Chance-constrained games with mixture distributions

Abstract

In this paper, we consider an n-player non-cooperative game where the random payoff function of each player is defined by its expected value and her strategy set is defined by a joint chance constraint. The random constraint vectors are independent. We consider the case when the probability distribution of each random constraint vector belongs to a subset of elliptical distributions as well as the case when it is a finite mixture of the probability distributions from the subset. We propose a convex reformulation of the joint chance constraint of each player and derive the bounds for players’ confidence levels and the weights used in the mixture distributions. Under mild conditions on the players’ payoff functions, we show that there exists a Nash equilibrium of the game when the players’ confidence levels and the weights used in the mixture distributions are within the derived bounds. As an application of these games, we consider the competition between two investment firms on the same set of portfolios. We use a best response algorithm to compute the Nash equilibria of the randomly generated games of different sizes.

Appendix

1.1 Proof of Lemma 2

Proof

Let $h_1:[0, 1]\rightarrow [\bar{\alpha }_{k,i}, 1]$ such that $h_1(z_k^i)=\alpha _i^{z_k^i}$, and $h_2:[\bar{\alpha }_{k,i}, 1]\rightarrow {\mathbb {R}}$ such that $h_2(p)=\log {\left( \varPsi _k^i\right) ^{-1}}(p)$ be two functions. Then, the function composition $(h_2\circ h_1)(z_k^i)= \log {\left( \varPsi _k^i\right) ^{-1}\left( \alpha _i^{z_k^i}\right) }$. It is easy to see that $h_1(z_k^i)$ is a convex function of $z_k^i$ and $h_2(p)$ is a nondecreasing function. Then, from Lemma 3.11 of Peng et al. (2018), it suffices to show that $h_2(p)$ is a convex function of p in order to show $\log {\left( \varPsi _k^i\right) ^{-1}\left( \alpha _i^{z_k^i}\right) }$ is a convex function. The second order derivative of $h_2(p)$ is given by

$$\begin{aligned} -\frac{u g'^i_k(u)+g_k^i(u)}{c^2 u^2 (g_k^i(u))^3}, \end{aligned}$$

where $u=\left( \varPsi _k^i\right) ^{-1}(p)$, $g^i_k(u)$ is the radial density function corresponding to distribution function $\varPsi ^i_k(u)$, $g'^i_k(u)$ is the derivative of $g^i_k(u)$. Therefore, the function $h_2(p)$ is convex if

$$\begin{aligned} \left( \varPsi _k^i\right) ^{-1}(p) g'^i_k\left( \left( \varPsi _k^i\right) ^{-1}(p)\right) +g_k^i\left( \left( \varPsi _k^i\right) ^{-1}(p)\right) \le 0. \end{aligned}$$

(22)

The verification of (22) for each distribution listed in Table 1 is as follows:

Normal distribution: Using the radial density function of normal distribution given in Table 1, the condition (22) can be written as

$$\begin{aligned} e^{-\frac{u^2}{2}}(1-u)(1+u)\le 0, \end{aligned}$$

(23)

where $u=\left( \varPsi _k^i\right) ^{-1}(p)$. From Table 2, the value of $\bar{\alpha }_{k,i}$ associated with normal distribution is $\varPsi ^i_k(1)$. Since, the function $g_2(p)$ is defined on $[\bar{\alpha }_{k,i}, 1]$, $p\ge \bar{\alpha }_{k,i}= \varPsi ^i_k(1)$. Hence, the condition (23) is satisfied.

t distribution: Using the radial density function of t distribution given in Table 1, the condition (22) can be written as

$$\begin{aligned} \left( 1+\frac{u^2}{\nu }\right) ^{-\frac{1+\nu }{2}-1}(1+u)(1-u)\le 0, \end{aligned}$$

(24)

where $u=\left( \varPsi _k^i\right) ^{-1}(p)$. From Table 2, the value of $\bar{\alpha }_{k,i}$ associated with t distribution is $\varPsi ^i_k(1)$. From the similar arguments used in case of normal distribution, we have $p\ge \varPsi ^i_k(1)$. Hence, the condition (24) is satisfied.

