The 6th MEETING ON GAME THEORY AND PRACTICE

Zaragoza, Spain 10-12 July

Fuzzy Access:

Modeling Grazing Rights in Sub-Saharan Africa

Rachael GOODHUE†

Department of Agricultural and Resource Economics

University of California, Davis‡

Nancy MCCARTHY§

International Food Policy Research Institute**

Abstract. Increasing populations and agricultural encroachments upon rangelands are

stressing traditional land management institutions in sub-Saharan Africa. We use fuzzy

set theory to model key features of the traditional grazing rights system: the incomplete

nature of grazing access and the flexibility of these rights. In contrast to the findings of

the existing common property literature, we find that under some conditions the

imprecision which characterizes access rights and use can result in higher social returns

than a conventional common property regime characterized by well-defined user groups

and well-defined user rights.

JEL references: Q15, O13

Key Words: Grazing rights, fuzzy logic, property rights, mobility

† goodhue@primal.ucdavis.edu ‡ University of California, Davis, One Shields Avenue, Davis, CA, 95616 USA § n.mccarthy@cgiar.org ** International Food Policy Research Institute, Washington, D.C. USA

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1. Introduction

Pastoralist systems in sub-Saharan Africa have three important features: mobility,

common use of land, and the imprecise nature of the rules determining grazing access

and the underlying property rights. These features enable pastoralists to deal with the

highly variable rainfall and resulting uneven forage distribution characteristic of the

region. Economists have analyzed the effects of mobility on the efficiency of traditional

pastoralist systems. Mobility and the resulting ability to respond to shocks ex post may

dominate private property rights for an individual herder (or single herder group) (van

den Brink, Bromley and Chavas), while the externalities resulting from common use may

partially or completely mitigate these gains (Goodhue and McCarthy, 1998). In this

paper, we focus on the third characteristic, imprecisely defined grazing access and

property rights.

Conclusions regarding the role of mobility and common use are dependent upon

the assumed nature of the underlying property regimes. In most analyses of private and

common property regimes, property rights are assumed to be complete. As a pastoralist,

if you have the right to graze a particular area, you may freely access it, regardless of

whether or not others have this same right. Rights are usually assumed to be

noncontingent, as well as complete. Under either regime, your right to graze the pasture

is unaffected by states of nature, such as weather shocks to other grazing areas accessed

by you or by other users.

In contrast to these institutional settings, grazing rights and resource access in

sub-Saharan Africa are frequently described as incomplete and contingent. Rights are

incomplete ex ante and ex post. Ex ante, when pastoralists make their stocking decisions,

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they recognize that their access to certain resources is incomplete; that is, they are not

entitled to fully exploit the available forage. Incompleteness may be reflected in a

number of ways ex post; for example, once animals are grazing the area, pastoralists may

leave the area before the forage is fully exploited.1 Their stocking decision, in terms of

livestock days on pasture, is below the level they would choose if they had a complete

right to the available forage. Rights are also contingent in the sense that traditional

institutions recognize the importance of relative weather shocks. Pastoralists who

sustain relatively large negative shocks to their primary grazing areas may obtain greater

access to certain resources; that is to say, access in relatively bad years is large relative to

the underlying incomplete right.2 This institutional flexibility serves as a mutual

insurance mechanism.

We model the incomplete, contingent nature of sub-Saharan African grazing

rights using fuzzy sets. We then compare social returns for two non-cooperative

pastoralist groups sharing a single common pasture under the traditional land access

system modeled with fuzzy sets to a conventional common property regime. Under some

conditions, the traditional fuzzy access system results in a higher level of social welfare.

Generally, these conditions are such that the system behaves as if the pastoralist with

relatively strong access rights acted as a primary “manager”, while the other has rights

that are weaker than the first, but not so weak that it is never profitable to access the

pasture.

1 Such incomplete rights are not unique to sub-Saharan Africa. In Australia, herders have the right to graze their animals along commonly recognized stock trails, provided they keep the animals moving. (We are indebted to Richard Howitt for this example.) 2 Rights are contingent in other senses, although we focus on relative weather shocks. For instance, herds composed of milking cows and young animals generally receive higher priority than the main herds, which are more mobile.

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2. Describing Fuzzy Access Rights

Access to grazing land in sub-Saharan Africa does not closely correspond to the

conventional economic concept of common property, where some fixed number of

members of the common property user group have equal and complete access to the

available forage. Nor does it correspond to the conventional economic concept of an

open access resource, where all potential users may compete to exploit the resource until

returns from the resource are dissipated fully. An empirical regularity identified and

discussed by a number of authors is that grazing area boundaries and membership in the

access group are not well defined. Scoones (1995) writes the following:

“Overlapping claims to resources, shifting assertions of rights and continuous contestation and negotiation of access rules dominate tenurial arrangements in uncertain environments. The solution is not to impose particular tenure types on a variable setting; whether these are uniquely communal or private they are unlikely to work. Instead, the need for flexible tenure arrangement must be recognized. … Customary tenure systems operate shared, overlapping forms of tenure rights in such settings as maintaining strict boundaries is usually untenable.” (Scoones, Chapter 1, pg. 27, 1995).

