On the existence of Nash equilibrium in discontinuous games

Abstract

This paper offers an equilibrium existence theorem in discontinuous games. We introduce a new notion of very weak continuity, called quasi-weak transfer continuity that guarantees the existence of pure strategy Nash equilibrium in compact and quasiconcave games. We also consider possible extensions and improvements of the main result. We present applications to show that our conditions allow for economically meaningful payoff discontinuities.

Appendix

Proof of Theorem 1

Define a new game $G'$ that includes all of the original players $i\in I$ and a new player $i=0\notin I$. Thus, the new set of players is $N=I\cup \{0\}$. Player 0’s strategy set is X and the strategy set of player $i\in I$ is $X_i$. If player 0 chooses $z\in X$ and each player $i\in I$ chooses $x_i\in X_i$, then the payoff to player 0 is

$$\begin{aligned}v_0(z,x)=\left\{ \begin{array}{ll} 1, &{}\;\;\text {if }z=x\\ 0, &{}\;\;\text {otherwise} \end{array} \right. \end{aligned}$$

and the payoff to player $i\in I$ is

$$\begin{aligned} v_i(z,x)=\underline{u}_i(x)-\underline{u}_i(z). \end{aligned}$$

Suppose that the game G does not have an equilibrium in X. Let $\overline{x}\in X$ be a nonequilibrium of $G'$. Then, by quasi-weak transfer continuity, there exists a player i, a strategy $\overline{y}_i\in X_i$ and some neighborhood $\mathcal{N}_{\overline{x}}$ of $\overline{x}$ such that for every $z\in \mathcal{N}_{\overline{x}}$, we have $\underline{u}_i(\overline{y}_i,z_{-i})> \underline{u}_i(z)$. Therefore, for each $t\in \mathcal{N}_{\overline{x}}$, there is a player $j=i\in I$ for whom, we have for each $z\in \mathcal{N}_{\overline{x}}$,

$$\begin{aligned} v_j(z,(\overline{y}_j,z_{-j}))=\underline{u}_i(\overline{y}_i,z_{-i})-\underline{u}_i(z)>0=v_j(t,t). \end{aligned}$$

Hence $G'$ is point secure with respect to I. Consequently, since $\underline{u}_i(x_i,x_{-i})$ is quasiconcave in $x_i$, then this new game satisfies the hypotheses of Theorem 3.4 in Reny (2013) and therefore possesses a pure strategy Nash equilibrium $(\overline{z},\overline{x})$. Payoff of player 0 implies that $\overline{z}=\overline{x}$. For each player $j\in I$, his payoff implies that $\overline{x}_j$ maximizes $\underline{u}_j(y_j,\overline{x}_{-j})$. By quasi-weak transfer continuity, there exists a player i and a strategy $\overline{y}_i\in X_i$ such that $\underline{u}_i(\overline{y}_i,\overline{x}_{-i})> \underline{u}_i(\overline{x})$, which contradicts that $\overline{x}_i$ maximizes $\underline{u}_i(y_i,\overline{x}_{-i})$.$\square$

Proof of Proposition 1

(a) Suppose that G is strongly quasi-weakly transfer continuous. Then, if $x \in X$ is not an equilibrium, there exists a player i, $\overline{y}_i\in X_i$, $\epsilon >0$, and some neighborhood $\mathcal{V}$ of x such that for every $z\in \mathcal{V}$, we have $u_i(\overline{y}_i,z_{-i})\ge \underline{u}_i(z)+\epsilon$. Therefore, for each $z\in \mathcal{V}$ and $\mathcal{N}_z\subset \mathcal{V}$ as a neighborhood of z, we have

$$\begin{aligned} \underset{z'\in \mathcal{N}_z}{\inf }u_i(\overline{y}_i,z'_{-i})\ge \underset{z'\in \mathcal{N}_z}{\inf }\underline{u}_i(z_i,z'_{-i})+\epsilon . \end{aligned}$$

(2)

