The decider function D:
We want each actor to be selected as decider with frequency proportional to their effective stake. But each actor may not be reachable (online) all the time. They may also not be honest or accurate. So we want D to select not just one decider, but a sequence of candidate deciders of the next versions of history, so that actors can have a choice between alternative versions of history via the preference function. We also want D to be based on the version of history h you wish to extend with your current events, we write D(h) to reflect this.
One way for D(h) to satisfy this is via:
1) For each account with a positive effective stake, roll a deterministic die based on the history h you wish to extend. Call the result HIT
2) Order all such accounts in increasing HIT/EFFECTIVE_STAKE.
3) The first account that is reachable (online) is the first candidate, the next reachable account is the second, etc.
This D does indeed choose each actor with frequency proportional to their effective stake, as the number of accounts with an effective stake grows to infinity. (subject to simple niceness conditions; ask me if you're interested in the proof).
When you want to extend a version of history with your current events, you send a request to the first few candidates. Send to more to increase the chances of your events being included in some version of history.
I think the above is what's known so far about TF, from BCNext via CfB. This assumes all actors are honest and accurate, which not all will be. Below is one possible (untested) modification:
Each actor maintains a personal weighted blacklist of other actors that they believe to be dishonest, inaccurate, or otherwise do not wish to send their information to. The weights are probabilities of not choosing them as a candidate decider and so not sending your events to them when you would normally do so. So a weight of 1 would mean never send them any events. Weights are updated as new information is gained about other actors and your confidence in their honesty and accuracy changes.