Mostly looking for a name, or better even, reference to literature, for this type of problem in terms of game theory or economics. If no such thing, speculation welcome.

Intro below is more or less how I originally came to think of it and felt like writing, please skip unless you have already read the rest of the internet and found your way back here.

[intro]

Its been a dark month, turning upside down every last rock in every last alley in this god-forsaken city. "The things we see in this job", you think to yourself, as your phone starts to ring. But hey! You seem to have caught a break. You rush to the station

And there he is, right as you enter the interrogation cell, staring back across the room with those demented, demented eyes. The kind of look that sits like lead in the bottom of your guts. The kind of look only a man that has stolen 31 lollipops in broad daylight could summon. And right then, you know. You have him, dead to rights. The city can sleep at last. 

Or can it?

"Its not as easy as you put it, as per usual" Assistant DA Alice says, as every Assistant DA in every gritty cop story has said before her "We still haven't found the sticks for Lollipops number 6,16 , 25 and 30, and we owe those candy shop owners some closure, if nothing else" she says, as you think of Bob, who lost seven of his lollipops to the first citywide rampage of the very man you have in custody. Almost a decade later and you still think of him, of what has been taken from him.  You just noticed your fist has been closed the whole time.

"He says he can deliver the sticks we are missing. Wants only 5 years."

"5 years, down from life?" You shout, barely able to stand still.

"I told him 5 is a non starter. But there is a lot to consider here"

"Politics", you grunt, grinding your teeth in a very manly fashion

"If there is no plea, we still have a very good chance to put him in jail for good, but no lollipop sticks for us. However, in a suspiciously convenient turn of events, the judge has forgotten the maximum time allowed before pressing charges and we still have a month to negotiate."

[end of intro]

Okay, so, 

player A has (t, s) as resources. He wants to minimize t. He doesn't mind giving his s away. 

player B has (t, s) as resources. He wants to maximize t. He wants to maximize his s. 

If they don't agree on an arrangement after a certain number of iterations, t is maximized (originally, you would have a fixed, high chance of conviction, but seems simpler to solve first without this) and s lost. This is an acceptable situation for B but less so for A, as from his perspective any number of s he has to give away for any number of t is worth it, this is saying, any t:s exchange ratio B offers, A should take. (Similarity to the Ultimatum is discussed below)

A few things I think this is not: 

• Prisoner's Dilemma, for many reasons. There is a prisoner though.
• Chicken game, since one party suffers a harsher penalty if coordination fails
• Ultimatum, since its iterated. 
• Iterated Ultimatum (this one looks most similar , I think, but can't quite connect all the dots): main difference is, again, one party suffers a harsher penalty if coordination fails. If I offer you 10 dollars (I keep 90 for being the mastermind) and you decline, you could say I lost more than you, but seems an abuse of the term since I come out with the same money I came in the room. Also, here there isn't a second resource (s) in the equation, which might be superfluous but I think influences the negotiation

As I said, a name would be appreciated, as would any possible strategies you might come up with.