Indeed, if (2) is false for some $z\in \mathcal{V}$ and $\mathcal{N}_z\subset \mathcal{V}$, then there is $\tilde{z}\in \mathcal{N}_z$ so that

$$\begin{aligned} u_i(\overline{y}_i,\tilde{z}_{-i})<\underset{z'\in \mathcal{N}_z}{\inf }\underline{u}_i(z_i,z'_{-i})+\epsilon \le \underline{u}_i(z_i,\tilde{z}_{-i})+\epsilon \le u_i(\overline{y}_i,\tilde{z}_{-i}), \end{aligned}$$

which is impossible. Obviously, for each $z\in X$ and each neighborhood $\mathcal{N}_z$ of z, we have

$$\begin{aligned} \underset{z'\in \mathcal{N}_z}{\inf }u_i(t_i,z^{'}_{-i})=\underset{z'\in \mathcal{N}_z}{\inf }\underline{u}_i(t_i,z'_{-i}),\;\text { for each }t_i\in X_i. \end{aligned}$$

(3)

Combining (2) and (3), we obtain for each $\mathcal{N}_z\subset \mathcal{V}$,

$$\begin{aligned} \underline{u}_i(\overline{y}_i,z_{-i})\ge \underset{z'\in \mathcal{N}_z}{\inf }u_i(\overline{y}_i,z^{'}_{-i})\ge \underset{z'\in \mathcal{N}_z}{\inf }\underline{u}_i(z_i,z'_{-i})+\epsilon =\underset{z'\in \mathcal{N}_z}{\inf }u_i(z_i,z^{'}_{-i})+\epsilon . \end{aligned}$$

Consequently for each $z\in \mathcal{V}$, we have $\underline{u}_i(\overline{y}_i,z_{-i})\ge \underline{u}_i(z)+\epsilon > \underline{u}_i(z)$, which means G is quasi-weakly transfer continuous.

(b) Suppose that G is QWUSC and payoff secure. If $\overline{x}\in X$ is not a Nash equilibrium, then by quasi-weak upper semicontinuity, some player i has a strategy $\hat{x}_i\in X_i$, $\epsilon >0$ and a neighborhood $\mathcal{N}^1$ of $\overline{x}$ such that for every $z\in \mathcal{N}^1$ and every neighborhood $\mathcal{N}_z\subseteq \mathcal{N}^1$ of z, $u_i(\hat{x}_i,\overline{x}_{-i})>u_i(z_i,z_{-i}')+\epsilon$ for some $z'\in \mathcal{N}_z$. The payoff security of G implies that there exists a strategy $y_i$ and a neighborhood $\mathcal{N}^2$ of $\overline{x}$ such that $u_i(y_i,z_{-i})\ge u_i(\hat{x}_i,\overline{x}_{-i})-\frac{\epsilon }{2}$ for all $z\in \mathcal{N}^2$. Thus, for every $z\in \mathcal{N}=\mathcal{N}^1\cap \mathcal{N}^2$ and its neighborhood $\mathcal{N}_z\subseteq \mathcal{N}$, there exists $z'\in \mathcal{N}_z$ such that $u_i(y_i,z_{-i})>u_i(z_i,z_{-i}')+\frac{\epsilon }{2}$.

(c) Suppose that G is WTLSC and QUSC. Then, if $\overline{x}\in X$ is not a Nash equilibrium, by WTLSC, there exists a player i, $y_i\in X_i$, $\epsilon >0$ and a neighborhood $\mathcal{N}^1$ of $\overline{x}$ such that $u_i(y_i,z_{-i})> u_i(\overline{x})+\epsilon$ for all $z\in \mathcal{N}^1$. The QUSC implies that for $i\in I$, $\overline{x}\in X$, and $\epsilon >0$, there exists a neighborhood $\mathcal{N}^2$ of $\overline{x}$ such that for every $z\in \mathcal{N}^2$ and every neighborhood $\mathcal{N}_z\subseteq \mathcal{N}^2$ of z, $u_i(\overline{x})\ge u_i(z_i,z_{-i}')-\frac{\epsilon }{2}$ for some $z'\in \mathcal{N}_z$. Thus, for every $z\in \mathcal{N}=\mathcal{N}^1\cap \mathcal{N}^2$ and its neighborhood $\mathcal{N}_z\subseteq \mathcal{N}$, there exists $z'\in \mathcal{N}_z$ such that $u_i(y_i,z_{-i})>u_i(z_i,z_{-i}')+\frac{\epsilon }{2}$.

