Hervé Crès
email: herve.cres@sciences-po.fr.

Mich Tvede
email: mich.tvede@ncl.ac.uk.

Production externalities: internalization by voting *

Keywords: general equilibrium, majority voting, production externalities, shareholder governance vs. stakeholder democracy, social choice JEL-classification: D21, D51, D72, G39, L21

We study internalization of production externalities in perfectly competitive markets where production plans are decided by majority voting. Since shareholders want firms to maximize dividends of portfolios rather than profits, they are interested in some internalization. Two governances, namely the shareholder governance (one share, one vote) and the stakeholder democracy (one stakeholder, one vote), are compared. We argue that perfect internalization is more likely to be the outcome of the stakeholder democracy than the shareholder governance.

Introduction

Decisions on some activities, such as consumption, saving and work, are made individually while decisions on other activities, such as production and policy, are made collectively. In case of perfectly competitive and complete markets, the outcome of each type of decision has a positive impact on the outcome of the other type: (1) utility maximization results in equal marginal rates of substitution across consumers so shareholders agree unanimously on profit maximization as the right objective for collective decision making; and, (2) profit maximization results in the maximal dividends to shareholders making the outcome Pareto optimal.

However, in case of perfectly competitive markets and direct externalities between firms none of these two properties hold: (1) equalization of marginal rates of substitution does not make the shareholders agree on profit maximization; and on top of that, (2) the outcome of firm by firm profit maximization is typically not Pareto optimal. In order to understand why shareholders disagree with profit maximization, consider at one extreme a consumer with shares in only one firm: she wants that firm to maximize its profit irrespectively of the impact on profits of other firms, corresponding to no internalization. At the other extreme, consider a consumer with a fraction of the market portfolio, where the market portfolio is the sum of portfolios of all shareholders: she wants every firm to maximize aggregate profit1 , corresponding to perfect internalization.

The argument is illustrated in [START_REF] Hansen | Externalities and corporate objectives in a world with diversified shareholder/consumers[END_REF]. Indeed they emphasize that, besides the traditional benefits of risk reduction, portfolio diversification offers additional benefits for shareholders through helping internalize externalities. Investigating cross-ownership of stocks by institutions and domestic mutual funds in the computer and automobile industries, they provide evidence that externalities exist and shareholders are well diversified.

Hence for perfectly competitive markets and direct externalities between firms there is a genuine social choice problem in every firm as well as a genuine problem with efficiency in the economy. We study how these problems can be solved through majority voting in firms. In equilibrium consumers maximize utilities subject to budget constraints, production plans in firms are stable with respect to majority voting and markets clear. Our main findings are the following: (1) in terms of behavior of firms, voting is equivalent to maximizing weigted sums of profits with weights being in the intersection of the convex hulls of portfolios of majorities of voters; (2) equilibria exist provided the rate of majority is at least max{q/(q + 1), (n -1)/n}, where n is the number of firms and q is the dimension of the production set; and, (3) perfect internalization of externalities can be the outcome of voting in case the market portfolio is in the intersection of the convex hulls of portfolios of majorities of voters.

Two governances are considered: the shareholder governance (one share, one vote); and, the stakeholder democracy (one stakeholder, one vote), where all consumers are allowed to vote in all firms. The performance of the two governances is compared with respect to aggregation of preferences and market efficiency.

In general, for the shareholder governance a majority of shareholders in some firm tends to have more shares in that firm than in some other firm, so they put "too much" weight on the profit of that firm when voting over production plans in that firm. Therefore perfect internalization is typically not the outcome of voting for the shareholder governance. In the stakeholder democracy, majorities in one firm are also majorities all other firms. Therefore perfect internalization can be the outcome of voting for the stakeholder democracy. At first sight the stakeholder democracy appears unrealistic. However, public regulation in democracies can be seen as a proxy of the stakeholder democracy.

The decision making in firms in case of market failures has received some attention for quite many years. One strand of research has focused on the link between equilibrium and (constrained) Pareto optimality. Contributions include [START_REF] Drèze | Investment under private ownership: optimality, equilibrium and stability[END_REF] and [START_REF] Grossman | A theory of competitive equilibrium in stock market economies[END_REF], who emphasize the role of the mean shareholder, or the mean gradient of the shareholders to be more precise, in the context of incomplete financial markets. In [START_REF] Drèze | Uncertainty and) The firm in general equilibrium theory[END_REF] and [START_REF] Dierker | Ownership structure and control in incomplete market economies with transferable utility[END_REF] the role of the control group is emphasized. In [START_REF] Dierker | The price normalization problem in imperfect competition and the objective of the firm[END_REF], [START_REF] Bejan | The objective of a privately owned firm under imperfect competition[END_REF], Bejan & Bidian (2010) and [START_REF] Magill | Reforming capitalism[END_REF] among others imperfect competition is considered. In particular [START_REF] Bejan | The objective of a privately owned firm under imperfect competition[END_REF] and Bejan & Bidian (2010) consider properties of equilibria where firms maximize the wealth of shareholders.

Another strand of research has focused on majority voting. Contributions include [START_REF] Demarzo | Majority voting and corporate control: the rule of the dominant shareholder[END_REF], who emphasizes the role of the dominant shareholder, [START_REF] Gevers | Competitive equilibrium of a stock exchange and Pareto efficiency[END_REF], [START_REF] Kelsey | the existence of equilibrium in incomplete markets and the objective function of the firm[END_REF], [START_REF] Ritzberger | Shareholder voting[END_REF], [START_REF] Tvede | Voting in assemblies of shareholders and incomplete markets[END_REF], [START_REF] Crès | A geometric study of shareholders' voting in incomplete markets: multivariate median and mean shareholder theorems[END_REF] and [START_REF] Demichelis | A general equilibrium analysis of corporate control and the stock market[END_REF]. In [START_REF] Tvede | Voting in assemblies of shareholders and incomplete markets[END_REF] multidimensional median voter/shareholder theorems à la [START_REF] Greenberg | Consistent majority rules over compact sets of alternatives[END_REF] are obtained. In the present paper we bridge the two strands of research: equilibria are based on majority voting; and, the link between equilibrium and optimality is studied.

The problems arising in the governance of firms because of incomplete markets or imperfect competition are ultimately problems of (indirect) externalities. In case of imperfect competition, decisions on production plans impact shareholders through income and prices and consumers through prices. In case of incomplete markets, decisions on production plans impact shareholders through income and spanning and consumers through spanning. In contrast with the cases of incomplete markets and imperfect competition, the case of direct production externalities makes it possible to fully characterize the relation between the behavior of firms and the distribution of portfolios. Moreover the link between equilibrium and optimality becomes transparent and directly related to the distribution of portfolios. Characterizations parametrized by the distribution of portfolios rather than gradients of consumers have the advantage of being based on available information as can be seen in [START_REF] Hansen | Externalities and corporate objectives in a world with diversified shareholder/consumers[END_REF].

The paper is organized as follows: in Section 2, the set-up including definitions of equilibria and assumptions is presented; in Section 3 the results are stated; and, in Section 4 some concluding remarks are offered.

The model

In the present section the economy is described and the notion of equilibrium is introduced.

Set-up

Consider an economy with ℓ goods, m consumers and n firms. Markets are perfectly competitive and there are direct externalities between firms. In general perfect competition makes sense in case all agents are small relative to the rest of the economy. In case of direct externalities between firms, both production plans and externalities should be small for firms to be small. Indeed actions in one firm should influence at most a few other firms a lot or a lot of other firms a little.

Let p = (p 1 , . . . , p ℓ ), where p k ≥ 0 for all k, be a price vector. Price vectors are normalized such that prices sum to one. Let S = { p ∈ R ℓ + | k p k = 1 } be the set of normalized prices.

Consumers are characterized by their identical consumption sets X = R ℓ , endowment vectors ω i ∈ R ℓ , utility functions u i : X → R and portfolios δ i = (δ i1 , . . . , δ in ) where δ ij ≥ 0 for all i and j and i δ ij = 1 for all j. Let e n = i δ i be the market portfolio, so e n j = 1 for all j. Consumption sets are assumed to be unbounded from below to ensure that for all lists of individual production plans, consumers are able to finance consumption plans in their consumption sets. Since firms are not necessarily maximizing their profits, the value of a firm can be negative.

There are direct externalities between firms: in every firm an action is taken and the production plan of every firm depends on the actions taken in all firms. Firms are described by their sets of action A j ⊂ R q and production functions f j : A → R ℓ , where A = j ′ A j ′ , such that y j = f j (a), where a = (a 1 , . . . , a n ), is the production plan of firm j. Actions could include choice of some inputs or outputs. As an example suppose firms choose inputs while output in every firm depends on aggregate inputs: if K j ≥ 0 is the input of firm j, then the output of firm j is ( j ′ =j K j ′ ) α K β j as in [START_REF] Romer | Increasing returns and long-run growth[END_REF]. Thus the approach in the present paper is in line with the approach used in endogenous growth.

Traditionally in general equilibrium direct externalities between firms are described by correspondences Y j : (R ℓ ) n-1 → R ℓ such that if y -j = (y 1 , . . . , y j-1 , y j+1 , . . . , y n ) is a list of individual production plans for all firms but firm j, then the production set of firm j is Y j (y -j ). In the traditional approach, a list of production plans y is an equilibrium for the production sector if y j ∈ Y j (y -j ) for all j and no y ′ j ∈ Y j (y -j ) is more attractive than y j for any j. Hence, in the traditional approach it is not taken into account that if the production plan of firm j is changed, then the production set of firm j ′ is changed too. Consequently the production plan of firm j ′ can become impossible or more attractive production plans can become possible.

