Externalities in economies with endogenous sharing rules


Endogenous sharing rules were introduced by Simon and Zame (Econometrica 58(4):861–872, 1990) to model payoff indeterminacy in discontinuous games. They prove the existence in every compact strategic game of a mixed Nash equilibrium and an associated sharing rule. We extend their result to economies with externalities (Arrow and Debreu in Econometrica 22(3):265–290, 1954) where, by definition, players are restricted to pure strategies. We also provide a new interpretation of payoff indeterminacy in Simon and Zame’s model in terms of preference incompleteness.

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  1. 1.

    For simplicity, we use the same letter N for the set of players or the number of players.

  2. 2.

    A preorder is a reflexive and transitive binary relation.

  3. 3.

    Every preorder \(\lesssim \) on X admits a multi-utility representation (see Evren and Efe 2011), that is, there exists a family \((v_j)_{j \in J}\) of real-valued functions defined on X such that: \(x \lesssim y\) \(\Leftrightarrow \) for every \(j \in J\), \(v_j(x) \le v_j(y)\). Thus, there is no loss of generality in working with a cardinal multi-representation.

  4. 4.

    Here, \(\lnsim _i\) denotes the strict preorder associated to \(\lesssim _i\), that is: for every \((x,y) \in X^2\), \(x \lnsim _i y\) if and only if \(x \lesssim _i y\) and not (\(y \lesssim _i x\)).

  5. 5.

    Remark that conversely, our paper does not generalize the existence results of these two papers.

  6. 6.

    Here, to simplify the exposition, we do not allow externalities, that is \({\mathcal {U}}_i\) depends only of player i’s strategies.

  7. 7.

    The set \(\Delta ({\mathbf {R}}^m_+)\) denotes the unit simplex of \({\mathbf {R}}^m_+\).

  8. 8.

    A Kakutani-type multivalued mapping is a multivalued mapping with nonempty convex values and a closed graph.

  9. 9.

    This function was introduced by Carmona (see Carmona 2011).


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Corresponding author

Correspondence to Philippe Bich.

Additional information

Laraki’s work was supported by grants administered by the French National Research Agency as part of the Investissements d’Avenir program (Idex [Grant Agreement No. ANR-11-IDEX-0003-02/Labex ECODEC No. ANR11- LABEX-0047] and ANR-14-CE24-0007-01 CoCoRICo-CoDec).


Appendix A: Proof of Theorem 2

The proof consists of several steps: first, we turns G into an auxiliary discontinuous strategic game \(G'\). Second (steps 2 and 3), we prove the existence of a relaxed Nash equilibrium of \(G'\). This is used to construct in step 4 a sharing rule solution of \(G'\) that satisfies some desirable properties. Finally, such a sharing rule solution is used to build a solution of G. This methodology follows the one developed in Bich and Laraki (2017) and Bich and Laraki (2012) to prove existence of Nash, approximate and sharing rule equilibria in discontinuous games. But it is more complicated because of the externalities. Importantly, the existence results contained in steps 3 and 4 are valid for any quasiconcave compact discontinuous strategic game \(((X_{i})_{i\in N},(u_i)_{i\in N})\).

By assumption, \({\mathcal {U}}\) admits a single-valued selection \(\phi =(\phi _i)_{i \in I}\) where each \(\phi _i\) is quasiconcave in player i’s strategy.

Step 1. Associate to G a discontinuous game \(G'\).

Following an idea of Reny (1999), we associate to the economy with externalities \(G=((X_{i})_{i\in N},(\phi _i)_{i\in N}), {\mathcal {B}})\) a strategic game \(G'\) as follows. Because \({\mathcal {U}}\) is bounded, there exists \(\Lambda \in {\mathbf {R}}\) such that \(\phi _i(x) \ge \Lambda +1\) for every \(i \in N\) and every profile \(x \in X\). The game \(G'\) has N players. For every \(i \in N\), strategy set of player i is \(X_i\), and his payoff is

$$\begin{aligned} u_i(x)= \left\{ \begin{array}{ll} \phi _i(x) &{} \quad \hbox {if } x_i \in {\mathcal {B}}_i(x_{-i}), \\ \Lambda &{} \quad \hbox {otherwise}. \end{array} \right. \end{aligned}$$

These new payoff functions are also quasiconcave.

