In a Stackelberg Pricing Game a distinguished player, the leader , chooses prices for a set of items, and the other players, the followers , each seek to buy a minimum cost feasible subset of the items. The goal of the leader is to maximize her revenue, which is determined by the sold items and their prices. Most previously studied cases of such games can be captured by a combinatorial model where we have a base set of items, some with fixed prices, some priceable, and constraints on the subsets that are feasible for each follower. In this combinatorial setting, Briest et al. and Balcan et al. independently showed that the maximum revenue can be approximated to a factor of $$H_k\sim \log k$$ H k ∼ log k , where k is the number of priceable items. Our results are twofold. First, we strongly generalize the model by letting the follower minimize any continuous function plus a linear term over any compact subset of $${\mathbb {R}}^n_{\ge 0}$$ R ≥ 0 n ; the coefficients (or prices ) in the linear term are chosen by the leader and determine her revenue. In particular, this includes the fundamental case of linear programs. We give a tight lower bound on the revenue of the leader, generalizing the results of Briest et al. and Balcan et al. Besides, we prove that it is strongly NP-hard to decide whether the optimum revenue exceeds the lower bound by an arbitrarily small factor. Second, we study the parameterized complexity of computing the optimal revenue with respect to the number k of priceable items. In the combinatorial setting, given an efficient algorithm for optimal follower solutions, the maximum revenue can be found by enumerating the $$2^k$$ 2 k subsets of priceable items and computing optimal prices via a result of Briest et al., giving time $$O(2^k|I|^c)$$ O ( 2 k | I | c ) where | I | is the input size. Our main result here is a W[1]-hardness proof for the case where the followers minimize a linear program, ruling out running time $$f(k)|I|^c$$ f ( k ) | I | c unless $$\mathsf {FPT} =\mathsf {W[1]} $$ FPT = W [ 1 ] and ruling out time $$|I|^{o(k)}$$ | I | o ( k ) under the Exponential-Time Hypothesis.

中文翻译：

---

Stackelberg 定价游戏中的收入最大化：超越组合设置

在 Stackelberg 定价游戏中，一位杰出的玩家，领导者，为一组物品选择价格，而其他玩家，即追随者，每个人都试图购买这些物品的最小成本可行子集。领导者的目标是最大化她的收入，这是由售出的物品及其价格决定的。大多数以前研究过的此类游戏案例可以通过组合模型来捕获，其中我们有一组基本的项目，一些具有固定价格，一些价格合理，以及对每个追随者可行的子集的约束。在这种组合环境中，布里斯特等人。和巴尔坎等人。独立表明，最大收入可以近似为 $$H_k\sim \log k$$ H k ∼ log k 的因子，其中 k 是可定价物品的数量。我们的结果是双重的。第一的，我们通过让跟随者在 $${\mathbb {R}}^n_{\ge 0}$$ R ≥ 0 n 的任何紧凑子集上最小化任何连续函数加上线性项来强烈概括模型；线性项中的系数（或价格）由领导者选择并决定她的收入。特别是，这包括线性规划的基本情况。我们给出了领导者收入的严格下限，概括了 Briest 等人的结果。和巴尔坎等人。此外，我们证明了决定最优收入是否超过任意小因素的下界是非常 NP-hard 的。其次，我们研究了计算与可定价物品的数量 k 相关的最佳收入的参数化复杂性。在组合设置中，给定一个用于最优跟随器解决方案的有效算法，最大收入可以通过枚举 $$2^k$$ 2 k 的有价物品子集并通过 Briest 等人的结果计算最优价格来找到，给出时间 $$O(2^k|I|^c)$ $ O ( 2 k | I | c ) 其中 | 我| 是输入大小。我们这里的主要结果是 W[1]-hardness 证明，用于跟随者最小化线性程序的情况，排除运行时间 $$f(k)|I|^c$$ f ( k ) | 我| c 除非 $$\mathsf {FPT} =\mathsf {W[1]} $$ FPT = W [ 1 ] 并排除时间 $$|I|^{o(k)}$$ | 我| o ( k ) 在指数时间假设下。排除运行时间 $$f(k)|I|^c$$ f ( k ) | 我| c 除非 $$\mathsf {FPT} =\mathsf {W[1]} $$ FPT = W [ 1 ] 并排除时间 $$|I|^{o(k)}$$ | 我| o ( k ) 在指数时间假设下。排除运行时间 $$f(k)|I|^c$$ f ( k ) | 我| c 除非 $$\mathsf {FPT} =\mathsf {W[1]} $$ FPT = W [ 1 ] 并排除时间 $$|I|^{o(k)}$$ | 我| o ( k ) 在指数时间假设下。