We systematically study pairwise counter-monotonicity, an extremal notion of negative dependence. A stochastic representation and an invariance property are established for this dependence structure. We show that pairwise counter-monotonicity implies negative association, and it is equivalent to joint mix dependence if both are possible for the same marginal distributions. We find an intimate connection between pairwise counter-monotonicity and risk sharing problems for quantile agents. This result highlights the importance of this extremal negative dependence structure in optimal allocations for agents who are not risk averse in the classic sense.

中文翻译：

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成对反单调性

我们系统地研究了成对反单调性，这是一种负相关的极端概念。为该依赖结构建立了随机表示和不变性。我们表明成对反单调性意味着负关联，如果对于相同的边际分布两者都是可能的，则它等同于联合混合依赖。我们发现成对反单调性和分位数代理的风险分担问题之间存在密切联系。这一结果突出了这种极值负依赖结构在对经典意义上不厌恶风险的代理人进行优化分配时的重要性。