# A Game of Competition for Risk

Louis Abraham<EMAIL_ADDRESS>Université Paris 1 Panthéon–Sorbonne
Association Française d’Épargne et de Retraite (AFER)

###### Abstract

In this study, we present models where participants strategically select their
risk levels and earn corresponding rewards, mirroring real-world competition
across various sectors. Our analysis starts with a normal form game involving
two players in a continuous action space, confirming the existence and
uniqueness of a Nash equilibrium and providing an analytical solution. We then
extend this analysis to multi-player scenarios, introducing a new numerical
algorithm for its calculation. A key novelty of our work lies in using regret
minimization algorithms to solve continuous games through discretization. This
groundbreaking approach enables us to incorporate additional real-world
factors like market frictions and risk correlations among firms. We also
experimentally validate that the Nash equilibrium in our model also serves as
a correlated equilibrium. Our findings illuminate how market frictions and
risk correlations affect strategic risk-taking. We also explore how policy
measures can impact risk-taking and its associated rewards, with our model
providing broader applicability than the Diamond-Dybvig framework. We make our
methodology and code open-source111available at
https://github.com/louisabraham/cfrgame. Finally, we contribute
methodologically by advocating the use of algorithms in economics, shifting
focus from finite games to games with continuous action sets. Our study
provides a solid framework for analyzing strategic interactions in continuous
action games, emphasizing the importance of market frictions, risk
correlations, and policy measures in strategic risk-taking dynamics.

††journal: Games and Economic Behaviour

##  1 Introduction

Risk-taking during competition is an everyday occurrence, spanning numerous
scenarios from financial markets to environmental policies. In these settings,
individuals and organizations must balance the lure of potential rewards
against the potential for negative outcomes such as bankruptcy or ecological
disasters. Understanding and predicting behaviors in these contexts is crucial
for an array of parties, including policymakers, regulators, and investors.

Game theory provides a compelling lens for analyzing these situations. It
helps model strategic interactions among players and outlines the incentives
prompting their actions. Our work focuses on normal form games - situations
where each player selects a strategy and earns a payoff based on the
collective actions of all players. In this article, we explore continuous
models of competition, where players can choose their level of risk, receiving
higher rewards for taking on more risk.

Nash equilibrium is a key concept for our study. It’s a state of stability in
the game, where no player sees an advantage in deviating from their chosen
strategy. In a normal form game, a Nash equilibrium consists of strategies
where each player’s strategy is the best response to the strategies of others.
This concept is fundamental to game theory and has been widely used in various
fields like economics and political science to model strategic behavior
(Moulin, 1986; Varoufakis, 2008).

Our exploration starts with a straightforward normal form game involving just
two players. For this setup, we provide solid proof for both the existence and
uniqueness of a Nash equilibrium, and we go further by presenting an
analytical solution. This simple model serves as our fundamental building
block, a starting point that offers a solid base of understanding.

Subsequently, we enhance our model to incorporate the complexity of multiple
players. This extension allows us to probe deeper into the strategic dynamics
in more realistic, multi-actor competitive environments. Even with the
additional complexity, we manage to maintain the uniqueness of the Nash
equilibrium and solve the game analytically.

The third stage of our investigation introduces two vital real-world
components: market frictions and risk correlations among firms. We begin by
defining these elements in a two-player context, paving the way for more
complex scenarios.

The final phase of our study marks a significant departure from conventional
approaches. Given the complexities introduced by market frictions and risk
correlations, we adopt a novel technique—using regret minimization algorithms
to discretize and solve our game. This innovation, which opens new vistas in
the study of strategic interactions, proves especially valuable in the face of
the potentially intractable analytical solutions that these intricate
scenarios might present.

Our experimental validation establishes that the Nash equilibria in our model
also function as correlated equilibria, endorsing the use of correlated
equilibria to model strategic behavior. To compute these equilibria, we employ
an array of algorithms, prominently featuring regret matching and
counterfactual regret minimization, thus highlighting the expanding potential
of algorithmic solutions for tackling complex strategic interactions.

Next, we examine the impact of penalties and market frictions on strategic
behavior and results in our continuous model. We find that penalties reduce
both the average risk taken by players and their total rewards. Market
frictions, on the other hand, lower average risk but increase total rewards.
These frictions have a more significant effect on total rewards in high-
penalty environments. In especially inefficient markets with high market
frictions, raising penalties can promote cooperation and increase total
rewards.

We also assess the effects of risk correlations among firms on strategic
behavior and performance. We find that players take more risks in negative
correlation situations, which boosts their payoff compared to a no-correlation
scenario. On the flip side, in positively correlated settings, risk-taking is
reduced. The impact on performance varies, being negative in efficient markets
but potentially positive in less predictable markets.

Our model interestingly aligns with the Diamond-Dybvig framework, where
financial institutions can choose a parameter affecting their utility function
and their likelihood of bankruptcy. This parallel allows our model to explore
situations such as competition among banks over deposit contract interest
rates, akin to the dynamic modeled by Diamond and Dybvig. But our model is
distinct and more generalized, focusing not on specific financial metrics, but
on a broader notion of failure probability, enabling us to explore strategic
competition dynamics in a broader array of scenarios beyond baking.

Our findings offer valuable insights for policymakers, regulators, and
investors who need to understand behavior in competitive, risk-laden
situations. We highlight the significant influence of penalties and market
frictions on strategic behavior and outcomes, and show how risk correlations
can considerably alter strategic behavior and performance in competitive
dynamics. By clarifying these elements, we contribute to the discussion on how
to design effective interventions and policies that encourage cooperation and
improve outcomes in competitive situations involving risk.

##  2 A simple model of competition for risk

###  2.1 Description

In this section, we introduce a simple model of competition for risk that
serves as a backbone for our study. We consider a situation where two actors,
denoted as Player 1 and Player 2, engage in competition by taking actions that
make them more attractive to customers but also increase their risk of
failure. For example, firms may choose to lower their prices to attract more
customers but in doing so, they increase the likelihood of not being able to
repay their loans. Similarly, insurance companies may lower their premiums to
attract more customers but this comes at the cost of a higher risk of failure.
Banks may increase their deposit rates to attract more customers but this also
increases their vulnerability to liquidity crises.

In our model, each player directly sets their failure probability, denoted as
$r_{p}$. While this assumption may not be realistic in practice, we note that
in many situations, firms use models that map real-world actions, such as
setting prices or premiums, to failure probabilities. This mapping is often a
monotonous function that can be inverted to yield real-world actions from
failure probabilities, making our model practical. Based on the failure
probabilities set by each player, the players can randomly “lose” the game. In
our simple model, this translates into being applied a penalty, denoted as
$P$. We assume $P>0$. We assume that the failure events are independent,
meaning that each player draws a uniform random variable $f_{p}$ from the
interval $[0,1]$ and fails if $f_{p}<r_{p}$. We will later introduce
correlations between the variables $f_{p}$ to model real-life situations where
correlations may be positive or negative.

After the failure events are determined, the players that did not fail compare
their risk levels, and the player that played the highest risk level is
rewarded with a payoff, denoted as $R$. Since the game is unchanged when
scaling both $P$ and $R$, we assume $R=1$. In the case of ties between risk
levels, we consider several ways of resolving them, such as none of the
players receiving the reward, the reward being shared equally between them, or
the reward being randomly given to one of them.

We note that, as shown later, the optimal strategies in our model are modeled
by real distributions, which means that the probability of ties is zero.
However, when we use discrete action sets to compute approximate Nash
equilibria, the action sets can overlap, and we implement the first two
variations of resolving ties (the last two are equivalent in expectation).
This simple model serves as a foundation for our study, and we will extend it
by introducing correlations between the players’ failure probabilities and
market frictions in subsequent sections. For two players, assuming
$r_{1}>r_{2}$, the outcome matrix will be:

| $f_{1}\geq r_{1}$ | $f_{1}<r_{1}$
---|---|---
$f_{2}\geq r_{2}$ | $R=1,0$ | $-P,1$
$f_{2}<r_{2}$ | $1,-P$ | $-P,-P$

Each cell contains the rewards to each player. For example, in the upper left
cell, no failure happens. Since we assumed $r_{1}>r_{2}$, player 1 gets $R=1$
and player 2 gets $0$.

###  2.2 Equivalence to a normal-form game

We can represent the game described above in the framework of extensive-form
games (Hart, 1992) by modeling the drawing of the random variables $f_{p}$
using Chance nodes. Since the outcomes are subject to randomness, it is
natural to assume that the actors operate under the expected utility
hypothesis, which implies that they possess a von Neumann–Morgenstern utility
function (Neumann et al., 1944). Consequently, we can define a normal-form
game with payoffs equal to the expected payoffs of the corresponding
extensive-form game. By doing so, we can leverage the theory of normal-form
games and apply various solution concepts, such as Nash equilibria, to analyze
the competition between the actors.

######  Proposition 2.1.

The expected utilities $u_{p}$ are computed as follows in the 2-player game:

$\displaystyle u_{2}(r_{1},r_{2})$
$\displaystyle=u_{1}(r_{2},r_{1})\text{~{}(symmetry)}$ $\displaystyle
u_{1}(r_{1},r_{2})$
$\displaystyle=r_{2}(1-r_{1})R-r_{1}P+[r_{1}>r_{2}](1-r_{1})(1-r_{2})R$

where $[\cdot]$ is the Iverson bracket.

###### Proof.

Player 1 can fail with probability $r_{1}$, in which case they lose $P$. If
Player 2 loses and Player 1 does not, which happens with probability
$r_{2}(1-r_{1})$, Player 1 wins $R$. Finally, if none of the players fails,
when $r_{1}>r_{2}$, Player 1 can win $R$. ∎

It is possible to encompass the shared payoff in case of ties by defining the
Iverson bracket to be $\frac{1}{2}$ when $r_{1}=r_{2}$. Figure 1 shows what
the reward function of Player 1 looks like when Player 2 adopts the fixed
strategy $r_{2}=0.2$.

