We provide a necessary and sufficient condition under which a convex set is approachable in a game with partial monitoring, i.e.\ where players do not observe their opponents' moves but receive random signals. This condition is an extension of Blackwell's Criterion in the full monitoring framework, where players observe at least their payoffs. When our condition is fulfilled, we construct explicitly an approachability strategy, derived from a strategy satisfying some internal consistency property in an auxiliary game. We also provide an example of a convex set, that is neither (weakly)-approachable nor (weakly)-excludable, a situation that cannot occur in the full monitoring case. We finally apply our result to describe an $ε$-optimal strategy of the uninformed player in a zero-sum repeated game with incomplete information on one side.
We consider Blackwell approachability, a very powerful and geometric tool in game theory, used for example to design strategies of the uninformed player in repeated games with incomplete information. We extend this theory to "generalized quitting games" , a class of repeated stochastic games in which each player may have quitting actions, such as the Big-Match. We provide three simple geometric and strongly related conditions for the weak approachability of a convex target set. The first is sufficient: it guarantees that, for any fixed horizon, a player has a strategy ensuring that the expecte…
Studying continuous time counterpart of some discrete time dynamics is now a standard and fruitful technique, as some properties hold in both setups. In game theory, this is usually done by considering differential games on Euclidean spaces. This allows to infer properties on the convergence of values of a repeated game, to deal with the various concepts of approachability, etc. In this paper, we introduce a specific but quite abstract differential game defined on the Wasserstein space of probability distributions and we prove the existence of its value. Going back to the discrete time dynamic…
Blackwell approachability is a framework for reasoning about repeated games with vector-valued payoffs. We introduce predictive Blackwell approachability, where an estimate of the next payoff vector is given, and the decision maker tries to achieve better performance based on the accuracy of that estimator. In order to derive algorithms that achieve predictive Blackwell approachability, we start by showing a powerful connection between four well-known algorithms. Follow-the-regularized-leader (FTRL) and online mirror descent (OMD) are the most prevalent regret minimizers in online convex optim…
在完全监控的可逼近性中，已知有两种类型的条件对于凸集是等价的：原始条件和对偶条件。原始的形式是：一个集合C是可接近的当且仅当所有包含的半空间在一次博弈中是可接近的;而对偶的形式是：一个凸集C是可接近的当且仅当它与所有具有某种形式的支付集相交。我们考虑部分监控博弈中的可接近性。在以前的工作（Perchet 2011; Mannor et al. 2011）中，我们提供了可逼近凸集的对偶特征;我们还展示了C是多胞形的情况下的有效策略。本文给出了凸集可部分监控逼近的基本条件。它们依赖于修改后的报酬函数，并导致可接近性策略，基于修改后的支付函数，进行类似于Blackwell（1956）的策略的预测;这是在对比与以前研究的策略，在这种情况下，主要依赖于信号结构，旨在估计以及所收到的信号的分布。我们的结果推广了Kohlberg 1975的经典结果（也可参见Mertens et al. 1994），并适用于具有任意信号结构的博弈以及任意凸集。
We consider repeated games where the players behave according to cumulative prospect theory (CPT). We show that, when the players have calibrated strategies and behave according to CPT, the natural analog of the notion of correlated equilibrium in the CPT case, as defined by Keskin, is not enough to capture all subsequential limits of the empirical distribution of action play. We define the notion of a mediated CPT correlated equilibrium via an extension of the stage game to a so-called mediated game. We then show, along the lines of the result of Foster and Vohra about convergence to the set …
We consider an attacker-operator game for monitoring a large-scale network that is comprised on components that differ in their criticality levels. In this zero-sum game, the operator seeks to position a limited number of sensors to monitor the network against an attacker who strategically targets a network component. The operator (resp. attacker) seeks to minimize (resp. maximize) the network loss. To study the properties of mixed-strategy Nash Equilibria of this game, we first study two simple instances: (i) When component sets monitored by individual sensor locations are mutually disjoint; …
In this work, we introduce and study the \emph{$(α, β)$-Monitoring} game on networks. Our game is composed of two parties an attacker and a defender. The attacker can launch an attack by distributing a limited number of seeds (i.e., virus) to the network, and/or manipulate the propagation probabilities on a limited number of edges. Under our $(α, β)$-Monitoring game, we say an attack is successful if and only if the following two conditions are satisfied: (1) the outbreak/propagation reaches $α$ individuals, and (2) it has not been detected before reaching $β$ individuals. On the other end, th…
Disentangling trust from cooperation: Evolution of trust as reduced monitoring in social dilemmas
University of Lausanne（洛桑大学）；Teesside University（蒂斯赛德大学）；Université libre de Bruxelles（布鲁塞尔自由大学）；Vrije Universiteit Brussel（布鲁塞尔自由大学）；University of Stirling（斯特灵大学）
We study pure Nash equilibria in games on graphs with an imperfect monitoring based on a public signal. In such games, deviations and players responsible for those deviations can be hard to detect and track. We propose a generic epistemic game abstraction, which conveniently allows to represent the knowledge of the players about these deviations, and give a characterization of Nash equilibria in terms of winning strategies in the abstraction. We then use the abstraction to develop algorithms for some payoff functions.
