...orrrr you could complicate your reasoning processes to the point where they were no longer valid, and add a slight dash of motivated reasoning, that might also work for crossing the gap.

"I'm likelier than that to cooperate!" objects Tonia. "I think I'll probably cooperate though not if we stop talking right this second because if we stop talking right this second you just said there's only a one in eight chance you'll cooperate. But I expect we'll keep having this stupid argument in front of Keltham until I'm convinced. ...and a dath ilani would be convinced already from predicting that. So I guess I'm convinced. How likely are you to cooperate right now."

"One in four, let's say?  You don't really believe that but you're trying to convince yourself, which is something I want to go on encouraging."

...okay, his being here is probably influencing this, if only through his not being able to fully conceal his horrified expressions at their version of the Algorithm.  So he is going to go check and see if there's a Silent Proctor waiting for him, and if Security explained the instructions already.

Paxti is repeatedly spinning a coin and noting down the results, while a (visible) Security officer watches with a bored expression.

Keltham will wait until Paxti stops spinning, then speak.  "Yo.  Your bid on the job?"

"Two coppers.  I was bored."

"Pilar and Meritxell finished before I got here, Sevar and Peranza just left.  I don't tell you the particular results, right?"

"Correct.  You tell me the total number of Defections and Cooperations, unless that number was one or zero for either, in which case you tell me nothing.  You also tell me the total payout, and an amount of copper and silver you can use to make those payouts."

"Which means I tell you nothing right now, right?"

"Right.  Well, repeat back to me your scoring instructions so I can make sure you got those right."

"Everyone starts with three copper plus one copper for each of ten coinspins that comes up Queen plus one if they Defected minus three if their partner Defected."  Never mind flunking out, anyone who can't follow instructions on that level doesn't get to go to wizard school.

"Yup.  Carry on, then, sorry for interrupting."

"Keltham."

Keltham manages not to startle.  Over the last days of sitting quietly in a far corner of the room not saying anything, Broom has achieved something tantamount to cognitive invisibility.

Or, actually...

"Broom, are you using magic to make people not much notice or think about you?"

"It is not magic.  It is a talent of my people.  We are simply good at being unobtrusive."

Broom knows, speaking, that these are not the words of a slave.  He sets aside old anxieties at the thought.  It has been claimed to Broom that he is not a slave anymore, at least not of Cheliax; and in alterCheliax he has never been a slave.  He has been requested to conduct himself accordingly.

(He knows of alterCheliax now, the divergent mini-reality they are constructing to contain Keltham.  Broom spoke for a time with the one called Asmodia, who, despite her strange mission, seemed harried and exasperated in a way that made Broom feel oddly sympathetic towards her.  In her own way Asmodia seemed like a slave with too much to clean up, much like he once was; he would not, without reason, make her work any harder.)

"Is your people's talent, by any chance, a sort that gets blocked by Mind Blank."

"No," Broom says, Security having swiftly instructed him that this is the true answer so far as the Security knows.

"Right.  Well, what's up, then?"

"I wish to play against you the game that you are having these others play."

"...why."

"In search of understanding."

"Understanding of the game, or understanding of Keltham?"

"- don't say 'yes', either."

"Understanding of Keltham."

All right, that seems honest enough.  "Okay then.  We'll get paper and find a breakout room."

"I want both of those two idiots humbled, and by humbled I mean raped."