadline. The veto threat compels strategic agents to share
utility with their counterparties. Hence, the trade deadline reduces the gap
between strategic and honest agents. Experiments on simulated as well as real-
world datasets validate these ideas.

There are several directions for future work. This paper focuses on a
sequential process for contract negotiations. It begins with a single strategy
phase, followed by forming the contract network and ending with reward
collection. But contract negotiations are often iterative processes, with
repeated interactions between agents. Each iteration offers all agents the
chance to update their negotiation positions. In such negotiations, we would
seek feedback Nash equilibria for the resulting dynamic game [BO98].

Another direction is to find the Nash equilibrium under the trade deadline
mechanism. Given the computational intractability of the Robust Veto problem,
we only analyze Nash equilibria in the trade deadline setting for specific
networks. Analyzing the equilibria for more general networks is an interesting
direction for future work.

Our work can also be extended to include all private beliefs of agents in the
strategy. Our work only considers negotiating positions based on mean rewards.
But an agent can also deceive others about their risk aversion parameter
$\gamma$ or their correlation matrix $\Sigma$. For example, an agent may
induce counterparties to offer better terms by appearing to be risk-averse.
Similarly, an agent can try to convince others that she is negatively
correlated with other agents, so contracts with her serve to reduce risk.

Finally, we have only considered mean-variance utility functions. However,
many other utility functions have been studied in the literature. While mean-
variance utility leads to tractable Nash equilibria, other utility functions
may be more appropriate in some settings.

##  9 Proofs

###  9.1 Proof of Theorem 3.4

######  Proposition 9.1.

Let $k\in[n]$. Let $\bm{\delta}_{-k}:=\\{\bm{\delta}_{i}:i\neq k\\}$ and let
$(W,P)$ be the stable point if $k$ reports honestly and all others report
according to $\bm{\delta}_{-k}$. Next, consider some
$\bm{\delta}_{k}\neq\bm{0}$ and let $(W^{\prime},P^{\prime})$ be the stable
point if $k$ reports $\bm{\delta}_{k}$ and all others report according to
$\bm{\delta}_{-k}$. Then $W^{\prime}\bm{e}_{k}=W\bm{e}_{k}+B\bm{\delta}_{k}$
for a matrix $B$ defined as follows. Let
$\Gamma^{-1/2}\Sigma\Gamma^{-1/2}=V\Lambda V^{T}$ be the eigendecomposition of
$\Gamma^{-1/2}\Sigma\Gamma^{-1/2}\succ 0$. Let $A\in\RR^{n\times n}$ be a
symmetric matrix such that:

$A_{ij}=\begin{cases}\frac{V_{ki}^{2}}{4\lambda_{i}}+\sum\limits_{\ell=1}^{n}\frac{V_{k\ell}^{2}}{2(\lambda_{i}+\lambda_{\ell})}&i=j\\\
\frac{V_{ki}V_{kj}}{2(\lambda_{i}+\lambda_{j})}&i\neq j\end{cases}$

Then $B=\gamma_{k}^{-1}\Gamma^{-1/2}VAV^{T}\Gamma^{-1/2}$.

######  Proof 9.2.

Let $\Delta_{W}=W^{\prime}-W$. By Theorem 2.1,
$2(\Sigma\Delta_{W}\Gamma+\Gamma\Delta_{W}\Sigma)=\bm{e}_{k}\bm{\delta}_{k}^{T}+\bm{\delta}_{k}\bm{e}_{k}^{T}$.
Therefore
$\textrm{vec}(\Delta_{W})=\frac{1}{2}(\Sigma\otimes\Gamma+\Gamma\otimes\Sigma)^{-1}(\bm{e}_{k}\bm{\delta}_{k}^{T}+\bm{\delta}_{k}\bm{e}_{k}^{T})$.

Next, let $\bm{v}_{i}:=V\bm{e}_{i}$. Using the eigendecomposition properties
of Kronecker sums [HJ08] as in Corollary 1 of [JCS24],

$\displaystyle\Delta_{W}$
$\displaystyle=\Gamma^{-1/2}\bigg{(}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\frac{\bm{v}_{i}^{T}\Gamma^{-1/2}\big{(}\bm{e}_{k}\bm{\delta}_{k}^{T}+\bm{\delta}_{k}\bm{e}_{k}^{T}\big{)}\Gamma^{-1/2}\bm{v}_{j}}{2(\lambda_{i}+\lambda_{j})}\bm{v}_{i}\bm{v}_{j}^{T}\bigg{)}\Gamma^{-1/2}$

Let $G:=\Gamma^{-1/2}V$ and $\bm{g}_{i}:=G\bm{e}_{i}$. Then
$\bm{v}_{i}^{T}\Gamma^{-1/2}\bm{e}_{k}=G_{ki}$ and
$\bm{e}_{k}^{T}\Gamma^{-1/2}\bm{v}_{j}=G_{kj}$. Hence:

$\displaystyle\Delta_{W}\bm{e}_{k}$
$\displaystyle=\Gamma^{-1/2}\bigg{(}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\frac{G_{ki}\bm{g}_{j}^{T}\bm{\delta}_{k}+G_{kj}\bm{g}_{i}^{T}\bm{\delta}_{k}}{2(\lambda_{i}+\lambda_{j})}\bm{v}_{i}\bm{v}_{j}^{T}\bigg{)}\Gamma^{-1/2}\bm{e}_{k}$
$\displaystyle=\Gamma^{-1/2}\bigg{(}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\frac{G_{ki}\bm{g}_{j}^{T}\bm{\delta}_{k}+G_{kj}\bm{g}_{i}^{T}\bm{\delta}_{k}}{2(\lambda_{i}+\lambda_{j})}G_{kj}\bm{v}_{i}\bigg{)}$
$\displaystyle=\Gamma^{-1/2}\bigg{(}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\frac{G_{ki}G_{kj}}{2(\lambda_{i}+\lambda_{j})}\bm{v}_{i}\bm{v}_{j}^{T}+\sum\limits_{j=1}^{n}G_{kj}^{2}\cdot\bigg{(}\sum\limits_{i=1}^{n}\bigg{(}\frac{1}{2(\lambda_{i}+\lambda_{j})}\bm{v}_{i}\bm{v}_{i}^{T}\bigg{)}\bigg{)}\bigg{)}\Gamma^{-1/2}\bm{\delta}_{k}$

Hence $W^{\prime}\bm{e}_{k}-W\bm{e}_{k}=B\bm{\delta}_{k}$ for a matrix $B$
defined as above. We can further simplify $B$ as

$\displaystyle B$
$\displaystyle=\Gamma^{-1/2}\bigg{(}\sum\limits_{i=1}^{n}\bigg{(}\frac{G_{ki}^{2}}{4\lambda_{i}}+\sum\limits_{j=1}^{n}\frac{G_{kj}^{2}}{2(\lambda_{i}+\lambda_{j})}\bigg{)}\bm{v}_{i}\bm{v}_{i}^{T}+\sum\limits_{i=1}^{n}\sum\limits_{j\neq
i}\frac{G_{ki}G_{kj}}{2(\lambda_{i}+\lambda_{j})}\bm{v}_{i}\bm{v}_{j}^{T}\bigg{)}\Gamma^{-1/2}$

Notice that $G_{ki}=\gamma_{k}^{-1/2}V_{ki}$, and that $B$ depends only on the
$k^{th}$ row of $G$ so we can factor out $\gamma_{k}^{-1}$ and replace the
entries $G_{ki}$ with $V_{ki}$.

