Tight Guarantees in the Commons

In our context-free model of a commons, the function$\mathcal{W}$ transforms the profile of the agents’ types $(x_1,\mathrm{.}\mathrm{.},x_n)$ to a freely transferable output $\mathcal{W}(x_1,\mathrm{.}\mathrm{.},x_n)$ that they must share fairly. We expand the ubiquitous concept of $\mathit{\text{endogenous}}\mathit{\text{fair}}\mathit{\text{shares}}$ to include both a lower and an upper bound on agent $i$’s share at the interim stage where $i$ only knows its own type $x_i$. Two functions $(g^{-},g^{+})$ form a pair of tight guarantees if 1) they satisfy the system of inequalities $$ for all profiles, and 2) the interval $[g^{-}(x_i),g^{+}(x_i)]$ is inclusion minimal across all types. For super (resp sub) modular functions 1) the $\mathit{\text{Unanimity}}$share% $$$\frac{1}{n}\mathcal{W}(x_i,x_i,\mathrm{.}\mathrm{.},x_i)$ is the unique tight upper (resp lower) guarantee, 2) two $\mathit{\text{Stand}}\mathit{\text{Alone}}$ shares $$ (where $x_0$ is the smallest or largest type) bracket all tight guarantees on the other side of Unanimity, 3) serial cost sharing implements the Unanimity and Stand Alone guarantees. In applications to specific microeconomic models, tight guarantees vindicate or dismiss familiar deterministic sharing rules and suggest new ones with a clear normative interpretation. Our examples include joint production with substitute or complementary inputs, allocating an indivisible good and cash transfers, sharing the cost (or benefit) of the variance or the spread of types, the waiting cost in a queue, and more.