Is the Nash product really maximised ex post? In my game theory class this term, we studied Nash bargaining. It is only now when starting to prepare for the exam that I have come to realise there is something I fundamentally don't understand, and hope that somebody here might be able to help. :)
This is a question highlighting my issue:



The Nash bargaining solution to this is (confirmed with the lecturer) the payoff pair (3/2,3/2). This is achieved by player 1 playing T and player 2 mixing L and C with 1/4, 3/4 probability. The perhaps simplest way to find this solution is to find the midpoint of a tangent to the convex set of possible utilities.

My issue is, that I am not convinced that this maximises the Nash product s_1*s_2 (the disagreement pair is (0,0)).

IF one assumes that the payoff of player 1 and player 2 vary independently, then the above is indeed the solution because E[s_1] = 1/4*3 + 3/4*1 = 3/2 and E[s_2] = 1/4*0 + 3/4*2 = 3/2, which gives E[s_1*s_2] = 9/4 (which can be shown to be maximum).

However, the payoffs do not vary independently! (T,L) corresponds to Nash product 0 and (T,C) corresponds to 2. Hence the true expected value for the Nash product under this randomisation is 1/4*0+3/4*2=3/2. This is not optimal, as purely playing (T,C) gives a higher Nash product.

Hence, how comes that the Nash bargaining solution isn't actually maximising the Nash product? Where do I go wrong in my thinking?

Very many thanks in advance! :)
The Nash bargaining solution DOES maximize the Nash product. You have to separate the playing of the game from the bargaining problem. If the players negotiate a binding agreement they will realize that their maximal total payoff from playing the game is $3$. This can be achieved by playing $(T,L)$ or $(T,C)$, or any mixture of those two profiles.

The bargaining solution just asks: Given a total payoff of $3$, how will the players divide this payoff among them, given a threat point of $(0,0)$? So the efficient frontier is (a part of) the line $u_1+u_2=3$, and by symmetry (or by maximizing the Nash product $u_1(3-u_1)$) you find $u_1=u_2=3/2$. The only remaining question is how to play the game (under a binding agreement) in such a way that these are the expected payoffs. But you already solved this part.