Two‐Aggregate Games: Demonstration Using a Production–Appropriation Model

DOIhttp://doi.org/10.1111/sjoe.12257
AuthorRichard Cornes,Yuji Tamura,Roger Hartley
Date01 January 2019
Published date01 January 2019
Scand. J. of Economics 121(1), 353–378, 2019
DOI: 10.1111/sjoe.12257
Two-Aggregate Games: Demonstration
Using a Production–Appropriation Model*
Richard Cornes
Australian National University, Canberra ACT 2601, Australia
Roger Hartley
University of Manchester, Manchester M13 9PL, UK
roger.hartley@manchester.ac.uk
Yuji Tamura
La Trobe University, Melbourne VIC 3086, Australia
y.tamura@latrobe.edu.au
Abstract
We expand the scope of the two-aggregate method by applying it to a situation in which many
heterogeneous players are free to contribute to both aggregates. Such situations naturally arise
in various resource allocation problems. Hence, our method is useful in many applications. A
production–appropriation model is employed to illustrate how the problem of establishing the
Nash equilibrium can be reduced from solving n>2 best-response functions in nunknowns
to solving two consistency conditions in two unknowns.We then conduct a comparative static
exercise that the conventionalapproach could not handle easily, if at all, to demonstrate the power
of our method.
Keywords: Conflict; insecure private property rights; non-cooperative game
JEL classification:C72; D74
I. Introduction
The analysis of several economic problems has benefited from exploiting
the observation that, in each case, a single aggregate provides a
sufficient statistic for solving the implied non-cooperative game with
many heterogeneous players. Dickson and Hartley (2008, 2013) made an
important contribution to the study of aggregate games using the analysis
of market games in which a single aggregate is not sufficient to describe
each player’s optimal behavior, but just two aggregates can describe it.
Sadly,Richard Cor nes passed away on 22 August 2015.
*We are grateful to Tim Hatton, Alex Dickson, Yvette Kirby, the audiences at ANU, Galway,
Manchester,Massey, and twoanonymous referees for useful comments. The research of R. Cornes
was supported by the F. H. Gruen endowment. Anyremaining er rors are ours.
C
The editors of The Scandinavian Journal of Economics 2017.
354 Two-aggregate games
In their models, each player is allowed to contribute to only one of the
two aggregates, which is not suitable for examining many situations of
interest. We show that the two-aggregate method can readily accommodate
situations in which each player is free to contribute to both aggregates,
thereby expanding the scope of the method and opening up new analytical
possibilities for many applications. A model of production and appropriation
is used to illustrate the usefulness of the method.
Many economic models involve non-cooperative games with an
aggregate structure that can be exploited to facilitate their analysis. By
conditioning every player’s behavior on a common aggregate, instead of
the sum of choices of all others, it is possible to avoid the proliferation
of dimensions associated with the use of best-response functions. As a
result, the presence of many heterogeneous players does not necessarily
complicate the analysis. Early use of the single-aggregate method appears
in Szidarovszky and Yakowitz (1977) and Novshek (1985) in the context
of proving the existence of Cournot equilibrium. Corch´on (2001) points
out this structure in a wide range of applications. Cornes and Hartley have
exploited the method in analyzing contests (2003, 2005, 2012) and public
goods (2007a, 2007b).1
The scope of this analytically attractive approach was considerably
broadened by Dickson and Hartley (2008, 2013), who examined the Nash
equilibrium of market games that not one but just two aggregates can
describe, regardless of how many heterogeneous players there are. In their
models, there are two types of goods. Each player is endowed with only one
of these two and decides how many units of the endowment to trade for the
other good. Each market participant can thus contribute to only one of the
two aggregate supplies, and the initial endowment allocation exogenously
determines the group partition.
We expand the scope of the aggregate-game approach further by
showing how to apply the two-aggregate method to a situation in which
every one of many heterogeneous players is allowed to contribute to both
aggregates. In our model, unlike the model in Dickson and Hartley (2008,
2013), the initially allocated endowment does not prevent each player from
contributing to both aggregates. Players are free to contribute to either one
of the two, or both, according to their payoff maximization. This extension
is significant because many situations of interest indeed involve a resource
allocation to multiple activities. Furthermore, this feature enables us to sort
players into groups endogenously according to their resource allocation
decisions. Our model for illustration is of production and appropriation,
1Other applications of the single-aggregate method include Ihori and McGuire (2007), Kotchen
(2007), Karaivanov (2009), andVicary (2009).
C
The editors of The Scandinavian Journal of Economics 2017.

To continue reading

Request your trial

VLEX uses login cookies to provide you with a better browsing experience. If you click on 'Accept' or continue browsing this site we consider that you accept our cookie policy. ACCEPT