Abstract
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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Notes
 1.
For simplicity, we use the same letter N for the set of players or the number of players.
 2.
A preorder is a reflexive and transitive binary relation.
 3.
Every preorder \(\lesssim \) on X admits a multiutility representation (see Evren and Efe 2011), that is, there exists a family \((v_j)_{j \in J}\) of realvalued 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 multirepresentation.
 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.
Remark that conversely, our paper does not generalize the existence results of these two papers.
 6.
Here, to simplify the exposition, we do not allow externalities, that is \({\mathcal {U}}_i\) depends only of player i’s strategies.
 7.
The set \(\Delta ({\mathbf {R}}^m_+)\) denotes the unit simplex of \({\mathbf {R}}^m_+\).
 8.
A Kakutanitype multivalued mapping is a multivalued mapping with nonempty convex values and a closed graph.
 9.
This function was introduced by Carmona (see Carmona 2011).
References
Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22(3), 265–290 (1954)
Aumann, R.J.: Utility theory without the completeness axiom. Econometrica 30(3), 445–462 (1962)
Barelli, P., Meneghel, I.: A note on the equilibrium existence problem in discontinuous games. Econometrica 81(2), 813–824 (2013)
Bich, P., Laraki, R.: A unified approach to equilibrium existence in discontinuous strategic games. Preprint: halshs00717135 (2012)
Bich, P., Laraki, R.: On the existence of approximate equilibria and sharing rule solutions in discontinuous games. Theor. Econ. 12(1), 79–108 (2017)
Carmona, G.: An existence result for discontinuous games. J. Econ. Theory 144(3), 1333–1340 (2009)
Carmona, G.: Understanding some recent existence results for discontinuous games. Econ. Theory 48(1), 31–45 (2011)
Carmona, G., Podczeck, K.: Existence of Nash equilibrium in ordinal games with discontinuous preferences. Econ. Theory 61(3), 457–478 (2016)
Evren, Ö., Ok, E.A.: On the multiutility representation of preference relations. J. Math. Econ. 47(4–5), 554–563 (2011)
Gale, D., MasColell, A.: An equilibrium existence theorem for a general model without ordered preferences. J. Math. Econ. 2(1), 9–15 (1975)
He, W., Yannelis, N.C.: Existence of Walrasian equilibria with discontinuous, nonordered, interdependent and pricedependent preferences. Econ. Theory 61(3), 497–513 (2016)
Jackson, M.O., Simon, L.K., Swinkels, J.M., Zame, W.R.: Communication and equilibrium in discontinuous games of incomplete information. Econometrica 70(5), 1711–1740 (2002)
Reny, P.J.: On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67(5), 1029–1056 (1999)
Reny, P.J.: Nash equilibrium in discontinuous games. Working paper, University of Chicago (2013)
Reny, P.: Nash equilibrium in discontinuous games. Econ. Theory 61(3), 553–569 (2016)
Shafer, W., Sonnenschein, H.: Equilibrium in abstract economies without ordered preferences. J. Math. Econ. 2(3), 345–348 (1975)
Simon, L.K., Zame, W.R.: Discontinuous games and endogenous sharing rules. Econometrica 58(4), 861–872 (1990)
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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. ANR11IDEX000302/Labex ECODEC No. ANR11 LABEX0047] and ANR14CE24000701 CoCoRICoCoDec).
Appendices
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 singlevalued 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
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 Kakutanitype^{Footnote 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 regularization^{Footnote 9} of the utility function \(u_i\)
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:
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 betterreply secure. Recall that \(G'\) is generalized betterreply 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 Kakutanitype 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, (x, v) 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 betterreply security. Consequently, from Barelli and Meneghel (2013), since \(G'\) is generalized betterreply secure, it admits a Nash equilibrium. But this is a contradiction, since if \(x \in X\) is a Nash equilibrium, (x, u(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:

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

(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
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 Kakutanitype 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
Since \(x^*\) is a Nash equilibrium of \(G''\), we have:
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
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.
There are \((N+1)\) players.

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

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.
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 (x, p) 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 \(v_{N+1}\), \({\tilde{q}}_{N+1}(x,p)=v_{N+1}(x,p)=p\cdot \sum _{i \in N} (x_ie_i).\)

2.

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

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

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

(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
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 \(\sum _{i \in N} (x^*_ie_i) \le 0\). Then, let us define \(p=(p(1),\ldots ,p(k),\ldots ,p(m)) \in \Delta ({\mathbf {R}}^m_+)\) with \(p(k)=0\) when \(\sum _{i \in N} (x^*_ie_i)(k) \le 0\) (where \(\sum _{i \in N} (x^*_ie_i)(k)\) denotes kcomponent of \(\sum _{i \in N} (x^*_ie_i)\)), and \(p(k)=\lambda .\sum _{i \in N} (x^*_ie_i)(k)\) otherwise (where \(\lambda >0\) is a normalization coefficient that insures that \(p \in \Delta ({\mathbf {R}}^m_+)\)). By definition, we get \(p.\sum _{i \in N} (x^*_ie_i)>0\), thus from (iii) above, \(p^*.\sum _{i \in N} (x^*_ie_i)>0\). 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 \(p^*.\sum _{i \in N} (x^*_ie_i) \le 0\), a contradiction. This proves \(\sum _{i \in N} (x^*_ie_i) \le 0\).
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/s4050501701183
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Keywords
 Abstract economies
 Generalized games
 Endogenous sharing rules
 Walrasian equilibrium
 Incomplete and discontinuous preferences
 Better reply security
JEL Classification
 C02
 C62
 C72
 D50