Weakly continuous security and nash equilibrium

Abstract

This paper investigates the existence of pure strategy Nash equilibria in discontinuous and nonquasiconcave games. We introduce a new notion of continuity, called weakly continuous security, which is weaker than the most known weak notions of continuity, including the surrogate point secure of SSYM game of Carbonell-Nicolau and Mclean (Econ Theory, 2018a), the continuous security of Barelli and Meneghel (Econometrica 81:813–824, 2013), C-security of McLennan et al. (Econometrica 79:1643–1664 2011), generalized weakly transfer continuity of Nessah (Economics 47:659–662, 2011), generalized better-reply security of Carmona (Econ Theory 48:1–4, 2011), Barelli and Soza (On the existence of Nash equilibria in discontinuous and qualitative games, 2009), Barelli and Meneghel (Econometrica 81:813–824, 2013), lower single deviation property of Reny (Further results on the existence of Nash equilibria in discontinuous games. University of Chicago, 2009), better-reply security of Reny (Econometrica 67:1029–1056, 1999) and the results of Prokopovych (Econ Theory 48:5–16, 2011, Econ Theory 53:383–402, 2013) and Carmona (J Econ Theory 144:1333–1340, 2009). We show that a compact, convex and weakly continuous secure Hausdorff locally convex topological vector space game has a pure strategy Nash equilibrium. Moreover, it holds in a large class of discontinuous games.

Appendix

Proof of Theorem 3.1

Define a new game $\tilde{G}$ 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

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

Suppose that the game G does not have an equilibrium in X. Let ${\overline{x}}\in X$ be a nonequilibrium of $\tilde{G}$. Then by the weakly continuity secure of G, $\tilde{G}$ is correspondence secure with respect to I. Consequently, $\tilde{G}$ game satisfies the hypotheses of Theorem 5.6 in Reny (2016a) 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 ${\overline{u}}_j(y_j,{\overline{x}}_{-j})$. By weakly continuity secure of G, there exists a player i and a neighborhood ${{\mathcal {N}}}$ of ${\overline{x}}$, a well-behaved correspondence $\phi :\mathcal{N}\twoheadrightarrow X$ such that

${\overline{u}}_i(y_i,{\overline{x}})> {\overline{u}}_i({\overline{x}}_i,{\overline{x}})$, for each $y_i\in \phi _{i}({\overline{x}})$, which contradicts that ${\overline{x}}_i$ maximizes ${\overline{u}}_i(.,{\overline{x}})$. $\square$

Proof of Proposition 3.1

Assume that G is weakly reciprocal upper semicontinuous at ${\overline{x}}$ (${\overline{x}}$ is not an equilibrium) and it is not pseudo upper semicontinuous at ${\overline{x}}$. Then, by definition, for each neighborhood ${{\mathcal {N}}}$ of ${\overline{x}}$ and $\epsilon >0$, there exists $z\in {{\mathcal {N}}}$, so that $\sup _{y_i\in X_i} u_i(y_i,{\overline{x}}_{-i})< {\underline{u}}_i(z)+\epsilon$, for each $i\in I$. Thus, for a directed system of neighborhoods $\{\mathcal{N}^{k}\}_{k}$ of ${\overline{x}}$ and a sequence $\{\epsilon ^k\}_k$ that converges to 0, there exists a sequence $\{z^{k}\}_k$ with $z^{k}\in {{\mathcal {N}}}^k$ so as $\{z^{k}\}_k$ converges to ${\overline{x}}$ and for each $i\in I$, we have

$$\begin{aligned} \underset{y_i\in X_i}{\sup }u_i(y_i,{\overline{x}}_{-i})< {\underline{u}}_i(z^k)+\epsilon ^k\le u_i(z^k)+\epsilon ^k. \end{aligned}$$

Assume that $\{u({z}^k)\}_k$ converges and let ${\overline{u}}= {\lim }_{k\rightarrow \infty }u(z^{k})$. Then $({\overline{x}},{\overline{u}})$ is in the closure of the graph of G. If $({\overline{x}},{\overline{u}})$ is not in the frontier of G, then ${\overline{u}}=u({\overline{x}})$ and consequently $\sup _{y_i\in X_i} u_i(y_i,{\overline{x}}_{-i})\le u_i({\overline{x}})$, for each $i\in I$, which is a contradiction, because ${\overline{x}}$ is not a Nash equilibrium. Then $({\overline{x}},{\overline{u}})$ is on the frontier of G. By weakly reciprocal upper semicontinuity of G, there exists a player j who has a strategy ${\hat{y}}_j\in X_j$ such that $u_j({\hat{y}}_j,{\overline{x}}_{-j})>{\overline{u}}_j\ge \sup _{y_j\in X_j} u_j(y_j,{\overline{x}}_{-j})$, which is impossible. $\square$

