Electronic copy available at: http://ssrn.com/abstract=789768

Strategic Exploitation of

a Common-Property Resource

under Uncertainty

Elena Antoniadou, Christos Koulovatianos,1

Leonard J Mirman2 3

May 16, 2009

1Corresponding author: Department of Economics, University of Exeter, Streatham

Court, Rennes Drive, Exeter, EX4 4PU, UK, Tel: +44-1392-26-3219, Fax: +44-1392-

26-3242.2Contacts: Elena Antoniadou: Australian National University,

elena.antoniadou@anu.edu.au; Christos Koulovatianos: University of Exeter;

C.Koulovatianos@exeter.ac.uk; Leonard J Mirman: University of Virginia,

lm8h@virginia.edu.3We thank Gerhard Sorger, seminar participants in Vienna, and an anonymous

referee for useful comments. Koulovatianos thanks the Austrian Science Fund under

project P17886, for financial support.

Electronic copy available at: http://ssrn.com/abstract=789768

Abstract

We study the impact of uncertainty on the strategies and dynamics of symmet-

ric non-cooperative games among players who exploit a non-excludable resource

that reproduces under uncertainty. We focus on a particular class of games that

deliver a Nash equilibrium in linear-symmetric strategies of resource exploita-

tion. We show that, for this class of games, the tragedy of the commons is

always present. For various changes in the riskiness of the random primitives

of the model we provide general characterizations of features of the model that

explain links between the degree of riskiness and strategic exploitation decisions.

Finally, we provide a specific example that demonstrates the usefulness of our

general results and, within this example, we study cases where increases in risk

amplify or mitigate the tragedy of the commons.

Keywords: Renewable resource exploitation, Stochastic non-cooperative dy-

namic games, Tragedy of the commons

JEL Classification Codes C73, C72, C61, Q20, O13, D90, D43

1 Introduction

Games of common-property renewable resource exploitation have the feature

that each player chooses their current level of exploitation while (partly) con-

trolling the future evolution of the resource, given the strategies of other players.

Models in which there is a dynamic element and in which resources are shared,

play an important role in economics, e.g., industrial organization models or mod-

els with natural resources. The fundamental, infinite-horizon setup, where all

players have full information about the economic environment, has been studied

in the economics literature almost exclusively within the deterministic frame-

work. The main finding of this literature is that the equilibrium is characterized

by a ‘tragedy of the commons.’ Namely, the higher the number of players, the

higher the aggregate exploitation rate, so the lower the level of the resource in

the long run.1

The tragedy of the commons results after a complex strategic accounting

by players in a perfect-foresight environment. In particular, it is transparent

to each player that overexploitation implies future trajectories of the common

resource that increase their present and future individual costs. Yet, given what

other players do, and what they will be doing throughout the infinite horizon,

it is often individually optimal to support a strategic path of aggregate overex-

ploitation of the resource, when the number of players increases. How does such

strategic behavior change if, instead of acting in a deterministic environment,

players act in an environment of uncertainty and rational expectations? Our

study contributes some answers to this general question, focusing on infinite-

horizon games where the law of reproduction of the resource is subject to random

shocks and players have rational expectations.2

The scope and aim of our analysis is to contribute to the understanding of

common-property resource games under uncertainty. Yet, stochastic dynamic

games can be particularly complex and very difficult to characterize with general

primitives, i.e. with general payoff functions of players, general resource or

renewal laws of motion and general distributions of random disturbances.3At the

1See, for example, Mirman [16] and Levhari and Mirman [13], Levhari, Michener and

Mirman [12], Benhabib and Radner [2], Dockner and Sorger [6], Sorger [18] and [19] and

Koulovatianos and Mirman [11].2Our focus on randomness in resource reproduction is a natural starting point for the

study of uncertainty in resource games. In the real world, resources evolve according to

stochastic laws of motion. Especially when in the context of natural resources, as is the

case with biological populations such as forests and fish species, these evolve subject to the

existence of predators or climate, that are affected by random disturbances. In cases of

governmental provisions of infrastructure for companies, such as railroads, electricity grids,

telecommunication networks, etc., financing and maintenance is also subject to random shocks,

such as business cycles or political cycles.3An example of a study examining the link between extraction decisions and uncertain

reproduction outcomes under perfect competition, and also optimal resource preservation

policies is the fishery application of Mirman and Spulber [17]. For a paper studying uncertainty

and games see Amir [1]. For studies pointing out technical issues in deterministic differential

resource games, such as multiplicity of equilibrium strategies, arising even in setups with some

simplifying assumptions on primitives, see Dockner and Sorger [6], and Sorger [18], while for

fundamental proofs of equilibrium existence see Sundaram [23] and Dutta and Sundaram [7].

1

same time, the task of characterizing decisions in the presence of uncertainty in

a general framework can be very demanding even when there is only one player.4

We discuss several technical problems that arise in general dynamic games. In

particular, in resource games, part of the objective function of each player is

the other players’ strategies. When the Markov strategies of other players are

strictly concave, a player’s objective function may lose key properties, such

as concavity, differentiability, and continuity. These technical difficulties are

discussed in Mirman [16].

One special class of dynamic games where such difficulties can be avoided

is where there is a symmetric Markov-Nash equilibrium of the game which is

in linear Markov strategies.5 A game that falls in this class is the parametric

example of Levhari and Mirman [13]. Our paper is devoted to the study of this

tractable class of games and equilibria, a natural starting point for undestanding

players’ strategic behavior under uncertainty. In section 3 we provide necessary

and sufficient conditions, on the payoff functions of players and on the law of

resource reproduction, for the existence of an equilibrium in linear-symmetric

strategies.6 In section 8 we provide a parametric example that falls into this

class of games.

We deal with two basic issues concerning all models in this special class

of dynamic games, where there is a Nash equilibrium is in linear-symmetric

Markov strategies. The first is whether the tragedy of the commons holds in this

equilibrium. In Section 4 we show that all games, stochastic or deterministic,

that have a Markov-Nash equilibrium in linear-symmetric Markov strategies,

always imply a tragedy of the commons in this equilibrium. This means, in

this class of games, adding one more player always leads to a higher aggregate

exploitation rate in the unique equilibrium. Secondly, we study the effect of

the riskiness of the random primitives of the model on the players’ strategies.

In particular, we highlight those features of the model that explain the links

between the degree of riskiness and strategic exploitation decisions. In Section

4 we show that uncertainty may increase or decrease the equilibrium aggregate

level of exploitation, relative to the deterministic case.