Demand, supply and equilibrium

The demand of every consumer is assumed to be small relative to the rest of the economy and the portfolio of every consumer is assumed to be small relative to the market portfolio e n . Therefore consumers consider prices to be fixed and suppose their votes have no impact on the decisions in the firms. For a price vector p and a list of individual actions a the problem of consumer i is max

x i u i (x i ) s.t. p • x i ≤ p • ω i + j δ ij p • f j (a).
Firms are assumed to be small relative to the rest of the economy. Hence consumers consider the decisions in the firms to have no impact on prices. Describing the decision process in firm j takes a few steps. For a price vector p and a list of individual actions a let P ij (p, a) ⊂ A j be the set of actions in firm j that make consumer i better off or equivalently wealthier

P ij (p, a) = { a ′ j ∈ A j | j ′ δ ij ′ p • f j ′ (a ′ j , a -j ) > j ′ δ ij ′ p • f j ′ (a) }.
For a price vector p, a list of individual actions a and another action a ′ j for firm j let M j (p, a, a ′ j ) ⊂ {1, . . . , m} be the set of consumers who are better off with a ′ j than with a j

M j (p, a, a ′ j ) = { i ∈ {1, . . . , m} | a ′ j ∈ P ij (p, a) }.
Let ρ ∈ [0, 1[ be the rate of majority needed to change actions in firms and let θ = (θ 1 , . . . , θ n ), where θ j = (θ 1j , . . . , θ mj ), θ ij ≥ 0 and i θ ij = 1, be the voting weights. For a price vector p, a list of actions a and another action a ′ j for firm j, a change of actions from a j to a ′ j in firm j is adopted if and only if

i∈M j (p,a,a ′ j ) θ ij > ρ.
Two cases of voting weights are considered: the shareholder governance where θ ij = δ ij (one share, one vote); and, the stakeholder democracy where θ ij = 1/m (one stakeholder, one vote). For the shareholder governance consumer i can vote in firm j if and only if δ ij > 0 and for the stakeholder democracy all consumers can vote in all firms. However consumers without shares have no interest in voting because they receive no dividends. For a price vector p and a list of individual actions a, let Q ρ j (p, a) ⊂ A j be the set of actions preferred to a j in firm j in the sense that every action in Q ρ j (p, a) is preferred to a j by some majority in firm j

Q ρ j (p, a) = { a ′ j ∈ A j | i∈M j (p,a,a ′ j ) θ ij > ρ }.
For a price vector p and a list of individual actions a -j for all firms but firm j the problem of firm j is to find stable actions, i.e., actions a j such that

Q ρ j (p, a j , a -j ) = ∅.
In a ρ-majority stable equilibrium (or ρ-MSE) consumers maximize utilities, actions in firms are stable for unilateral changes and markets clear.

Definition 1 A ρ-majority stable equilibrium is a price vector, a list of individual consumption bundles and a list of individual actions (p, x, ā) such that:

(C) xi is a solution to the problem of consumer i given p and ā for all i.

(F) āj is a solution to the problem of firm j given p and ā-j for all j such that Q ρ j (p, āj , ā-j ) = ∅.

(E) i xi = i ω i + j f j (ā).
Example: For ℓ = 1, m = 2 and n = 2 suppose that the portfolios are δ 1 = (1, 0) and δ 2 = (0, 1), the set of actions A 2 = A 1 = [0, 1], and the production functions

f 1 (a 1 , a 2 ) = 1 -a 1 + 2a 2 and f 2 (a 1 , a 2 ) = 1 -a 2 + 2a 1 . Let ρ ∈ [0, 1[. Then for the shareholder governance there is a unique ρ-MSE with ā = (ā 1 , ā2 ) = (0, 0). Indeed f 1 (a ′ 1 , a 2 ) > f 1 (a)
for all a with a 1 > 0 and a ′ 1 < a 1 so consumer 1 votes for a ′ 1 . Let ρ ∈ [1/2, 1[. Then for the stakeholder democracy there is a continuum of ρ-MSEs because for every pair of actions a = (a 1 , a 2 ) there exists a ρ-MSE with ā = a. Indeed for all a and a ′ 1 with a

′ 1 = a 1 , if a ′ 1 < a 1 , then f 2 (a ′ ) < f 2 (a) so consumer 2 votes against a ′ 1 and if a ′ 1 > a 1 , then f 1 (a ′ ) < f 1 (a) so consumer 1 votes against a ′ 1 .

Multilateral changes of actions in firms

Consumers can typically vote in several firms: for the shareholder governance in all firms they have shares in; and, for the stakeholder governance in all firms. Therefore multilateral changes of actions in firms, where voters coordinate changes of actions in several firms, are considered. Perfectly competitive markets make sense for multilateral changes of actions in case firms can be partitioned into groups of firms such that externalities are restricted to firms in the same groups and groups are small relative to the rest of the economy. The partition of firms could reflect location, technology or other characteristics.

For multilateral changes of actions in firms the problem of the firm has to be changed into a problem of the production sector. For a price vector p and a list of individual actions a, let P i (p, a) ⊂ A be the set of actions that make consumer i better off or equivalently wealthier

P i (p, a) = { a ′ ∈ A | j δ ij p • f j (a ′ ) > j ′ δ ij p • f j (a) }.
For a price vector p and a pair of lists of individual actions a and a ′ , let M (p, a, a ′ ) ⊂ {1, . . . , m} be the set of consumers who are better off with a ′ than with a

M (p, a, a ′ ) = { i ∈ {1, . . . , m} | a ′ ∈ P i (p, a) }.
For a change of actions from a to a ′ the change is adopted if and only if

i∈M (p,a,a ′ ) θ ij > ρ
for all j with a ′ j = a j . For a price vector p and a list of individual actions a, let Q ρ (p, a) ⊂ A be the set of actions preferred to a in the sense that every list of individual actions in Q ρ (p, a) is preferred to a by majorities in all firms where actions are changed

Q ρ (p, a) = { a ′ ∈ A | i∈M (p,a,a ′ ) θ ij > ρ for all j with a ′ j = a j }.
For a price vector p the problem of the production sector is to find stable actions a, i.e., actions a such that Q ρ (p, a) = ∅.

In a strong ρ-MSE consumers maximize utilities, actions are stable for multilateral changes and markets clear.

Definition 2 A strong ρ-majority stable equilibrium is a price vector, a list of individual consumption bundles and a list of individual actions (p, x, ā) such that:

(C) xi is a solution to the problem of consumer i given p and ā for all i.

(P) ā is a solution to the problem of the production sector given p such that Q ρ (p, ā) = ∅.

(E) i xi = i ω i + j f j (ā).
For the shareholder governance economies need not have strong ρ-MSEs because having a majority of the votes in one firm does not imply having a majority of the votes in another firm as the following example shows.

Example: For ℓ = 1, m = 2 and n = 2 suppose the portfolios are δ 1 = (1, 0) and δ 2 = (0, 1), the set of actions A 2 = A 1 = [0, 1], and the production functions

f 1 (a 1 , a 2 ) = 1 -a 1 + 2a 2 and f 2 (a 1 , a 2 ) = 1 -a 2 + 2a 1 . Let ρ ∈ [0, 1[. Then for the shareholder governance there is no strong ρ-MSE: a = (a 1 , a 2 ) with a 1 , a 2 < 1 is not part of a strong ρ-MSE, because f 1 (a ′ ) > f 1 (a) and f 2 (a ′ ) > f 2 (a) for a ′ = a + (1 -max{a 1 , a 2 })(1, 1) so both consumers vote against a; and, a = (1, a 2 ) is not part of a strong ρ-MSE, because f 1 (a ′ ) > f 1 (a) for a ′ = (0, a 2 ) so consumer 1 votes against a.
Let ρ ∈ [1/2, 1[. Then for the stakeholder democracy there is a continuum of strong ρ-MSEs: a = (a 1 , a 2 ) is part of a strong ρ-MSE if and only if

a 1 = 1 or a 2 = 1. Indeed if a 1 = 1 or a 2 = 1, then f 1 (a ′ ) < f 1 (a) or f 2 (a ′ ) < f 2 (a)
for every a ′ = a so one of the consumers votes against a ′ and if both a 1 < 1 and a 2 < 1, then f 1 (a ′ ) > f 1 (a) and f 2 (a ′ ) > f 2 (a) for a ′ = a + (1 -max{a 1 , a 2 })(1, 1), so both consumers vote against a.

Assumptions

Consumers are supposed to satisfy the following assumptions:

(A.1) u i is continuous. (A.2) u i is strongly monotone, so z k i ≥ x k i for all k and z i = x i imply u i (z i ) > u i (x i ), and quasi-concave, so u i ((1-τ )x i +τ z i ) ≥ min{ u i (x i ), u i (z i ) } for all τ ∈ [0, 1]. (A.3) The set u -1 i (r) = { x i ∈ X | u i (x) = r } is bounded from below for all r ∈ R.
All assumptions are standard.

Firms are supposed to satisfy the following assumptions:

(A.4) A j is convex and compact.

(A.5) f j is continuous.

(A.6) f j is concave, so f k j ((1 -τ )a + τ a ′ ) ≥ (1 -τ )f k j (a) + τ f k j (a ′ ) for all k and τ ∈ [0, 1].
The assumptions ensure that the free disposal hull of the production set

{ y ∈ R ℓn | ∃a : y k j ≤ f k j (a) for all j and k } is convex.
3 Results

In the present section results are presented and discussed.

Objectives of firms

Below it is shown that the outcome of voting over actions in firms can be viewed as solutions to firms maximizing weighted sums of profits. The issue is identification of the weights. Clearly, coalitions of voters with majorities of votes are decisive in the process of choosing weights. Therefore the weights are related to the portfolios of voters in these coalitions. Before stating the results some notation is needed.