Toulmin (1995) also notes that:

“Such movement is possible where secondary rights of access exist for herders to bring their animals into areas they do not usually use…. In addition, drought conditions are likely to increase competition for scarce fodder between local animals and herds from elsewhere, further constraining access in such periods for those with weak claims to exploit local pastures.” (Toulmin, Chapter 7, pg. 101, 1995)

In these analyses, the ill-defined or “fuzzy” nature of resource boundaries and

resource access rights is considered a positive factor in the functioning of the pastoral

system. A belief in the importance of this characteristic is shared by local observers.

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Recently in the Daily Nation (Kenya), an editorialist wrote of the pastoralist situation in

NorthEastern Province of Kenya that: “Mobility, which is part and parcel of nomadic

pastoralism, has been reduced, the resource use cycle has been shortened and individual

and at times community territorial claims are becoming more specific"(italics added).

The author believes that the specificity of the claims is counter-productive to the

functioning of the pastoral system. This consensus starkly contrasts with the standard

common property finding that a well-defined membership and a resource with well-

defined boundaries are required ingredients for the successful management of the

commons (See for example Ostrom, 1990; Thompson & Wilson, 1994; Oakerson, 1992).

Each property rights system has relative advantages. When access rights and the

resource are well-defined, 1) each individual can be assured that any collective benefits

provided by the group will accrue to the group alone, and 2) the actual undertaking of

management activities should be easier with a relatively small, well-defined group.

There are also two reasons why flexibly-defined, or fuzzy, access may be preferable to

well-defined grazing areas: 1) the ability for pastoralists to improve their income

realizations by mutually adjusting access to common areas in response to relative

outcomes on other areas of their grazing ranges, and 2) the risk-reducing role of mobility.

The following example from Casimir (1987) neatly clarifies the incomplete nature

of access rights. The author notes that conflict between groups may arise due to the

failure of a group seeking access to an area primarily grazed by another group to ask

permission before moving, and may also arise due to the failure of the primary user to

grant this permission when requested. The fact that both these situations may lead to

conflict suggests that access is defined imprecisely, or fuzzily. If the primary user group

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had clearly defined rights that allowed it to exclude other groups arbitrarily, as in a

crisply defined common property or private property regime, the refusal of permission to

graze would not lead to conflict, within the context of the institutional system. Similarly,

there would be no need for indirect signals that pastoralists with weaker rights had

exhausted their claim to the resource. The behavioral norm of other groups requesting

access from the primary group indicates that, even though they expect to be allowed to

graze, they do not feel that their right to do so is complete ex ante, as would be the case

in a more traditional common property regime. Further, pastoralist norms allow primary

users to encourage other users to move on, through direct communication and by tactics

such as restricting their animals’ access to water as is done on the Borana Plateau in

Ethiopia. Such measures illustrate the incompleteness of grazing rights ex post. The

acceptance of these behaviors coexists with strong social disapproval (and the threat of

sanctions) of groups who attempt to close their primary grazing areas to other users

entirely (Coppock, 1994). Thus, observed patterns of behavior indicate that pastoralists’

ex ante grazing rights and ex post realized access are incomplete and contingent.

The fuzzy access rights regime is a regime of incomplete contingent markets. We

focus initially on the incomplete nature of the fuzzy access regime in section 4, and

evaluate the net effects of incompleteness and contingency in section 5.3 In contrast to

the regime we model, private and common property are complete unconditional market

regimes. Mineral rights are an example of an incomplete unconditional right. Complete

contingent markets place us in the world analyzed by Debreu and others. In this

framework, it is possible to analyze a contingent property rights system, where two or

3 We are indebted to Jean-Paul Chavas and Lowell Jarvis for stressing the importance of this distinction.

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more pastoralists are each randomly assigned exclusive ownership of a given pasture.

Such an analysis, however, would conflate two essential features of sub-Saharan Africa

pastoralist systems, by defining incompleteness in terms of uncertainty. 4 The objective

of this paper is to develop a model that incorporates these important features,

incompleteness and contingency, using fuzzy set theory, and consider under what

conditions a flexible, partial access regime dominates a conventional common property

regime. We then consider the implications of our findings for sub-Saharan African

policymakers.

