Top

Journal of Economic Interaction and Coordination

Published in:

07-03-2019 | Regular Article

Convergence to Walrasian equilibrium with minimal information

Author: Ratul Lahkar

Published in: Journal of Economic Interaction and Coordination | Issue 3/2020

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

next article Macroprudential regulation for a dynamic Chinese banking system with a scale-free network

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Gintis (2007, 2012) provides simulation based results on such convergence using methods from evolutionary game theory in a general equilibrium framework. We discuss those papers in more detail towards the end of this section.

Such models in which each agent is of measure zero are also called nonatomic games (Aumann and Shapley 1974).

The method of applying the aggregate form of a best response dynamic has also been applied by Ely and Sandholm (2005) in the context of Bayesian population games. Their best response dynamic is defined for any finite strategy game with a continuum of types. In our case, the BR dynamic is defined only for an aggregative game with a continuous strategy set and a finite number of types.

The standard evolutionary dynamics that have been extended to continuous strategy games are the replicator dynamic (Oechssler and Riedel 2001, 2002; Cheung 2016), the Brown–von Neumann–Nash (BNN) dynamic (Hofbauer et al. 2009), the Smith dynamic (Cheung 2014) and the logit dynamic (Perkins and Leslie 2014; Lahkar and Riedel 2015).

Cheung and Lahkar (2018) define a continuous strategy potential game with a single population and apply that definition to a large population Cournot competition model in which all firms have the same cost function. Lahkar and Mukherjee (2018) extend that definition to multi-type aggregative games. Lahkar (2017) also analyzes a single population Cournot model, but with a finite strategy approximation. We note here that we need a multi-type model in order for the issue of firms lacking information about the type of other firms to be meaningful.

Here, \(\delta _{x_p}\) is the Dirac distribution with probability 1 on \(x_p\).

These derivatives will be required in characterizing the Nash equilibrium of the Cournot competition model.

These assumptions imply that agents are not aware of the population state \(\mu _p\) of any \(p\in \mathcal {P}\), including their own population.

Recall that we have assumed that \(v(x,A(\mu ))\) is concave with respect to x. In (3), \(v(x,A(\mu ))=x\beta (A(\mu ))\) is linear with respect to x. We will use the assumption of bounded derivatives in “Appendix A.2”.

It is possible that best responses are not well defined at every social state in certain games with continuous strategy sets. However, as we will see, this problem does not occur in an aggregative game like (4).

This is because v is concave with respect to x and \(c_p\) is strictly convex.

This is because \(\beta (\bar{x})\) and \(\beta (\underline{x})\) are, respectively, the lowest and highest values of \(\beta (\alpha )\). Hence, \(\beta (\alpha )\ge \beta (\bar{x})\), which rules out the first case of (7) since, by Assumption 2.2(3), \(\beta (\bar{x})>c^{\prime }(\underline{x})\). Similarly, \(\beta (\alpha )\le \beta (\underline{x})\), which rules out the third case of (7) since, by Assumption 2.2(3), \(\beta (\underline{x})<c^{\prime }(\bar{x})\).

This is unlike the case of the best response dynamic for finite strategy games where the best response may not be uniquely defined (Gilboa and Matsui 1991). In that case, the best response dynamic needs to be defined as a differential inclusion.

This is because if \(\mu ^{*}\) is a Nash equilibrium of the aggregative game (4), then by Proposition 3.1, \(A(\mu ^{*})=\alpha ^{*}\) is the solution to (6). In that case, by (10), \(\alpha ^{*}\) is the rest point of the ABR dynamic.

As mentioned earlier, the BR dynamic cannot be generally extended to the continuous strategy case due to the possibility that the best response may not exist. Only in special cases like aggregative games with continuous strategy sets can it be defined.

For the definition of the variational norm, see (17) in “Appendix A.3”.

These extensions are required because the domain of the potential function f is \(\mathcal {M}\). With such an extension, it is possible that \(A(\mu )<0\), which requires us to extend \(\beta \) to \(\mathbf {R}\).

This result has been established for the BNN dynamic and the Smith dynamic by Hofbauer et al. (2009) and Cheung (2014) respectively.

Convergence may not happen from boundary states because, as is well known, non-Nash monomorphic states are also rest points of the replicator dynamic.

Sandholm (2010a) provides a detailed discussion of the informational requirements of revision protocols that generate different evolutionary dynamics.

The average payoff in population p at social state \(\mu \) is \(\bar{F}_p(\mu )=\int _{\mathcal {S}}F_{x,p}(\mu )\mu _p(dx)\).

See Cheung (2016) for continuous strategy versions of these revision protocols.

Of course, as mentioned earlier, we could have used the potential game argument even to the BR dynamic. The relevant result establishing convergence in aggregative potential games under this dynamic is in Lahkar and Mukherjee (2018).

Consider \(\mu ,\nu \in \mathscr {M}\). Then, \(|A(\mu )-A(\nu )|=\left| \sum _p\int _\mathcal {S}x\mu _p(dx)-\sum _p\int _\mathcal {S}x\nu _p(dx)\right| \le \sum _p\int _\mathcal {S}x\left| \mu _p-\nu _p\right| (dx)=\Vert \mu -\nu \Vert \).

