Correlated Equilibrium and the Pricing of Public Goods∗

Joseph M. Ostroy† Joon Song‡

Department of Economics Department of Economics

UCLA University of Essex

April 2007

Abstract

The duality underlying correlated equilibrium is based on public goods. That du-

ality is combined with the view that the outcome of a game is implemented by an

organizer managing a team of players whose actions are unobservable and unenforce-

able. Competition among organizers leads to a price-quantity description of efficient

correlated equilibria, called incentive constrained Lindahl equilibria. Duality charac-

terizations of the sets of (i) all correlated equilibria, (ii) incentive constrained Lindahl

equilibria, and (iii) non-incentive constrained Lindahl equilibria for games in normal

form are compared.

Keywords: correlated equilibrium; incentive constraints; public goods; Lindahl equi-

librium

∗Thanks to Neil Gretsky and David Rahman for helpful discussions.†Contact: ostroy@ucla.edu‡Contact: joonsong@essex.ac.uk

1 Introduction

Demonstrations by Hart and Schmeidler (1989) and Nau and McCardle (1990) that the

existence of correlated equilibrium (Aumann [1974]) can be achieved via duality theory

raises the possibility that the language of prices and quantities may be applicable to

non-cooperative equilibrium. Additionally, there is a question of how such prices and

quantities are related to competitive equilibrium. These two issues are the focus of this

paper.

The connection between prices and quantities typically presumes that goods are

private, i.e., the allocation of a commodity to one person precludes its availability to

others. But a game in normal form is predicated on the logic of public goods since

every player “consumes” the play of the game. Duality theory for public goods is

known in economics as Lindahl equilibrium. In comparison to private goods, the roles

of prices and quantities are reversed: prices are personalized while the quantity of the

public good is impersonal since it is the same for everyone. Personalization of pricing

is often linked to the observation that Lindahl equilibrium is vulnerable to strategic

manipulation. Therefore the attempt to connect correlated equilibrium — which takes

strategic behavior into account— with Lindahl pricing might appear incongruous. The

disparity is resolved by noting that Lindahl pricing is vulnerable to misrepresentation

of utilities while we shall deal only with games of complete information where the

players’ payoffs are known. The strategic behavior addressed here is limited to actions

and our goal will be to show that “hidden actions” can be priced.

That non-cooperative outcomes can be described as correlated rather than Nash

equilibrium has been accompanied by an increased willingness to view a game from

the perspective of an outsider. For example, Nau and McCardle motivate correlated

equilibrium by saying that players’ behavior “should be coherent in the sense of not

presenting opportunities for arbitrage (“Dutch Books”) to an outside observer who

serves as a betting opponent.” Myerson (1990) uses the metaphor of a mediator who

makes recommendations to the players that must be self-enforcing because they are

1

non-binding. This invites the interpretation that a normal form game can be regarded

as a team managed by an organizer facing a moral hazard problem. Our point of

departure is to view the equilibrium of a game as resulting from competition among

organizers in managing the team.

2 Preliminaries

The following concepts and results from convex analysis (e.g., Rockafeller [1970]) sum-

marize the duality used below.

A polyhedral convex cone K ⊆ Rm is the intersection of a finite number of closed

half-spaces whose boundary points contain the origin, i.e., there exists b1, . . . , bq ∈ Rm

such that K = {z : bk · z ≤ 0, k = 1, . . . , q}. If Ki, i = 1, . . . , n, are polyhedral convex

cones, so is∑

i Ki and ∩iKi. If K contains a non-zero z ≥ 0, then by appropriate

rescaling, it contains a z ≥ 0 whose components sum to 1.

The polar of K is the convex cone K∗ = {p : p · K ≤ 0} that is also polyhedral.

In fact, K∗ = {p =∑

k αkbk : αk ≥ 0, k = 1, . . . , q}. Starting with K∗, its polar is

(K∗)∗ = K. The polar of ∩iKi is∑

i K∗i and therefore (

∑i K

∗i )∗ = ∩iKi. In a model

with private goods, if Ki were the technology of producer i, the relevant prices for the

aggregate technology∑

i Ki would be ∩iK∗i . Here, the roles of prices and quantities

are reversed.