Figure 1: Reward function

The discontinuity of our game is similar to two games: the War of Attrition
game from Smith (1974) and the visibility game from Lotker et al. (2008). In
the War of Attrition game, each player independently chooses a time to quit
the game. The player who stays in the game for the longest time wins a prize.
However, both players incur a cost that increases over time while they are
still in the game. In the visibility game, the payoff of each player is the
difference with the next player, or 1 for the player that plays the largest
move. A major difference between our game and those two games is that we model
the probability of failure. This means, for example, that the player taking
less risk can still win the reward if the first player fails.

However, the structure of our problem and the analytical solution of the Nash
equilibrium are similar to Lotker et al. (2008). We name our game the
Competition for Risk game and will write it CfR in the rest of the article.

###  2.3 Nash equilibrium

A Nash equilibrium is a set of strategies, one for each player, such that no
player can improve their payoff by unilaterally changing their strategy, given
the strategies of the other players. In other words, each player’s strategy is
the best response to the strategies chosen by the other players. Nash
equilibria are important because they provide a way to predict the outcome of
a game if each player acts rationally and selfishly. They can also help
explain why certain outcomes occur in real-world situations.

In our model of competition for risk, finding Nash equilibria can help us
understand how firms, banks, and insurance companies behave when they compete
for prices and take different levels of risk. By analyzing the Nash equilibria
of our model, we can predict how different players will act and what the
resulting outcomes will be. Moreover, we can compare the efficiency of
different equilibria and use them as a benchmark to evaluate the performance
of different strategies.

As in the game of Lotker et al. (2008), we can prove that there is no pure
Nash equilibrium, that is, a deterministic optimal strategy.

######  Theorem 2.2.

The CfR game does not admit any pure Nash equilibrium.

###### Proof.

Suppose the existence of an equilibrium $s_{1},s_{2}$. Suppose that
$s_{1}>s_{2}$. Then Player 1 can improve their payoff by playing
$s_{1}-\varepsilon$ since they still get the reward and take less risk. By
symmetry, this implies that $s_{1}=s_{2}$. If $s_{1}<1$, then player 1 can
improve their situation by playing $s_{1}+\varepsilon$ since they get $R$ (or
$\frac{R}{2}$ if the reward is shared). If $s_{1}=1$ then the payoff is $-P<0$
with probability $1$ so it is better to play $0$ which gives payoff $0$ with
probability $1$. ∎

######  Definition 2.1.

A strategy $s$ (a couple of strategies) is Pareto optimal if there is no other
strategy $s^{\prime}$ such that $\forall p,u_{p}(s)\leq u_{p}(s^{\prime})$ and
$\exists p,u_{p}(s)<u_{p}(s^{\prime})$. It is $\varepsilon$-Pareto optimal if
there is no strategy $s^{\prime}$ such that $\forall p,u_{p}(s)\leq
u_{p}(s^{\prime})$ and $\exists p,u_{p}(s)+\varepsilon<u_{p}(s^{\prime})$.

######  Remark.

If the reward is shared in case of tie, the pure strategy $(0,0)$ gives reward
$\frac{R}{2}$ to each player. This strategy is Pareto-optimal.

######  Theorem 2.3.

For every $\varepsilon$, there is a $\varepsilon$-Pareto optimal strategy that
gives $\frac{R-\varepsilon}{2}$ to each player.

###### Proof.

Let us consider the joint mixed strategy where each player plays uniformly at
random in the interval $[0,2\varepsilon]$. The payoff is

$\displaystyle\mathbb{E}[u_{1}]$
$\displaystyle=\mathbb{E}\left[r_{2}(1-r_{1})R-r_{1}P+[r_{1}>r_{2}](1-r_{1})(1-r_{2})R\right]$
$\displaystyle=\varepsilon(1-\varepsilon)R-\varepsilon
P+\frac{(1-\varepsilon)^{2}}{2}R$
$\displaystyle\rightarrow_{\varepsilon\rightarrow 0}\frac{R}{2}$

so by taking $\varepsilon$ small enough we can get as close to $\frac{R}{2}$
as we want.

If each player gets payoff $\frac{R-\varepsilon}{2}$then no player can get
$\varepsilon$ without degrading the other’s performance else the total payoff
would be more than $R$. ∎

However, the $\varepsilon$-Pareto strategy is highly concentrated around $0$,
incentivizing players to deviate and increase their chances of winning $R$
without taking on additional risk. Thus, this strategy fails to form a Nash
equilibrium.

Fortunately, the CfR game possesses a unique Nash equilibrium, a powerful
property that showcases the strength of our approach. Moreover, this
equilibrium is symmetric.

For finite games, Nash (1950) proved the existence of mixed Nash equilibria,
while Glicksberg’s theorem (Glicksberg, 1952) extended this result to
continuous reward functions. Dasgupta and Maskin (1986) established conditions
under which discontinuous games can possess Nash equilibria and symmetric
games can admit symmetric equilibria.

The uniqueness of the Nash equilibrium is a highly desirable property, with
most models using concave reward functions to ensure it. Therefore, it is
noteworthy that the CfR game exhibits a unique Nash equilibrium.

We recall Theorem 2.1 from Lotker et al. (2008):

######  Theorem 2.4.

Let $(f_{1},\ldots,f_{n})$ be a Nash equilibrium point, with expected payoff
$u_{i}^{*}$ to Player $i$ at the equilibrium point. Let $u_{i}(x)$ (as an
abuse of notation) denote the expected payoff for Player $i$ when he plays the
pure strategy $x$ and all other players play their equilibrium mixed strategy.
Then $u_{i}(x)\leq u_{i}^{*}$ for all $x\in[0,1]$, and furthermore, there
exists a set $\mathcal{Z}$ of measure $0$ such that $u_{i}(x)=u_{i}^{*}$ for
all $x\in support(f_{i})\setminus\mathcal{Z}$.

This theorem means that at the Nash equilibrium, almost any move that is in
the support of a player’s strategy should give them the same (maximal) payoff.
This theorem is crucial to find the equilibrium in the CfR game.

theoremnashcor Up to a set of measure zero, the CfR game admits a unique Nash
equilibrium. This equilibrium is symmetric and its distribution is
$f(x)=\left[x<1-\sqrt{\frac{k-1}{k+1}}\right]\frac{k-1}{(1-x)^{3}}$ with
$k:=\sqrt{(P+1)^{2}+1}$. The the average move is $\bar{r}=k-(P+1)$ and the
utility of each player is $u^{*}=\bar{r}$.

###### Proof.

See A for a full proof. For a less rigorous treatment, refer to the proof of
the more general Theorem 3.1. ∎

At $P=1$, the cutoff value is
$1-\sqrt{\frac{\sqrt{5}-1}{\sqrt{5}+1}}=2-\phi\approx 0.382$ with $\phi$ the
Golden ratio. We plot the distribution in Figure 2.

Figure 2: Nash equilibrium Figure 3: The cutoff goes to zero when
$P\rightarrow\infty$.

The behavior of the cutoff ${r_{max}}$ is displayed in Figure 3.
Unsurprisingly, when $P\rightarrow\infty$, the penalty becomes much larger
than the reward and the players play closer to $0$.

The case when $P\rightarrow 0$ is more surprising: the maximal cutoff value at
$P=0$ is $h=1-\sqrt{\frac{\sqrt{2}}{\sqrt{2}+2}}\approx 0.356$. This is
because even if the penalty is $0$, the players cannot get the reward if they
“lose”, which prevents them from taking too much risk. We plot the
distribution in Figure 2.

##  3 Generalization to multiple players

Quite naturally, we wonder what the Nash equilibrium looks like for multiple
players. The visibility game of Lotker et al. (2008) probably does not admit
an analytical solution and they instead give an algorithm to produce
approximate solutions. We show that the CfR game for multiple players admits a
unique symmetric equilibrium and present a new numerical algorithm to compute
it. We finally study the asymptotic behavior of the equilibrium.

###  3.1 Nash equilibrium

Interestingly, our Correlation for Risk game admits an analytical solution
even for multiple players. More precisely:

######  Theorem 3.1.

There is a unique symmetric Nash equilibrium for in the CfR game with $n$
players defined by

$f(x)=\frac{P+w}{(n-1)(1-x)^{2+\frac{1}{n-1}}(Px+w)^{1-\frac{1}{n-1}}}$

for some constants $r_{max}$ and $w:=\bar{r}^{n-1}$ (the probability of
winning when taking no risk) such that

$\displaystyle\int_{0}^{r_{max}}f(x)dx$ $\displaystyle=1$
$\displaystyle\int_{0}^{r_{max}}xf(x)dx$ $\displaystyle=\bar{r}$

###### Proof.

We adapt the proof of Theorem 2.4 and start by assuming the existence of a
symmetric mixed equilibrium defined by the probability density $f$. First we
derive a nice expression for $u(x)$, defined as the utility of one player
choosing move $x$ while the others play according to $f$. For all $x\in
support(f)$:

$u(x)=-xP+(1-x)\left(\int_{0}^{x}f(y)dy+\int_{x}^{1}yf(y)dy\right)^{n-1}$

This equation is quite natural: the player loses $P$ with probability $x$. If
they survive, with probability $1-x$, they need the $n-1$ other players to
either play a lower value or play a higher value and fail. We can suppose as
previously that $0$ is in the support to subtract $u(0)$. We write $\bar{r}$
for the expectation of the action $r$ under $f$.

$\left(\frac{\bar{r}^{n-1}+xP}{1-x}\right)^{\frac{1}{n-1}}=\int_{0}^{x}f(y)dy+\int_{x}^{1}yf(y)dy$

We define $w:=\bar{r}^{n-1}$ to be the probability of winning when taking no
risk, we derivate and divide by $1-x$ to obtain:

$f(x)=\frac{P+w}{(n-1)(1-x)^{2+\frac{1}{n-1}}(Px+w)^{1-\frac{1}{n-1}}}$

Finally we can solve $\int_{0}^{r_{max}}f(x)dx=1$ and
$\int_{0}^{r_{max}}xf(x)dx=\bar{r}$. We relegate the description of the
numerical estimation of ${r_{max}}$ and $w$ to B. ∎

We display the behavior of the solution for multiple players in Figure 4.

