机制设计与风险规避
Mechanism Design and Risk Aversion
摘要：
我们开发了有效的算法来构建效用最大化机制中存在的风险厌恶的球员（买家和卖家）在贝叶斯设置。我们通过凹效用函数来模拟风险规避，玩家策略性地发挥作用以最大化他们的预期效用。贝叶斯机制设计通常集中在{ \em风险中性}环境中的期望收益最大化，即使对于单参数多单位拍卖，也没有简洁的期望效用最大化机制的特征。我们首先考虑在多单位拍卖的情况下，为风险厌恶的卖方设计最优DSIC机制的问题，我们给出了一个多时间可计算SPM，即$(1-1/e-\eps)$ -近似卖方在最优DSIC机制的期望效用。我们的结果是基于一个新的应用程序的相关性差距的界限，随着{ \em分裂}和{ \em合并}的随机变量，以重新分配概率质量的买家。这使我们能够将问题简化为检查少量不同配置的可行性，每个配置都对应于一个覆盖LP。LP的一个可行解为我们提供了每个买家在随机SPM中使用的价格分布。接下来，我们考虑的设置时，买方以及卖方的风险厌恶，目标是最大化卖方的预期效用。我们设计了一个真实的预期机制，其效用是一个$(1-1/e -\eps)^3$近似的最佳BIC机制下两个温和的假设。我们的机制由多轮组成，在一轮中以小概率处理每个买家。最后，我们考虑了风险中性卖方在存在风险厌恶买方的情况下的收益最大化问题，并给出了一个多时间算法来设计卖方的最优机制。
英文摘要：
We develop efficient algorithms to construct utility maximizing mechanisms in the presence of risk averse players (buyers and sellers) in Bayesian settings. We model risk aversion by a concave utility function, and players play strategically to maximize their expected utility. Bayesian mechanism design has usually focused on maximizing expected revenue in a {\em risk neutral} environment, and no succinct characterization of expected utility maximizing mechanisms is known even for single-parameter multi-unit auctions. We first consider the problem of designing optimal DSIC mechanism for a risk averse seller in the case of multi-unit auctions, and we give a poly-time computable SPM that is $(1-1/e-\eps)$-approximation to the expected utility of the seller in an optimal DSIC mechanism. Our result is based on a novel application of a correlation gap bound, along with {\em splitting} and {\em merging} of random variables to redistribute probability mass across buyers. This allows us to reduce our problem to that of checking feasibility of a small number of distinct configurations, each of which corresponds to a covering LP. A feasible solution to the LP gives us the distribution on prices for each buyer to use in a randomized SPM. We next consider the setting when buyers as well as the seller are risk averse, and the objective is to maximize the seller's expected utility. We design a truthful-in-expectation mechanism whose utility is a $(1-1/e -\eps)^3$-approximation to the optimal BIC mechanism under two mild assumptions. Our mechanism consists of multiple rounds that processes each buyer in a round with small probability. Lastly, we consider the problem of revenue maximization for a risk neutral seller in presence of risk averse buyers, and give a poly-time algorithm to design an optimal mechanism for the seller.
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