In Section 5 we study the applicability of a recursive procedure for calcu-

lating linear-symmetric strategies. Levhari and Mirman [13] used a method for

solving dynamic games that started from the one-period problem and contin-

ued recursively for the -period problem, in order to obtain the infinite-horizon

solution asymptotically. We find that for the class of games that lead to lin-

4For example, Mirman [15] analyzes uncertainty in a model with a single controller, pro-

viding a general result about the role of uncertainty on decisions in two-period models, and

discussing issues arizing in the infinite-horizon setup.5We thank an anonymous referee for highlighting the need to clarify this issue. Our analysis

does not restrict the search for optimal strategies within the linear class. Rather, it restricts

attention to those dynamic games which have a symmetric equilibrium in linear Markov

strategies, among all possible Markov strategies.6This is not a global uniqueness result. However, linear-symmetric equilibrium strategies,

where they exist, are unique amongst linear strategies. To the best of our knowledge there

are no known sufficient conditions for global uniqueness of a Markov-Nash equilibrium, for

the general game we study.

2

ear symmetric Markov strategies the calculation method used by Levhari and

Mirman [13] always converges to the infinite-period solution, as long as linear-

symmetric solutions for the -period problem exist.7

We use these results in Section 6 to examine the link between the degree

of riskiness and strategic decisions. For games with the same payoff functions

and the same natural law of resource reproduction, we examine how different

shocks that are linked through second- or first-order stochastic dominance affect

strategies. In all cases we identify simple conditions on the payoff function of

players that explain the direction of the change in exploitation rates as the

stochastic structure of the shocks changes.

We then study specific parametric models in order to demonstrate the useful-

ness of our results. In section 7 we show the Levhari-Mirman [13] in the presence

of uncertainty, and in Section 8 we present an extended parametric model that

nests the Levhari-Mirman [13] example while maintaining the property of a

unique Markov-Nash equilibrium which has linear-symmetric strategies, but in

which uncertainty plays a crucial role. We show that the Levhari-Mirman [13]

model is a knife-edge case where the presence of uncertainty has no impact on

the strategies. Our parametric model is a key contribution of this paper, indi-

cating a new benchmark for initiating interesting questions about uncertainty

in games. One question that becomes possible to analyze with this parametric

model, is the effect of changes in risk on the intensity of the ‘tragedy of the

commons.’ In the context of this model, we show that, indeed, there are cases

where the ‘tragedy of the commons’ is mitigated, and cases where it is amplified,

as the random parameter becomes more risky.

We begin in Section 2 with a description of the general framework for our

analysis.

2 The general framework

Time is discrete and the horizon is infinite, i.e. = 0 1 . Let the state

variable, , evolve naturally (when no consumption occurs) according to the

law of motion,

+1 = () (1)

We assume 0 0, 00 ≤ 0. The random variable is i.i.d., independent of and,

∼ Θ () , = 0 1

We assume that Θ, the distribution function of , has support ⊆ R+ andmean () ∞ for all .

We consider ≥ 1 identical players. In period , each player ∈ 1 consumes ≥ 0 units of the available stock, and then a realization of the

7The standard Inada condition on the payoff (utility) functions and a simple condition

on the distribution function of the random production shock are sufficient to guarantee the

existence of solutions to the associated -period problem.

3

random shock takes place. Next period’s level of is given by

+1 =

à −

X=1

! (2)

Each player ∈ 1 maximizes his expected discounted utility over aninfinite-period horizon,

0

" ∞X=0

()

#

We assume : R+ → R is twice continuously differentiable with 0 0, 00 0.All players have the same state utility function, , and discount factor ∈ (0 1).We study Markov-Nash equilibria, i.e. equilibria with strategies of the form

= ()=1. The problem of player ∈ 1 can be expressed inBellman equation form as,

() = max≥0

⎧⎪⎨⎪⎩ () +

⎡⎢⎣⎛⎜⎝

⎛⎜⎝− −X=1 6=

()

⎞⎟⎠⎞⎟⎠⎤⎥⎦⎫⎪⎬⎪⎭ (3)

We focus on two issues: (a) the strategic behavior of players, in particular,

the “tragedy of the commons,” and (b) the effect of uncertainty, or change in

risk, on consumption strategies. We study the effects of “changing risk” using

the concepts of first- and second order stochastic dominance. We also compare

decisions made in the stochastic model with decisions made in a version of the

deterministic model in which the shock is always equal to its mean, () = .8

In the deterministic model, player ’s problem, given the strategies of all other

players is,

() = max≥0

⎡⎢⎣ () +

⎛⎜⎝

⎛⎜⎝− −X=1 6=

()

⎞⎟⎠⎞⎟⎠⎤⎥⎦ (4)

We distinguish between the stochastic model carrying the subscript “,” and

the deterministic model with subscript “”. Let S (resp. S) denote the Markovequlibrium strategies of the stochastic (resp. deterministic) game, i.e.

S =½©

∗ ()ª=1

¯ ∀ ∈ 1 ∗ () solves problem (3)

given () = ∗ () ∈ 1 with 6=

¾ and

S =½©

∗ ()ª=1

¯ ∀ ∈ 1 ∗ () solves problem (4)

given () = ∗ () , ∈ 1 with 6=

¾

8 See Hahn [9], Stiglitz [22] and Mirman [15].

4

With the assumptions imposed so far, it is not possible to analyze the complete

equilibrium sets, nor to give conditions for the existence of globally unique

equilibrium in either the stochastic or deterministic game. The presence of other

players’ strategies in problems 3 and 4 gives rise to various potential problems.

Suppose, for the moment, that the value function is twice continuously dif-

ferentiable.9 The first-order condition, from the dynamic program (3), is

0 () = 0 ()£ 0

( ())¤, with = − −

X=16=

() (5)

Using the envelope theorem on (3) yields,

0 () =

⎡⎢⎣1− X=16=

0 ()

⎤⎥⎦ 0 () £ 0 ( ())

¤ (6)

Combining (6) with (5),

0 () = 0 ()

⎡⎢⎣1− X=16=

0 ()

⎤⎥⎦ (7)

A crucial technical difficulty is revealed in (7). While (·) is strictly concave,the strategies (·) of the other players need not be convex, and the valuefunction (·) need not be concave. In fact, the possible non-concavity of thevalue function is not the only problem that may arise in this setup, from the

presence of the strategies of other players in a player’s value function. Mirman

[16] provides several examples of games using functions and , which meet

our general assumptions, but lead to value functions that are not concave, not

differentiable, not even continuous, and may admit multiple solutions.10 What

is revealing about the examples in Mirman [16] is that, if the same functions

and were used in a single-controller optimization problem, all the resulting

value functions would be twice continuously differentiable and strictly concave.

3 Games with a Markov-Nash equilibrium in

linear-symmetric strategies

Our parametric example in section 8 motivates our analysis. In that exam-

ple problems associated with the value function, as identified in section 2, are

avoided due to the linearity of the optimal symmetric strategies. Therefore, in

line with our parametric example, from this point onward we restrict attention

9This is not a valid assumption for the general problem, as we discuss below.10 Such problems do not preclude the existence of equilibria.

