A Mechanism for Inducing Cooperation in Non-Cooperative

Environments: Theory and Applications.

Christopher J. Ellis and Anne van den Nouweland

Department of Economics, University of Oregon, Eugene,

OR 97403, USA.

E-mail: cjellis@uoregon.edu, annev@uoregon.edu.

May 2004.

For many helpful discussions and comments we wish to thank Emilson Silva, Bill Harbaugh, Richard Cornes,

Art Snow, Jurgen Eichberger, and the participants of the Public Economic Theory Conference held at the Uni-

versity of Alabama in May 1998.

1

Abstract.

We construct a market based mechanism that induces players in a non-cooperative game to make the same

choices as characterize cooperation. We then argue that this mechanism is applicable to a wide range of

economic questions and illustrate this claim using the problem of ”The Tragedy of the Commons”.

JEL: C72, D62, H40.

Keywords: Cooperation, Externalities, Spillovers, Efficiency.

Running Head: Inducing Cooperation

Proofs: Professor Christopher Ellis, Department of Economics, University of Oregon, Eugene, OR 97403, USA.

2

1 Introduction.

There are many examples in economics where, in either a Paratian or welfare sense, cooperative behavior by

economic agents yields outcomes that are significantly superior to those achieved under non-cooperation. However,

the problem that is frequently encountered is that cooperation cannot be straightforwardly sustained in a non-

cooperative environment. When others are behaving cooperatively non-cooperative behavior often yields an

individual agent significant gains. Perhaps the most well-known example of this problem is the prisoners’ dilemma,

although any economic activity that generates external effects, or has some degree of ”publicness” attached to it

typically suffers from similar problems. There are of course many well known methods by which agents may be

induced to internalize an externality, or achieve an efficient allocation of goods some of which are characterized

by a degree of publicness. These solutions include; (i) Pigouvian taxes or subsidies; (ii) quantity rationing; (iii)

tradable quotas; (iv) limiting the number of participants in an activity or market; (v) creating appropriate private

property rights; and (vi) promoting collusive welfare maximization. Each of these solutions work in the right

circumstances1. However, each has well-known problems.

The purpose of this paper is to propose a new market based mechanism that induces rational self-interested

agents in a non-cooperative environment to make the same choices as characterize the cooperative outcome. The

mechanism, once introduced, does not promote overt cooperation, no explicit collusion or joint decision making

takes place, rather it provides the appropriate incentives for non-cooperative agents to act exactly as if such

activities had occurred. The intuition behind the mechanism we propose is straightforward. Suppose we view an

economic activity as a game in which a number of players choose actions which then generate payoffs according to

a given payoff matrix. If the mechanism has the characteristic that each player’s final individual payoff is strictly

increasing in the sum of the payoffs from the economic activity, then each has an incentive to choose the action

that is jointly maximizing. Thus each behaves as if they are cooperating. To construct a mechanism that can

achieve this result is quite simple to do. Suppose we view the economic activity as a two stage game2. In the

first stage the players make choices that generate payoffs, these payoffs are then redistributed in the second stage

1See for example the discussion in Cornes and Sandler [3].2Guttman and Schnytzer [6] propose that reciprocal externality problems may be solved in a two stage strategic matching game.

In the first stage the players precommit to matching rates that linearly link their externality causing actions. In the second theychoose the actions. Both stages are played non-cooperatively. The resulting allocation is Pareto efficient.

3

according to some prespecified mechanism. Clearly any mechanism with the property that each player receives a

proportion of the total payoff will generate the incentives necessary for total payoff maximization in the first stage.

The purpose of this paper is to propose one specific mechanism that has this requisite property. The mechanism

we propose is not operated by an outside party determining a redistribution of the payoffs. Rather, it is a game

played by agents who are trying to ’claim’ as much of the payoffs as possible. We argue that non-cooperative

agents may be given the incentives to make the cooperative choices by the creation of a new type of property

right, termed a ”pooling property right”. The key characteristic of a pooling property right is that it may be

used to claim a share of any payoff in a game. This might work as follows. Consider a two stage game in which

two players each hold pooling property rights prior to any play of the game. In the first stage the players play a

classic prisoners’ dilemma. In the second stage the players play a non-cooperative game in which they allocate

their pooling property rights to the payoffs generated in stage 1. Each payoff is then redistributed between the

players in the same proportions as the share of the property right they have allocated to that payoff. We are

able to show that the Nash equilibrium in this second stage of the game has the property of proportionality. It

follows that in a subgame perfect equilibrium the players have an incentive to maximize total payoffs in the first

stage. The outcome described as cooperative in a one-shot prisoners dilemma game becomes a non-cooperative

equilibrium in this two stage game.

We concentrate on pooling property rights as the mechanism for achieving the cooperative outcome both

because of their theoretical appeal, and because they may have a wide range of policy implications. Theoreti-

cally any problem of externalities may be thought of as arising because there is no market for the externality.

Interestingly in our analysis the second stage of the game operates by introducing an extra market, however

this is not directly a market for the externality. Further, we are able to show that even with a small number

of players our model works as if there is a market for the externality which is perfectly arbitraged. This is the

key theoretical contribution of our paper. It is well known that if the players of a game receive fixed shares

of the total payoff then cooperation and thus efficiency will result, our paper provides a decentralized (market)

mechanism for achieving this. From a policy perspective our results may be applicable to real world externality

problems. Suppose that a pair of firms that imposed external costs or benefits on each other swapped standard

equity for pooling equity, shares that represent claims on either (but not both) firms. This, which amounts to

4

little more than a relabeling exercise, would generate the same setup as in our theoretical model. Notice that this

mechanism may be introduced without the need to create or purchase any new property right or to agree on any

type of sharing scheme, thus it may avoid many of the pitfalls of alternative solutions. We view this as the major

advantage of our mechanism over simpler ones such as those that involve specifying for each player a fixed share

of the first stage payoffs.

The rest of our paper is organized as follows. In section 2 we present a general model of an N player two

stage non-cooperative game and demonstrate formally the results discussed above. In section 3 we discuss our

mechanism and how it fits into the literature. In section 4 we apply our mechanism to the well known problem of

the tragedy of the commons. We demonstrate how our mechanism induces cooperation and produces an efficient

solution. Finally in section 5 we supply a conclusion.

2 The Model.

In this section we develop a formal model of our cooperation inducing mechanism. This consists of a two-stage

game in which each of the two stages consists of a (non-cooperative) game in strategic form.

