Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Dynamic Games and Applications
Published in: Dynamic Games and Applications 3/2017

23-04-2016

Large Population Aggregative Potential Games

Author: Ratul Lahkar

Published in: Dynamic Games and Applications | Issue 3/2017

Log in

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

search-config
loading …

Abstract

We consider population games in which payoff depends upon the aggregate strategy level and which admit a potential function. Examples of such aggregative potential games include the tragedy of the commons and the Cournot competition model. These games are technically simple as they can be analyzed using a one-dimensional variant of the potential function. We use such games to model the presence of externalities, both positive and negative. We characterize Nash equilibria in such games as socially inefficient. Evolutionary dynamics in such games converge to socially inefficient Nash equilibria.
previous article Approachability in Stackelberg Stochastic Games with Vector Costs
next article Cooperative and Noncooperative Extraction in a Common Pool with Habit Formation

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 "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

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
Appendix
Available only for authorised users
Footnotes
1
See Sandholm [15] for a review of such results.
 
2
The assumptions that m and n are positive integers ensure that the dimension \(mn+1\) in \(\mathbf {R}^{mn+1}_+\) makes sense.
 
3
It is possible to define more general versions of the strategy aggregate, as in Alós-Ferrer and Ania [2]. See the last paragraph of Sect. 3 for more on this.
 
4
Note that we define the cost function with respect to the domain [0, m] and not the finite set \(S_n\). This is primarily because when we define the quasi-potential function in Sect. 4, we need the continuous version of the cost function.
 
5
See Sandholm [14] for an alternative definition of potential games. Definition 3.1 extends the domain of F from X to \(\mathbf {R}^{mn+1}_+\) in order to ensure that partial derivatives of the type \(\frac{\partial f}{\partial x_i}\) does exist. The alternative way is to confine oneself to X and define a potential game using affine calculus, as in Sandholm [14].
 
6
The converse is not true. It is possible for Nash equilibria to be local minimizers of the potential function. For example, completely mixed equilibria of pure coordination games minimize the corresponding potential function.
 
7
Also note that in defining g, we use the fact that c has domain [0, m]. See also footnote 4.
 
8
As an example of a game with multiple equilibria, consider a tragedy of the commons model with production function \(\pi (z)=5\sqrt{z}\), strategy set \(S_n=\{0,1,2,3,4,5\}\) and linear cost function \(c(i)=3i\). The average product, and so the aggregate benefit function, is \(AP(z)=\frac{5}{\sqrt{z}}\). Hence, the resulting quasi-potential function is \(g(\alpha )=\int _0^{\alpha }AP(z)dz-c(\alpha )=10\sqrt{\alpha }-3\alpha \). This function is maximized at \(\alpha ^{*}=\frac{25}{9}\). The linearity of the cost function implies that if \(a(x)=\alpha \), then \(f(x)=g(\alpha )\). Hence, the potential function f is maximized at any state \(x^{*}\) such that \(a(x^{*})=\alpha ^{*}=\frac{25}{9}\). The set of such states is convex which, by Proposition 4.3, coincides with the set of Nash equilibria.
 
9
The large population model of Cournot competition is, therefore, actually a model of perfect competition. Formally, this follows from the fact that each producer has zero weight in the whole population. See also the related discussion in the Introduction and comparison with results in Vega-Redondo [19] and Alós-Ferrer and Ania [2].
 
10
For example, \(\beta (z)=\frac{3}{2}\sqrt{z}\) and \(c(\alpha )=\alpha ^{\frac{3}{2}}\). In that case, \(g(\alpha )=0\) for all \(\alpha \), and there is no unique maximizer.
 
11
The converse of Proposition 4.4(1) does not hold if \(S_n\) is not sufficiently dense. For example, consider F with \(S_n=\{0,1\}\), \(\beta (\alpha )=\frac{1}{2}(1+\alpha )\) and \(c(i)=i^2\). Then, \(e_0\) is a Nash equilibrium as \(F_0(e_0)=0>F_1(e_0)=-\frac{1}{2}\).
 
12
We note that if c is linear, then a necessary (but not sufficient condition) for g to be as assumed in Proposition 4.4 is \(\beta (0)>0\). If \(\beta (0)=0\) and c is linear, then \(\beta ^{\prime }(0)<0\) and we will be in the case considered in Proposition 4.5 below.
 
13
We refer the reader to Sect. 6 for a more detailed discussion on how evolutionary dynamics behave in potential games.
 
14
Monomorphic states that minimize the potential function in Proposition 4.4 need not be Nash equilibria. For example, consider the aggregative potential game F with \(\beta (\alpha )=(\alpha +1)^{0.5}\), \(c(\alpha )=\alpha ^2\) and \(S_n=\{0,0.25.0.5,0.75,1\}\). The resulting potential function is globally minimized at the state \(e_0\), which corresponds to \(\alpha =0\) that minimizes the quasi-potential function. But, \(e_0\) is not a Nash equilibrium of F as \(F_{0.5}(e_0)=0.25>F_0(e_0)=0\).
 