5

to games that deliver a symmetric Markov-Nash equilibrium in linear strategies,

i.e. games with Markov equilibrium strategies of the form

∗ () = , with ∈ (0 1) for all 0 all ∈ 1 . (8)

This class of games is non-empty. A well-studied model with a Markov-Nash

equilibrium in linear-symmetric strategies is the model of Levhari and Mir-

man [13], where () = ln (), () = , and ∈ (0 1).11 Our parametric

example in section 8 gives an extended pair of functions and that nests

the Levhari-Mirman example, and delivers an equilibrium in linear-symmetric

Markov strategies in the stochastic and corresponding deterministic game.12

For a game h Θi in this class of games,13 strategies (8) solve the dynamicprogram (3). The linearity of the optimal strategies allows us to avoid problems

associated with the non-differentiability of the value function. These strategies

satisfy conditions (5) and (6), and therefore condition (7), which becomes,

0 () = [1− ( − 1)]0 () (9)

Therefore, in this equilibrium 0 0, while 00 exists and is strictly negative.In other words, a Markov-Nash equilibrium with linear symmetric strategies en-

sures that is twice continuously differentiable, strictly increasing and strictly

concave. Notice that, since we focus on symmetric strategies, the index is

dropped from the value function.

Since models that deliver linear-symmetric Markov equilibrium strategies

are identified, we proceed with a general analysis for this class of models. First

we give conditions for the existence of a linear-symmetric equilibrium.

A game h Θi can have at most one linear-symmetric strategy in theequilibirum set S and similarly a game

­

®can have at most one linear-

symmetric strategy in the equilibirum set S Consider the stochastic case first.Fix 0, and consider (5) assuming (interior) linear symmetric strategies

() = . This necessary condition implies14

() = −0 ()+ 0 ((1−)) [ 0 ( ((1−)))] = 0 for all .

(10)

Similarly in the case of­

® the necessary condition from (4) implies

11Levhari and Mirman [13] also examine cases of non-symmetric strategies by allowing the

discount factors of players to differ.12We believe that this equilibrium is globally unique but we provide no explicit proof of this

result, which in any case is not needed for our analysis.13 Since players are identical and we focus on state-dependent (Markov) strategies, it suffices

to denote a game only by h Θi for simplicity.14Notice that since (10) must be met for all 0 in the case of linear-symmetric strategies,

the function (·) does not depend on in equilibrium (i.e., when = ). Even if the left-

hand side of (10) depends on whenever (10) is not met with equality (i.e., when 6= ),

this potential dependence on does not affect our analysis, so we discard for the sake of

simplicity.

6

() = −0 ()+ 0 ((1−)) 0

¡ ((1−))

¢= 0 for all .

(11)

Notice that

0 () 0 and 0 () 0 for all ∈µ0

1

¶, (12)

so there can be at most one solution in each case.

In the games we study, the equilibrium strategies, and of the games

h Θi and ­ ® respectively, are the unique solutions to:15 () = 0 and () = 0. (13)

While functions and are convenient to work with in order to characterize

properties of the Markov-Nash equilibrium in linear strategies, they do not pro-

vide the best way to verify the existence of a solution, since the value functions

themselves enter the expressions in 10 and 11. Theorem 1 gives necessary and

sufficient conditions so that there is precisely one such symmetric equilibrium

in linear strategies in the stochastic and the deterministic games respectively,

based on the data of the corresponding game.16

Theorem 1 Game h Θi has a unique Nash equilibrium in linear symmetricstrategies, if and only if there exists only one ∈ (0 1) such that

Φ ( ) = some ∈ R, for all 0 (14)

where

Φ ( ) = ()− 1− ( − 1)1−

[ ( ((1−)))]

and Θ is such that all players’ value functions are well-defined.17 Game­

®has a unique Nash equilibrium in linear symmetric strategies if and only if there

exists only one ∈ (0 1) such thatΦ ( ) = some ∈ R, for all 0 (15)

15Amir [1] shows that for some games an equilibrium does not exist in the deterministic

case, while there is at least one equilibrium for the stochastic version of the same model. In

our extended example, presented in Section 8, linear-symmetric equilibrium strategies exist

in both the deterministic and the stochastic game.16For the approach followed in the proof of Theorem 1, see Chang [5], Xie [24], and Shimo-

mura and Xie [20]. Note that this result is not an equilibrium global uniqueness result.17Whether a player’s optimization problem is well-defined in a stochastic Markovian game

can depend on the nature of the shock. For example, if the support of the shock is unbounded,

conditions must be placed on the distribution of the shock in order to guarantee that value

functions of players exist. In the context of the general single-controller stochastic growth

model (which is the same as the model of Brock and Mirman [4]), Stachurski [21] identifies

a simple condition on the mean of some monotonic transformation of the random shock that

is sufficient to guarantee a well-behaved optimization problem and a well-defined long-run

stationary distribution. Unlike Stachurski [21], we do not provide such a condition for the

general game, but we do so in the context of our more specific analysis in Section 8.

7

where

Φ ( ) = ()− 1− ( − 1)1−

¡ ((1−))

¢and is such that all players’ value functions are well-defined.

Proof. For the necessity part in the case of the stochastic game, suppose game

h Θi has an equlibrium in linear-symmetric strategies, ∗ () = all ,©∗ ()

ª=1∈ S Consider the first order condition (5). By hypothesis this

becomes:

0 ()− [1− ( − 1)] 0 ((1−)) [0 ( ((1−)) )] = 0

(16)

Condition (16) holds for all 0. Therefore, integrating with respect to ,

yields the expression corresponding to Φ ( ) on the left hand side Hence

Φ ( ) = for some ∈ R, for all 0.

For the sufficiency part, assume that there is only one ∈ (0 1) suchthat Φ ( ) = for some ∈ R, for all 0, and let Θ be such that

the value function of each player is well-defined. Then, differentiating both

sides of Φ ( ) = with respect to leads to (16), which is the necessary

condition for a linear-symmetric optimum. Differentiating both sides of (9)

with respect to , yields 00 () = [1− ( − 1)]00 () 0, due to that ∈ (0 1). So, ∗ () = for all ∈ 1 is the only Nashequilibrium in symmetric linear strategies.

For the deterministic case suppose that game­

®has an equilibrium

in linear-symmetric strategies, ∗ () = all ,n∗ ()

o=1∈ S The

necessary first order condition becomes:

0 () = [1− ( − 1)] 0 ((1−)) 0 ¡ ((1−))

¢. (17)

The remainder of the proof is analogous to that in the stochastic game and

is omitted.

In Section 8 we use Theorem 1 in order to identify appropriate functions ,

, Θ, and the level of in our parametric example. For the general results we

concentrate our analysis on the unique equilibrium in linear-symmetric Markov

strategies for games which satisfy the conditions of theorem 1. We use the short-

hand notation h Θi∞ () and­

®∞

() to denote the game

and the equilibrium we are studying in the stochastic and deterministic game

respectively.

4 The tragedy of the commons and uncertainty:

a first look

Theorem 2 demonstrates a global result about the strategic behavior of play-

ers. We show that for all games h Θi∞ () and­

®∞

() the

tragedy of the commons always holds.

