Summary: Based on a dream that I just woke up from: Prisoner's Dilemma AU, but make it Cooperate or Die (technically, cooperate or the other dies; see below).
Explanation:
The Prisoner's Dilemma is a more or less hypothetical scenario from game theory: two people (agents in one or the other sense, so to say) who cannot communicate among each other can either decide to cooperate for mutual gain, or betray the other for their own gain; except when both betray, no one gets anything/both lose. Most often in this game, the individual gain for the cooperative approach is less than it would be for the non-cooperative one, but the summed-up gain for the cooperative one is higher than for the individual, ergo you're constantly tempted to betray the other, knowing that you're losing out too if you're betrayed in turn. Once you play this "game" multiple rounds, it becomes more attractive in the long term to cooperate because you know the other can react to your decisions.
(see https://ncase.me/trust/ which was my first introduction to this concept.)
Imagine Harry and Severus, captured in the same trap, with no way to communicate to each other, the fate and ultimate survival of one relying on the other, but constantly being tempted by the villain / the system to "betray" the other for basic comforts / necessities.
For added angst, you can make it so that
1) they don't know in the beginning who it is they're "playing" with.
2) the answers "cooperate" and "betray" are not immediately visible as such; or there are multiple options in each round where only one or few of those lead to cooperation.
3) adjacent to that, game theory also introduces the element of randomness where a defined fraction of the responses is flipped to account for "transmission errors".
(it also means that there's no direct way of communicating one's intention / that you only see the result of what the other decided.)
4) each round, the stakes are differently high, but they don't know how high.
From there, we have a couple questions. There are many ways to play it (like always deciding to cooperate, cooperate according to pre-determined fixed pattern, only betray if the other betrayed in the previous round, always betray, etc.). I assume Severus could be using the predetermined pattern one to try to communicate / send a message (e.g. in Morse), but which one could Harry play?
How high are the stakes of each round / is there a way to anticipate how high they are?
If you take the added angst elements, how do they eventually find out who the other is?
How do they recover (if at all) from getting off on the wrong foot, so to say, if they got caught into a spiral of 'betray's early in the game?
(end result is of course that they both survive and build trust against all possible odds, and beat the game while they're at it.)
Anyway, some playing around in the sandbox mode of the link above leads me to believe that, even in a (rather inadequate) simulation of those harsher conditions, on a population basis, the more "forgiving" strategies / the more cooperative ones still hold out the best if the cooperation reward is above 0. If below, the winning mode seems to be "simpleton" which is "start with cooperate, copy you if you cooperated last time, do the opposite as you did if you cheated the last time".
So basically, your innate survival at all costs instinct is your worst enemy here.
(oh, and for some reason, Ezkridis was the main antagonist / creator of this setting.)
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