Finally, let $A\in\RR^{n\times n}$ be defined as in the statement of this
Proposition. Then we conclude that

$B=\gamma_{k}^{-1}\Gamma^{-1/2}\big{(}\sum\limits_{i=1}^{n}A_{ii}\bm{v}_{i}\bm{v}_{i}^{T}+\sum\limits_{i=1}^{n}\sum\limits_{j\neq
i}A_{ij}\bm{v}_{i}\bm{v}_{j}^{T}\big{)}\Gamma^{-1/2}=\gamma_{k}^{-1}\Gamma^{-1/2}VAV^{T}\Gamma^{-1/2}$

######  Proposition 9.3.

Let $K=\Lambda^{1/2}A\Lambda^{1/2}$. The eigenvalues of $K$ are all real and
contained in $(0,1)$.

######  Proof 9.4.

Notice that $K$ is symmetric, since $\Lambda$ is diagonal and $A$ is
symmetric. By the Spectral Theorem, $K$ has an eigendecompsition with real
eigenvalues.

Next, notice $A=C+D$ for
$C_{ij}=C_{ji}=\frac{V_{ki}V_{kj}}{2(\lambda_{i}+\lambda_{j})}$ and $D$ a
diagonal matrix with
$D_{ii}=\sum\limits_{\ell=1}^{n}\frac{V_{k\ell}^{2}}{2(\lambda_{i}+\lambda_{\ell})}$.
Then $K=\tilde{C}+\tilde{D}$ for $\tilde{C}=\Lambda^{1/2}C\Lambda^{1/2}$ and
$\tilde{D}=\Lambda^{1/2}D\Lambda^{1/2}$.

Let $\bm{x}=\frac{1}{\sqrt{2}}\Lambda^{1/2}V^{T}\bm{e}_{k}$. Then $\tilde{C}$
satisfies the Lyapunov equation:

$\Lambda\tilde{C}+\tilde{C}\Lambda=\bm{x}\bm{x}^{T}$

Since $\tilde{C}$ is self-adjoint it has an eigenbasis with real eigenvalues.
Let $\bm{y}$ be an eigenvector of $\tilde{C}$ with eigenvalue $\mu$. Then
$(\bm{y}^{T}\bm{x})^{2}=\bm{y}^{T}(\tilde{C}\Lambda+\Lambda\tilde{C})\bm{y}=2\mu\bm{y}^{T}\Lambda\bm{y}$.
By the Cauchy-Schwarz inequality,

$\displaystyle\mu$
$\displaystyle=\frac{(\bm{y}^{T}\bm{x})^{2}}{2\bm{y}^{T}\Lambda\bm{y}}=\frac{1}{4}\frac{(\sum\limits_{i=1}^{n}\sqrt{\lambda_{i}}y_{i}V_{ki})^{2}}{\sum\limits_{i=1}^{n}\lambda_{i}y_{i}^{2}}\leq\frac{1}{4}\frac{(\sum\limits_{i=1}^{n}\lambda_{i}y_{i}^{2})(\sum\limits_{i=1}^{n}V_{ki}^{2})}{\sum\limits_{i=1}^{n}\lambda_{i}y_{i}^{2}}\leq\frac{1}{4}$

Further, since $\Lambda\succ 0$, $\mu\geq 0$. So the eigenvalues of
$\tilde{C}$ are all within $[0,\frac{1}{4}]$. Next, the eigenvalues of
$\tilde{D}$ are simply its diagonal entries. Recall that

$\displaystyle\tilde{D}_{ii}$
$\displaystyle=\sum\limits_{j}\frac{\lambda_{i}V_{kj}^{2}}{2(\lambda_{i}+\lambda_{j})}>\frac{\lambda_{i}}{2(\lambda_{i}+\max_{j}\lambda_{j})}\sum_{j}V_{kj}^{2}>0,$

since $V$ is orthonormal and $\lambda_{j}>0$. By similar reasoning,
$\tilde{D}_{ii}<\frac{1}{2}$. We conclude that the eigenvalues of $K$ are
contained in $(0,\frac{3}{4})$.

We are ready to prove Theorem 3.4.

######  Proof 9.5 (Proof of Theorem 3.4.).

Fix the index $k$ of the strategic actor. Suppose $k$ reports
$\bm{\delta}_{k}$ and each $i\neq k$ reports some $\bm{\delta}_{i}$. Let
$(W^{\prime},P^{\prime})$ be the resulting stable point.

By Lemma 3.7, the utility of $k$ at $(W^{\prime},P^{\prime})$ is
$-\la\bm{\delta}_{k},\bm{w}_{k}^{\prime}\ra+\gamma_{k}\la\bm{w}_{k}^{\prime},\Sigma\bm{w}_{k}^{\prime}\ra$
where $\bm{w}_{k}^{\prime}$ is the $k^{th}$ column of $W^{\prime}$.

By Proposition 9.1, $\bm{w}_{k}^{\prime}=\bm{w}_{k}+B\bm{\delta}_{k}$ for
$B=\gamma_{k}^{-1}\Gamma^{-1/2}VAV^{T}\Gamma^{-1/2}$ with $V$ and $A$ as in
Proposition 9.1. Hence the utility of $k$ is quadratic in $\bm{\delta}_{k}$,
and the quadratic term is $-\bm{\delta}_{k}^{T}\big{(}B-\gamma_{k}B\Sigma
B\big{)}\bm{\delta}_{k}$. A straightforward calculation gives:

$B-\gamma_{k}B\Sigma
B=\gamma_{k}^{-1}\Gamma^{-1/2}V\Lambda^{-1/2}\bigg{(}\Lambda^{1/2}A\Lambda^{1/2}-\big{(}\Lambda^{1/2}A\Lambda^{1/2}\big{)}^{2}\bigg{)}\Lambda^{-1/2}V^{T}\Gamma^{-1/2}$

Let $K:=\Lambda^{1/2}A\Lambda^{1/2}$. To show that the Hessian of the utility
of $k$ is negative definite in $\bm{\delta}_{k}$, we need to show
$K-K^{2}\succ 0$. Since the spectrum of $K$ is contained in $(0,1)$ by
Proposition 9.3, $K\succ K^{2}$ and the conclusion follows.

