Convergence to Walrasian equilibrium with minimal information

Abstract

We consider convergence to Walrasian equilibrium in a situation where firms know only market price and their own cost function. We term this a situation of minimal information. We model the problem as a large population game of Cournot competition. The Nash equilibrium of this model is identical to the Walrasian equilibrium. We apply the best response (BR) dynamic as our main evolutionary model. This dynamic can be applied under minimal information as firms need to know only the market price and the their own cost to compute payoffs. We show that the BR dynamic converges globally to Nash equilibrium in an aggregative game like the Cournot model. Hence, it converges globally to the Walrasian equilibrium under minimal information. We extend the result to some other evolutionary dynamics using the method of potential games.

Appendix

1.1 Nash equilibrium

Proof of Proposition 3.1

First, consider the only if part. Let $\mu ^{*}$ be a Nash equilibrium of F. We need to show that $\mu _p^{*}=m_p\delta _{b_p(\alpha ^{*})}$, where $\alpha ^{*}$ is a solution to (6). Since $\mu ^{*}$ is a Nash equilibrium, Definition 2.1 implies that every agent in every population p plays a payoff maximizer. By (5), the unique payoff maximizer for population p is $b_p(\alpha ^{*})$, where $\alpha ^{*}=A(\mu ^{*})$. Therefore, $\mu _p^{*}=m_p\delta _{b_p(\alpha ^{*})}$. But this means $\int _{\mathcal {S}}x\mu _p^{*}(dx)=m_pb_p(\alpha ^{*})$, which means

$$\begin{aligned} \alpha ^{*}=A(\mu ^{*})=\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}x\mu _p^{*}(dx)=\sum _{p\in \mathcal {P}}m_pb_p(\alpha ^{*}). \end{aligned}$$

Hence, $\alpha ^{*}$ is a solution to (6).

Now consider the if part. We need to show that if $\alpha ^{*}$ is a solution to (6), then $\mu ^{*}$ such that $\mu ^{*}_p=m_p\delta _{b_p(\alpha ^{*})}$ is a Nash equilibrium of F. If $\mu _p^{*}=m_p\delta _{b_p(\alpha ^{*})}$, then

$$\begin{aligned} A(\mu ^{*})=\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}x\mu _p^{*}(dx)=\sum _{p\in \mathcal {P}}m_pb_p(\alpha ^{*}). \end{aligned}$$

But since $\alpha ^{*}$ is a solution to (6), this must mean $A(\mu ^{*})=\alpha ^{*}$. Hence, by the definition of $b_p(\alpha )$ in (5), if $\mu _p^{*}=m_p\delta _{b_p(\alpha ^{*})}$, then at $\mu _p^{*}$, every agent in every population p is playing the unique best response to $\mu ^{*}$. But this must mean that $\mu ^{*}$ is a Nash equilibrium of F. $\square$

1.2 Best response dynamic

Proof of Lemma 4.1

Consider $\mu ^{*}$ and denote $A(\mu ^{*})=\alpha ^{*}$. For the if part, let $\mu ^{*}$ be a Nash equilibrium of F. Hence, every agent plays the unique best response to $\mu ^{*}$, which is $b_p(\alpha ^{*})$. Therefore, $\mu _p^{*}=m_p\delta _{b_p(\alpha ^{*})}$ so that $\dot{\mu }_p^{*}=0$.

For the only if part, let $\mu ^{*}$ be a rest point of the BR dynamic. Hence, $\mu _p^{*}=m_p\delta _{b_p(\alpha ^{*})}$. Therefore,

$$\begin{aligned} \sum _{p\in \mathcal {P}}m_p\int _{\mathcal {S}}x\delta _{b_p(\alpha ^{*})}(dx)=\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}x\mu _p^{*}(dx)\Rightarrow \sum _{p\in \mathcal {P}}m_pb_p(\alpha ^{*})=A(\mu ^{*})=\alpha ^{*}. \end{aligned}$$

Hence, $\alpha ^{*}$ is a solution to (6). Hence, by Proposition 3.1, $\mu ^{*}$ is a Nash equilibrium of F. $\square$

To show that the BR dynamic is well defined in an aggregative game like (4), we first need to impose a norm on $\mathcal {M}(\mathcal {S})$. This will allow us to establish Lipschitz continuity of the dynamic. Due to the aggregative nature of the underlying population game, the appropriate norm turns out to be

$$\begin{aligned} \Vert \mu \Vert =\sum _{p\in \mathcal {P}}\int _\mathcal {S}x|\mu _p|(dx)=\sum _{p\in \mathcal {P}}a(|\mu _p|)=A(|\mu |), \end{aligned}$$

(14)

where $\mu \in \mathcal {M}(\mathcal {S})$. We refer to this norm concisely as the AS norm. The following theorem states the relevant result. In writing this theorem, we use $\tau$ to denote time.

