Is the Nash product really maximised ex post?

Question

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! :)

Answer 1

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.