3. Defining Fuzzy Access Rights

In Goodhue and McCarthy (1998), we demonstrate that non-cooperative use of a

common-pool resource reduces the expected benefit of flexibility to individual groups

identified in van den Brink, Bromley, and Chavas (1995). As stated in the introduction,

access to various pastures, even those referred to as being held in common, is generally

more complicated than that captured in a conventional common property model, or an

open access model.5 One particularly important case is that of group, lineage, or

subgroup territories. While boundaries are not fixed, and subgroups may cross them

particularly in times of stress, users recognize that different subgroups have different

levels of access (Casimir, 1987; Coppock, 1994). We will focus on sub-group level

access rights’ rather than those of a tribe as a whole or of all users. Some subgroups

may use a pasture consistently from year to year but for different lengths of time,

4 We would like to thank an anonymous reviewer for suggesting we clarify this point. 5 We follow the distinction elucidated in Dasgupta and Heal (1979) between a common property resource freely accessed by a finite number of individuals and the open access case where the number of accessing individuals expands until profits are driven to zero.

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whereas others may use it only occasionally. Further, a subgroup’s use of the pasture

may depend on conditions in other parts of its grazing range. Hence, the effective

pasture available to a given subgroup is imprecisely defined. We utilize fuzzy sets to

model these attributes.

Fuzzy Sets

Fuzzy mathematics examines imprecise phenomena that lack clearly defined class

criteria. Rather than a simple 0-1 definition of set membership, each object in a fuzzy set

possesses a degree of membership in that set from the closed 0-1 interval. Standard sets

are a special “crisp” case of fuzzy sets, where every element of the set has a degree of

membership of 1, and every element of its complement has a degree of membership of 0.

Zadeh (1965) introduced fuzzy mathematics as a way of modeling extremely complex

systems, where precise set definitions are either absent or extremely costly to model.

Rather than requiring immense numbers of specific rules to define precisely the nature of

a system, the use of fuzzy sets allows for imprecision in how each exact situation is

described. SSA grazing rights are defined in an extremely complex traditional system

that is difficult to characterize fully. Factors such as kinship, other relationships such as

age groups, relative shocks to primary grazing areas, and overall access rights to other

areas all appear to affect access to specific areas. Rather than seek to precisely define all

these relationships, which are site- and user-specific, we use fuzzy set theory to model

the system and its consequences for users.

Standard economic modeling treats uncertainty as due to an underlying random

variable. Rather than using probability to model the chance that a pastoralist will be the

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user of a specific area for a specific period of time, we wish to model how pastoralists

manage their herds given their highly complex access system which governs their use of

specific areas. Fuzzy set theory describes the extent to which items possess a specific

property, using the degree of membership. Essentially, fuzzy sets define the property by

assigning linguistic variables to partitions of continuous values. For example, seasonal

rainfall may be usefully characterized as high or low. Fuzzy sets gain descriptive power

relative to classical, or crisp, sets because the partitions are not mutually exclusive. That

is, an item may be a member of more than one partition. This property allows fuzzy set

theory to model the important features of the sub-Saharan Africa pastoralist grazing

access system in a consistent way.6,7 In contrast, a probabilistic representation of access

rights would force an “either/or” interpretation of access that is not necessarily consistent

with empirical reality.

A pastoralist’s (or a pastoralist group’s), i’s, access to pasture F, denoted PI, is

the degree of membership of that pasture in the fuzzy set “areas accessed by pastoralist

i”. In fuzzy set notation, this may be written more formally as “area accessed by

A={PA/F}”, and similarly for B. Under a private property regime or a common property

regime, i’s degree of access is 1. If a pasture is the private property of another

pastoralist, i’s degree of access is 0. Thus, if we wished to represent two private

6 An alternative method of modeling the observed system of partial access would be to allow herders to have beliefs over the percentage access that they have to forage in a given area. This alternative, however, will be internally consistent only in cases where the pasture is perfectly managed. If the idea of percentage access is defined more broadly, then it becomes functionally similar to fuzzy access, but lacks an equivalent theoretical basis. 7 In addition to many other sources, Kaufmann and Gupta (1991), Driankov, Hellendoorn and Reinfrank (1993), Jamshidi, Vadiee and Ross (1993) develop basic fuzzy set theory. Dompere (1995), Mansur (1995),Greenhut, Greenhut and Mansur (1995), and Goodhue (1998) use fuzzy logic to model features of economic systems.

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property pastoralists, we would write “area accessed by A={1/F1, 0/F2},” and “area

accessed by B={0/F1, 1/F2}”.

4. Model

We model two pastoralist groups, A and B, who allocate time within a single season for

their animals to graze two pastures. Each pastoralist group has a “home” pasture, to

which it has complete rights. A’s home pasture is denoted 1, and B’s home pasture is

denoted 2. Each pastoralist group also has incomplete, “fuzzy” access to the other

group’s home pastures. Empirically, this setup represents cases where range areas are

predominantly used by one group, but where another subgroup has partial access or a

weaker claim. The group with the high level of access rights can be considered to

implicitly function as the primary “managers” of the area, and the group with the low

access may access the area either through lineage ties or other links between the two.

