In J. Schwenk.(2018) ['What is the Correct Way to Seed a Knockout Tournament?' Retrieved from The American Mathematical Monthly], Schwenk identified a surprising weakness in the standard method of seeding a single elimination (or knockout) tournament. In particular, he showed that for a certain probability model for the outcomes of games it can be the case that the top seeded team would be less likely to win the tournament than the second seeded team. This raises the possibility that in certain situations it might be advantageous for a team to intentionally lose a game in an attempt to get a more optimal (though possibly lower) seed in the tournament. We examine this question in the context of a four team league which consists of a round robin "regular season" followed by a single elimination tournament with seedings determined by the results from the regular season [4]. Using the same probability model as Schwenk we show that there are situations where it is indeed optimal for a team to intentionally lose. Moreover, we show how a team can make the decision as to whether or not it should intentionally lose. We did two detailed analysis. One is for the situation where other teams always try to win every game. The other is for the situation where other teams are smart enough, namely they can also lose some games intentionally if necessary. The analysis involves computations in both probability and (multi-player) game theory.
参见J. Schwenk.(2018)[什么是为淘汰赛播种的正确方法?]从《美国数学月刊》(The American Mathematical Monthly)中，Schwenk发现了单一淘汰赛(或淘汰赛)的标准播种方法的一个令人惊讶的弱点。他特别指出，对于比赛结果的某个概率模型，可能会出现头号种子队比第二种子队更不可能赢得比赛的情况。这就增加了一种可能性，即在某些情况下，对于一支球队来说，故意输掉一场比赛以获得更优(尽管可能是更低)的参赛资格是有利的。我们在一个四队联赛的背景下研究这个问题，这个联赛包括一个循环赛“常规赛”，然后是一个单淘汰赛，由常规赛的结果决定种子[4]。使用与Schwenk相同的概率模型，我们证明了在某些情况下，球队故意输球确实是最优的。此外，我们还展示了一个团队如何做出是否应该故意输球的决定。我们做了两个详细的分析。一个是其他球队总是想赢下每一场比赛。另一种情况是，其他球队足够聪明，也就是说，如果有必要，他们也会故意输掉一些比赛。分析涉及概率和(多人)博弈论的计算。
{"title":"Best Strategy for Each Team in The Regular Season to Win Champion in The Knockout Tournament","authors":"Zijie Zhou","doi":"10.1137/20s1340460","DOIUrl":"https://doi.org/10.1137/20s1340460","url":null,"abstract":"In J. Schwenk.(2018) ['What is the Correct Way to Seed a Knockout Tournament?' Retrieved from The American Mathematical Monthly], Schwenk identified a surprising weakness in the standard method of seeding a single elimination (or knockout) tournament. In particular, he showed that for a certain probability model for the outcomes of games it can be the case that the top seeded team would be less likely to win the tournament than the second seeded team. This raises the possibility that in certain situations it might be advantageous for a team to intentionally lose a game in an attempt to get a more optimal (though possibly lower) seed in the tournament. We examine this question in the context of a four team league which consists of a round robin \"regular season\" followed by a single elimination tournament with seedings determined by the results from the regular season [4]. Using the same probability model as Schwenk we show that there are situations where it is indeed optimal for a team to intentionally lose. Moreover, we show how a team can make the decision as to whether or not it should intentionally lose. We did two detailed analysis. One is for the situation where other teams always try to win every game. The other is for the situation where other teams are smart enough, namely they can also lose some games intentionally if necessary. The analysis involves computations in both probability and (multi-player) game theory.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80771446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A single jump filtration $({mathscr{F}}_t)_{tin mathbb{R}_+}$ generated by a random variable $gamma$ with values in $overline{mathbb{R}}_+$ on a probability space $(Omega ,{mathscr{F}},mathsf{P})$ is defined as follows: a set $Ain {mathscr{F}}$ belongs to ${mathscr{F}}_t$ if $Acap {gamma >t}$ is either $varnothing$ or ${gamma >t}$. A process $M$ is proved to be a local martingale with respect to this filtration if and only if it has a representation $M_t=F(t){mathbb{1}}_{{t 0}$. This result seems to be new even in a special case that has been studied in the literature, namely, where ${mathscr{F}}$ is the smallest $sigma$-field with respect to which $gamma$ is measurable (and then the filtration is the smallest one with respect to which $gamma$ is a stopping time). As a consequence, a full description of all local martingales is given and they are classified according to their global behaviour.