Let Λ n-1 + be the unit simplex in R n + Λ n-1 + = { λ ∈ R n + | j λ j = 1 }.
For I = {1, . . . , m} let Γ : 2 I → Λ n-1 + , where 2 I is the set of all subsets of I, be a correspondence that maps coalitions of consumers to convex hulls of normalized portfolios

Γ(M ) =    { λ ∈ Λ n-1 + | ∃α ∈ R |M | + : i∈M α i δ i = λ } for i∈M δ i = 0 Λ n-1 + for i∈M δ i = 0.
For the shareholder governance every coalition M with i∈M δ ij > ρ is decisive in firm j. Let I ρ j ⊂ 2 I be the set of decisive coalitions so M ∈ I ρ j if and only if i∈M δ ij > ρ. For the stakeholder democracy every coalition M with |M | > ρm is decisive in every firm. Let I ρ ⊂ 2 I be the set of decisive coalitions so M ∈ I ρ if and only if |M | > ρm.

For a finite set of vectors v 1 , . . . , v k let co {v 1 , . . . , v k } be the convex hull of the vectors v 1 , . . . , v k .

The outcome of voting is equivalent to maximizing a weighted sum of profits. Indeed for every decisive coalition there exist weights λ M in the convex hull of portfolios Γ(M ) such that the outcome is equivalent to maximizing a weighted sum of profits for all weights in the convex hull of the weights of the decisive coalitions co {λ M } M .

Theorem 1 Consider an economy.

• Suppose (p, x, ā) is a ρ-majority stable equilibrium for the shareholder governance. Then for all j and M ∈ I ρ j there exists λ j M ∈ Γ(M ) such that āj maximizes j ′ λ j j ′ p • f j ′ (a j , ā-j ) for all λ j ∈ co {λ j M } M ∈I ρ j . • Suppose (p, x, ā) is a ρ-majority stable equilibrium for the stakeholder democracy. Then for all j and M ∈ I ρ there exists

λ j M ∈ Γ(M ) such that āj maximizes j ′ λ j j ′ p • f j ′ (a j , ā-j ) for all λ j ∈ co {λ j M } M ∈I ρ .
• Suppose (p, x, ā) is a strong ρ-majority stable equilibrium for the stakeholder democracy. Then for all M ∈ I ρ there exists

λ j M ∈ Γ(M ) such that ā maximizes j ′ λ j ′ p • f j ′ (a) for all λ ∈ co {λ j M } M ∈I ρ .
Proof: The proof is by contradiction. For the shareholder governance suppose (p, x, ā) is a ρ-MSE and there exist j and M ∈ I ρ j such that āj does not maximize j ′ λj ′ p • f j ′ (a j , ā-j ) for any λ ∈ Γ(M ).

According to Theorem 21.1 in [START_REF] Rockafellar | Convex Analysis[END_REF] either: (1) there exists a j ∈ A j such that

j ′ δ ij ′ p • (f j ′ (ā j , ā-j ) -f j ′ (a j , ā-j )) < 0
for all i ∈ M ; or alternatively, (2) there exists (α i ) i , where α i ≥ 0 for all i ∈ M and i α i > 0, such that

i α i j ′ δ ij ′ p • (f j ′ (ā j , ā-j ) -f j ′ (a j , ā-j )) ≥ 0 for all a j ∈ A j .
Suppose a j ∈ A j is a solution to (1), then a j ∈ Q ρ j (p, ā). This contradicts that (p, x, ā) is a ρ-MSE.

Suppose (α i ) i , where α i ≥ 0 for all i and i α i > 0, is a solution to (2), then āj maximizes j ′ α ′ j ′ p • f j ′ (a j , ā-j ) for α ′ = i α i δ i , but this contradicts that āj does not maximize j ′ λj ′ p • f j ′ (a j , ā-j ) for any λ ∈ Γ(M ).

Suppose for all j and M ∈ I ρ j there exists λ j M such that āj maximizes j ′ λ j M j ′ p • f j ′ (a j , ā-j ). Then clearly āj maximizes j ′ λ j j ′ p • f j ′ (a j , ā-j ) for all λ j ∈ co {λ j M } M ∈I ρ j . The proofs for the stakeholder democracy are identical to the proof for the shareholder governance.

Q.E.D.

For a ρ-MSE or a strong ρ-MSE (p, x, ā) assume the sets of actions are smooth manifolds with boundary, the production functions are differentiable and the n × q-matrix of derivatives of profits

   pT D a j f 1 (ā) . . . pT D a j f n (ā)   
has rank n for all j. Then the outcome of voting is equivalent to maximizing a weighted sum of profits for weights being in the intersection of the convex hulls of portfolios of decisive coalitions ∩ M Γ(M ).

Corollary 1 Assume A j is a smooth q-dimensional manifold with boundary and f j : A → R ℓ is a differentiable function for all j. For (p, ā) assume the

n × q-matrix    pT D a j f 1 (ā) . . . pT D a j f n (ā)   
has rank n for every j.

• Suppose (p, x, ā) is a ρ-majority stable equilibrium for the shareholder governance. Then for all j there exists λ j ∈ ∩ M ∈I ρ j Γ(M ) such that āj maximizes j ′ λ j j ′ p • f j ′ (a j , ā-j ).

• Suppose (p, x, ā) is a ρ-majority stable equilibrium for the stakeholder democracy. Then for all j there exists λ j ∈ ∩ M ∈I ρ Γ(M ) such that āj maximizes j ′ λ j j ′ p • f j ′ (a j , ā-j ).

• Suppose (p, x, ā) is a strong ρ-majority stable equilibrium for the stakeholder democracy. Then there exists λ ∈ ∩ M ∈I ρ Γ(M ) such that ā maximizes j ′ λ j ′ p • f j ′ (a).

Proof: The proof is by contradiction. For the shareholder governance suppose (p, x, q) is a ρ-MSE and there exists j such that ∩ M ∈I ρ j Γ(M ) = ∅ or āj does not maximize j ′ λ j ′ p • f j ′ (a j , ā-j ) for any λ ∈ ∩ M ∈I ρ j Γ(M ). It follows from Theorem 1 that if (p, x, ā) is a ρ-MSE, then for all j and M ∈ I ρ j , there exists λ j M ∈ Γ(M ) such that āj maximizes j ′ λ M j ′ p • f j ′ (a j , ā-j ). Therefore suppose there exist M, M ′ ∈ I ρ j and λ ∈ Γ(M ) and λ ′ ∈ Γ(M ′ ), where λ ′ = λ, such that āj maximizes both j ′ λ j ′ p • f j ′ (a j , ā-j ) and j ′ λ ′ j ′ p • f j ′ (a j , ā-j ). By assumption if āj is in the interior of A j , then āj + ∆a j ∈ A j provided ∆a j is sufficiently small. If āj is in the boundary of A j , then there exists v = 0 such that for all ∆a j ∈ R q with ∆a j = 0, if v • ∆a j ≤ 0, then there exists a sequence (∆a k j ) k∈N , where lim k→∞ ∆a k j = 0 and lim k→∞ 1 ∆a k j ∆a k j -1 ∆a j ∆a j = 0, such that āj + ∆a k j ∈ A j for all k. Let v = 0 in case āj is in the interior of A j . Then āj maximizing j ′ λ j ′ p T f j ′ (a j , ā-j ) implies there is no solution to

( j ′ λ j ′ p T D a j f j ′ (ā j , ā-j )) • ∆a j > 0 v • ∆a j < 0 (1)
and āj maximizing j ′ λ ′ j ′ p T f j ′ (a j , ā-j ) implies there is no solution to

( j ′ λ ′ j ′ p T D a j f j ′ (ā j , ā-j )) • ∆a j > 0 v • ∆a j < 0. (2)
The two vectors j ′ λ j ′ p T D a j f j ′ (ā j , ā-j ) and j ′ λ ′ j ′ p T D a j f j ′ (ā j , ā-j ) are not collinear because by assumption the n × q-matrix

   pT D a j f 1 (ā) . . . pT D a j f n (ā)   
has rank n for every j and λ and λ ′ are not collinear. Hence there exists a solution to (1) or ( 2), but this contradicts that āj maximizes both j ′ λ j ′ p T f j ′ (a j , ā-j ) and j ′ λ ′ j ′ p T f j ′ (a j , ā-j ). Thus (p, x, ā) is not a ρ-MSE. The proofs for the stakeholder democracy are identical to the proof for the shareholder governance.

Q.E.D.

A list of individual actions a is productively efficient if and only if there does not exist another list of individual actions a ′ such that f k j (a ′ ) ≥ f k j (a) for all j and k and f (a ′ ) = f (a). As far as productive efficiency of actions is concerned, the stakeholder democracy is likely to perform better than the shareholder governance. For the shareholder governance, suppose that there exist firms j and j ′ such that (∩ M ∈I ρ j Γ(M )) ∩ (∩ M ′ ∈I ρ j ′ Γ(M ′ )) = ∅. Then the outcome of the production sector is productively inefficient because firms are maximizing weighted sums of profits using different weights. For the stakeholder democracy, the outcome of the production sector can be productively efficient because firms can be maximizing weighted sums of profits using identical weights (assuming

∩ M ∈I ρ Γ(M ) ∩ R n ++ = ∅).
Moreover the outcome of the production sector is productively efficient in strong ρ-MSEs (assuming ∩ M ∈I ρ Γ(M ) ⊂ R n ++ ) because firms are using identical weights.

Existence of equilibrium

Consider a society consisting of some individuals who must choose an alternative from a set of alternatives. Suppose the set of alternatives is compact and convex and has dimension k and individuals have convex and continuous preferences. Then, as shown in [START_REF] Greenberg | Consistent majority rules over compact sets of alternatives[END_REF], there exists a majority stable equilibrium provided the rate of majority is at least k/(k + 1).

