Add ERR to ranking evaluation documentation (#32314)

This change adds a section about the Expected Reciprocal Rank metric (ERR) to
the Ranking Evaluation documentation.

docs/reference/search/rank-eval.asciidoc

@@ -263,6 +263,56 @@ in the query. Defaults to 10.
 |`normalize` | If set to `true`, this metric will calculate the https://en.wikipedia.org/wiki/Discounted_cumulative_gain#Normalized_DCG[Normalized DCG].
 |=======================================================================
 
+[float]
+==== Expected Reciprocal Rank (ERR)
+
+Expected Reciprocal Rank (ERR) is an extension of the classical reciprocal rank for the graded relevance case
+(Olivier Chapelle, Donald Metzler, Ya Zhang, and Pierre Grinspan. 2009. http://olivier.chapelle.cc/pub/err.pdf[Expected reciprocal rank for graded relevance].)
+
+It is based on the assumption of a cascade model of search, in which a user scans through ranked search
+results in order and stops at the first document that satisfies the information need. For this reason, it
+is a good metric for question answering and navigation queries, but less so for survey oriented information 
+needs where the user is interested in finding many relevant documents in the top k results.
+
+The metric models the expectation of the reciprocal of the position at which a user stops reading through
+the result list. This means that relevant document in top ranking positions will contribute much to the
+overall score. However, the same document will contribute much less to the score if it appears in a lower rank,
+even more so if there are some relevant (but maybe less relevant) documents preceding it. 
+In this way, the ERR metric discounts documents which are shown after very relevant documents. This introduces 
+a notion of dependency in the ordering of relevant documents that e.g. Precision or DCG don't account for.
+
+[source,js]
+--------------------------------
+GET /twitter/_rank_eval
+{
+    "requests": [
+    {
+        "id": "JFK query", 
+        "request": { "query": { "match_all": {}}},
+        "ratings": []  
+    }],
+    "metric": {
+       "expected_reciprocal_rank": {
+            "maximum_relevance" : 3,
+            "k" : 20
+       }
+    }
+}
+--------------------------------
+// CONSOLE
+// TEST[setup:twitter]
+
+The `expected_reciprocal_rank` metric takes the following parameters:
+
+[cols="<,<",options="header",]
+|=======================================================================
+|Parameter |Description
+| `maximum_relevance` | Mandatory parameter. The highest relevance grade used in the user supplied
+relevance judgments.
+|`k` | sets the maximum number of documents retrieved per query. This value will act in place of the usual `size` parameter
+in the query. Defaults to 10.
+|=======================================================================
+
 [float]
 === Response format