###  9.2 Proof of Theorem 5.3.

######  Proof 9.6 (Proof of Theorem 5.3. ).

Let
$\Sigma_{i}:=\gamma_{i}\Sigma,\bm{\mu}_{i}=M\bm{e}_{i},\bm{w}_{i}^{\prime}=W^{\prime}\bm{e}_{i},\bm{w}_{i}=W\bm{e}_{i}$,
and $\bm{y}=\bm{w}_{i}-\bm{w}_{i}^{\prime}$. Then:

$\displaystyle g_{i}(W,P)$
$\displaystyle=(\bm{\mu}_{i}-P\bm{e}_{i})^{T}\bm{w}_{i}-\bm{w}_{i}^{T}\Sigma_{i}\bm{w}_{i}$
$\displaystyle=(\bm{\mu}_{i}-P\bm{e}_{i})^{T}\bm{w}_{i}^{\prime}+(\bm{\mu}_{i}-P\bm{e}_{i})^{T}\bm{y}-(\bm{w}_{i}^{\prime})^{T}\Sigma_{i}\bm{w}_{i}^{\prime}-2(\bm{w}_{i}^{\prime})^{T}\Sigma_{i}\bm{y}-\bm{y}^{T}\Sigma_{i}\bm{y}$
$\displaystyle=g(W^{\prime},P)+(\bm{\mu}_{i}-P\bm{e}_{i})^{T}\bm{y}-\bm{y}^{T}\Sigma_{i}(2\bm{w}_{i}^{\prime}+\bm{y})$
$\displaystyle=g(W^{\prime},P)+(\bm{\mu}_{i}-P\bm{e}_{i})^{T}\bm{y}-\bm{y}^{T}\Sigma_{i}(2\bm{w}_{i}-\bm{y})$
$\displaystyle=g(W^{\prime},P)+(\bm{\mu}_{i}-P\bm{e}_{i})^{T}\bm{y}-\bm{y}^{T}\Sigma_{i}\bigg{(}\Sigma_{i}^{-1}(\bm{\mu}_{i}-P\bm{e}_{i})\bigg{)}+\bm{y}^{T}\Sigma_{i}\bm{y}$
$\displaystyle=g(W^{\prime},P)+\bm{y}^{T}\Sigma_{i}\bm{y}$

Since $\Sigma_{i}\succ 0$, we conclude that $g(W^{\prime},P)\leq g(W,P)$, with
equality if and only if $\bm{w}_{i}^{\prime}=\bm{w}_{i}$.

###  9.3 Proof of Theorem 5.4

Our proof is based on L2-ShortestVectorProblem, which concerns finding the
shortest vector in a discrete lattice within $\RR^{d}$ given a basis. We will
show a series of deterministic polynomial time reductions that go from
L2-ShortestVectorProblem to the problem of finding a Robust Veto (Definition
5.2). Combining these reductions, we prove Theorem 5.4 at the end of this
section.

######  Theorem 9.7 ([Ajt98]).

There is a reverse-unfaithful-randomized (RUR) reduction from SubsetSum on $n$
variables to L2-ShortestVectorProblem on $(n+2)$ variables.

We treat RUR reductions as a black box; for details on randomized reductions,
we refer the reader to [AB09]. Next, we will need the following definitions.

######  Definition 9.8 (L2-IntegerQuadraticMinimization).

Given $Q\succ 0$ and $\bm{c}\in\RR^{n}$, find $\bm{x}^{*}$ such that:

$\displaystyle\bm{x}^{*}=\arg\min\limits_{\bm{x}\in\ZZ^{n}}\bm{x}^{T}Q\bm{x}+\bm{c}^{T}\bm{x}$

######  Definition 9.9 (L2-BinaryQuadraticMinimization).

Given $Q\succ 0$ and $\bm{c}\in\RR^{n}$, find $\bm{x}^{*}$ such that:

$\displaystyle\bm{x}^{*}=\arg\min\limits_{\bm{x}\in\\{0,1\\}^{n}}\bm{x}^{T}Q\bm{x}+\bm{c}^{T}\bm{x}$

The following reduction is standard.

######  Proposition 9.10.

There is a deterministic polynomial time reduction from
L2-ShortestVectorProblem to L2-IntegerQuadraticMinimization.

Next, we will reduce from L2-BinaryQuadraticMinimization to the Robust Veto
5.2.

######  Proposition 9.11.

There is a deterministic polynomial time reduction from
L2-BinaryQuadraticMinimization to Robust Veto.

######  Proof 9.12.

Given an instance of L2-BinaryQuadraticMinimization with some $Q,\bm{c}$, let
$\theta=0$, $\bm{w}_{i}=\bm{v}_{-i}=(1,1,\dots,1)^{T}$ and $P=0$. Let
$\bm{\mu}_{i}=-\bm{c}$ and $\Sigma_{i}=Q,\gamma_{i}=1$. Then
$\la\bm{v}_{i}\odot\bm{v}_{-i}\odot\bm{w}_{i},\bm{\mu}_{i}-P\bm{e}_{i}\ra=\bm{c}^{T}\bm{v}_{i}$
and
$-\gamma\la\bm{v}_{i}\odot\bm{v}_{-i}\odot\bm{w}_{i},\Sigma_{i}(\bm{v}_{i}\odot\bm{v}_{-i}\odot\bm{w}_{i})\ra=\bm{v}_{i}^{T}Q\bm{v}_{i}$.
Hence minimizing the L2-BinaryQuadraticMinimization objective is equivalent to
the objective of Definition 5.2. Moreover, we obtain a valid problem instance
since $\Sigma_{i}=Q\succeq 0$ since $Q\succ 0$ by assumption, and
$\gamma_{i}>0,P^{T}=-P$.

Finally, we will use a recent reduction from a constrained version of
L2-IntegerQuadraticMinimization to L2-BinaryQuadraticMinimization.

######  Proposition 9.13 ([Alb+23]).

Consider the following constrained version of
L2-IntegerQuadraticMinimization on $n$ variables. Given data $Q\succeq
0,\bm{c}\in\RR^{n}$, and $a>0$, find $\bm{x}^{*}$ such that:

$\displaystyle\bm{x}^{*}=\arg\min\limits_{\bm{x}\in\ZZ^{n}}$
$\displaystyle\bm{x}^{T}Q\bm{x}+\bm{c}^{T}\bm{x}$
$\displaystyle\text{s.t.}\quad\forall i\quad$
$\displaystyle\abs{\bm{x}_{i}}\leq a$

There is a deterministic polynomial time reduction from the constrained
problem on $n$ variables to an instance of L2-BinaryQuadraticMinimization on
$m$ variables, for $m\leq n\log(2a)$.

We are ready to prove Theorem 5.4.