8

Theorem 2 Suppose games h Θi and ­ ® satisfy the conditions of the-orem 1 and consider h Θi∞ () and

­

®∞

() As the num-

ber of players, , increases, the aggregate exploitation rates, Ω ≡ and

Ω ≡ increase.

Proof. Consider h Θi∞ (). For any 0, a player’s necessary con-

dition (10) can be expressed as a function of the aggregate exploitation rate

Ω ≡ , as

Ψ (Ω ) = −0µΩ

¶+ (Ω) = 0, where

(Ω) = 0 ((1−Ω)) [ 0 ( ((1−Ω)))]

Given that 00 0, and 00 ≤ 0, 0 (Ω) 0. Applying the implicit function

theorem on the equilibrium condition

−0µΩ

¶+ (Ω) = 0,

we get

Ω

=

−00 ¡Ω¢

−00 ¡Ω¢+ 0 (Ω)

Ω

2 0, (18)

which proves the result. The argument for the deterministic case is the analo-

gous and is omitted.

Theorem 2 is sharp for the class of games we examine. It indicates that the

‘tragedy of the commons’ is robust for games of joint exploitation.18 However,

Theorem 2 does not address the effect of the introduction of uncertainty on

the tragedy of commons. Theorem 3 shows that uncertainty may increase or

decrease the equilibrium aggregate level of exploitation, by comparing the equi-

libria of pairs of games h Θi∞ () and­

®∞

() Theorem 3

is similar to Theorem 2 in Mirman [15], adjusted to accommodate a symmetric

equilibrium in linear strategies.

Theorem 3 Suppose games h Θi and ­ ® satisfy the conditions of the-orem 1 and consider h Θi∞ () and

­

®∞

() Then, for all

0

T ⇐⇒ [Λ ()] S Λ¡¢

(19)

where,Λ () ≡ 0 (), Λ () ≡ 0

() and ≡ ((1−)).

Proof. Fix 0. Given that both the deterministic and the stochastic strate-

gies are interior, ∈ (0 1), then () = () = 0. Since is

strictly increasing on (0 1) (see (12)),

T ⇐⇒ () S 0⇐⇒ () S () (20)

18Other studies that examine the issue of the tragedy of the commons include Dutta and

Sundaram [8], Sorger [18], [19], and Dockner and Sorger [6].

9

Moreover, from (10)

() = −0 ()+ 0 ((1−))

((1−)) [ ((1−))

0 ( ((1−)))]

or

() = −0 () + 0 ((1−))

[Λ ()] . (21)

Similarly, from (11),

() = −0 () + 0 ((1−))

Λ¡¢. (22)

Combining (21) and (22) with (20), (19) holds, for all 0.

Theorem 3 gives necessary and sufficient conditions for the direction of the

impact of uncertainty on consumption. However, these are based on character-

istics (relative curvature) of the two value functions and not on the primitives

of the games.19 In order to find the link to the primitives of the games, notice

that from (9)

0 () = [1− ( − 1)]0 () (23)

and

0 () = [1− ( − 1)]0 () . (24)

Hence

[Λ ()] = [1− ( − 1)] 0 ((1−)) [0 ( ((1−)) )]

(25)

and

Λ¡¢= [1− ( − 1)] 0 ((1−))

0 ¡ ((1−)) ¢. (26)

This connection, between the value functions and the primitives of the model

proves useful in identifying the role of the model’s primitives in the mechanics

behind the effects of uncertainty on the players’ strategies.

Some of our proofs in the remainder of the paper rely on the existence of a

well-behaved recursive mapping for calculating the equilibrium. This calculation

procedure is the solution technique suggested by Levhari and Mirman [13]. We

next present some key results about this procedure.

19Within the class of games we discuss, explicit value functions can be derived, thus making

the application of theorem 3 straightforward. In our parametric examples in Section 8, we

provide such explicit formulas for the value functions. The necessity part of theorem 3

proves useful in characterizing parameter choices that determine the effect of uncertainty on

strategies.

10

5 The Levhari-Mirman recursive procedure

Levhari and Mirman [13] start from the static, one period, symmetric equi-

librium, where the consumption rates of players are equal to 1 . They use

this strategy in order to form the value function and then they continue with

the two-period problem, calculate the symmetric strategies again, generalizing

the process to the -period problem. In general, if a linear symmetric strat-

egy exists for the -period problem, then a tractable recursive mapping on the

consumption rates () can be constructed, where () denotes the symmetric-

equilibrium consumption strategy of the -period problem. Characterizing the

evolution of (), as increases, is sufficient to characterize the evolution of the

linear-symmetric consumption functions in our class of models.

For = 2 3 , the -period problem of player ∈ 1 in Bellmanform, is

() () = max

≥0

⎧⎪⎨⎪⎩ () +

⎡⎢⎣ (−1)

⎛⎜⎝

⎛⎜⎝− −X=16=

() ()

⎞⎟⎠⎞⎟⎠⎤⎥⎦⎫⎪⎬⎪⎭ ,(27)

for the stochastic case, where () and

(−1) are the - and (− 1)-period

value functions of player , and () is the -period strategy of player . Simi-

larly, in the deterministic case, for = 2 3 , the -period problem of player

∈ 1 is

()

() = max≥0

⎡⎢⎣ () + (−1)

⎛⎜⎝

⎛⎜⎝− −X=16=

()

()

⎞⎟⎠⎞⎟⎠⎤⎥⎦ . (28)

Again, we focus on (interior) linear-symmetric Markov-Nash strategies. For

the stochastic game, ()∗

() = () for all 0, with

() ∈ (0 1)

all and for the deterministic game, ()∗

() = ()

for all 0, with

()

∈ (0 1) all .Assuming

()∗

() = () in 27, the necessary first order condition of the

(+ 1)-period problem of player ∈ 1 is,

−0 () + h1− ( − 1)()

i 0 ()

h0

³() ()

´i= 0 (29)

where = − −P=16=

() and () is the strategy of a player 6= .20

20This necessary condition is derived from (27) following the same steps as in the infinite-

horizon case, i.e., after applying the envelope theorem on (27). Notice that 29 does not restrict

the player to linear strategies in + 1.

11

Therefore, linear symmetric equilibrium strategies, (+1) , for the (+ 1)-

period problem, solve,21

Ψ³(+1) ()

´= 0, (30)

where

Ψ³ ()

´= () +

³ ()

´, (31)

with () = −0 (), and

³ ()

´=

h1− ( − 1)()

i 0 ((1−))

h0

³() ((1−))

´i

It follows that, due to the assumptions 00 0 and 00 ≤ 0,

Ψ1

³ ()

´ 0 , Ψ2

³()

´ 0 ∀ 0, ∈

µ01

¶ () ∈

µ01

¸.

(32)

Therefore, if (+1) , the solution to (30) exists, and lies in the open interval

(0 1), after applying the implicit function theorem to (30),

(+1)

()

= −Ψ2

³(+1)

()

´Ψ1

³(+1)

()

´ 0 , ∀ 0, () ∈µ01

¸. (33)

The same remarks hold for the deterministic case, for the deterministic game.

