# Are all allocation games equivalent to a negotiation game?

Definitions

An allocation game is a pair $(N, v)$, where $N = \{1, \dots, n\}$ and represents the set of players in the game, and $v$ is a function from $2^N$ to $\mathbb{R}$ that represents the $\textbf{v}$alue any subset of players (a coalition) is capable of producing. A solution to an allocation game is a pair $(C, d)$, where $C \subset N$ is a participating coalition and $d \in \mathbb{R}^n$ is a $\textbf{d}$ivision of the value produced among the coalition: $d \ge 0$, $\sum d_i = v(C)$, and $d_j = 0$ whenever $j \notin C$.

A negotiation game is a triple $(N, W, u)$ where $N = \{1, \dots, n\}$ is again the players, $W = \{1, \dots, w\}$ is a set representing different types of $\textbf{w}$ork available to the group, and $u$ is a continuous $\textbf{u}$tility function from $\mathbb{R}^{n \times w}_{\ge 0}$ to $\mathbb{R}^n$. The game is played by each player simultaneously choosing a level of work (i.e. a number from $0$ to $\infty$) for each available job, jointly producing a matrix in $\mathbb{R}^{n \times w}_{\ge 0}$; then the utility function decides how much each player benefits at the chosen solution. Assume that players always weakly like it when other players do work: i.e. $\frac{\partial u_i(a)}{\partial a_{jw}} \ge 0$ for any $w$ whenever $i \ne j$.

My Question

Intuitively, it would seem that negotiation games are much broader than allocation games. I wonder if this can be made precise. Given an arbitrary allocation game $(N, v)$, can we produce a negotiation game $(N, W, u)$ along with a transformation function $t : 2^N \times \mathbb{R}^n \to \mathbb{R}^{n \times w}_{\ge 0}$ such that whenever $(C, d)$ is a solution of the allocation game, $t(C, d)$ gives corresponding work levels in the negotiation game such that $d = u(t(C, d))$?

Your solution gets extra credit if the $W$ we create is only polynomial in the size of $N$.

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