Abstract
We propose a model where agents located in a social network decide whether or not to exert effort to provide a local public good. We assume that they have strong incentives to free-ride on their neighbors’ effort decisions. We characterize the equilibria of the induced game. We also study a mean-field dynamics in which agents choose in each period the best response to the last period's decisions of their neighbors. We characterize the fraction of free-riders in the stable state of such a dynamics and show how it depends on properties of the degree distribution.
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Notes
There is a wide empirical literature analyzing how a behavior spreads in a social network in various contexts some concerning coordination and others anti-coordination [see e.g., Foster and Rosenzweig 1995; Glaeser et al. 1996; Conley and Udry 2007].
In particular, Bramoullé and Kranton [2007] assume that agents can choose a level of effort in [0, +∞).
General random networks provide a natural benchmark to study mean-field approximations because they lack any kind of degree or structural correlations.
The degree distribution in a population is a key property crucial for studying diffusion processes in social networks. Thus, there is relevant work measuring the degree distribution for real large networks [e.g. Newman 2003; Goyal et al. 2006; Jackson and Rogers 2007].
The results can easily be extended to account for the effect on free-riding behavior of a FOSD and MPS of the degree distribution of a neighboring agent (see Propositions 7 and 8).
In fact, the incomplete information approach has a problem of interpretation when the game is repeated on a fixed network, since in those cases one would expect agents to converge to the complete information Nash equilibrium by learning from the actions of neighbors and the payoffs the agents receive.
Although the results obtained in Galeotti et al. [2008] and this paper are of a similar nature (in both cases results on comparative statics of networks are derived) they are fundamentally different. For instance, Galeotti et al. [2008] derive a threshold result in terms of the agents’ degree (i.e., agents exert effort only if their degrees are below a certain threshold), whereas in the mean-field model presented in this paper we obtain that in equilibrium there is a fraction of agents exerting effort for any given degree. I thank an anonymous referee for raising this point.
This proposition is not trivial since the dynamics may enter a cycle and never reach an absorbing state.
See, for example, Pastor-Satorrás and Vespignani [2001].
The mean-field approximation proposed in this paper can also be interpreted as a setting in which the network is randomly redrawn every period. This interpretation, although formally correct, seems artificial when put together with the fact that agents are choosing a myopic best response to the current actions of the current neighbors. Therefore, we prefer to focus on the idea that the mean-field approach analyzed in this paper is simply an approximation of the dynamics on fixed networks, but for networks that have been randomly generated.
Note that dH̃(θ)/dθ=−1/〈k〉∑ k ⩾1 kP(k)k(1−θ)k−1−1 is negative for all 0⩽θ⩽1.
Notice that we are not claiming any welfare implications derived from a free-riding behavior.
Condition (9) can be written as
A necessary condition for this to hold is that
, which is a decreasing function of k, must be above 1. Note that if k=k
m
this holds whenever 
which provides a lowerbound for k m . Moreover, in order for the condition to hold, the probability of selecting an agent from the population with a degree close to k m must be high.
To actually compare the values ρ * and ρ̄ * one would have to compare ∑ k ⩾1 P(k)(1−θ *)k and ∑ k ⩾1 P̄(k)(1−θ̄ *)k. Since P̄(k) FOSD P(k) and (1−θ *)k is decreasing with respect to k then ∑ k ⩾1 P(k)(1−θ *)k⩾∑ k ⩾1 P̄(k)(1−θ *)k but notice that (1−θ̄ *)k⩾(1−θ *)k , which is why the comparison between ρ * and ρ̄ * might depend on further details of P(k) and P̄(k).
To actually compare the values ρ * and ρ̄ * one would have to compare ∑ k ⩾1 P(k)(1−θ *)k and ∑ k ⩾1 P̄(k)(1−θ̄ *)k. Since P(k) is an MPS of P̄(k) and (1−θ *)k is convex with respect to k then ∑ k ⩾1 P(k)(1−θ *)k⩾∑ k ⩾1 P̄(k)(1−θ *)k but notice that (1−θ̄ *)k⩾(1−θ *)k , which is why the comparison between ρ * and ρ̄ * might depend on further details of P(k) and P̄(k).
These networks were generated using the program Pajek, a software package for Large Network Analysis.
, which is a decreasing function of k, must be above 1. Note that if k=k
m
this holds whenever 
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Acknowledgements
This paper was mostly completed while I was a post-doc in the Department of Economics at Universidad Autónoma de Barcelona, Spain. I am grateful to Juan D. Moreno-Ternero and anonymous referees for valuable comments and suggestions. Thanks are also due to the audience at the Summer School in Political Economy and Social Choice (Torremolinos, 2007) and at Málaga Weekly Economic Seminars. Financial support from the Spanish Ministry of Education and Science through grant SEJ2006-27589-E, FEDER and Barcelona Economics-XREA is gratefully acknowledged.
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López-Pintado, D. The Spread of Free-Riding Behavior in a Social Network. Eastern Econ J 34, 464–479 (2008). https://doi.org/10.1057/eej.2008.30
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DOI: https://doi.org/10.1057/eej.2008.30