In the present paper the conflicts between voters on the objectives of firms can be formulated as conflicts over actions or conflicts over relative weights on profits in firms. For ρ-MSEs the dimension of the set of actions A j is q and the dimension of the set of relative weights Λ n-1 + is n -1. For strong ρ-MSEs the dimension of the set of actions A is qn and the dimension of the set of relative weights Λ n-1

+ is n -1.
Theorem 2 Consider an economy.

• Suppose

ρ ≥ min q q + 1 , n -1 n .
Then there exist ρ-majority stable equilibria for both governances.

• Suppose

ρ ≥ n -1 n .
Then there exist strong ρ-majority stable equilibria for the stakeholder democracy.

The bound on the rate of majority in Theorem 2 is very high even for economies with few firms, but the bound is binding. Indeed as the following example shows, there exists an economy such that if the rate of majority is lower than the bound, then the economy does not have a ρ-MSE. However in Section 4 it is discussed how the bound can be lowered to 1 -1/e ≈ 0.64 or even 0.5 for some classes of distributions of portfolios.

Example: For ℓ = 1 and n = m suppose consumer i is the sole owner of firm j = i and has 1/m of the shares in firm j = m for i < m and consumer i has 1/m of the shares in firm j = m for i = m and the rate of majority is ρ < (n -1)/n. Suppose A j = Λ n-1 + for all j and the production functions are f j (a m , a -m ) = a mj for all j so output in every firm only depends on the action of firm m.

Let ρ < q/(q + 1) = (n -1)/n. If a mj > 0, then all consumers but consumer i = j are better off with a ′ m where a ′ mj ′ = a mj ′ + a mj /(m -1) for all j ′ = j and a ′ mj ′ = 0 for j ′ = j. Therefore for all a m ∈ A m it is possible to make at least n -1 out of n consumers better off. Hence there is no ρ-MSE for neither the shareholder governance nor the stakeholder democracy.

Proof of Theorem 2

For ρ-MSEs under the shareholder governance the proof consists of two parts: min{q/(q + 1), (n -1)/n} = q/(q + 1); and, min{q/(q + 1), (n -1)/n} = (n -1)/n. In the first part the conflict between shareholders in firm j is over actions. The first part is a straightforward application of the theorem in [START_REF] Shafer | Equilibrium in abstract economies without ordered preferences[END_REF] or Corollary A.1 in [START_REF] Won | Equilibrium theory with unbounded consumption sets and non-ordered preferences. Part I. Non-satiation[END_REF]. Alternatively In the second part the conflict between shareholders in firm j is over weights on profits. The second part consists of two steps: first artificial equilibria are introduced and shown to be equivalent to ρ-MSEs; and, second existence of artificial equilibria is established.

The proof for ρ-MSEs under the stakeholder democracy is identical to the proof for the shareholder governance. The proof for strong ρ-MSEs under the stakeholder democracy is identical to the second part of the proof for the shareholder governance.

The min{q/(q + 1), (n -1)/n} = q/(q + 1) part Assume ρ ≥ q/(q + 1). Then Q ρ j has open graph and a j / ∈ co Q ρ j (p, a) for all (p, a) according to the proof of Theorem 2 in [START_REF] Greenberg | Consistent majority rules over compact sets of alternatives[END_REF].

Let

v L , v U ∈ R ℓ be such that if a ∈ A, then v k L < f k j (a) < v k U for all j and k. Let w L ∈ R ℓ be such that if u i (x i ) ≥ u i (ω i + j δ ij v L ), then x k i > w k
L for all i and k. Let the truncated consumption set X T ⊂ R ℓ be defined by

X T = { x ∈ X | w k L ≤ x k ≤ i ω k i + nv k U -(m -1)w k L for all k }.
Let the budget correspondence C i : S × A → X T be defined by

C i (p, a) = { x ′ i ∈ X T | p • x ′ i ≤ p • ω i + j δ ij f j (a) }.
Then C i is continuous with non-empty, convex and compact values. Let the preference correspondence Q i : X T → X T be defined by

Q i (x i ) = { x ′ i ∈ X T | u i (x ′ i ) > u i (x i ) }.
Then Q i has open graph with convex values and x i / ∈ Q i (x i ) for all x i . Let the preference correspondence of the auctioneer Q 0 : S×(X T ) m ×A → S be defined by

Q 0 (p, x, a) = { p ′ ∈ S | p ′ • ( i x i -i ω i -j f j (a)) > p • ( i x i -i ω i -j f j (a)) }.
Then Q 0 has open graph with convex values and p / ∈ Q 0 (p, x, a) for all (p, x, a).

According to the theorem in [START_REF] Shafer | Equilibrium in abstract economies without ordered preferences[END_REF] or Corollary A.1 in [START_REF] Won | Equilibrium theory with unbounded consumption sets and non-ordered preferences. Part I. Non-satiation[END_REF]

there exists (p, x, ā) ∈ S × (X T ) m × A such that xi ∈ C i (p, ā) and Q i (x i ) ∩ C i (p, ā) = ∅ for all i, Q ρ j (p, ā) = ∅ for all j and Q 0 (p, x, ā) = ∅.
Next it is shown that an equilibrium of the truncated economy is a ρ-MSE of the original economy. Suppose pk = 0 for some k, then xk

i = i ′ ω k i ′ + nv k U -(m -1)w k
L for all i because u i is strongly monotone for all i. Therefore

i xk i > i ω k i + j f k j (ā) because i xk i - i ω k i - j f k j (ā) = (m -1) i (ω i + j δ ij v k U -w k L ) + j (v k U -f k j (ā)) > (m -1) i (ω i + j δ ij v k L -w k L ) + j (v k U -f k j (ā)) Hence i xk ′ i > i ω k ′ i + j f k ′ j (ā) for all k ′ because Q 0 (p, x, ā) = ∅. Thus there exists i such that p • xi > p • ω i + j δ ij p • f j (ā), but this contradicts that xi ∈ C i (p, ā). All in all, pk > 0 for all k. Condition (E) in Definition 1 is satisfied because Q 0 (p, x, ā) = ∅ and pk > 0 for all k imply i xk i = i ω k i + j f k j (ā) for all k. Condition (F) in Definition 1 is satisfied because Q ρ j (p, ā) = ∅ for all j.
It takes a few steps to show that condition (C) in Definition 1 is satisfied.

Firstly xk i > w k L because ω k i + j δ ij f k j (ā) > ω k i +nv k L for all k. Secondly xk i < i ′ ω k i ′ +nv k U -(m-1)w k L
for all i and k because xk i ′ > w k L for all i ′ and k, f j (ā) < v k U for all j and k and

i xi = i ω i + j f j (ā). Thirdly if u i (x i ) > u i (x i ), then u i ((1-τ )x i +τ xi ) > u i (x i ) for all τ ∈]0, 1]. Indeed if there exists τ ∈]0, 1[ such that u i ((1 -τ )x i + τ xi ) = u i (x i ),
then for e ℓ ∈ R ℓ being the vector with all coordinates equal to one there exists ε > 0 such that u i (x i -εe ℓ ) > u i (x i ), because u i is continuous, and

u i ((1 -τ )(x i + εe ℓ ) + τ xi ) < u i (x i ), because u i is strongly monotone, but this contradicts that u i is quasi-concave. Therefore if there exists x i ∈ X such that u i (x i ) > u i (x i ) and p • x i ≤ p • ω i + j δ ij p • f j (ā), then there exists x ′ i ∈ X T such that u i (x ′ i ) > u i (x i ) and p • x ′ i ≤ p • ω i + j δ ij p • f j (ā). Hence xi ∈ C i (p, ā) and Q i (x i ) ∩ C i (p, ā) = ∅ for all i imply that condition (C) in Definition 1 is satisfied. All in all, (p, x, ā) is a ρ-MSE.
Definition 3 An artificial equilibrium is a list of individual vectors, a price vector, a list of individual consumption bundles and a list of individual actions (μ, p, x, ā), where μ = (μ 1 , . . . , μn ) and μj ∈ Λ ℓn-1 for all j, such that:

(C) xi is a solution to the problem of consumer i given p and ā for all i.

(F') āj maximizes the profit of firm j given μj and ā-j , so āj is a solution to max

a j j ′ μj ′ j • f j ′ (a j , ā-j ) s.t. a j ∈ A j . (F") W ρ j (p, μj ) = ∅. (E) i xi = i ω i + j f j (ā).
The problems of the firms (F') and (F") in Definition 3 are artificial in the sense that they are not related to the preferences of the consumers. However as shown in Lemma 1 if (μ, p, x, ā) is an artificial equilibrium, then (p, x, ā) is a ρ-MSE.

Lemma 1 Suppose (μ, p, x, ā) is an artificial equilibrium, then (p, x, ā) is a ρ-majority stable equilibrium.

Proof: Suppose (μ, p, x, ā) is an artificial equilibrium. Then (C) and (E) are satisfied. Therefore it suffice to show that (F) is satisfied. The strategy of the proof is to show that if Q ρ j (p, ā) = ∅, then W ρ j (p, μ) = ∅. Suppose Q ρ j (p, ā) = ∅, then there exists a j ∈ A j such that

i∈M j (p,ā,a j ) θ ij > ρ.
Let ȳj ′ = f j ′ (ā) and y j ′ = f j ′ (a j , ā-j ) for all j ′ , then

j ′ μj ′ j • (y j ′ -ȳj ′ ) ≤ 0 and j ′ φ i j ′ (p) • (y j ′ -ȳj ′ ) > 0
for all i ∈ M j (p, ā, a j ). Let µ j ∈ Λ ℓn-1 be defined by

µ j = μj + τ       y 1 -ȳ1 . . . y n -ȳn    - j ′ (y k j ′ -ȳk j ′ ) • e ℓn    e . . . e       .
Then tedious and straightforward calculations show that µ j ∈ V i (p, μj ) if and only if

j ′ (2φ i j ′ (p) -(µ j ′ j + μj ′ j )) • (µ j ′ j -μj ′ j ) > 0.
Therefore there exists τ > 0 such that µ j ∈ V i (p, μj ) for all j ∈ M j (p, ā, a j ) because

j ′ (2φ i j ′ (p) -(µ j ′ j + μj ′ j )) • (µ j ′ j -μj ′ j ) = 2τ j ′ (φ i j ′ (p) -μj ′ j ) • (y j ′ -ȳj ′ ) -τ 2 j ′ (y j ′ -ȳj ′ ) - j ′′ (y j ′′ -ȳj ′′ ) • e ℓn e 2 .
Hence if Q ρ j (p, ā) = ∅, then W ρ j (p, μ) = ∅. Thus if (μ, p, x, ā) is an artificial equilibrium, then (p, x, ā) is a ρ-MSE.