This is a relatively common arrangement of property rights in our study area.

Because the access to pasture is sometimes incomplete, we use the concept of

“fuzzy” property rights. The maximum number of stock-days a group can use a pasture

is determined by its property right. Each group’s fuzzy property right to the other

group’s home pasture is a function of absolute and relative rainfall on the two pastures.

Rainfall realizations are independent and identically distributed across pastures. Herders

observe rainfall, 0<rj≤ 1, for both pastures. Regarding the role of absolute rainfall, each

pasture’s rainfall is used to define its membership in the fuzzy set corresponding to the

following statement: “Pasture j is in a drought year.”

drought={j/max(0,1-γrj), j={1,2}},

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where 1>γ . We restrict γ in order to ensure that there is some closed interval of rainfall

realizations that are classified as non-drought conditions under our fuzzy rule. If we did

not do so, then either all possible rainfall realizations would be members of the drought

set to some extent ( 1<γ ), or only 1 would be a non-member of the drought set ( 1−γ ).

Regarding the role of relative rainfall, the two rainfall realizations are used to define each

pasture’s membership in the fuzzy set corresponding to the following statement: “Pasture

j is in more of a drought year than pasture ≠j is.”

Worse={1/max(0, r2-r1), 2/max(0, r1-r2)}

These two rules are integrated into rules defining pastoralist group i’s access to

the other group’s home area. For A, the rule provides the membership of pasture 2 in the

fuzzy set defining A’s access to the pastures:

AccessA={1/1, 2/ min(drought,worse,XA2)}

where X A2 is defined by the following “crisp” rule:

If α(r2-r1)+βa>0, then arr

baX A βαβ

+−+

−=)(

)(112

2 , else X A2=0.

For convenience, we will denote the membership of pasture j in i’s access set as Pij. PA2

increases whenever A has a relatively or absolutely bad rainfall realization. Defining the

membership using a minimum reflects that in practice a group’s access to another group’s

home territory is dependent on the latter group having forage to share. The third

component of the fuzzy property right rule recognizes that under a fuzzy access property

rights regime A would never choose to use B’s home pasture more than it would if the

groups were in a standard common property regime and both had complete rights to the

pasture, and vice versa.

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Each group has a herd of fixed size, denoted a and b, respectively. Each group

maximizes its animals’ total weight gain, which we assume is a linear-quadratic function

of the stock-days on the pasture (following McCarthy (1996)). The assumption of fixed

herd size and the focus on weight gain corresponds to a pastoralist group’s short-run

decision, when they are not investing in growing their herd or realizing gains by selling.

While this is an approximation, it is broadly consistent with pastoralists’ reluctance to

sell animals during droughts simply because there is inadequate pasture, and not because

they need the income. For convenience, we assume the output price is determined

exogenously and normalize it to 1, and abstract from production costs, such as

pastoralists’ opportunity cost of time , so that total weight gain equals total profits. Each

group allocates its fixed number of stock-days across two pastures in order to maximize

its profits as a function of rainfall, which is independently realized on the two pastures.

The pastures are physically identical, and are characterized by the forage productivity

parameter, α, and the externality parameter, β. Pasture size is normalized to 1. Total

forage on pasture j is equal to αrj. However, placing livestock on the pasture reduces the

available forage. This stocking externality is a non-linear function of the number of

animal-days, and is defined as 21 AP

s−β

for pasture 2, where },{ babs +∈ . Intuitively,

the externality per animal (21 AP−

β) increases as A spends more animal-days on 2. If A

does not elect to move to 2, the externality per animal becomes simply β.8

8 Our static model ignores any effects of stocking decision in one period on the forage available in future periods. Whether or not current stocking densities have an effect on future forage productivity has been much debated in the range ecology literature, especially for the arid and semi-arid environments of Sub-Saharan Africa. Consensus is building that except at high densities, dynamic externalities are in fact likely to be zero or very small. (See, for example, the discussions in Scoones (1994).)

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Clearly, there are two inputs into the production function: animal days and

hectares; however, total hectares and the number of animals are fixed, and thus only the

allocation of stock-days across pastures is a choice variable. Under the fuzzy access

property rights system, A chooses whether to move to pasture 2, or remain entirely on

pasture 1. B chooses whether to move to pasture 1, or remain entirely on pasture 2. A

will choose to exercise its fuzzy right and move to pasture 2 if

arP

barA

βαβα −>−+

− 12

2 1)( .