Thus, by Theorem 1, the game possesses a pure strategy Nash equilibrium. $\square$

Proof of Theorem 2

Define a new two-player game $G'$ by a simple player’s strategy set Z. If player 1 chooses $t\in Z$ and player 2 chooses $z\in Z$, player 1’s preferences are presented by

$$\begin{aligned}v_1(t,z)=\left\{ \begin{array}{ll} 1, &{}\;\;\text {if }z=t\\ 0, &{}\;\;\text {otherwise} \end{array} \right. \end{aligned}$$

and player 2’s preference relation is defined by $v_2(t,z)=\underline{f}(z,t)-\underline{f}(t,t)$. The game $G'$ is point secure with respect to $I'=\{2\}$. Then, by Theorem 3.4 in Reny (2013), $G'$ possesses a Nash equilibrium $(\overline{x},\overline{x})$ which is also a symmetric Nash equilibrium of G (because G is diagonally quasi-weak transfer continuous). $\square$

Proof of Theorem 3

Suppose that the considered game does not have a symmetric Nash equilibrium. Then, by diagonal weak transfer continuity, for each $x\in X$, there exists an open neighborhood $\mathcal{N}_x$ and $y\in X$ so that $f(z,y)>f(z,z)$, for each $z\in \mathcal{N}_x$. Thus, we obtain a collection $\{(\mathcal{N}_x,y^x)\}_{x\in X}$ where $\{\mathcal{N}_x\}_{x\in X}$ forms an open cover of X. Since X is compact, then one can extract a finite subcollection $\{(\mathcal{N}_{x^k},y^k)\}_{k\in K}$ (K is a finite set), so as for each $k\in K$ and for each $z\in \mathcal{N}_{x^k}$, $f(z,y^k)>f(z,z)$ for each $z\in \mathcal{N}_{x^k}$. Let $\{\beta _k\}_{k\in K}$ be a partition of the unity subordinate to $\{\mathcal{N}_{x^k}\}_{k\in K}$. By the diagonal transfer quasiconcavity, for the finite set $\{y^k,\;k\in K\}\subseteq X$, there is a corresponding finite subset $\{\tilde{x}^k,\;k\in K\}$ such that for any subset J of K and every $\overline{x}\in \text {co}\{\tilde{x}^j,\;j\in J\}$, we have

$$\begin{aligned} f(\overline{x},\overline{x})\ge \underset{j\in J}{\min }f(\overline{x},y^k). \end{aligned}$$

(4)

Let us consider the following function $g:X\rightarrow X$ defined by $g(z)=\sum _{k\in K}\beta _k(z)\tilde{x}^{k}$. Brouwer fixed point theorem implies that there is $\overline{x}\in X$ so that $\overline{x}=g(\overline{x})=\sum _{j\in J}\beta _j(\overline{x})\tilde{x}^{j}$ where $J=\{j\in K:\;\beta _j(\overline{x})>0\}$. Then, for each $j\in J$, $\overline{x}\in \mathcal{N}_{x^j}$ and consequently $f(\overline{x},y^j)>f(\overline{x},\overline{x})$. Hence $f(\overline{x},\overline{x})\ge \min _{j\in J} f(\overline{x},y^k)>f(\overline{x},\overline{x})$, which is impossible. $\square$

Proof of Theorem 4

Suppose that there is no equilibrium in X. Define the following two-player game by a single player’s strategy set X. If player 1 chooses $t\in X$ and player 2 chooses $z\in X$, player 1’s preferences are presented by

$$\begin{aligned} u_1(t,z)=\left\{ \begin{array}{ll} 1, &{}\;\;\text {if }z=t\\ 0, &{}\;\;\text {otherwise} \end{array} \right. \end{aligned}$$

and player 2’s preference relation is defined by $(t,z)\succeq (t',z')$ if and only if $(t,z)\in P_i^{\succeq }(t',z')$, for each $i\in I$.