Stage 1: The players play a non-cooperative game Γ =­M ; (Ak)k∈M ; (pk)k∈M

®with player set M = {1, 2, ...,m} in

which each player k ∈ M has an action set Ak and a positive payoff function pk. If the players choose an

action tuple a = (ak)k∈M ∈ (Ak)k∈M in this game, then every player k ∈M generates a payoff pk(a) > 0.

Stage 2: There is a player set ofN = {1, 2, ..., n}, where n ≥ m and n ≥ 2. We assume each player i ∈ N holds positive

property rights (shares) Si, which may be used as claims on the m positive payoffs p1(a), p2(a), ..., pm(a)

that result from the first stage of the game. Denote the shares contributed by player i to payoff pk(a) by

sik. A strategy for player i in the second stage is a set of contributions¡sik¢mk=1

such that sik ≥ 0 for each

k andPm

k=1 sik = Si. If the players all determine their contributions, then for each payoff pk(a) player i

gets the share sikPj∈N sjk

of pk(a) ifP

j∈N sjk > 0, i.e. if a positive amount of property right is contributed to

pk(a), and ifP

j∈N sjk = 0 then payoff pk(a) is divided among the players in some predetermined way (we

will show that in equilibrium there will never be a payoff to which no property rights are contributed, no

matter what the division). At the end of the game, every player i ∈ N has a payoff ofPm

k=1sikPj∈N sjk

pk(a).

5

Notice that M ⊆ N, all players who play the first stage also play the second, but some players play only the

second stage. Those that play both stages may be thought of as both owners and producers, those that play only

the second stage are pure owners.3 Having defined the game, we may now proceed to solve it for its subgame

perfect Nash equilibria.4 Consequently we apply backward induction and analyze first the second stage of the

game, and then use the equilibria from the second-stage (sub)games to solve for optimal behavior in the first

stage of the game.

2.1 The Second Stage.

In this subsection, we solve for the equilibria of the second stage of the game. We show that for every outcome

of the first stage of the game, the second-stage game has a unique Nash equilibrium.

Suppose that in the first stage the action tuple a was played. Then, there would be the payoffs p1(a), p2(a), ..., pm(a)

to which property rights can be applied. To simplify notation and to try and avoid confusion, we will denote

pk(a) by Pk for every k ∈ M . Lower indices correspond to the different payoffs of the first stage that can be

(partially) claimed and upper indices correspond to players in the second stage. We will see later that in a Nash

equilibrium there will be no payoff Pk such thatP

i∈N sik = 0. Anticipating this, we will ignore the possibility

thatP

i∈N sik = 0 for some k in our formulation of the maximization problem that each player faces.

A Nash equilibrium is obtained if all players i simultaneously solve the following maximization problem:

MaximizemXk=1

sikPj∈N sjk

Pk

s.t.mXk=1

sik = Si

and sik ≥ 0 for each k

3We assume all players that play the first stage also play the second. This seems to accord well with the idea of managersrepresenting the interests of their shareholders. In a companion paper Ellis and Van den Nouweland [5] we allow some players to playonly the first stage as agents of those that play the second. The agents exclusively care about their own ”effort”. We find that theoptimal agency contracts combined with the cooperation inducing mechanism induce an efficient allocation.

4Actually, as will become clear in the analysis of the first stage of the game, we consider a subset of the set of subgame perfectequilibria.

6

If a strategy¡sik¢mk=1

solves the maximization problem of player i ∈ N , then there exists a multiplier λi such that

for every k = 1, 2, ...,m

Pj∈N sjk − sik³Pj∈N sjk

´2 Pk =

Pj∈N\{i} s

jk³P

j∈N sjk

´2Pk ≤ λi, with equality if sik > 0 (1)

We shall proceed in two stages, first we shall demonstrate what a Nash equilibrium looks like, provided one

exists, then we shall demonstrate that there exists a unique Nash equilibrium.

Theorem 1. If¡¡sik¢mk=1

¢i∈N is a Nash equilibrium. Then it has the following properties.

1. Each player makes a positive claim on each payoff, i.e. sik > 0 for every i ∈ N and k = 1, 2, ....,m.

2. Each player divides their property rights between the payoffs such that their share in each payoff is in the

same proportions as their share of total property rights, i.e. sikPj∈N sjk

= SiPj∈N Sj for every i ∈ N and

k = 1, 2, ....,m.

3. Each player divides their property rights between the payoffs such that the proportion of their shares allocated

to each payoff is the same as the proportion of that payoff to total payoffs, i.e. sikSi =

PkPml=1 Pl

for every i ∈ N

and k = 1, 2, ....,m.

The proof of this and all subsequent theorems, propositions and lemmas may be found in the appendix.

Theorem 1 may be most easily understood by examining the players incentives to allocate their property rights

across the payoffs. Consider first the third part of the theorem and notice that this may be rewritten

Pml=1 PlSi

=Pksik

.

Cross multiplying this expression, summing over the i ∈ N and then cross multiplying again we obtain

Pml=1 PlPi∈N Si

=PkPi∈N sik

(2)

Expression (2) tells us that in the second stage the payoff per unit of property right is equalized across all first

stage payoffs, and is essentially an arbitrage condition. Given the number of property rights allocated to each

7

payoff no player may reallocate their shares and raise their total payoff. This has the further implication that

more shares are placed on the larger payoffs, and in strict proportion to their size.

Next consider part 2, this tells us that each player holds the same percentage of the shares allocated to

each payoff. This is an optimality condition that tells us that the reallocation of a property right by a player

between first stage payoffs cannot raise their total payoff, this recognizes the effect of the reallocation both on the

numerators and denominators of the relevant terms sikPj∈N sjk

. This is the proportionality property that we have

already claimed will induce cooperative behavior.

Now we know what a Nash equilibrium looks like if one exists. It remains to be shown that there exists a

unique Nash equilibrium.

Theorem 2. Let¡¡sik¢mk=1

¢i∈N be the set of strategies defined by

sik = Si

⎛⎜⎜⎝ PkmPl=1

Pl

⎞⎟⎟⎠ (3)

for every i ∈ N and k = 1, 2, ....,m. This set of strategies is the unique Nash equilibrium of the second stage

of the game. Moreover, for every player i ∈ N his payoff according to the Nash equilibrium is

SiPj∈N Sj

ÃmXk=1

Pk

!.

The unique Nash equilibrium to the second stage of the game is characterized by each player receiving a share

of the sum of the payoffs from the first stage. Further, this share is equal to the ratio of each players’ property

rights to total property rights.