Given ()

, (+1)

is the solution to

Ψ³(+1)

()

´= 0, (34)

where

Ψ³

()

´= () +

³

()

´, (35)

with

³

()

´=

h1− ( − 1)()

i 0 ((1−)) 0

³()

((1−))´.

The two conditions, (30) and (34) define two recursive mappings.

Definition 4 Consider h Θi and ­ ® The mapping : [0 1 ] →[0 1 ] is given by Ψ ( () ) = 0. The mapping : [0 1 ]→ [0 1 ]

is given by Ψ ( () ) = 0.

21As above, with function (·), (30) must be met for all 0 in the case of linear-

symmetric strategies, so the function Ψ (·) does not depend on in equilibrium (i.e., when the

expression given by (31) is evaluated at(+1)

()

, = 1 2 ). Even if the expression

given by (31) depends on whenever (30) is not met with equality (i.e., when this expression

is evaluated at some

()

, with 6=

(+1) , = 1 2 ), this potential dependence on

does not affect our analysis, so we discard for the sake of simplicity.

12

The following results provide a characterization for these two mappings. In

particular, Lemma 5 shows the importance of imposing an Inada condition on

the utility function for obtaining interior solutions (this Inada condition is a

sufficient condition).

Lemma 5 If lim→0

0 () =∞, for all (1) ∈ (0 1 ] and all (1) ∈ (0 1 ], thesequences

n()

o∞=2

generated by the mapping , andn()

o∞=2

generated

by , are such that () and

()

are unique with ()

()

∈ (0 1), =2 3 .

Proof. See Appendix.

Lemma 5 leads to a proof of the following result:

Theorem 6 Consider h Θi∞ () and­

®∞

() and suppose

the one-period games have a symmetric equilibrium strategy, (1) =

(1)

= 1

If lim→0

0 () =∞, then starting from (1) =

(1)

= 1 , the recursive mappings

and are convergent with lim→∞

() = and lim

→∞()

= .

Proof. See Appendix.

Theorem 6 shows that the procedure that Levhari and Mirman [13] sug-

gested and implemented in their example, leads to recursive computability

of the infinite-horizon strategies for a more general class of games, i.e., the

class of games h Θi∞ () and­

®∞

() which in addition

have a unique symmetric equilibrium in the stage game, and which satisfy

lim→0 0 () = ∞. While this result is interesting in its own right (as we haveidentified a reliable calculation procedure), some properties of the mappings

and are useful in the analysis of uncertainty. Lemma 7 states these

properties of and .

Lemma 7 Consider h Θi∞ () and­

®∞

() and suppose the

one-period games have a symmetric equilibrium strategy, (1) =

(1)

= 1 If

lim→0

0 () =∞, then for any interval f = [ ] ⊆ (0 1 ],

() ≥ and () ≤ ⇒ ∈ f , ⇒ ∈ ( )

and

lim→∞

() = for all (1) ∈ f .

Moreover, for any interval f = [ ] ⊆ (0 1 ], () ≥ and () ≤ ⇒ ∈ f ,

⇒ ∈ ( ) ,and

lim→∞

()

= (1) ∈ f

13

Proof. See Appendix.

Lemma 7 is crucial for the comparisons that follow. Specifically, in com-

paring two distinct models (e.g. the stochastic and the deterministic, or two

models with different stochastic structures), we can view the solution to one

model as a starting point for calculating the solution to the other model. If this

starting point drives the necessary condition of the second model to be positive

or negative, then we can identify the direction in which the strategy must be

updated. The results in Lemma 7, yield a method for identifying where the

fixed point (infinite-horizon equilibrium) of the second model lies.

6 Uncertainty and strategic decisions

With Lemma 7 we are able to identify the primitive features of the model

that are responsible for the impact of uncertainty on strategies. For games that

satisfy the conditions of the lemma, we show that the result of Theorem 3 hinges

on features of the utility function, , alone, and not on . Of course, plays

an implicit role, since features of both and interact for the game to belong

to this class of games. Nevertheless, our results identify simple conditions on

that lead to specific effects of uncertainty on strategies.

In addition to analyzing the comparison between the stochastic and the

deterministic game, we study the effect of changes in risk under (i) second-order

stochastic dominance (SOSD) change, and, (ii) first-order stochastic dominance

(FOSD) change in the distribution of the shock. We find that, for the class of

games we study, only the structure of the utility function is needed to explain

the impact of changing risk on strategies. In particular, in the case of first-

order stochastic dominance, it is the coefficient of relative risk aversion that is

responsible for the effect of changes in risk on strategies.

6.1 The tragedy of commons and uncertainty: a second

look

Theorem 8 Suppose games h Θi and ­ ® satisfy the conditions of the-orem 1 and consider h Θi∞ () and

­

®∞

() Further suppose

the one-period games have a symmetric equilibrium strategy, (1) =

(1)

= 1

If lim→0

0 () = ∞, then: (i) if and only if () is strictly convex, (ii)

if and only if () is strictly concave, and (iii) = if and only if

() is affine, where

() = 0 () (36)

Proof. See Appendix

Theorem 8 implies that, for the class of games with linear symmetric equi-

librium strategies, the model’s characteristics behind the result of Theorem 3

are given solely by a condition that pertains to the utility function, . This does

14

not mean that does not play any role in the linkup between uncertainty and

strategic behavior. Together with , the function is crucial for placing a game

h Θi in the class of games with linear strategies. The implication of Theorem8 is that, in this class of games, the effect of uncertainty on strategic behavior is

determined by a condition on the utility function alone. Moreover, notice that

Theorem 8 does not require that (·) be thrice continuously differentiable.22We provide two additional characterizations based on comparisons of games

using the notions of second- and first-order stochastic dominance.

6.2 Resource Exploitation and SOSD

We examine two games, h Θi andD Θ

E, that have linear-symmetric

equilibrium strategies, and , with shocks denoted by ∼ Θ () and ∼Θ³´, and such that one shock is riskier than the other, in the sense that

one distribution dominates the other with respect to Second Order Stochastic

Dominance. Recall:

Definition 9 Let two random variables, and ,defined on a common prob-

ability space, with both supports being subsets of ⊆ R+. Then second-order

stochastically dominates , or ¹ , if and only if

[ ()] ≥ h³´i

for all concave functions .

Theorem 10 provides conditions that dictate the effect of increasing risk on

strategic decisions.

Theorem 10 Suppose games h Θi andD Θ

Esatisfy the conditions of

theorem 1 and consider h Θi∞ () andD Θ

E∞

() Further sup-

pose the one-period games have a symmetric equilibrium strategy, (1) =

(1)

=

1 and lim→0

0 () =∞. If ¹ then: (i) if () is strictly convex, then

, (ii) if () is strictly concave, then , and, (iii) if () is

affine, then = , where () is as defined in 36.

