The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Ernesto Mordecki - One of the best experts on this subject based on the ideXlab platform.
-
a finite exact algorithm to solve a dice game
Journal of Applied Probability, 2016Co-Authors: Fabian Crocce, Ernesto MordeckiAbstract:AbstractWe provide an algorithm to find the value and an optimal strategy of the solitairevariant of the Ten Thousand dice game in the framework of Markov Control Processes.Once an optimal critical threshold is found, the set of non-stopping states of the gamebecomes finite, and the solution is found by a backwards algorithm that gives thevalues for each one of these states of the game. The algorithm is finite and exact.The idea to find the critical threshold comes from the continuous pasting conditionused in optimal stopping problems for continuous-time processes with jumps. 1 Introduction The emergence of Probability Theory is closely related to the practice of dice games, asmasterly exposed by Hald [5]. Inspired by this fact, and by the power of the mathematicalmethod, we consider a popular dice game known as “Ten Thousand”.In this game, played with five dice, several players, by turns, roll the dice several times.Each possible outcome of a roll has an assigned score, which may be zero. If after rollingthe dice, a strictly positive score is obtained, the player can roll again some dice to increasehis turn account, or he can stop rolling to bank his turn account into his general account,ending his turn; otherwise, if null score is obtained, the player looses the accumulated turnscore, also ending his turn.In consequence, each turn consists in a sequence of rolls, ending either when the playerobtains no score or when he decides to stop and bank his accumulated turn score. Herearises the problem of taking an optimal decision. The goal of the game is to be the firstplayer in reaching a certain amount of points, usually 10000. This game, also known asZilch and Farkle, among other names, has several versions with minor differences (someof them played with six dice).In this paper we consider the problem that faces an individual player who aims tomaximize his turn score, what can be considered a solitaire variant of the game. Modelingthis optimization problem in the framework of Markov Control Processes (MCP), weprovide a finite exact algorithm that gives the value function and an optimal strategy forthe considered game. We expect that this algorithm can constitute a first step in findingoptimal strategies for the original multi-player game.Despite the popularity of dice games and the interest of the probabilistic communityin the topic, only a few references can be found concerning the family of dice games weconsider. Roters [9] solves the solitaire version of the Pig game (a simpler variant with
-
a finite exact algorithm to solve a dice game
arXiv: Optimization and Control, 2014Co-Authors: Fabian Crocce, Ernesto MordeckiAbstract:We provide an algorithm to find the value and an optimal strategy of the solitaire variant of the Ten Thousand dice game in the framework of Markov Control Processes. Once an optimal critical threshold is found, the set of non-stopping states of the game becomes finite, and the solution is found by a backwards algorithm that gives the values for each one of these states of the game. The algorithm is finite and exact.The idea to find the critical threshold comes from the continuous pasting condition used in optimal stopping problems for continuous-time processes with jumps.
Fabian Crocce - One of the best experts on this subject based on the ideXlab platform.
-
a finite exact algorithm to solve a dice game
Journal of Applied Probability, 2016Co-Authors: Fabian Crocce, Ernesto MordeckiAbstract:AbstractWe provide an algorithm to find the value and an optimal strategy of the solitairevariant of the Ten Thousand dice game in the framework of Markov Control Processes.Once an optimal critical threshold is found, the set of non-stopping states of the gamebecomes finite, and the solution is found by a backwards algorithm that gives thevalues for each one of these states of the game. The algorithm is finite and exact.The idea to find the critical threshold comes from the continuous pasting conditionused in optimal stopping problems for continuous-time processes with jumps. 1 Introduction The emergence of Probability Theory is closely related to the practice of dice games, asmasterly exposed by Hald [5]. Inspired by this fact, and by the power of the mathematicalmethod, we consider a popular dice game known as “Ten Thousand”.In this game, played with five dice, several players, by turns, roll the dice several times.Each possible outcome of a roll has an assigned score, which may be zero. If after rollingthe dice, a strictly positive score is obtained, the player can roll again some dice to increasehis turn account, or he can stop rolling to bank his turn account into his general account,ending his turn; otherwise, if null score is obtained, the player looses the accumulated turnscore, also ending his turn.In consequence, each turn consists in a sequence of rolls, ending either when the playerobtains no score or when he decides to stop and bank his accumulated turn score. Herearises the problem of taking an optimal decision. The goal of the game is to be the firstplayer in reaching a certain amount of points, usually 10000. This game, also known asZilch and Farkle, among other names, has several versions with minor differences (someof them played with six dice).In this paper we consider the problem that faces an individual player who aims tomaximize his turn score, what can be considered a solitaire variant of the game. Modelingthis optimization problem in the framework of Markov Control Processes (MCP), weprovide a finite exact algorithm that gives the value function and an optimal strategy forthe considered game. We expect that this algorithm can constitute a first step in findingoptimal strategies for the original multi-player game.Despite the popularity of dice games and the interest of the probabilistic communityin the topic, only a few references can be found concerning the family of dice games weconsider. Roters [9] solves the solitaire version of the Pig game (a simpler variant with
-
a finite exact algorithm to solve a dice game
arXiv: Optimization and Control, 2014Co-Authors: Fabian Crocce, Ernesto MordeckiAbstract:We provide an algorithm to find the value and an optimal strategy of the solitaire variant of the Ten Thousand dice game in the framework of Markov Control Processes. Once an optimal critical threshold is found, the set of non-stopping states of the game becomes finite, and the solution is found by a backwards algorithm that gives the values for each one of these states of the game. The algorithm is finite and exact.The idea to find the critical threshold comes from the continuous pasting condition used in optimal stopping problems for continuous-time processes with jumps.
