This work contains the mathematical exploration of a few prototypical games
in which central concepts from statistics and probability theory naturally
emerge. The first two kinds of games are termed Fisher and Bayesian games,
which are connected to Frequentist and Bayesian statistics, respectively.
Later, a more general type of game is introduced, termed Statistical game, in
which a further parameter, the players' relative risk aversion, can be set. In
this work, we show that Fisher and Bayesian games can be viewed as limiting
cases of Statistical games. Therefore, Statistical games can be viewed as a
unified framework, incorporating both Frequentist and Bayesian statistics.
Furthermore, a philosophical framework is (re-)presented -- often referred to
as minimax regret criterion -- as a general approach to decision making.
The main motivation for this work was to embed Bayesian statistics into a
broader decision-making framework, where, based on collected data, actions with
consequences have to be made, which can be translated to utilities (or
rewards/losses) of the decision-maker. The work starts with the simplest
possible toy model, related to hypothesis testing and statistical inference.
This choice has two main benefits: i.) it allows us to determine (conjecture)
the behaviour of the equilibrium strategies in various limiting cases ii.) this
way, we can introduce Statistical games without requiring additional stochastic
parameters. The work contains game theoretical methods related to two-player,
non-cooperative games to determine and prove equilibrium strategies of Fisher,
Bayesian and Statistical games. It also relies on analytical tools for
derivations concerning various limiting cases.

数理探索了几种原型游戏，其中涉及统计学和概率论中的核心概念，通过引入统计游戏作为一个统一框架，将 Fisher 游戏和 Bayesian 游戏视为极限情况，并提出了用于决策制定的最小后悔准则。

统计游戏

Statistical Games

We study repeated first-price auctions and general repeated Bayesian games
between two players, where one player, the learner, employs a no-regret
learning algorithm, and the other player, the optimizer, knowing the learner's
algorithm, strategizes to maximize its own utility. For a commonly used class
of no-regret learning algorithms called mean-based algorithms, we show that (i)
in standard (i.e., full-information) first-price auctions, the optimizer cannot
get more than the Stackelberg utility -- a standard benchmark in the
literature, but (ii) in Bayesian first-price auctions, there are instances
where the optimizer can achieve much higher than the Stackelberg utility.
On the other hand, Mansour et al. (2022) showed that a more sophisticated
class of algorithms called no-polytope-swap-regret algorithms are sufficient to
cap the optimizer's utility at the Stackelberg utility in any repeated Bayesian
game (including Bayesian first-price auctions), and they pose the open question
whether no-polytope-swap-regret algorithms are necessary to cap the optimizer's
utility. For general Bayesian games, under a reasonable and necessary
condition, we prove that no-polytope-swap-regret algorithms are indeed
necessary to cap the optimizer's utility and thus answer their open question.
For Bayesian first-price auctions, we give a simple improvement of the standard
algorithm for minimizing the polytope swap regret by exploiting the structure
of Bayesian first-price auctions.

我们研究了重复的一阶售价拍卖和一般重复贝叶斯博弈的情况，在这种情况下，一个参与者（学习者）采用了一个无悔学习算法，而另一个参与者（优化者）在了解学习者的算法的情况下，策略化地追求自己的效用最大化。 对于一类被称为基于均值的无悔学习算法，我们证明：（i）在标准（即完全信息）的一阶售价拍卖中，优化者不能获得超过 Stackelberg 效用的效用 -- 这是文献中的标准基准，但是（ii）在贝叶斯一阶售价拍卖中，存在优化者可以获得远高于 Stackelberg 效用的实例。 另一方面，Mansour 等人（2022）证明了一类更复杂的算法，称为无多面体交换后悔算法可以将优化者的效用限制在任意重复贝叶斯博弈（包括贝叶斯一阶售价拍卖）的 Stackelberg 效用上，并提出是否有必要使用无多面体交换后悔算法来限制优化者的效用。对于一般的贝叶斯博弈，在一个合理且必要的条件下，我们证明了无多面体交换后悔算法确实是将优化者的效用限制在 Stackelberg 效用上的必要条件，从而回答了他们的开放性问题。对于贝叶斯一阶售价拍卖，我们通过利用贝叶斯一阶售价拍卖的结构给出了一个简单的改进标准算法来最小化多面体交换后悔。

对首价拍卖中的无懊悔学习者进行策略规划

Strategizing against No-Regret Learners in First-Price Auctions

This paper explores equilibrium concepts for Bayesian games, which are
fundamental models of games with incomplete information. We aim at three
desirable properties of equilibria. First, equilibria can be naturally realized
by introducing a mediator into games. Second, an equilibrium can be computed
efficiently in a distributed fashion. Third, any equilibrium in that class
approximately maximizes social welfare, as measured by the price of anarchy,
for a broad class of games. These three properties allow players to compute an
equilibrium and realize it via a mediator, thereby settling into a stable state
with approximately optimal social welfare. Our main result is the existence of
an equilibrium concept that satisfies these three properties.
Toward this goal, we characterize various (non-equivalent) extensions of
correlated equilibria, collectively known as Bayes correlated equilibria. In
particular, we focus on communication equilibria (also known as coordination
mechanisms), which can be realized by a mediator who gathers each player's
private information and then sends correlated recommendations to the players.
We show that if each player minimizes a variant of regret called untruthful
swap regret in repeated play of Bayesian games, the empirical distribution of
these dynamics converges to a communication equilibrium. We present an
efficient algorithm for minimizing untruthful swap regret with a sublinear
upper bound, which we prove to be tight up to a multiplicative constant. As a
result, by simulating the dynamics with our algorithm, we can efficiently
compute an approximate communication equilibrium. Furthermore, we extend
existing lower bounds on the price of anarchy based on the smoothness arguments
from Bayes Nash equilibria to equilibria obtained by the proposed dynamics.

本文探讨了贝叶斯博弈的均衡概念，包括相关均衡、通信均衡，推导出基于均衡对策的博弈稳定状态的实现方法，提出一种满足稳定、高效、优化多个博弈均衡的新均衡概念。