Discussion Paper No. 1010

AN EVOLUTIONARY ANALYSISOF THE ASSIGNMENTOF PROPERTY RIGHTS

Atsushi Tsuneki

September 2017

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

An Evolutionary Analysis of the Assignment of Property Rights
Abstract: We develop an evolutionary game model to reveal the theoretical basis for theassignment of property right, where both plaintiff and defendant argue for their rights
by claiming their reliance investment. We allow for the possibility that the value of thetotal product depend not only on the investment conferred by the owner but also on thereliance investment provided by the trespasser. The resulting evolutionary stable set ofpreferences shows that the endowment effect hardwired to the owners and trespassers
depends on the difference of productivities among both parties and the density ofowners within the population.
Keywords: Evolutionary Game, Property Right, Reliance Investment, Endowment
Effect.
JEL Classification Code:C73, K11

Institute of Social and Economic Research, Osaka University. The author thanks
Mukesh Eswaran and Hugh Neary for valuable suggestions on the idea of this paper.
Remaining errors are the responsibility of the author. This research is supported by
JSPS Grant-in-Aid for Scientific Research for Challenging Exploratory Research (No.
26590046).
1

1.

Introduction
As pointed out by North (1990), the basic premise of the institutionalist school of

economics is the critical role of the system of property rights in the process of economic
development.1 Because the delineation and protection of the property right are roles oflaw, this suggests an inalienable relationship between law and economics in the study ofsocial science.
Traditional legal studies and economic analysis of law have taken different paths on
the issue. The traditional school of law emphasizes the role of moral principles, such as
first possession or reliance reflected in customs and precedents, in the resolution of
disputes over property rights, thus promoting procedural fairness and distributive
justice among conflicting parties. Unfortunately, traditional analyses lack thetheoretical basis of legal principles and, hence, legal rules and principles are sometimes
applied by courts in unpredictable ways.
Coase (1960), who provided the basis of law and economics emphasized the efficiency
aspect of the assignment of legal rights. His celebrated Coase Theorem shows that theclear and explicit determination of legal rights and private bargaining among private
parties can resolve the externality problem, without government intervention, if notransaction costs exist. Furthermore, he emphasizes the reciprocal nature of theexternality problem, and shows that the efficient outcome is attained by private
bargaining, without depending on the distribution of legal rights assigned to each party.
Therefore, for Coase, the assignment of legal rights was a problem of enforcing the law
in order to minimize aggregate transaction costs. Coase’s idea was succeeded by theclassic economic analysis of property law by Posner (1986), who argued that property
law should be structured to minimize the transaction costs of private bargaining andthe market exchange of properties.2
The recent contribution of Eswaran and Neary (2014) proposed an evolutionary
approach to reveal the theoretical basis that supports the institution of property rights.
According to their analysis, the legal system of property rights is preceded by theevolutionarily developed human psyche, which is attributed either to John Locke’s labor
theory or to the principle of first possession. These two reasons are quite often adopted
within legal discourse to stipulate the existence and assignment of property rights inorder to resolve specific disputes. As such, they provide the evolutionary basis for thelegal institution of property rights, and its associated legal doctrines, as a sense of
See Besley and Ghatak (2010) for a more recent survey on this subject.
See also Cooter (1982), who points out the other efficiency aspect of property law:
minimizing the cost of the failure of private bargaining over natural rights. This aspect
is attributed originally to Hobbes (1651/1973).
1
2

