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My setting is the following.

I have a sequence of games $\lbrace G_n \rbrace$ in which the strategy space is $S=[0,1]$, there are two players $(I=\lbrace 1,2 \rbrace)$, and payoff functions are given by the continuous (no other restrictions so far) functions $U_i^n(x_1,x_2)$ for each player $i=1,2$. That is, each game is defined by $G_n = (U^n, S,I)$.

Now, I know that the payoff functions $U_i^n$ converge pointwise to a (discontinuous) limit $U_i$. I therefore write down the 'limit game' $G = (U,S,I). $ We can write down a correspondence that maps a game to its nash equilibria. Call that correspondence $EQ$, and we define, for a particular game $G_n$, that $EQ(G_n) = \lbrace (s_{n1}^*, s_{n2}^*), (s_{n1}^{**},s_{n2}^{**}) \dots\rbrace $ where the right hand side is the set of (potentially infiite) nash equilibria for the game $G_n$ and, generically, we allow each equilibrium strategy $s_{ni}$ to be a mixed strategy. That is, a probability measure over $[0,1].$

I then analysed the limit game, and have found a unique mixed strategy nash equilibria in that game. That is, $EQ(G) = \lbrace (s_1^*, s_2^*) \rbrace$.

I do not have closed form solutions for the equilibria, $\lbrace (s_{n1}^*, s_{n2}^*), (s_{n1}^{**},s_{n2}^{**}) \dots\rbrace $, of the games in the sequence $\lbrace G_n \rbrace$. Naturally, though, I would like to say something about how the equilibria converge to the equilibrium of the limit game.

My questions are (to start) as follows.

• Given the pointwise convergence of payoff functions $U^n \rightarrow U$, in what sense can I say the games $\lbrace G_n \rbrace $ converge to the limit game $G$?
• What conditions do I need on the equilibrium correspondence $EQ$ so that I can say something about the convergence of the equilibrium strategies? That is, am I able, with some argument, to establish that there exists a sequence of equilibria that converges to the equilibrium of the limit game in some sense. That is, $\lbrace (s_{n1}^*, s_{n2}^*)\rbrace_{n=1}^\infty \rightarrow \lbrace (s_{1}^*, s_{2}^*)$, perhaps in the weak sense, perhaps in the strong sense, or perhaps in some other sense.

Any help or references are greatly appreciated.

Edit: clarified the notation and question.

My setting is the following.

I have a sequence of games $\lbrace G_n \rbrace$ in which the strategy space is $S=[0,1]^2$, there are two players $(I=\lbrace 1,2 \rbrace)$, and payoff functions are given by the continuous (no other restrictions so far) functions $U_i^n(x_1,x_2)$ for each player $i=1,2$. That is, each game is defined by $G_n = (U^n, S,I)$.

Now, I know that the payoff functions $U_i^n$ converge pointwise to a (discontinuous) limit $U_i$. I therefore write down the 'limit game' $G = (U,S,I). $ We can write down a correspondence that maps a game to its nash equilibria. Call that correspondence $EQ$, and we define, for a particular game $G_n$, that $EQ(G_n) = \lbrace (s_{n1}^*, s_{n2}^*), (s_{n1}^{**},s_{n2}^{**}) \dots\rbrace $ where the right hand side is the set of (potentially infiite) nash equilibria for the game $G_n$ and, generically, we allow each equilibrium strategy $s_{ni}$ to be a mixed strategy. That is, a probability measure over $[0,1].$

I then analysed the limit game, and have found a unique mixed strategy nash equilibria in that game. That is, $EQ(G) = \lbrace (s_1^*, s_2^*) \rbrace$.

I do not have closed form solutions for the equilibria, $\lbrace (s_{n1}^*, s_{n2}^*), (s_{n1}^{**},s_{n2}^{**}) \dots\rbrace $, of the games in the sequence $\lbrace G_n \rbrace$. Naturally, though, I would like to say something about how the equilibria converge to the equilibrium of the limit game.

My questions are (to start) as follows.