Q.E.D.

The min{q/(q + 1), (n -1)/n} = (n -1)/n part If (μ, p, x, ā) is an artificial equilibrium, then μj is in the set co {φ i (p)} i∈M δ =0 for all j. The set co {φ i (p)} i∈M δ =0 has dimension n -1 or less. Indeed

µ ∈ co {φ i (p)} i∈M δ =0 if and only if there exists λ ∈ Γ(M δ =0 ) such that µ =    λ 1 p . . . λ n p    .
Assume ρ ≥ (n -1)/n. Then ∩ M ∈I ρ j Γ(M ) is non-empty according to the proof of Theorem 2 in [START_REF] Greenberg | Consistent majority rules over compact sets of alternatives[END_REF]. For any λ j ∈ ∩ M ∈I ρ j Γ(M ), let the function ψ j : S → Λ ℓn-1 be defined by

ψ j (p) =    λ j 1 p . . . λ j n p    .
Then ψ j (p) ∈ ∩ M ∈I ρ j co {φ i (p)} i∈M for all p. Therefore W ρ j (p, ψ j (p)) = ∅ for all p. Hence (F") is satisfied for all p ∈ S.

For p and a -j the problem of firm j is max

a j j ′ ψ j j ′ (p) • f j ′ (a j , a -j ) s.t. a j ∈ A j .
It follows from Berge's maximum theorem that the solution correspondence α j : S × A -j → A j of firm j is upper hemi-continuous and convex valued.

For p and a the truncated problem of consumer i is max

x i u i (x i ) s.t.    p • x i ≤ p • ω i + j δ ij f j (a)
x i ∈ X T .

It follows from Berge's maximum theorem that the demand correspondence β i : S × A → X T is upper hemi-continuous and convex valued. For x and a the price problem is

max p p • i x i - i ω i - j f j (a) s.t. p ∈ S.
It follows from Berge's maximum theorem that the price correspondence γ : (X T ) m × A → S is upper hemi-continuous and convex valued.

Let the correspondence g : S × (X T ) m × A → S × (X T ) m × A be defined by g(p, x, a) = (γ(x, a), β 1 (p, a), . . . , β m (p, a), α 1 (p, a -1 ), . . . , α n (p, a -n )).

It follows from Kakutani's fixed point theorem that the correspondence has a fixed point (p, x, ā). Clearly, (μ, p, x, ā), where μ = (ψ 1 (p), . . . , ψ n (p)), is an artificial equilibrium so (p, x, ā) is a ρ-MSE according to Lemma 1.

Indeterminacy of equilibria

From the proof of Theorem 2 it follows that the outcome of maximizing a weighted sum of profits for weights in the intersection of the convex hulls of portfolios of decisive coalitions ∩ M Γ(M ) corresponds to voting.

Corollary 2 Consider an economy.

• Suppose ∩ M ∈I ρ j Γ(M ) = ∅ for all j. Then for every (λ 1 , . . . , λ n ) with λ j ∈ ∩ M ∈I ρ j Γ(M ) for all j there exists a ρ-majority stable equilibrium (p, x, ā), where āj maximizes j ′ λ j j ′ p • f j ′ (a j , ā-j ) for all j, for the shareholder governance.

• Suppose ∩ M ∈I ρ Γ(M ) = ∅. Then for every (λ 1 , . . . , λ n ) with λ j ∈ ∩ M ∈I ρ Γ(M ) for all j there exists a ρ-majority stable equilibrium (p, x, ā), where āj maximizes j ′ λ j j ′ p • f j ′ (a j , ā-j ) for all j, for the stakeholder democracy.

• Suppose ∩ M ∈I ρ Γ(M ) = ∅. Then for every λ ∈ ∩ M ∈I ρ Γ(M ) there exists a strong ρ-majority stable equilibrium (p, x, ā), where ā maximizes

j ′ λ j ′ p • f j ′ (a)
, for the stakeholder democracy.

According to Corollary 2 for all weights in the intersections of the convex hulls of normalized portfolios of decisive coalitions there exist equilibria where firms maximize weighted sums of profits for these weights. Suppose that for some firm j the intersection of the convex hulls of normalized portfolios of decisive coalitions contains more than one weight. Then there is a continuum of weights in the intersection of the convex hulls of normalized portfolios of decisive coalitions. Moreover, for the same firm j suppose that the actions maximizing weighted sums of profits vary with the weights. Then there is indeterminacy of equilibria because there is a continuum of equilibria.

As shown in the proof of Theorem 2 if ρ ≥ (n-1)/n, then ∩ M ∈I ρ j Γ(M ) = ∅ for all j and ∩ M ∈I ρ Γ(M ) = ∅. However depending on the distribution of portfolios ∩ M ∈I ρ j Γ(M ) = ∅ for all j and ∩ M ∈I ρ Γ(M ) = ∅ is possible for lower values of ρ. Therefore Corollary 2 rests on assumptions on ∩ M ∈I ρ j Γ(M ) and ∩ M ∈I ρ Γ(M ) rather than assumptions on ρ.

Corollary 2 is a kind of converse to Theorem 1 and Corollary 1. On the one hand the results in Corollary 2 and Corollary 1 are converses, but Corollary 1 rests on stronger assumptions than Corollary 2. On the other hand the result in Theorem 1 is weaker than the converse of Corollary 2, but Corollary 2 and Theorem 1 rest on identical assumptions.

Internalization in equilibrium

Consider a ρ-MSE or a strong ρ-MSE (p, x, ā). Then actions are outcomes of voting in firms, but they can be viewed as solutions to firms maximizing weighted sums of profits as shown in Theorem 1 and Corollary 1. Therefore it is natural to consider the extreme cases of no internalization and perfect internalization as solutions to problems of maximizing weighted sums of profits for different weights.

There is no internalization in case actions in firms corresponds to firms not taking externalities into account. Hence the action of every firm is a solution to the problem of maximzing its profit max

a j p • f j (a j , ā-j ) s.t. a j ∈ A j .
Typically the equilibrium allocation is not Pareto optimal.

There is perfect internalization in case actions in firms corresponds to firms taking externalities into account. Hence the action of every firm is a solution to the problem of maximizing aggregate profit max

a j j ′ p • f j ′ (a j , ā-j ) s.t. a j ∈ A j .
If the sets of actions are convex and the production functions are concave, then the equilibrium allocation is Pareto optimal. Perfect internalization is possible in equilibrium if the normalized market portfolio (1/n)e n is in the convex hull of normalized portfolios of every decisive coalition.

Theorem 3 Consider an economy.

• Suppose

1 n e n ∈ j M ∈I ρ j Γ(M ).
Then for the shareholder governance there exist ρ-majority stable equilibria with perfect internalization.

• Suppose

1 n e n ∈ M ∈I ρ Γ(M ).
Then for the stakeholder democracy there exist both ρ-majority stable equilibria and strong ρ-majority stable equilibria with perfect internalization.

Proof: For the shareholder governance assume The proofs for the stakeholder democracy are identical to the proof for the shareholder governance.

1 n e n ∈ j M ∈I ρ j Γ(M ).
Q.E.D.

Concluding remarks

In the present paper we have studied general equilibrium economies with perfectly competitive markets and direct externalities between firms. Actions in firms are decided by majority voting. Since there are externalities between firms, shareholders typically do not agree on objectives of firms: they want firms to maximize dividends of portfolios rather than profits. We found that: (1) voting is equivalent to maximizing weigted sums of profits for weights in the intersection of the convex hulls of portfolios of majorities of voters; (2) ρ-majority stable equilibria exist in case the rate of majority is at least min{q/(q + 1), (n -1)/n}, where n is the number of firms and q is the dimension of the set of actions; and, (3) an efficient outcome can be the outcome of voting in case the market portfolio is in the convex hull of portfolios of majorities of voters. Moreover two governances, namely the shareholder governance (one share, one vote) and the stakeholder democracy (one stakeholder, one vote) were compared. The outcome of the production sector is more likely to be productively efficient for the stakeholder democracy than for shareholder governance.

In relation to (2), it is possible to lower the rate of majority needed to ensure existence of ρ-MSEs. Assume the distribution of portfolios is symmetric around the diagonal (the line going through the zero portfolio and the market portfolio). Symmetric distributions of portfolios correspond to no wealth effects in portfolios: the relative distribution of shares does not depend on the amount of shares. Then for the stakeholder democracy there exist ρ-MSEs and strong ρ-MSEs for the rate of simple majority ρ = 0.5. Indeed according to [START_REF] Grandmont | Intermediate preferences and majority-rule[END_REF] and the second part of the proof of Theorem 2 the mean portfolio is stable in the sense that actions, which maximize a weighted sum of profits with the mean portfolio as weights, are stable for the rate of simple majority ρ = 0.5.

In relation to (3), since the mean portfolio is the market portfolio for the stakeholder democracy, there exist ρ-MSEs with perfect internalization for symmetric distributions of portfolios for ρ = 0.5. However for the shareholder governance the mean portfolio in firm j is i δ ij δ i which typically is not the market portfolio. Therefore ρ-MSEs with perfect internalization need not exist. Actually, since the distribution of portfolios for firm j need not be symmetric for voting weights θ ij = δ ij , ρ-MSEs need not exist for the shareholder governance, unless the rate of majority is increased to (n -1)/n.