2

justice, hardwired by the evolutionary process of natural selection.
The chief goal of Eswaran and Neary (2014) was to show that the preference
intensities of the original owner and an interloper with regard to a property become
different when the former has a production opportunity to enhance the value of theproperty, or has an incumbency advantage over the latter in terms of securing the fruits
from the property. In particular, their study showed that the owner develops a stronger
preference intensity on his property than that of the interloper within the evolutionary
process when only the owner has an opportunity to enhance the value of the property
using his labor input. This is true even if they are ex ante identical with respect to theirpreference and ability to keep the fruits from the property. This proposition provided arigorous evolutionary foundation to Locke’s labor theory of value (Locke (1689/1988)).
However, when the legal right to a property is contested among people, it is often thecase that both the plaintiff and the defendant argue their rights by claiming theirreliance investment in the property. For example, when the ownership of land iscontested between the legal owner and a de facto owner, both parties often lay claim totheir ownership by asserting their contributions to the development and protection of
the land.
The purpose of this study is to extend the propositions of Eswaran and Neary (2014)
to the above-mentioned case. We start with a model where individuals are identical exante, except that some individuals, who we call owners, naturally have their own
property (sometimes referred to as land). The remaining individuals are called
trespassers, who survive by using some fixed resource with which they are naturally
endowed. Owners meet with trespassers with a fixed probability, which depends on thedensity of owners within the total population, as determined by nature. Once they meet,a struggle for the distribution of the product inevitably occurs, owing to the trespasser's
demand to consume the product from the land.
Our model deviates from that of Eswaran and Neary (2014) in that we allow for the
possibility that the value of the total product depends not only on the labor investment
conferred by the owner, but also on the reliance investment provided by the trespasser
to protect and enhance the value of the land.
Consider the case where the owner is a farmer who cultivates corn for his survival.
The amount of corn produced depends on the labor of the farmer. However, it also
depends on the level of effort supplied by the trespasser when he met with the owner ofthe property. In order to provide an appropriate level of care to protect and improve theland, the trespasser needs a reliance investment ex ante, as in the case of the owner,
who chooses the level of labor to apply to his land. The resulting evolutionary stable set3

of preferences differs from that of Eswaran and Neary (2014) in that the relative
endowment effect hardwired to the owner becomes smaller than it does in their case.
Furthermore, in some cases, trespassers have more intense preferences for the productgenerated from the property than the owners do.
The rest of this paper proceeds as follows. Section 2 presents our model, and section
3 presents our principal results, along with intuitive explanations of the results. Then,
section 4 discusses the implications of our results for law and economics. Lastly, section
5 concludes the paper.
2. ModelWe consider an evolutionary model where people live for one period. Their genes are
inherited by their offspring, and the biological cycle repeats itself. The naturalenvironment offers a random share θ of the total population of each generation thepossibility of owning some resource, say land, for their survival. These people, who wecall owners, invest effort in the resource (e.g., by growing corn on the land). The
remaining people (the 1 − θ share of the total population) are not originally endowedwith land and, thus, they survive using some outside option, such as collecting fruit in a1

forest. When θ ≥ 2, any trespasser must meet with one owner on his land. In contrast,1

some trespassers need not meet with an owner when θ < 2 . We assume that any
trespasser who meets with an owner must lose his original outside option, and must
share the land with the owner and grow corn to survive. This reflects the inevitable
nature of the externality we envision in our model. In both cases, once the owner andtrespasser meet, there begins a contest over the share of the corn each should receive inorder to survive.
We assume that the production function of farming corn on the land is as follows.
When the landowner does not meet with a trespasser, he follows a production processrepresented by the usual production function,(1)

𝑞(𝐾) = 𝐴𝐾 𝑎 , 0 ≤ 𝑎 ≤ 1,

where 𝐾 denotes the level of labor conferred by the owner on his land to produce corn.
In contrast, when the landowner meets with a trespasser, we assume that theaggregate production level of the corn produced by the two people is given by thefollowing production function:(2)

