Quasi- Concave Social Welfare Functions

功利主义认为，社会应该关注个人效用总和的最大化。现代对功利主义的反对意见是，它意味着对随机化漠不关心。 当社会对把一种不可分割的物品给予几个人中的一个人漠不关心时，通过抽签选择接受者什么也得不到。 非线性替代功利主义可以解决这个问题。这些替代方案使用了技术上称为“准凹社会福利函数”的方法。“在这里，我们提出了功利主义和这些替代品之间的另一个区别。我们没有从社会冷漠的情况开始，而是集中在极端不平等的情况下，并研究功利主义和非功利主义在这些情况下的含义。假设社会面临着减少更糟糕的人的数量和改善他们的整体地位之间的权衡。这两个目标都是需要的，但往往一个是以另一个为代价的。我们的理论推测是，在对称的情况下，社会越倾向于随机化，它就越倾向于以增加底层人口为代价来改善他们的地位。在本研究的理论性较低的部分，我们想建立以下主张：自由、自由的社会与准凹偏好所暗示的方向相同。慈善事业假设社会由两个群体组成，富人和穷人。“我们有一个给定的预算来帮助第二组，选择是两个政策：“个人帮助”，从第二组中取出个人并将他们转移到第一组，“社区帮助”，其中的钱用于改善第二组的设施。我们也可以选择两者的任何可行的组合。增加用于社区帮助的金额意味着用于个人帮助的资金减少，因此处于较低群体的概率更高，但该群体中所有个人的福利随着用于此类计划的资金而增加。我们预测，一个准凹进的社会将更倾向于社区而不是个人的帮助。残酷而异常的惩罚严厉的惩罚比温和的惩罚更有威慑力。通过使用这样的惩罚，我们可以减少受惩罚的人数，提高那些没有犯罪的人的地位，而代价是减少那些处于底层的人的福利。 我们的推测是，功利社会将使用更严厉的惩罚比准凹社会。安全和预防有时，准凹社会选择的结果，其最优性是值得怀疑的。假设社会可以在改善公共安全和花钱拯救事故受害者之间做出选择。进一步假设，第一种方法可以拯救更多的人。 结果表明，在一定的预算下，准凹型社会比功利型社会在事故发生后的预防支出更少，而在事故发生后的救助支出更多。

Utilitarianism suggests that societies should be interested in maximizing the sum of individual utilities. Modern objections to utilitarianism are that it implies indifference to randomization. When society is indifferent between giving an indivisible good to one of several individuals, nothing is gained by choosing the recipient by a lottery. Nonlinear alternatives to utilitarianism can solve this problem. Such alternatives use what is technically labeled as 'quasi concave social welfare functions.'Here we suggest another difference between utilitarianism and these alternatives. Rather than starting with situations of social indifference, we concentrate on situations of extreme inequality, and investigate the implications utilitarian and non-utilitarian have in these situations.Suppose society is facing trade-off between reducing the number of those who are worse off and improving their overall position. Both objects are desired, but often one comes at the cost of the other. Our theoretical conjecture is that the more society prefers randomization in symmetric situations, the more it will prefer improving the position of those who are at the bottom at the cost of increasing their numbers.In the less theoretical part of this research, we would like to establish the following claim: free, liberal societies move in the same direction as the one suggested by quasi concave preferences. CharitySuppose society is composed of two groups, `haves' and `have nots.' We have a given budget to help the second group, and the choice is between two policies: "Individual help" that takes individuals from the second group and transfer them into the first, and "community help," where the money is spent on improving facilities for the second group. We can also choose any feasible combination of the two.Increasing the amount spent on community help means less money for individual help, therefore higher probability of being at the lower group, but the welfare of all individuals of this group is increasing with the money spent on such programs. We predict that a quasi-concave society will prefer the community to the individual help.Cruel and Unusual PunishmentHarsh punishment is more deterring than mild one. By using such punishments we can reduce the number of people who will be punished, and improve the position of those who do not commit crimes, at the cost of reducing the welfare of those who are at the bottom. Our conjecture is that the utilitarian society will use harsher penalties than the quasi-concave society.Safety and PreventionSometimes, quasi-concave societies choose outcomes whose optimality is questionable. Suppose society can choose between improving public safety, or spending money on saving people who had an accident. Suppose further that more people could be saved the first way. It follows from our conjectures that for a given budget, the quasi concave society will spend less on prevention and more on saving people after an accident happens than the utilitarian society.