Proof. See Appendix

6.3 Resource Exploitation and FOSD

We examine two games, h Θi andD Θ

E, that have linear-symmetric

equilibrium strategies, and , with shocks denoted by ∼ Θ () and ∼Θ³´, and such that one distribution dominates the other with respect to First

Order Stochastic Dominance. Recall:22For a discussion of this point see Mirman [15] (page 182) in his analysis of a two-period

problem.

15

Definition 11 Let two random variables, and , in a common probability

space, with both supports being subsets of ⊆ R+. Then first-order stochas-

tically dominates , or ¹ , if and only if23

[ ()] ≥ h³´i

for all non-decreasing functions .

Theorem 12 Suppose games h Θi andD Θ

Esatisfy the conditions of

theorem 1 and consider h Θi∞ () andD Θ

E∞

() Further sup-

pose the one-period games have a symmetric equilibrium strategy, (1) =

(1)

=

1 and lim→0

0 () =∞. If ¹ then:

0 () T 0 for all 0⇒ T . (37)

Proof. See Appendix

Remark 13

0 () T 0 0⇔ −00 ()

0 ()S 1 0

i.e. the sufficient condition (37) is tightly linked with the value of the coeffi-

cient of relative risk aversion.

For the remainder of the paper we study our results in the context of specific

parametric examples. We first review the Levhari-Mirman [13] example in the

presence of uncertainty. Then we present our extended example which shows

that the Levhari-Mirman example can, not only, be extended but also can be

shown to have a knife-edge property which is not always valid in the extended

example.

7 The stochastic Levhari-Mirman [13] model

Using the Levhari-Mirman [13] functions, namely () = ln () and () = ,

the value function of each player in the stochastic model is of the form,

() =

1− ln () + , (38)

whereas the value function in the deterministic case is,

() =

1− ln () + , (39)

where and are constants. So,

Λ () = Λ () =

1− for all 0 .

23Letting and be the distribution functions of and ,respectively, then ¹

if and only if () ≥ () for all ∈ .

16

Theorem 3 implies that = . In fact,

= =1−

(1− ) + . (40)

Therefore, the presence of uncertainty does not alter the rate of consumption

in the Levhari-Mirman [13] model. The difference between the stochastic and

the deterministic model is that in the stochastic case the state variable evolves

randomly and approaches a long-run stationary distribution. Consequently, the

payoffs of players are also random in each period.

With the aid of Theorem 3 we show that the Levhari-Mirman model is a

knife-edge case. We do this by extending the Levhari-Mirman model to a class of

models for which uncertainty affects strategies and then showing that it is only

in the Levhari-Mirman model that strategies are not affected by the presence of

uncertainty, i.e., the Levhari-Mirman model is a knife edge. This extension to a

more general “Great Fish War” model allow us to study the effect of uncertainty

on the strategies in a game with uncertainty. This task is undertaken below.

8 An extended example

Consider the case where,

() =1−

1 − 1

1− 1

, (41)

and

() =h1−

1 + (1− )1−

1

i −1

, (42)

with 0, ≥ 0 and ∈ (0 1]. Notice that for = 1, () = ln () and

() = 1−, i.e. the Levhari-Mirman model. A similar example, applied

to problems of Cournot oligopoly, has been presented in Koulovatianos and

Mirman [11].24

24For a game with linear symmetric strategies, take (41) and consider a general CES pro-

duction function

() =

1− 1 + (1− )

1− 1

−1

.

From the general first-order conditions of a game with linear strategies,

0 () = [1− ( − 1)] 0 ((1−))0 ( ((1−)) )

(see (5) and (9)). Using (41) and the CES production function,

− 1 = [1− ( − 1)]

1(1−)

− 1

− 1

1− 1

,

with ≡ ((1−)). Setting = is the only way to obtain linear strategies with these

functions. The single-controller version of this example with uncertainty (i.e. = 1), is

similar to this presented by Benhabib and Rustichini [3].

17

8.1 Existence of linear-symmetric equilibrium strategies

As in Stachurski [21], we identify a single sufficient condition on the mean of

the distribution of the transformed random variable, 1− 1

, in order that the

stochastic equilibrium be well-defined.

Proposition 14 If and are given by (41) and (42) respectively, and Θ is

such that,

³1−

1

´≡

1

and [ ()]

1− 1 ≡

1

, (43)

then satisfying,

=1

h(1−)

1 − (1−)

i, (44)

and satisfying,

=1

h(1−)

1 − (1−)

i, (45)

define unique linear-symmetric strategies that are Markov-Nash equilibrium strate-

gies of the infinite-horizon stochastic game and the deterministic game, respec-

tively.

Proof. Theorem 1, by simple substitution we can see that Φ ( ) takes the

form,

Φ ( ) =1−

1 (1−)

− 1

1− 1

n(1−)

1 − [(1− ( − 1))]

o1−

1+ () ,

where () does not depend on . Therefore, if (1−)1− [(1− ( − 1))] =

0 for a unique ∈ (0 1) then the sufficiency part of Theorem 1 can be ap-

plied. This condition is clearly equivalent to 44. By analogous argument, in the

deterministic case the sufficiency part of theorem 1 is satisfied if there is a unique

∈ (0 1) satisfying condition 45. Moreover, it is easy to verify that (44)is equivalent to Ψ ( ) = 0 and that (45) is equivalent to Ψ

( ) = 0.

Now it remains to show that and are unique, and also that the optimiza-

tion problem of each player is well-defined. From the two value functions,

() =

1− 1

1− (1−)1− 1

1−1 − 1

1− 1

+ , (46)

() =

1− 1

1− (1−)1− 1

1−1 − 1

1− 1

+ , (47)

18

where and are constants, it is immediate that if = Ω ∈ (0 1) and = Ω ∈ (0 1), then the problem of each player is well-defined. To show

that Ω ∈ (0 1), we examine two cases.Case 1: 1

Focusing on aggregate consumption rates, Ω ≡ , we express (44) as,

=Ω

(1−Ω) 1 − (1−Ω)≡ (Ω) . (48)

Here,

(0) = 1 and (1) =∞ ,

while

0 (Ω) =

1−(1− 1 )Ω

(1−Ω)1−1

− h(1−Ω) 1 − (1−Ω)

i2 0 , for all Ω ∈ (0 1) .

To see the last statement, notice that 1−³1− 1

´Ω ≥ (1−Ω)1− 1

for all Ω ∈[0 1), with equality if and only if Ω = 0. So, applying the intermediate-value

theorem to (48) shows that Ω ∈ (0 1), a unique symmetric equilibrium in linearstrategies. This case is depicted in Figure 1. For Ω ∈ (0 1) and uniqueness,replace with in the above argument.

Case 2: 1

For this case it is useful to express (44) as,

Γ (Ω) ≡

Ω=

(1−Ω) 1 − (1−Ω)≡ Ξ (Ω) . (49)

Where Γ (0) =∞ and Ξ (0) = (1− ), and

Ξ (Ω) 0 if Ω ∈h0 1− ()

1−i.

