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Dynamic Games and Applications
Erschienen in: Dynamic Games and Applications 4/2018

22.11.2017

An Evolutionary Analysis of Growth and Fluctuations with Negative Externalities

verfasst von: Anindya S. Chakrabarti, Ratul Lahkar

Erschienen in: Dynamic Games and Applications | Ausgabe 4/2018

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Abstract

We present an evolutionary game theoretic model of growth and fluctuations with negative externalities. Agents in a population choose the level of input. Total output is a function of aggregate input and a productivity parameter. The model, which is equivalent to a tragedy of the commons, constitutes an aggregative potential game with negative externalities. Aggregate input at the Nash equilibrium is inefficiently high causing aggregate payoff to be suboptimally low. Simulations with the logit dynamic reveal that while the aggregate input increases monotonically from an initial low level, aggregate payoff may decline from the corresponding high level. Hence, a positive technology shock causes a rapid initial increase in aggregate payoff, which is unsustainable as agents increase aggregate input to the inefficient equilibrium level. Aggregate payoff, therefore, declines subsequently. A sequence of exogenous shocks, therefore, generates a sustained pattern of growth and fluctuations in aggregate payoff.
Vorheriger Artikel Discrete and Continuous Distributed Delays in Replicator Dynamics
Nächster Artikel On Distributed Scheduling of Flexible Demand and Nash Equilibria in the Electricity Market

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Fußnoten
1
Production capacity refers to the ability of a sector to produce the relevant output. In the case of railways and the dot-com sector, the production capacity may be interpreted as the relevant infrastructure like railway lines and fiber-optic cables, respectively. In the case of finance, we interpret producers as banks and the production capacity as the resources devoted by these banks to create financial products. The output is the service that such products provide.
 
2
An example of over-accumulation of production capacity is the fiber-optics crash during the telecommunication boom accompanying the dot-com bubble, which led to a large amount of unused ‘dark’ fiber cable laid all over the USA [12].
 
3
Railway lines are clearly interlinked with each other. Dot-com companies provide services through the Internet and are, therefore, linked with each other through the common network that is the Internet. Developments in the financial sector have created interlinkages between different financial institutions by allowing them to hold each other’s assets and liabilities. See, for example, Elliot et al. [3] for an analysis of such financial interlinkages from a network theory perspective.
 
4
Apart from the difference in the evolutionary approach and the learning approach, we also note that Eusepi and Preston’s [4] model is a general equilibrium model, while ours is a partial equilibrium model.
 
5
Formally, let \(\alpha _2>\alpha _1>0\). Hence, \(\alpha _2=\lambda \alpha _1\), for some \(\lambda >1\). Since \(\pi \) satisfies decreasing returns to scale, \(\pi (\lambda \alpha _1)<\lambda \pi (\alpha _1)\Rightarrow \frac{\pi (\lambda \alpha _1)}{\lambda \alpha _1}<\frac{\pi (\alpha _1)}{\alpha _1}\Rightarrow \frac{\pi (\alpha _2)}{\alpha _2}<\frac{\pi (\alpha _1)}{\alpha _1}\). Therefore, average product is strictly declining.
 
6
Multiplying the individual share with the share of agents using strategy i and summing over all strategies, we obtain \(\sum _{i\in S_{(m,n)}}i\theta AP(a(x))x_i=\theta AP(a(x))a(x)=\theta \pi (a(x))\).
 
7
Intuitively, (3) implies that adding new agents has a negative impact on the payoff of existing agents. We note that this interpretation of negative externalities is common in large population games and evolutionary game theory. See, for example, Sandholm [18] for this interpretation. This is distinct from the more usual interpretation of negative externalities in economics where given a fixed number of agents, individual agents make a choice that is higher than what other agents would like.
 
8
Since \(AP(\alpha )\) is strictly decreasing, \(\pi ^{\prime }(\alpha )=MP(\alpha )<AP(\alpha )\). Recall our assumption that \(\theta \pi ^{\prime }(\alpha )>c^{\prime }(\alpha )\) in the neighborhood of 0. Hence, \(\theta AP(\alpha )>c^{\prime }(\alpha )\) in the neighborhood of 0, which implies that 0 cannot be the maximizer of \(g^{\theta }(\alpha )\). We also note that the condition \(\theta AP(\alpha ^{*}(\theta ))\ge c^{\prime }(\alpha ^{*}(\theta ))\) would hold with strict inequality only if m, the highest strategy, is sufficiently small relative to \(\theta \) that \(\theta AP(m)>c^{\prime }(m)\), which would imply that \(\theta AP(\alpha )>c^{\prime }(\alpha )\) for all \(\alpha \in [0,m]\) due to the fact that \(AP(\alpha )\) is a decreasing function. In that case, (6) is strictly increasing on all of [0, m] so that \(\alpha ^{*}(\theta )=m\). Since m can be arbitrarily high, we would expect the maximization condition \(\theta AP(\alpha ^{*}(\theta ))\ge c^{\prime }(\alpha ^{*}(\theta ))\) to typically hold with equality.
 
9
Lahkar [13] also considered aggregative potential games with positive externalities. Such games do not necessarily have unique Nash equilibrium.
 