Proof of Proposition 3.2

Let $x\in X$ be such that it is not an equilibrium. Then, by pseudo upper semicontinuity of G, there exist a neighborhood ${{\mathcal {N}}}$ of x and $\epsilon >0$ such that for each $z\in {{\mathcal {N}}}$ nonequilibrium, there exists a player $i\in I$ so that $\sup _{y_i\in X_i} u_i(y_i,x_{-i})> {\underline{u}}_i(z)+2\epsilon$. For $\epsilon >0$, there exists ${\hat{y}}$ so as $u_i({\hat{y}}_i,x_{-i})\ge \sup _{y_i\in X_i} u_i(y_i,x_{-i})-\epsilon$. By generalized payoff security in $({\hat{y}}_i,x_{-i})$, for $\epsilon >0$, and $i\in I$, there exist a neighborhood ${{\mathcal {N}}}^i$ of $x_{-i}$, and a well-behaved correspondence $\phi _{x,i}:{{\mathcal {N}}}\twoheadrightarrow X_i$ so that $u_i(y_i,t_{-i})\ge u_i({\hat{y}}_i,{\overline{x}}_{-i})-\epsilon$, for each $(t_{-i},y_i)\in \text {Graph}(\phi _{x,i})$. We also have ${\underline{u}}_i(y_i,t_{-i})\ge \inf _{(t'_{-i},y'_i)\in \text {Graph}(\phi _{x,i})}\;u_i(y'_i,t'_{-i})$, for each $(t_{-i},y_i)\in \text {Graph}(\phi _{x,i})$.

Let $\overline{{{\mathcal {N}}}}=\bigcap _{i\in I}\left( X_i\times {{\mathcal {N}}}^i\right) \bigcap {{\mathcal {N}}}$ be a neighborhood of x. Then for each $z\in \overline{{{\mathcal {N}}}}$ nonequilibrium, there exists a player $i\in I$ so that for each $(t_{-i},y_i)\in \text {Graph}(\phi _{x,i})$:

$$\begin{aligned} {\underline{u}}_i(y_i,t_{-i})+2\epsilon \ge u_i({\hat{y}}_i,x_{-i})+\epsilon \ge \sup _{y_i\in X_i} u_i(y_i,x_{-i})> {\underline{u}}_i(z) +2\epsilon . \end{aligned}$$

Since the game is quasiconcave, then it is g-weakly continuous secure. $\square$

Proof of Proposition 3.3

See the proof of Theorem 3.1. $\square$

Proof of Proposition 4.1

Suppose that G is continuous secure at x, where x is not an equilibrium. Then there is a neighborhood ${{\mathcal {N}}}$ of x, $\alpha \in {\mathbb {R}}^n$ and a well-behaved¹⁴ correspondence $\phi _{x}:\mathcal{N}\twoheadrightarrow X$, so that

• (1) for each $t\in {{\mathcal {N}}}$, and $i\in I$, we have $\phi _{x,i}(t)\subseteq B_i(t,\alpha _i)$, and

• (2) for each $z\in {{\mathcal {N}}}$, there exists a player j for whom, $z_j\notin co B_j(z,\alpha _j)$.

where $B_i(x,\alpha _i)=\{y_i\in X_i\text { such that }u_i(y_i,x_{-i})\ge \alpha _i\}$.

Condition (1) implies that for each $i\in I$, $t\in {{\mathcal {N}}}$ and $y_i\in \phi _{x,i}(t)$, $u_i(y_i,t_{-i})\ge \alpha _i$. Therefore, ${\inf }_{(t,y_i)\in \text {Graph}(\phi _{x,i})}\;u_i(t_i,z_{-i})\ge \alpha _i$. We have for each $(t,y_i)\in \text {Graph}(\phi _{x,i})$,

$$\begin{aligned} {\underline{u}}_i(y_i,t_{-i})\ge \underset{(t',y'_i)\in \text {Graph}(\phi _{x,i})}{\inf }\;u_i(t'_i,z'_{-i})\ge \alpha _i. \end{aligned}$$

Assume that for some $\tilde{z}\in {{\mathcal {N}}}$ nonequilibrium and for each player $i\in I$, there exists $(\tilde{x},\tilde{y}_i)\in \text {Graph}(\phi _{x,i})$ so that

$$\begin{aligned} \tilde{z}_i\in co \{t_i\in X_i\text { such that }{\underline{u}}_i(t_i,\tilde{z}_{-i})\ge {\underline{u}}_i(\tilde{y}_i,\tilde{x}_{-i})\}. \end{aligned}$$

There is a finite set $A_i=\{t_i^1,...,t_i^{p_i}\}\subseteq \{t_i\in X_i,\;{\underline{u}}_i(t_i,\tilde{z}_{-i})\ge {\underline{u}}_i(\tilde{y}_i,\tilde{x}_{-i})\}$ so that $\tilde{z}_i\in co A_i$. Therefore, for each $i\in I$, and $h=1,...,p_i$, we have

$$\begin{aligned} {\underline{u}}_i(t^h_i,\tilde{z}_{-i})\ge {\underline{u}}_i(\tilde{y}_i,\tilde{x}_{-i}) \ge \alpha _i. \end{aligned}$$