If in a signaling game the receiver expects to gain no information by monitoring the signal of the sender, then when a cost to monitor is implemented he will never pay that cost regardless of his off-path beliefs. This is the argument of a recent paper by T. Denti (2021). However, which pooling equilibrium does a receiver anticipate to gain no information through monitoring? This paper seeks to prove that given a sufficiently small cost to monitor any pooling equilibrium with a non-zero index will survive close to the original equilibrium.
Arizona State University（亚利桑那州立大学）；WS AI Labs（WS人工智能实验室）
We consider scenarios where a worker robot, who may be unaware of the human's exact expectations, may have the incentive to deviate from a preferred plan (e.g. safe but costly) when a human supervisor is not monitoring it. On the other hand, continuous monitoring of the robot's behavior is often difficult for humans because it costs them valuable resources (e.g., time, cognitive overload, etc.). Thus, to optimize the cost of monitoring while ensuring the robots follow the {\em safe} behavior and to assist the human to deal with the possible unsafe robots, we model this problem in a game-theore…
This paper studies repeated games where two players play multiple duopolistic games simultaneously (multimarket contact). A key assumption is that each player receives a noisy and private signal about the other's actions (private monitoring or observation errors). There has been no game-theoretic support that multimarket contact facilitates collusion or not, in the sense that more collusive equilibria in terms of per-market profits exist than those under a benchmark case of one market. An equilibrium candidate under the benchmark case is belief-free strategies. We are the first to construct a …
Partial monitoring games are repeated games where the learner receives feedback that might be different from adversary's move or even the reward gained by the learner. Recently, a general model of combinatorial partial monitoring (CPM) games was proposed \cite{lincombinatorial2014}, where the learner's action space can be exponentially large and adversary samples its moves from a bounded, continuous space, according to a fixed distribution. The paper gave a confidence bound based algorithm (GCB) that achieves $O(T^{2/3}\log T)$ distribution independent and $O(\log T)$ distribution dependent re…
Grim-trigger strategies are a fundamental mechanism for sustaining equilibria in iterated games: the players cooperate along an agreed path, and as soon as one player deviates, the others form a coalition to play him down to his minmax level. A precondition to triggering such a strategy is that the identity of the deviating player becomes common knowledge among the other players. This can be difficult or impossible to attain in games where the information structure allows only imperfect monitoring of the played actions or of the global state. We study the problem of synthesising finite-state s…
Electric power grid components, such as high voltage transformers (HVTs), generating stations, substations, etc. are expensive to maintain and, in the event of failure, replace. Thus, regularly monitoring the behavior of such components is of utmost importance. Furthermore, the recent increase in the number of cyberattacks on such systems demands that such monitoring strategies should be robust. In this paper, we draw inspiration from work in Moving Target Defense (MTD) and consider a dynamic monitoring strategy that makes it difficult for an attacker to prevent unique identification of behavi…
Through a stochastic control theoretic approach, we analyze reputation games where a strategic long-lived player acts in a sequential repeated game against a collection of short-lived players. The key assumption in our model is that the information of the short-lived players is nested in that of the long-lived player. This nested information structure is obtained through an appropriate monitoring structure. Under this monitoring structure, we show that, given mild assumptions, the set of Perfect Bayesian Equilibrium payoffs coincide with Markov Perfect Equilibrium payoffs, and hence a dynamic …
Zero-determinant (ZD) strategies, a recently found novel class of strategies in repeated games, has attracted much attention in evolutionary game theory. A ZD strategy unilaterally enforces a linear relation between average payoffs of players. Although existence and evolutional stability of ZD strategies have been studied in simple games, their mathematical properties have not been well-known yet. For example, what happens when more than one players employ ZD strategies have not been clarified. In this paper, we provide a general framework for investigating situations where more than one playe…
Test-then-Punish: A Statistical Approach to Repeated Games
CMAP, CNRS, École polytechnique（巴黎综合理工学院）；Inria Saclay, Université Paris Saclay（萨克雷大学）；Inria Paris, Ecole Normale Supérieure, PSL, France, University of California, Berkeley（巴黎高等师范学院）；University of California, Berkeley（加州大学伯克利分校）
Smart Jammer and LTE Network Strategies in An Infinite-Horizon Zero-Sum Repeated Game with Asymmetric and Incomplete Information
Intel Corporation（英特尔公司）；Georgia Institute of Technology（佐治亚理工学院）；Florida A&M University - Florida State University College of Engineering（佛罗里达农工大学-佛罗里达州立大学工程学院）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