𝑞(𝐾, 𝐿) = 𝐴𝐾 𝑎 𝐿𝑏 , 0 ≤ 𝑎, 𝑏 ≤ 1, 𝑎 + 𝑏 ≤ 1,

where K and L denote the input levels of the owner and trespasser, respectively, toproduce corn 𝑞. It may appear to be more natural to assume that the level of production
depends on the aggregate level of input 𝐾 + 𝐿, as long as the type and quality of the4effort of the two parties are homogeneous and substitutable. Instead, we assume that
there exists some complementarity between 𝐾 and 𝐿 by assuming heterogeneous
aspects of the inputs used to produce the corn, which are necessary for both parties toprotect the land from harm during the production process. If the usage of land is not
appropriate (e.g., using incorrect machines for production, or supplying toxic chemicals),the land is devastated and the corn production level becomes very low, irrespective of
the amount of labor used. To protect the land from such harm, both parties must put
additional effort into protecting the land. In particular, if either side supplies anegligible input as a protective investment, the level of production approaches zero,irrespective of the labor input by the other party. Our assumption on production reflects
the cooperative and heterogeneous aspects of the productive inputs supplied by the twoparties that become necessary to protect the common resource from fatal deterioration.
Our model consists of three stages. In Stage 1, nature chooses the preference
parameters 𝑣 and 𝑤 of the owner and trespasser, respectively, as an evolutionary
process. Note that we sometimes refer to the owner as Player 1, and the trespasser as
Player 2. Then, in Stage 2, Player 1 and Player 2 choose the levels of prior effort 𝐾 and
𝐿 , respectively, required to increase the level of production, given the preference
parameters 𝑣 and 𝑤. Finally, in Stage 3, both players contest the distribution of thecorn produced in Stage 2. Because we solve the model in a backward manner, we begin
by analyzing Stage 3.
2.1 Distribution Contest
By Stage 3, the levels of productive effort, 𝐾 and 𝐿, and the output 𝑞 are already
fixed. With natural probability, a distribution contest occurs between the two players to
divide the fixed aggregate production 𝑞. We denote the levels of effort required byPlayers 1 and 2 to receive more of the product for consumption as 𝑒 and 𝑓, respectively.
We assume that the preference parameters 𝑣 and 𝑤, which are predetermined in Stage1, are observable to each player in the distribution contest in Stage 3. Our model closely
follows Eswaran and Neary (2014), as does the justification for this assumption.3
We further assume that the effectiveness of the effort for distribution is equal forboth players in order to reflect the equality of people in the state of nature. Therefore,
the shares of corn produced by Players 1 and 2 (s and t, respectively) are determined asfollows:
(3)

3

s=

𝑒
𝑒+𝑓

,𝑡 = 1 − 𝑠 =

𝑓
𝑒+𝑓

.

See Eswaran and Neary (2014:220–221).
5

In Stage 3, both players maximize their utility functions:(4)max𝑒 𝑣𝑙𝑛(𝑠𝑞) − (𝑒 + 𝐾)

(5)

max𝑓 𝑤𝑙𝑛(𝑡𝑞) − (𝑓 + 𝐿),

where 𝑙𝑛(𝑐) is the biological level of the fitness of the consumption, common to all
generations and types of players. In the process of choice in one generation, people
deviate from the natural fitness level of consumption 𝑙𝑛(𝑐) , depending on theparameters that reflect the intensity of the players’ preferences, 𝑣𝑙𝑛(𝑐) and 𝑤𝑙𝑛(𝑐).
These parameters are the result of an evolutionary process, which we analyze in thenext section.
We can explicitly solve for the Nash equilibrium effort levels for both players, as
follows.
(6)𝑒 ∗ (𝑣, 𝑤) =𝑣√𝑤√𝑣𝑤
, 𝑓 ∗ (𝑣, 𝑤) = 𝑣+ 𝑤
√𝑣+√𝑤
√ √

(7)

𝑠 ∗ (𝑣, 𝑤) =√𝑣
, 𝑡 ∗ (𝑣, 𝑤) = 1 − 𝑠 ∗ (𝑣, 𝑤).4√𝑣+√𝑤

It is easy to check that𝜕𝑒 ∗ 𝜕𝑓 ∗
𝜕𝑣

,

𝜕𝑤

,

𝜕𝑠∗
𝜕𝑣

, and

𝜕𝑡 ∗
𝜕𝑤

> 0, which means that the players’

equilibrium effort levels and their shares of the output increase as their preference
intensities increase in the evolutionary process.
2.2 Stage 2: Choice of Production Inputs
Denote μ = min [1,1−θθ] and φ = min [1,θ1−θ] , where μ is the probalitity that theowner meets with a trespasser, and φ is the probability that the trespasser meets with
an owner.
The production decisions of the owner and trespasser are formalized respectively as
follows:(8)

max𝐾 𝜇{𝑣𝑙𝑛(𝑠 ∗ 𝐴𝐾 𝑎 𝐿𝑏 ) − 𝑒 ∗ − 𝐾} + (1 − 𝜇){𝑣𝑙𝑛(𝐴𝐾 𝑎 ) − 𝐾},

and
(9)

