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! :)