Figure 4: We observe a clear difference between the cases $P=0$ (no penalty)
and $P=1$ (presence of a penalty). In both cases, the cutoff increases.
However, the average risk seems to decrease sharply when there is a nonzero
penalty with a mode at $r=0$.

###  3.2 Asymptotic behavior

We are interested in studying the equilibrium when the number of players goes
to infinity. For fixed $P$, we have the following:

######  Proposition 3.2.

When $n\rightarrow\infty$, $\lim{r_{max}}=\frac{1}{1+P}$ and
$\bar{r}\sim\frac{1}{nP}$

###### Proof.

We verify experimentally that $r_{max}$ is never close to 0 or 1 and that
$\bar{r}\rightarrow 0$. Equation 3 gives

$\displaystyle\frac{w+nP(1-{r_{max}})+P{r_{max}}}{n(1-{r_{max}})(P+w)}\sqrt[n-1]{\frac{P{r_{max}}+w}{1-{r_{max}}}}$
$\displaystyle=1+\frac{w+nP}{n(P+w)}\bar{r}$
$\displaystyle\sqrt[n-1]{\frac{P{r_{max}}+w}{1-{r_{max}}}}$
$\displaystyle\rightarrow 1$

Equation 4 gives

$\displaystyle\frac{w-nw(1-{r_{max}})+P{r_{max}}}{n(1-{r_{max}})(P+w)}\sqrt[n-1]{\frac{P{r_{max}}+w}{1-{r_{max}}}}$
$\displaystyle=\bar{r}\frac{w+nP}{n(P+w)}$
$\displaystyle\frac{w}{P}+\frac{{r_{max}}}{n(1-{r_{max}})}\sim\frac{{r_{max}}}{n(1-{r_{max}})}$
$\displaystyle\sim\bar{r}$

using $w=\bar{r}^{n-1}=o(\bar{r})$.

$\sqrt[n-1]{\frac{P{r_{max}}+w}{1-{r_{max}}}}\rightarrow 1$ implies
$\frac{{r_{max}}}{1-{r_{max}}}\rightarrow\frac{1}{P}$ and
${r_{max}}\rightarrow\frac{1}{1+P}$.
Finally, $\bar{r}\sim\frac{1}{nP}$. ∎

We illustrate this behavior in Figure 5.

Figure 5: The plot of $r_{max}$ to $\log(P)$ is very similar to the function
$\frac{1}{1+\exp(\cdot)}$. This is because $r_{max}\sim\frac{1}{1+P}$.

A common concept in game theory is the price of anarchy $PoA$ (Koutsoupias and
Papadimitriou, 1999). The price of anarchy is the ratio between the Pareto
optimum and the Nash equilibrium. It is easy to generalize Theorem 2.3 for
multiple players and show that the reward can be split almost perfectly to
obtain a Pareto optimal utility $\frac{R}{n}$. The utility of our symmetric
equilibrium is $R\bar{r}^{n-1}=R\bar{r}^{n-1}=Rw$. Hence, $PoA=1/nw$. We will
instead compute the efficiency $E=\frac{1}{PoA}=nw\in[0,1]$.

We observe that when $P=\frac{1}{n^{e}}$ with $e\geq 0$, the efficiency $E=nw$
of the Nash equilibrium goes to $0$ if $e\leq 1$ and it goes to $1$ if $e>1$.
We plot the behavior of $E$ in Figure 6. We interpret this as an indication
that resources, here modeled by the ratio $\frac{1}{P}=\frac{R}{P}$ of rewards
to penalties, need to scale faster than the number of players for them to
adopt an efficient behavior. Scarcity of resources creates an inefficient Nash
equilibrium.

Figure 6: The efficiency clearly goes to $0$ even when $e=1$. When $e>1$, it
seems that $E\rightarrow 1$. Values for greater values of $n$ suffer of
numerical precision issues as $\log w\rightarrow 0$.

##  4 Extensions of the Competition for Risk Game

###  4.1 Market frictions

One limitation of our model is the assumption that the utility functions are
discontinuous at a certain threshold. While this is appropriate for certain
scenarios such as call for bids, it may not hold in other real-life situations
that involve noisy evaluations or aggregate many individual choices. To
address this limitation, we propose replacing the threshold $[r_{1}>r_{2}]$
with a smooth choice model using the logistic function,
$\sigma_{\tau}(r_{1}-r_{2})$, where $\sigma_{\tau}$ is the scaled sigmoid:

$\sigma_{\tau}(x):=\frac{1}{1+\exp\left(-\frac{x}{\tau}\right)}$

Recall that the failure events are $f_{p}<r_{p}$. For a game between two
players, the outcome matrix can be represented as follows:

| $f_{1}\geq r_{1}$ | $f_{1}<r_{1}$
---|---|---
$f_{2}\geq r_{2}$ | $R~{}\sigma_{\tau}(r_{1}-r_{2}),R~{}\sigma_{\tau}(r_{2}-r_{1})$ | $-P,R$
$f_{2}<r_{2}$ | $R,-P$ | $-P,-P$

######  Proposition 4.1.

The expected utilities $u_{p}$ for the 2-player game with frictions are
computed as follows:

$\displaystyle u_{2}(r_{1},r_{2})$
$\displaystyle=u_{1}(r_{2},r_{1})~{}\text{(symmetry)}$ $\displaystyle
u_{1}(r_{1},r_{2})$
$\displaystyle=r_{2}(1-r_{1})R-r_{1}P+(1-r_{1})(1-r_{2})\sigma_{\tau}(r_{1}-r_{2})R$

As $\tau\rightarrow 0$, $\sigma_{\tau}$ approaches the Heaviside step function
and market frictions disappear.

###  4.2 Correlation between risks

In the real world, risks are often correlated, which is not accounted for in
our current model. To incorporate correlation between risks, we can introduce
joint distributions for the failure events $f_{p}$, which occur according to
latent variables.

In our model, we assume that $f_{p}$ follows a uniform distribution. To
introduce correlation between $f_{1}$ and $f_{2}$, we use the well-known NORTA
(NORmal To Anything) method (Cario and Nelson, 1997). This method allows us to
create a joint distribution $(f_{1},f_{2})$ such that the marginals are
uniform distributions and the Pearson correlation between $f_{1}$ and $f_{2}$
can be set to any arbitrary value.

Following NORTA, we define $f_{p}=\Phi(z_{p})$, where $\Phi$ is the cumulative
distribution function of the Normal distribution, and

$\begin{pmatrix}z_{1}\\\
z_{2}\end{pmatrix}\sim\mathcal{N}\left(\mu,\Sigma\right)$

with $\mu=\begin{pmatrix}0\\\ 0\end{pmatrix}$ and
$\Sigma=\begin{pmatrix}1&\rho(z_{1},z_{2})\\\
\rho(z_{1},z_{2})&1\end{pmatrix}$. Here, $\rho$ is the Pearson correlation
coefficient between $z_{1}$ and $z_{2}$, which determines the correlation
between $f_{1}$ and $f_{2}$. As shown in Cario and Nelson (1997), specifying a
correlation between $z_{p}$ or $f_{p}$ is equivalent to specifying
$\rho(f_{1},f_{2})$. Specifically, we have

$\rho(f_{1},f_{2})=\frac{6}{\pi}\sin^{-1}\left(\frac{\rho(z_{1},z_{2})}{2}\right)$

Hence, we use $\rho$ to denote $\rho(z_{1},z_{2})$ throughout the rest of the
document. This model is well-suited to real-world scenarios, such as financial
portfolios, where $z_{p}$ can represent the returns on investments. In such
cases, joint distributions of portfolios are typically modeled as multivariate
normal distributions, and $r_{p}$ corresponds to the Value at Risk
$v_{p}=\Phi^{-1}(r_{p})$ through the bijective function $\Phi$, such that the
failure event $z_{p}<v_{p}$ is equivalent to $f_{p}<r_{p}$.

######  Proposition 4.2.

The expected utilities $u_{p}$ for the 2-player game with frictions and
correlated risks are computed as follows:

$\displaystyle u_{2}(r_{1},r_{2})$
$\displaystyle=u_{1}(r_{2},r_{1})~{}\text{(symmetry)}$ (1) $\displaystyle
u_{1}(r_{1},r_{2})$
$\displaystyle=(r_{2}-\tilde{r})R-r_{1}P+(1-r_{1}-r_{2}+\tilde{r})\sigma_{\tau}(r_{1}-r_{2})R$
(2)

where $\tilde{r}:=\Phi_{\rho}(\Phi^{-1}(r_{1}),\Phi^{-1}(r_{2}))$ is the
probability of joint failure, with

$\Phi_{\rho}(v_{1},v_{2})=\frac{1}{2\pi\sqrt{1-\rho^{2}}}\int_{-\infty}^{v_{1}}\int_{-\infty}^{v_{2}}\exp\left(-\frac{x^{2}-2\rho
xy+y^{2}}{2(1-\rho)^{2}}\right)dy~{}dx$

the cumulative distribution of the bivariate normal distribution with
correlation $\rho$.

In the absence of noise, when $\rho=\pm 1$, it is also possible to calculate
the Nash equilibrium analytically:

theoremnashwithcor For $\rho=1$, the equilibrium is given by:

$p(x)=\frac{1+P}{1-x}\left[x<1-\exp\left(-\frac{1}{P+1}\right)\right]$

We have

$\bar{r}=1-(P+1)\left(1-\exp\left(-\frac{1}{P+1}\right)\right)$

For $\rho=-1$, the equilibrium is given by:

$p(x)=\frac{P}{(1-2x)^{3/2}}\left[x<\frac{1}{2}-\frac{P^{2}}{2(P+1)^{2}}\right]$

We have

$\bar{r}=\frac{1}{2P+2}$

###### Proof.

See C. ∎

##  5 Computing approximate Nash equilibrium

###  5.1 Approximations to games and equilibria

In this section, we define some key concepts and metrics related to games and
equilibria.