######  Proof 9.14 (Proof of Theorem 5.4.).

Let $n$ be a positive integer and $\mathcal{A}$ be an instance of SubsetSum on
$n$ variables. The reduction of Theorem 9.7 gives a lattice $\mathcal{L}$ in
$n+2$ dimensions with basis vectors $\bm{v}_{1},\dots,\bm{v}_{n+2}$. Call this
instance $\mathcal{B}$. $\mathcal{L}$ has the property that if
$\sum\limits_{i\in[n+2]}\gamma_{i}\bm{v}_{i}$ is a shortest vector (with
$\gamma_{i}\in\ZZ$), then $\gamma_{i}\in\\{0,1\\}$ for $i\leq n$, and
$\gamma_{i+1}=1$. Moreover, the set of such coefficient sequences
$(\gamma_{1},\dots,\gamma_{n+2})$ for shortest vectors of $\mathcal{L}$ has
size at most exponential in $n$. Therefore, by Prop 9.10, there exists a
deterministic reduction from $\mathcal{B}$ to the following constrained
version of L2-IntegerQuadraticMinimization for $\bm{c}\in\RR^{n+2},Q\succ 0$.

$\displaystyle\min\limits_{\bm{x}\in\ZZ^{n+2}}$
$\displaystyle\bm{x}^{T}Q\bm{x}+\bm{c}^{T}\bm{x}$
$\displaystyle\text{s.t.}\quad\forall i\leq n\quad$
$\displaystyle\abs{\bm{x}_{i}}\in\\{0,1\\}$ $\displaystyle\bm{x}_{n+1}=1$
$\displaystyle\log(\abs{\bm{x}_{n+2}})\leq n^{O(1)}$

Call this constrained instance $\mathcal{C}$. By Prop 9.13, there is a
deterministic reduction from $\mathcal{C}$ to an instance $\mathcal{D}$ of
L2-BinaryQuadraticMinimization on $(n+2)\cdot n^{O(1)}=n^{O(1)}$ variables.

Next, by Prop 9.11, there is a deterministic reduction from $\mathcal{D}$ to
an instance $\mathcal{E}$ of the Robust Veto problem 5.2.

Since all of our reductions were deterministic, our overall reduction from
$\mathcal{A}$ to $\mathcal{E}$ has the same one-sided error as the reduction
from $\mathcal{A}$ to $\mathcal{B}$.

We conclude that there is a reverse unfaithful random (RUR) reduction from an
instance $\mathcal{A}$ of SubsetSum on $n$ variables to an instance
$\mathcal{E}$ of the Robust Veto problem on $n^{O(1)}$ variables. Since
SubsetSum is NP-hard, if $\textbf{NP}\neq\textbf{RP}$ then Robust Veto problem
is not in RP.

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##  Appendix A Additional Proofs

###  A.1 Proof of Lemma 3.7

######  Proof A.1 (Proof of Lemma 3.7).

Let $W^{\prime},P^{\prime}$ be the new stable point and
$\bm{w}_{k}^{\prime}=W^{\prime}\bm{e}_{k}$. Let
$\bm{\mu}_{k}^{\prime}=\bm{\mu}_{k}+\bm{\delta}_{k}$. Then:

$\displaystyle g_{k}$
$\displaystyle=\la\bm{w}_{k}^{\prime},\bm{\mu}_{k}-P^{\prime}\bm{e}_{k}\ra-\gamma_{k}\la\Sigma\bm{w}_{k}^{\prime},\bm{w}_{k}^{\prime}\ra$
$\displaystyle=\la\bm{w}_{k}^{\prime},\bm{\mu}_{k}-\bm{\mu}_{k}^{\prime}\ra+\la\bm{w}_{k}^{\prime},\bm{\mu}_{k}^{\prime}-P^{\prime}\bm{e}_{k}\ra-\gamma_{k}\la\Sigma\bm{w}_{k}^{\prime},\bm{w}_{k}^{\prime}\ra$

Notice that since
$\bm{w}_{k}^{\prime}=\frac{1}{2\gamma_{k}}\Sigma^{-1}(\bm{\mu}_{k}^{\prime}-P^{\prime}\bm{e}_{k})$
that
$\la\bm{w}_{k}^{\prime},\bm{\mu}_{k}^{\prime}-P^{\prime}\bm{e}_{k}\ra=2\gamma_{k}\la\Sigma\bm{w}_{k}^{\prime},\bm{w}_{k}^{\prime}\ra$.
The conclusion follows.

###  A.2 Non-Convexity of Constraints

In this section, we will show that the constraint $-g_{j;i}\leq\theta$ is not
convex in $\bm{\delta}_{i}$ in general.

######  Proposition A.2.

Let $\bm{\delta}_{-i}$ be fixed. If $j\neq i$, then the individual utility
$g_{j;i}$ is quadratic in $\bm{\delta}_{i}$ and its Hessian with respect to
$\bm{\delta}_{i}$ is of the form $\bm{v}\bm{z}^{T}+\bm{z}\bm{v}^{T}$ for
vectors $\bm{v},\bm{z}$. Moreover, $\bm{z}\neq\bm{v}$ unless
$(2\gamma_{i}B\Sigma+B+\gamma_{i}\Sigma_{ii}I)\bm{e}_{j}=\bm{e}_{j}$, where
$B$ is as in Proposition 9.1. Therefore the Hessian has a positive eigenvalue
unless $(2\gamma_{i}B\Sigma+B+\gamma_{i}\Sigma_{ii}I)\bm{e}_{j}=\bm{e}_{j}$

######  Proof A.3.

From the proof of Theorem 3.4, we know that if $(W,P)$ is the stable point at
$(\bm{0},\bm{\delta}_{-i})$ and $(W^{\prime},P^{\prime})$ is the stable point
at $(\bm{\delta}_{i},\bm{\delta}_{-i})$ then
$\bm{w}_{i}^{\prime}=\bm{w}_{i}+B\bm{\delta}_{i}$ for a symmetric $B$.
Moreover, we know that if $M^{\prime}$ is the matrix of reported means at
$(\bm{\delta}_{i},\bm{\delta}_{-i})$ then $P^{\prime}=-2\Sigma
W^{\prime}\Gamma+M^{\prime}$. In particular, $i$ gives $j$ a payment per unit
contract of
$P_{ji}^{\prime}=-2\gamma_{i}\bm{e}_{j}^{T}\Sigma(W\bm{e}_{i}+B\bm{\delta}_{i})+\bm{e}_{j}^{T}M\bm{e}_{i}+\bm{e}_{j}^{T}\bm{\delta}_{i}$.
Therefore the individual utility $g_{j;i}$ can be written as:

$\displaystyle g_{j;i}$
$\displaystyle=w_{ij}^{\prime}(\bm{\mu}_{j;i}+P_{ji}^{\prime})-\gamma_{i}\Sigma_{ii}(w_{ij}^{\prime})^{2}$