Cauchy distribution: The Cauchy distribution is a special case of t distribution when $\nu =1$. Therefore, the condition (22) holds in this case using the similar arguments used in the case of t distribution.

Laplace distribution: Using the radial density function of Laplace distribution given in Table 1, the condition (22) can be written as

$$\begin{aligned} e^{- u}(1- u) \le 0, \end{aligned}$$

(25)

where $u=\left( \varPsi _k^i\right) ^{-1}(p)$. From Table 2, the value of $\bar{\alpha }_{k,i}$ associated with Laplace distribution is $\varPsi ^i_k(1)$ which gives $p\ge \varPsi ^i_k(1)$. Hence, the condition (25) is satisfied.

Kotz type distribution: Using the radial density function of Kotz type distribution given in Table 1, the condition (22) can be written as

$$\begin{aligned} u^{2(N-1)}e^{-ru^{2s}}(2N-1-2rs u^{2s}) \le 0, \end{aligned}$$

(26)

where $u=\left( \varPsi _k^i\right) ^{-1}(p)$. From Table 2, the value of $\bar{\alpha }_{k,i}$ associated with Kotz type distribution is $\varPsi _k^i\left( \left( \frac{2N-1}{2sr}\right) ^{\frac{1}{2s}}\right)$ which gives $p\ge \varPsi _k^i\left( \left( \frac{2N-1}{2sr}\right) ^{\frac{1}{2s}}\right)$. Hence, the condition (26) is satisfied.

Pearson type VII distribution: Using the radial density function of Pearson Type VII distribution given in Table 1, the condition (22) can be written as

$$\begin{aligned} \left( 1+\frac{u^2}{m}\right) ^{-N-1}\left( 1-\frac{2N-1}{m}u^2\right) \le 0, \end{aligned}$$

(27)

where $u=\left( \varPsi _k^i\right) ^{-1}(p)$. From Table 2, the value of $\bar{\alpha }_{k,i}$ associated with Pearson type VII distribution is $\varPsi _k^i\left( \sqrt{\frac{m}{2N-1}}\right)$ which gives $p\ge \varPsi _k^i\left( \sqrt{\frac{m}{2N-1}}\right)$. Hence, the condition (27) is satisfied. $\square$

1.2 Proof of Lemma 5

Proof

The second order derivative of $\varPsi _{k,m}^i(e^ {\tau _{k,m}^i})$ is

$$\begin{aligned}\varPsi ^{'' i}_{k,m}(e^ {\tau _{k,m}^i}) (e^ {2\tau _{k,m}^i})+\varPsi ^{' i}_{k,m}(e^ {\tau _{k,m}^i}) (e^ {\tau _{k,m}^i}),\end{aligned}$$

where $\varPsi ^{' i}_{k,m}(\cdot )$ and $\varPsi ^{'' i}_{k,m}(\cdot )$ are the first and second order derivative functions of $\varPsi _{k,m}^i(\cdot )$, respectively. Let $g_{k,m}^i(\cdot )$ be a radial density associated with the distribution function $\varPsi _{k,m}^i(\cdot )$. Then, the concavity of $\varPsi _{k,m}^i(e^ {\tau _{k,m}^i})$ is equivalent to

$$\begin{aligned} g_{k,m}^{' i}(e^ {\tau _{k,m}^i}) (e^ {\tau _{k,m}^i})+g_{k,m}^i(e^ {\tau _{k,m}^i}) \le 0. \end{aligned}$$

(28)

The verification of (28) for each distribution listed in Table 3 is as follows:

