You forgot to select child divs:
HtmlDocument doc = new HtmlDocument();
doc.LoadHtml(matches);
string matchResultDivId = "match_results";
string xpath = String.Format("//div[@id='{0}']/div", matchResultDivId);
var people = doc.DocumentNode.SelectNodes(xpath).Select(p => p.InnerText);
foreach(var person in people)
Console.WriteLine(person);
Output:
person1
person2
person3