• Given the pointwise convergence of payoff functions $U^n \rightarrow U$, in what sense can I say the games $\lbrace G_n \rbrace $ converge to the limit game $G$?
• What conditions do I need on the equilibrium correspondence $EQ$ so that I can say something about the convergence of the equilibrium strategies? That is, am I able, with some argument, to establish that there exists a sequence of equilibria that converges to the equilibrium of the limit game in some sense. That is, $\lbrace (s_{n1}^*, s_{n2}^*)\rbrace_{n=1}^\infty \rightarrow \lbrace (s_{1}^*, s_{2}^*)\rbrace$, perhaps in the weak sense, perhaps in the strong sense, or perhaps in some other sense.

Any help or references are greatly appreciated.

Edit: clarified the notation and question.

My setting is the following.

I have a sequence of zero sum games $\lbrace G_n \rbrace$ in which the strategy space is $S=[0,1]$, there are two players $(I=\lbrace 1,2 \rbrace)$, and payoff functions are given by the continuous (no other restrictions so far) functions $U_i^n(x_1,x_2)$ for each player $i=1,2$. That is, each game is defined by $G_n = (U^n, S,I)$.

Now, I know that the payoff functions $U_i^n$ converge pointwise to a (discontinuous) limit $U_i$. I therefore write down the 'limit game' $G = (U,S,I). $ We can write down a correspondence that maps a game to its nash equilibria. Call that correspondence $EQ$, and we define, for a particular game $G_n$, that $EQ(G_n) = \lbrace (s_{n1}^*, s_{n2}^*), (s_{n1}^{**},s_{n2}^{**}) \dots\rbrace $ where the right hand side is the set of (potentially infiite) nash equilibria for the game $G_n$ and, generically, we allow each equilibrium strategy $s_{ni}$ to be a mixed strategy. That is, a probability measure over $[0,1].$

I then analysed the limit game, and have found a unique mixed strategy nash equilibria in that game. That is, $EQ(G) = \lbrace (s_1^*, s_2^*) \rbrace$.

I do not have closed form solutions for the equilibria, $\lbrace (s_{n1}^*, s_{n2}^*), (s_{n1}^{**},s_{n2}^{**}) \dots\rbrace $, of the games in the sequence $\lbrace G_n \rbrace$. Naturally, though, I would like to say something about how the equilibria converge to the equilibrium of the limit game.

My questions are (to start) as follows.

• Given the pointwise convergence of payoff functions $U^n \rightarrow U$, in what sense can I say the games $\lbrace G_n \rbrace $ converge to the limit game $G$?
• What conditions do I need on the equilibrium correspondence $EQ$ so that I can say something about the convergence of the equilibrium strategies? That is, am I able, with some argument, to establish that there exists a sequence of equilibria that converges to the equilibrium of the limit game in some sense. That is, $\lbrace (s_{n1}^*, s_{n2}^*)\rbrace_{n=1}^\infty \rightarrow \lbrace (s_{1}^*, s_{2}^*)$, perhaps in the weak sense, perhaps in the strong sense, or perhaps in some other sense.

Any help or references are greatly appreciated.

Edit: clarified the notation and question.

My setting is the following.

I have a sequence of games $\lbrace G_n \rbrace$ in which the strategy space is $S=[0,1]$, there are two players $(I=\lbrace 1,2 \rbrace)$, and payoff functions are given by the continuous (no other restrictions so far) functions $U_i^n(x_1,x_2)$ for each player $i=1,2$. That is, each game is defined by $G_n = (U^n, S,I)$.

Now, I know that the payoff functions $U_i^n$ converge pointwise to a (discontinuous) limit $U_i$. I therefore write down the 'limit game' $G = (U,S,I). $ We can write down a correspondence that maps a game to its nash equilibria. Call that correspondence $EQ$, and we define, for a particular game $G_n$, that $EQ(G_n) = \lbrace (s_{n1}^*, s_{n2}^*), (s_{n1}^{**},s_{n2}^{**}) \dots\rbrace $ where the right hand side is the set of (potentially infiite) nash equilibria for the game $G_n$ and, generically, we allow each equilibrium strategy $s_{ni}$ to be a mixed strategy. That is, a probability measure over $[0,1].$

I then analysed the limit game, and have found a unique mixed strategy nash equilibria in that game. That is, $EQ(G) = \lbrace (s_1^*, s_2^*) \rbrace$.

I do not have closed form solutions for the equilibria, $\lbrace (s_{n1}^*, s_{n2}^*), (s_{n1}^{**},s_{n2}^{**}) \dots\rbrace $, of the games in the sequence $\lbrace G_n \rbrace$. Naturally, though, I would like to say something about how the equilibria converge to the equilibrium of the limit game.

