We study the price of anarchy (PoA) of simultaneous 2nd price auctions (S2PA) under a natural condition of {\em no underbidding}. No underbidding means that an agent's bid on every item is at least its marginal value given the outcome. In a 2nd price auction, underbidding on an item is weakly dominated by bidding the item's marginal value. Indeed, the no underbidding assumption is justified both theoretically and empirically. We establish bounds on the PoA of S2PA under no underbidding for different valuation classes, in both full-information and incomplete information settings. To derive our results, we introduce a new parameterized property of auctions, namely $(\gamma,\delta)$-revenue guaranteed, and show that every auction that is $(\gamma,\delta)$-revenue guaranteed has PoA at least $\gamma/(1+\delta)$. An auction that is both $(\lambda,\mu)$-smooth and $(\gamma,\delta)$-revenue guaranteed has PoA at least $(\gamma+\lambda)/(1+\delta+\mu)$. Via extension theorems, these bounds extend to coarse correlated equilibria in full information settings, and to Bayesian PoA (BPoA) in settings with incomplete information. We show that S2PA with submodular valuations and no underbidding is $(1,1)$-revenue guaranteed, implying that the PoA is at least $\frac{1}{2}$. Together with the known $(1,1)$-smoothness (under the standard no overbididng assumption), it gives PoA of $2/3$ and this is tight. For valuations beyond submodular valuations we employ a stronger condition of {\em set no underbidding}, which extends the no underbidding condition to sets of items. We show that S2PA with set no underbidding is $(1,1)$-revenue guaranteed for arbitrary valuations, implying a PoA of at least $1/2$. Together with no overbidding we get a lower bound of $\frac{2}{3}$ on the Bayesian PoA for XOS valuations, and on the PoA for subadditive valuations.

中文翻译：

---

不低于竞价的同步第二价格项目拍卖

我们研究了在 {\em no underbidding} 的自然条件下同步第二价格拍卖 (S2PA) 的无政府状态 (PoA) 的价格。没有低价意味着代理人对每个项目的出价至少是给定结果的边际价值。在第二价格拍卖中，出价过低的物品主要受出价物品的边际价值的影响。事实上，没有低价假设在理论上和经验上都是合理的。我们在全信息和不完整信息设置下，在不低于不同估值类别的情况下建立了 S2PA 的 PoA 界限。为了得出我们的结果，我们引入了一个新的拍卖参数化属性，即 $(\gamma,\delta)$-收入保证，并表明每个 $(\gamma,\delta)$-收入保证的拍卖的 PoA 为至少 $\gamma/(1+\delta)$。$(\lambda,\mu)$-smooth 和 $(\gamma,\delta)$-收入保证的拍卖至少有 $(\gamma+\lambda)/(1+\delta+\mu)$ . 通过扩展定理，这些界限扩展到完整信息设置中的粗相关均衡，以及不完整信息设置中的贝叶斯 PoA (BPoA)。我们表明，具有子模块估值且没有低价的 S2PA 是 $(1,1)$-收入保证，这意味着 PoA 至少为 $\frac{1}{2}$。加上已知的 $(1,1)$-smoothness（在标准没有出价过高的假设下），它给出了 $2/3$ 的 PoA，这是很紧的。对于超出子模块估值的估值，我们采用更强的 {\em set no underbidding} 条件，将无低于投标条件扩展到项目集。我们表明，没有设置低价的 S2PA 是 $(1,1)$ - 保证任意估值的收入，意味着至少 1/2 美元的 PoA。在没有出价过高的情况下，我们在 XOS 估值的贝叶斯 PoA 和次可加估值的 PoA 上得到了 $\frac{2}{3}$ 的下限。