By condition (2), for $\tilde{z}\in {{\mathcal {N}}}$, there exists a player $j\in I$ so as $\tilde{z}_j\notin co B_j(\tilde{z},\alpha _j)$. If $A_j\subseteq B_j(\tilde{z},\alpha _j)$, then we obtain a contradiction $\tilde{z}_j\in co A_j\subseteq coB_j(\tilde{z},\alpha _j)$. Thus $A_j\nsubseteq B_j(\tilde{z},\alpha _j)$, and therefore, there is $h_0=1,...,p_j$ such that $t_j^{h_0}\notin B_j(\tilde{z},\alpha _j)$, i.e., $\alpha _j>u_j(t_j^{h_0},\tilde{z}_{-j})$. Let $\epsilon =\frac{1}{2}[\alpha _j-u_j(t_j^{h_0},\tilde{z}_{-j})]>0$. Then for $\epsilon >0$, there exists a neighborhood $\mathcal{N}_{\tilde{z}}$ so that

$$\begin{aligned} u_j(t_j^{h_0},\tilde{z}_{-j})\ge \alpha _j-\epsilon =\frac{1}{2}\alpha _j+\frac{1}{2}u_j(t_j^{h_0},\tilde{z}_{-j}). \end{aligned}$$

Thus $u_j(t_j^{h_0},\tilde{z}_{-j})\ge \alpha _j>u_j(t_j^{h_0},\tilde{z}_{-j})$ which is a contradiction. $\square$

Proof of Proposition 4.3

For each $x\in X$, we have ${\underline{u}}^u_j(x_j,x)\le 0$. Indeed, if ${\underline{u}}^u_j(x_j,x)>0$ for some $i\in I$ and $x\in X$, choose $\epsilon >0$ such that ${\underline{u}}^u_i(x,x_i)>2\epsilon$. Then there exists a neighborhood ${{\mathcal {N}}}$ of x and a well-behaved correspondence $\phi :{{\mathcal {N}}}\twoheadrightarrow X_i$ with $x_i\in \phi _i(x)$ such that for each $(z,y_{i})\in \text {Graph}(\phi _i)$, we have $u_i(y_i,z_{-i})>u_i(z_i,z_{-i})+\epsilon$. Then for $z=x$ and $y_i=x_i$, we obtain $u_i(x)>u_i(x)+\epsilon$, i.e., $\epsilon <0$ which is impossible.

Now, assume that G is generalized weakly transfer continuous. Then if $x\in X$ is not an equilibrium, there exist a player i, a neighborhood ${{\mathcal {N}}}$ of x and a well-behaved correspondence $\varphi _i:{{\mathcal {N}}}\twoheadrightarrow X_i$ such that $\inf _{(x^{'},y^{'}_i)\in \text {Graph}(\varphi _i)}\;\left[ u_i(y^{'}_i, x'_{-i})-u_i(x')\right] >0$. Thus for each $z\in {{\mathcal {N}}}$, there exists a player $j=i$ for whom, ${\overline{u}}^u_j(y_j,t)\ge \inf _{(x^{'},y^{'}_j)\in \text {Graph}(\varphi _j)}\;\left[ u_j(y^{'}_j, x'_{-j})-u_j(x')\right] >0\ge {\overline{u}}^u_j(z_j,z)$, for each $(t,y_j)\in \text {Graph}(\phi _{x,j})$. $\square$

Proof of Proposition 4.4

For each $x\in X$, we have ${\underline{u}}^{{\underline{u}}}_j(x_j,x)\le 0$. Now, assume that G is quasi-weakly transfer continuous. Then if $x\in X$ is not an equilibrium, there exist a player i, a neighborhood ${{\mathcal {N}}}$ of x, $\epsilon >0$ and a strategy $y_i\in X_i$ such that $u_i(y_i, t_{-i})>{\underline{u}}_i(t)+\epsilon$, for each $t\in {{\mathcal {N}}}$. Let us consider the following well-behaved correspondence $\phi _x:\mathcal{N}\twoheadrightarrow X$ defined by $\phi _x(z)=\{(y_1,y_2)\}$. Since for each $(t,y^{'}_j)\in \text {Graph}(\phi _{x,j})$, we have ${\underline{u}}^{{\underline{u}}}_j(y^{'}_j,t)\ge \inf _{(t',y^{''}_j)\in \text {Graph}(\phi _{x,j})} [u_j(y^{''}_j,t'_{-j})-{\overline{u}}_j(t')]$. Thus for each $z\in {{\mathcal {N}}}$ nonequilibrium, there exists a player $j=i$ for whom, ${\underline{u}}^{{\underline{u}}}_j(y^{'}_j,t)>0\ge {\overline{u}}^{{\overline{u}}}_j(z_j,z)$, for each $(t,y^{'}_j)\in \text {Graph}(\phi _{x,j})$. $\square$