max𝐿 𝜑{𝑤𝑙𝑛(𝑡 ∗ 𝐴𝐾 𝑎 𝐿𝑏 ) − 𝑓 ∗ − 𝐿} + (1 − 𝜑){𝑤𝑙𝑛𝑐̅ − 𝐿},

where 𝑐̅ denotes the outside consumption opportunity endowed upon trespassers when
they do not meet with an owner. Note that from the analysis in the previous subsection,𝑒 ∗ , 𝑓 ∗ , 𝑠 ∗ , and 𝑡 ∗ are all independent of 𝐾 and 𝐿. The optimal levels of the inputs for
production in the non-cooperative Nash equilibrium can be calculated in a
The results in (6) and (7) are special cases of the formulae derived in Eswaran and
Neary (2014:220); thus, the proof is omitted.
4

6

straightforward manner, as follows:(10)

𝐾 ∗ = 𝑎𝑣, 𝐿∗ = 𝜑𝑏𝑤.

3. Analysis of the Evolutionary Process
Here, the preference parameters are chosen by the process of natural selection. This
process is governed by the fitness function Ψ(c, i, J), defined as(11)

Ψ(c, i, J) = ln(𝑐) − (𝑖 + 𝐽),

where 𝑐 denotes consumption, and 𝑖 and 𝐽 denote the levels of the distribution effort
and production input, respectively. Furthermore, we denote the ratio of the expectedconsumption by the owner and that by the trespasser as Π, indicating the level of theproperty right distributed to the owner relative to that of trespasser.
Using (10) as the subgame perfect level of the production input by the two players,
we can write the fitness function (11) for the owner σ(𝑣, 𝑤) and for the trespasser
τ(𝑣, 𝑤) as(12)

σ(𝑣, 𝑤) = μ[ln{𝑠 ∗ (𝑣, 𝑤)A(a𝑣)𝑎 (𝜑𝑏𝑤)𝑏 } − 𝑒 ∗ (𝑣, 𝑤) − 𝑎𝑣] + (1 −μ)[lnA(a𝑣)𝑎 − 𝑎𝑣]=μ𝑔(𝑣, 𝑤) + ln A(a𝑣)𝑎 + μln(𝜑𝑏𝑤)𝑏 − 𝑎𝑣,(13)τ(𝑣, 𝑤) = 𝜑[ln{𝑡 ∗ (𝑣, 𝑤)A(a𝑣)𝑎 (𝜑𝑏𝑤)𝑏 } − 𝑓 ∗ (𝑣, 𝑤) − 𝜑𝑏𝑤] +(1 − 𝜑)[𝑙𝑛𝑐̅ − 𝜑𝑏𝑤]
= 𝜑ℎ(𝑣, 𝑤) + (1 − 𝜑)𝑙𝑛𝑐̅+ 𝜑𝑙𝑛 A(a𝑣)𝑎 + 𝜑𝑙𝑛 A(𝜑𝑏𝑤)𝑏 − 𝜑𝑏𝑤,

where we define(14)𝑔(𝑣, 𝑤) = ln(𝑠∗ (𝑣, 𝑤)) − 𝑒∗ (𝑣, 𝑤), ℎ(𝑣, 𝑤) = ln(𝑡∗ (𝑣, 𝑤)) −𝑓∗ (𝑣, 𝑤).

The evolution of preferences is characterized by the following two maximization
problems:(15)

max𝑣 σ(𝑣, 𝑤), max𝑤 τ(𝑣, 𝑤).

The first-order conditions for the two maximization problems are(16)∂σ(𝑣,𝑤)(17)∂τ(𝑣,𝑤)

𝜕𝑣

𝜕𝑤

=𝜇
=

∂𝑔(𝑣,𝑤)𝜕𝑣∂ℎ(𝑣,𝑤)
𝜕𝑤

1+ 𝑎 (𝑣 − 1) = 0
1+ 𝑏 (𝑤 − 1) = 0.

Equations (16) and (17) form the two best-response functions 𝑣 𝑏𝑟 (𝑤) and 𝑤 𝑏𝑟 (𝑣),respectively.
Our principal results are summarized as follows. The proofs of all propositions are
given in the Appendix.