2.2 The First Stage.

We now know that for every action tuple a played in the first stage the subgame played in the second stage has

a unique Nash equilibrium. We next analyze the first stage and solve for the subgame-perfect equilibria of the

two-stage game. The payoffs that were exogenous in the second stage, are determined in the first stage. As we

have seen in our analysis of the second stage, in equilibrium these payoffs will be re-distributed among the players

8

in proportion to their share of total property rights. Hence, after the second stage, player i ∈ N will end up with

a payoff of

SiPj∈N Sj

ÃXk∈M

pk(a)

!.

Since every player in the first stage is also a player in the second, the expression above gives us the payoff that

every player in the first stage expects to get at the end of the second stage. Hence, the players in the first stage

are playing a weighted potential game (cf. Monderer and Shapley [9]). Each gets a share of the total payoff

obtained in the first stage where their shares are determined by their property rights, and are independent of the

(relative) payoffs in the first stage. The incentive therefore is for each to maximize total payoffs rather than their

own payoff in the first stage.

Now, there may be Nash equilibria that do not result in the maximal total payoff obtainable (due to the fact

that it might take more than one player deviating to get to an action tuple with a higher total payoff), but it

seems reasonable to restrict attention to the Nash equilibria that do result in the maximal total payoff possible.

In the terminology of Monderer and Shapley (op. cit.): we consider Nash equilibria that are in the argmax set of

the weighted potential. Monderer and Shapley (op. cit.) point out that the argmax set of a weighted potential

does not depend on the particular weighted potential chosen to represent the game. They also show that every

action tuple that maximizes total payoffs is a Nash equilibrium of the weighted potential game. This shows

that the argmax set of a weighted potential constitutes a well-defined unambiguous Nash equilibrium refinement.

Restricting the set of Nash equilibria to those maximizing total payoffs brings about a sensible refinement of

Nash equilibrium that is well-established in the literature and that has been supported by theoretical as well as

empirical arguments.5

In many important economic examples, the payoff structure of the first-stage game is such that for any action

tuple that does not maximize total payoffs there exists a player who can strictly increase total payoffs through a

unilateral deviation. This is true, for example, for voluntary contribution games and for common pool resource

games. If the first-stage game satisfies this condition, then every Nash equilibrium of the weighted potential game

in which players’ payoffs are given by SiPj∈N Sj

¡Pk∈M pk(a)

¢, results in the maximal total payoff possible. This

5See. e.g., Ui ”Robust Equilibria of Potential Games” Econometrica 69 (2001), 1373-1380, Ermoliev and Flam ”Learning inPotential Games” IIASA Interim Report IR-97-022 (1997), and Van Huyck, Battalio, and Beil ”Tactic Coordination Games, StrategicUncertainty, and Coordination Failure” AER 80 (1990), 234-248.

9

is easily seen as any action profile in which total payoffs are not maximal is not a Nash equilibrium, because

there exists a player who can strictly increase total payoffs and thereby his or her share of these payoffs through

unilateral deviation. Hence, if the first-stage game satisifes the condition mentioned above, then all Nash equilibria

maximize total payoffs and we do not have to resort to an equilibrium refinement to obtain maximal total payoffs.

Stated differently, the refinement of the argmax set of the weighted potential does not have any bite for such

games and selects all Nash equilibria.

2.3 Equilibria of the Two-Stage Game.

Combining the results we obtained so far, we conclude that in potential-maximizing subgame-perfect equilibria of

the two-stage game the players in the first stage choose an action tuple that maximizes the total payoffs from the

game played in the first stage and in which the players in the second stage then play the strategies as described

in Theorem 2. Hence, in such a subgame-perfect equilibrium, the total payoffs from the first-stage game will be

maximized, and in the second stage each player will receive a share of this amount as determined by his proportion

of the property rights. Using total payoff maximization as our definition of cooperation, we have thus described

a mechanism that, once introduced, induces the cooperative outcome in a non-cooperative environment without

any communication, explicit collusion, agreement on or imposition of payoff shares, or joint decision making. This

mechanism is fully decentralized, at each stage the players simply choose their own best replies, cooperation is

not imposed, but rather arises as a consequence of the private incentives generated.6

2.4 Participation in the Scheme.

Participation in the pooling equity scheme is both ex ante and ex post optimal. That is, each firm has an

incentive to join the scheme when decisions on adoption of pooling property rights are simultaneous, and, once

all firms are participating, none has any incentive to quit. Suppose representatives of each firm are told that

all the others are participating and that they too may do so by converting their current equity into pooling

equity on a dollar-for-dollar basis. Consider the problem then faced by the mth firm. Denote as a = (ak)k∈M

6 In a related work Roemer [11] examines how the levels of provision of a public bad are related to how egalitarian is public shareownership. Our work differs from this line of inquiry in that our ”solution” to externality problems has the Coasian property thatthe outcome is independent of the initial distribution of property rights.

10

the actions taken in the first stage of the game in a subgame perfect Nash equilibrium when the mth firm has

not joined the scheme but all others have, and denote as b = (bk)k∈M the actions taken in the first stage of

the game in a subgame perfect Nash equilibrium that maximizes the weighted potential when the mth firm has

joined the scheme. Clearly,P

k∈M pk(a) ≤P

k∈M pk(b), as b maximizes joint payoffs. For the mth firm with

Sm units of outstanding equity before it joins the scheme, the price per unit of this equity is qm = pm(a)Sm

while

the price of each of the S−m pooling equities in the other firms is q−m =P

k∈M\{m} pk(a)S−m

. Should the mth firm

decide to opt into the pooling equity scheme, each of its shares would receive³

qmSmqmSm+q−mS−m

´ Pk∈M pk(b)

Sm. Hence,

participation is ex ante optimal if³

qmSmqmSm+q−mS−m

´ Pk∈M pk(b)

Sm≥ pm(a)

Sm. Now since qmSm = pm(a), this inequality

holds if³

1qmSm+q−mS−m

´Pk∈M pk(b) ≥ 1. But, by definition, qmSm + q−mS−m =

Pk∈M pk(a) ≤

Pk∈M pk(b).

Hence participation is ex ante optimal, and each firm has an incentive to opt in. The intuition is straightforward;

assuming efficient equity markets, the prices of the firms’ shares prior to joining the scheme reflect the market

rate of return. When the mth firm joins the scheme, its shareholders receive pooling equity of value equal to

that which they held in the mth firm. However, since the participation of the last firm brings about actions that

maximize the weighted potential, it must be that there is an appreciation in the price of pooling equity and hence

the equity holders of the mth firm achieve a capital gain from participating.