Theorem A.1

Consider the aggregative game F defined in (4). From every initial state $\mu (0)\in \Delta$, there exists a unique solution trajectory $\{\mu (\tau )\in \Delta \}_{\tau \ge 0}$ under the BR dynamic (9). Further, the solutions to the dynamic are continuous in the AS norm (14) with respect to initial conditions.

We provide a review of the proof of this result in Lahkar and Mukherjee (2018). First, we extend the domain of the payoff function (4) from $\Delta$ to $\mathscr {M}$. With this extension, the range of $A(\mu )$ extends from $[\underline{x},\bar{x}]$ to $\mathbf {R}$. The extended payoff function is

$$\begin{aligned} \tilde{F}_{x,p}(\mu )=\left\{ \begin{array}{l l} v(x,\bar{x})-c_p(x), &{} \quad \text {for}~\mu ~\hbox {such that}~A(\mu )>\bar{x},\\ v(x,A(\mu ))-c_p(x), &{} \quad \text {for}~ \mu ~\hbox {such that}~A(\mu )\in [\underline{x},\bar{x}],\\ v(x,\underline{x})-c_p(x), &{} \quad \text {for}~ \mu ~ \hbox {such that}~ A(\mu )<\underline{x}. \end{array} \right. \end{aligned}$$

(15)

We then obtain the following best response function for $\alpha \in \mathbf {R}$ from (15).

$$\begin{aligned} \tilde{b}_p(\alpha )=\left\{ \begin{array}{l l} b_p(\bar{x}), &{} \quad \text {for}~ \alpha > \bar{x}\\ b_p(\alpha ), &{} \quad \text {for}~ \alpha \in [\underline{x},\bar{x}]\\ b_p(\underline{x}), &{} \quad \text {for}~ \alpha < \underline{x}. \end{array} \right. \end{aligned}$$

(16)

The proof of Theorem A.1 is based on two additional lemmas. Lemma A.2 in Lahkar and Mukherjee (2018) establishes that the aggregate strategy function $A(\cdot )$ is Lipschitz continuous with respect to the AS norm (14) with Lipschitz constant 1. Thus, for $\mu ,\nu \in \mathscr {M}$,

$$\begin{aligned} |A(\mu )-A(\nu )|\le \Vert \mu -\nu \Vert . \end{aligned}$$

This follows from (2) and (14).²⁴

The second lemma, Lemma A.3 in Lahkar and Mukherjee (2018), establishes the $\tilde{b}_p(\alpha )$ defined in (16), is Lipschitz continuous with respect to $\alpha \in \mathbf {R}$. That is, there exists a constant $K_{B,p}$ such that for every $\alpha ^1,\alpha ^2\in \mathbf {R}$

$$\begin{aligned} |\tilde{b}_p(\alpha ^1)-\tilde{b}_p(\alpha ^2)|\le K_{B,p}|\alpha ^1-\alpha ^2|. \end{aligned}$$

The proof of this result requires us to show that $\tilde{b}_p(\alpha )$ has a bounded derivative for almost all $\alpha \in \mathbf {R}$. For this, note from (16) that since $\tilde{b}_p(\alpha )\in [\underline{x},\bar{x}]$, it must be either (i) $\underline{x}$ or (ii) $\bar{x}$ of (iii) $x_p\in (\underline{x},\bar{x})$ such that $v_1(x_p,\alpha )=c_p^{\prime }(x_p)$. This case is possible only if $\alpha \in [\underline{x},\bar{x}]$. In the first two cases, it is obvious that $\tilde{b}_p(\alpha )$ has a bounded derivative. In the third case, $v_1(\tilde{b}_p(\alpha ),\alpha )=c_p^{\prime }(\tilde{b}_p(\alpha ))$. We can then calculate

$$\begin{aligned} \tilde{b}_p^{\prime }(\alpha )=\frac{v_{12}(\tilde{b}_p(\alpha ),\alpha )}{c_p^{\prime \prime }(\tilde{b}_p(\alpha ))-v_{11}(\tilde{b}_p(\alpha ),\alpha )}, \end{aligned}$$

which is bounded by our assumptions that v and $c_p$ have bounded first and second derivatives on $[\underline{x},\bar{x}]$.

Lemmas A.2 and A.3 then lead to the Lipschitz continuity of the BR dynamic on $\mathscr {M}$. Standard results on ODE systems in Banach spaces then imply Theorem A.1. Details are in Appendix A.3 in Lahkar and Mukherjee (2018).