Similarly, B will choose to exercise its fuzzy right and move to pasture 1 if

brP

barB

βαβα −>−+

− 21

1 1)( .

Based on the pastoralist groups’ decision rules, we offer the following two comments.

Comment 1: It will never be the case that A will choose to move to 2 and B will choose to move to 1. At most, one pastoralist group will move.

To see this, begin by recalling the definition of Pij. It will never be the case that both PA2

and PB1 are greater than zero. For PA2>0, it must be the case that r2-r1>0, while for PB1>0,

it must be the case that r1-r2>0. Now, rearrange the decision rules. For A, if 012 <− rr ,

then A will never wish to move.

01

)(1

)()(2

2

212 >

−+

=−−+

>−A

A

A PbaP

aP

barrβ

ββα .

Similarly, B will never wish to move if r1-r2<0.

01

)(1

)()(1

1

121 >

−+

=−−+

>−B

B

B PbPab

Pbarr β

ββα

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Comment 2: A will never choose to spend a smaller share of its animal-days on 2 than it is entitled to under the fuzzy access property rights system.

This comment follows from the combination of the fuzzy set defining A’s access to the

pastures and A’s decision rule for moving. Because PA2 can be no larger than the non-

cooperative common property stocking day choice A would make in the absence of the

fuzzy access right regime, there is no incentive for A to reduce the share of animal-days it

spends on 2.

Based on these comments we offer the following proposition regarding the Nash

equilibrium for the stocking decision game between the two pastoralist groups, when

each chooses whether or not to spend part of the season on the other group’s home

pasture.

Proposition 1: The Nash equilibrium is the outcome of the game where taking the rules governing fuzzy access rights as given, each pastoralist group chooses whether or not to spend part of the season on the other group’s home pasture based on the following decision rules: A will choose to exercise its fuzzy right and move to pasture 2 if

arP

barA

βαβα −>−+

− 12

2 1)( . Similarly, B will choose to exercise its fuzzy

right and move to pasture 1 if brP

barB

βαβα −>−+

− 21

1 1)( .

The following expressions are profits for the pastoralist groups under the fuzzy access

property rights system:

))(1()1

)(()1

)(( 1121

112

22 arPPP

barPP

barP BAB

BA

AFA βαβαβαπ −−−+−+

−+−+

−=

))(1()1

)(()1

)(( 2121

112

22 brPPP

barPP

barP BAB

BA

AFB βαβαβαπ −−−+−+

−+−+

−= .

5. Alternative Property Rights Regimes

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In this section, we derive the outcomes of two property rights regimes in order to

compare them to the outcomes of the fuzzy access property rights regime. The first

regime is common property. Under this regime, both pastoralist groups have complete

access to both pastures. In fuzzy set notation, AccessA={1/1, 2/1} and AccessB={1/1,

2/1}. Notice, however, that the membership values no longer correspond to A’s use of 2,

or B’s use of 1. The second regime is private property. Under this regime, each

pastoralist group has complete rights to its own home pasture and no rights to the other

group’s home pasture. In fuzzy set notation, AccessA={1/1, 2/0} and AccessB={1/0, 2/1}.

Common Property. Under the common property regime, we assume that A begins the

season on pasture 1, and B begins the season on pasture 2. Now, rather than exercising a

fuzzy access right when its home rainfall realization is poor, each pastoralist group

optimizes its total returns by choosing the share of the season it will spend on the other

group’s home pasture: mA2 and mB1. Either group will move if it increases total profits to

do so.

Proposition 2: Under the common property regime, the Nash equilibrium depends on the rainfall realizations. When r2-r1>0,

arrbamA βα

β+−

+−=

)()(1

122 and mB1=0. When r2-r1<0, mA2=0 and

brrbamB βα

β+−

+−=

)()(1

212 . When r2=r1, mA2=0 and mB1=0.

Proof: The proof is in the appendix. Private Property. Under the private property regime, A remains on pasture 1 and B

remains on pasture 2 regardless of the rainfall realizations. In this case, returns take the

following form:

=PAπ ar βα −1 and

=PBπ br βα −2 .

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6. Results

We compare individual and total expected returns under the three property rights

regimes, in order to examine whether or not the fuzzy access system is preferable to

either or both of our benchmark systems in terms of expected value. We begin by noting

the comparison between common property and private property established in the

literature.

Remark 1: Assuming rainfall realizations are independently and uniformly distributed on the [0,1] interval, A. expected returns to each pastoralist group may be higher or lower under the common property regime than under the private property regime, depending on parameter values. B. expected total returns may be higher or lower under the common property regime than under the private property regime, depending on parameter values.

Intuitively, this finding is due to a tradeoff between the two regimes. Under private

property, each pastoralist group benefits fully from the effective forage on its home

pasture, and the externality due to stocking days is determined only by its own herd size.

