A Gulag strategy is not only about knowing who is loyal and who is disloyal, it's also about knowing who knows who is loyal and who is disloyal. Without this knowledge, your gameplay will be limited to guessing. Let's look at some situations where you use your knowledge.
You're loyal and play the commissar: Your challenge is to find either three loyal workers or two loyal workers and one disloyal worker to be supervised by one of the loyal workers. In the latter case, the supervising worker should ideally know the supervised worker but not be known by him or her.
You're loyal and are selected to work: Your challenge is to know whether you were selected because of your loyalty (the commissar trusts you to work) or in spite of your loyalty (the commissar intends to have you supervised). In addition, you need to know if the supervisor knows you and is with you (and won't supervise you) or against you (and will supervise you).
You're loyal and are selected to supervise: Your challenge is to know if the other workers are with you (in which case they shouldn't be supervised) or against you (in which case they should be supervised).
Let's look at a typical opening game to illustrate this. When the game starts, the commissar (1) knows one other player (2) who in turn knows one third player (3). If the commissar is loyal and player 2 is also loyal, that player should be selected as a supervisor. Why? Because the supervisor will then play according to your objective and either supervise player 3 (if player 2 knows player 3 is disloyal) or refrain from supervision (if player 2 knows player 3 is loyal).
Player 3 may seem to have a disadvantage here but it's only partly true. Yes, she'll likely lose the "poker game", since she has no knowledge of player 1 and 2. On the other hand, given their actions, she will end the turn with knowledge of two additional players. If player 3 is disloyal and gets supervised, she can deduce that player 1 and player 2 are loyal, and if she doesn't get supervised, she can deduce that player 1 and 2 are disloyal as well. This knowledge will be very valuable in coming turns.
What if we change the opening example and let player 2 be disloyal then? This puts the commissar in a more difficult situation. One option is to exclude player 2 from the selected workers but this gives her no guarantee that the other players are loyal and will also make player 2 suspicious to her. She may also accept the short-term loss and include player 2 to give her the impression that she's also disloyal. If this helps her into future disloyal teams, she'll be able to create confusion and regain the loss. I generally favor the latter, particularly in longer games.
In the opening game, the knowledge is equal with each player knowing one other player. The longer the game, the more unequal does the knowledge get and the more challening the decisions get. It's not enough to know which players are on your side, you must also know how well they know each other. If you're loyal and select three other loyal player, the result may be disastrous if they don't trust each other and supervise or play against their objectives. Similarly, if most other players know you're loyal and you select three disloyal player, they may start suspecting each other (why else would they be selected by a loyal player?).
In the endgame, most players should know the loyalty of each other. This is where the guessing game starts. If the supervisor knows you and you know the supervisor and you both have different objectives, what do you do? Do you expect to be supervised and play against your objective (to be "forced" to play differently) or do you expect someone else to be supervised and play according to your objective? There is no definite answer but the better you can answer, the greater your chance of winning!