Since $W^{\prime},P^{\prime}$ are linear in $\bm{\delta}_{i}$, it is clear
that $g_{j;i}$ is quadratic in $\bm{\delta}_{i}$. Next, let
$\nabla^{2}g_{j;i}$ be the Hessian with respect to $\bm{\delta}_{i}$. Only the
terms of $g_{j;i}$ that are quadratic in $\bm{\delta}_{i}$ appear in the
Hessian. Therefore,

$\displaystyle\nabla^{2}g_{j;i}$
$\displaystyle=\nabla^{2}(w_{ij}^{\prime}P_{ji}^{\prime})-\gamma_{i}\Sigma_{ii}\nabla^{2}(w_{ij}^{\prime})^{2}$
$\displaystyle=\nabla^{2}\bigg{(}\bm{e}_{j}^{T}B\bm{\delta}_{i}\cdot\big{(}-2\gamma_{i}\bm{e}_{j}^{T}\Sigma(W\bm{e}_{i}+B\bm{\delta}_{i})+\bm{e}_{j}^{T}M\bm{e}_{i}+\bm{e}_{j}^{T}\bm{\delta}_{i}\big{)}\bigg{)}-\gamma_{i}\Sigma_{ii}\nabla^{2}\bigg{(}(\bm{e}_{j}^{T}B\bm{\delta}_{i})^{2}\bigg{)}$
$\displaystyle=\nabla^{2}\bigg{(}\bm{\delta}_{i}^{T}B^{T}\bm{e}_{j}\bm{e}_{j}^{T}\bm{\delta}_{i}\bigg{)}-\nabla^{2}\bigg{(}\bm{\delta}_{i}^{T}B^{T}\bm{e}_{j}\big{(}2\gamma_{i}\bm{e}_{j}^{T}\Sigma
B\bm{\delta}_{i}\big{)}\bigg{)}-\gamma_{i}\Sigma_{ii}\nabla^{2}\bigg{(}\bm{\delta}_{i}^{T}B^{T}\bm{e}_{j}\bm{e}_{j}^{T}B\bm{\delta}_{i}\bigg{)}$
$\displaystyle=B^{T}\bm{e}_{j}\bm{e}_{j}^{T}+\bm{e}_{j}\bm{e}_{j}^{T}B-2\gamma_{i}B^{T}\bm{e}_{j}\bm{e}_{j}^{T}\Sigma
B-2\gamma_{i}B^{T}\Sigma\bm{e}_{j}\bm{e}_{j}^{T}B-2\gamma_{i}\Sigma_{ii}B^{T}\bm{e}_{j}\bm{e}_{j}^{T}B$
$\displaystyle=B\bm{e}_{j}\bm{e}_{j}^{T}+\bm{e}_{j}\bm{e}_{j}^{T}B-2\gamma_{i}B\bm{e}_{j}\bm{e}_{j}^{T}\Sigma
B-2\gamma_{i}B\Sigma\bm{e}_{j}\bm{e}_{j}^{T}B-2\gamma_{i}\Sigma_{ii}B\bm{e}_{j}\bm{e}_{j}^{T}B$

Let $\bm{v}=B\bm{e}_{j}$ and
$\bm{z}=\bm{e}_{j}-2\gamma_{i}B\Sigma\bm{e}_{j}-\gamma_{i}\Sigma_{ii}\bm{v}$.
Then $\nabla^{2}g_{j;i}=\bm{v}\bm{z}^{T}+\bm{z}\bm{v}^{T}$.

###  A.3 Proof of Theorem 4.3

Throughout this section, we will refer to the investor as P1 and the hedge
funds as P2, P3. Hence P1 has beliefs according to $M\bm{e}_{1}$, and so on.

######  Proposition A.4.

Assume that P1 reports $M_{21},M_{31}$ as $\tilde{m}$, P2 reports $M_{12}$ as
$\tilde{a}$, and P3 reports $M_{13}$ as $\tilde{b}$. Then
$w_{21}=0.5\cdot(\tilde{\alpha}+\tilde{a}\nu-\tilde{b}\eta)$ and
$w_{31}=0.5\cdot(\tilde{\alpha}+\tilde{b}\nu-\tilde{a}\eta)$ for
$\tilde{\alpha}=\frac{\tilde{m}}{2+\rho}$.

######  Proof A.5.

Notice that $\Sigma$ has eigenvalues
$\lambda_{1}=1,\lambda_{2}=1+\rho,\lambda_{3}=1-\rho$ and corresponding
eigenvectors
$\bm{v}_{1}=(1,0,0)^{T},\bm{v}_{2}=\frac{1}{\sqrt{2}}(0,1,1)^{T},\bm{v}_{3}=\frac{1}{\sqrt{2}}(0,1,-1)^{T}$.
Therefore $2^{-1/2}(\bm{w}_{2}+\bm{w}_{3})=W\bm{v}_{2}$. Let $\tilde{M}$ be
the matrix of reported values, so $\tilde{M}_{21}=\tilde{M}_{31}=\tilde{m}$,
$\tilde{M}_{12}=\tilde{a}$, and $\tilde{M}_{13}=\tilde{b}$. All other entries
of $\tilde{M}$ are zero.

From Theorem 2.1, the resulting network is
$W=\sum\limits_{i,j\in[3]}\frac{\bm{v}_{i}^{T}(\tilde{M}+\tilde{M}^{T})\bm{v}_{j}}{2(\lambda_{i}+\lambda_{j})}\bm{v}_{i}\bm{v}_{j}$.
By orthogonality of eigenvectors, we have:

$W\bm{v}_{2}=\sum\limits_{i}\frac{\bm{v}_{i}^{T}(\tilde{M}+\tilde{M}^{T})\bm{v}_{2}}{2(\lambda_{i}+\lambda_{2})}\bm{v}_{i}$

Only the term at $i=1$ is nonzero, and therefore
$2^{-1/2}(\bm{w}_{2}+\bm{w}_{3})=\frac{2\tilde{m}+\tilde{a}+\tilde{b}}{2\sqrt{2}(2+\rho)}\bm{e}_{1}$.
Similarly,
$2^{-1/2}(\bm{w}_{2}-\bm{w}_{3})=\frac{\tilde{a}-\tilde{b}}{2\sqrt{2}(2-\rho)}\bm{e}_{1}$.
The conclusion follows.

######  Proposition A.6.

Let $\tilde{\alpha}$ be as in Proposition A.4. Assume P1 reports
$\tilde{\mu}$, and P3 reports $\tilde{b}$. The optimal choice of reported
$\tilde{a}$ for P2 is:

$\displaystyle\tilde{a}^{*}=c_{a}+s\tilde{b}$

For $c_{a}=\frac{a\nu-\tilde{\alpha}(1-\nu)}{\nu(2-\nu)}$ and
$s=\frac{\eta(1-\nu)}{\nu(2-\nu)}$.