My questions are (to start) as follows.

• Given the pointwise convergence of payoff functions $U^n \rightarrow U$, in what sense can I say the games $\lbrace G_n \rbrace $ converge to the limit game $G$?
• What conditions do I need on the equilibrium correspondence $EQ$ so that I can say something about the convergence of the equilibrium strategies? That is, am I able, with some argument, to establish that there exists a sequence of equilibria that converges to the equilibrium of the limit game in some sense. That is, $\lbrace (s_{n1}^*, s_{n2}^*)\rbrace_{n=1}^\infty \rightarrow \lbrace (s_{1}^*, s_{2}^*)$, perhaps in the weak sense, perhaps in the strong sense, or perhaps in some other sense.

Any help or references are greatly appreciated.

Edit: clarified the notation and question.

my setting is the following.

I have a sequence of zero sum games $G_n $ in which the strategy space is $[0,1]$, there are two players, and payoff functions are given by the continuous (no other restrictions so far) functions $U_i^n(x_1,x_2)$ for each player $i=1,2$.

Now, I know that the payoff functions $U_i^n$ converge pointwise to a (discontinuous) limit $U_i$. We can write down a correspondence $$EQ: G_n \rightarrow M([0,1])$$ that maps a game to its nash equilibrium. $M([0,1])$ are the probability measures on $[0,1]$.

I am struggling to understand the following. Help and appropriate references are much appreciated.

• Given the convergence of payoff functions, in what sense can I say the games $\lbrace G_n \rbrace $ converge to a limit game $G$?
• I have studied the limit game, and found it to have a unique (mixed strategy =proababilistic) equilibrium. What conditions do I need on the correspondence so that some sequence of equilibria of $\lbrace G_n \rbrace$ converge (uniformly) to the equilibrium of the limit game?

Any help or references are greatly appreciated.

My setting is the following.

I have a sequence of zero sum games $\lbrace G_n \rbrace$ in which the strategy space is $S=[0,1]$, there are two players $(I=\lbrace 1,2 \rbrace)$, and payoff functions are given by the continuous (no other restrictions so far) functions $U_i^n(x_1,x_2)$ for each player $i=1,2$. That is, each game is defined by $G_n = (U^n, S,I)$.

Now, I know that the payoff functions $U_i^n$ converge pointwise to a (discontinuous) limit $U_i$. I therefore write down the 'limit game' $G = (U,S,I). $ We can write down a correspondence that maps a game to its nash equilibria. Call that correspondence $EQ$, and we define, for a particular game $G_n$, that $EQ(G_n) = \lbrace (s_{n1}^*, s_{n2}^*), (s_{n1}^{**},s_{n2}^{**}) \dots\rbrace $ where the right hand side is the set of (potentially infiite) nash equilibria for the game $G_n$ and, generically, we allow each equilibrium strategy $s_{ni}$ to be a mixed strategy. That is, a probability measure over $[0,1].$

I then analysed the limit game, and have found a unique mixed strategy nash equilibria in that game. That is, $EQ(G) = \lbrace (s_1^*, s_2^*) \rbrace$.

I do not have closed form solutions for the equilibria, $\lbrace (s_{n1}^*, s_{n2}^*), (s_{n1}^{**},s_{n2}^{**}) \dots\rbrace $, of the games in the sequence $\lbrace G_n \rbrace$. Naturally, though, I would like to say something about how the equilibria converge to the equilibrium of the limit game.

My questions are (to start) as follows.

• Given the pointwise convergence of payoff functions $U^n \rightarrow U$, in what sense can I say the games $\lbrace G_n \rbrace $ converge to the limit game $G$?
• What conditions do I need on the equilibrium correspondence $EQ$ so that I can say something about the convergence of the equilibrium strategies? That is, am I able, with some argument, to establish that there exists a sequence of equilibria that converges to the equilibrium of the limit game in some sense. That is, $\lbrace (s_{n1}^*, s_{n2}^*)\rbrace_{n=1}^\infty \rightarrow \lbrace (s_{1}^*, s_{2}^*)$, perhaps in the weak sense, perhaps in the strong sense, or perhaps in some other sense.

Any help or references are greatly appreciated.

Edit: clarified the notation and question.