Efficient Defection: Overage-Proportional Rationing Attains the Cooperative Frontier

Abstract

We study a noncooperative $n$-player game of slack allocation in which each player $j$ has entitlement $L_j>0$ and chooses a claim $C_j\ge0$. Let $v_j=(C_j-L_j)_+$ (overage) and $s_j=(L_j-C_j)_+$ (slack); set $X=\sum_j v_j$ and $I=\sum_j s_j$. At the end of the period an overage-proportional clearing rule allocates cooperative surplus $I$ to defectors in proportion to $v_j$; cooperators receive $C_j$. We show: (i) the selfish outcome reproduces the cooperative payoff vector $(L_1,\dots,L_n)$; (ii) with bounded actions, defection is a weakly dominant strategy; (iii) within the $\alpha$-power family, the linear rule ($\alpha=1$) is the unique boundary-continuous member; and (iv) the dominant-strategy outcome is Strong Nash under transferable utility and hence coalition-proof (Bernheim et al., 1987). We give a policy interpretation for carbon rationing with a penalty collar.