In relation to (2) it is shown in [START_REF] Caplin | Aggregation and social choice -a mean voter theorem[END_REF] that ρ-MSEs exist for rates of majority lower than 64 percent. It would be a useful exercise to substitute our framework with a finite number of agents with their framework with a continuum of agents. Very interestingly, in [START_REF] Caplin | Aggregation and social choice -a mean voter theorem[END_REF] it is shown that the mean portfolio is a ρ-MSE. In relation to (3), once again, for the stakeholder democracy the mean portfolio is the market portfolio so there exist ρ-MSEs with perfect internalization. However, for the shareholder governance the mean portfolio is typically not the market portfolio so ρ-MSEs with perfect internalization need not exist.

We therefore conjecture that internalization is more likely to be the outcome of voting for the stakeholder democracy than for the shareholder governance.

  i (p)} i∈M for all p. Therefore it follows from second part of proof of Theorem 2 that there exists a ρ-MSE with perfect internalization.

In models with representative consumers such as most macroeconomic models as well as all models with agents having some fraction of the market portfolio such as the CAPM model of finance, all shareholders agree that the aggregate profit should be maximized.

Artificial economies

Following [START_REF] Tvede | Voting in assemblies of shareholders and incomplete markets[END_REF] an artificial economy is used. The problem of a firm is decomposed into a problem of selecting a price vector for profit maximization and a problem of maximizing profit.

Describing the problem of selecting a price vector takes a few steps. Let M δ =0 ⊂ {1, . . . , m} be the set of consumers with shares in some firm

Then φ i (p) is the ideal point of consumer i in the sense that consumer i wants firm j to maximize j ′ φ i j ′ (p) • f j ′ (a j , a -j ) for every j. For every i ∈ M δ =0 let the correspondence V i : S × Λ ℓn-1 → Λ ℓn-1 associate every price vector p in S and vector µ in Λ ℓn-1 with the set of vectors µ ′ in Λ ℓn-1 closer to φ i (p) than µ

Let the correspondence N : S × Λ ℓn-1 × Λ ℓn-1 → M δ =0 associate every price vector p in S and pair of vectors µ and µ ′ in Λ ℓn-1 with the set of consumers with

Let the correspondence W ρ j : S × Λ ℓn-1 → Λ ℓn-1 associate every price vector p and vector µ in Λ ℓn-1 with the set of vectors µ ′ in Λ ℓn-1 closer to φ i (p) than µ for a majority of consumers


C Mbodji

E Jutant

Zaza
email: ezaza@tetraedge.com

Michel Simatic

Isabelle Astic

Coline Aunis

Annie Gentes
email: gentes@telecom-paristech.fr|audeguyot@gmail.com

Aude Guyot-Mbodji

Camille Jutant

Emmanuel Zaza

"Plug: Secrets of the Museum": A Pervasive Game Taking Place in a Museum

HAL is

Introduction

Museums intend to arouse visitors' interest in their pieces. Computer games can contribute to this mission [START_REF] Gupta | The End of the Game, a Mystery in Four Parts[END_REF]. [START_REF] Heumer | Via Mineralia -a pervasive museum exploration game[END_REF][START_REF] Laurillau | Supporting museum co-visits using mobile devices[END_REF] present examples of such games. None of them meet simultaneously all of the following requirements: 1) the game is educational but also entertaining; 2) installation and operating costs are limited; 3) it can be played by anyone (whatever their age and their social category); 4) it is a family game; 5) the same player can play several times without getting bored.

This is why we have developed "Plug4 : Secrets of the Museum" (PSM). Section 2 presents its game design and how it matches all of the requirements. Section 3 concludes.

Game Description

In PSM, 8 teams play with NFC-enabled handsets. They manipulate virtual playing cards, which represent specific objects of the museum. These cards are stored either on handsets, or on RFID tags located besides these objects.

The goal is to get the best score throughout the game session. To gain points, the teams may prove: their collector's ability, by gathering on their handset 4 cards of the same family (through exchanges "card stored in their handset ⇔ card stored in a RFID tag" or "card stored in their handset ⇔ card stored in another teams handset"); their public-spiritedness, by storing a card to its reference RFID tag (thanks to an exchange "card stored in their handset ⇔ card stored in a RFID tag"); their generosity, by exchanging one of their cards with another team; their curiosity, by answering quiz related to the objects of the Museum.

PSM is educational as, in particular, players discover parts of the museum they would not have paid attention to, if they had not played PSM. Nevertheless players considered it as very entertaining.

Installation of PSM consists in the deployment of tags throughout the museum. Moreover no network is needed at game time. This results in limited installation and operating costs.

Because there are several ways of gaining points, PSM reaches a broad audience (e.g. young children like its collector aspect whereas seniors enjoy quiz).

PSM suits a family visit. Family members can choose to play either together (in the same team), or against each other (in different teams).

A player can play several sessions of PSM without getting bored: each time, they can decide to choose a different strategy (focus on generosity points, focus on curiosity points. . . ). Moreover their competitors will always be different and this will give a different thrill to each session.

Conclusion

12 public game sessions (with 150 players) showed that PSM makes players take up the whole museum. PSM gives them the desire to stay after their session, in order to focus on some parts of the museum discovered during the game.

During ICEC 2009, we intend to demonstrate PSM by making participants play PSM sessions in the Musée des arts et métiers museum.

PSM key concept is to collect cards through exchanges with RFID tags or other teams. This concept can be generalized easily to any kind of museums, with limited installation and operating costs. Thanks to PSM's educational but also entertaining features, one of the main missions of museums (arouse visitors' interest in their contents) is fulfilled.

PLUG is a research project funded by ANR and labelled by Cap Digital. Its contributors are academics (CNAM, Institut TELECOM, Université de La Rochelle), companies (Net Innovations, Orange, TETRAEDGE Games), a museum (Musée des arts et métiers) and an association (Dune-Aventure).


Gilles Armand

HYDROMECHANICAL RESPONSE TO A MINE BY TEST EXPERIMENT IN DEEP CLAYSTONE

In order to demonstrate the feasibility of radioactive waste repository in deep geological formation, an underground research laboratory is being constructed by Andra (French national radioactive waste management agency) in eastern France, in a Callovo-Oxfordian claystone. 15 boreholes were drilled from a drift at -447 m to install sensors around the shaft (6 m diameter) at depth -460 m to -474 m in order to record the hydromechanical behaviour of the claystone during a shaft sinking (drill and blast method). This paper is devoted to the analysis of the mechanical and hydromechanical behaviour observed during the shaft sinking. Analytical approach used herein allows to realistically evaluate the undrained response of the shaft neighboring with agreement with the in situ measurements. For the transient phase, prediction is qualitatively in accordance with measurement. In addition, deformation and displacement measurements are successfully compared to a simple 3D elastic calculation performed with the real face advancing. This emphasizes the quality of the data set which would be compared in the Modex-Rep European project with complex numerical modelling (poro-elasto-plasic-damage models, creep behaviour, …).

RÉSUMÉ

En France, l'Andra (Agence Nationale de gestion des déchets radioactifs) est en charge des études pour la faisabilité d'un stockage de déchet radioactif haute activité à vie longue, dans une formation géologique profonde comme des argiles. Pour cela, l'Andra a construit le laboratoire de recherche souterrain de Meuse Haute Marne dans le Nord-Est de la France (à 300 km environs de Paris) dans une formation d'argilite du Callovo-Oxfordien qui se trouve entre 420 m et 550 m au niveau du puits principal. La première expérimentation géomécanique réalisée dans le laboratoire est un « mine by test » autour du puits principal (diamètre 6 m). A partir d'une galerie se trouvant à -445 m, 15 forages ont été réalisés pour installer des capteurs et mesurer le comportement hydromécanique de l'argilite entre -460 et -470 m lors du creusement du puits principal. Les évolutions des mesures in situ sont présentées et comparées avec la solution analytique poro-élastique du creusement d'un puits infini. Les mesures de déplacement sont comparées avec les résultats d'un calcul élastique en 3D. Ces analyses simples montrent la cohérence et la qualité des différentes mesures in situ qui serviront de données de référence dans le projet Européen ModexRep, où le creusement du puits est simulé avec des modèles 3D complexes (poro-elasto-endo-plasticité, fluage, ….). 1.

In order to demonstrate the feasibility of a radioactive waste repository in the claystone formation, the French national radioactive waste management agency (Andra) started in 2000 to build an underground research laboratory (URL) in the village of Bure (near the boundary between the Meuse and Haute-Marne Departments) at nearly 300 km East of Paris. The host formation consists of a claystone (Callovo-Oxfordian argillites) and is approximately 500 m deep and 130 m thick (Andra 2005a). The Callovo-Oxfordian argillites are overlaid and underlain by relatively impermeable carbonate formations. The thermohydromechanical behaviour of the argillites and the EDZ are ones of the key issues being investigated in many underground experiments carried out or planned in the Meuse/Haute-Marne URL (Su et al. 2000, Piguet 2001, Delay et al. 2005). Figure 1 presents the drifts network in the Meuse Haute-Marne URL.

Among those experiments, the first geomechanical experiment (REP experiment) was carried out during the excavation of the main access shaft. It is a vertical mineby-test designed to follow in real time the mechanical and hydromechanical responses of the argillite formation to shaft sinking.

The investigated zone is located between 460 and 474 m in depth, where the secant modulus of the argillites is about 5600 MPa and the mean value of the uniaxial compressive strength is 21 MPa. The strain at failure during compression tests on argillite core samples is about 1 to 1.5%. In-situ stress in the argillite layer is: σz = γZ; σh ≈ σv, σH /σh close to 1.2, with the exact value varying with depth and with the rheological characteristics of the respective layers. The horizontal major stress is oriented NE155° (Wileveau et al. 2005).