A Basic Separation Theorem stated as two mutually exclusive alternatives says:

(i) Either K = Rm, or (ii) K is contained in a half-space, i.e., there is a non-zero

p ∈ K∗ such that p ·K ≤ 0.

The normal cone to z ∈ K is NK(z) = {p : p·K ≤ p·z}. Note that NK(0) = K∗. If z

belongs to the boundary of K, there is a non-zero p ∈ NK(z) such that p ·K ≤ p ·z = 0,

i.e., p defines a supporting hyperplane to K passing through z. Conversely, if there is

a non-zero p ∈ NK(z), then z defines a supporting hyperplane to K∗ passing through

p, i.e., z ∈ NK∗(p) = {z′ : K∗ · z′ ≤ p · z′}. However, when p = 0 the existence of a

non-zero z ∈ NK∗(0), while implying that 0 is on the boundary of K∗ and 0 ∈ NK(z),

2

does not necessarily imply that z belongs to the boundary of K. This is a feature,

exhibited in Proposition 1, of the possible multiplicity of correlated equilibria that is

the focus of the paper.

3 Characterization of correlated equilibrium

The purpose of this Section is to apply the duality of the previous Section to correlated

equilibrium.

A game in normal form is described by A = ×iAi, where ki is the number of

elements of Ai, the set of actions or pure strategies available to i = 1, · · · , n and (vi),

where vi : A → R is the payoff to i from each a ∈ A. Denote by ∆m the set of non-

negative vectors of length m whose sum is 1. The set of possible randomized plays of

the game is ∆m, where m = k1 × · · · × kn is the number of elements in A. The payoff

to i from z ∈ ∆m is vi · z =∑

a vi(a)z(a).

The values of vi determine the matrix Di of deviation gains for i: Di has (ki)2 rows,

one for each pair (di, ai) ∈ Ai ×Ai, and m columns, one for each a′ ∈ A. The entry in

row (di, ai) and column a′ ∈ A is

Di[(di, ai); a′] =

vi(di, a

′−i)− vi(a′i, a

′−i) if ai = a′i,

0 otherwise

The matrix (D1, . . . , Dn) has H = (k1)2 + · · ·+ (kn)2 rows and m columns. Of course,

when di = ai = a′i, Di[(di, ai); a′] is also 0. Throughout the following z ∈ RK+ and

yi ∈ R(ki)2

+ . Therefore, Diz is dimension (ki)2 and yiDi is dimension m. Multiplying z

against row (di, ai) of Di yields

hi(di, ai|z) =∑a′−i

[vi(di, a′−i)− vi(ai, a

′−i)]z(ai, a

′−i).

When z is a probability, hi(di, ai|z)/∑

a′−iz(ai, a

′−i) is the expected value to i of devi-

ating from ai to di.

3

In a 2× 2 game, m = 4 and (ki)2 = 22 = 4, so each Di is 4× 4. The pattern of D1

and D2 are

D1 =

v1(a1, b1)− v1(a1, b1) 0 v1(a1, b2)− v1(a1, b2) 0

v1(a2, b1)− v1(a1, b1) 0 v1(a2, b2)− v1(a1, b2) 0

0 v1(a1, b1)− v1(a2, b1) 0 v1(a1, b2)− v1(a2, b2)

0 v1(a2, b1)− v1(a2, b1) 0 v1(a2, b2)− v1(a2, b2)

D2 =

v2(a1, b1)− v2(a1, b1) v2(a2, b1)− v2(a2, b1) 0 0

v2(a1, b2)− v2(a1, b1) v2(a2, b2)− v2(a2, b1) 0 0

0 0 v2(a1, b1)− v2(a1, b2) v2(a2, b1)− v2(a2, b2)

0 0 v2(a1, b2)− v2(a1, b2) v2(a2, b2)− v2(a2, b2)

Example 1. In the prisoners’ dilemma game

b1 b2

a1 5, 5 0, 6

a2 6, 0 1, 1

the matrices Di are

D1 =

5− 5 0 0− 0 0

6− 5 0 1− 0 0

0 5− 6 0 0− 1

0 6− 6 0 1− 1

, D2 =

5− 5 0− 0 0 0

6− 5 1− 0 0 0

0 0 5− 6 0− 1

0 0 6− 6 1− 1

D1z and D2z are

0

z(a1, b1) + z(a2, b1)

−z(a1, b2)− z(a2, b2)

0

,

0

z(a1, b1) + z(a1, b2)

−z(a2, b1)− z(a2, b2)

0

Therefore, D1z ≤ 0, D2z ≤ 0 imply

z(a1, b1) + z(a2, b1) ≤ 0,−z(a1, b2)− z(a2, b2) ≤ 0

z(a1, b1) + z(a1, b2) ≤ 0,−z(a2, b1)− z(a2, b2) ≤ 0.

