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A widely employed solution concept for such generalized Nash games is the generalized Nash equilibrium (GNE). The variational equilibrium (VE) [<span class="zbmathjax-textit">F. Facchinei, Ch. Kanzow</span>, 4OR 5, No. 3, 173–210 (2007; <a href="/1211.91162">Zbl 1211.91162</a>)] is a specific kind of GNE characterized by the solution of the variational inequality formed from the common constraint and the mapping of the gradients of player objectives. Our contribution is a theory that provides sufficient conditions for ensuring that the existence of a GNE implies the existence of a VE; in such an instance, the VE is said to be a refinement of the GNE. For certain games, these conditions are shown to be necessary. This theory rests on a result showing the equality of the Brouwer degree of two suitably defined functions, whose zeros are the GNE and VE, respectively. This theory has a natural extension to the primal-dual space of strategies and Lagrange multipliers corresponding to the shared constraint. Our results unify some known results pertaining to such equilibria and provide mathematical substantiation for ideas that were known to be appealing to economic intuition.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6071671">Cited in <strong>51</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A91A10" title="MSC2020">91A10</a> </td> <td class="space"> Noncooperative games </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A91B52" title="MSC2020">91B52</a> </td> <td class="space"> Special types of economic equilibria </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ageneralized+Nash+games">generalized Nash games</a>; <a href="/?q=ut%3Ashared+constraints">shared constraints</a>; <a href="/?q=ut%3Arefinement+of+an+equilibrium">refinement of an equilibrium</a>; <a href="/?q=ut%3Avariational+equilibrium">variational equilibrium</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1211.91162">Zbl 1211.91162</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1245.91006" data-ciurl="/ci/06071671" data-biburl="/bibtex/06071671.bib" data-amsurl="/amsrefs/06071671.bib" data-xmlurl="/xml/06071671.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06071671.pdf" title="Zbl 1245.91006 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.automatica.2011.09.042" aria-label="DOI for “On the variational equilibrium as a refinement of the generalized Nash equilibrium”" title="10.1016/j.automatica.2011.09.042">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Arrow, K.; 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