For a given game with $n$ players, we use $u_{i}(\sigma)$ to denote the reward
of player $i$ when all players follow the strategy
$\sigma=(\sigma_{1},\ldots,\sigma_{n})$222This $\sigma$ is not to be confused
with the scaled sigmoid $\sigma_{\tau}$ defined earlier.. A strategy $\sigma$
is said to be a Nash equilibrium if it satisfies the following condition for
all players $i$ and all alternative strategies
$\sigma_{i}^{\prime}\in\Sigma_{i}$:

$u_{i}(\sigma)\geq u_{i}(\sigma_{i}^{\prime},\sigma_{-i})$

where $\sigma_{-i}$ is the strategy of all players but $i$, and $\Sigma_{i}$
is the set of actions available to player $i$.

A game is said to be continuous if the action space $\Sigma_{i}$ is compact
and $u_{i}$ is continuous. In such games, it is possible to approximate the
Nash equilibria using a sequence of games over a reduced finite support, which
leads to Glicksberg’s theorem, without relying on Kakutani’s theorem (Myerson,
1997).

In our CfR game, which has a few points of discontinuity, it is also possible
to approximate the Nash equilibria using a similar method. However, we do not
provide a proof of this here, as the introduction of frictions makes our game
continuous anyway.

To measure the closeness of a strategy $\sigma$ to a Nash equilibrium, we use
the NashConv metric (Lanctot et al., 2017):

$\textsc{NashConv}(\sigma)=\sum_{i=1}^{n}\max_{s_{i}\in\Sigma_{i}}u_{i}(s_{i},\sigma_{-i})-u_{i}(\sigma)$

Note that this metric only considers pure strategies $s_{i}\in\Sigma_{i}$, due
to the linearity of the payoff function for mixed strategies.

The NashConv metric satisfies $\textsc{NashConv}(\sigma)\geq 0$, with equality
holding only for a Nash equilibrium. This implies that
$\textsc{NashConv}(\sigma)$ corresponds to the notion of $\varepsilon$-Nash
equilibrium, where a $\varepsilon$-Nash equilibrium $\sigma$ has
$\textsc{NashConv}(\sigma)=n\varepsilon$.

For a finite action space, NashConv is easy to compute since $\Sigma_{i}$ is
finite. However, for a continuous action space, no such metric is known.
Nonetheless, we can approximate NashConv by taking the maximum over a finite
sample of points from $\Sigma_{i}$. This sample can be chosen randomly, or if
$\Sigma_{i}$ is an interval or a product of intervals of $\mathbb{R}$, we can
use a grid.

In our CfR game, the action space is $[0,1]$. Here, we use quasi-random
numbers to measure the closeness to a Nash equilibrium, inspired by the
literature on hyperparameter sampling (Bousquet et al., 2017) and the
efficiency of quasi-Monte-Carlo methods (Sobol’, 1990). Specifically, we
define the QuasiNashConv metric as:

$\textsc{QuasiNashConv}(\sigma,m)=\sum_{i=1}^{n}\max_{s_{i}\in\textsc{Sobol}(m)}u_{i}(s_{i},\sigma_{-i})-u_{i}(\sigma)$

where $\textsc{Sobol}(m)$ is a set of $m$ quasi random numbers drawn using
Sobol’s method (Sobol’, 1967).

###  5.2 Correlated Equilibria

A Nash equilibrium is a set of strategies where no player can improve their
payoff by unilaterally changing their strategy, assuming that all other
players’ strategies remain unchanged. However, in some games, players may
benefit from coordinating their actions in ways not captured by traditional
Nash equilibrium. This is where the concept of correlated equilibrium comes
in.

A correlated Nash equilibrium is a set of correlated strategies where no
player can improve their expected payoff by unilaterally changing their
strategy, given that they observe the correlation signal. This correlation
signal is not necessarily a message or communication between the players, but
rather a shared random variable that affects each player’s strategy
consistently.

######  Definition 5.1.

A correlated Nash equilibrium is a joint distribution $\sigma$ over all moves
$\Sigma_{1}\times\Sigma_{2}\times\ldots\times\Sigma_{n}$ such that for any
player $i$ and any strategy modification
$\phi:\Sigma_{i}\rightarrow\Sigma_{i}$,

$u_{i}(\sigma_{i},\sigma_{-i})\geq u_{i}(\phi(\sigma_{i}),\sigma_{-i})$

Thus, a Nash equilibrium can be viewed as a correlated Nash equilibrium that
can be decomposed into independent strategies for each player. It is evident
that any Nash equilibrium is a correlated Nash equilibrium.

Correlated equilibria are more suitable for the real world because they allow
for a broader range of possible outcomes that can arise through coordination
among the players, without necessarily requiring communication or binding
agreements between them.

In many real-world scenarios, it is challenging or impossible for players to
communicate and make binding agreements, or they may not have complete
information about the strategies of the other players. Correlated equilibria
provide a way for players to achieve coordination and cooperation without
requiring such communication or information, by relying on shared random
variables that affect each player’s strategies consistently.

Finally, correlated equilibria can also capture situations where players have
some degree of trust or social norms that encourage them to coordinate their
actions in a specific way. For instance, in a repeated game where players
interact with each other over a long period, they may develop a sense of
reciprocity or reputation that encourages them to follow a certain coordinated
strategy.

###  5.3 Finding Correlated Equilibria with Linear Solvers

Correlated equilibria are of interest because they can be computed more easily
for a finite action set.

A joint strategy can be represented by a mapping of probabilities:

$\Pr\nolimits_{\sigma}(s_{1},s_{2},\ldots,s_{n}):=\Pr[\sigma=(s_{1},s_{2},\ldots,s_{n})]$

for all joint actions $(s_{1},s_{2},\ldots,s_{n})$. Therefore, the equation
from Definition 5.1 is linear in these probabilities. An additional equation
is that probabilities must sum to 1, and all probabilities are constrained to
be positive. For two players, the equations are:

$\forall(s_{1},s^{\prime}_{1}),\sum_{s_{2}}\Pr\nolimits_{\sigma}(s_{1},s_{2})u_{1}(s_{1},s_{2})\geq\sum_{s_{2}}\Pr\nolimits_{\sigma}(s_{1},s_{2})u_{1}(s^{\prime}_{1},s_{2})$

$\forall(s_{2},s^{\prime}_{2}),\sum_{s_{1}}\Pr\nolimits_{\sigma}(s_{1},s_{2})u_{1}(s_{1},s_{2})\geq\sum_{s_{1}}\Pr\nolimits_{\sigma}(s_{1},s_{2})u_{1}(s_{1},s^{\prime}_{2})$

$\forall(s_{1},s_{2}),\Pr\nolimits_{\sigma}(s_{1},s_{2})\geq 0$

$\sum_{s_{1},s_{2}}\Pr\nolimits_{\sigma}(s_{1},s_{2})=1$

The set of correlated equilibria is thus a convex polytope $P$. It is possible
to find the boundary in any direction using a linear programming solver.

It is also possible to check that the correlated equilibrium is unique and is
a Nash equilibrium by trying to maximize and minimize each variable over the
polytope. If the maximum and minimum are equal for each variable, then the
polytope only contains one point. Another method described in Appa (2002)
checks the uniqueness of a solution to a linear program by solving a new
linear program. However, that method requires a reformulation of the linear
program as $\max cx\text{ s.t. }Ax=b,x\geq 0$, which is cumbersome in our
case. We propose a simple randomized method (algorithm 1) that can produce
confidence intervals for any confidence level (or p-value).

theorempvalue Given a polytope $P$ defined by constraints $c_{1},\ldots,c_{m}$

$\Pr\left[\textsc{SumDiamSquared}(K,c_{1},\ldots,c_{m})<\varepsilon\right]\leq
F_{\chi^{2}}\left(\frac{\varepsilon}{diam(P)},K\right)$

with $F_{\chi^{2}}(\cdot,K)$ the cumulative distribution function of the
$\chi^{2}$ distribution with $K$ degrees of freedom.

Algorithm 1 Confidence interval on $diam(P)$

Iterations $K$, constraints $c_{1},\ldots,c_{m}$ defining a polytope $P$ in
$\mathbb{R}^{n}$

function SumDiamSquared($K,c_{1},\ldots,c_{m}$)

for $i\leftarrow 1,\ldots,K$ do

Sample $v_{j}\sim\mathcal{N}(0,1)$ for $j=1,\ldots,n$

$a_{i}\leftarrow\textsc{LinProg}(v,c)$ $\triangleright$ $\min\limits_{x\in
P}v\cdot x$

$b_{i}\leftarrow\textsc{LinProg}(-v,c)$ $\triangleright$ $\max\limits_{x\in
P}v\cdot x$

$d_{i}\leftarrow b_{i}-a_{i}$

end for

return $\sum_{i}d_{i}^{2}$

end function

p-value $p$

function MaxDiameter($p,K,c_{1},\ldots,c_{m}$)

$\varepsilon\leftarrow$ SumDiamSquared($K,c_{1},\ldots,c_{m}$)

$q\leftarrow$ Chi2.ppf($p,K$)

$d\leftarrow\varepsilon/q$

return $d$

end function

We used the HiGHS solver (Huangfu and Hall, 2018) to solve the linear
optimization subproblems (calls to LinProg). In numerical experiments, we use
$K=5$, confidence $p=0.95$, and report

$d_{max}:=\textsc{MaxDiameter}(p,K,c_{1},\ldots,c_{m})=\frac{\textsc{SumDiamSquared}(K,c_{1},\ldots,c_{m})}{Q_{\chi^{2}}\left(1-p,K\right)}$

where $Q_{\chi^{2}}(\cdot,K)$ is the quantile function of the $\chi^{2}$
distribution with $K$ degrees of freedom. When the polytope describe
probability distributions, we have the bound $d_{max}\leq 2$.

Finally, we make the following trivial remark:

######  Proposition 5.1.

A correlated equilibrium $\sigma$ is a Nash equilibrium iff the matrix
$(\Pr\nolimits_{\sigma}(i,j))_{i,j}$ has rank 1.