Due to (43) this interval is nonempty. Moreover,

Ξ0 (Ω) =

1(1−Ω) 1−1 − h

(1−Ω) 1 − (1−Ω)i2 0 for all Ω ∈

h0 1− ()

1−i

as 0 (1−Ω) 1 − (1−Ω) h1(1−Ω) 1−1 −

i(1−Ω) for all Ω ∈h

0 1− () 1−i. Finally,

Ξ³1− ()

1−´=∞ .

Given that Γ³1− ()

1−´∞ and Γ0 (Ω) 0 for all Ω ∈

h0 1− ()

1−i,

it follows that Ω ∈³0 1− ()

1−´⊂ (0 1), and it is unique. This is shown

19

in Figure 2. For Ω ∈ (0 1) and the uniqueness of it replace with in the

above argument.

For the last case of = 1, see Section 7 above, to verify that (40) satisfies

both (44) and (45), and also that Ω = Ω ∈ (0 1).Note that Proposition 14 shows that the Levhari-Mirman model is indeed a

knife edge case for = 1 in our extended example. According to our analysis

in Section 7, in the case of = 1 of the extended example, uncertainty plays no

role.25 However, for other values of , uncertainty plays a major role .

Clearly our paremetric example satisfies the conditions identified in Section

5, i.e. lim→0

0 () = ∞, and also the one-period games have a unique symmetricequilbrium strategy,

(1) =

(1)

= 1 Furthermore, substituting (41) and

(42) into the necessary conditions of the finite-horizon problem given by (30)

and (34), we find,

() =1

+ () [1−(−1)]

(50)

() =1

+¡

¢ [1−(−1)]

(51)

It is straightforward to verify from (50) and (51) that () () ∈ (0 1)for all ∈ (0 1 ]. Therefore, we can apply Theorems 3, 8, 10 and 12 to ourmodel.

8.2 Impact of uncertainty on strategies

We can apply Theorem 3 as follows. Let = ((1−)), and recall the

definitions of Λ and Λ given in Theorem 3. Notice that

[Λ ()] =

1− 1

1− (1−)1− 1

1− 1

. (52)

while

Λ¡¢=

1− 1

1− (1−)1− 1

1− 1

. (53)

From (44),

1− 1

1− (1−)1− 1

=

µ1

−

¶ 1

,

so substituting this expression into (52),

[Λ ()] =

µ1

−

¶ 1

1− 1

. (54)

25Notice also that, for = 1, (46) and (47) yield (38) and (39).

20

After using (45), as was done with (44), (53) becomes,

Λ¡¢=

µ1

−

¶ 1

1− 1

. (55)

Proposition 15 identifies the parameters that are behind the comparison given

in Theorem 3.

Proposition 15 If and are given by (41) and (42), and Θ is such that

(43) is met, then

S 1⇔ S .

Proof. By applying the implicit function theorem to (48) and (49) and with

the aid of Figures 1 and 2, it follows that

= () and = ¡¢,

where 0 () 0 for all ∈ (0 1 ()). So,

S ⇔ ³1−

1

´T [ ()]1−

1 ⇔ S 1 , (56)

as the relationship between and 1 determines whether the function () =

1−1 is concave, convex or affine (in the present setting, when = 1, () is

equal to unity for all 0). The result follows from Jensen’s inequality.

In our example,

() = 0 () = 1−1 ⇒

(0 () T 0⇔ T 100 () S 0⇔ T 1 . (57)

Since is thrice differentiable, the concavity of can be examined through

the sign of its second derivative. Notice that, (57) illustrates the connection

between Theorems 8, 10 and 12 and Theorem 3, as well as, with the result of

Proposition 15. Most importantly, it shows how the properties of can affect

the role of uncertainty on strategies.

8.3 Changes in risk and the tragedy of the commons

In this subsection we investigate whether increasing risk has an impact on the

rate at which aggregate exploitation rates increase with the number of players.

This investigation requires a comparison of aggregate exploitation rates along

two dimensions, the number of players and the degree of riskiness. In perform-

ing such a comparison, if we employ two random variables, and , with a

discrete difference in the degree of riskiness, the comparison becomes unclear,

as the initial aggregate exploitation rates can also be discretely different before

increasing the number of players. For this reason, we employ a concept of chang-

ing risk that allows for marginal increases in riskiness. Consider a lognormally

distributed shock,

ln () ∼

µ− 2

2 2¶. (58)

21

The expectation of is, () = , whereas its variance is, () = 2³

2 − 1´,

i.e., the parameter has an impact only on the variance of and not on its

mean (but the parameter has an impact on both the mean and the variance

of ). So, any two distributions given by (58) with different values of parameter

are linked through second-order stochastic dominance (increase in risk). In

particular, if ∼ Θ () with parameters ( ) and ∼ Θ³´with parameters

( ), then implies that ¹ ( represents an increase in risk from

).

Proposition 16 examines the impact of increases in risk on the intensity of

the tragedy of the commons.

Proposition 16 If and are given by (41) and (42), Θ obeys (58) and

condition (43) is met, then, (i) the tragedy of the commons is amplified by an

increase in riskiness if and only if 1, (ii) the tragedy of the commons is

mitigated by an increase in riskiness if and only if 1, (iii) the tragedy of the

commons is unaffected by an increase in riskiness if and only if = 1.

Proof. See Appendix

Proposition 16 shows that, if, 1, the overexploitation tendency is mit-

igated as both riskiness and the number of players increase. Note, however,

Theorem 12 states that when 1, for a fixed number of players, all play-

ers would tend to increase their consumption rates as uncertainty increases.

These results indicate the complex, yet interesting, strategic behavior in Nash

equilibrium outcomes.

9 Appendix

Proof. [Lemma 5] Fix any 0 and any () ∈ (0 1 ]. Then, from (31),

Ψ³

()

´= () + () where

() 0 and () 0 for all ∈ (0 1) , andlim→0

() = −∞ and lim→1

() =∞

so Ψ³

()

´intersects the zero axis within the interval (0 1) at least

once. Yet, the fact that, for all ∈ (0 1), 0 () 0 and 0 () 0, impliesthat Ψ

³

()

´= 0 has a unique solution, (+1) ∈ (0 1). The same

argument can be used for the deterministic analogue of the stochastic model.

Proof. [Theorem 6] Fix 0 and consider the stochastic game. Using the

same argument as in the proof of Lemma 5, starting from (1) there exists a

unique (2) ∈ (0 1), i.e. (2)

(1) . Moreover,

(2) (1) =1

. (59)

22

In order to show (59), suppose that (2) ≤ . Since Ψ

1

³(+1)

()

´ 0,

Ψ2

³(+1)

()

´ 0, and Ψ ( ) = 0, 0 = Ψ

( ) Ψ³

(1)

´≥

Ψ³(2)

(1)

´, which contradicts that Ψ

³(2)

(1)

´= 0. From Lemma 5,

(1) = 1 gives rise to a unique sequence

n()

o∞=1

that is generated from

(30). Given that ∈ (0 1) is unique, and the mapping is a continuous

function with 0

³()

´= − Ψ

2(

(+1) () )

Ψ1

(+1)

()

0 for all 0, (59) and the

intermediate value theorem imply that (+1)

() , = 1 2 . There-

fore the sequencen()

o∞=1

converges and lim→∞

() = The same argument

holds for the deterministic analogue of the stochastic model.