4

Of course, the only z ∈ ∆4 satisfying the inequalities is z(a2, b2) = 1. Uniqueness is

not, however, typical.

For the following, it is helpful to consider z ≥ 0. Let

ci(z) = sup{yiDiz : yi ∈ R(ki)

2

+

},

be the cost that i would impose on the organizer if the organizer were to choose z.

Hence, whenever z is such that hi(di, ai|z) > 0 for some (di, ai), ci(z) = ∞. Let

Zi = {z : ci(z) = 0}.

Therefore, ci is an indicator function taking value 0 whenever z ∈ Zi and ∞ when

z 6∈ Zi.

The set of correlated equilibria for the game [(vi), A] are those z ∈ ∆m that are also

in⋂

i Zi.

Example 2. As another illustration, consider a variation of an example in Aumann

(1974) given by Myerson (1990).

b1 b2

a1 5, 1 0, 0

a2 4, 4 1, 5

The D1 and D2 matrices are:

D1 =

0 0 0 0

4− 5 0 1− 0 0

0 5− 4 0 0− 1

0 0 0 0

, D2 =

0 0 0 0

0− 1 5− 4 0 0

0 0 1− 0 4− 5

0 0 0 0

D1z and D2z are:

D1z =

0

−z(a1, b1) + z(a2, b1)

z(a1, b2)− z(a2, b2)

0

, D2z =

0

−z(a1, b1) + z(a1, b2)

z(a2, b1)− z(a2, b2)

0

5

Figure 1: Tetrahedron

z11

z22

z21

z12

A

B

CD

E

F

G

Figure 2: Utility Image

1 2 3 4 5

1

2

3

4

5

AB

G

D

v2 z2

v1 z1

E

C

F

.

.0

H

The conditions that ci(z) = 0, or z ∈ Zi, i = 1, 2, are

Z1 = {z : −z(a1, b1) + z(a2, b1) ≤ 0, z(a1, b2)− z(a2, b2) ≤ 0, }

Z2 = {z : −z(a1, b1) + z(a1, b2) ≤ 0, z(a2, b1)− z(a2, b2) ≤ 0}.

The tetrahedron ∆4 is illustrated in Figure 1. The intersection of Z1 ∩ Z2 with ∆4 is

the set of convex combinations of the following extreme points: A = (1/3, 0, 1/3, 1/3);

G = (1/4, 1/4, 1/4, 1/4); E = (1, 0, 0, 0); F = (0, 0, 0, 1); D = (1/3, 1/3, 0, 1/3). Call

this set Z. The points E, F and G are the Nash equilibria of the game. The utility

image of the set of correlated equilibria, U [Z] = {(v1 · z, v2 · z) : z ∈ Z}, is the area

FDEA illustrated in Figure 2. Note that while G is a Nash equilibrium and an extreme

point of Z, the utility image of G is in the interior of U [Z]. The utility image of A

illustrates the well-known property that correlated equilibrium expands (in the positive

direction) the set of utility outcomes over and above those in the convex hull of the set

of payoffs associated with Nash equilibria (the area of the triangle GEF ). The utility

image of D shows that correlated equilibrium also expands the set of utility outcomes

in the negative direction. The utility image of the set of all possible plays, U [∆4], is

the area 0ECF . The utility of C = (0, 0, 1, 0) is (4, 4), an extreme point of U [∆4],

whose boundary includes ECF . Thus, the area between ECF and EAF represents

the utility consequences of the incentive constraints.

6

It is readily seen that Zi = {z : Diz ≤ 0(ki)2} is a polyhedral convex cone. The

polar of Zi is

Pi = {pi : pi · Zi ≤ 0}

Note that for each i, Zi ∩ ∆m 6= ∅: set zi(a) = 1, where vi(a) = max{vi(a) : a ∈ A}.