This gives us another numerical method to check that a correlated equilibrium
is a Nash equilibrium: compute the second highest eigenvalue and check that it
is $0$. In numerical experiments, we define $\lambda_{1}$ and $\lambda_{2}$ as
the highest and second highest eigenvalues and report the value

$\lambda:=\frac{\lambda_{2}}{\lambda_{1}}$

###  5.4 Related works solving continuous games

Our exploration of regret-minimization algorithms in the context of continuous
games has led us to a wide range of approaches. From these, a few distinct
groups emerge, each characterized by their unique methods, assumptions, and
requirements.

One group includes the work of Perkins and Leslie (2014), Ganzfried (2021),
and Kroupa and Votroubek (2023). Perkins and Leslie (2014) extended stochastic
fictitious play to the continuous action space framework, showing convergence
to an equilibrium point in two-player zero-sum games. However, this method
assumes specific linear or quadratic utility functions, which limits its
applicability to a narrow set of games. Similarly, Ganzfried (2021) proposed a
novel algorithm for approximating Nash equilibria in continuous games, yet the
scalability of their method is a concern due to the storage of all previous
moves and the use of mixed integer linear programs for best response
computation. Lastly, Kroupa and Votroubek (2023) presented an iterative
strategy generation technique for finding mixed strategy equilibria in
multiplayer general-sum continuous games, but this method requires an oracle
for best response computation, a requirement that may not be met in all
practical scenarios.

Another group consists of Raghunathan et al. (2019) and Dou et al. (2019).
Raghunathan et al. (2019) introduced the Gradient-based Nikaido-Isoda
function, a merit function providing error bounds to a stationary Nash point.
They showed that gradient descent converges sublinearly to a first-order
stationary point of this function, making it a potential method for steady
convergence towards equilibrium. Extending this work, Dou et al. (2019)
offered a deep learning-based approach for approximating Nash equilibria in
continuous games, allowing the finding of mixed equilibria. They utilized the
pushforward measure technique to represent mixed strategies in continuous
spaces and applied gradient descent for convergence to a stationary Nash
equilibrium. However, we found this method to be slow and unable to converge
in our game.

The third group encompasses the works of Bichler et al. (2021) and Martin and
Sandholm (2022). Bichler et al. (2021) proposed using artificial neural
networks to learn equilibria in symmetric auction games through gradient
dynamics in self-play. This work was later extended by Martin and Sandholm
(2022), who used the same pushforward trick as Dou et al. (2019), adding
random noise as input. They applied zeroth-order optimization techniques to
compute approximate Nash equilibria in continuous-action games without access
to gradients. However, we found that the success of this method is very
sensitive to hyperparameters, and its dynamics can be unstable.

###  5.5 Applying regret-minimization algorithms to continuous games

Our study delves into the application of regret-minimization algorithms to
continuous games, a distinct approach given that it does not demand any
differentiability assumptions on the reward function. This approach pivots
around algorithms known for identifying correlated equilibria, such as regret
matching (Hart and Mas-Colell, 1997, 2001a, 2001b), Counterfactual Regret
Minimization (CFR) (Neller and Lanctot, 2013), and stochastic fictitious play
(Fudenberg and Kreps, 1993).

Regret matching is a notable algorithm in the realm of game theory for finding
correlated equilibria. It operates by having each player select a distribution
of moves at each step of the algorithm, wherein this selection is geared
towards maximizing their expected utility, given the past moves of their
opponent. When run for an ample number of iterations, the distributions arrive
at a convergence, forming a correlated equilibrium.

Building on this, we bring into play the CFR algorithm. This algorithm
accumulates the expected regrets of each action played by a player at every
information set. It then updates the regrets based on the counterfactual
outcomes of the game - that is, what the result would have been if a different
action had been taken. As these regrets are iteratively updated and actions
chosen based on these revised regrets, CFR converges to a Nash equilibrium in
extensive form games. In our normal form game, we incorporate CFR as a
deterministic variant of Regret Matching.

Further enriching our methodology is stochastic fictitious play, another
variant of regret matching. It involves the computation of probability
distributions using the softmax function. We also test this variant along with
CFR in our experiments, thereby evaluating their performance in approximating
the Nash equilibrium of our game.

To approximate the CfR game, we resort to a finite grid of actions, which are
evenly distributed within the interval $[0,1]$ for both players. In doing so,
we differentiate between two settings, governed by a boolean variable, shift.
When shift = false, both players are offered the same set of actions. However,
when shift = true, the two players are presented with non-intersecting sets of
actions, with each action in the interval $[0,1]$ alternately attributed to
each player.

Despite the fact that the action set of the resulting approximate Nash
equilibrium is necessarily a subset of a predefined finite action set, our
method brings a fresh perspective to the field, with its emphasis on regret
minimization for both players and the removal of any differentiability
assumption on the reward function. Through this research, we hope to enrich
our understanding of the behavior of continuous games and lay a sturdy
foundation for future exploration and improvements in this domain.

##  6 Experimental results

We employ a Numba (Lam et al., 2015) implementation of the numerical method
proposed by Genz (2004) to compute the bivariate normal probability in the
utility function, which is the computational bottleneck in equation 1.

Our experiments show that the Nash equilibrium in finite approximations of the
CfR game can be obtained by computing a correlated Nash equilibrium. Moreover,
we prove that either the correlated equilibrium is unique or the correlated
equilibrium maximizing the total reward is a Nash equilibrium.

We present the results of our experiments with different values of $P$,
$\tau$, and $\rho$, as well as different numbers of actions, in Figure 7,
where we plot the maximum distance $d_{max}$ and the parameter $\lambda$
computed according to Section 5.3 with shift = true. We also present the
results with shift = false in Section E.

Figure 7: Our experiments with different values of $P$, $\tau$, and $\rho$, as
well as different numbers of actions, for the setting with shift = true show
that, for almost all values of the parameters, $d_{max}$ is numerically zero.
We cross out cases where the solver failed to find a solution, most probably
due to rounding errors which caused the small solution set to disappear. In
the few cases where $d_{max}$ was not observed to be zero, $\lambda$ is
clearly zero, indicating that the best correlated equilibrium is a Nash
equilibrium.

We implement and evaluate all algorithms to solve a discrete version of our
game where the action space is reduced to a grid, with and without a shift. We
then report the value of NashConv QuasiNashConv as defined in Section 5.1. We
display the solution found by Regret Matching in Figure 8 and check that it
matches the actual solution from Figure 2. We observe on Figure 9 that vanilla
Regret Matching outperforms all methods. In general, Regret Matching performs
much better than CFR, even when using less actions, using softmax is
detrimental, and shifting the action space does not seem to impact performance
much. CFR runs faster as it does not involve any random sampling. Therefore,
we use vanilla Regret Matching with shifting in the rest of the experiments
for the sake of simplicity. Figure 10 shows that the number of sampled actions
is the main factor that drives the quality of solutions.

Figure 8: Approximate and analytical solutions for $P=1,\rho=0,\tau=0$.
Actions were discretized to a grid of $2^{12}$ with shift = true. The
approximate solution was computed by $10^{4}$ iterations of regret matching.

For our experiments on equilibria in various settings, we use 500 actions and
2000 iterations (hence $10^{6}$ steps) of vanilla Regret Matching with
shifting. Our efficient implementation computes the equilibrium in just a few
seconds, allowing us to explore the effects of penalties, market frictions,
and correlation on risk-taking behavior and performance, as well as to
evaluate the effectiveness of various interventions and policies in a
competitive environment.

Figure 9: We evaluate the performance of different algorithms in finding the
Nash equilibrium of the CfR game in the standard setting with $P=R=1$,
$\tau=0$, and $\rho=0$, without sharing. We reduce the game to a grid of
$a=2^{12}$ actions, and evaluate the QuasiNashConv metric on $2^{15}$ points.
The algorithms are run for $10^{4}$ iterations, where an iteration is defined
as an update to the strategy. For CFR, an iteration involves updating the
regrets based on the counterfactual outcomes of the game, while for regret
matching, an iteration is defined as $a$ steps of the sampling, play, and
update process. Both CFR and regret matching do $\mathcal{O}(a^{2})$
operations per iteration. We do not include the computation of the utility
matrix in the computation time.

Figure 10: We evaluate the performance of the vanilla regret matching
algorithm in finding the Nash equilibrium of the CfR game in the standard
setting with $P=R=1$, $\tau=0$, and $\rho=0$, without sharing. The quality of
the resulting equilibrium depends mainly on the number of actions $a$. We run
the algorithm for $t$ iterations, where the time complexity of the algorithm
is $\mathcal{O}(ta^{2})$ and the memory complexity is $\mathcal{O}(a^{2})$ as
it requires computing the reward matrix. We evaluate the QuasiNashConv metric
on $8a$ points.

Our results demonstrate that penalties and market frictions have a significant
impact on the strategic behavior of actors in competition. Specifically, we
find that penalties $P$ decrease both the average risk taken $\bar{r}$ by the
players and their total reward $u$, while market frictions $\tau$ not only
decrease the average risk taken but also increase the total reward. We also
find that market frictions have a greater impact on the total reward in high-
penalty environments. In particular, in particularly inefficient markets with
high $\tau$, increasing penalties can improve cooperation and the total
reward, as illustrated in Figure 11.

Figure 11:  The two figures depict the relationship between the total reward
$u$ and the average risk taken $\bar{r}$ under different penalty $P$ and
market friction $\tau$ settings. The left figure shows constant $P$ level
lines, while the right figure shows constant $\tau$ level lines. We observe a
linear relationship between $u$ and $\bar{r}$ when changing $\tau$ at constant
$P$. We find that this behavior holds for all values of $\rho$. Interestingly,
there seems to exist a linear relationship between $\bar{r}$ and $u$ when
changing $\tau$ for fixed levels of $P$.

The correlation between firm risks also has a significant impact on risk-
taking behavior and performance. Specifically, players take more risks in
environments of negative correlation, which can improve their payoff, while
they take less risks in environments of positive correlation. The impact of
correlation on performance is negative in efficient markets, but it can become
positive in noisy markets, as illustrated in Figure 12. These findings provide
valuable insights into the complex interplay between market structure, risk-
taking behavior, and performance in a competitive environment.