Uwe Helmke - One of the best experts on this subject based on the ideXlab platform.
-
network flows that solve linear equations
IEEE Transactions on Automatic Control, 2017Co-Authors: Brian D O Anderson, Uwe HelmkeAbstract:We study distributed network flows as solvers in continuous time for the linear algebraic equation $\mathbf{z}=\mathbf{H}\mathbf{y}$ . Each node $i$ has access to a row ${\mathbf{h}}_{i}^{\mathrm{T}}$ of the matrix $\mathbf{H}$ and the corresponding entry $z_i$ in the vector $\mathbf{z}$ . The first “consensus + projection” flow under investigation consists of two terms, one from standard consensus dynamics and the other contributing to projection onto each affine subspace specified by the $\mathbf{h}_{i}$ and $z_i$ . The second “projection consensus” flow on the other hand simply replaces the relative state feedback in consensus dynamics with projected relative state feedback. Without dwell-time assumption on switching graphs, we prove that all node states converge to a common solution of the linear algebraic equation, if there is any. The convergence is global for the “consensus + projection” flow while local for the “projection consensus” flow in the sense that the initial values must lie on the affine subspaces. If the linear equation has no exact solutions, we show that the node states can converge to a ball around the least-squares solution whose radius can be made arbitrarily small through selecting a sufficiently large gain for the “consensus + projection” flow for a fixed bidirectional graph. Semi-global convergence to approximate least-squares solutions is also demonstrated for switching balanced directed graphs under suitable conditions. It is also shown that the “projection consensus” flow drives the average of the node states to the least-squares solution with a complete graph. Numerical examples are provided as illustrations of the established results.
-
network flows that solve linear equations
arXiv: Systems and Control, 2015Co-Authors: Brian D O Anderson, Uwe HelmkeAbstract:We study distributed network flows as solvers in continuous time for the linear algebraic equation $\mathbf{z}=\mathbf{H}\mathbf{y}$. Each node $i$ has access to a row $\mathbf{h}_i^{\rm T}$ of the matrix $\mathbf{H}$ and the corresponding entry $z_i$ in the vector $\mathbf{z}$. The first "consensus + projection" flow under investigation consists of two terms, one from standard consensus dynamics and the other contributing to projection onto each affine subspace specified by the $\mathbf{h}_i$ and $z_i$. The second "projection consensus" flow on the other hand simply replaces the relative state feedback in consensus dynamics with projected relative state feedback. Without dwell-time assumption on switching graphs as well as without positively lower bounded assumption on arc weights, we prove that all node states converge to a common solution of the linear algebraic equation, if there is any. The convergence is global for the "consensus + projection" flow while local for the "projection consensus" flow in the sense that the initial values must lie on the affine subspaces. If the linear equation has no exact solutions, we show that the node states can converge to a ball around the least squares solution whose radius can be made arbitrarily small through selecting a sufficiently large gain for the "consensus + projection" flow under fixed bidirectional graphs. Semi-global convergence to approximate least squares solutions is demonstrated for general switching directed graphs under suitable conditions. It is also shown that the "projection consensus" flow drives the average of the node states to the least squares solution with complete graph. Numerical examples are provided as illustrations of the established results.
Ranjit Jhala - One of the best experts on this subject based on the ideXlab platform.
-
DSolve safety verification via liquid types
Computer Aided Verification, 2010Co-Authors: Ming Kawaguchi, Patrick M Rondon, Ranjit JhalaAbstract:We present DSolve, a verification tool for OCaml DSolve automates verification by inferring “Liquid” refinement types that are expressive enough to verify a variety of complex safety properties.
Liwei Chen - One of the best experts on this subject based on the ideXlab platform.
-
learn to solve algebra word problems using quadratic programming
Empirical Methods in Natural Language Processing, 2015Co-Authors: Lipu Zhou, Shuaixiang Dai, Liwei ChenAbstract:This paper presents a new algorithm to automatically solve algebra word problems. Our algorithm solves a word problem via analyzing a hypothesis space containing all possible equation systems generated by assigning the numbers in the word problem into a set of equation system templates extracted from the training data. To obtain a robust decision surface, we train a log-linear model to make the margin between the correct assignments and the false ones as large as possible. This results in a quadratic programming (QP) problem which can be efficiently solved. Experimental results show that our algorithm achieves 79.7% accuracy, about 10% higher than the state-of-the-art baseline (Kushman et al., 2014).