######  Proof A.7.

Let $w:=w_{21}$ for shorthand. The utility of firm $2$, by Lemma 3.7, is given
by:

$\displaystyle g_{2}$ $\displaystyle=-w(\tilde{a}-a)+w^{2}$
$\displaystyle=w(a-\tilde{a}+w)$ $\displaystyle 2g_{2}$
$\displaystyle=(\tilde{\alpha}+\tilde{a}\nu-\tilde{b}\eta)(a+(0.5\nu-1)\tilde{a}+0.5\tilde{\alpha}-0.5\tilde{b}\eta)$

The coefficient of $\tilde{a}^{2}$ in $g_{2}$ is $\nu(0.5\nu-1)$. Since
$\nu>0$ and $0.5\nu<1$ for all $\rho\in(-1,1)$, the Hessian of $g_{2}$ with
respect to $\tilde{a}$ is negative definite, and so the optimal choice of $a$
is at the critical point $\frac{\del g_{2}}{\del\tilde{a}}=0$. Solving for
$\tilde{a}$ gives:

$\tilde{a}^{*}=c_{a}+s\tilde{b}$

With $c_{a},s$ as in the statement.

A symmetric argument gives the following.

######  Proposition A.8.

Let $\tilde{\alpha}$ be as in Proposition A.4. Assume P1 reports
$\tilde{\mu}$, and P2 reports $\tilde{a}$. The optimal choice of reported
$\tilde{b}$ for P3 is:

$\tilde{b}^{*}=c_{b}+s\tilde{a}$

For $c_{a}=\frac{b\nu-\tilde{\alpha}(1-\nu)}{\nu(2-\nu)}$ and
$s=\frac{\eta(1-\nu)}{\nu(2-\nu)}$.

Next, we can solve for the Nash equilibria given the reported $\tilde{m}$ of
the investor.

######  Proposition A.9.

If $M_{12}=M_{13}=a$, then let $c:=c_{a}=c_{b}$ and $s$ be as in Proposition
A.6 and A.8. Assume P1 reports $\tilde{\mu}$. The Nash equilibrium for P2, P3
is to report:

$\displaystyle M_{12}^{\prime}=M_{13}^{\prime}$ $\displaystyle=\frac{c}{1-s}$

######  Corollary A.10 (Theorem 4.3 Part 1).

Assume P1 reports $\tilde{m}$. If both hedge funds are strategic, then the
Nash equilibrium for P2, P3 is to report:

$\displaystyle M_{12}^{\prime}=M_{13}^{\prime}$
$\displaystyle=\frac{a\nu-\tilde{m}(1-\nu)(\nu-\eta)}{\nu+(1-\nu)(\nu-\eta)}$

Hence if $\tilde{m}=m$, then $M_{12}^{\prime}=M_{13}^{\prime}$ are as in
Theorem 4.3.1.

######  Proof A.11.

We simplify $a_{NS}=\frac{c}{1-s}$ as follows.

$\displaystyle\frac{c}{1-s}$
$\displaystyle=\frac{a\nu-\tilde{\alpha}(1-\nu)}{\nu(2-\nu)(1-s)}$
$\displaystyle=\frac{a\nu-\tilde{\alpha}(1-\nu)}{\nu(2-\nu)\big{(}1-\frac{\eta(1-\nu)}{\nu(2-\nu)}\big{)}}$
$\displaystyle=\frac{a\nu-\tilde{m}(1-\nu)(\nu-\eta)}{\nu(2-\nu)-\eta(1-\nu)}$
$\displaystyle=\frac{a\nu-\tilde{m}(1-\nu)(\nu-\eta)}{\nu+(1-\nu)(\nu-\eta)}$

In the setting of Theorem 4.3.1, the investor reports $\tilde{m}=m$. The
conclusion follows.

Next, we solve for the optimal report of the investor if all agents are
strategic.

######  Proposition A.12 (Theorem 4.3 Part 2).

Let $y=\frac{1}{2(2+\rho)}$. If all agents are strategic, then the optimal
reported $\tilde{m}$ for the investor is:

$M_{21}^{\prime}=M_{31}^{\prime}=\frac{m-a\zeta}{1+\zeta}$

For $\zeta=\frac{\nu-\eta}{\nu+(\nu-\eta)(1-\nu)}=(1-2y(1+\rho))$. The optimal
report for the hedge funds is:

$M_{12}^{\prime}=M_{13}^{\prime}=\frac{a\nu-
M_{21}^{\prime}(1-\nu)(\nu-\eta)}{\nu+(1-\nu)(\nu-\eta)}$

######  Proof A.13.

From Proposition A.9 and A.4, we know $w_{12}=w_{13}$. Therefore by Lemma 3.7,
if P1 reports $\tilde{m}$ then the investor utility is:

$\displaystyle g_{1}(\tilde{m})$
$\displaystyle=2(m-\tilde{m})w_{12}+w_{12}^{2}(2+2\rho)$

Let $w_{12}=(c_{2}+\tilde{m}y)$ for shorthand. Then $g_{1}$ is quadratic in
$\tilde{m}$ and the coefficient of $\tilde{m}^{2}$ is $2y^{2}+2\rho
y^{2}-2y=\frac{-(\rho+3)}{2(\rho+2)^{2}}<0$. Therefore, the optimal
$\tilde{m}$ is at the critical point $\frac{\del g_{1}}{\del\tilde{m}}=0$.
This is given as:

$\displaystyle\tilde{m}$
$\displaystyle=\frac{-c_{2}+my+2c_{2}y(1+\rho)}{2y(1-y(1+\rho)}$
$\displaystyle=\frac{m-(c_{2}/y)(1-2y(1+\rho))}{2(1-y(1+\rho))}$
$\displaystyle=\frac{m-(c_{2}/y)(1-2y(1+\rho))}{1+(1-2y(1+\rho))}$

We simplify the terms $(c_{2}/y),2y(1+\rho)$ as follows. First, notice that
$c_{2}=\frac{\nu-\eta}{2}\cdot\frac{a\nu}{\nu(1-\nu)(1-s)}$, where $a$ is true
value of $M_{12}$ and $M_{13}$ for the hedge funds. Let
$c_{1}:=\frac{a\nu}{\nu(1-\nu)(1-s)}$ for shorthand, so that
$c_{2}=\frac{\nu-\eta}{2}c_{1}$. Next,

