An Extension of Ramsey\u27s Theorem to Multipartite Graphs

Ramsey Theorem, in the most simple form, states that if we are given a positive integer l, there exists a minimal integer r(l), called the Ramsey number, such any partition of the edges of K_r(l) into two sets, i.e. a 2-coloring, yields a copy of K_l contained entirely in one of the partitioned sets, i.e. a monochromatic copy of Kl. We prove an extension of Ramsey\u27s Theorem, in the more general form, by replacing complete graphs by multipartite graphs in both senses, as the partitioned set and as the desired monochromatic graph. More formally, given integers l and k, there exists an integer p(m) such that any 2-coloring of the edges of the complete multipartite graph K_p(m);r(k) yields a monochromatic copy of K_m;k . The tools that are used to prove this result are the Szemeredi Regularity Lemma and the Blow Up Lemma. A full proof of the Regularity Lemma is given. The Blow-Up Lemma is merely stated, but other graph embedding results are given. It is also shown that certain embedding conditions on classes of graphs, namely (f , ?) -embeddability, provides a method to bound the order of the multipartite Ramsey numbers on the graphs. This provides a method to prove that a large class of graphs, including trees, graphs of bounded degree, and planar graphs, has a linear bound, in terms of the number of vertices, on the multipartite Ramsey number

Sequential correlated equilibrium in stopping games

In many situations, such as trade in stock exchanges, agents have many instances to act even though the duration of interactions take a relatively short time. The agents in such situations can often coordinate their actions in advance, but coordination during the game consumes too much time. An equilibrium in such situations has to be sequential in order to handle mistakes made by players. In this paper, we present a new solution concept for infinite-horizon dynamic games, which is appropriate for such situations: a sequential constant-expectation normal-form correlated approximate equilibrium. Under additional assumptions, we show that every such game admits this kind of equilibrium

Perfect correlated equilibria in stopping games

We define a new solution concept for an undiscounted dynamic game - a perfect uniform normal-form constant-expectation correlated approximate equilibrium with a canonical and universal correlation device. This equilibrium has the following appealing properties: (1) “Trembling-hand” perfectness - players do not use non-credible threats; (2) Uniformness - it is an approximate equilibrium in any long enough finite-horizon game and in any discounted game with a high enough discount factor; (3) Normal-form correlation - The strategy of a player depends on a private signal he receives before the game starts (which can be induced by “cheap-talk” among the players); (4) Constant expectation - The expected payoff of each player almost does not change when he receives his signal; (5) Universal correlation device - the device does not depend on the specific parameters of the game. (6) Canonical - each signal is equivalent to a strategy. We demonstrate the use of this equilibrium by proving its existence in every undiscounted multi-player stopping game

Perfect correlated equilibria in stopping games

In many situations, such as trade in stock exchanges, agents have many instances to act even though the duration of interactions take a relatively short time. The agents in such situations can often coordinate their actions in advance, but coordination during the game consumes too much time. An equilibrium in such situations has to be sequential in order to handle mistakes made by players. In this paper, we present a new solution concept for infinite-horizon dynamic games, which is appropriate for such situations: a sequential constant-expectation normal-form correlated approximate equilibrium. Under additional assumptions, we show that every such game admits this kind of equilibrium

Sequential correlated equilibrium in stopping games

In many situations, such as trade in stock exchanges, agents have many instances to act even though the duration of interactions take a relatively short time. The agents in such situations can often coordinate their actions in advance, but coordination during the game consumes too much time. An equilibrium in such situations has to be sequential in order to handle mistakes made by players. In this paper, we present a new solution concept for infinite-horizon dynamic games, which is appropriate for such situations: a sequential constant-expectation normal-form correlated approximate equilibrium. Under additional assumptions, we show that every such game admits this kind of equilibrium

Perfect correlated equilibria in stopping games

We prove that every undiscounted multi-player stopping game in discrete time admits an approximate correlated equilibrium. Moreover, the equilibrium has five appealing properties: (1) “Trembling-hand” perfectness - players do not use non-credible threats; (2) Normal-form correlation - communication is required only before the game starts; (3) Uniformness - it is an approximate equilibrium in any long enough finite-horizon game and in any discounted game with high enough discount factor; (4) Universal correlation device -the device does not depend on the specific parameters of the game. (5) Canonical - the signal each player receives is equivalent to the strategy he plays in equilibrium