Step 2. Generalized regularization of payoff functions of \(G'\).

Throughout this proof, for every \(i \in N\), \(x \in X\), and U in \( {\mathcal {V}}(x_{-i})\) (the set of open subsets of \(X_{-i}\)), denote by \(W_{U}(x_i,x_{-i})\) the set of Kakutani-typeFootnote 8 multivalued mappings \(d_i\) from U to \(X_{i}\) such that \(x_i \in d_i(x_{-i})\) for every \(x_{-i} \in U\). Let \(\underline{\underline{u_{i}}}: X \rightarrow {\mathbf {R}}\) be the following regularizationFootnote 9 of the utility function \(u_i\)

$$\begin{aligned} \underline{\underline{u_{i}}}(x):=\mathop {\sup }\nolimits _{U \in {\mathcal {V}}(x_{-i})} \mathop {\sup }\nolimits _{d _{i}\in W_{U}(x)} \mathop {\inf }\nolimits _{x'_{-i} \in U, x'_i \in d_i(x'_{-i})} u_{i}(x'). \end{aligned}$$

Remark that \(\underline{\underline{u_i}}(x) \le {u_i}(x)\) for every \(x \in X\), since in the infimum above one can take \(x'=x\).

Step 3. Existence of a refined Reny solution of \(G'\).

Let us prove that there exists a pair \((x^*,v^*) \in \overline{\Gamma }\) (where \(\Gamma :={\{(x,u(x)): x \in X\}}\)) such that:

$$\begin{aligned} \forall i \in N, \sup _{x_i \in X_i} \underline{\underline{u_i}}(x_i,x^*_{-i}) \le v_i^*. \end{aligned}$$

Such pair \((x^*,v^*)\) refines the Reny solution concept introduced in Bich and Laraki (2017). When \(u_i\) is continuous for every \(i \in N\), \(x^*\) is a Nash equilibrium and \(v^*=u(x^*)\) is the associated payoff vector.

By contradiction, assume that there is no such pair, and let us prove that \(G'\) is generalized better-reply secure. Recall that \(G'\) is generalized better-reply secure (Barelli and Meneghel 2013) if whenever \((x,v) \in \overline{\Gamma }\) and x is not a Nash equilibrium, there exists a player i and a triple \((d_i,V_{x_{-i}},\alpha _i)\), where \(V_{x_{-i}}\) is an open neighborhood of \({x_{-i}}\), \(d_i\) is a Kakutani-type multivalued function from \(V_{x_{-i}}\) to \(X_i\) and \(\alpha _i>v_i\) is a real number such that for every \({x'_{-i}} \) in \(V_{x_{-i}}\) and \(x'_i \in d_i(x'_{-i})\), one has \(u_i(x'_i,x'_{-i}) \ge \alpha _i.\)

For, consider \((x,v) \in \overline{\Gamma }\) such that x is not a Nash equilibrium. By assumption, (xv) does not satisfy inequality (3), thus there exists some player \(i \in N\) such that \(\sup _{y_i \in X_i} \underline{\underline{u_i}}(y_i,x_{-i}) > v_i.\) From the definition of \(\underline{\underline{u_i}}\), there is \(\varepsilon >0\), \(U \in {\mathcal {V}}(x_{-i})\), \(d _{i} \in W_{U}(x)\) such that for every \(x'_{-i} \in U\) and every \(x_i' \in d_i(x'_{-i})\), \(u_{i}(x_i',x'_{-i}) \ge v_i+\varepsilon :\) this implies generalized better-reply security. Consequently, from Barelli and Meneghel (2013), since \(G'\) is generalized better-reply secure, it admits a Nash equilibrium. But this is a contradiction, since if \(x \in X\) is a Nash equilibrium, (xu(x)) satisfies inequality (3) (because \(\underline{\underline{u_i}}(x) \le {u_i}(x)\) for every \(x \in X\)). By contradiction, this proves the existence of \((x^*,v^*) \in \overline{\Gamma }\) satisfying inequality (3).