Normal distribution: The radial density function of normal distribution is $g_{k,m}^i(u)=e^{-\frac{u^2}{2}}$ and its first derivative is $g_{k,m}^{' i}(u)=-ue^{-\frac{u^2}{2}}$. The condition (28) can be written as

$$\begin{aligned} e^{-\frac{u^2}{2}}(1-u)(1+u)\le 0, \end{aligned}$$

(29)

where $u=e^{\tau _{k,m}^i}$. From Table 3, the weight $w_{k,m}^i$ associated with normal distribution is such that $w_{k,m}^i\ge \frac{1}{1- \varPsi _{k,m}^{i}(1)}(1-\alpha _i)$. Then, from Lemma 4, $r^i_{k,m}\ge 1$ which in turn implies that $e^{\tau _{k,m}^i} \ge 1$. Hence, the condition (29) is satisfied.

t distribution: Using the radial density function of t distribution given in Table 1, the condition (28) can be written as

$$\begin{aligned} \left( 1+\frac{u^2}{\nu }\right) ^{-\frac{1+\nu }{2}-1}(1+u)(1-u)\le 0, \end{aligned}$$

(30)

where $u=e^{\tau _{k,m}^i}$. As similar to the case of normal distribution, the weight $w_{k,m}^i$ associated with t distribution is such that $w_{k,m}^i\ge \frac{1}{1- \varPsi _{k,m}^{i}(1)}(1-\alpha _i)$. From the similar arguments used in the case of normal distribution, the condition (30) is satisfied.

Cauchy distribution: The Cauchy distribution is a special case of t distribution when $\nu =1$. Therefore, the condition (28) holds in this case using the similar arguments used in the case of t distribution.

Laplace distribution: Using the radial density function of Laplace distribution given in Table 1, the condition (28) can be written as

$$\begin{aligned} e^{- u}(1- u) \le 0, \end{aligned}$$

(31)

where $u=e^{\tau _{k,m}^i}$. From Table 3, the weight $w_{k,m}^i$ associated with Laplace distribution is such that $w_{k,m}^i\ge \frac{1}{1- \varPsi _{k,m}^{i}(1)}(1-\alpha _i)$. Then, from Lemma 4, $r^i_{k,m}\ge 1$ which in turn implies that $e^{\tau _{k,m}^i} \ge 1$. Hence, the condition (31) is satisfied.

Kotz type distribution: Using the radial density function of Kotz type distribution given in Table 1, the condition (28) can be written as

$$\begin{aligned} u^{2(N-1)}e^{-ru^{2s}}(2N-1-2rs u^{2s}) \le 0, \end{aligned}$$

(32)

where $u=e^{\tau _{k,m}^i}$. From Table 3, the weight $w_{k,m}^i$ associated with Kotz type distribution is such that $w_{k,m}^i\ge \frac{1}{1- \varPsi _{k,m}^{i}\left( \frac{2N-1}{2sr}\right) ^{\frac{1}{2s}}}(1-\alpha _i)$. Then, from Lemma 4, $r^i_{k,m}\ge \left( \frac{2N-1}{2sr}\right) ^{\frac{1}{2s}}$ which in turn implies that $e^{\tau _{k,m}^i} \ge \left( \frac{2N-1}{2sr}\right) ^{\frac{1}{2s}}$. Hence, the condition (32) is satisfied.

Pearson type VII distribution: Using the radial density function of Pearson Type VII distribution given in Table 1, the condition (28) can be written as

$$\begin{aligned} \left( 1+\frac{u^2}{m}\right) ^{-N-1}\left( 1-\frac{2N-1}{m}u^2\right) \le 0, \end{aligned}$$

(33)

where $u=e^{\tau _{k,m}^i}$. From Table 3, the weight $w_{k,m}^i$ associated with Pearson Type VII is such that $w_{k,m}^i\ge \frac{1}{1- \varPsi _{k,m}^{i}\left( \sqrt{\frac{m}{2N-1}}\right) }(1-\alpha _i)$. Then, from Lemma 4, $r^i_{k,m}\ge \sqrt{\frac{m}{2N-1}}$ which in turn implies that $e^{\tau _{k,m}^i} \ge \sqrt{\frac{m}{2N-1}}$. Hence, the condition (33) is satisfied. $\square$