Figure 12: We plot the average risk taken $\bar{r}$ and the total utility $u$
as functions of the correlation $\rho$ between firm risks.

##  7 Economic Policy Implications and Real-World Case Studies

In this section, we aim to connect the insights gleaned from our model with
broader economic literature and real-world policy implications. Drawing on
established economic theories, particularly the Diamond-Dybvig model, we offer
a new perspective on how financial institutions manage risk and competition.
Further, we delve into how our model’s findings can inform current policy
debates, such as the regulation of financial advisory commissions in the
European Union and contract transferability among life insurers in France.
Through this discussion, we illustrate the practical applicability and
potential impact of our theoretical framework on shaping economic policies and
practices.

###  7.1 Ties with the Diamond-Dybvig model

The Diamond-Dybvig model (Diamond and Dybvig, 1983) plays a central role in
our understanding of financial intermediaries, especially banks. This seminal
model represents a situation where banks provide liquidity transformation
services. In a nutshell, it portrays a setting in which depositors need to
decide whether to withdraw their money early (impatient depositors) or keep
their money in the bank until a later period (patient depositors). Banks offer
a contract that allows depositors to withdraw their money early, but at a
lower return than if they wait until the end. This can lead to a bank run if
too many depositors decide to withdraw their money early.

By allowing financial institutions in our model to “choose” a parameter that
influences both a utility function and their likelihood of bankruptcy, our
approach shares similarities with the Diamond-Dybvig framework. This feature
makes our model particularly useful for studying phenomena such as competition
among banks over deposit contract interest rates, a scenario that mirrors the
dynamic modeled by Diamond and Dybvig.

Despite its parallels with the Diamond-Dybvig framework, our model is
distinct, original, and more expansive in scope. It is unconcerned with
specific financial metrics or measures, making it more generalizable across
different settings. Instead, our model pivots around the broadly applicable
concept of “failure probability”. This fundamental characteristic allows us to
abstract away from the complexities of real-world financial instruments and
focus on the core strategic interactions of players. By viewing the
competition through the lens of failure probability, we can derive insights
that are not confined to specific financial instruments or markets, but
instead provide a versatile theoretical tool that can be applied across
various sectors and scenarios. This innovative feature enhances the relevance
and applicability of our model in analyzing strategic risk-taking behavior.

Under certain circumstances, as we show in our study, market frictions, such
as customers not optimizing their choice of bank, can increase bank
profitability. Although this finding may seem intuitive, our formal model
provides a rigorous foundation for this result, focusing not on explicit
interest rates but rather on failure probabilities.

Interestingly, our results also shed light on the implications of risk
correlation for the equilibrium. In cases where the failure events are
negatively correlated, market frictions appear to have little effect on the
equilibrium. In contrast, when risks are positively correlated – mirroring the
reality of endogenously generated financial risks tied to financial markets –
a lack of market friction can spur banks to take on greater risk. This
suggests that the optimal situation for both customers and banks arises when
the correlation between endogenous risks is zero or negative. Under these
conditions, lower frictions can benefit customers without harming banks. On
the other hand, if risks are positively correlated, market frictions can harm
customers while benefiting banks, creating an incentive for banks to increase
these frictions (e.g., by fostering customer loyalty or withholding
information).

Moreover, the impact of failure penalties (or the absence of a safety net) can
exacerbate this dynamic, suggesting that regulators could mitigate perverse
incentives by reducing the penalty parameter $P$ in a positively correlated
environment, such as through bailouts or other protective measures (e.g.,
deposit insurance, liquidity provision, and bank resolution mechanisms). This
perspective aligns with a strand of literature that extrapolates from the
Diamond-Dybvig model to inform policy making (Bhattacharya et al., 1985; Ennis
and Keister, 2009).

###  7.2 Practical applications in current policy debates

Outside of the banking sector, our framework has implications for other policy
debates. In the European Union, for instance, the Financial Services
Commissioner, Mairead McGuinness, recently proposed banning “inducements” –
commissions paid by banks or insurers to financial advisors who sell their
products (Jones, 2023). Proponents argue this would enhance transparency and
reduce costs. Critics, however, fear it could inhibit access to financial
advice. Within our model, increasing transparency equates to reducing the
friction parameter $\tau$, which alters the equilibrium, potentially
heightening risk for financial institutions but also augmenting returns for
consumers. Our findings indicate that regulators might safeguard consumers and
influence prices simply by mandating transparency. However, we caution that
the impact of such a policy on market frictions is uncertain, as it could
inadvertently limit access to information via advisors.

Furthermore, in France, a legislative proposal by MPs Husson and Montgolfier
(Husson and de Montgolfier, 2023) seeks to enhance the transferability of
contracts between life insurers, effectively reducing market frictions. Our
framework suggests that, in response, insurers could take on more risk due to
increased competition. However, the relationship between risk and interest
rates may also change, as performance becomes costlier in terms of risk when
the duration of contracts decreases because of enhanced transferability.
Consequently, while our model predicts an uptick in the risks assumed by
insurers, the ultimate benefit to savers remains ambiguous.

##  8 Conclusion

In this study, we thoroughly examined competition models where participants
strategically choose their risk levels, with those who take on more risk
potentially outperforming their competitors.

We devised and tested multiple algorithms to solve our game in its discrete
form, with vanilla Regret Matching proving to be the most effective. We
utilized this efficient implementation to delve into the impacts of penalties,
market frictions, and risk correlations on strategic behavior and overall
performance. Additionally, we scrutinized the effectiveness of diverse
interventions and policies within this competitive landscape.

Our research revealed that market frictions tend to lower the average risk
taken while boosting the total reward. Moreover, we found that enhancing
failure penalties can foster cooperation and augment the total reward,
particularly in inefficient markets. Our exploration also showed that negative
correlations among failure events stimulate risk-taking, while positive
correlations may discourage it in efficient markets but potentially encourage
it in less predictable, noisy markets.

One noteworthy aspect of our study is its parallel with the Diamond-Dybvig
framework. Our model, similar to Diamond-Dybvig, examines how financial
institutions select parameters influencing their utility functions and
likelihood of failure. However, our model is more generalized, focusing on
universally applicable notions of failure probabilities, thereby enabling us
to study the dynamics of strategic competition in a broader array of
scenarios.

We also showcased the adaptability of our model for policy exploration. By
imposing policy measures such as transparency requirements or facilitating
contract transferability, we demonstrated how policy changes can influence the
equilibrium of risk-taking and consequently, the rewards.

Our findings offer substantial insights for economics, finance, and
policymaking. By understanding how market frictions and penalties influence
competition, firms and governments can make more informed strategic decisions
leading to more efficient markets. Moreover, our use of algorithmic solvers
for games with continuous action sets illustrates the potential for handling
more intricate models lacking closed-form solutions.

In conclusion, our work provides a robust framework for modeling and analyzing
strategic interactions in continuous action games, extending its implications
far beyond to enrich economic research and practice.

## Acknowledgement

We thank Aurélie Coursimault, Professor Hélyette Geman, Marc Lanctot, Jules
Pondard and Professor Philippe Raimbourg for helpful discussions and useful
comments.

