scholarly journals Equilibria Existence in Bayesian Games: Climbing the Countable Borel Equivalence Relation Hierarchy

Author(s):  
Ziv Hellman ◽  
Yehuda John Levy

The solution concept of a Bayesian equilibrium of a Bayesian game is inherently an interim concept. The corresponding ex ante solution concept has been termed a Harsányi equilibrium; examples have appeared in the literature showing that there are Bayesian games with uncountable state spaces that have no Bayesian approximate equilibria but do admit a Harsányi approximate equilibrium, thus exhibiting divergent behaviour in the ex ante and interim stages. Smoothness, a concept from descriptive set theory, has been shown in previous works to guarantee the existence of Bayesian equilibria. We show here that higher rungs in the countable Borel equivalence relation hierarchy can also shed light on equilibrium existence. In particular, hyperfiniteness, the next step above smoothness, is a sufficient condition for the existence of Harsányi approximate equilibria in purely atomic Bayesian games.

Sensors ◽  
2021 ◽  
Vol 21 (16) ◽  
pp. 5300
Author(s):  
Antonia Nisioti ◽  
George Loukas ◽  
Stefan Rass ◽  
Emmanouil Panaousis

The use of anti-forensic techniques is a very common practice that stealthy adversaries may deploy to minimise their traces and make the investigation of an incident harder by evading detection and attribution. In this paper, we study the interaction between a cyber forensic Investigator and a strategic Attacker using a game-theoretic framework. This is based on a Bayesian game of incomplete information played on a multi-host cyber forensics investigation graph of actions traversed by both players. The edges of the graph represent players’ actions across different hosts in a network. In alignment with the concept of Bayesian games, we define two Attacker types to represent their ability of deploying anti-forensic techniques to conceal their activities. In this way, our model allows the Investigator to identify the optimal investigating policy taking into consideration the cost and impact of the available actions, while coping with the uncertainty of the Attacker’s type and strategic decisions. To evaluate our model, we construct a realistic case study based on threat reports and data extracted from the MITRE ATT&CK STIX repository, Common Vulnerability Scoring System (CVSS), and interviews with cyber-security practitioners. We use the case study to compare the performance of the proposed method against two other investigative methods and three different types of Attackers.


1992 ◽  
Vol 12 (2) ◽  
pp. 283-295 ◽  
Author(s):  
Alexander S. Kechris

AbstractIt has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete measurable section. This is extended here to the purely Borel theoretic category, consisting of a Borel action of such a group on an analytic Borel space (without any measure). Characterizations of when an arbitrary Borel equivalence relation admits a countable complete Borel section are also established.


Author(s):  
Yuhu Wu ◽  
Shuting Le ◽  
Kuize Zhang ◽  
Xi-Ming Sun
Keyword(s):  

2011 ◽  
Vol 76 (2) ◽  
pp. 461-476 ◽  
Author(s):  
Greg Hjorth ◽  
André Nies

AbstractWe show that there is a complete, consistent Borel theory which has no “Borel model” in the following strong sense: There is no structure satisfying the theory for which the elements of the structure are equivalence classes under some Borel equivalence relation and the interpretations of the relations and function symbols are uniformly Borel.We also investigate Borel isomorphisms between Borel structures.


Sensors ◽  
2019 ◽  
Vol 19 (12) ◽  
pp. 2664 ◽  
Author(s):  
Jingxian Wang ◽  
Yongmei Sun ◽  
Yuefeng Ji ◽  
Shuyun Luo

According to the IEEE 802.15.6 standard, interference within each wireless body area network (WBAN) can be well addressed by the time division multiple access (TDMA)-based media access control (MAC) protocol. However, the inter-WBAN interference will be caused after multiple WBANs are gathered together. This paper proposes a priority-aware price-based power control (PPPC) scheme for mitigating the inter-WBAN interference. Specifically, to maximize the transmission data rate of sensors and control the aggregate interference suffered by coordinators, a Stackelberg game is established, in which the coordinators issue interference prices and the active sensors adjust their transmission power accordingly. On the other hand, since the information about the identities of the active sensors in a specific time slot is kept private, a Bayesian game is designed to model the interaction among sensors. Moreover, the timeliness and reliability of data transmission are guaranteed by designing the sensors’ priority factors and setting a priority-related active probability for each sensor. At last, a power control algorithm is designed to obtain optimal strategies of game players. Simulation results show that compared with other existing schemes, the proposed scheme achieves better fairness with a comparable network sum data rate and is more energy efficient.


2016 ◽  
Vol 37 (7) ◽  
pp. 2084-2098
Author(s):  
CLINTON T. CONLEY ◽  
BENJAMIN D. MILLER

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\unicode[STIX]{x1D707}$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $\unicode[STIX]{x1D6E4}$, there is a Borel sequence $(\cdot _{r})_{r\in \mathbb{R}}$ of free actions of $\unicode[STIX]{x1D6E4}$ on $X$, generating subequivalence relations $E_{r}$ of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic, with the further property that $(E_{r})_{r\in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $\unicode[STIX]{x1D707}$-reducibility. In particular, we show that if $E$ satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on $X$, generating a subequivalence relation of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic.


2008 ◽  
Vol 28 (5) ◽  
pp. 1615-1633 ◽  
Author(s):  
BENJAMIN MILLER

AbstractGiven a Polish space X, a countable Borel equivalence relation E on X, and a Borel cocycle $\rho : E \rightarrow (0, \infty )$, we characterize the circumstances under which there is a probability measure μ on X such that ρ(ϕ−1(x),x)=[d(ϕ*μ)/dμ](x) μ-almost everywhere, for every Borel injection ϕ whose graph is contained in E.


2005 ◽  
Vol 70 (3) ◽  
pp. 979-992 ◽  
Author(s):  
Greg Hjorth

This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:Theorem 0.1. There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we haveThe proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the followingMessy Fact 0.2. There are countable Borel equivalence relationssuch that:(i) eachExis defined on a standard Borel probability space (Xx, μx); each Ex is μx-invariant and μx-ergodic;(ii) forx1 ≠ x2 and A μxι -conull, we haveExι/Anot Borel reducible toEx2;(iii) if f: Xx → Xxis a measurable reduction ofExto itself then(iv)is a standard Borel space on which the projection functionis Borel and the equivalence relation Ê given byif and only ifx = x′ andzExz′ is Borel;(V)is Borel.We first prove the theorem granted this messy fact. We then prove the fact.(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.


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