Because Zi 6= {0}, it follows that Pi 6= Rm and each non-zero z ∈ Zi is a supporting

hyperplane for Pi, i.e., Pi ·z ≤ 0. The difficulty is the public goods nature of the choice:

the same z must be chosen for all i.

Since each Pi is a polyhedral convex cone, so is∑

Pi. The polar of∑

Pi is

(∑

Pi)∗ = {z : (∑

Pi) · z ≤ 0}

= {z ∈ ∩iZi}

There is a half-space containing∑

Pi if and only if it satisfies the condition of the

Basic Separation Theorem that∑

Pi 6= Rm or, equivalently, the dual condition that

∩iZi 6= {0}. Moreover, since Pi contains Rm− , any non-zero z such that (

∑Pi) · z ≤ 0

can be normalized to z ∈ ∆m. Hart and Schmeidler showed ∩iZi 6= {0} while Nau and

McCardle showed∑

Pi 6= RK .

Proposition 1 Let z ∈ ∆m. The following are equivalent:

• z is a correlated equilibrium, i.e., z ∈ ∩iZi

• z ∈ NPiPi

(0)

By construction, NPiPi

(0) = (∑

i Pi)∗. Since (∑

i Pi)∗ = ∩iZi, we have z ∈ ∩iZi.

A correlated equilibrium z acts like a “price” for the “quantities”∑

i Pi, i.e., z

defines a supporting hyperplane to∑

i Pi at its 0 boundary point. From z ∈ NPiPi

(0),

it follows that 0 ∈ N∩iZi(z). But as indicated below, 0 is not an informative price for

z.

7

4 Pricing equilibrium in games

The purpose of this Section is to give three definitions of equilibrium for games in

normal form using the profit- and utility-maximizing language of general equilibrium.

The first is a restatement of correlated equilibrium, the second is a restriction of cor-

related equilibrium that resembles the price-taking description of Lindahl equilibrium,

while the third is the naive Lindahl equilibrium for games. Each has unique as well

as overlapping properties. The second definition combines the incentive constraints of

the first with the utility-maximizing considerations of the third and is the focus of the

paper.

Think of z as being supplied by an organizer facing the personalized prices pi for

each i. The net profit from supplying z at (pi) is (∑

i pi) · z − c(z), where

c(z) =∑

i

ci(z).

Hence, individual incentive constraints impute private costs to a publicly supplied z.

Define

π(∑

pi) = sup{∑

i[pi · z − ci(z)] : z ≥ 0}

as the profit function based on prices (pi).

Borrowing from the notion of decentralization via prices, the organizer could del-

egate the responsibility for dealing with each individual to a separate agent. The

profit-function for supplying i based on prices pi is

πi(pi) = sup{pi · zi − ci(zi) : zi ≥ 0}.1

If pi ∈ Pi, then πi(pi) = 0, while if pi 6∈ Pi, πi(pi) = ∞. Consequently, πi is the indicator

function of Pi. If the sup of πi(pi) is achieved at zi, then pi ∈ NZi(zi) = {p : p·Zi ≤ p·zi}

and pi · zi = 0. [Note: 0 ∈ NZi(zi) for all zi ∈ Zi.]

Because the profit function π(∑

pi) restricts attention to a single z, it is readily

seen that π(∑

pi) ≤∑

i πi(pi). However, if pi · zi − ci(zi) = πi(pi) and for all i, zi = z,

1πi(pi) is the conjugate function of ci

8

then π(∑

i pi) =∑

i πi(pi). I.e., the organizer’s profits are maximized when each of his

agents maximizes while choosing the same z.

We use these results to convert the definition of correlated equilibrium into a joint

statement about prices, quantities and maximization. The profit maximizing supply

of the organizer is

η(∑

ipi) = {z ∈ Rm+ : (

∑ipi) · z −

∑ici(z) = π(

∑ipi)}.

Definition 1 z ∈ ∆m is a correlated equilibrium if and only if there exists (pi) such

that for all i,

• z ∈ η(∑

i pi) (profit maximization)

The exclusive attention to profit maximization without a corresponding concern

for utility maximization differs from the traditional notion of economic equilibrium.

The reason, of course, is that the definition of correlated equilibrium depends on (ci)

which, in turn, depends only on the matrices of deviation gains (Di), not on the utility

functions (vi). In fact, prices need not play any meaningful role because, as noted

above, it suffices to set pi = 0.