To explore ex-post optimality we need to consider first what it means for a firm to opt out of the scheme.

Given that there has been a change in property rights such that all shareholders own identical shares that are

claims on any firm, opting out in the simple ownership sense is not possible. However, the following could be

proposed; a production unit might be selected to opt out of the scheme, and any pooling equity holders that

choose may convert their equity to standard claims on the break-away firm. Clearly, in this case equity holders

will convert their claim from pooling equity to claims on the new firm up to the point where the expected value of

claims on those firms remaining in the scheme is equal to the expected value of a claim on the new firm. However,

since the new configuration will not be potential maximizing, all shareholders will lose from this reconfiguration

and so the opt-out option will be rejected. Participating in the pooling equity scheme is thus ex-post optimal.

We also note that a potential-maximizing subgame perfect Nash equilibrium with the pooling equity scheme

Pareto dominates the Nash equilibrium obtained in its absence. This follows because industry profits are maximal

when the scheme is implemented, so they are no higher prior to the implementation of the scheme. Provided that

11

equity markets are efficient, the value of a dollar of equity is equalized across all equity prior to the implementation

of the scheme, as it is after its implementation. So, if equity is converted on a dollar-for dollar basis, each

shareholder will gain from all firms participating in the scheme.

3 Discussion.

The mechanism developed above is perhaps best described as a hybrid of the Coasian and Pigouvian/Mechanism

Design approaches to the problem of externalities. Coase [1] argues that externalities may be dealt with by

recognizing that they are goods over which property rights may be established and whose ultimate disposition

may then be determined by a market mechanism. Clearly our mechanism has Coasian features. The introduction

of pooling property rights that may be used as claims on the payoffs generated as a consequence of the players’

actions creates a market on which the players may trade.

The Pigouvian or Mechanism Design approach, as surveyed for example in Jackson [8], suggests that the

problem of externalities may be solved by designing a mechanism (a game form) such that the incentives of

the players are to choose the efficient outcome. The mechanism chosen needs to be imposed and administered

by the mechanism designer. Typically this involves the players sending ”messages” to the designer who then

responds by requiring actions on the part of the players and enforcing transfers between them. If the transfers are

constructed appropriately, it is an equilibrium for the players to send messages that reveal their true preferences

to the designer, who can then implement desirable outcomes. The exact nature of the mechanism depends on

the specific features of the problem, most particularly the information structure. One strand of the literature,

and the one closest to our analysis, assumes that there is complete information between the players in terms of

knowing each others’ preferences, but the mechanism designer only knows that each player is drawn from some

set of preference types. A classic example of a mechanism designed for this sort of environment is supplied by

Varian [13], which cleverly exploits the designer’s knowledge that the players are well informed.

In our mechanism we similarly assume the players each know each others’ preferences and that the mechanism

designer does not. Indeed, in our mechanism the designer needs only know that the players are playing a

transferable utility game, no knowledge of players’ preferences is required. Further the designer is not required

to administer transfers, these occur automatically (from the point of view of the designer) in the second stage of

12

the game. The role of the designer is to introduce the pooling property rights and then provide for their legal

enforcement. Surprisingly it is not even clear that the designer needs to impose the mechanism, simply proposing

the mechanism may be sufficient. Typically in mechanism design this is not the case, the mechanism needs to be

imposed to avoid the problem of free-riding. Here, as we have shown, the adoption of pooling property right is

a Nash equilibrium to a simultaneous participation game. It is intuitively clear why the incentive to free-ride is

diminished under our mechanism. In the standard set-up, if a player decides to free-ride he receives significant

benefits from the players that choose to participate and incurs none of the costs of participation. However,

because of the sharing nature of our scheme a player who decides to join still faces reduced individual benefits,

but shares in the gains his joining implies for others and gets to share both his own costs and loss of individual

benefits with others. The incentive to free-ride is thus greatly reduced and with it the need for the designer to

impose the mechanism on the players.

It appears we have the best of both worlds. Our mechanism is Coasian with all the advantages that decen-

tralization entails, but is also Pigouvian in that the second stage generates a system of transfers that provide the

incentives for players to behave efficiently.

There are also some interesting connections between the problem we investigate and the problem of moral

hazard in teams as initially analyzed by Holmstrom [7] and subsequently extended by Rasmussen [10] and Strausz

[12]. Consider first the problem in the absence of our mechanism; each player knows that the action tuple a they

choose will not be Pareto efficient, and thus they recognize there are potential gains from cooperation. They

realize they are a team. The actions of each affect the payoffs of others. Without our mechanism we may

think of the team consisting of a group of agents each with their own principal (or each may be considered their

own principal). Each principal recognizes the potential gains from cooperation, but doesn’t find it individually

rational to direct his agent to cooperate. Even if each agreed to choose the efficient action and even if there

was no uncertainty, moral hazard remains because of the team nature of the problem. Once our mechanism is

introduced it as if there is now one principal and many agents (or more precisely many identical principals with

many common agents). Each principal’s payoff is a share of total team production. The problem is then reduced

to one of coordinating on the efficient equilibrium. Given the incentives of the players we assume that this is

achieved via the equilibrium refinement of choosing the Nash equilibrium that maximizes the weighted potential.

13

While we believe our mechanism is in itself theoretically interesting, we also believe that it can be practically

applied to a range of real world problems in which there is some kind of external effect or spillover that requires

internalization. We explore this issue in the next section by applying our theory to Cornes, et. al’s [2] well-known

model of the ”Tragedy of the Commons.”

4 Applications.

Below we illustrate the usefulness of our mechanism by analyzing it’s application to Cornes, Mason and Sandler’s

influential analysis of the ”Common Pool Resource Problem” In this example we demonstrate how in a problem

where there is only one distortion preventing the achievement of an efficient allocation, the application of our

mechanism can achieve a first best equilibrium. In this example the cooperative outcomes may be achieved by a

simple modification of the property rights system. If we assume that the players in the game are shareholder owned

firms, then the problem may be transformed into one where our mechanism operates by the simple expedient of

an equity swap. Holders of standard equity are offered the opportunity to swap their existing holdings dollar-for-

dollar for ”pooling equity” which has the characteristic that it may be presented to any firm for a share in profits.