7

PROPOSITION 1(1) The best-response functions 𝑣 𝑏𝑟 (𝑤) and 𝑤 𝑏𝑟 (𝑣) have negative slopes; 𝑣 and 𝑤 are
strategic substitutes.
(2) The Nash equilibrium (𝑣̃, 𝑤̃) is unique and locally uninvadable.
(3) The evolutionary stable preference parameters (𝑣̃, 𝑤̃) satisfy the following
relationship:
(18)

≥
≥
𝑣̃ 𝑤
̃ 𝑖𝑓𝑓 𝑎 𝜇𝑏.
<<(4) The index of the relative ratio of property rights assigned to the two parties, s,satisfies the relationship
(19)

s

≥1
≥
𝑖𝑓𝑓 𝑎 𝜇𝑏.
<2<

In our model, the key parameters are (𝑣̃, 𝑤
̃), which reflect the preference intensitiesof the owner and the trespasser. When one parameter is larger than the other, theperson with the larger parameter puts more effort into distribution and more input into
production. Considering these effects, the above evolutionary process fixes the relative
magnitudes of the two preference parameters. By our assumption on the equality of
power for the distribution of the output, this aspect does not influence the difference inthe preference parameters. In contrast, if the productivity levels of the players differ,nature has a stronger preference for the consumption of the more productive player, inwhich case, he is more motivated to produce, as in (18). As a result, the more productive
player receives a greater share of the product. Note that the productivity of the owner ismeasured by the standard productivity parameter 𝑎 of the Cobb–Douglas production
function. However, the substantial productivity of the trespasser is usually measured by
the similar parameter 𝑏, and is discounted by 𝜇. This is because when the ratio ofowners to the total population is more than 50%, not all owners can meet with atrespasser. Those owners who do not meet with a trespasser can engage in production
independently of the effort of a trespasser. Therefore, the production potential of thetrespasser is considered less useful than his technical productivity, from the viewpointof nature.
To determine the implications of the above results, we focus on the case of constantreturns to scale, a + b = 1. From the definition of μ, we have the following corollary.
Corollary to Proposition 1
Assume that a + b = 1. The evolutionary stable preference parameters (𝑣̃, 𝑤̃) and therelative ratio of property rights, s, satisfy the following relationships:
8

(20)

1
1
≥
≥1
≥
𝑣̃ 𝑤
̃ iff 𝑎 2 (if 𝜃 ≤ 2) , 𝑎 1 − 𝜃 (if 𝜃 > 2).
<
<
<

(21)

s

1
1
≥1
≥1
≥
iff 𝑎 2 (if 𝜃 ≤ 2) , 𝑎 1 − 𝜃 (if 𝜃 > 2).
<2<<

When the ratio of owners in the population is less than one-half, the intensity of thepreferences and the ratio of the right to the property depend only on the relative
magnitude of the productivity, 𝑎, between the two players. However, if the ratio isgreater than one-half, they also depend on the fraction of owners within the
population, 𝜃. In particular, as 𝜃 approaches one and the fraction of trespassers within
1

the total population become negligibly small, 𝑣̃ > 𝑤
̃ and s > 2 apply for almost any
productivity level of the owner. In this case, the productivity of the trespassers is ofnegligible importance.
Returning to the general case, we next derive the comparative static results.
PROPOSITION 2(1) When the productivity of the owner increases, his preference intensity increases,
while that of the trespasser decreases.
(2) When the productivity of the trespasser increases, his preference intensity increases,
while that of the owner decreases.
(3) When the density of the population of owners is less than 1/2, the preference
intensity parameters of the two players remain unchanged when the density ofowners increases. However, when the density of owner is more than 1/2, and thenincreases, the preference intensity parameter of the owner increases, while that of
the trespasser decreases.
4．Application to Legal Discourse
In order to exploit the implications of the above propositions, we apply the
propositions to the legal reasoning often adopted by the court to resolve disputes. Whentwo people contest a property right in a legal dispute, one method of legal reasoning
compares the interests of the two parties in terms of utility, and then divides the right tothe property accordingly. 5 The most difficult problem with this decision is how toNote that this method should not be confused with the wealth-maximization criterion
for legal judgment adopted by Posner (1986), which is based on the hypotheticalcompensation principle (or the Kaldor–Hicks criterion, in Posner’s notation). While
Posner’s criterion implies that a legal judgment should be made exclusively based on
efficiency, the method explained in the text explicitly introduces distributive equity
among the contesting parties by comparing their utility.
59