This paper recalls the concept and instrumentation of the REP experiment, and presents the hydromechanical in situ measurements and the first analysis of the results focusing on pore pressure and displacements in the adjacent massif. First analysis are based on analytical solution of the sinking of an infinite shaft in an anisotropic stress field and simple elastic 3D calculations 2.

DESIGN OF THE REP EXPERIMENT Figure 2. Layout of the REP experiment

In order to monitor the hydromechanical behaviour of the argillites around the REP zone (-460 to -476 m), 15 instrumentation boreholes (20 to 30 m depth) were drilled from an experimental drift (-445 m) in December 2004 (Figure 2). About 120 hydromechanical sensors are placed in these boreholes as follows: 20 for pore pressure, 20 for displacements, 4 CSIRO 12-gauges cells for strains, 10 for inclination and 27 piezoelectric sensors (figure 2). Furthermore, two radial boreholes were specifically drilled at -467 m to measure radialdisplacement behind the excavation face. At this depth the convergence of the shaft has although been measured along 6 diametral positions. These sensors constitute a monitoring matrix for the hydromechanical behaviour of the argillites. In this paper, displacements and pore pressures are discussed. Further in situ data, like evolution of velocity during the shaft sinking could be found in Armand and Su (2006).

Shaft-excavation method

A 6.1 m diameter shaft was sunk with a drill-and-blast method. Blast pattern was designed to excavate rock over a height of 2 to 3.1 m. Figure 3 shows a shaft crosssection in the REP study zone.

The standard support system before application of concrete lining consists of 20, 2.2 m length, bolts with wire mesh and shotcrete in order to stabilise the rock wall and to prevent spalling.

In the main REP zone (between -464.7 and -469.7 m), the support system is replaced by six TH21/548 arches. Each arch is made up of five elements measuring 4.5 m in length. The arches are designed to slide with a friction force of 120 kN.

The final lining consists of a layer of at least 0.5 m of B45 concrete injected by 6-m plots at a time. The distance between the final lining and the working face ranges from 18 m to 24 m, or even 30 m in certain cases.

RESULTS OF THE EXPERIMENT

Displacement field in the rock mass

Figure 4 shows the evolution of displacements within the rock as measured in borehole REP2202 compared with the head of the REP2202 extensometer. Before sinking resumed, measurements were stable, which means that there were no relative displacements between anchoring points. When sinking resumed, the shaft depth was 453.2 m and all anchorings of the REP2202 extensometer laid at least 6.20 m below that level. As the working face progressed towards the REP zone, an initial compression phase is observed, followed by an extension phase. As long as shaft sinking has not reached the anchoring level, the extensometer remains in compression. As soon as the shaft passes the anchoring level, the extensometer starts to expand. This behaviour was expected because the displacement field located below the shaft-sinking face includes movements that are rather vertical due to void attraction and compress the extensometer, whereas anchorings located at right angle with the sunk shaft are subject to the convergence effects of the shaft and generate an expansion of the extensometer. At every blast, an instantaneous displacement is measured over the extensometer. During mucking, the recorded displacements are lower in amplitude. At -480.3 m, increments of instantaneous displacements start to be very low, but an increase varying between 0.5 and 0.8 mm was observed between 15 July and 1 September 2005. The displacements measured during that period are not induced by the sinking of the shaft, but rather to its convergence. It is interesting to point out that the radial distance between the first (REP2202_DFO_01) and the last (REP2202_DFO_10) measurement points is 3.7 m. The last point is located at 3 cm from the shaft wall. By projecting the relative displacement horizontally along the borehole, it is possible to observe, after the passage of the working face, that the relative horizontal displacement between points REP2202_DFO_01 and REP2202_DFO_10 stabilises at 0.17 mm in compression. After the passage of the working face, a maximum relative displacement in compression reaching 3 mm between the two sensors is recorded. By analysing the relative displacement between points REP2202_DFO_09 and REP2202_DFO_10, a compressive deformation of 3.4e-4•on the horizontal plane is reached after the passage of the working face. When the face has passed the deepest anchoring level, a relative displacement in extension is observed.

After the passage of the working face, measurement points do not return to their initial positions. A permanent extension is observed on all sensors. However, only very slight variations were detected concerning the relative positions between sensors REP2202_DFO_1 to 8, which reflects a quasi-elastic behaviour of the rock where these sensors are located. The behaviour of REP2202_DFO_9 and 10 is different from the others. Their positions close to the shaft wall (0.48 and 0.83 m, respectively) suggest that anelastic deformations occur within the argillite close to the shaft wall.

The measured displacements in borehole REP2201, oriented in the σh direction, are similar to those for borehole REP2202. A significant deferred displacement of that borehole head is observed, thus complicating considerably the interpretation of measurements collected in that borehole.

3.2

Evolution of pore pressure during shaft sinking

Figure 5 shows the evolution of pore pressure in borehole REP2101 during the sinking of the access shaft. The borehole is oriented in the direction of the major horizontal stress. The distances between the measurement sensors and the wall range from 1.8 to 5 m. Before sinking resumed, pore pressures were almost stable at 3.4 to 3.9 MPa. On 8 July, a pulse test was The evolution of pore pressure in borehole REP2103 is also recorded. The borehole is oriented rather in the direction of the minor horizontal stress. This borehole exhibits the same pore pressure evolution. Chambers which are oriented in the direction of the minor horizontal stress versus the shaft exhibit drop down of the pore pressure with the face advancing Chambers which are oriented in the direction of the major horizontal stress versus the shaft exhibit over pressure before the face reaches the level of the chambers and a decrease after.

Borehole REP2104 is vertical and located at 13 m from the shaft. Successive blasts induce an instantaneous decrease in the pore pressure. Between March to beginning of September, the decrease in pore pressure is near 0.5 MPa (figure 7).

ANALYSIS OF THE IN SITU DATA

It should be noticed that the REP experiment has generated many numerical modelling before and after its completion, especially in the framework of the European Modex-Rep project. This section does not present this sophisticated numerical modelling, but provide simplest analysis in order to check the consistency of the in situ data base. A 3D elastic calculation is used to investigate the displacement field as a function of the face advancing.

3D elastic modelling of the shaft sinking

Due to the anisotropy of the initial stress field, the first numerical computations are performed in 3D with the use of the three-dimensional finite difference code, FLAC 3D . Calculations have been performed with horizontal stresses anisotropy of 1.3 (maximum of the anisotropy found in the layer). In order to better represent all of the measurement points, a full geometry including the REP zone is modelled (top of model at -414 m; and bottom of model at -514 m). In addition, the lateral boundary of the geometrical model is located at 75 m from the shaft axis. This geometry includes the three main units (A, B&C) encountered in the argillites (Andra 2005a and b).

Both unit A and units B&C are modelled as linear-elastic, transversely isotropic (with horizontal plane of isotropy) whose properties, used for simulation, are derived from laboratory and in situ characterisations: Unit A : ° E1=E//=8620 MPa; E3=E ⊥ =9450 MPa; ν12=ν23=0.3 Units B&C : ° E1=E//=6960 MPa; E3=E ⊥ =8700 MPa; ν12=0.22; ν23=0.2 For all units, the shear modulus G12 is evaluated based on the well-known Saint Venant's formula and valid for most of published experimental data (Worotnicki 1993).

The modelling sequence is performed as follows : firstly, the model without excavation was consolidated under in situ stresses depending on the depth (i.e. model loaded using a combination of gravity, in situ stresses and boundary stresses in order to produce the in situ stress field) and secondly, the excavation of shaft was carried out (roller boundaries were applied to the lateral sides of the model) according to the real advancing of the shaft sinking. That corresponds to 12 steps of excavation between -451.4 and -481.3 m level.

Figure 8 shows the comparison between the in situ measurements and the calculated convergence at depth -467 m. On base 6-12 in the direction of the major horizontal stress, a good agreement between the model and the in situ measurement is find until 30 of June (shaft face at -480.3 m). After this date, the increase of the calculated convergence is small with the face advancing.

The measurement shows that the convergence continues to increase versus time even if there are just two steps of excavation in one month. The convergence exhibits long term behaviour due to creep and/or pore pressure diffusion. The elastic model couldn't catch this type of behaviour, but predicted well the short term convergence in the major horizontal stress direction. In the minor horizontal stress, at depth -480.3 m, the convergence (base 2-8) is under estimated with the elastic model showing that plastic deformation occurred in this direction.

Figure 9 presents the measured and calculated displacements of 3 anchors in the extensometer REP2202 and confirms that the 3D elastic model well predicts the deformation in the direction of the major horizontal stress.

When the working face is before the anchors location, compression is seen in the extensometer and predicted by the numerical model. Every steps of excavation imply a jump in displacement. When the working face reaches the sensor depth, extension is measured and calculated. The displacement magnitude is well predicted until the end of June.

Poroelastic approach

Problem position

Consider an infinite cylindrical borehole excavated in a saturated porous rock subject to an anisotropic in situ stress field with isotropic component P0 and deviatoric component S0. The borehole boundary is free to drain and is exposed to atmospheric pressure. The initial pore pressure field is p0. This problem can be analysed by assuming plane strain conditions and instantaneous drilling of the borehole.