The two-player game is correspondence secure with respect to $I'=\{2\}$. Indeed, assume that it is not correspondence secure with respect to $I'=\{2\}$ in nonequilibrium $\overline{x}$. Then combining it with the $P^{\succeq }$-correspondence security, there exists a neighborhood $\mathcal{N}\subseteq X$ of $\overline{x}$, a well-behaved correspondence $\varphi :\mathcal{N}\rightrightarrows X$, $\tilde{z}\in \mathcal{N}$, $\tilde{y}\in \varphi (\tilde{x})$ and a player j such that

$$\begin{aligned}&\tilde{z}\in co\;\{w:\;(\tilde{z},w)\succeq (\tilde{x},\tilde{y})\}\end{aligned}$$

(5)

$$\begin{aligned}&\tilde{z}\notin co \left\{ w\in X\text { such that }(\tilde{z},w)\in P_j^{\succeq }(\tilde{x},\tilde{y})\right\} . \end{aligned}$$

(6)

By definition of $\succeq$, we obtain

$$\begin{aligned}&\tilde{z}\in co\;\{w:\;(\tilde{z},w)\in P_i^{\succeq }(\tilde{x},\tilde{y}), \text { for each }i\in I\}\\&\quad =co\left( \underset{i\in I}{\bigcap }\{w:\;(\tilde{z},w)\in P_i^{\succeq }(\tilde{x},\tilde{y})\}\right) . \end{aligned}$$

Since $co\left( \bigcap _{i\in I} A_i\right) \subseteq \bigcap _{i\in I}\left( co A_i\right)$, then for each $i\in I$,

$$\begin{aligned} \tilde{z}\in co \left\{ w\in X\text { such that }(\tilde{z},w)\in P_i^{\succeq }(\tilde{x},\tilde{y})\right\} , \end{aligned}$$

which contradicts (5). It is straightforward to show that the preference $\succeq$ is complete, reflexive, and transitive. Then, by Theorem 5.6 in Reny (2013), this game has an equilibrium $(\overline{x},\overline{x})$, i.e., for each $y\in X$, we have $(\overline{x},\overline{x})\succeq (\overline{x},y)$ which is equivalent to

$$\begin{aligned} (\overline{x},\overline{x})\in P_i^{\succeq }(\overline{x},y), \quad \text {for each}~i\in I. \end{aligned}$$

(7)

Or $\overline{x}$ is not a Nash equilibrium of G. Then by $P^{\succeq }$-correspondence security of G, there is a neighborhood $\mathcal{N}\subseteq X$ of $\overline{x}$, a well-behaved correspondence $\varphi :\mathcal{N}\rightrightarrows X$ and a player $j\in I$ so that

$$\begin{aligned} \overline{x}\notin co \left\{ t\in X:\;(\overline{x},t)\in P_j^{\succeq }(\overline{x},y)\right\} \end{aligned}$$

holds for each $y\in \varphi (\overline{x})$, which contradicts (7). $\square$

Proof of Theorem 5

Let $\varOmega (x)$ be the set of all open neighborhoods $\mathcal{N}$ of x. For each player $i\in I$ and every $x\in X$, define the following correspondence $C_i:X\rightrightarrows X_i$ by

$$\begin{aligned} C_i(x)=\{y_i\in X_i:\;\exists \mathcal{N}\in \varOmega (x),\;\forall z\in \mathcal{N},\; \underset{z'\in \mathcal{N}}{\inf }u_i(y_i,z'_{-i})> \underset{z''\in \mathcal{N}}{\inf }u_i(z_i,z''_{-i})\}. \end{aligned}$$