determine and compare the intensity of the utility of each party from the property. Even
though this is observable information for the conflicting players, it is usually
understood as being unverifiable by the court. In usual legal practice, this evaluation isdeferred to the deliberation of the court, despite this limitation.
Our propositions in the previous section give important insights in construing thepreference intensities of the two disputing players from observable parameters insociety. Proposition 1 connects the difference between the preference intensities of thetwo players to more objective parameters a, b, and μ , where a and b represent therespective productivities in developing the land of the two players contesting the rights
to the land. From the above proposition, the more productive player can be construed ashaving a stronger preference for the property. Because μ denotes the probability of theowner meeting a trespasser, the original owner tends to have a stronger preference forthe property. This is true even if his productivity for the land is relatively smaller thanthat of the trespasser, when the likelihood of the owner meeting a trespasser issufficiently small. When μ is small, the investment of the trespasser is less useful, evenif his productivity is high, and nature gives the owner a greater chance to develop
stronger preference for the property even when his productivity to the land is small.
Information on a, b can be collected by the court with the help from the expertson the subject or can be verified to the court through contesting parties by presenting
sufficient proof. In contrast, it is more difficult to interpret μ. We understand that it
shows if the right to the property is naturally well-determined or ambiguously
delineated. In the latter case where μ is close to unity, therefore, most owners are sued
for his right to the property, while in the former case where μ deviates from unity
sufficiently, only small part of the owners is contested their right by the trial in court. As
the Proposition 2 states, the more a gets larger than b, or μ gets closer to zero,
divergence of the preference intensity by the two players gets larger and hence court
can confer more property right on the original owner.
5. Concluding Remarks
Our analysis extends the evolutionary analysis of the emergence of property right by
Eswaran and Neary (2014) to the case where both natural owner and trespasser haveopportunity to improve the value of the property by investing on it using his resource.
This change of the natural environment critically make difference to the evolutionary
determination of preference intensity parameters by the two players, an owner and atrespasser. Furthermore, our environment can be more naturally applied to the legal
reasoning exerted by judges in court. In this sense, we hope that our analysis shed some10light on the nature of legal reasoning and its associated legal studies by providing with
its evolutionary-theoretical foundations.

11

REFERENCES
Besley, T. and M. Ghatak (2010), Property Rights and Economic Development,
4525-4595 in Handbook of Development Economics vol. 5, D. Rodrik and M.
Rosenzweig eds., New York: Elsevier.
Coase, R.H. (1960), The Problem of Social Cost, Journal of Law and Economics 3, 1-44.
Cooter, R.C. (1982), The Cost of Coase, Journal of Legal Studies 1, 1-33.
Cressman, R. (2009), Continuously Stable Strategies, Neighborhood Superiority and
Two-Player Games with Continuous Strategy Space, International Journal of Game

Theory 38, 221-247.
Eswaran, M. and H.M. Neary (2014), An Economic Theory of the Evolutionary
Emergence of Property Rights, American Economic Journal: Microeconomics 6,203-226.
Hobbes, T. (1651/1973), Leviathan, New York: Collier-MacMillan.
Locke, J. (1689/1988), Two Treatises of Government, ed. by P. Laslett, Cambridge, UK:
Cambridge University Press.
Maynard-Smith, J. (1982), Evolution and the Theory of Games, Cambridge, UK:
Cambridge University Press.
North, D.C. (1990), Institutions, Institutional Change and Economic Performance,
Cambridge, UK: Cambridge University Press.
Posner, R.A. (1986), The Economic Analysis of Law, 3rd ed., Boston: Little, Brown & Co.