As the intersection of convex sets ∆m ∩ (⋂

i Zi), the set of correlated equilibria is

convex. If the set were a singleton, as in the Prisoners’ Dilemma, the following analysis

would be otiose.

We shall regard the players in a game as members of a team managed by an orga-

nizer competing with other organizers. And we shall think of z as a contract offered

by an organizer of the team. As in the metaphor adopted by Myerson, the stipulations

in the contract are recommendations because the organizer cannot observe or enforce

the actions undertaken by a team member.

The point of departure is that instead of writing π(∑

i pi) =∑

i πi(pi), where each

individual is as-if made to pay his own incentive costs, the organizer can engage in more

sophisticated pricing. In particular, while adhering to the principle that∑

i pi ∈∑

i Pi,

the organizer need not confine himself to the restriction that pi ∈ Pi. To increase profit

9

opportunities, the organizer can exploit the fact that in choosing z(a) the losses imposed

by some individuals can be offset by the gains received from others.

The profits from the team are revenues∑

i pi · z minus costs∑

i ci(z). Because∑i ci(z) = ∞, when z /∈ ∩iZi, such contracts will be avoided. Starting with a status

quo correlated equilibrium z, suppose there is another z ∈ ∆m such that∑

i ci(z) = 0,

but vi · z ≥ vi · z for all i, with at least one inequality strict. An organizer of z could

make a profit by offering to split the gains from z (by a side-payment). Competition

among organizers, all with the same opportunities, leads to the conclusion that profits

will be driven to zero, but only after all the gains from offering contracts has been

exhausted. We want to show how prices can guide this process.

Define a budget constraint for i as γi(qi; 0) = {z : z ∈ ∆m, qi · z = 0}. Regarding

z ∈ ∆m as a random contract, or lottery, the restriction qi · z = 0 means that its

price-weighted value is zero. To be non-empty, prices qi must be such that purchases

(those a such that qi(a)z(a) ≥ 0) are financed by sales (qi(a′)z(a′) < 0). Faced with

such a hypothetical constraint, i’s utility maximizing demands for lotteries are

ξi(qi) = {z ∈ γi(qi; 0) : vi · z − qi · z ≥ vi · z′ − qi · z′,∀z′ ∈ ∆m}.

The conditions for z ∈ ξi(qi) include the usual restriction that any z′ that is preferred

costs more, i.e., vi · z′ > vi · z implies qi · z′ > 0. But it also exhibits the added

restriction that the utility gain from any departure from z is outweighed by the costs,

i.e., qi · (z′ − z) ≥ vi · (z′ − z). Define the inverse demand ξ−1i (z).

The following result says that for any play of the game there are prices for which

it would be utility maximizing.

Lemma 1 For all z ∈ ∆m, ξ−1i (z) 6= ∅.

Proof: A qi ∈ ξ−1i (z) must fulfill two conditions: (A) there is a qi such that

qi · (z′− z) ≥ vi · (z′− z) for all z′ ∈ ∆m, i.e., qi is in the subdifferential of vi(z) ≡ vi · z

regarded as a concave function on ∆m; and (B) qi is orthogonal to z. Because vi is

linear, it is readily verified that there are qi satisfying the inequality in (A). Suppose

10

qi · z 6= 0. Let qi(a) = qi(a) − qi · z. Then,∑

a qi(a)z(a) =∑

a[qi(a) − qi · z]z(a) =

qi · z − qi · z = 0 since z ∈ ∆m. Therefore, qi satisfies (A). In addition, qi · (z′ − z) =∑a[qi(a)− qi · z][z′(a)− z(a)] = qi · z′− qi · z− qi · z + qi · z = qi · (z′− z) and therefore

qi satisfies the subdifferential inequality in (A). �

Remark: The proof of Lemma 1 relies on the particular utility representation of pref-

erences over lotteries, vi. Nevertheless, once qi ∈ ξ−1i (z) is obtained, the only use we

make of this information is that qi is a supporting hyperplane to the convex set

Ri(z) = {z′ : vi · z′ ≥ vi · z},

passing through the origin. Therefore, if vi = αvi + β, where α > 0, Ri(z) would

remain unchanged. If qi ∈ ξ−1i (z), then qi can be used as a ordinal measure of utility

gain as follows: if z′ ∈ ∆m and qi · z′ < 0, then z′ /∈ Ri(z); and if z′ ∈ Ri(z) and z′

does not belong to the boundary of Ri(z), then qi · z′ > 0.