By the strong quasiconcavity, $C_i(x)$ is convex for all $x\in X$ and $i\in I$. The correspondence $C_i$ has the lower open section. Now suppose, by way of contradiction, that for each $x\in X$, there exists a player $i\in I$ such that $C_i(x)\ne \emptyset$. Then, by Theorem 3a in Deguire and Lassonde (1995), there exists a point $\widetilde{x}\in X$ and $i\in I$ such that $\widetilde{x}_i\in C_i(\widetilde{x})$, which is impossible. Thus, there exists $\overline{x}\in X$ such that for each $i\in I$, we have $C_i(\overline{x})=\emptyset$. If $\overline{x}$ is not a Nash equilibrium, then as the game G is quasi-weakly transfer continuous, there exists a player i, a strategy $\overline{y}_i\in X_i$ and some neighborhood $\mathcal{N}_{\overline{x}}$ of $\overline{x}$ such that for every $z\in \mathcal{N}_{\overline{x}}$, we have $\inf _{z'\in \mathcal{N}} u_i(\overline{y}_i,z'_{-i})> \inf _{z''\in \mathcal{N}} u_i(z_i,z''_{-i})$. Then, $\overline{y}_i\in C_i(\overline{x})$, which is a contradiction to $C_i(\overline{x})=\emptyset$. $\square$

Proof of Proposition 2

Let $G=(X_i,u_i)_{i\in I}$ be better-reply secure. Suppose, by way of contradiction, that the game is not pseudo quasi-weakly transfer continuous. Then, there exists a nonequilibrium $x^{*}\in X$ such that for all player j, $\epsilon >0$, every neighborhood $\mathcal{N}$ of $x^{*}$, and all $y_j$, there exists $x'\in \mathcal{N}$ satisfying

$$\begin{aligned} u_j(y_j,x'_{-j})\le u_j(x'_j,x^{\prime \prime }_{-j})+\epsilon ,\; \text { for all }x^{\prime \prime }\in \mathcal{N}. \end{aligned}$$

Letting $\overline{u}$ be the limit of the vector of payoffs corresponding to some sequence of strategies converging to $x^*$, and $U^*$ be the set of all such points, which is a compact set by the boundedness of payoffs, we have $(x^{*}, \overline{u}) \in \text {cl}(\varGamma )$ for all $\overline{u} \in U^*$. Then, for each $(x^{*}, \overline{u}) \in \text {cl}(\varGamma )$ with $\overline{u} \in U^*$, there exists a player i, a strategy $\overline{y}_i$, $\epsilon >0$ and a neighborhood $\overline{\mathcal{N}}$ of $x^{*}$ such that $u_i(\overline{y}_i,x'_{-i})>\overline{u}_i+\epsilon$ for all $x'\in \overline{\mathcal{N}}$. Then ${\inf }_{x'\in \overline{\mathcal{N}}}\;u_i(\overline{y}_i,x'_{-i})\ge \overline{u}_i+\epsilon$. Let $U^*_i$ be the projection of $U^*$ to coordinate i and

$$\begin{aligned} u^*_i=\sup \left\{ \overline{u}_i\in U^*_i: \underset{x'\in \overline{\mathcal{N}}}{\inf }\;u_i(\overline{y}_i,x'_{-i})\ge \overline{u}_i+\epsilon \right\} . \end{aligned}$$

Then, for $\epsilon /2 >0$, there is a neighborhood $\mathcal{N}^{i,*}$ of $x^{*}$ and a strategy $y^{*}_i$ such that

$$\begin{aligned} \underset{x'\in \mathcal{N}^{i,*}}{\inf }\;u_i(y^{*}_i,x'_{-i})\ge (u_i^{*} + \epsilon ) -\epsilon /2 = u_i^{*} + \epsilon /2. \end{aligned}$$

(8)

Now, since the game is not pseudo quasi-weakly transfer continuous, then for a directed system of neighborhoods $\{\mathcal{N}^{k}\}_{k}$ of $x^{*}$, a sequence $\{\epsilon ^k\}_k$ converging to 0, and every $j\in I$, there exists a sequence $\{x^{j,k}\}_k$ with $x^{j,k}\in \mathcal{N}^k$ so that $\{x^{j,k}\}_k$ converges to $x^{*}$ and

$$\begin{aligned} u_j(y^{*}_j,x^{j,k}_{-j})\le u_j(x^{j,k}_j,x'_{-j})+\epsilon ^k,\;\text { for each }x'\in \mathcal{N}^k. \end{aligned}$$