12

Appendix
We first summarize preliminary results on the functions 𝑔(𝑣, 𝑤) = ln(𝑠 ∗ (𝑣, 𝑤)) −𝑒 ∗ (𝑣, 𝑤) and ℎ(𝑣, 𝑤) = ln(𝑡 ∗ (𝑣, 𝑤)) − 𝑓 ∗ (𝑣, 𝑤), defined in (14). From (6) and (7), we have
(A.1)
(A.2)

√𝑤(1−𝑣−𝑒 ∗ )∂𝑔∂ℎ√𝑣(1−𝑤−𝑓 ∗)= 2𝑣( 𝑣+ 𝑤) , ∂𝑤 = 2𝑤( 𝑣+ 𝑤) ,
∂𝑤
√

∂2 𝑔

√

√

1−𝑣−2𝑒 ∗√∂2 ℎ

1−𝑤−2𝑓 ∗= 4 𝑣𝑤( 𝑣+ 𝑤)2 , ∂𝑣 ∂𝑤 = 4 𝑣𝑤( 𝑣+ 𝑤)2.
∂𝑣 ∂𝑤
√

√

√

√

√

√

As pointed out in Eswaran and Neary (2014:222),
(A.3)

∂2 𝑔
∂𝑣 2

5

∂2 ℎ

5

< 0, 𝑖𝑓 𝑣 ≤ 3, ∂𝑤 2 < 0, 𝑖𝑓 𝑤 ≤ 3,

and
(A.4)

∂2 𝑔

∂2 ℎ

∂2 𝑔

∂2 ℎ

Δ = (∂𝑣2 ) ( ∂𝑤 2) − (∂𝑣 ∂𝑤) (∂𝑣 ∂𝑤) > 0, if 𝑣 < 1 and 𝑤 < 1.

Proof of PROPOSITION 1
Substituting (A.1) into the first-order conditions (16) and (17), we have
(A.5)

𝜇√𝑤
1−𝑣
1−𝑤
√𝑣
=− 1−𝑣−𝑒 ∗ > 0, 2𝑏( 𝑣+ 𝑤)=− 1−𝑤−𝑓∗ > 0,2𝑎(√𝑣+√𝑤)√ √which in turn implies the inequality conditions(A.6)

1 − 𝑣 > 0, 1 − 𝑣 − 𝑒 ∗ < 0, 1 − 𝑤 > 0, 1 − 𝑤 − 𝑓 ∗ < 0.

Therefore, the values (𝑣̃, 𝑤̃) that satisfy (16) and (17) are both less than unity.
∂2 𝑔

∂2 ℎ

(1) From the inequalities (A.6) and (A.2), ∂𝑣 ∂𝑤 < 0 and ∂𝑣 ∂𝑤 < 0 at (𝑣, 𝑤) = (𝑣̃, 𝑤
̃).
From this result and (A.3), the slopes of the best-response functions are∂2 𝜎− ∂𝑣∂2∂𝑤< 0 and𝜎∂𝑣2

d𝑤 𝑏𝑟 (𝑣)𝑑𝑣

d𝑣 𝑏𝑟 (𝑤)
𝑑𝑤

=

∂2 𝜏

= − ∂𝑣∂2∂𝑤< 0 at (𝑣, 𝑤) = (𝑣̃, 𝑤̃) . Therefore, 𝑣 and 𝑤 are𝜏∂𝑤2strategic substitutes at (𝑣, 𝑤) = (𝑣̃, 𝑤
̃).
(2) We can readily show from (16), (17), and (A.3) that the second-order conditions forthe maximization problem (15) are satisfied for (𝑣̃, 𝑤
̃). Hence, it is a locally uniqueequilibrium. Then, (𝑣̃, 𝑤̃) is locally uninvadable because this concept requires onlythat (𝑣̃, 𝑤̃) is a neighborhood strict Nash equilibrium, as proved in Cressman(2009).
(3) Using (16) and (17), and substituting in (A.1) we have
(A.7)

𝑎
𝜇

−𝑏 =

1

𝑓∗

𝑒∗

{√𝑣 (1 −) − √𝑤 (1 − 1−𝑣)}.
2(√𝑣+√𝑤)
1−𝑤

13

Substituting (6) into (A.7) shows that
𝑎

(A.8)

𝜇

−𝑏 =

1
2(√𝑣+√𝑤)(√𝑣 − √𝑤 +𝑣𝑤(𝑣−𝑤)(1−𝑣)(1−𝑤)(√𝑣+√𝑤)

must be satisfied at (𝑣, 𝑤) = (𝑣̃, 𝑤̃).
𝑎

If 𝜇 > 𝑏, suppose that 𝑣̃ ≤ 𝑤
̃. Then, the right-hand side of (A.8) becomes non-positive𝑎and the contradiction follows. Therefore, 𝑣̃ > 𝑤
̃ must be met. Similarly, 𝜇 < 𝑏 → 𝑣̃ < 𝑤̃.
𝑎