Step 4. Existence of a sharing rule solution of \(G'\).

We now prove that there exists some new payoff functions \((q_i)_{i \in I}\) and a pure Nash equilibrium \(x^* \in X\) of \(G''=((X_{i})_{i\in N},(q_{i})_{i\in N})\), with the additional properties:

  1. (i)

    for every i and \(d_{i} \in X_{i}\), \( {q_{i}}(d_{i},{x^*}_{-i}) \ge \underline{\underline{u_{i}}}(d_{i},{x^*}_{-i})\).

  2. (ii)

    For every \(y \in X\) , there exists some sequence \((y^n)\) converging to y such that \(u(y^n)\) converges to q(y).

For every \(i \in N\), denote by \(\underline{\underline{\mathcal{S}_i}}(y)\) the space of sequences \((y^n)_{n \in {\mathbf {N}}}\) of X converging to y such that \(\lim _{n \rightarrow +\infty } u_{i}(y^n)=\underline{\underline{u_i}}(y)\). Then, define \(q:X \rightarrow {\mathbf {R}}^N\) by

$$\begin{aligned} q(y)= \left\{ \begin{array}{lll} v^* &{} \quad \hbox {if } y=x^*, \\ \hbox {any limit point of } u(x^n)_{n \in {{\mathbf {N}}}} &{} \quad \hbox {if } y=(d_{i},x^*_{-i}) \hbox { for some }i \in N,\\ &{}\qquad d_{i} \ne x^*_{i}, (x^n)_{n \in {\mathbf {N}}} \in \underline{\underline{\mathcal{S}_i}}(d_i,x^*_{-i}), \\ q(y)=u(y) &{} \quad \hbox {otherwise.} \end{array} \right. \end{aligned}$$

Since \((x^*,v^*) \in \overline{\Gamma }\), and by definition of q, condition (ii) above is satisfied at \(x^*\). Clearly, by definition, it is also satisfied at every y different from \(x^*\) for at least two components, and finally also at every \((d_{i},x^*_{-i})\) with \(d_{i} \ne x^*_{i}\) (for some \(i \in N\)), from the definition of \(q(d_{i}, x^*_{-i})\) in this case. Condition (i) is true at every y different from \(x^*\) for at least two components (from \(\underline{\underline{u_i}} \le {u_i}\)), is true at every \((d_{i},x^*_{-i})\) with \(d_{i} \ne x^*_{i}\) by definition, and is finally true at \(x^*\) from inequality (3). This ends the proof of Step 4.

Step 5. Existence of a solution of G.

Now, we finish the proof of Theorem 2. Take \(d_i \in {\mathcal {B}}_i(x^*_{-i}) \ne \emptyset \). For every \(x_{-i}'\) in some neighborhood of \(x^*_{-i}\) and every \(x_i' \in {\mathcal {B}}_i(x_{-i}'),\) we have, by definition, \(u_i(x_i',x_{-i}')=\phi _i(x_i',x_{-i}') \ge \Lambda +1\). Since \({\mathcal {B}}_i\) is a Kakutani-type mapping, this implies, by definition, \(\underline{\underline{u_{i}}}(d_i,x^*_{-i}) \ge \Lambda +1\) (where \(\underline{\underline{u_{i}}}\) is the regularization of \({u_{i}}\), defined in the beginning of this proof). Thus, from condition (i) in step 4 above, we get

$$\begin{aligned} \forall d_i \in {\mathcal {B}}_i(x^*_{-i}),\ q_i(d_i,x^*_{-i}) \ge \underline{\underline{u_{i}}}(d_i,x^*_{-i}) \ge \Lambda +1. \end{aligned}$$

Since \(x^*\) is a Nash equilibrium of \(G''\), we have:

$$\begin{aligned} \forall i \in N, {q_i(x^*) \ge \sup _{d_i \in X_i} q_i(d_i,x^*_{-i}) \ge \Lambda +1}. \end{aligned}$$