Alós-Ferrer C, Ania A (2005) The evolutionary stability of perfectly competitive behavior. Econ Theory 26:497–516CrossRef

Arrow KJ, Hurwicz L (1958) On the stability of the competitive equilibrium, I. Econometrica 26:522–552CrossRef

Askari H, Cummings JT (1977) Estimating agricultural supply response with the nerlove model: a survey. Int Econ Rev 18:257–292CrossRef

Aumann RJ, Shapley LS (1974) Values of non-atomic games. Princeton University Press, Princeton

Björnerstedt J, Weibull JW (1996) Nash equilibrium and evolution by imitation. In: Arrow KJ et al (eds) The rational foundations of economic behavior. St. Martins Press, New York, pp 155–181

Brown GW, von Neumann J (1950) Solutions of games by differential equations. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games I. Annals of mathematics studies, vol 24. Princeton University Press, Princeton, pp 73–79

Cheung MW (2014) Pairwise comparison dynamics for games with continuous strategy space. J Econ Theory 153:344–375CrossRef

Cheung MW (2016) Imitative dynamics for games with continuous strategy space. Games Econ Behav 99:206–223CrossRef

Cheung MW, Lahkar R (2018) Nonatomic potential games: the continuous strategy case. Games Econ Behav 108:341–362CrossRef

Corchón L (1994) Comparative statics for aggregative games the strong concavity case. Math Soc Sci 28:151–165CrossRef

Ely JC, Sandholm WH (2005) Evolution in Bayesian games I: theory. Games Econ Behav 53:83–109CrossRef

Fisher FM (1983) Disequilibrium foundations of equilibrium economics. Cambridge University Press, CambridgeCrossRef

Fudenberg D, Levine DK (1998) The theory of learning in games. MIT Press, Cambridge

Gilboa I, Matsui A (1991) Social stability and equilibrium. Econometrica 59:859–867CrossRef

Gintis H (2007) The dynamics of general equilibrium. Econ J 117:1289–1309CrossRef

Gintis H (2012) The dynamics of pure market exchange. In: Aoki M, Binmore K, Deakin S, Gintis H (eds) Complexity and Institutions: norms and corporations. Palgrave, London

Hofbauer J (2000) From Nash and Brown to Maynard Smith: equilibria, dynamics, and ESS. Selection 1:81–88CrossRef

Hofbauer J, Oechssler J, Riedel F (2009) Brown–von Neumann–Nash dynamics: the continuous strategy case. Games Econ Behav 65(2):406–429CrossRef

Kirman AP (1992) Whom or what does the representative individual represent? J Econ Perspect 6:117–136CrossRef

McKenzie LW (1960) Stability of equilibrium and value of positive excess demand. Econometrica 28:606–617CrossRef

Lahkar R (2017) Large population aggregative potential games. Dyn Games Appl 7:443–467CrossRef

Lahkar R, Riedel F (2015) The logit dynamic for games with continuous strategy sets. Games Econ Behav 91:268–282CrossRef

Lahkar R, Mukherjee S (2018) Evolutionary implementation in a public goods game. Working paper, https://sites.google.com/site/rlahkar/home. Accessed 6 Mar 2019

Monderer D, Shapley L (1996) Potential games. Games Econ Behav 14:124–143CrossRef

Nax HH, Pradelski BSR (2015) Evolutionary dynamics and equitable core selection in assignment games. Int J Game Theory 44:903–932CrossRef

Nerlove M (1958) Estimates of the elasticities of supply of selected agricultural commodities. J Farm Econ 38:496–508CrossRef

Nikaido H, Uzawa H (1960) Stability and nonnegativity in a Walrasian Tâtonnement process. Int Econ Rev 1:50–59CrossRef

Oechssler J, Riedel F (2001) Evolutionary dynamics on infinite strategy spaces. Econ Theory 17:141–162CrossRef

Oechssler J, Riedel F (2002) On the dynamic foundation of evolutionary stability in continuous models. J Econ Theory 107:223–252CrossRef

Perkins S, Leslie D (2014) Stochastic fictitious play with continuous action sets. J Econ Theory 152:179–213CrossRef

Sandholm WH (2001) Potential games with continuous player sets. J Econ Theory 97:81–108CrossRef

Sandholm WH (2010a) Population games and evolutionary dynamics. MIT Press, Cambridge

Sandholm WH (2010b) Pairwise comparison dynamics and evolutionary foundations for Nash equilibrium. Games 1:3–17CrossRef

Schlag KH (1998) Why imitate, and if so, how? A boundedly rational approach to multi-armed bandits. J Econ Theory 78:130–156CrossRef

Smith MJ (1984) The stability of a dynamic model of traffic assignment: an application of a method of Lyapunov. Transp Sci 18:245–252CrossRef

Solymosi T, Raghavan TES (1994) An algorithm for finding the nucleolus of assignment games. Int J Game Theory 23:119–143CrossRef

Taylor PD, Jonker L (1978) Evolutionarily stable strategies and game dynamics. Math Biosci 40:145–156CrossRef

Walras L (1954[1874]) Elements of pure economics. George Allen and Unwin, London

Vega-Redondo F (1997) The evolution of Walrasian behavior. Econometrica 65:375–384CrossRef

Title: Convergence to Walrasian equilibrium with minimal information
Author: Ratul Lahkar
Publication date: 07-03-2019
Publisher: Springer Berlin Heidelberg
Published in: Journal of Economic Interaction and Coordination / Issue 3/2020
Print ISSN: 1860-711X
Electronic ISSN: 1860-7128
DOI: https://doi.org/10.1007/s11403-019-00243-8

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 3/2020

Macroprudential regulation for a dynamic Chinese banking system with a scale-free network

Information versus imitation in a real-time agent-based model of financial markets

Does social learning promote cooperation in social dilemmas?

Margin trade, short sales and financial stability

Innovation, finance, and economic growth: an agent-based approach

Epidemiology of inflation expectations and internet search: an analysis for India