Notice that no equity holder has an incentive to unilaterally refuse this swap as it allows them to exactly mimic

their previous wealth holding position, or, if they desire, costlessly switch their assets to another firm. Notice

also that the simplicity of how the mechanism might be introduced is one of its primary attractions, property

rights are redefined not redistributed, and no shares need be agreed upon. In this sense its introduction poses no

distributional issues.

4.1 Cornes, Mason, and Sandler’s Model of The Commons.

Cornes, Mason and Sandler’s (op. cit.) model of the ”Tragedy of the Commons” provides a tractable transparent

exposition of the problem of the over exploitation of a common pool resource. In their analysis there are two

sources of distortions, the externality associated with the commons problem, and the distortion associated with an

imperfectly competitive output market. They demonstrate that these two distortions can be offsetting, such that

14

if the ”correct” number of firms extract from the resource then the efficient rate of extraction may be achieved7.

We wish to focus on our mechanism rather than market structure as a potential solution to the commons problem

and so follow Weitzman [14] in assuming that the output market is competitive. With this modification we are

able to illustrate how our cooperation inducing mechanism can induce an efficient rate of extraction from the

resource. We shall follow Cornes, Mason and Sandler’s example and discuss the problem of the exploitation of

a common access fishery, applications to similar problems such as extraction from an oil pool should be fairly

obvious to the reader.

4.1.1 The Model of a Common Access Fishery.

The industry consists of k = 1, ...,m fishing firms who’s objective is to maximize the profit received by their hk

shareholders. The firms are assumed to sell their output on a perfectly competitive market at the price P . The

total catch, or output of the commons, is denoted C and is determined by the total size of the fleet, R, according

to the production technology

C = F (R)

where F (R) is assumed to be strictly increasing and strictly concave while the input R is assumed to be essential,

i.e. F (0) = 0. Further the total catch is bounded above by the fish population. These assumptions ensure that

F (R)/R > F 0(R) and limR→∞

F (R)/R = 0.

We examine the symmetric or ”pure” commons case in which the fish population is distributed evenly across

the commons, so that the catch per vessel is equal. It then follows that each firm’s catch can be represented by

ck =

µrk

rk + eRk

¶F (rk + eRk)

where rk is the number of vessels of any given firm, k, and eRk = R− rk is the size of the rest of the fleet. Under

the assumption of non-cooperative Nash behavior each firm chooses its fleet size to maximize profit per equity

7 It is not exactly clear how the correct number of firms is achieved or how this number is varied over time to achieve the efficienttime path for extraction.

15

taking fRk as given, that is

Maxrk

πkhk=

ÃPF (rk + eRk)

rk + eRk

− w

!rkhk

where w is the rental rate per vessel. With free access entry drives profit to zero

PF (R)

R− w = 0

Denote the solution to this problem Rf .

4.1.2 Socially Efficient Fishing.

Given that the firms’ output sells on a competitive market at a given price P , and if we assume social welfare

to be given by the sum of consumer and producer surplus then the efficient level of fishing simply involves the

maximization of industry profit, or

MaxR

W = PF (R)− wR

with the first order condition

PF 0(R)− w = 0

Denote the solution to this problem RW . This immediately reveals the classic commons problem.

Proposition 1. With free access the commons is overexploited Rf > RW .

4.1.3 Introduction of the Cooperation Inducing Mechanism.

Suppose now that instead of equity representing a fixed claim on a particular firm it may instead be used as a

claim on any firm in the industry. Note that this equity is converted not created at this stage. Adopting the same

notation for this pooling equity as before we write sk as the claims made on the profits of firm k, and S =P

k

sk as total equity claims. The optimization of an individual firm k now involves

Maxrk

πksk=

ÃPF (rk + eRk)

R− w

!rksk.

16

Proposition 2. The introduction of pooling equities induce a socially efficient level of fishing.

In this model the only deviation from the efficient rate of extraction from the resource arises as a consequence

of the crowding externality that fishing vessels impose on each other8. Introducing the pooling equities causes

profits per share to be equalized across firms. When choosing the number of vessels in its fleet each firm knows

that the negative external effects it imposes on other firms will cause a decline in their total profits and thus lead

to a redistribution of equity across firms. The increase in equity claims on an expanding firm’s profits reduce

profit per share and thus cause it to fully internalize the external effects it has on the rest of the industry9.

5 Conclusion.

In this paper we have proposed a novel mechanism for inducing agents playing a non-cooperative game to choose

the cooperative outcome. Our mechanism adds a second non-cooperative stage to the game. In the unique

Nash equilibrium of this second stage, the payoffs generated in the first stage are reallocated between the players

according to the allocation of shares. We show that this effectively converts the first stage into a weighted potential

game (cf. Monderer and Shapley (op.cit.)), the players of which have incentives to maximize the total payoff. If

we follow Monderer and Shapley (op. cit.) further, and restrict attention to those Nash equilibria that lie in the

argmax set of the weighted potential then the mechanism implements the cooperative outcome10.

This mechanism has applications to a wide range of economic problems, as any situation in which external

effects or spillovers are present may be viewed as one where cooperative behavior can potentially produce welfare

improvements. We believe the mechanism is both of theoretical interest and raises some interesting possibilities

for policy.

Theoretically, as the arbitrage conditions (2) indicate, it is as if a new market has been created. Establishing

property rights and a competitive market on which an external effect may be traded is a well known solution to

an externality problem. What is perhaps of interest is that the ”market” in the current paper is not directly for

8 It can be shown (see Ellis [4]) that in a dynamic model our mechanism induces the internalization of both the dynamic and staticexternalities associated with the commons problem.

9Our mechanism causes firms to behave as if they have merged. However, this solution may be superior to a merger. If individualfirms productions technologies were concave then a merger may well increase marginal and average costs. ie. if f(x0) = f(x1)+f(x1) =A, then f 0(x0) < f 0(x1).10The reader can probably think of several stories that might justify this refinement. However, these typically involve specifying

the beliefs of the players prior to play. This issue is not our focus in this paper.

17

the external effect but for the returns generated by the activity that produces it. In at least some examples the

arbitrage of profits perfectly substitutes for a competitive market in the externality.

An alternative way to view our theoretical contribution is that it decentralizes a payoff sharing scheme. In the

subgame perfect equilibrium each player receives a proportion of the total payoff. It is as if fixed proportionate

payoff shares have somehow been agreed in advance of play. However, such prior agreements are unnecessary

precisely because proportionate payoffs are a characteristic of the non-cooperative equilibrium.