From condition (ii) in step 4 above, there is a sequence \((x^n)\) converging to \(x^*\) such that \(u(x^n)\) converges to \(q(x^*)\). Since \(q_i(x^*) \ge \Lambda +1\) for every \(i \in N\), we cannot have \(u_i(x^n)=\Lambda \) for n large enough. Consequently, from the definition of \(u_i\), we get \(u_i(x^n)=\phi _i(x^n)\) and \(x_i^n \in {\mathcal {B}}_i(x^n_{-i})\) for n large enough. Passing to the limit, we get \(x^*_i \in {\mathcal {B}}_i(x^*_{-i})\) for every \(i \in I\) (because \({\mathcal {B}}_i\) has a closed graph). A similar argument can be applied to any \((y_i,x^*_{-i}) \in X\) for which \(y_i \in {\mathcal {B}}_i(x^*_{-i})\): there is a sequence \((x^n)\) converging to \((y_i,x^*_{-i})\) such that \(u(x^n)\) converges to \(q(y_i,x^*_{-i})\). Since \(q_i(y_i,x^*_{-i}) \ge \Lambda +1\) (from inequality (4)), we cannot have \(u_i(x^n)=\Lambda \) for n large enough. Consequently, \(u_i(x^n)=\phi _i(x^n)\) and \(x_i^n \in {\mathcal {B}}_i(x^n_{-i})\) for n large enough. In particular, since \(\phi \) is a selection of U and since U has a closed graph, we get

$$\begin{aligned} \forall y_i \in {\mathcal {B}}_i(x^*_{-i}), \ q(y_i,x^*_{-i}) \in U(y_i,x^*_{-i}). \end{aligned}$$

Now, define \({\tilde{q}}({y_i,x^*_{-i}})=q({y_i},x^*_{-i})\) whenever \(y_i \in {\mathcal {B}}_i(x^*_{-i})\) for some \(i \in N\), and \({\tilde{q}}(y)=\phi (y)\) elsewhere. The proof that \(x^*\) is a equilibrium of \(((X_i)_{i \in N},(\tilde{q}_i)_{i \in N}, B)\) is a straightforward consequence of \(x^*\) being a Nash equilibrium of \(((X_i)_{i \in N},(q_i)_{i \in N})\). Last, we have to prove that \({\tilde{q}}(y) \in U(y)\) for every \(y \in X\). This is clear at \(y=({y_i,x^*_{-i}})\) for \(y_i \in {\mathcal {B}}_i(x^*_{-i})\), from (5) above. For others y, we have \({\tilde{q}}(y)=\phi (y) \in U(y)\) by definition. This ends the proof of Theorem 2.

Appendix B: Proof of the statements in Sect. 3.3

From the exchange economy, define an economy with externalities and discontinuous payoffs \(({\mathcal {G}},{\mathcal {B}})\) as follows:

  1. 1.

    There are \((N+1)\) players.

  2. 2.

    For \(i=1,\ldots ,N\), player i’s convex compact strategy space is \(X_i=\{x_i \in {\mathbf {R}}^m_+: x_i \le \sum _{i=1}^N e_i+(1,\ldots ,1)\}\) and his payoff function is \(u_{i}\).

  3. 3.

    The strategy space of player \((N+1)\) (called the auctioneer) is \(X_{N+1}=\Delta ({\mathbf {R}}^m_+)\), and his payoff function is \(v_{N+1}(x,p)=p\cdot \sum _{i \in N} (x_i-e_i)\).

  4. 4.

    Last, define the strategy correspondences as follows: for every \(i \in N\), \({\mathcal {B}}_i(x,p)=B_i(p)=\{x_i \in X_i: p \cdot x_i \le p \cdot e_i\}\), and finally define \({\mathcal {B}}_{N+1}(x,p)=X_{N+1}.\)

Following Sect. 3.2, this economy has a solution \((x^*,p^*,{\tilde{q}})\). This means that:

  1. 1.