{"title":"Single jump filtrations and local martingales","authors":"A. Gushchin","doi":"10.15559/20-VMSTA153","DOIUrl":"https://doi.org/10.15559/20-VMSTA153","url":null,"abstract":"A single jump filtration $({mathscr{F}}_t)_{tin mathbb{R}_+}$ generated by a random variable $gamma$ with values in $overline{mathbb{R}}_+$ on a probability space $(Omega ,{mathscr{F}},mathsf{P})$ is defined as follows: a set $Ain {mathscr{F}}$ belongs to ${mathscr{F}}_t$ if $Acap {gamma >t}$ is either $varnothing$ or ${gamma >t}$. A process $M$ is proved to be a local martingale with respect to this filtration if and only if it has a representation $M_t=F(t){mathbb{1}}_{{t 0}$. This result seems to be new even in a special case that has been studied in the literature, namely, where ${mathscr{F}}$ is the smallest $sigma$-field with respect to which $gamma$ is measurable (and then the filtration is the smallest one with respect to which $gamma$ is a stopping time). As a consequence, a full description of all local martingales is given and they are classified according to their global behaviour.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78095595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-01DOI: 10.9734/BPI/TPMCS/V1/5056D
Lo Gane Samb, N. Babacar, S. Harouna
In this note, we combine the two approaches of Billingsley (1998) and Csőrgő and Revesz (1980), to provide a detailed sequential and descriptive for creating s standard Brownian motion, from a Brownian motion whose time space is the class of non-negative dyadic numbers. By adding the proof of Etemadi's inequality to text, it becomes self-readable and serves as an independent source for researches and professors.
{"title":"Study on A Direct Construction of the Standard Brownian Motion","authors":"Lo Gane Samb, N. Babacar, S. Harouna","doi":"10.9734/BPI/TPMCS/V1/5056D","DOIUrl":"https://doi.org/10.9734/BPI/TPMCS/V1/5056D","url":null,"abstract":"In this note, we combine the two approaches of Billingsley (1998) and Csőrgő and Revesz (1980), to provide a detailed sequential and descriptive for creating s standard Brownian motion, from a Brownian motion whose time space is the class of non-negative dyadic numbers. By adding the proof of Etemadi's inequality to text, it becomes self-readable and serves as an independent source for researches and professors.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73343467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a solution is given to reflected backward doubly stochastic differential equations when the barrier is not necessarily right-continuous, and the noise is driven by two independent Brownian motions and an independent Poisson random measure. The existence and uniqueness of the solution is shown, firstly when the coefficients are stochastic Lipschitz, and secondly by weakening the conditions on the stochastic growth coefficient.
{"title":"Irregular barrier reflected BDSDEs with general jumps under stochastic Lipschitz and linear growth conditions","authors":"M. Marzougue, Yaya Sagna","doi":"10.15559/20-VMSTA155","DOIUrl":"https://doi.org/10.15559/20-VMSTA155","url":null,"abstract":"In this paper, a solution is given to reflected backward doubly stochastic differential equations when the barrier is not necessarily right-continuous, and the noise is driven by two independent Brownian motions and an independent Poisson random measure. The existence and uniqueness of the solution is shown, firstly when the coefficients are stochastic Lipschitz, and secondly by weakening the conditions on the stochastic growth coefficient.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84734560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we introduce and study a higher-dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant k-dimensional cycles (with 0-cycles being connected components). Considering a continuum percolation model in the flat d-dimensional torus, we show that all the giant k-cycles (k=1,...,d-1) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant k-cycles are increasing in k and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.
{"title":"Homological Percolation: The Formation of Giant k-Cycles","authors":"O. Bobrowski, P. Skraba","doi":"10.1093/imrn/rnaa305","DOIUrl":"https://doi.org/10.1093/imrn/rnaa305","url":null,"abstract":"In this paper we introduce and study a higher-dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant k-dimensional cycles (with 0-cycles being connected components). Considering a continuum percolation model in the flat d-dimensional torus, we show that all the giant k-cycles (k=1,...,d-1) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant k-cycles are increasing in k and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91042134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Florent Nzissila, O. Moutsinga, Fulgence Eyi Obiang
In fluid dynamics governed by the one dimensional inviscid Burgers equation $partial_t u+upartial_x(u)=0$, the stirring is explained by the sticky particles model. A Markov process $([Z^1_t,Z^2_t],,tgeq0)$ describes the motion of random turbulent intervals which evolve inside an other Markov process $([Z^3_t,Z^4_t],,tgeq0)$, describing the motion of random clusters concerned with the turbulence. Then, the four velocity processes $(u(Z^i_t,t),,tgeq0)$ are backward semi-martingales.
{"title":"Backward semi-martingales into Burgers turbulence","authors":"Florent Nzissila, O. Moutsinga, Fulgence Eyi Obiang","doi":"10.1063/5.0036721","DOIUrl":"https://doi.org/10.1063/5.0036721","url":null,"abstract":"In fluid dynamics governed by the one dimensional inviscid Burgers equation $partial_t u+upartial_x(u)=0$, the stirring is explained by the sticky particles model. A Markov process $([Z^1_t,Z^2_t],,tgeq0)$ describes the motion of random turbulent intervals which evolve inside an other Markov process $([Z^3_t,Z^4_t],,tgeq0)$, describing the motion of random clusters concerned with the turbulence. Then, the four velocity processes $(u(Z^i_t,t),,tgeq0)$ are backward semi-martingales.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73517639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study linearly edge-reinforced random walks on $mathbb{Z}_+$, where each edge ${x,x+1}$ has the initial weight $x^{alpha} vee 1$, and each time an edge is traversed, its weight is increased by $Delta$. It is known that the walk is recurrent if and only if $alpha leq 1$. The aim of this paper is to study the almost sure behavior of the walk in the recurrent regime. For $alpha 0$, we obtain a limit theorem which is a counterpart of the law of the iterated logarithm for simple random walks. This reveals that the speed of the walk with $Delta>0$ is much slower than $Delta=0$. In the critical case $alpha=1$, our (almost sure) bounds for the trajectory of the walk shows that there is a phase transition of the speed at $Delta=2$.