10
This conclusion follows from an application of results in Friedman [5] to a one-input production function.
 
11
Recall that negative externalities arise because the production function satisfies decreasing returns to scale. This provides an alternative and more direct argument to prove Proposition 4.2, instead of the proof in “Appendix A.2” which relies on more general results from Lahkar [13]. Assuming interior solutions, the efficient state tends to equate the marginal product of aggregate input to marginal cost, while the Nash equilibrium tends to equate the average product to marginal cost. Since \(\pi \) satisfies decreasing returns to scale, \(AP(\alpha )\) is strictly decreasing on [0, m] due to which, \(MP(\alpha )<AP(\alpha )\). Hence, the result follows.
 
12
See Lahkar and Sandholm [14] for details of the strategy revision process.
 
13
The precise notion of “closeness” is as follows. Consider a general population game F(x) and suppose that \(\tilde{x}_{\eta }\) is a sequence of logit equilibria of F such that \(\lim _{\eta \rightarrow 0}\tilde{x}_{\eta }=x^{*}\). Then, \(x^{*}\) is a Nash equilibrium of F. This result follows from an adaptation of Theorem 2 in McKelvey and Palfrey [15] to the context of population games.
 
14
In fact, all well-known evolutionary dynamics converge to the set of Nash equilibria in potential games. We refer the reader to Sandholm [20] for a review of such results.
 
15
Without negative externalities, the aggregative potential game \(F^\theta \) may have had multiple Nash equilibria. In that case, while there would be convergence to some Nash equilibrium depending upon the initial state, it is not necessarily the case that there would be global convergence to the Nash equilibrium approximated by \(\alpha ^{*}(\theta )\). See also footnote 9 in this context.
 
16
Alternatively, the Nash equilibrium and efficient state can be calculated directly by maximizing the potential function and the aggregate payoff function for this game. The potential function takes the form \(f(x)=2\theta \sqrt{a(x)}-\sum _{i\in S_{(m,n)}}i^2x_i\) and the aggregate payoff function is defined in (12).
 
17
We follow this norm of presenting only the significant state variables in all the figures in this subsection.
 
18
Strictly speaking, we have generated this simplex by taking the initial state \(e_{0.2}\) in the game with three strategies, \(\{0.2,0.4,0.6\}\). In the simulation we are actually concerned with, there are four strategies but \(x_0\approx 0\) throughout the trajectory. Hence, heuristically speaking, we may ignore \(x_0\). Therefore, we are able to project the actual trajectory that lies in the interior, but arbitrarily close to the boundary, of the three-dimensional simplex onto its two-dimensional face where \(x_0=0\).
 
19
To see this, note that along the trajectory, the values of \(x_0\) and \(x_{0.4}\) remain very close to zero. If these are exactly zero, then, given the strategy set, \(a(x)=0.4\) only if \(x_{0.2}=x_{0.6}=0.5\).
 
20
It is evident from (4) that if two states generate the same aggregate strategy level, then any difference between their aggregate payoffs can only arise due to difference in total cost, \(\sum _{i\in S_{(m,n)}}c(i)x_i\).
 
21
The derivation of the state \(\tilde{x}\) is similar to the derivation of \(\hat{x}\) in footnote 19. Along the solution trajectory in Fig. 4, \(x_{0.2}\) and \(x_{0.4}\) remain very close to zero. If they are indeed zero, then \(a(\tilde{x})=0.4\).
 
22
Our model can easily accommodate negative shocks as well. Since the focus is on the effects of technological breakthroughs rather than regress, we discuss positive shocks.
 
23
We note that in presenting this figure, we have normalized the time period required for each round of evolution to 1. Thus, we may interpret the shock process (17) as implying that the value of \(\theta \) increases by 1 after every time period of 1.
 
24
As in Fig. 6, the decline, which is not very apparent in Fig. 7 due to the scale of the vertical axis, occurs due to the presence of strategies significantly higher than the Nash equilibrium strategy of 0.6.
 
25
As noted previously, this would happen provided the aggregate input has not hit the upper limit of m.
 
26
We note that the failure to coordinate that is alluded to in this paper is distinct from the more usual notion of coordination failure in economics. Coordination failure in that sense usually refers to the existence of a “good” and a “bad” equilibrium but where agents coordinate on the bad equilibrium. In our model, there is only one equilibrium. Failure to coordinate, therefore, refers to the inability of agents to sustain the efficient state which is not an equilibrium.
 
27
Recall that \(\bar{F}^{\theta }(x)=\sum _{j\in S_{(m,n)}}x_jF_j^{\theta }(x)\). Hence, \(\frac{\partial \bar{F}^{\theta }(x)}{\partial x_i}=F_i^{\theta }(x)+\sum _{j\in S_{(m,n)}}x_j\frac{\partial F_j^{\theta }(x)}{\partial x_i}=\hat{F}_i^{\theta }(x)\).
 
28
The resulting model will have payoff function \(F_i^{\theta }(x)=i\frac{\pi (\theta a(x))}{a(x)}-c(i)\). This is an aggregative potential game with potential function \(f^{\theta }(x)=\int _0^{a(x)}\frac{\pi (\theta z)}{z}\hbox {d}z-\sum _{i\in S_{(m,n)}}c(i)x_i\). The quasi-potential function is \(g^{\theta }(\alpha )=\int _0^{\alpha }\frac{\pi (\theta z)}{z}\hbox {d}z-c(\alpha )\).
 
29
See Lahkar [13] for an analysis of such a model.
 
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Metadaten
Titel
An Evolutionary Analysis of Growth and Fluctuations with Negative Externalities
verfasst von
Anindya S. Chakrabarti
Ratul Lahkar
Publikationsdatum
22.11.2017
Verlag
Springer US
Erschienen in
Dynamic Games and Applications / Ausgabe 4/2018
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-017-0234-6

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