    For every \((x,p) \in \prod _i X_i \times \Delta ({\mathbf {R}}^m_+)\) such that \(x_i \in B_i(p)\) for every \(i \in N\), there is a sequence \((x^n,p^n)_{n \in \tiny {\mathbbm {N}}}\) converging to (xp) such that \(x_i^n\in B_i (p^n)\) for every \(i\in N\) and \({\tilde{q}}_i(x,p)=\lim _{n \rightarrow +\infty } u_i(x_i^n)\). In particular, from the continuity of \(v_{N+1}\), \({\tilde{q}}_{N+1}(x,p)=v_{N+1}(x,p)=p\cdot \sum _{i \in N} (x_i-e_i).\)

  2. 2.
    1. (i)

      For every \(i \in N\), \(x^*_i \in B_i(p^*)\).

    2. (ii)

      For every \(i \in N\), for every \(x_i \in B_i(p^*)\), \({\tilde{q}}_i(x_i,x^*_{-i},p^*) \le {\tilde{q}}_i(x^*,p^*).\)

    3. (iii)

      For every \(p \in \Delta ({\mathbf {R}}^m_+)\), \(p.\sum _{i \in N} (x^*_i-e_i) \le p^*\cdot \sum _{i \in N} (x^*_i-e_i).\)

Let us now define \(q_i(x_i):={\tilde{q}}_i(x_i,x_{-i}^*,p^*)\) for every \(x_i \in B_i(p^*)\), and \(q_i(x_i)=u_i(x_i)\) otherwise. From 1 and 2 above, there is a sequence \((x^n,p^n)_{n \in \tiny {\mathbbm {N}}}\) converging to \((x_i,x_{-i}^*,p^*)\) such that \(x_i^n\in B_i (p^n)\) for every \(i\in N\) and

$$\begin{aligned} {\tilde{q}}_i(x_i,x_{-i}^*,p^*)=\lim _{n \rightarrow +\infty } u_i(x_i^n)=q_i(x_i). \end{aligned}$$

Thus condition (b) and (c) in Sect. 3.3 hold. Let us prove that condition (a) also holds, that is, \((x^*, p^*)\) is a Walrasian equilibrium of the economy with payoff functions \((q_i)_{i\in N}\).

First, assume that we do not have \(\sum _{i \in N} (x^*_i-e_i) \le 0\). Then, let us define \(p=(p(1),\ldots ,p(k),\ldots ,p(m)) \in \Delta ({\mathbf {R}}^m_+)\) with \(p(k)=0\) when \(\sum _{i \in N} (x^*_i-e_i)(k) \le 0\) (where \(\sum _{i \in N} (x^*_i-e_i)(k)\) denotes k-component of \(\sum _{i \in N} (x^*_i-e_i)\)), and \(p(k)=\lambda .\sum _{i \in N} (x^*_i-e_i)(k)\) otherwise (where \(\lambda >0\) is a normalization coefficient that insures that \(p \in \Delta ({\mathbf {R}}^m_+)\)). By definition, we get \(p.\sum _{i \in N} (x^*_i-e_i)>0\), thus from (iii) above, \(p^*.\sum _{i \in N} (x^*_i-e_i)>0\). But from condition (i) above, the budget constraint yields \(p^*(x_i^*-e_i) \le 0\) for every \(i \in N\), and summing these inequalities, we get \(p^*.\sum _{i \in N} (x^*_i-e_i) \le 0\), a contradiction. This proves \(\sum _{i \in N} (x^*_i-e_i) \le 0\).

From (ii) above, for every \(x_i \in B_i(p^*)\), we have \(q_i(x_i)={\tilde{q}}_i(x_i,x^*_{-i},p^*) \le {\tilde{q}}_i(x^*,p^*)=q_i(x_i^*).\) Thus, for every \(i \in N\), \(x_i^*\) maximizes \(q_i\) in \(B_i(p^*)\), which ends the proof.

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Bich, P., Laraki, R. Externalities in economies with endogenous sharing rules. Econ Theory Bull 5, 127–137 (2017). https://doi.org/10.1007/s40505-017-0118-3

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  • Abstract economies
  • Generalized games
  • Endogenous sharing rules
  • Walrasian equilibrium
  • Incomplete and discontinuous preferences
  • Better reply security

JEL Classification

  • C02
  • C62
  • C72
  • D50