From a policy perspective our proposed mechanism has several advantages over alternative solutions to ex-

ternality problems. Once implemented it requires no regulatory body to oversee it, there are no information

requirements such as those needed to implement tax or quota based solutions, and no new property rights are es-

tablished so there are no equity issues such as those that arise when, for example, pollution permits are introduced.

Further our solution requires no monitoring.

Despite our proposed mechanism’s advantages there are of course some caveats and issues that require further

study. While it seems to us the very nature of a proportional sharing scheme mitigates against the free-rider

problem (see our discussion in section 3), it may though be the case that there are situations in which this problem

does arise. If this is the case there are several possible ways to ensure full participation. The firms involved might

write a contract that specifies that the scheme comes into force if and only if all firms participate. Each firm is then

pivotal and cannot free ride. Alternatively, the introduction of the scheme may require government involvement.11

This is an issue that we hope to investigate in greater depth in the future.

In other applications cooperation may benefit the participants but harm a third party, such as consumers.

Obviously this is not a direct caveat to our mechanism, but a familiar type of warning that if there are potentially

multiple distortions in an economy then limited cooperation may be worse in a welfare sense than none at all.

This is a standard problem frequently encountered in a second-best world where if two adjustments are required to

move the economy to the Pareto frontier, one of the two may actually move the economy away from it. However,

from a policy perspective this is still a very interesting issue. Again this suggests possibilities for future study.

11This seems a minimal role for public policy. The government requires no specific information, but simply has to mandate theadoption of the scheme.

18

References

[1] Coase, R. (1960), ”The Problem of Social Cost,” Journal of Law and Economics, 3, 1-44.

[2] Cornes, R. C., C. F. Mason, and T. Sandler (1986), ”The Commons and the Optimal Number of Firms,”

Quarterly Journal of Economics 100, 641-646.

[3] Cornes, R. C. and T. Sandler. (1986), The Theory of Externalities, Public Goods, and Club Goods, Cambridge

[Cambridgeshire] ; New York : Cambridge University Press.

[4] Ellis, C. J. (1998), ”Common Pool Equities: An Arbitrage Based Non-Cooperative Solution to the Common

Pool Resource Problem,” Working Paper, University of Oregon.

[5] Ellis, C. J., and A. van den Nouweland (1999), ”Efficiency in a Model of Agency with Reciprocal Externali-

ties,” Mimeo, University of Oregon.

[6] Guttman, J. M. and A. Schnytzer. (1992), ”A Solution of the Externality Problem using Strategic Matching,”

Social Choice and Welfare, 8, 73-88.

[7] Holmstrom, B. (1982), Moral Hazard in Teams; Bell Journal of Economics, 13(2), 324-340.

[8] Jackson, M.(2001), ”Mechanism Theory,” Forthcoming in The Encyclopedia of Life Support Systems.

[9] Monderer, D. and L. Shapley. (1996), ”Potential Games,” Games and Economic Behavior, 14, 124-143.

[10] Rasmusen, E. (1987), “Moral Hazard in Risk Averse Teams,” Rand Journal of Economics, 18: 428-435.

[11] Roemer, J. E. (1995), ”Would Economic Democracy Decrease the Amount of Public Bads?” Scandinavian

Journal of Economics, 2, 227-238.

[12] Strausz, R. (1999), ”Efficiency in Sequential Partnerships;” Journal of Economic Theory 85, 140-156.

[13] Varian, H. R. (1994), ”A Solution to the Problem of Externalities When Agents Are Well-Informed,” The

American Economic Review, 84, No. 5., 1278-1293.

[14] Weitzman, M. L. (1974), ”Free Access vs. Private Ownership as Alternative Systems for Managing Common

Property,” Journal of Economic Theory, VIII, 225-34.

19

Appendix

Proof of Theorem 1. We prove our first theorem via a sequence of three parts that correspond to the parts of

the theorem.

Let¡¡sik¢mk=1

¢i∈N be a Nash equilibrium, then the following hold.

Part 1. Each player makes a positive contribution to each payoff, i.e. sik > 0 for every i ∈ N and k = 1, 2, ....,m.

P roof. If there is only one payoff (m = 1), then the unique Nash equilibrium is for every player to contribute

his entire endowment of property rights to this payoff and the lemma is true. Hereafter, assume that m ≥ 2.

First we prove that for any payoff there is at least one player who contributes a positive amount to that payoff,

and that this is true for each payoff.

Suppose k ∈ {1, 2, ...,m} is such that sik = 0 for every i ∈ N . Then payoff Pk is divided among the players

in some predetermined way. Since there are at least 2 players (n ≥ 2), there is at least one player i ∈ N

who gets a part P ik < Pk from this payoff. Take such a player i. If player i would make an arbitrarily small

positive contribution µi to payoff Pk, then he would get the entire payoff and increase his payoff by Pk−P ik. The

contribution µi would have to be taken away from some other payoff. SincePm

l=1 sil = Si > 0, there exists an

l ∈ {1, 2, ...,m}, l 6= k, such that sil > 0. Take such an l. Player i gets a proportion silPj∈N sjl

> 0 of payoff Pl.

Notice that the proportion that i gets from payoff Pl is a continuous function of i0s contribution to this payoff as

long as his contribution is positive. Hence, player i can reduce his contribution to payoff Pl by an amount µi in

such a way that he loses less than Pk − P ik from payoff Pl, i.e.

silPj∈N sjl

Pl − sil−µiPj∈N\{i} s

jl+(s

il−µi)

Pl < Pk − P ik. So,

if player i reduces his contribution to payoff Pl by such an amount µi and increases his contribution to payoff Pk

from 0 to µi, then he increases his total payoff. Hence, the initial contributions did not form a Nash equilibrium.

Next we show that for every payoff at least two players contribute a positive property rights to that payoff.

Suppose k ∈ {1, 2, ...,m} and i ∈ N are such that sik > 0 and sjk = 0 for every j ∈ N\{i}. Then player i

gets the entire payoff Pk, because he is the only player contributing to this payoff. However, if he reduces his

contribution to payoff Pk by an amount µi < sik, then he would still get the entire payoff Pk. Then he can increase

his contribution sil to some other payoff Pl by the amount µi. If he chooses a payoff Pl for which it holds that

there is some other player j ∈ N\{i} such that sjl > 0 (note that such a payoff exists), then increasing sil will

20

increase the proportion that i gets of payoff Pl (note that the share that i gets from payoff Pl is a continuously

increasing function of i0s contribution to this payoff). Hence, the initial contributions of the players did not form

a Nash equilibrium.

We now are ready to prove that sik > 0 for every i ∈ N and k = 1, 2, ...,m.