The two-dimensional poroelastic solution of this problem can be obtained by the transforms space of Laplace. In the general case, there is no analytical solution in real space: results in time domain can be derived using numerical inversion techniques. Nevertheless, for small values of time (short term response where the time is much smaller than the characteristic time) the general relations developed by Cheng and Detournay (1988) tend towards the short-time asymptotic solution. This solution is formulated by superposition of asymptotic solutions for three fundamental loading modes : (1) a far field isotropic stress (classical Lamé solution) where rock strain is entirely associated with deviatoric strain, then without mechanism of pore pressure;

(2) a virgin pore pressure distribution; and (3) a far field deviatoric stress. For each of these three loading modes (denoting by superscript i), the short-time asymptotic solution for pore pressure generation is given by equation (1) where ν is the undrained Poisson's ratio; Bs : Skempton 's coefficient; erfc : complementary error function; r : radial distance; a : radius of borehole; c : fluid diffusivity; θ : angular position with respect to x-axis, and t : time.

For loading mode 1, the stress field and displacements are independent of the time and given by well-known Lamé solution, whereas the asymptotic solution for stresses and displacements (modes 2 and 3) are detailed in Detournay and Cheng (1988).

( )

(2) 0

2 0 2 2 (3) 0 2 0 1 4 4 ( ) - 1 exp 8 4 - 1 4 4 1 cos 2 3 4 s p a r a p p erfc r ct p a a ct r a r r a ct r r a erfc a ct a r a a p S B erfc r r ct π ν θ   =     -    = -               -     - -               -      -               -   = + - +            (1)
The equations relative to these fundamental solutions (stress, pore pressure, displacements and derived strain) were programmed with the formal computation software Mathematica, for a quick and efficient sensitive analysis of poroelastic problems. Comparison with a numerical computation with the code FLAC allows us to validate our programming.

Application to REP experiment

Like all the chambers of pore pressure measurements (REP2101, REP2102, REP2103 and REP2104) are located between -462 and -470 m level (figure 2), we will consider a section at the average depth of -467 m, and corresponding to the section SMGR2. With respect to this section, we assume that the shaft is infinite and instantaneously excavated.

In consequence, these assumptions are identical to those of the 2D poroelastic problem previously described. So, the corresponding constitutive equations were thus used for a first critical analysis of pore pressure measurements.

The poromechanical properties used for reference case are based on a large laboratory characterisation (Andra 2005b) and tabled below (Table 1). Figure 10 shows the distribution of pore pressure according to the radial distance in the directions of initial principal stresses for time ranged between 1 sec and 60 days. Just after the shaft drilling (t=1s) in the previously described anisotropic initial stress, we note the instantaneous response induces overpressures in the direction of the minor horizontal stress (θ = 0°). Indeed, the instantaneous solution is accompanied by high variations of mean stress (increase here). Except in the vicinity of shaft wall where null pore pressure is prescribed, the response of material is not drained and induces a peak of pore pressure located inside the rock mass. Location of this peak progresses toward the rock mass and decreases in magnitude with time. These overpressures dissipate in the time in relation to the fixed pore pressure applied at the shaft wall which will impose itself in front of the undrained disturbance.

The instantaneous response induces underpressures in the direction of the major horizontal stress (θ = 90°).in relation with the reduction in the induced mean stress in this direction. This tendency to develop some overpressures in the direction of σh or some underpressures in the direction of σH was also observed on the site measurements at the passage of the face. More precisely, as in the previous analytical results, pressure sensors oriented with an angle ranged between -45° and +45° with respect to the σHdirection do not record some overpressures. Overpressures are observed in all the chambers with an angle less than 45° of the σh-direction. This character of development of overpressures or underpressures inside the rock mass, also highlighted by the poroelastic analysis, manifest the obviousness of the anisotropy of the in situ stresses. Table 2 summarizes the comparison between measurements and the results of poroelastic analysis of the induced overpressures and underpressures respectively near σh-direction and σHdirection. The evolution of the measured pore pressure is therefore consistent with the poroelastic model, because the highest overpressures occur in the closest measurement chambers to the direction of the minor horizontal stress and the strongest underpressures occur in the closest measurement chambers to the major horizontal stress, thus confirming the sound operation of the experimental system.

Figure 11 presents the evolution of pore pressure at 30 days after the shaft excavation. Analytical solution has been used with two different permeability of the rock mass (5×10 -20 m 2 and 5×10 -19 m 2 ). 30 days after the shaft sinking if the permeability is equal to the lower one, overpressures have to be seen in the direction of the minor horizontal stress. If the permeability is higher, overpressures have been dissipated and pressures are nearly equal in all directions. The in situ data shows clearly that after 30 days pore pressures are higher in the direction of the minor stress proving rock mass permeability value measured in situ. However, after 30 days, the analytical solution exhibits higher pore pressure as a function of the distance to the shaft. The discrepancy between the analytical solution and the in situ measurement could be explained by different factors:

the rock mass behaviour is not purely elastic around the shaft (see section 4.1) -the permeability is affected by the sinking in the vicinity of the shaft due to the EDZ (excavated damage zone) ) the pore pressure just after the excavation, p(30d) the pore pressure one month after the excavation 5. CONCLUSIONS Among the 120 sensors installed for the experiment, 112 worked during the shaft-sinking, thus providing a important data base to understand to coupled hydromechanical behaviour of the claystone argillite of the Callovo-Oxfordien. Measured parameters (mainly displacements and pore pressures) are consistent and their evolutions confirm that: -measured deformations are low. Argillites react almost elastically. Irreversible low-amplitude strains are only observed nearby the shaft wall in the direction of the minor horizontal stress in the REP zone; -the evolution of pore pressure depends on the advance of the excavation work, the radial distance from the wall and the orientation of the measurement chamber in relation to the anisotropy of the in-situ stress field; -the location of excavation-induced overpressures and underpressures is consistent with the stress concentrations generated by the sinking of a shaft and show the obviousness of the anisotropy of the initial stress field. Simple analysis shows the consistency of the data set measured during the shaft sinking and emphasizes a strong hydromechanical coupling of the argillite rock behaviour (especially mechanical on hydraulic).

In addition to the data base provided by the REP experiment and the present simple data analysis, a numerical modelling program with more sophisticated models (taking into account HM coupling, plasticity, creep, …) and the real sequences of shaft excavation (in time and space) is undertaken within the framework of the European Modex-Rep project. Publications concerning these models and there predictions are under way.

INTRODUCTIONFigure 1 .

 1 Figure 1. 3D view of the Meuse/Haute-Marne Underground Research Laboratory

Figure 3 .

 3 Figure 3. Cross-section of the access shaft (6 m diameter) between -460 m and -475 m 2.1 Step of the experiment The REP experiment consists of following steps: -Step 1: Drift excavation at the -445 m level and instrumentation of the REP zone (July-December 2004): The drift is the starting point from which REP experiments are run. Sensors and measuring systems were installed in December 2004. Pore pressure chambers were emplaced first in order to have the longer pressure build up; -Step 2: Initial characterisation through dilatometer tests and velocity measurements and stabilisation of sensors (December 2004-February 2005): a sensor-stabilisation phase was required for all sensors. Especially in the case of pore pressure measurements, the build up of pressure was quick and the pressures were nearly stabilized before the beginning of the shaft sinking; -Step 3: Resumption of shaft sinking: shaft advancing resumed on 10 March 2005, with the instrumentation matrix monitoring in real-time the effects of the approaching working face. In situ measurements described in this paper occurred during the shaft sinking from -451 to -483.3 m level. At 483.3 m the sinking operations were stopped middle of August 2005 in order to install other extensometers and to perform measurement. After the sinking started again to reach 510 m, but the hydromechanical parameters in the REP zone, are less affected and are not presented in this paper.

Figure 4 .

 4 Figure 4. Measured displacements in borehole REP2202

Figure 5 .Figure 7 .

 57 Figure 5. Evolution of pore pressures in borehole REP2101 (chambers at 1.8 m to 5 m from the wall shaft)

Figure 9 .

 9 Figure 8. Comparison between measured and calculated convergences

Figure 10 .

 10 Figure 10. Evolution of pore pressure : (a) along σhdirection; (b) along σH-direction

Table 1

 1

	-Poromechanical properties used
	Undrained Young modulus, E [MPa] Undrained Poisson ration, ν [-]	5600 0.3
	Porosity, n [-] Intrinsic permeability, k [m Biot coefficient, b [-]	2 ]	0.15 5x10 0.6	-20
	Biot modulus, M [MPa]		6542
	The initial in situ stress (at -467 m) is : σv=σh=-11.3 MPa ;
	σH=-14.7 MPa (compressive stresses are negative)
	leading to : P0= (σH + σh)/2 and S0 = (σH -σh)/2 with σh
	oriented in the direction of x-axis.

Table 2

 2

	(MPa)	-Induced overpressures and underpressures
	Chamber-Distance to the shaft (m)	In situ measurement	Analytical solution
	-Angle to σH (°)	Dp1	Dp2	Dp1	Dp2
	REP2101	1-4.85-21.8 2-3.69-24.3 3-2.68-26.6 4-2.25-28.9 5-1.71-31.7	-0.29 -0.46 -0.45 -0.52 -0.69		0.50 0.66 1.22 1.60 2.80	-0.31 -0.38 -0.44 -0.48 -0.51	0 0 0.02 0.10 0.36
	REP2102	1-2.26-58.1 2-1.89-63.8 3-1.41-70.2 4-1.36-77.4 5-1.09-91.9	0.06 0.31 0.60 0.88 1.58		1.58 2.47 3.02 1.52 5.12	0.40 0.62 0.88 1.13 1.36	0.18 0.40 0.74 1.15 1.63
	REP2103	1-2.11-55.5 2-2.24-30.0 3-2.5-16.8 4-3.55-6.4 5-4.13-1.9	0.24 -1.55 -0.68 -0.53 -0.24		2.42 1.69 2.15 1.06 1.37	0.36 -0.47 -0.65 -0.59 -0.52	0.33 0.13 0.03 0 0
	with Dp1=p(0	+ ) -p0 and Dp2=p(0	+ ) -p(30d)
	p0 the initial pore pressure p(0 +