In contrast, under common property each pastoralist group does not benefit fully from the

effective forage on its home pasture. Whenever its home pasture has more rainfall than

the other group’s home pasture, the other group moves at some point during the season.

This increases the externality and reduces the effective forage consumed by the home

group’s animals. However, the off-setting benefit is that the pastoralist group can move

and use the other pasture when its home pasture has relatively low rainfall, and increase

the forage consumed by its animals.

The same tradeoff exists when comparing the fuzzy access property right regime

and the private property regime.

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Proposition 3: Assuming rainfall realizations are independently and uniformly distributed on the [0,1] interval, A. expected returns to each pastoralist may be higher or lower under the fuzzy access regime than under the private property regime, depending on parameter values. B. expected total returns may be higher or lower under the fuzzy access regime than under the private property regime, depending on parameter values. Proof: The proof is in the appendix.

It is interesting to note that the fuzzy access property regime dominates the private

property regime provided the resource base parameter α is sufficiently large, relative to

other parameters. Essentially, the insurance provided by fuzzy access rights does not

have any value if it does not empower one to actually use many resources. In that

instance, it is preferable to protect what few resources one has, rather than share them.

The tradeoff between insurance against poor home pasture realizations and using

all of the effective forage on one’s home pasture determines the relative returns from

private property and the two non-exclusive property rights regimes. The comparison of

the fuzzy access property regime and the common property regime introduces another

element: in order for A to access 2, the rainfall realization on 1 must be absolutely

(“Pasture j is in a drought year”) as well as relatively (“Pasture j is in more of a drought

year than pasture ≠j is”) low, while for the common property regime only the relative

rainfalls matter. Prior to comparing returns under the two regimes, we compare expected

mobility by examining the pairs of rainfall realizations for which each pastoralist group

will move.

Proposition 4: Assuming rainfall realizations are independently and uniformly distributed on the [0,1] interval, expected mobility is lower under the fuzzy access property regime than under the common property regime. That is, each pastoralist group moves less often in expectation.

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Proof: We need only to demonstrate that there exist pairs of rainfall realizations for which neither group moves under the fuzzy access property regime but does move under the common property regime. We will do so for a case where r2-r1>0, when PA2>0 and PB1=0. Given the definition of the fuzzy set denoting A’s access rights and the fuzzy set denoting drought

(drought={j/max(0,1-γrj), j={1,2}}), A will not move if γ1

1 ≥r . Because

1>γ , this will be true for some interval [r*,1], where 0<r*. In contrast, under the common property regime A will always move when r2-r1>0 (Proposition 2).

The difference in mobility occurs because fuzzy access rights are a function of absolute

and relative rainfall, while under common property both groups have complete rights,

and make the decision to move purely as a function of absolute rainfall. We now compare

expected returns under the two regimes.

Proposition 5: Assuming rainfall realizations are independently and uniformly distributed on the [0,1] interval, provided γ is not too large then A. expected returns to each pastoralist are at least as high under the fuzzy access regime as under the common property regime. B. expected total returns are at least as high under the fuzzy access regime as under the common property regime. Proof: The proof is in the appendix.

Under some conditions, the fuzzy access right regime increases returns relative to a

common property regime.

7. Discussion and Policy Relevance

Pastoralist mobility, common use of pastures, and the imprecise nature of grazing rights

affect the relative desirability of policy options for sub-Saharan African grazing systems.

Modeling traditional grazing rights with fuzzy sets, we find that under some conditions

the traditional system results in higher total returns than conventional common property

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does. Both private and conventional common property rights limit herders’ ability to

respond to weather shocks ex post. While there is anecdotal evidence supporting this sort

of risk shifting, its importance remains to be verified empirically.

The partial access model may aid policymakers in evaluating alternative

interventions. If the fuzzy representation captures facets of empirical reality that causes

the system to behave differently than do systems with well-defined boundaries and

groups, then the fuzzy representation will provide a better theoretical basis for evaluating

possible policy initiatives and their predicted effects. Land reform is an extremely

important policy issue in sub-Saharan Africa. Historically, policy efforts have effectively

discriminated against nomadic pastoralists. Land privatization, which restricted access to

a single individual, was the favored means of assigning property rights until fairly

recently. Now, however, policymakers are questioning the appropriateness of individual

property rights, especially in regions of extensive livestock production. While

privatization is no longer assumed to be the best policy response, policymakers are still

concerned with groups’ abilities to manage common property resources. In conventional

common property research, well-defined boundaries and well-defined access are

considered essential for successful management of the commons. If this were true in the

sub-Saharan pastoral context, then one option would be an access clarification policy that

specifies groups’ access to given pastures, and assigns the management of these pastures

to specified groups. If, on the other hand, benefits of the fuzziness of the traditional land

rights system would be lost under such an access-clarification policy, then perhaps it

would be better to support the traditional land access system.