$\displaystyle\frac{c_{2}}{y}$ $\displaystyle=\frac{c_{1}}{\nu-\eta}$
$\displaystyle=\frac{a}{(2-\nu)(1-s)}\cdot\frac{2(\nu-\eta)}{2(2+\rho)^{-1}(1-(\nu-\eta)x)}$
$\displaystyle=\frac{a}{(2-\nu)(1-s)}\cdot\bigg{(}1-(\nu-\eta)\cdot\frac{1-\nu}{\nu(2-\nu)(1-s)}\bigg{)}^{-1}$
$\displaystyle=\frac{a\nu}{\nu(2-\nu)(1-s)-(\nu-\eta)(1-\nu)}$
$\displaystyle=\frac{a\nu}{\nu(2-\nu)-\eta(1-\nu)-(\nu-\eta)(1-\eta)}$
$\displaystyle=\frac{a\nu}{\nu}$ $\displaystyle=a$ $\displaystyle
1-2y(1+\rho)$ $\displaystyle=1-2(c_{2}/a)(1+\rho)$
$\displaystyle=1-(1+\rho)\frac{c_{1}(\nu-\eta)}{a}$
$\displaystyle=1-(1+\rho)\cdot\frac{a}{(2-\nu)(1-s)}\frac{(\nu-\eta)}{a}$
$\displaystyle=1-\frac{(\nu-\eta)(1+\rho)\nu}{\nu(2-\nu)-\eta(1-\nu)}$
$\displaystyle=1-\frac{\nu(\nu-\eta)(1+\rho)}{\nu+(\nu-\eta)(1-\nu)}$
$\displaystyle=1-\frac{\nu(\nu-\eta)((2+\rho)-1)}{\nu+(\nu-\eta)(1-\nu)}$
$\displaystyle=1-\frac{\nu-\nu(\nu-\eta)}{\nu+(\nu-\eta)(1-\nu)}$
$\displaystyle=\frac{\nu-\eta}{\nu+(\nu-\eta)(1-\nu)}$

Let $\zeta:=\frac{\nu-\eta}{\nu+(\nu-\eta)(1-\nu)}=1-2y(1+\rho)$. The optimal
$\tilde{m}^{*}=\frac{m-a\zeta}{1+\zeta}$.

Finally, substituting this $\tilde{m}^{*}$ into Corollary A.10 gives the
optimal reports for the hedge funds.

###  A.4 Exact Formulas for Theorem 5.8

Throughout this section, we will use the notation $M_{21}=a,M_{31}=b$ and
$M_{12}=M_{13}=m$ for the true beliefs of the hedge funds and investor, and
$\tilde{a},\tilde{b},\tilde{m}$ for the reported beliefs.

######  Proposition A.14.

Assume that P1, P2 report
$M_{21}^{\prime}=\tilde{a},M_{31}^{\prime}=\tilde{b}$ and the investor
honestly reports $M_{12}^{\prime}=M_{13}^{\prime}=m=\tilde{m}$. Then there
exist constants
$\ell_{\theta},h_{\theta},\ell_{\theta}^{\prime},h_{\theta}^{\prime}$ such
that the optimal $\tilde{a},\tilde{b}$ under the trade deadline with penalty
$\theta>0$ are:

$\displaystyle\ell_{\theta}<\tilde{a}<h_{\theta}$
$\displaystyle\ell_{\theta}^{\prime}<\tilde{b}<h_{\theta}^{\prime}$

######  Proof A.15.

Let $(W,P)$ be the stable network if all report as in the statement of this
Proposition. Let $w=w_{21}$ for shorthand. Notice that
$g_{2}=w(a-P_{1;2})-w^{2}=w^{2}$ since, given the prices $P$, P2 will select
$w=1/2(a-P_{1;2})$. Therefore at the stable point, $P_{2;1}=-P_{1;2}=2w-a$.

For P2 to retain their contract with P1, they must report $\tilde{a}$ such
that the individual utility of the investor $g_{1;2}>-\theta$. If
$g_{1;2}=-\theta$, then we can solve for the roots of the resulting polynomial
in $a$. Notice $g_{1;2}=w(m-P_{2;1})-w^{2}=w(m-2w+a)-w^{2}$. We obtain:

$g_{1;2}>-\theta\iff-3w^{2}+mw+aw+\theta>0$

Let $g_{1;2}+\theta=f(\tilde{a})=w^{2}+mw-aw+\theta$ where
$w=0.5(\alpha+\tilde{a}\nu-\tilde{b}\eta)$ by Proposition A.4.

Let $\ell_{\theta}<h_{\theta}$ be the roots at which $f(\tilde{a})$ vanishes.
Since $\frac{\del^{2}f}{\del\tilde{a}^{2}}<0$, it follows that $g_{1;2}>0$ iff
$\ell_{\theta}<\tilde{a}<h_{\theta}$. The argument for $\tilde{b}$ is
identical.

We can give explicit formulas for the bounds as follows.

######  Proposition A.16.

Assume that P1, P2 report
$M_{21}^{\prime}=\tilde{a},M_{31}^{\prime}=\tilde{b}$ and the investor
honestly reports $M_{12}^{\prime}=M_{13}^{\prime}=m=\tilde{m}$. Then
$\ell_{\theta}<\tilde{a}<h_{\theta}$ and
$\ell_{\theta}^{\prime}<\tilde{b}<h_{\theta}^{\prime}$, where:

$\displaystyle\ell_{\theta}$
$\displaystyle=\frac{a-3\tilde{\alpha}+3\tilde{b}\eta+m-\sqrt{(a+m)^{2}+12\theta}}{3\nu}$
$\displaystyle h_{\theta}$
$\displaystyle=\frac{a-3\tilde{\alpha}+3\tilde{b}\eta+m+\sqrt{(a+m)^{2}+12\theta}}{3\nu}$
$\displaystyle\ell_{\theta}^{\prime}$
$\displaystyle=\frac{b-3\tilde{\alpha}+3\tilde{a}\eta+m-\sqrt{(b+m)^{2}+12\theta}}{3\nu}$
$\displaystyle h_{\theta}^{\prime}$
$\displaystyle=\frac{b-3\tilde{\alpha}+3\tilde{a}\eta+m+\sqrt{(b+m)^{2}+12\theta}}{3\nu}$

If $a=b$ then $\ell_{\theta}=\ell_{\theta}^{\prime}$ and
$h_{\theta}=h_{\theta}^{\prime}$, where:

$\displaystyle\ell_{\theta}$
$\displaystyle=\frac{3\tilde{\alpha}-(a+m)-\sqrt{(a+m)^{2}+12\theta}}{3\nu}$
$\displaystyle h_{\theta}$
$\displaystyle=\frac{3\tilde{\alpha}-(a+m)+\sqrt{(a+m)^{2}+12\theta}}{3\nu}$

######  Proposition A.17.

Let $\tilde{a}_{NS}=M_{12}^{\prime}=M_{13}^{\prime}=\frac{c}{1-s}$ be as in
Theorem 4.3.1. Suppose $a=M_{12}=M_{13}=b$. Under the trade deadline mechanism
with penalty $\theta$, the Nash-optimal choice of $\tilde{a}$ for P2 when both
hedge funds are strategic is:

$\displaystyle\tilde{a}_{NS}(\theta)=\textrm{clip}(\tilde{a}_{NS},\ell_{\theta}^{\dagger},h_{\theta}^{\dagger})$

Where $\ell_{\theta}^{\dagger}=\max\\{\ell_{\theta},A_{min}\\}$,
$h_{\theta}^{\dagger}=\min\\{h_{\theta},A_{max}\\}$, and $A_{min},A_{max}$ are
the min and max of $\\{c_{a}+s\ell_{\theta},c_{a}+sh_{\theta}\\}$. Further,
the Nash-optimal choice for P3 is
$\tilde{b}_{NS}(\theta)=\tilde{a}_{NS}(\theta)$.