Consider a status quo z ∈ ∆m for which∑

ci(z) < ∞ and qi ∈ ξ−1i (z). Suppose

there is another z ∈ ∆m for which∑

ci(z) < ∞ such that for all i, vi · z ≥ vi · z for

all i and for at least one j, vj · z > vi · z. A competing organizer could then offer z

with the knowledge that qi · z ≥ qi · z and∑

i qi · z >∑

i qi · z = 0. Because z is a

correlated equilibrium, there are prices (pi) such that∑

i pi · z −∑

i ci(z) = π(∑

i pi).

So, an organizer can supply z at prices (pi) and receive prices (qi) such that∑

i qi · z >∑i pi · z = 0.

The following notion of equilibrium is based on the elimination of profit opportu-

nities through competition among organizers.

Definition 2 〈z, (qi)〉 is an incentive compatible Lindahl equilibrium if

• z ∈⋂

i ξi(qi) (utility maximization)

• z ∈ η(∑

i qi) (profit maximization)

In each of the Definitions in this section, z denotes an equilibrium play of the game. The

differences among them is indicated by the different notation for the prices which with

11

they are paired. Also, in each of the Definitions, equilibrium prices are “homogeneous

of degree zero,” e.g., if (qi) are equilibrium prices, so are (λqi) for λ > 0.

In contrast to Definition 1 where prices are uninformative, in Definition 2 they

achieve a remarkable balance. From profit-maximization,∑

i qi · z = 0. But the

fact that z ∈ ξi(qi) precludes (except in trivial cases) the possibility that qi = 0.2

Therefore, (∑

i qi)(a) will be positive for some a and (∑

i qi)(a′) negative for other a′

reflecting the fact that employing some plays of the game increase the organizer’s profits

while others do the opposite. Alternatively put, the gains from supplying z ∈ ∩iZi

to some individuals cannot be separated from the costs imposed by others. On the

buyers’ side, the individual components qi(a) of the sum (∑

i qi)(a) are such that if the

individual were restricted to the hypothetical budget constraint γi(qi; 0) each individual

could no better than choosing z. Moreover, since profit maximization implies z ∈ Zi,

i’s incentive unconstrained price-taking behavior z ∈ ξi(qi) is as-if his choices were

restricted to ∆m ∩ Zi.

The enforcement of the budget constraint γi(qi; 0) is an essential difference between

the way z ∈ ξi(qi) is defined as utility maximizing demand, above, and the standard

definition. In the latter, purchases must be financed by sales. The tacit assumption

supporting this conclusion is that such transactions are assumed to be observable and

enforceable. However, in a game, each individual is free to deviate because his actions

are neither observable nor enforceable; hence, incentives must be provided to make the

recommended lottery contract self-enforcing.

To illustrate incentive constrained Lindahl equilibrium, consider Example 2 and Fig-

ures 1 and 2. The point A = (z(a1, b1), z(a1, b2), z(a2, b1), z(a2, b2)) = (1/3, 0, 1/3, 1/3)

is associated with the following prices.

2In an incentive constrained Lindahl equilibrium 〈z, (qi)〉, qj = 0 would occur only if vj · z = max{vj(a) :

a ∈ A}.

12

Payoffs Prob. Price for 1 Price for 2 Price for Seller

A (10/3, 10/3)1/3 0

1/3 1/3

5/3 −10/3

2/3 −7/3

−7/3 −10/3

2/3 5/3

−2/3 −20/3

4/3 −2/3

At prices q1 = (5/3,−10/3, 2/3,−7/3) and q2 = (−7/3,−10/3, 2/3, 5/3), A ∈

ξ1(q1)∩ξ2(q2). At these prices, the sum of the values for supplying each of the 4 possible

plays of the game is q1 +q2 = (−2/3,−20/3, 4/3,−2/3). The choice of A = z ∈ Z1∩Z2

maximizes (q1 + q2) · z among those z ∈ Z1 ∩ Z2.

