Existence of equilibria in constrained discontinuous games

Abstract

We establish the existence of pure strategy equilibria in games with discontinuous payoffs where the set of feasible actions of each player varies, also in a discontinuous fashion, as a function of the actions of the other players. Such games are used in modeling abstract economies and other games where players share common constraints. Our approach circumvents the difficulties that arise from the presence of discontinuities by modifying the original problem and allowing the players to use strategies that possibly lie outside their feasible sets. We then show that each modified game has $\varepsilon$-equilibria points. Under certain conditions, and as the extent of modification becomes smaller and smaller and $\varepsilon$ approaches zero, the $\varepsilon$-equilibria points of the modified games will converge to a strategy profile that is an equilibrium of the original game. Hence, we obtain a set of sufficient conditions for the existence of pure equilibria of the original game. We apply our results to a number of classic games that have discontinuous payoffs and discontinuous constraint correspondences.

Appendix

Example 7

Consider a two-player game $G=(X_i,u_i,S_i,2)$ where $X_1=X_2=[0,1]$. The payoff of player one is

$$\begin{aligned}u_1(x_1,x_2)= {\left\{ \begin{array}{ll} 0 &{}\quad \text {if}~ x_1\not =x_2 ~\text {and}~ x_1\cdot x_2\not =0, \quad \text {or if }~ (x_1,x_2)= (0,0)\\ -1&{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

The payoff for the second player is

$$\begin{aligned}u_2(x_2,x_1)= {\left\{ \begin{array}{ll} 0 &{}\quad \text {if}~ x_1\cdot x_2\not =0\\ 1&{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

These payoffs are shown in Fig. 1 below

Fig. 1

The payoffs in Example 7

Let $S_1(x_{2})=\{x_{2}\}$ for any $x_2\in [0,1]$. Let $S_2(x_1)=\{0\}$ for $x_1\not = 0$ and $S_2(0)=[0,1]$.

This game is variationally continuous at every point in $[0,1]\times [0,1]$. However, we verify variational continuity in details only at (0, 0). Note that (0, 0) is the most problematic point in the game since neither $u_1$ nor $u_2$ is continuous, $S_2$ is not continuous, and despite the continuity of $S_1$, $V_1$ is not lsc. Let $(x^n_1,x^n_2)\longrightarrow (0,0)$ in $[0,1]\times [0,1]$. Starting with player 1, if $x^n_2 \not = 0$, let $C^n_1=(0,\delta ^n)$ for $\delta ^n \in (0,x^n_2)$ that is monotonically decreasing to 0. If $x^n_2=0$, let $C^n_1=\{0\}$. Clearly, $\mathop {{\hbox {Li}}}\nolimits BR_1(x^n_2)=\mathop {{\hbox {Ls}}}\nolimits C^n_1=S_1(0)=\{0\}$. For the second player, let $C^n_2=S_2(x^n_1)$. Again it is clear that $\mathop {{\hbox {Li}}}\nolimits BR_2(x^n_1)=\mathop {{\hbox {Ls}}}\nolimits C^n_2\subseteq S_2(0)$ since $C^n_2(x^n_1)=BR_2(x^n_1)$ for every $x^n_1 \in [0,1]$. Therefore, conditions (I) is satisfied.

Furthermore, we have $u_1(y^n_1,x^n_2)=0$ and $u_2(y^n_2,x^n_1)=1$ for any $y^n_1\in C^n_1$ and for any $y^n_2\in C^n_2$. Hence, $((y_1,y_2),(0,0),\alpha _1, \alpha _2)\in \mathop {{\hbox {Ls}}}\nolimits \Upsilon (x^n,C^n)$ implies that $\alpha _1=0$, $\alpha _2=1$, $y_1=0$. Moreover, $y_2=0$, if $x^n_1\not = 0$ for all n larger than some $n_0$. If $x^n$ contains a subsequence $x^{n_k}= 0$, then $y_2$ can be any point in [0, 1]. Therefore, any point in $\mathop {{\hbox {Ls}}}\nolimits \Upsilon (x^n,C^n)$ is also a point in $\mathop {{\text {hypo}}}\nolimits \tilde{u}$, and condition (III) of Definition 2 holds vacuously.

For all n, we have $V_1(x^n_2,C^n_1)=0=V_1(0)$ and $V_2(x^n_1, C^n_2)=1=V_2(1)$, and condition (II) also holds. Hence, the game G is variationally continuous at (0, 0). Note that as we mentioned in Remark 6, our assumptions do not imply the continuity of $V_i$, and in fact $V_1$ is not lsc at $x_2=0$. For comparison, note that both the constrained and the unconstrained games in Example 1 are not variationally continuous at (0, 0).

Note that $BR_1$ and $BR_2$ are convex-valued (despite the fact that $u_1(\cdot ,x_2)$ is not quasi-concave). Therefore, Theorem 3 implies that the game in this example has a pure strategy equilibrium despite the fact that the aggregator function in this game is not lsc, $S_2$ is not continuous, and neither the assumptions of Theorem 1 in Section 9.3 in Aubin (2007) nor the assumptions of Theorems 2 and 6, respectively in Tian (1992) and Tian and Zhou (1993), are satisfied. Moreover, assumption 2.2.6 in Balder (2002) is not satisfied, and therefore Theorem 2.2.1 in Balder (2002) does not apply.

Example 8

Let the correspondence be defined as follows:

$$\begin{aligned}F(z)= {\left\{ \begin{array}{ll} (z,1/4) &{}\quad \text {if}~ 0\le z< 1/2\\ 1/2\times [0,0.4]\bigcup 1/2\times [0.6,1] &{}\quad \text {if}~ z=1/2\\ (z,3/4) &{}\quad \text {if}~ 1/2 < z\le 1. \end{array}\right. } \end{aligned}$$

Clearly, F is usc but it is neither lsc nor convex valued, Michael’s selection theorem does not apply, and in fact F does not have a continuous selection. Let $\theta _1(z)=z$ and let

$$\begin{aligned}\theta _2(z)= {\left\{ \begin{array}{ll} 1/4 &{}\quad \text {if}~ 0\le z< 1/2\\ 1 &{}\quad \text {if}~ z=1/2\\ 3/4 &{}\quad \text {if}~ 1/2 \le z\le 1. \end{array}\right. } \end{aligned}$$

Then, $\theta (z)=(\theta _1(z), \theta _2(z))$ is a selection of F that satisfies the second half of condition (i) in Theorem 7. Similarly, let

$$\begin{aligned}\tilde{\theta }_2(z)= {\left\{ \begin{array}{ll} 1/4 &{}\quad \text {if}~ 0\le z< 1/2\\ 0 &{}\quad \text {if}~ z=1/2\\ 3/4 &{}\quad \text {if}~ 1/2 \le z\le 1. \end{array}\right. } \end{aligned}$$

Then, $\theta (z)=(\theta _1(z), \tilde{\theta }_2(z))$ is a selection of F that satisfies the second half of condition (ii) in Theorem 7.