Proof. [Lemma 7] Fix any 0 and let f = [ ] ⊆ (0 1 ] with () ≥ and () ≤ . Since is a continuous function with

0

³()

´= − Ψ

2(

(+1) () )

Ψ1

(+1)

()

0 for all 0, the intermediate value theorem

implies that ∈ f, since is by assumption unique. As () ≥ ,

strict equality holds only if = . If () , then From

Lemma 5, any (1) ∈ (0 1 ] gives rise to a unique sequence

n()

o∞=1

, so

()

(+1) , = 1 2 , for all

(1) ∈ [ ). Thus, is stable for

all (1) ∈ [ ). By a similar argument, if () ,

()

(+1) ,

= 1 2 , for all (1) ∈ ( ], so is stable for all

(1) ∈ ( ]. Finally,

for () ≤ , equality holds only if = , completing the proof. The

same argument can be used for the deterministic analogue of the stochastic

model.

Proof. [Theorem 8] Fix any 0. Ψ¡ ()

¢and Ψ

¡()

¢can be

expressed as,

Ψ³ ()

´= −0 () (60)

+

£1− ( − 1)()¤

() 0 ((1−))

((1−))h³() ((1−))

´iand

Ψ³ ()

´= −0 () (61)

+

£1− ( − 1)()¤

() 0 ((1−))

((1−))³() ((1−))

´From (60), (61), and Jensen’s inequality,

(a) Ψ¡ ()

¢ Ψ

¡ ()

¢for all () ∈ (0 1), if and only if, (·)

is strictly convex,

23

(b) Ψ¡ ()

¢ Ψ

¡ ()

¢for all () ∈ (0 1), if and only if, (·)

is strictly concave,

(c) Ψ¡()

¢= Ψ

¡ ()

¢for all () ∈ (0 1), if and only if, (·)

is affine.

So, in case (a), Ψ ( ) Ψ ( ) = 0, so () . In the proof

of Theorem 6 it is shown that (1) 1 . Then, by Lemma 7, ∈( 1), which proves statement (i). In case (b), Ψ

( ) Ψ ( ) =

0, so () . Since (1) 1 (from the proof of Theorem 6),

Lemma 7 implies that ∈ ( 1), which proves statement (ii). Finally,statement (iii) is straightforward.

Proof. [Theorem 10] Fix any 0. The necessary conditions of the two

problems, Ψ¡ ()

¢and Ψ

¡ ()

¢, can be expressed as,

Ψ³ ()

´= −0 () (62)

+

£1− ( − 1)()¤

() 0 ((1−))

((1−))h³() ((1−))

´iand

Ψ³ ()

´= −0 () (63)

+

£1− ( − 1)()¤

() 0 ((1−))

((1−))h³() ((1−))

´i.

Using the expressions (62) and (63), and Definition 9,

(a) if (·) is strictly convex, then Ψ ¡ ()¢ Ψ ¡ ()¢ for all () ∈(0 1),

(b) if (·) is strictly concave, then Ψ ¡ ()¢ Ψ ¡ ()¢ for all () ∈(0 1),

(c) if (·) is affine, then Ψ ¡ ()¢ = Ψ ¡ ()¢ for all () ∈ (0 1).So, in case (a), Ψ ( ) Ψ

( ) = 0, so () . In the proof

of Theorem 6 it was shown that (1) 1 . By Lemma 7, ∈ ( 1),which proves statement (i). In case (b), Ψ ( ) Ψ ( ) = 0, so

() . Since (1) 1 (see the proof of Theorem 6), Lemma

7 implies that ∈ ( 1), which proves statement (ii). Finally, statement(iii) is straightforward.

Proof. [Theorem 12] Integration by parts yields,

[ ()]−h³´i=

Z

hΘ ()−Θ ()

i0 ()

for all differentiable functions . So, setting () = ¡() ((1−))

¢,

for any () ∈ (0 1 ] and any ∈ (0 1), the fact that ¹ implies,

0 () T 0 ∀ 0⇒ h³() ((1−))

´iS

h³() ((1−))

´i.

(64)

24

Based on (64), (62), and (63),

(a) 0 () 0 for all 0⇒ Ψ ( ) Ψ ( ) = 0,(b) 0 () 0 for all 0⇒ Ψ ( ) Ψ ( ) = 0,(c) 0 () = 0 for all 0 ⇒ Ψ

¡ ()

¢= Ψ

¡ ()

¢= 0 for all

() ∈ (0 1).The rest of the proof follows exactly as in Theorem 10.

Proof. [Proposition 16] We denote the aggregate exploitation rate as

Ω () = (). Equation (18) implies (after some algebra) that

Ω ()

=

() () ()−1

h1− −1

() ()

−1i . (65)

where

() ≡h

³ ()

1− 1

´i,

and

() ≡ [1− ( − 1) ()] .Equation (65) can be re-written as,

Ω()

Ω ()=

ln [Ω ()]

=

() ()−1

2

h1− −1

() ()

−1i . (66)

Taking the partial derivative of this last expression with respect to , yields,

2 ln [Ω ()]

=

0 () ()−2h () + ( − 1) () 0()

0()

i2

h1− −1

() ()

−1i2 . (67)

Since,0 () 0 ()

= ()

(),

we find () (), by applying the implicit function theorem to (44). In

particular, (44) can be written as,

− 1

() ()+1

− () = 0 , (68)

so,

0 () 0 ()

=−1

()

1− −1

() ()−1 . (69)

After substituting (69), the expression () + ( − 1) () 0()0() on the RHS of

(67), becomes,

() + ( − 1) () 0 () 0 ()

= ()− −1

() ()

1− −1

() ()−1 .

25

Yet, (68) implies that, ()− −1

() ()= 1

, so,

() + ( − 1) () 0 () 0 ()

=1

h1− −1

() ()

−1i ,

and (67) gives,

2 ln [Ω ()]

=

0 () ()−2

3

h1− −1

() ()

−1i3 . (70)

As we have found in Theorem 2,

Ω ()

=

() () ()−1

h1− −1

() ()

−1i 0⇒ 1− − 1

() ()

−1 0 ,

so, (70) implies that,

2 ln [Ω ()]

T 0⇔ 0 () T 0⇔ S 1 ,

which proves the Proposition.

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28

Figure 1 Equilibrium strategies when 1>η

0 χ1

N

( )NNg ** =χ

( )χH

Figure 2 Equilibrium strategies when 1<η

0 χ1( )NNg ** =χ

( )χΞ

( ) ηη

αδζ −− 11

( )χ

χ N=Γ