Because the seller recognizes that z′ = (0, 0, 1, 0) /∈ Z1∩Z2, the fact that q1(a2, b1)+

q2(a2, b1) > 0 does not tempt the seller to want to supply z′. (The seller knows that

z′ is unenforceable.) Also, note that profit maximization is based on the sum of the

prices paid by the individuals for each play of game. It does not satisfy the separate

conditions that z is a solution to

maxz

q1 · z − c1(z) and maxz

q2 · z − c2(z).

In other words, the supplier typically relies on cross-subsidization.3

To illustrate a play of the game that is not an incentive constrained Lindahl equi-

librium, consider B = (1/2, 0, 0, 1/2). The following prices would induce both players

to choose B.

Payoffs Prob. Price for 1 Price for 2 Price for Seller

B (3, 3)1/2 0

0 1/2

2 −3

1 −2

−2 −3

1 2

0 −6

2 0

However, given the prices (q1 + q2) = (0,−6, 2, 0), the seller would not choose B.

Rather, the seller would want to increase the quantity z(a2, b1) because the profit from

selling more of it is positive, and (1/2− ε, 0, 2ε, 1/2− ε) ∈ Z1 ∩Z2. Since all the sellers

see the profit opportunity, this is not an equilibrium.

In addition to A, the following Table illustrates two other incentive constrained

Lindahl equilibrium plays of the game, H and E, along with their prices.

3This is a familiar conclusion from pricing with joint supply, e.g., of beef and hides.

13

Payoffs Prob. Price for 1 Price for 2 Price for Seller

H (13/6,25/6)1/6 0

1/6 2/3

17/6 −13/6

11/6 −7/6

−133/30 −35/6

−7/30 7/6

−8/5 −8

8/5 0

E (1, 5)0 0

0 1

4 −1

3 0

−28/5 −7

−7/5 0

−8/5 −8

8/5 0

Definitions 1 and 2 have incentive constraints in common but differ with respect

to utility maximization. The following definition reverses the comparison with 2 by

having utility maximization in common but not incentive constraints.

To capture the idea that that incentive constraints are ignored, let c0(z) = 0 if

z ∈ Z0 ≡ Rm+ and c0(z) = ∞, otherwise. Profit maximization is now

π0(∑

iri) = sup{(∑

iri) · z −∑

ic0(z)},

and profit maximizing supply is

η0(∑

iri) = {z : (∑

iri) · z −∑

ic0(z) = π0(

∑iri)}.

Definition 3 〈z, (ri)〉 is a Lindahl equilibrium if for all i,

• z ∈ ∩iξi(ri) (utility maximization)

• z ∈ η0(∑

i ri) (profit maximization without incentive constraints)

Because the feasible set for profit-maximizing supply is now Rm+ (instead of ∩iZi),

the polar is∑

i Pi = Rm− (instead of (∩iZi)∗). Consequently, attention can be restricted

to those (ri) such that∑

i ri = 0, instead of (∑

i qi) · z.

We reiterate that the usual deficiencies of Lindahl equilibrium — personalized prices

depend on the preferences of the individuals that they would be reluctant to reveal—

does not apply here because in a game with complete information there are no hidden

characteristics. Hence, (ri) could be determined. The problem with Definition 3 is

hidden actions —the z associated with (ri) could not be enforced.

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Referring to Example 2, again, to illustrate, if there is no incentive compatibility

constraint, the seller would try to sell more of z(a2, b1) beyond point A. One possible

equilibrium without incentive is C = (0, 0, 1, 0) with the following prices.

Payoffs Prob. Price for 1 Price for 2 Price for Seller

C (4, 4)0 0

1 0

1 −4

0 −3

−3 −4

0 1

−2 −8

0 −2

5 Existence and Efficiency

From the parallels with general equilibrium theory, both incentive constrained Lindahl

equilibria and Lindahl equilibria are efficient, although different standards of efficiency

are required for each. But the public goods feature of a normal form game adds an

unusual level of indeterminacy. Changes in the distribution of wealth are typically

required in a general equilibrium model to demonstrate that price-taking equilibria

fill out the entire set of efficient allocations. In this setting, however, incentive con-

strained Lindahl equilibria for a given game will coincide with the set of all incentive

constrained efficient allocations. Similarly, Lindahl equilibria will coincide with the set

of all efficient allocations.