## References

  * Appa (2002) Appa, G., 2002.  On the uniqueness of solutions to linear programs.  Journal of the Operational Research Society 53, 1127–1132.
  * Bhattacharya et al. (1985) Bhattacharya, S., Gale, D., Barnett, W., Singleton, K., 1985\.  Preference shocks, liquidity, and central bank policy.  Liquidity and crises 35.
  * Bichler et al. (2021) Bichler, M., Fichtl, M., Heidekrüger, S., Kohring, N., Sutterer, P., 2021.  Learning equilibria in symmetric auction games using artificial neural networks.  Nature machine intelligence 3, 687–695.
  * Bousquet et al. (2017) Bousquet, O., Gelly, S., Kurach, K., Teytaud, O., Vincent, D., 2017.  Critical hyper-parameters: No random, no cry.  arXiv preprint arXiv:1706.03200 .
  * Cario and Nelson (1997) Cario, M.C., Nelson, B.L., 1997\.  Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix .
  * Dasgupta and Maskin (1986) Dasgupta, P., Maskin, E., 1986\.  The existence of equilibrium in discontinuous economic games, i: Theory.
  * Diamond and Dybvig (1983) Diamond, D.W., Dybvig, P.H., 1983\.  Bank runs, deposit insurance, and liquidity.  Journal of political economy 91, 401–419.
  * Dou et al. (2019) Dou, Z., Yan, X., Wang, D., Deng, X., 2019.  Finding mixed strategy nash equilibrium for continuous games through deep learning.  arXiv preprint arXiv:1910.12075 .
  * Ennis and Keister (2009) Ennis, H.M., Keister, T., 2009\.  Bank runs and institutions: The perils of intervention.  American Economic Review 99, 1588–1607.
  * Fudenberg and Kreps (1993) Fudenberg, D., Kreps, D.M., 1993\.  Learning mixed equilibria.  Games and economic behavior 5, 320–367.
  * Ganzfried (2021) Ganzfried, S., 2021.  Algorithm for computing approximate nash equilibrium in continuous games with application to continuous blotto.  Games 12, 47.
  * Genz (2004) Genz, A., 2004.  Numerical computation of rectangular bivariate and trivariate normal and t probabilities.  Statistics and Computing 14, 251–260.
  * Glicksberg (1952) Glicksberg, I.L., 1952.  A further generalization of the kakutani fixed theorem, with application to nash equilibrium points.  Proceedings of the American Mathematical Society 3, 170–174.
  * Hart (1992) Hart, S., 1992.  Games in extensive and strategic forms.  Handbook of game theory with economic applications 1, 19–40.
  * Hart and Mas-Colell (1997) Hart, S., Mas-Colell, A., 1997\.  A simple adaptive procedure leading to correlated equilibrium.  Game Theory and Information .
  * Hart and Mas-Colell (2001a) Hart, S., Mas-Colell, A., 2001a.  A reinforcement procedure leading to correlated equilibrium.  Springer.
  * Hart and Mas-Colell (2001b) Hart, S., Mas-Colell, A., 2001b.  A reinforcement procedure leading to correlated equilibrium: A correction .
  * Huangfu and Hall (2018) Huangfu, Q., Hall, J.J., 2018\.  Parallelizing the dual revised simplex method.  Mathematical Programming Computation 10, 119–142.
  * Husson and de Montgolfier (2023) Husson, J.F., de Montgolfier, A., 2023\.  Proposition de loi tendant à renforcer la protection des épargnants.  https://www.senat.fr/rap/l22-272/l22-272.html.  Rapport n° 272 (2022-2023) fait au nom de la commission des finances. Accessed: 2023-05-29.
  * Jones (2023) Jones, H., 2023.  Eu financial services chief favours banning sales inducements on investment products.  Reuters URL: https://www.reuters.com/business/finance/eu-financial-services-chief-favours-banning-sales-inducements-investment-2023-01-06/. accessed: 2023-05-29.
  * Koutsoupias and Papadimitriou (1999) Koutsoupias, E., Papadimitriou, C., 1999\.  Worst-case equilibria, in: Stacs, Springer. pp. 404–413.
  * Kroupa and Votroubek (2023) Kroupa, T., Votroubek, T., 2023\.  Multiple oracle algorithm to solve continuous games, in: Decision and Game Theory for Security: 13th International Conference, GameSec 2022, Pittsburgh, PA, USA, October 26–28, 2022, Proceedings, Springer. pp. 149–167.
  * Lam et al. (2015) Lam, S.K., Pitrou, A., Seibert, S., 2015.  Numba: A llvm-based python jit compiler, in: Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC, pp. 1–6.
  * Lanctot et al. (2017) Lanctot, M., Zambaldi, V., Gruslys, A., Lazaridou, A., Tuyls, K., Perolat, J., Silver, D., Graepel, T., 2017\.  A unified Game-Theoretic approach to multiagent reinforcement learning.  Adv. Neural Inf. Process. Syst. 30\.
  * Lotker et al. (2008) Lotker, Z., Patt-Shamir, B., Tuttle, M.R., 2008.  A game of timing and visibility.  Games and Economic Behavior 62, 643–660.
  * Martin and Sandholm (2022) Martin, C., Sandholm, T., 2022\.  Finding mixed-strategy equilibria of continuous-action games without gradients using randomized policy networks.  arXiv preprint arXiv:2211.15936 .
  * Moulin (1986) Moulin, H., 1986.  Game theory for the social sciences.  NYU press.
  * Myerson (1997) Myerson, R.B., 1997.  Game theory: analysis of conflict.  Harvard university press.
  * Nash (1950) Nash, J.F., 1950.  Equilibrium points in N-Person games.  Proc. Natl. Acad. Sci. U. S. A. 36, 48–49.
  * Neller and Lanctot (2013) Neller, T.W., Lanctot, M., 2013\.  An introduction to counterfactual regret minimization, in: Proceedings of Model AI Assignments, The Fourth Symposium on Educational Advances in Artificial Intelligence (EAAI-2013), pp. 1602–1603.
  * Neumann et al. (1944) Neumann, J.v., Morgenstern, O., et al., 1944\.  Theory of games and economic behavior .
  * Perkins and Leslie (2014) Perkins, S., Leslie, D.S., 2014\.  Stochastic fictitious play with continuous action sets.  Journal of Economic Theory 152, 179–213.
  * Raghunathan et al. (2019) Raghunathan, A., Cherian, A., Jha, D., 2019.  Game theoretic optimization via gradient-based nikaido-isoda function, in: International Conference on Machine Learning, PMLR. pp. 5291–5300.
  * Smith (1974) Smith, J.M., 1974.  The theory of games and the evolution of animal conflicts.  Journal of theoretical biology 47, 209–221.
  * Sobol’ (1990) Sobol’, I., 1990.  Quasi-monte carlo methods.  Progress in Nuclear Energy 24, 55–61.
  * Sobol’ (1967) Sobol’, I.M., 1967.  On the distribution of points in a cube and the approximate evaluation of integrals.  Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki 7, 784–802.
  * Varoufakis (2008) Varoufakis, Y., 2008.  Game theory: Can it unify the social sciences?  Organization Studies 29, 1255–1277.

##  Appendix A Proof of Theorem 2.4

We follow broadly the same scheme as Theorem 3.6 of Lotker et al. (2008),
mutatis mutandis since the equations are different. Their proof proceeds by
supposing the existence of a Nash equilibrium and deriving properties to
characterize it. Contrary to what they claim, we can prove the existence of a
Nash equilibrium without any computation using Dasgupta and Maskin (1986). Let
$(f_{1},f_{2})$ be the density functions of Player 1 and 2 in a Nash
equilibrium. We note $S_{1},S_{2}$ their support.

######  Proposition A.1.

For almost all $x\in S_{1}$, $f_{2}(x)\sim\frac{1}{(1-x)^{3}}$ and conversely.

###### Proof.

Let us simplify the notations by noting $a\sim f_{1}$ and $b\sim f_{2}$ the
random moves of players 1 and 2. We note $u_{1}^{*}$ the utility of Player 1
and $\bar{b}$ the expectation of $b$. Then, by Theorem 2.4, for almost all
$a\in S_{1}$,

$\displaystyle u_{1}^{*}=u_{1}(a)$ $\displaystyle=\mathbb{E}_{b}[u_{1}(a,b)]$
$\displaystyle=\mathbb{E}_{b}\left[Rb(1-a)-aP+[a>b](1-a)(1-b)R\right]$
$\displaystyle=R\bar{b}(1-a)-aP+(1-a)\int_{0}^{a}(1-b)Rf_{2}(b)db$
$\int_{0}^{a}(1-b)Rf_{2}(b)db=\frac{u_{1}^{*}+aP}{1-a}+R\bar{b}$

We derivate to obtain $f_{2}(a)=\frac{u_{1}^{*}+P}{R(1-a)^{3}}$. ∎

######  Proposition A.2.

With the exception of a set measure zero, $S_{1}=S_{2}$.

###### Proof.

We apply the previous result that implies $x\in S_{1}\implies f_{2}(x)\neq
0$.∎

######  Proposition A.3.

$\inf S_{1}=\inf S_{2}=0$

###### Proof.

Suppose $\inf S_{1}=\inf S_{2}=l>0$. By Theorem 2.4,
$u_{1}(l)=u_{1}^{*}=\max_{x}u_{1}(x)$. But $u_{1}(0)>u_{1}(l)$ since playing
$0$ decreases the risk of Player 1 without compromising its chances to get the
reward. By contradiction, $\inf S_{1}=\inf S_{2}=0$. ∎

######  Proposition A.4.

For all intervals $[x_{1},x_{2}]$ with $0<x_{1}<x_{2}<\sup S_{1}$ we have that
$\int_{x_{1}}^{x_{2}}f_{1}(x)dx>0$

###### Proof.

Suppose that there is an interval $[x_{1},x_{2}]$ such that
$\int_{x_{1}}^{x_{2}}f_{1}(x)dx=0$. Assume that this interval is maximal so
that $x_{1},x_{2}\in S_{1}$. We also have that
$\int_{x_{1}}^{x_{2}}f_{2}(x)dx=0$ since $S_{1}=S_{2}$. Hence Player 2 never
plays between $x_{1}$ and $x_{2}$ and this implies that
$u_{1}(x_{1})>u_{1}(x_{2})$ since Player 1 can decrease their risk without
compromising its chances to get the reward. This contradicts the fact that
$u_{1}(x_{1})=u_{1}(x_{2})$ since they are both in the support of the Nash
equilibrium. ∎

######  Proposition A.5.

There is no point $x$ with positive probability.

###### Proof.

Suppose the existence of a point $x$ with positive probability (which means
that $f_{1}(x)$ is a Dirac. Since we determined the expression of $f_{1}(x)$
for almost every $x$, there is a $\varepsilon>0$ such that there is no other
point with positive probability in $[x,x+\varepsilon]$. Hence, there is
$0<\varepsilon^{\prime}<\varepsilon$ such that
$u_{2}(x+\varepsilon^{\prime})>u_{2}(x)$ since any positive
$\varepsilon^{\prime}$ improves the probability of winning the reward by
$\mathbb{P}[\text{Player 1 plays }x]$ and the risk increment goes to $0$ with
$\varepsilon^{\prime}$. ∎

*

###### Proof.

Up to a set of measure 0 and on which the probability is 0, we have determined
the expression of $f_{1}$ and $f_{2}$ above up to a constant. This means that
$f_{1}=f_{2}=f$. We still have to find the constant and the upper limit of the
support. Let us reiterate the calculations, this time using the fact that
$0\in S$ with $S$ the support of $f$. This means that for any $a\in S$:

$\displaystyle 0$ $\displaystyle=u_{1}(a)-u_{1}(0)$
$\displaystyle=\mathbb{E}_{b}[u_{1}(a,b)-u_{1}(0,b)]$
$\displaystyle=\mathbb{E}_{b}\left[Rb(1-a)-aP+[a>b](1-a)(1-b)R-Rb\right]$
$\displaystyle=-aR\bar{b}-aP+(1-a)\int_{0}^{a}R(1-b)f(b)db$

$\int_{0}^{a}R(1-b)f(b)db=(R\bar{b}+P)\frac{a}{1-a}$

We derivate to obtain:

$f(a)=\left(\bar{b}+\frac{P}{R}\right)\frac{1}{(1-a)^{3}}$

We now have two unknowns and two unknowns:

$\left\\{\begin{array}[]{ll}h:=\sup S\text{ such that }\int_{0}^{h}f(x)dx=1\\\
\bar{b}=\int_{0}^{h}xf(x)dx\end{array}\right.$

We define $P:=\frac{P}{R}$.