######  Proof A.18.

By Proposition A.14, P2 should choose $a$ to maximize $g_{2}$ given
$\tilde{b},\tilde{m}$ and such that $\ell_{\theta}<a<h_{\theta}$. The
Lagrangian of the constrained problem is:

$\max\limits_{\tilde{a}}g_{2}(\tilde{a})-\lambda_{1}(\ell_{\theta}-\tilde{a})-\lambda_{2}(\tilde{a}-h_{\theta})$

For Lagrange multipliers $\lambda_{1},\lambda_{2}\geq 0$. The stationary point
with respect to $\tilde{a}$ is the solution to $\frac{\del
g_{2}}{\del\tilde{a}}=\lambda_{2}-\lambda_{1}$.

Next, given a fixed $\tilde{b}$ that is reported by the other hedge fund, the
optimal
$\tilde{a}^{*}=\textrm{clip}(c_{a}+s\tilde{b},\ell_{\theta},h_{\theta})$. A
similar formula holds for $\tilde{b}^{*}$. Hence, if both P2 and P3 are
strategic, we solve for the fixed point for $\tilde{a}^{*}$ via subsitution.

$\displaystyle\tilde{a}^{*}$
$\displaystyle=\textrm{clip}(c_{a}+s\tilde{b}^{*},\ell_{\theta},h_{\theta})$
$\displaystyle=\textrm{clip}(c_{a}+s\cdot\textrm{clip}(c_{b}+s\tilde{a}^{*},\ell_{\theta},h_{\theta}),\ell_{\theta},h_{\theta})$
$\displaystyle=\textrm{clip}(\textrm{clip}(c_{a}+sc_{b}+s^{2}\tilde{a}^{*},A_{min},A_{max}),\ell_{\theta},h_{\theta})$
$\displaystyle=\textrm{clip}(c_{a}+sc_{b}+s^{2}\tilde{a}^{*},\ell_{\theta}^{\dagger},h_{\theta}^{\dagger})$

The second step follows because $a=M_{12}=M_{13}=b$ and so
$\ell_{\theta}^{\prime}=\ell_{\theta}$ and $h_{\theta}^{\prime}=h_{\theta}$ in
Prop A.16.

Next, let $z(\tilde{a}^{*})=c_{a}+sc_{b}-(1-s^{2})(\tilde{a}^{*})$. Then
$0=\textrm{clip}(z(\tilde{a}^{*}),\ell_{\theta}^{\dagger}-\tilde{a}^{*},h_{\theta}^{\dagger}-\tilde{a}^{*})$.
Notice that $\abs{s}<1/2$, so $z(\tilde{a}^{*})$ is monotonically decreasing
in $\tilde{a}^{*}$. Moreover, if
$\tilde{a}^{*}=\frac{c_{a}+sc_{b}}{1-s^{2}}:=\tilde{a}_{NS}$ then
$z(\tilde{a}^{*})=0$. Since $z$ is linear in $\tilde{a}^{*}$, we know
$\tilde{a}_{NS}$ is its only root.

We conclude that the formula for $\tilde{a}^{*}$ is valid in all cases.

Case 1. If $\tilde{a}_{NS}<\ell_{\theta}^{\dagger}$, then
$z(\ell_{\theta}^{\dagger})<z(\tilde{a}_{NS})$.

Hence $\tilde{a}^{*}=\ell_{\theta}^{\dagger}$ is a solution, since
$\textrm{clip}(z(\ell_{\theta}^{\dagger}),\ell_{\theta}^{\dagger}-(\ell_{\theta}^{\dagger}),h_{\theta}^{\dagger}-(\ell_{\theta}^{\dagger}))=0$.

Case 2. If $\tilde{a}_{NS}>h_{\theta}^{\dagger}$, then
$z(h_{\theta}^{\dagger})>z(\tilde{a}_{NS})$.

Hence $\tilde{a}^{*}=h_{\theta}^{\dagger}$ is a solution, since
$\textrm{clip}(z(h_{\theta}^{\dagger}),\ell_{\theta}^{\dagger}-(h_{\theta}^{\dagger}),h_{\theta}^{\dagger}-(h_{\theta}^{\dagger}))=0$.

Case 3. If $\ell_{\theta}^{\dagger}<\tilde{a}_{NS}<h_{\theta}^{\dagger}$, then
$\tilde{a}^{*}=\tilde{a}_{NS}$ is a solution, since
$\textrm{clip}(z(\tilde{a}_{NS}),\ell_{\theta}^{\dagger}-\tilde{a}_{NS},h_{\theta}^{\dagger}-\tilde{a}_{NS})=0$.

Finally, since $\tilde{b}_{NS}=\tilde{a}_{NS}$ and the bounds
$\ell_{\theta}^{\prime}=\ell_{\theta}$, $h_{\theta}^{\prime}=h_{\theta}$ are
the same, we conclude that $\tilde{b}_{NS}(\theta)=\tilde{a}_{NS}(\theta)$.

As before, we can derive explicit formulas by combining the above result with
Propositions A.9 and A.16.

######  Proposition A.19.

In the setting of Proposition A.17, if the investor reports $\tilde{m}$ and
$\tilde{\alpha}=\frac{\tilde{m}}{2+\rho}$, then:

$\displaystyle\ell_{\theta}^{\dagger}$
$\displaystyle=\max\bigg{\\{}\ell_{\theta},\bigg{(}\frac{a\nu-\tilde{\alpha}}{\nu(2-\nu)}+\min\bigg{\\{}\frac{\nu(1-\nu)}{\nu(2-\nu)}\cdot\ell_{\theta},\frac{\nu(1-\nu)}{\nu(2-\nu)}\cdot
h_{\theta}\bigg{\\}}\bigg{)}\bigg{\\}}$ $\displaystyle h_{\theta}^{\dagger}$
$\displaystyle=\min\bigg{\\{}h_{\theta},\bigg{(}\frac{a\nu-\tilde{\alpha}}{\nu(2-\nu)}+\max\bigg{\\{}\frac{\nu(1-\nu)}{\nu(2-\nu)}\cdot\ell_{\theta},\frac{\nu(1-\nu)}{\nu(2-\nu)}\cdot
h_{\theta}\\}\bigg{\\}}\bigg{)}\bigg{\\}},$

where $\ell_{\theta},h_{\theta}$ are as in Proposition A.16.