¶1. In classical game theory, a normal form game between $k$ players is given by finite sets of options $A_1,\ldots,A_k$ and payoff functions $u_1,\ldots,u_k : \prod_i A_i \to \mathbb{R}$ (all of these data are known to all the players). Each player $i$ chooses an option $a_i\in A_i$ independently of all the others (following a — in general mixed — strategy), and receives the payoff $u_i(a)$ where $a = (a_1,\ldots,a_k)$, which they seek to maximize. The theory of such games is well developed and I need not say any more.
¶2. In contrast, I wonder if there is some kind of precise mathematical theory of negotiation games. I don't know exactly what I mean by “negotiation” because if I knew how to define it rigorously I wouldn't need to ask the question, but the idea is that the players, instead of choosing their options independently of each other, get around a table and negotiate, and ultimately make a choice with full knowledge of every other player's choice. Equivalently, they try to define an option $a$ that is agreeable to everyone, knowing that if they fail to agree on one they will receive a “default” payoff for failure of negotiations. The whole issue is whether we can give a rigorous meaning to this and to what “negotiation” can achieve.
¶3. Here is one possible setup: fix a finite set $A$ of options and payoff functions $u_1,\ldots,u_k : A \to \mathbb{R}$; and also fix a $k$-tuple of reals $(d_1,\ldots,d_k)$ called the “default payoffs” in case the negotiations fail (again, all of these data are known to all the players). The players negotiate and try to find some $a \in A$ which they all agree on — if they all agree to a common $a \in A$ (or a probability distribution on such) then their payoffs are given by the $u_i(a)$, whereas if they cannot come to an agreement then they will be given payoffs $d_i$ (of course we should think that $d_i \leq u_i(a)$ for each $i$ and each $a\in A$, otherwise the option $a$ will be unacceptable to player $i$ and will never be assented to). Is there a theory that tells us what happens for such games?
(Maybe a better definition for $k>2$ would be to allow partial negotiations between subsets of players, and thus partial defaults; but I don't know what kind of constraints make sense, so maybe assume $k=2$ because this case is already interesting on its own.)
¶4. Again, I don't know if such a theory exists or even makes sense, but here is one example of a simple game of this sort for which we can predict the outcome if a theory exists. Let $k=2$ and let $c,d_1,d_2$ be three real numbers such that $c > d_1+d_2$. The game is the following: the two players can either agree to share a common payoff $c$ between themselves (meaning that one player will receive $s$ and the other will receive $c-s$ for some $s$ that must be agreed upon by both players) or, alternatively, they can refuse the split and receive $d_1,d_2$ respectively. After discretizing the ways to split the common $c$ into $u_1(s) = s$ and $u_2(s) = c-s$ for some sufficiently large set $A$ of values $s$, this is a game of the form considered in ¶3.
¶5. But it is easy to see how the negotiation of the game in ¶4 should resolve: any sensible theory must be invariant under translation of the each player's payoff independently, so (by subtracting $d_1+d_2$ to $c$) we can assume $d_1=d_2=0$, in which case the two players have a symmetric role so the negotiation should arrive at $s = \frac{c}{2}$; so in the general form, the negotiations should arrive at $\frac{c + d_1 - d_2}{2}$ for player 1 and $\frac{c + d_2 - d_1}{2}$ for player 2. In other words, any mathematical theory of games such as in ¶2–3 which satisfies translation-independence of payoffs and symmetry of players must predict this outcome of negotiations for the example described in ¶4.
So, question: does there exist a precise mathematical theory of games such as in ¶2–3 which at least supports the analysis of the example in ¶4–5?