$\int_{0}^{h}f(x)dx=\frac{\bar{b}+P}{2}\left(\frac{1}{(h-1)^{2}}-1\right)=1$
$\int_{0}^{h}xf(x)dx=\bar{b}=\frac{\bar{b}+P}{2}\frac{h^{2}}{(h-1)^{2}}$

We get

$\bar{b}=\frac{h^{2}}{1-(h-1)^{2}}$
$2\frac{(h-1)^{2}}{1-(h-1)^{2}}-P=\frac{h^{2}}{1-(h-1)^{2}}$
$2(h-1)^{2}-P+P(h-1)^{2}-h^{2}=0$ $(P+1)h^{2}-(2P+4)h+2=0$

Hence, $h=\frac{2+P\pm\sqrt{P^{2}+2P+2}}{1+P}$
Since $h<1$, we get

$\displaystyle h$ $\displaystyle=\frac{2+P-\sqrt{P^{2}+2P+2}}{1+P}$
$\displaystyle=1-\frac{\sqrt{(P+1)^{2}+1}-1}{1+P}$

We remark that $(\sqrt{(P+1)^{2}+1}+1)(\sqrt{(P+1)^{2}+1}-1)=(P+1)^{2}$
Hence $h=1-\sqrt{\frac{\sqrt{(P+1)^{2}+1}-1}{\sqrt{(P+1)^{2}+1}+1}}$

$\displaystyle\bar{b}+P$ $\displaystyle=\frac{2}{\frac{1}{(h-1)^{2}}-1}$
$\displaystyle=\frac{2}{\frac{\sqrt{(P+1)^{2}+1}+1}{\sqrt{(P+1)^{2}+1}-1}-1}$
$\displaystyle=\sqrt{(P+1)^{2}+1}-1$

Finally, $f(x)=\left[x<1-\sqrt{\frac{k-1}{k+1}}\right]\frac{k-1}{(1-x)^{3}}$
with $k:=\sqrt{(P+1)^{2}+1}$
From the expressions above, we get $\bar{b}=k-1-P$ and $u^{*}=R\bar{b}$. ∎

##  Appendix B Estimation of ${r_{max}}$ and $w$ in the multiplayer setting

We recall the equations:

$f(x)=\frac{P+w}{(n-1)(1-x)^{2+\frac{1}{n-1}}(Px+w)^{1-\frac{1}{n-1}}}$
$\displaystyle\int_{0}^{r_{max}}f(x)dx$ $\displaystyle=1$
$\displaystyle\int_{0}^{r_{max}}xf(x)dx$ $\displaystyle=\bar{r}$

with

$w:=\bar{r}^{n-1}$

We could estimate the integrals by numerical integration. However $\bar{r}$
becomes smaller when increasing $n$, which causes $f$ to have a peak at $0$
and renders the integral estimates unreliable. Computations 333We used Wolfram
Alpha with ReplaceAll[Integrate[(1/(1 - x)) D[((w + x P)/(1 - x))^(1/(n - 1)),
x], x], n -> 42] for various values of $n$, found a pattern and checked that
the derivatives match. give us

$\int f(x)=\frac{w+nP(1-x)+Cx}{n(1-x)(P+w)}\sqrt[n-1]{\frac{Cx+w}{1-x}}$ $\int
xf(x)=\frac{w-nw(1-x)+Cx}{n(1-x)(P+w)}\sqrt[n-1]{\frac{Cx+w}{1-x}}$

Hence

$\frac{w+nP(1-{r_{max}})+P{r_{max}}}{n(1-{r_{max}})(P+w)}\sqrt[n-1]{\frac{P{r_{max}}+w}{1-{r_{max}}}}-\frac{w+nP}{n(P+w)}\sqrt[n-1]{w}=1$
(3)
$\frac{w-nw(1-{r_{max}})+P{r_{max}}}{n(1-{r_{max}})(P+w)}\sqrt[n-1]{\frac{P{r_{max}}+w}{1-{r_{max}}}}-\frac{w-nw}{n(P+w)}\sqrt[n-1]{w}=\sqrt[n-1]{w}$
(4)

The value of integrals are increasing in ${r_{max}}$ since $f(x)$ is positive,
hence for any value of $w$ we can find the corresponding value of ${r_{max}}$
by binary search using (3). Then we are left with finding the value of $w$
using (4) . We observe experimentally that the function
$w\rightarrow\int_{0}^{{r_{max}}(w)}xf(x)dx-\sqrt[n-1]{w}$ seems to have only
one root and that it is positive before that root and negative after that
root. We therefore use binary search.

Equation (4) also defines $w$ as a fixed point and the iterative algorithm
$w\leftarrow\left(\int_{0}^{{r_{max}}(w)}xf(x)\right)^{n-1}$ also converges,
although it seems to sometimes loop between a few close values because of
numerical errors.

We remark that $w$ goes very quickly to $0$ as $n$ increases. This causes
numeric errors in the computations of the primitives at $0$ that we fix by
storing $\log w$ instead of $w$. For the same reason, we compute the integrals
in log space.

Equation (4) is more interesting as the indefinite integral $F({r_{max}})$ can
be negative. Depending on the sign, we store either $\log F({r_{max}})$ or
$\log-F({r_{max}})$.

Finally, we apply the Log-Sum-Exp Trick to compute the value of the integral
in log space. The final algorithm to find $w$ and ${r_{max}}$ takes less than
1 ms on a laptop for any value of the parameters.

##  Appendix C Proof of Theorem 4.2

*

###### Proof.

We reuse the proof of A, only modifying the calculations. We note $a$ and $b$
the actions of the players, aka their individual probability of failure noted
$r_{1}$ and $r_{2}$ above and note $c$ their joint probability of failure
noted $\tilde{r}$ above. $c$ is a function $c(a,b,\rho)$. We note $p$ the
probability distribution corresponding to the Nash equilibrium. In both cases,
we have the equality

$\int_{0}^{1}p(b)c~{}db+aP=\int_{0}^{a}p(b)(1-a-b+c)~{}db$

When $\rho=1$, $c=\min(a,b)$

$\int_{0}^{a}p(b)b~{}db+a\int_{a}^{1}p(b)~{}db+aP=\int_{0}^{a}p(b)(1-a)~{}db$

We derivate wrt $a$:

$ap(a)+\int_{a}^{1}p(b)~{}db-ap(a)+P=p(a)(1-a)-\int_{0}^{a}p(b)~{}db$

$p(x)=\frac{1+P}{1-x}$

We solve $\int_{0}^{r_{max}}p(x)~{}dx=1$ as $\log(1-r_{max})=-\frac{1}{P+1}$,
thus

$r_{max}=1-\exp\left(-\frac{1}{P+1}\right)$

Finally, a simple computation gives

$\bar{r}=1-(P+1)\left(1-\exp\left(-\frac{1}{P+1}\right)\right)$

When $\rho=-1$, $c=\max(0,a+b-1)$. In other terms, $c=0$ if $b<1-a$ and
$c=a+b-1$ if $b>1-a$.

We first treat $a>\frac{1}{2}$, aka $a>1-a$, to show by contradiction that
$p(a)=0$.

$\int_{1-a}^{1}p(b)(a+b-1)~{}db+aP=\int_{0}^{1-a}p(b)(1-a-b)~{}db$
$aP=\int_{0}^{1}p(b)(1-a-b)~{}db=1-\bar{r}-a$

This is impossible, thus $p(a)=0$ for $a>1/2$.

We now suppose $a<1-a$:

$\int_{1-a}^{1}p(b)(a+b-1)~{}db+aP=\int_{0}^{a}p(b)(1-a-b)~{}db$

$\int_{1-a}^{1}p(b)(a+b-1)~{}db=0$ since $1-a>\frac{1}{2}$. We derivate twice
to get:

$(1-2a)p(a)=\int_{0}^{a}p(b)~{}db+P$ $1-2a)p^{\prime}(a)-3p(a)=0$

The first equation gives $p(0)=P$. Combined with the second differential
equation, we get:

$p(x)=\frac{P}{(1-2x)^{3/2}}$

Similarly, we solve

$r_{max}=\frac{1}{2}-\frac{P^{2}}{2(P+1)^{2}}$ $\bar{r}=\frac{1}{2P+2}$

∎

##  Appendix D Proof of Theorem 5.3

*

###### Proof.

We suppose that the diameter $d$ is obtained in some unit direction $\vec{a}$:

$\max\limits_{x\in P}a\cdot x-\min\limits_{x\in P}a\cdot x=d$

We define $D$ the line $\left[\operatorname*{arg\,min}\limits_{x\in P}a\cdot
x,\operatorname*{arg\,max}\limits_{x\in P}a\cdot x\right]$. Then, for any $v$,

$\max\limits_{x\in P}v\cdot x-\min\limits_{x\in P}v\cdot
x\geq\max\limits_{x\in D}v\cdot x-\min\limits_{x\in D}v\cdot x=d\times|v\cdot
a|$

Hence,

$\Pr\left[\max\limits_{x\in P}v\cdot x-\min\limits_{x\in P}v\cdot
x<\varepsilon\right]\leq\Pr\left[|v\cdot a|<\frac{\varepsilon}{d}\right]$

If $v$ is a vector of iid standard Gaussian variables, $v\cdot
a\sim\mathcal{N}(0,1)$ and $(v\cdot a)^{2}$ follows a $\chi^{2}$ distribution
with 1 degree of freedom. The sum of multiple iterations will follow:

$\Pr\left[\sum_{i}^{K}(\max\limits_{x\in P}v_{i}\cdot x-\min\limits_{x\in
P}v_{i}\cdot
x)^{2}<\varepsilon\right]\leq\Pr\left[\sum_{i}^{K}z_{i}^{2}<\frac{\varepsilon}{d}\right]$

where $z_{i}\sim\mathcal{N}(0,1)$ are independent normal variables.

$\sum_{i}^{K}z_{i}^{2}$ follows a $\chi^{2}$ distribution with $K$ degrees of
freedom. With $F_{\chi^{2}}(x,K)$ the cumulative distribution function, we
get:

$\Pr\left[\sum_{i}^{K}(\max\limits_{x\in P}v_{i}\cdot x-\min\limits_{x\in
P}v_{i}\cdot x)^{2}<\varepsilon\right]\leq
F_{\chi^{2}}\left(\frac{\varepsilon}{d},K\right)$

∎

##  Appendix E Linear equilibrium in finite approximations with shift = false