(9)

Consider the following sequence: for each k, let $\widetilde{x}^k=(x^{1,k}_1,\ldots ,x^{n,k}_n)$. Since for each $j\in I$, $x^{j,k}\in \mathcal{N}^k$ and $\{x^{j,k}\}_k$ converges to $x^{*}$, then $\widetilde{x}^{k}\in \mathcal{N}^k$ and the sequence $\{\widetilde{x}^k\}_k$ converges to $x^{*}$. Therefore, inequality (9) becomes

$$\begin{aligned} u_j\left( y^{*}_j,x^{j,k}_{-j}\right) \le u_j\left( x^{j,k}_j,\widetilde{x}^k_{-j}\right) = u_j(\widetilde{x}^k)+\epsilon ^k,\;\text { for each }k,\;j\in I. \end{aligned}$$

(10)

Assume that $\{u(\widetilde{x}^k)\}_k$ converges and $\widetilde{u}={\lim }_{k\rightarrow \infty } u(\widetilde{x}^{k})$. Hence, $(x^{*},\widetilde{u}) \in \text {cl}(\varGamma )$ with $\widetilde{u} \in U^*$, then there exists a player $i\in I$ such that $\widetilde{u}_i \le u^*_i$. Thus, for $\epsilon /3>0$, there exists $k_1$ such that whenever $k >k_1$, we have $u_i(y^{*}_i,x^{i,k}_{-i})\le u_i^{*}+\epsilon /3\le {\inf }_{x'\in \mathcal{N}^{i,*}}\;u_i(y^{*}_i,x'_{-i})-\epsilon /6$. Then for $k>k_1$, we obtain

$$\begin{aligned} u_i(y^{*}_i,x^{i,k}_{-i})\le u_i(y^{*}_i,x'_{-i})-\epsilon /6, \quad \text {for each }~x'\in \mathcal{N}^{i,*}. \end{aligned}$$

(11)

Since the sequence $\{x^{i,k}\}_{k}$ converges to $x^{*}$, then for $\mathcal{N}^{i,*}$, there exists $k_2$ such that for $k>k_2$, we have $x^{i,k}\in \mathcal{N}^{i,*}$. Thus, by (11) for $k>\max (k_1,k_2)$, we have $u_i(y^{*}_i,x^{i,k}_{-i})\le u_i(y^{*}_i,x_{-i}^{i,k} )-\epsilon /6$, which is impossible. Hence, the game must be pseudo quasi-weakly transfer continuous. $\square$

Proof of Theorem 6

Sufficiency. For each $y\in X$, let

$$\begin{aligned} F(y)=\{x\in X: u_i(y_i, x_{-i})\le u_i(x),\;\forall i\in I\}. \end{aligned}$$

It is clear that G is weakly transfer quasi-continuous if and only if F is transfer closed-valued. For $y \in X$, let $\bar{F}(y)=\text {cl } F(y)$. Then $\bar{F}(y)$ is closed, and by the strong diagonal transfer quasiconcavity, it is also transfer FS-convex. By Lemma 1 in Tian (1993), we know that ${\bigcap }_{y \in X} F(y)={\bigcap }_{y\in X} \bar{F}(y)\ne \emptyset$. Thus, there exists a strategy profile $\overline{x}\in X$ such that

$$\begin{aligned} u_i(y_i, \overline{x}_{-i})\le u_i(\overline{x}),\quad \text {for all} \; y \in X\; \text { and }\; i\in I. \end{aligned}$$

Thus $\overline{x}$ is a pure strategy Nash equilibrium of the game G.