Perfect correlated equilibria in stopping games

We prove that every undiscounted multi-player stopping game in discrete time admits an approximate correlated equilibrium. Moreover, the equilibrium has five appealing properties: (1) “Trembling-hand” perfectness - players do not use non-credible threats; (2) Normal-form correlation - communication is required only before the game starts; (3) Uniformness - it is an approximate equilibrium in any long enough finite-horizon game and in any discounted game with high enough discount factor; (4) Universal correlation device -the device does not depend on the specific parameters of the game. (5) Canonical - the signal each player receives is equivalent to the strategy he plays in equilibrium

Coherent approximation of distributed expert assessments

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 157-168).Expert judgments of probability and expectation play an integral role in many systems. Financial markets, public policy, medical diagnostics and more rely on the ability of informed experts (both human and machine) to make educated assessments of the likelihood of various outcomes. Experts however are not immune to errors in judgment (due to bias, quantization effects, finite information or many other factors). One way to compensate for errors in individual judgments is to elicit estimates from multiple experts and then fuse the estimates together. If the experts act sufficiently independently to form their assessments, it is reasonable to assume that individual errors in judgment can be negated by pooling the experts' opinions. Determining when experts' opinions are in error is not always a simple matter. However, one common way in which experts' opinions may be seen to be in error is through inconsistency with the known underlying structure of the space of events. Not only is structure useful in identifying expert error, it should also be taken into account when designing algorithms to approximate or fuse conflicting expert assessments. This thesis generalizes previously proposed constrained optimization methods for fusing expert assessments of uncertain events and quantities. The major development consists of a set of information geometric tools for reconciling assessments that are inconsistent with the assumed structure of the space of events. This work was sponsored by the U.S. Air Force under Air Force Contract FA8721- 05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government.by Peter B. Jones.Ph.D

Some Problems in Graph Theory and Scheduling

In this dissertation, we present three results related to combinatorial algorithms in graph theory and scheduling, both of which are important subjects in the area of discrete mathematics and theoretical computer science. In graph theory, a graph is a set of vertices and edges, where each edge is a pair of vertices. A coloring of a graph is a function that assigns each vertex a color such that no two adjacent vertices share the same color. The first two results are related to coloring graphs belonging to specific classes. In scheduling problems, we are interested in how to efficiently schedule a set of jobs on machines. The last result is related to a scheduling problem in an environment where there is uncertainty on the number of machines. The first result of this thesis is a polynomial time algorithm that determines if an input graph containing no induced seven-vertex path is 3-colorable. This affirmatively answers a question posed by Randerath, Schiermeyer and Tewes in 2002. Our algorithm also solves the list-coloring version of the 3-coloring problem, where every vertex is assigned a list of colors that is a subset of {1, 2, 3}, and gives an explicit coloring if one exists. This is joint work with Flavia Bonomo, Maria Chundnovsky, Peter Maceli, Oliver Schaudt, and Maya Stein. A graph is H-free if it has no induced subgraph isomorphic to H. In the second part of this thesis, we characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable H-free graphs. This solves a problem posed by Golovach et al. We also characterize all graphs H for which there are only finitely many H-free minimal obstructions for list 3-colorability. This is joint work with Maria Chudnovsky, Jan Goedgebeur and Oliver Schaudt. The last result of this thesis deals with a scheduling problem addressing the uncertainty regarding the machines. We study a scheduling environment in which jobs first need to be grouped into some sets before the number of machines is known, and then the sets need to be scheduled on machines without being separated. In order to evaluate algorithms in such an environment, we introduce the idea of an alpha-robust algorithm, one which is guaranteed to return a schedule on any number m of machines that is within an alpha factor of the optimal schedule on m machines, where the optimum is not subject to the restriction that the sets cannot be separated. Under such environment, we give a (5/3+epsilon)-robust algorithm for scheduling on parallel machines to minimize makespan, and show a lower bound of 4/3. For the special case when the jobs are infinitesimal, we give a 1.233-robust algorithm with an asymptotic lower bound of 1.207. This is joint work with Clifford Stein