我们在$mathbb{Z}_+$上研究线性边增强随机行走，其中每条边${x,x+1}$具有初始权值$x^{alpha} vee 1$，每遍历一条边，其权值增加$Delta$。众所周知，当且仅当$alpha leq 1$时，行走是复发性的。本文的目的是研究在循环状态下行走的几乎确定行为。对于$alpha 0$，我们得到了一个极限定理，它是简单随机游走的迭代对数定律的对应项。这表明用$Delta>0$行走的速度比$Delta=0$慢得多。在临界情况$alpha=1$中，我们的(几乎确定的)行走轨迹边界表明，在$Delta=2$处存在速度的相变。
{"title":"Almost sure behavior of linearly edge-reinforced random walks on the half-line","authors":"Masato Takei","doi":"10.1214/21-ejp674","DOIUrl":"https://doi.org/10.1214/21-ejp674","url":null,"abstract":"We study linearly edge-reinforced random walks on $mathbb{Z}_+$, where each edge ${x,x+1}$ has the initial weight $x^{alpha} vee 1$, and each time an edge is traversed, its weight is increased by $Delta$. It is known that the walk is recurrent if and only if $alpha leq 1$. The aim of this paper is to study the almost sure behavior of the walk in the recurrent regime. For $alpha 0$, we obtain a limit theorem which is a counterpart of the law of the iterated logarithm for simple random walks. This reveals that the speed of the walk with $Delta>0$ is much slower than $Delta=0$. In the critical case $alpha=1$, our (almost sure) bounds for the trajectory of the walk shows that there is a phase transition of the speed at $Delta=2$.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"462 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82986976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities. We call the introduced process as the convoluted fractional Poisson process (CFPP). The explicit expression for the Laplace transform of its state probabilities are obtained whose inversion yields its one-dimensional distribution. Some of its statistical properties such as probability generating function, moment generating function, moments etc. are obtained. A special case of CFPP, namely, the convoluted Poisson process (CPP) is studied and its time-changed subordination relationships with CFPP are discussed. It is shown that the CPP is a Levy process using which the long-range dependence property of CFPP is established. Moreover, we show that the increments of CFPP exhibits short-range dependence property.
{"title":"Convoluted Fractional Poisson Process","authors":"K. K. Kataria, M. Khandakar","doi":"10.30757/ALEA.v18-46","DOIUrl":"https://doi.org/10.30757/ALEA.v18-46","url":null,"abstract":"In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities. We call the introduced process as the convoluted fractional Poisson process (CFPP). The explicit expression for the Laplace transform of its state probabilities are obtained whose inversion yields its one-dimensional distribution. Some of its statistical properties such as probability generating function, moment generating function, moments etc. are obtained. A special case of CFPP, namely, the convoluted Poisson process (CPP) is studied and its time-changed subordination relationships with CFPP are discussed. It is shown that the CPP is a Levy process using which the long-range dependence property of CFPP is established. Moreover, we show that the increments of CFPP exhibits short-range dependence property.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"497 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80027484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-15DOI: 10.1142/s0219025720500241
Rico Heinemann
We prove that distribution dependent (also called McKean--Vlasov) stochastic delay equations of the form begin{equation*} mathrm{d}X(t)= b(t,X_t,mathcal{L}_{X_t})mathrm{d}t+ sigma(t,X_t,mathcal{L}_{X_t})mathrm{d}W(t) end{equation*} have unique (strong) solutions in finite as well as infinite dimensional state spaces if the coefficients fulfill certain monotonicity assumptions.
{"title":"Distribution-Dependent Stochastic Differential Delay Equations in finite and infinite dimensions","authors":"Rico Heinemann","doi":"10.1142/s0219025720500241","DOIUrl":"https://doi.org/10.1142/s0219025720500241","url":null,"abstract":"We prove that distribution dependent (also called McKean--Vlasov) stochastic delay equations of the form begin{equation*} mathrm{d}X(t)= b(t,X_t,mathcal{L}_{X_t})mathrm{d}t+ sigma(t,X_t,mathcal{L}_{X_t})mathrm{d}W(t) end{equation*} have unique (strong) solutions in finite as well as infinite dimensional state spaces if the coefficients fulfill certain monotonicity assumptions.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84702942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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