Necessity: Suppose the game $\varGamma$ has a pure strategy Nash equilibrium $x^{*} \in X$. We want to show that it is strongly diagonal transfer quasiconcave in y. Indeed, for any finite subset $\{y^{1}, \ldots , y^{m}\} \subset X$, let the corresponding finite subset $X^m = \{x^{1},\ldots ,x^{m}\} = \{x^{*}\}$. Then, for any subset $\{x^{k^{1}},x^{k^{2}},\ldots , x^{k^{s}}\} \subset X^m = \{x^{*}\}$, $1 \le s\le m$, $x \in co\, \{x^{k^{1}}, x^{k^{2}}, \ldots , x^{k^{s}}\}=\{x^{*}\}$, and $y \in \{y^{k^{1}}, y^{k^{2}}, \ldots , y^{k^{s}}\}$, we have

$$\begin{aligned} u_i(y_i, x_{-i})=u_i(y_i, x^{*}_{-i}) \le u_i(x^{*}_i, x^{*}_{-i}) =u_i(x_i, x_{-i}). \end{aligned}$$

Hence U is strongly diagonal transfer quasiconcave in x. $\square$

Proof of Proposition 3

Suppose x is not an equilibrium. Then, some player i has a strategy $y_i$ such that $u_i(y_i,x_{-i})>u_i(x)$, i.e., $F_i(y_i,S_i(y_i,x_{-i}))>F_i(x_i,S_i(x))$. If $(y_i,x_{-i})\in X\backslash D_i$, then by Assumption 3, there exists a player $j\in I$, a deviation strategy profile $\bar{y}$, and a neighborhood $\mathcal{V}$ of x such that for every $z\in \mathcal{V}$, we have $\underline{F}_j(\bar{y}_j,S_j(\bar{y}_j,z_{-j}))> \underline{F}_j(z_j,S_j(z))$, i.e., $\underline{u}_j(y_j' ,z_{-j})> \underline{u}_j(z)$. If $(y_i,x_{-i})\in D_i$, then by Assumption 2, there exists a player $j\in I$ and $\bar{y}_j$ such that $(\bar{y}_j, x_{-j})\in X\backslash D_j$ and $F_j(\bar{y}_j,S_j(\bar{y}_j, x_{-j}))>F_j(x_j,S_j(x))$. Thus, by Assumption 3, there exists a player $k\in I$, a deviation strategy profile $\tilde{y}$, and a neighborhood $\mathcal{V}$ of x such that for every $z\in \mathcal{V}$, we have $\underline{F}_k(\tilde{y}_k ,S_k(\tilde{y}_k,z_{-k}))>\underline{F}_k(z_k,S_k(z))$, i.e., $\underline{u}_k(\tilde{y}_k,z_{-k})> \underline{u}_k(z)$. Therefore, the game is quasi-weakly transfer continuous. Since it is also compact and quasiconcave, by Theorem 1, it has a pure strategy Nash equilibrium. $\square$

Proof of Proposition 4

Suppose p is not an equilibrium. Then, some player i has a strategy $q_i$ such that $\pi _i(q_i,p_{-i})>\pi _i(p)$. If $(q_i,p_{-i})\in P^n\backslash A_i$, then by Assumption 3, there exists a player $j\in I$, a deviation strategy profile $\bar{q}_j$ and a neighborhood $\mathcal{N}$ of p such that for every $r\in \mathcal{N}_p$, $\underline{\pi }_j(\bar{q}_j,r_{-i})> \underline{\pi }_j(r)$. If $(q_i,p_{-i})\in A_i$, then by Assumption 2, there exists a firm $j\in I$, and $\bar{q}_j$ such that $(\bar{q}_j,p_{-i})\in P^n\backslash A_j$ and $\pi _j(\bar{q}_j, p_{-i})>\pi _i(p)$. Thus, by Assumption 3, there exists a player $j\in I$, a deviation strategy profile $\bar{q}_j$ and a neighborhood $\mathcal{N}$ of p such that for every $r\in \mathcal{N}_p$, $\underline{\pi }_j(\bar{q}_j,r_{-i})> \underline{\pi }_j(r)$. Therefore, the game is weakly transfer quasi-continuous. Since the game is also compact, convex, and quasiconcave, by Theorem 6, it has a pure strategy Nash equilibrium. $\square$