I believe @jfriend00 answered the question very clearly. However, I do want to add a thought.
By throwing in a worst case (and improbable) scenario for Websockets vs. HTTP, you would clearly see that a Websocket connection will always have an advantage in regards to bandwidth (and probably all-round performance).
This is the worst case scenario for Websockets v/s HTTP:
your code uses Websocket connections the exact same way it uses HTTP requests, for polling.
(which isn't something you would do, I know, but it is a worst case scenario).
Every polling event is answered positively - meaning that no HTTP requests were performed in vain.
This is the worst possible situation for Websockets, which are designed for pushing data rather than polling... even in this situation Websockets will save you both bandwidth and CPU cycles.
Seriously, even ignoring the DNS query (performed by the client, so you might not care about it) and the TCP/IP handshake (which is expensive for both the client and the server), a Websocket connection is still more performant and cost-effective.
I'll explain:
Each HTTP request includes a lot of data, such as cookies and other headers. In many cases, each HTTP request is also subject to client authentication... rarely is data given away to anybody.
This means that HTTP connections pass all this data (and possibly perform client authentication) once per request.[Stateless]
However, Websocket connections are stateful. The data is sent only once (instead of each time a request is made). Client authentication occurs only during the Websocket connection negotiation.
This means that Websocket connections pass the same data (and possibly perform client authentication) once per connection (once for all polls).
So even in this worst case scenario, where polling is always positive and Websockets are used for polling instead of pushing data, Websockets will still save your server both bandwidth and other resources (i.e. CPU time).
I think the answer to your question, simply put, is "never". Websockets are never less efficient than polling.