19

Our findings indicate that fuzzy access rights will be preferred to well-defined

common property rights when absolute rainfall does not play too large of a role in

determining the fuzz access rights. Fuzzy access rights will be preferred to well-defined

private property rights when the resource base is large enough for the insurance feature of

fuzzy access rights to provide real value.

Given that there are conditions under which the traditional fuzzy system is

preferable, it is worthwhile to identify policies that may enhance the traditional system.

Linking funds for borehole improvement to specific groups, and supporting their control

of the improved water source may have spillover effects on the surrounding pasture, and

could reduce incentives to subvert the traditional system. Currently, the way such

investments are undertaken tends to make access rights more symmetric across users. It

may be possible to institutionalize some aspects of the traditional access system so that

groups that overgraze pastures given their access and alternatives can be sanctioned.

For the purpose of this work, we have abstracted from other factors that may influence

the relative desirability of property rights regimes. We did not allow for the possibility

of rainfall correlation across pastures. We expect that correlation will affect returns, but

will not cause one land tenure system to always dominate the other two alternatives.

Determining whether this is indeed the case requires further research.

For this paper, we have modeled fuzzy access rights as costless and the rules

determining these rights as exogenously specified, and examined the effects of these

rights on pastoralists’ stocking decisions for a fixed herd size over the course of a season.

Understanding the factors underlying groups’ access rights choices and maintenance

costs will provide a deeper understanding of the costs and benefits of the traditional

20

system. Factors such as distance, forage quantity, seasonal availability, rainfall amount

and correlation with other pastures, and the presence of other groups are all likely to

affect the value of maintaining fuzzy access rights to a given pasture for a given group.

21

Appendix

Proof of Proposition 2 A optimizes with respect to mA2, the share of the season it spends on pasture 2, holding B’s choice constant. Maximize

)()1

)()(1())(1)(1(

1

)()(1( 221

1121112

2212 ) armm

mba

mmarmmm

barmm AB

BrABBA

ABA βα

βαβα

βα −+

−+

−−+−−−−

+−− +

subject to .10 2 ≤≤ AmAs was the case under the fuzzy access property rights system, it will never be the case that both A and B desire to move. If the returns to B from moving to 1 and sharing it with A are greater than the returns to remaining alone on 2, then there is no incentive for A to move to 2, even if B moves to 1. Hence, we can simplify A’s decision problem considerably before solving it.

Max ))(1()1

)(( 122

22 armm

barm AA

A βαβα −−+−+

−

subject to .10 2 ≤≤ Am

There are two possible solutions: mA2=0 and arr

bamA βαβ

+−+

−=)(

)(112

2 . It is never the

case that mA2=0 and r2-r1>0, because of the externality due to the number of stocking days. A will always wish to exhaust the effective forage on 1 before moving to 2, and restricting rainfall realizations to be positive ensures that there will always be some forage on 1. If r2-r1<0, there is less effective forage available on 2 than on 1, holding stocking days constant. If B were to remain on 2, then the effective forage is even smaller. If B increases its profits by moving to 1 and sharing it with A, relative to remaining on 2, then A has no incentive to move to 2. Therefore, if r2-r1<0 then mA2=0.

By symmetry, if r2-r1<0, brr

bamB βαβ

+−+

−=)(

)(121

2 . If r2=r1, effective forage is the

same on pastures 1 and 2, holding stocking days constant. Neither group has an incentive to move, because if both are on the same pasture stocking days increase and effective forage is reduced. Returns under the common property regime are

))(1()1

)(()1

)(( 1121

112

22 armmm

barmm

barm BAB

BA

ACA βαβαβαπ −−−+−+

−+−+

−=

))(1()1

)(()1

)(( 2121

112

22 brmmm

barmm

barm BAB

BA

ACB βαβαβαπ −−−+−+

−+−+

−= ,

where mA2 and mB1 are defined as a function of rainfall in Proposition 2.

22

Proof of Proposition 3. Part A. When PA2>0, A’s returns under the fuzzy access property regime are higher than under the private property regime. To see this, begin with the expression for A’s returns under the fuzzy access property regime.

))(1()1

)(()1

)(( 1121

112

22 arPPP

barPP

barP BAB

BA

AFA βαβαβαπ −−−+−+

−+−+

−=

Because we are considering only cases where PA2>0, relying on Comment 1 we can simplify the expression to

))(1()1

)(( 122

22 arPP

barP AA

AFA βαβαπ −−+−+

−= .