Note that for each payoff at least two players make a positive contribution and that this is true for every

payoff, this implies thatP

j∈N\{i} sjk > 0 for every k = 1, 2, ...,m and i ∈ N . Suppose that there exists a payoff

to which not every player contributes a positive amount. Let k ∈ {1, 2, ...,m} and i ∈ N such that sik = 0 and let

l ∈ {1, 2, ...,m}, l 6= k, such that sil > 0. Define S ⊂ N by S = {h ∈ N | shk > 0}. Note that i /∈ S. Then we find

using condition (1) that Pj∈N\{i} s

jk³P

j∈N sjk

´2Pk ≤P

j∈N\{i} sjl³P

j∈N sjl

´2Pland P

j∈N\{h} sjk³P

j∈N sjk

´2 Pk ≥P

j∈N\{h} sjl³P

j∈N sjl

´2 Pl for every h ∈ S.

From this we derive that Pj∈N\{i} s

jkP

j∈N\{i} sjl

≤³P

j∈N sjk

´2³P

j∈N sjl

´2 PlPk

and Pj∈N\{h} s

jkP

j∈N\{h} sjl

≥³P

j∈N sjk

´2³P

j∈N sjl

´2 PlPk

for every h ∈ S.

Notice that the right-hand sides of the last two inequalities are identical. Hence, we find that

Pj∈N\{i} s

jkP

j∈N\{i} sjl

≤P

j∈N\{h} sjkP

j∈N\{h} sjl

for every h ∈ S,

which can be re-written as Pj∈N\{h} s

jlP

j∈N\{i} sjl

≤P

j∈N\{h} sjkP

j∈N\{i} sjk

for every h ∈ S.

21

We use this result to obtain12

(|S|− 1) +P

j∈N\S sjlP

j∈N\{i} sjl

<(|S|− 1)Pj∈N\{i} s

jl +

Pj∈N\S s

jl + (|S|− 1)silP

j∈N\{i} sjl

=

Ph∈S

Pj∈N\{h} s

jlP

j∈N\{i} sjl

=Xh∈S

ÃPj∈N\{h} s

jlP

j∈N\{i} sjl

!≤Xh∈S

ÃPj∈N\{h} s

jkP

j∈N\{i} sjk

!=

Ph∈S

Pj∈N\{h} s

jkP

j∈N\{i} sjk

=(|S|− 1)Pj∈N sjk +

Pj∈N\S s

jkP

j∈N sjk= (|S|− 1).

This shows that Pj∈N\S s

jlP

j∈N\{i} sjl

< 0.

However, we knowP

j∈N\{i} sjl > 0, i ∈ N\S, and sil > 0. Hence, we have a contradiction and conclude that

every player contributes a positive amount to each payoff. This proves part 1.

Part 2. Each player divides their property rights between the payoffs such that their share in each payoff is in

the same proportion as their share of total property rights, i.e. sikPj∈N sjk

= SiPj∈N Sj for every i ∈ N and

k = 1, 2, ....,m.

P roof. From part 1 it follows thatP

j∈N\{i} sjk > 0 for every k = 1, 2, ...,m and i ∈ N and, consequently,

that condition (1) implies that 0 <P

j∈N\{i} sjk

(P

j∈N sjk)2 Pk = λi for every i ∈ N . Dividing condition (1) corresponding to

player i and payoff Pk by that corresponding to player h and payoff Pk gives

µPj∈N\{i} s

jk

(P

j∈N sjk)2

¶µP

j∈N\{h} sjk

(P

j∈N sjk)2

¶ PkPk

=

Pj∈N\{i} s

jkP

j∈N\{h} sjk

=λi

λh.

From this we derive

Xj∈N\{i}

Sj =X

j∈N\{i}

mXk=1

sjk =mXk=1

Xj∈N\{i}

sjk =mXk=1

⎡⎣ λi

λh

Xj∈N\{h}

sjk

⎤⎦ = λi

λh

Xj∈N\{h}

mXk=1

sjk =λi

λh

Xj∈N\{h}

Sj

12We remind the reader that i /∈ S, sik = 0, sil > 0, and sjk = 0 for each j ∈ N\S.

22

and then Pj∈N\{i} s

jkP

j∈N\{i} Sj=

λi

λh

Pj∈N\{h} s

jk

λi

λh

Pj∈N\{h} Sj

=

Pj∈N\{h} s

jkP

j∈N\{h} Sj.

It follows that for each k = 1, 2, ...,m we can define a constant Ck such that

Pj∈N\{i} s

jkP

j∈N\{i} Sj= Ck for all i ∈ N.

Now, for each i ∈ N ,

(n− 1)sik =X

j∈N\{i}

⎛⎝ Xh∈N\{j}

shk

⎞⎠− (n− 2)⎛⎝ Xh∈N\{i}

shk

⎞⎠=

Xj∈N\{i}

⎛⎝Ck

Xh∈N\{j}

Sh

⎞⎠− (n− 2)⎛⎝Ck

Xh∈N\{i}

Sh

⎞⎠ = (n− 1)CkSi.

Hence,

sikSi= Ck for each i ∈ N

and

siksjk=

CkSi

CkSj=

Si

Sjfor all pairs i, j ∈ N.

Then, for each i ∈ N and k = 1, 2, ...,m,

sikPj∈N sjk

=

³siksik

´P

j∈N³sjksik

´ =³Si

Si

´P

j∈NSj

Si

=SiPj∈N Sj

.

This proves part 2.

Part 3. Each player divides their property rights between the payoffs in the same proportions as each has to total

payoffs, i.e. sikSi =

PkPml=1 Pl

for every i ∈ N and k = 1, 2, ....,m.

P roof. From the proof of part 2 we know that for every k = 1, 2, ...,m there exists a Ck such thatsikSi = Ck

for each i ∈ N . Let i ∈ N . The Nash-equilibrium strategy¡sik¢mk=1

satisfies condition (1) and, by part 1 we know

23

that sik > 0 for each k = 1, 2, ...,m. This implies that

Pj∈N\{i} s

jk³P

j∈N sjk

´2Pk =P

j∈N\{i} sjl³P

j∈N sjl

´2Plfor each k, l = 1, 2, ...,m. We use this to find

Pj∈N\{i} S

j³Pj∈N Sj

´2 PkCk=

Pj∈N\{i}CkS

j³Pj∈N CkSj

´2Pk =P

j∈N\{i} sjk³P

j∈N sjk

´2Pk=

Pj∈N\{i} s

jl³P

j∈N sjl

´2Pl =P

j∈N\{i}ClSj³P

j∈N ClSj´2Pl =

Pj∈N\{i} S

j³Pj∈N Sj

´2 PlCl

and, consequently, PkCk= Pl

Clfor all k, l ∈ {1, 2, ...,m}. Hence, Ck =

PkC1P1

for all k = 1, 2, ...,m Now we derive

1 =

Pml=1 s

il

Si=

mXl=1

silSi=

mXl=1

Cl =mXl=1

PlC1P1

=C1P1

mXl=1

Pl

and

C1 =P1Pml=1 Pl

.

