First: I cannot make much sense of your list. Is that supposed to be a JSON style object? if so then it is badly formed.
I will assume a POCO with id and value for this example.
You need a groupby followed by a min:
var result = list
.GroupBy(l=>l.id)
.Select(g=>new {id=g.key,value = g.min(l=>l.value);
this will return an anon type with id and value. You can select a new defined POCO type if that's what you need.