塔斯基定理、超模博弈与均衡的复杂性
Tarski's Theorem, Supermodular Games, and the Complexity of Equilibria
AI总结：
该研究分析了基于塔斯基定理的不动点与超模博弈均衡的计算复杂性，明确了相关问题的复杂度类归约，给出了不同场景下的求解时间并指出了开放问题。
摘要：
在经济学中，单调性与塔斯基定理被广泛用于均衡存在性证明，同时该定理也常被用于验证领域的类似目的，但针对该定理保证的不动点与均衡的寻找复杂性的分析却相对较少。我们研究基于边长为N的d维网格上的单调函数及其不动点的计算形式，以及联系紧密的超模博弈及其均衡相关主题。已知寻找单调函数的任意不动点可在log^d N时间内完成，我们证明即使是二维网格上的情况，该过程至少需要log² N次函数评估，即使对于随机算法也是如此。我们证明，当单调函数以布尔电路的形式简洁表示时，寻找任意不动点的一般塔斯基问题属于局部搜索可解的PLS类问题，且出人意料地同时属于PPAD类问题。然而，寻找塔斯基定理保证的最大或最小不动点需要d·N步，且在白盒模型中是NP难的。对于超模博弈，我们证明寻找均衡在计算上本质等价于塔斯基问题，寻找最大或最小均衡同样更难。有趣的是，当其中一个参与者的策略空间为一维时，两人超模博弈可在O(log N)步内求解。我们还证明，计算（近似）康登（沙普利）随机博弈的值可归约到塔斯基问题。本工作突出的一个重要开放问题是证明塔斯基问题（黑盒查询）复杂性的更优上界或下界。
英文摘要：
The use of monotonicity and Tarski's theorem in existence proofs of equilibria is very widespread in economics, while Tarski's theorem is also often used for similar purposes in the context of verification. However, there has been relatively little in the way of analysis of the complexity of finding the fixed points and equilibria guaranteed by this result. We study a computational formalism based on monotone functions on the $d$-dimensional grid with sides of length $N$, and their fixed points, as well as the closely connected subject of supermodular games and their equilibria. It is known that finding some (any) fixed point of a monotone function can be done in time $\log^d N$, and we show it requires at least $\log^2 N$ function evaluations already on the 2-dimensional grid, even for randomized algorithms. We show that the general Tarski problem of finding some fixed point, when the monotone function is given succinctly (by a boolean circuit), is in the class PLS of problems solvable by local search and, rather surprisingly, also in the class PPAD. Finding the greatest or least fixed point guaranteed by Tarski's theorem, however, requires $d\cdot N$ steps, and is NP-hard in the white box model. For supermodular games, we show that finding an equilibrium is essentially computationally equivalent to the Tarski problem, and finding the maximum or minimum equilibrium is similarly harder. Interestingly, two-player supermodular games where the strategy space of one player is one-dimensional can be solved in $O(\log N)$ steps. We also show that computing (approximating) the value of Condon's (Shapley's) stochastic games reduces to the Tarski problem. An important open problem highlighted by this work is proving better upper or lower bounds on the (blackbox query) complexity of the Tarski problem.
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