Ensuring global cooperation often poses governance problems shadowed by the tragedy of the commons, as wrong-doers enjoy the benefits set up by right-doers at no cost. Institutional punishment of wrong-doers is well-known to curtail their impetus as free-riders. However, institutions often have limited scope in imposing sanctions, more so when these are strict and potentially viewed as disproportionate. Inspired by the design principles proposed by the late Nobel Prize Elinor Ostrom, here we study the evolution and impact of a new form of institutional sanctioning, where punishment is graduated, growing with the incidence of free-riding. We develop an analytical model capable of identifying the conditions under which this design principle is conducive to the self-organization of stable institutions and cooperation. We employ evolutionary game theory in finite populations and non-linear public goods dilemmas in the presence of risk of global losses whose solution requires the self-organization of decision makers into an overall cooperative state. We show that graduated punishment is more effective in promoting widespread cooperation than conventional forms of punishment studied to date, being also less severe and thus, presumably, easier to implement. This effect is enhanced whenever the costs of its implementation are positively correlated with the severity of punishment. We frame our model within the context of the global reduction of carbon emissions, but the results are shown to be general enough to be applicable to other collective action problems, shedding further light into the origins of Human institutions.

确保全球合作通常会带来公地悲剧所掩盖的治理问题，因为做错了事的人会免费享受到做对了事的人的利益。众所周知，对过错者的制度惩罚是减少其作为搭便车者的动力。但是，机构实施制裁的范围通常有限，如果制裁严格且可能被视为不相称，则更是如此。受已故的诺贝尔奖埃里诺·奥斯特罗姆（Elinor Ostrom）提出的设计原则的启发，在这里，我们研究了一种新的制度制裁形式的演变和影响，这种形式的刑罚是逐渐增加的，随搭便车的发生而增加。我们开发了一种分析模型，该模型能够确定在哪些条件下该设计原则有利于稳定机构和合作的自组织。在存在全球损失风险的情况下，我们将演化博弈论应用于有限的种群和非线性公共物品困境，其解决方案需要决策者将自身组织为一个整体合作状态。我们表明，与迄今研究的传统惩罚形式相比，渐进式惩罚在促进广泛合作方面更为有效，而且程度也较轻，因此更容易实施。只要实施成本与惩罚的严重程度成正比，这种效果就会增强。我们在全球减少碳排放量的背景下构建了模型，但是结果显示出足够普遍，可适用于其他集体行动问题，这为人类机构的起源提供了更多的启示。