In order for PAFA ππ > , it must be the case that .

arP

barA

βαβα −>−+

− 12

2 1)( . This is precisely the condition under which A moves to

pasture 2 because it is more profitable to share 2 with B than to remain alone on 1, as A does under the private property regime. When PA2=0, A’s returns under the fuzzy access property regime are lower or the same as under the private property regime. If PB1=0, A’s returns are identical. If PB1>0, A’s returns are lower, because B shares pasture 1 with A for part of the season. The net effect of higher returns under the fuzzy access regime when PA2>0 and lower returns when PA2=0 depends on parameter values. For any set of parameter values {a=b,β,γ}, there is a sufficiently large α such that A’s returns under the fuzzy access property regime are larger than A’s returns under the private property regime. A symmetric argument holds for B’s returns. Part B. Part B follows from Part A and the assumptions of identical pastures and pastoralist groups.

Proof of Proposition 5 Part A. Because the previous proposition established that there are rainfall realizations for which a pastoralist group will move under the common property regime but not under the fuzzy access rights regime, expected returns are calculated for five cases. We will evaluate A’s profits for each case. By symmetry, results will be the same for B. Case 1. A moves under the common property regime, but not under the fuzzy access rights regime. A’s profits are lower under the fuzzy access regime in this case.

23

Case 2. B moves under the common property regime, but not under the fuzzy access rights regime. A’s profits are higher under the fuzzy access regime in this case. Notice that Cases 1 and 2 are mirror images of each other, given our assumption of identical pastoralist groups and pastures. For any pair r1=x, r2=y in case 1, the corresponding pair r1=y, r2=x will fall under Case 2. To evaluate the net effects of these two cases on expected returns, it is sufficient to compare A’s profits as the sum of one set of corresponding pairs of rainfall realizations (x,y) and (y,x). Because the common property regime allows A to benefit when its home pasture receives less rain, the net effect is that A’s profits are lower under the fuzzy access rights regime. Case 3. A moves under both regimes. For rainfall pairs in this region, the relative level of expected profits is determined by the share of the season A spends on pasture 2 under each regime. For CAFA ππ > , the following condition must hold:

))(1()1

)(())(1()1

)(( 122

22122

22 armm

barmarPP

barP AA

AAA

A βαβαβαβα −−+−+

−>−−+−+

−

Simplifying, this condition becomes

0))1)(1(

()(22

12 >−−

+−+−

AA Pmbaarr βα .

Substituting in the definition of mA2 when mA2>0 and rearranging, the condition can now be rewritten as

arP

barA

βαβα −>−

+− 12

22 )1(

)( .

Recall the condition under which A moves to pasture 2 under the fuzzy access

right regime: arP

barA

βαβα −>−+

− 12

2 1)( . Because 10 2 ≤≤ AP , .

Thus, it is possible in theory that there are cases where

222 )1(1 AA PP −>−

22

212

2 )1()(

1)(

AA Pbarar

Pbar

−+

−>−>−+

−βαβαβα . This would occur if .

This condition always holds, based on the definition of the fuzzy set describing A’s access. Thus, whenever A moves under both regimes, A’s profits are lower under the fuzzy access right regime.

22 AA Pm >

Case 4. B moves under both regimes. For rainfall pairs in this region, the relative level of expected profits is determined by the share of the season B spends on pasture 1 under each regime. The shorter the time B shares pasture 1 with A, the higher A’s returns. Thus, for CAFA ππ > , using algebra similar to the algebra in the previous case, the following condition must hold: . By the same 11 BB Pm >

24

logic, this condition always holds. Whenever B moves under both regimes, A’s profits are higher under the fuzzy access right regime. Notice that Cases 3 and 4 are mirror images of each other. For any pair r1=x, r2=y in Case 3, the corresponding pair r1=y, r2=x will fall under Case 4. To evaluate the net effects of these two cases on expected returns, it is sufficient to compare A’s profits as the sum of one set of corresponding pairs of rainfall realizations (x,y) and (y,x). Because , A gains more from a higher rainfall realization on 1 under the fuzzy access right regime than he loses from a lower rainfall realization on 2, relative to the common property regime.

11 BB Pm >

Case 5. Neither moves under either regime. Profits are identical across the two regimes. The net effect of Cases 1-4 depends on the importance of absolute rainfall compared to relative rainfall in the definition of fuzzy access rights. The negative net effect of Cases 1 and 2 increases with γ, because more rainfall realization pairs result in outcomes where neither pastoralist group moves under the fuzzy access right regime. The positive net effect of Cases 3 and 4 decreases with γ, because fewer rainfall realization pairs result in outcomes where one pastoralist group moves. Provided γ is not too large, relative to the other parameters, returns will be higher under the fuzzy access right regime than under the common property regime. Part B. Part B follows from Part A and the assumptions of identical pastoralist groups

and pastures.

25

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