When 𝜇 = 𝑏, suppose 𝑣̃ ≠ 𝑤
̃. Then, the left-hand side of (A.8) is zero and the right-handside is nonzero, and a contradiction follows. Therefore, 𝑣̃ = 𝑤
̃ is implied. In summary,𝑎𝑎𝑎

we have 𝜇 > 𝑏 → 𝑣̃ > 𝑤
̃, 𝜇 = 𝑏 → 𝑣̃ = 𝑤
̃, and 𝜇 < 𝑏 → 𝑣̃ < 𝑤
̃ . In proving the opposite
implications, we first check directly from (A.8) that 𝑣̃ = 𝑤
̃ →contrapositives of𝑣̃ < 𝑤
̃ →

𝑎
𝜇

𝑎
𝜇

> 𝑏 → 𝑣̃ > 𝑤
̃ and< 𝑏 and 𝑣̃ > 𝑤
̃ →

𝑎
𝜇

𝑎
𝜇

𝑎
𝜇

= 𝑏. By taking the< 𝑏 → 𝑣̃ < 𝑤
̃, and using 𝑣̃ = 𝑤̃→𝑎𝜇= 𝑏, we have> 𝑏. Then, (18) is established by summarizing all the

results.
(4) Using the definition of s in (7) and (18), (19) follows.
Proof of PROPOSITION 2
Denoting the second-order derivatives of σ and τ around (𝑣̃, 𝑤̃) as 𝜎𝑖𝑗 and 𝜏𝑖𝑗 ,respectively, where i, j = v, w, a, b, μ, the standard comparative static matrix using (16)σand (17) is [ 𝑣𝑣
τ𝑤𝑣

−σ𝑣𝑎 𝑑𝑎𝑑𝑣][ ] = [τ𝑤𝑤 𝑑𝑤
0
σ𝑣𝑤

1 τ
𝑑𝑣
side matrix, we have [ ] = [ 𝑤𝑤
Δ
𝑑𝑤
−τ

𝑤𝑣

0

−σ𝑣𝜇 𝑑𝜇−τ𝑤𝑏 𝑑𝑏0]. Inverting the left-hand0−σ𝑣𝑤 −σ𝑣𝑎 𝑑𝑎][
σ𝑣𝑣
0−τ𝑤𝑏 𝑑𝑏−σ𝑣𝜇 𝑑𝜇0]. From(16) and (17), we have σ𝑣𝑣 < 0 and τ𝑤𝑤 < 0, using (A.3). Similarly, from (A.2) and1(A.6), σ𝑣𝑤 < 0 and τ𝑤𝑣 < 0 . In addition, from (16) and (17), σ𝑣𝑎 = − 1 > 0 ,
𝑣

1

𝜕𝑔

τ𝑤𝑏 = 𝑤 − 1 > 0, and σ𝑣𝜇 = 𝜕𝑣 < 0.
Using the inverted matrix and the above inequalities, we have14(A.8)(A.9)(A.10)

∂𝑣

=−

∂𝑎
∂𝑣

=

∂𝑏
∂𝑣
∂μ

1
Δ

1
Δ

∂𝑤

τ𝑤𝑤 σ𝑣𝑎 > 0, ∂𝑎 =∂𝑤

σ𝑣𝑤 τ𝑤𝑏 < 0, ∂𝑏 = −

=−

1
Δ

1
Δ

τ𝑤𝑤 σ𝑣𝜇 > 0,

∂𝑤
∂μ

1
Δ

=

τ𝑤𝑣 σ𝑣𝑎 < 0,σ𝑣𝑣 τ𝑤𝑏 > 0,

1
Δ

From (A.8) and (A.9), results (1) and (2) follow. From

τ𝑤𝑣 σ𝑣𝜇 < 0.

∂μ
∂θ

1

= 0 if θ < 2, and∂μ∂θ< 0 if θ ≥1, result (3) follows.

2

Atsushi Tsuneki
Institute of Social and Economic Research, Osaka University,
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japantsuneki@iser.osaka-u.ac.jp15