Hence, Ck is uniquely determined for each k ∈ {1, 2, ...,m} by

Ck =PkC1P1

=PkPml=1 Pl

.

This yields

sikSi= Ck =

PkPml=1 Pl

,

which concludes the proof

Theorem 2. Let¡(sik)

mk=1

¢i∈N be the set of strategies defined by

sik = Siµ

PkPml=1 Pl

¶

for every i ∈ N and k = 1, 2, ....,m. This set of strategies is the unique Nash equilibrium of the second stage

24

of the game. Moreover, for every player i ∈ N his payoff according to the Nash equilibrium is

SiPj∈N Sj

ÃmXk=1

Pk

!.

P roof. Let¡(sik)

mk=1

¢i∈N be the set of strategies defined by

sik = Siµ

PkPml=1 Pl

¶

f or every i ∈ N and k = 1, 2, ....,m. This set of strategies is the unique Nash equilibrium of the second

stage of the game. Moreover, for every player i ∈ N his payoff according to the Nash equilibrium is

SiPj∈N Sj

ÃmXk=1

Pk

!.

From part 3 of the proof of theorem 1 we derive that if the strategies defined in (3) form a Nash equilibrium,

then this is the unique Nash equilibrium. To prove that the strategies defined in (3) form a Nash equilibrium,

let i ∈ N . We will prove that¡sik¢mk=1

maximizes player i’s payoff given the strategies³³

sjk

´mk=1

´j∈N\{i}

of

the other players. First we prove that the strategy¡sik¢mk=1

satisfies condition (1). It is immediately clear

that sik = Si³

PkPml=1 Pl

´> 0 for every k = 1, 2, ...,m. Therefore, it is sufficient to prove that

Pj∈N\{i} s

jk³P

j∈N sjk

´2Pk =P

j∈N\{i} sjl³P

j∈N sjl

´2Pl for all k, l ∈ {1, 2, ...,m}.

So let k, l ∈ {1, 2, ...,m}. Then

Pj∈N\{i} s

jk³P

j∈N sjk

´2Pk =P

j∈N\{i} Sj³

PkPmt=1 Pt

´³P

j∈N Sj³

PkPmt=1 Pt

´´2Pk =P

j∈N\{i} Sj³P

j∈N Sj´2 ³

PkPmt=1 Pt

´Pk=

Pj∈N\{i} S

j³Pj∈N Sj

´2 ³PlPmt=1 Pt

´Pl =P

j∈N\{i} Sj³

PlPmt=1 Pt

´³P

j∈N Sj³

PlPmt=1 Pt

´´2Pl =P

j∈N\{i} sjl³P

j∈N sjl

´2Pl.To simplify notation we define i’s objective function as f , where we collapse the contributions made by the

25

other players to a payoff Pk to s−ik =

Pj∈N\{i} s

jk: for all of player i’s strategies (µ

ik)mk=1 it holds that

f((µik)mk=1) :=

mXk=1

µiks−ik + µik

Pk.

Notice that it follows from definition 3 that s−ik > 0 for every k = 1, 2, ...,m, so that the objective function f

is well-defined and continuous in all (µik)mk=1, even the strategies with some µ

ik equal to 0. From the fact that

(sik)mk=1 satisfies condition (1), we know that (s

ik)mk=1 is either a local maximum or a local minimum location of

f . If we prove that the function f is strictly concave, then it follows that (sik)mk=1 is a unique global maximum

location. To show that f is strictly concave, we take two different strategies (µik)mk=1 and (ν

ik)mk=1 of player i

and an α ∈ (0, 1) and show that f(α(µik)mk=1+(1−α)(νik)mk=1) > αf((µik)mk=1)+(1−α)f((νik)mk=1). To prove

this, it is sufficient to prove that µiks−ik +µik

Pk is a strictly concave function of µik for every k ∈ {1, 2, ...,m}.

This is easily seen by taking the second derivative of this function with respect to µik, which is clearly

negative. This proves that the strategies defined in (3) form the unique Nash equilibrium.

To prove the second part of the theorem, let i ∈ N . Player i’s payoff according to the Nash equilibrium is

mXk=1

sikPj∈N sjk

Pk =mXk=1

Si³

PkPml=1 Pl

´P

j∈N Sj³

PkPml=1 Pl

´Pk = SiPj∈N Sj

ÃmXk=1

Pk

!.

As required.

Proof of Propositon 1: From the first order conditions F (Rf )Rf

= wP = F 0(RW ). Our earlier assumptions ensure

that for any given R, F (R)/R > F 0(R), so as F 00(R) < 0 it follows that F (Rf )Rf

= F 0(RW ) implies Rf > RW ,

which concludes the proof.

Proof of Proposition 2: We know from theorem 2 (and using the same notation as in the text) that

sk =πkSnP=1

π

26

so the individual firm’s optimization problem may be rewritten

Maxrk

πksk=

πk⎛⎝ πkSnP=1

π

⎞⎠ =

nP=1

π

S=1

S

nX=1

⎛⎜⎜⎜⎜⎝PF

ÃnPj=1

rj

!nPj=1

rj

− w

⎞⎟⎟⎟⎟⎠ r .

The first order condition to this optimization problem is now

∂³πksk

´∂rk

=1

S

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩PF

ÃnPj=1

rj

!nPj=1

rj

− w +nX=1

⎡⎢⎢⎢⎢⎢⎣

⎛⎜⎜⎜⎜⎜⎝PF 0

ÃnPj=1

rj

!ÃnPj=1

rj

!− PF

ÃnPj=1

rj

!Ã

nPj=1

rj

!2⎞⎟⎟⎟⎟⎟⎠ r

⎤⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭=1

S

∙PF (R)

R− w + PF 0(R)− PF (R)

R

¸= 0.

This reduces to

PF 0(R)− w = 0

which is precisely the first order condition for a social optimum.

27