MD Settings: Use paper-dropdown-menu for search engines.

chrome/browser/resources/settings/search_page/search_page.js

 // Copyright 2015 The Chromium Authors. All rights reserved.
 // Use of this source code is governed by a BSD-style license that can be
 // found in the LICENSE file.
 
 /**
  * @fileoverview
  * 'settings-search-page' is the settings page containing search settings.
  */
 Polymer({
   is: 'settings-search-page',
@@ skipping 33 matching lines @@
     this.browserProxy_.getSearchEnginesList().then(updateSearchEngines);
     cr.addWebUIListener('search-engines-changed', updateSearchEngines);
   },
 
   /** @private */
   onManageSearchEnginesTap_: function() {
     this.$.pages.setSubpageChain(['search-engines']);
   },
 
   /** @private */
-  onDefaultEngineChanged_: function() {
-    this.browserProxy_.setDefaultSearchEngine(this.$.searchEnginesMenu.value);
+  onIronSelect_: function() {
+    var searchEngine = this.searchEngines_[this.$$('paper-listbox').selected];
+    if (searchEngine.default) {
+      // If the selected search engine is already marked as the default one,
+      // this change originated in some other tab, and nothing should be done
+      // here.
+      return;
+    }
+
+    // Otherwise, this change originated by an explicit user action in this tab.
+    // Submit the default search engine change.
+    this.browserProxy_.setDefaultSearchEngine(searchEngine.modelIndex);
+  },
+
+  /**
+   * @return {number}
+   * @private
+   */
+  getSelectedSearchEngineIndex_: function() {
+    return this.searchEngines_.findIndex(function(searchEngine) {
+      return searchEngine.default;
+    });
   },
 });