1.3 Potential games

We first define the notion of the Fréchet derivative.

Definition A.2

Let X and Y are Banach spaces. We say that $g:X\rightarrow Y$ is Fréchet-differentiable at x if there exists a continuous linear map $T:X\rightarrow Y$ such that $g(x+\vartheta ) = g(x) + T\vartheta + o(\Vert \vartheta \Vert )$ for all $\vartheta$ in some neighborhood of zero in X. If it exists, this T is called the Fréchet-derivative of g at x, and is written as Dg(x).

In order to apply the Fréchet derivative, we impose the strong topology on $\mathcal {M}(\mathcal {S})$. This is the topology induced by the variational norm on $\mathcal {M}(\mathcal {S})$. For $\nu \in \mathcal {M}(\mathcal {S})$, the variational norm is given by $\Vert \nu \Vert =\sup _{g}|\int _\mathcal {S}g d\nu |$ where g is a measurable function $g:\mathcal {S}\rightarrow \mathbf {R}$ such that $\sup _{x\in S}|g(x)|\le 1$. The variational norm on $\mathscr {M}=\prod _{p=1}^n\mathcal {M}(\mathcal {S})$ is then given by (see, for example, Perkins and Leslie 2014)

$$\begin{aligned} \Vert \mu \Vert =\max \{\Vert \mu _1\Vert ,\ldots ,\Vert \mu _n\Vert \}\text { for }\mu =(\mu _1,\ldots ,\mu _n)\in \mathscr {M}. \end{aligned}$$

(17)

We now prove Proposition 5.3. Recall that in stating this proposition, we had extended the domain of $A(\cdot )$ and $C(\cdot )$ to $\mathcal {M}$ and that of $\beta$ to $\mathbf {R}$.

Proof of Proposition 5.3

Let f be defined as in (13) with appropriate extensions of $A(\cdot )$, $C(\cdot )$ and $\beta (\cdot )$. We need to show that for all $\mu \in \Delta$ and all $(x,p)\in \mathcal {S}\times \mathcal {P}$,

$$\begin{aligned} \nabla f(\mu )(x,p)=x\beta (A(\mu ))-c_p(x). \end{aligned}$$

(18)

Let $\zeta =(\zeta _1,\zeta _2,\ldots ,\zeta _n)\in \mathscr {M}$. Here, $\zeta _p$ represents the direction of change of $\mu _p$. Then,

$$\begin{aligned} Df(\mu )\zeta =\beta (A(\mu ))DA(\mu )\zeta -DC(\mu )\zeta , \end{aligned}$$

(19)

where $C(\mu )$ is as defined in (12). Note that

$$\begin{aligned} A(\mu +\zeta )=\sum _{p\in \mathcal {P}}a(\mu _{p}+\zeta _p)=\sum _{p\in \mathcal {P}}\int _\mathcal {S}x(\mu _p+\zeta _p)(dx)=A(\mu )+\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}\tilde{x}\zeta _p(d\tilde{x}). \end{aligned}$$

Therefore,

$$\begin{aligned} DA(\mu )\zeta =\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}\tilde{x}\zeta _p(d\tilde{x}). \end{aligned}$$

(20)

Further,

$$\begin{aligned} C(\mu +\zeta )&=\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}c_p(x)(\mu _p+\zeta _p)(dx)\\&=\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}c_p(x)\mu _p(dx)+\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}c_p(x)\zeta _p(dx)\\&=C(\mu )+\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}c_p(x)\zeta _p(dx). \end{aligned}$$

Hence,

$$\begin{aligned} DC(\mu )\zeta =\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}c_p(x)\zeta _p(dx). \end{aligned}$$

(21)

Inserting (20) and (21) into (19) and using (11), we obtain

$$\begin{aligned} \sum _{p\in \mathcal {P}}\int _\mathcal {S}\nabla f(\mu )(x,p)\zeta _p(dx)=\beta (A(\mu ))\sum _{p\in \mathcal {P}}\int _\mathcal {S}\tilde{x}\zeta _p(d\tilde{x})-\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}c_p(\tilde{x})\zeta _p(d\tilde{x}) \end{aligned}$$

This equation holds for all $\zeta \in \mathscr {M}$. In particular, it holds for $\zeta$ such that $\zeta _p=\delta _x$ and $\zeta _k=0$ for all $k\ne p$. With this $\zeta$, we obtain

$$\begin{aligned} \nabla f(\mu )(x,p)=x\beta (A(\mu ))-c_p(x), \end{aligned}$$

which gives us (18). Hence, $\nabla f(\mu )(x,p)=F_{x,p}(\mu )$, where $F_{x,p}(\mu )$ is as defined in (3). This establishes the result. $\square$