Even then, from the other player’s perspective, unless they have reason to trust you at your word that you’ll share, will have no incentive not to screw you over. In this variant there’s equally no incentive in favour of betrayal if you think your opponent will betray you, whereas in the original stating of the problem you still would have an incentive to choose betrayal – reducing your own penalty.

Basically the game’s not quite a full Prisoner’s Dilemma because betrayal only weakly dominates co-operation (you get an equal nothing no matter what you do if the other player betrays you, as opposed to getting royally screwed if you co-operate and they betray as against a minor penalty if both players betray each other)

Now I need to not use the word betray for a while, because it’s starting to sound like a nonsensical gibberish word. Much like elbow or spoon if you say them enough times.