15
Indeed, the similarity of the proof of this part of the proposition and the if part of Proposition 4.3 leads to a more general conclusion. If the cost function is linear and g has an interior maximizer or minimizer, say \(\alpha \), then there exists a convex set of Nash equilibria of F, with each such equilibrium having aggregate strategy level \(\alpha \). This is irrespective of the shape of \(\beta \).
 
16
Proposition 4.5 considers the case where a strictly convex g has minimizer \(\hat{\alpha }\in (0,m)\). If g is strictly convex with minimizer 0, then that case is covered by Proposition 4.4. If g is strictly convex with minimizer m, then a variant of Proposition 4.4 holds. In that case, \(e_0\) is a Nash equilibrium but \(e_m\) is not necessarily so. For example, if \(\beta (\alpha )=1+\sqrt{\alpha }\), \(c(\alpha )=2\alpha \) and \(m<1\), then the minimizer of g is m. In that case, \(F_m(e_m)=m(1+\sqrt{m})-2m<0=F_0(e_m)\).
 
17
We note, however, that \(x^{**}\) here is efficient only in the restricted sense of maximizing the aggregate profit of the population of producers, i.e., maximizing producer surplus. If, however, we also consider consumers as active agents, then the Nash equilibrium, which is the maximizer of the potential function (8), is indeed efficient in the wider sense of maximizing the sum producer and consumer surplus. Under a perfectly discriminating monopolist, the Nash equilibrium aggregate quantity level would have maximized producer surplus. But that requires that the monopolist obtains a different price for every unit of output sold, i.e., price p(i) for the ith unit of output. In the competitive scenario that we have modeled, that is not possible as producers get the same price p(a(x)) from every unit of output.
 
18
An example of such a game is the Cournot competition model with constant elasticity of demand function \(p(\alpha )=\alpha ^{-\frac{1}{\varepsilon }}\), \(\varepsilon >1\), and zero marginal cost. I thank one of the anonymous referees for suggesting this example.
 
19
For \(\beta \) to be increasing, \(r>0\) which means \(\beta (0)=0\). Hence, \(F_i(e_0)=-c\), for all \(i\in S_n\).
 
20
Positive homogeneity of \(\hat{F}\) means \(\hat{F}_i(tx)=t^k\hat{F}_i(x)\), \(k>-1\). The term positive homogeneity arises from the fact that if a potential game is homogeneous of degree \(k>-1\), then its potential function is homogeneous of degree \(l>0\). See Section 3.1.6, Sandholm [15] for details.
 
21
See, for example, Lahkar and Sandholm [9] for a more detailed exposition of behavioral rules, also called revision protocols, and the resulting evolutionary dynamics.
 
22
To obtain a congestion game of the type considered in Sandholm [12] with two parallel links from \(F_{iK}\), take \(b_K(i)=0\) and \(S_n=\{1\}\).
 
23
To see the existence of such \(\tilde{x}\), note that \(\{0,m\}\subseteq S_n\), for all n. Take the support of \(\tilde{x}\) to be \(\{0,m\}\) and let \(\tilde{x}=\{1-\frac{\tilde{\alpha }}{m},0,\ldots ,0,\frac{\tilde{\alpha }}{m}\}\). Then, \(a(\tilde{x})=\tilde{\alpha }\).
 
Literature
1.
2.
3.
4.
5.
Dindoš M, Mezzetti C (2006) Better-reply dynamics and global convergence to Nash equilibria in aggregative games. Games Econ Behav 54:261–292CrossRefMATH
6.
Fudenberg D, Levine D (1998) Theory of learning in games. MIT Press, CambridgeMATH
7.
Hofbauer J (2000) From Nash and Brown to Maynard Smith: equilibria, dynamics, and ESS. Selection 1:81–88CrossRef
8.
9.
10.
11.
Ostrom E (1990) Governing the commons: the evolution of institutions for collective action. Cambridge University Press, CambridgeCrossRef
12.
13.
14.
15.
Sandholm W (2010) Population games and evolutionary dynamics. MIT Press, CambridgeMATH
16.
Schaffer M (1988) Evolutionarily stable strategies for a finite population and a variable contest size. J Theor Biol 132:469–478MathSciNetCrossRef
17.
Smith M (1984) The stability of a dynamic model of traffic assignment. Transp Sci 18:245–252CrossRef
18.
19.
Metadata
Title
Large Population Aggregative Potential Games
Author
Ratul Lahkar
Publication date
23-04-2016
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 3/2017
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-016-0190-6

Other articles of this Issue 3/2017

Dynamic Games and Applications 3/2017 Go to the issue

OriginalPaper

The Continuous Time Infection–Immunization Dynamics

OriginalPaper

Cooperative and Noncooperative Extraction in a Common Pool with Habit Formation

OriginalPaper

Total Reward Semi-Markov Mean-Field Games with Complementarity Properties

OriginalPaper

On Nonzero-Sum Game Considered on Solutions of a Hybrid System with Frequent Random Jumps

OriginalPaper

Evasion Differential Game of Infinitely Many Evaders from Infinitely Many Pursuers in Hilbert Space

OriginalPaper

On Dynamic Games with Randomly Arriving Players

Premium Partner

    Image Credits