There is a society devoted to studying imprecise probabilities, http://www.sipta.org/. Many of the ideas in this field were introduced by Peter Walley in a 1991 book.
The literature on imprecise probabilities is not concerned explicitly with on-line prediction. It considers gambles that a person might consider acceptable, but unlike the literature on game-theoretic probability, it does not consider precisely defined games in which betting offers are made and accepted. But it has concepts of upper and lower probability similar to those in game-theoretic probability, and so we can ask how to relate it to game-theoretic probability. Gert de Cooman has done some work on this.
Here are some open problems concerning the relationship between game-theoretic probability and imprecise probabilities.
- Peter Walley. Statistical reasoning with imprecise probabilities. London; New York: Chapman and Hall, 1991. Monographs on statistics and applied probability: 42. ISBN: 0412286602.