To define efficiency, recall that Ri(z) = {z : z ∈ ∆m, vi · z ≥ vi · z, }. Define Si(z)

as those z ∈ Ri(z) that are strictly preferred. Let Z ⊂ Rm. Say that z is Z-efficient if

for each j,

Z⋂(

Sj(z) ∩i6=j Ri(z))

= ∅.

I.e., relative to Z, it is not possible to increase the utility of one individual without

decreasing the welfare of someone else. The efficiency criterion for incentive constrained

Lindahl equilibrium will be Z = ∩iZi, while for Lindahl equilibrium the criterion is

Z = Rm+ .

∆m is evidently non-empty and compact; hence, so is the efficient subset of those

z ∈ Rm+ ∩

(Sj(z) ∩i6=j Ri(z)

)= ∅ for all j. A similar conclusion applies to those

z ∈ (∩iZi)∩(Sj(z)∩i6=jRi(z)

)= ∅ for all j. The following Propositions demonstrate the

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existence of incentive constrained Lindahl equilibrium by showing that it coincides with

the incentive constrained efficient set and its dual, while the set of Lindahl equilibria

coincides with the ∆m-efficient set and its dual. Stated in the language of general

equilibrium theory, each of these Propositions combines the existence of equilibrium

with the conclusions of the First and Second Theorems of Welfare Economics.

Proposition 2 z is incentive constrained efficient if and only if there exists (qi) such

that 〈z, (qi)〉 is an incentive constrained Lindahl equilibrium.

Proposition 3 z is ∆m-efficient if and only if there exists (ri) such that 〈z, (ri)〉 is a

Lindahl equilibrium.

Proof: Suppose that 〈z, (qi)〉 is an incentive constrained Lindahl equilibrium that

is not incentive efficient. Then there exists i and z ∈ ∩(∩iZi) such that z ∈ Si(z) ∩i6=i

Ri(z). Therefore, qi · z > 0 and qi · z ≥ 0, so∑

i qi · z >∑

i qi · z = 0. From the

hypothesis that z is profit-maximizing at∑

i qi,∑

i qi · z ≥∑

i qi · z, a contradiction.

Conversely, suppose (∩iZi)⋂(

Sj(z)∩i6=j Ri(z))

= ∅ for all i. By Lemma 1, for each

i there exists a qi ∈ ξ−1i (z), so z ∈ ∩iξi(qi). By construction, qi · z = 0, so

∑i qi · z = 0.

Therefore, z ∈ η(∑

i qi).

The demonstrations that a Lindahl equilibrium 〈z, (ri)〉 is ∆m-efficient and that any

∆m-efficient z can be paired with some (ri) to form a Lindahl equilibrium are similar

to the above. �

6 Concluding Remark: the locus of competition

An implicit assumption in game theory is that interactions among the players occurs in

a self-contained environment. Consequently, the notions of competition and the pur-

suit of self-interest cannot be separated in non-cooperative equilibrium. Nevertheless,

implementation of correlated equilibrium has suggested a role for an outside observer

or mediator. That role is significantly expanded here, so that the locus of competition

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is redirected from the players themselves to competition among organizers for the right

to manage the behavior of a group of individuals. The one-sided perfect competition

among organizers does not eliminate the consequences of self-interest among the play-

ers. But it does reduce the set of outcomes to those lying on the upper utility boundary

of the correlated equilibrium set. The remaining indeterminacy is a reflection of the

fact that each player has no alternative but to play the game with the others. In

a world where individuals have to compete with each other for the opportunities to

participate in games, the indeterminacy would be reduced or eliminated, but incentive

constraints would continue to bind.

References

[1] Aumann, R. J., “Subjectivity and Correlation in Randomized Strategies,” Journal

of Mathematical Economics 1: 67-96 (1974)

[2] Hart, S., and D. Schmeidler, “Existence of Correlated Equilibria,” Mathematics

of Operations Research 14: 18-25 (1989)

[3] Myerson, R. B., Game Theory: Analysis of Conflict, Harvard University Press

(1991)

[4] Nau, R. F., and K. F. McCardle, “Coherent Behavior in Noncooperative Games,”

Journal of Economic Theory 50: 424-444 (1990)

[5] Rockafellar, R. T., Convex Analysis, Princeton University Press, (1970)

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