The niche graphs of interval orders

The niche graph of a digraph $D$ is the (simple undirected) graph which has the same vertex set as $D$ and has an edge between two distinct vertices $x$ and $y$ if and only if $N^+_D(x) \cap N^+_D(y) \neq \emptyset$ or $N^-_D(x) \cap N^-_D(y) \neq \emptyset$, where $N^+_D(x)$ (resp. $N^-_D(x)$) is the set of out-neighbors (resp. in-neighbors) of $x$ in $D$. A digraph $D=(V,A)$ is called a semiorder (or a unit interval order) if there exist a real-valued function $f:V \to \mathbb{R}$ on the set $V$ and a positive real number $\delta \in \mathbb{R}$ such that $(x,y) \in A$ if and only if $f(x) > f(y) + \delta$. A digraph $D=(V,A)$ is called an interval order if there exists an assignment $J$ of a closed real interval $J(x) \subset \mathbb{R}$ to each vertex $x \in V$ such that $(x,y) \in A$ if and only if $\min J(x) > \max J(y)$. S. -R. Kim and F. S. Roberts characterized the competition graphs of semiorders and interval orders in 2002, and Y. Sano characterized the competition-common enemy graphs of semiorders and interval orders in 2010. In this note, we give characterizations of the niche graphs of semiorders and interval orders.Comment: 7 page

Essays in organization formation and decision making

This thesis consists of three essays in microeconomic theory. The first two are about the formation of organizations, and the third is about individual or organizational decision making in ambiguous settings. In the first essay I explore the implications of costs associated with binding agreements on equilibrium agreement structures. Establishing binding agreements is often costly in real world economies. These contracting costs are usually regarded as harmful by economists as the costs decrease the gains from cooperation. They affect which agreements form by changing the incentives of agents, potentially prevent the establishment of efficient contracts. Using an alternating offers bargaining model of coalition formation I show that the presence of transaction costs can lead to an efficient outcome in situations where inefficiency arises in equilibrium without these costs. These results provide new insights for policies targeting transaction costs. There are many situations in Economics and Political Science that involve limited possibilities for firms or parties to organize themselves into groups, mostly due to regulatory restrictions. In addition, in these settings the surplus of a given group often depends on the organizational structures formed outside of the group. The second essay introduces a coalition formation model that is able to analyze markets with both restricted cooperation and externalities across coalitions. This concept allows a more realistic modeling, opening the possibility to use this framework to analyze the welfare effects of mergers. In the third essay I propose a new model of decision making under uncertainty with multiple priors that is, unlike the well-known model of Gilboa and Schmeidler (1989), able to express attitude towards ambiguity. In addition, the decision does not necessarily depend on the two extreme (worst case and best case) priors as in the model of Ghirardato et al. (2001). I use choice correspondences by lexicographic semiorders that are generalizations of the choice functions defined in Manzini and Mariotti (2012). I also provide a method constructing lexicographic semiorders for choosing from ambiguous acts

The competition hypergraphs of doubly partial orders

Since Cho and Kim (2005) showed that the competition graph of a doubly partial order is an interval graph, it has been actively studied whether or not the same phenomenon occurs for other variants of competition graph and interesting results have been obtained. Continuing in the same spirit, we study the competition hypergraph, an interesting variant of the competition graph, of a doubly partial order. Though it turns out that the competition hypergraph of a doubly partial order is not always interval, we completely characterize the competition hypergraphs of doubly partial orders which are interval.Comment: 12 pages, 6 figure

Essays in behavioural economics using revealed preference theory and decision theory

This thesis comprises three chapters that provide insights into consumer rationality and consideration sets using revealed preference theory, and a decision theoretic approach to status quo bias. Chapter 1 studies the presence of consideration sets through the lens of economic rationality from the perspective of revealed preference theory. In addition, I propose a new index of rationality (GAV Index) to accompany two commonly used measures (CCEI & MPI), which are applied to a scanner panel dataset and a simulated dataset. Under minimal restrictions, I detect the effects of exogenous consideration set formation on a household's ability to make rational bundle choices. There are also several key demographic factors that correlate well with rationality. This remains true when controlling for the (average) size of the consideration sets households use; these results suggest that a simpler decision-making process with fewer goods can lead to choices that are more rational. Overall, the use of consideration sets as a behavioural heuristic can seemingly benefit consumers by enhancing their decision-making process. Chapter 2 semi-parametrically estimates costs associated with consideration sets using revealed preference theory. The theorem provided ensures there are testable implications of a parsimonious model of consideration sets. Cost of consideration can be estimated in proportion to expenditure and is heterogeneous across consumers. Using the Stanford Basket Dataset, the model cannot reject the use of consideration sets in the presence of suitable restrictions. On average, the average consideration set cost is approximately 2% of monthly expenditure. Additionally, there appears to be a strong link between the consumer's cost of consideration and rationality level. Chapter 3 proposes a choice theory that explains status quo bias (SQB) with the concept of just-noticeable differences (JNDs). SQB comes from an inclination to choose a default option/current choice when decision-making, whereas a JND is the minimal stimulus required to perceive change. JND utility can be considered a general representation of SQB; it is shown that the SQB representation of Masatlioglu & Ok (2005) is a special case. As such, an agent will only move away from a current choice position if there exist other alternatives that are noticeably better, otherwise, the agent does not shift away, hence leading to a bias towards the status quo

Introduction to social choice and welfare

Social choice theory is concerned with the evaluation of alternative methods of collective decision-making, as well as with the logical foundations of welfare economics. In turn, welfare economics is concerned with the critical scrutiny of the performance of actual and/or imaginary economic systems, as well as with the critique, design and implementation of alternative economic policies. The Handbook of Social Choice and Welfare, which is edited by Kenneth Arrow, Amartya Sen and Kotaro Suzumura, presents, in two volumes, essays on past and on-going work in social choice theory and welfare economics. This paper is written as an extensive introduction to the Handbook with the purpose of placing the broad issues examined in the two volumes in better perspective, discussing the historical background of social choice theory, the vistas opened by Arrow's Social Choice and Individual Values, the famous "socialist planning" controversy, and the theoretical and practical significance of social choice theory.social choice theory, welfare economics, socialist planning controversy, social welfare function, Arrovian impossibility theorems, voting schemes, implementation theory, equity and justice, welfare and rights, functioning and capability, procedural fairness

d-반순서의 경쟁그래프의 연구

학위논문 (박사)-- 서울대학교 대학원 : 사범대학 수학교육과, 2018. 2. 김서령.The \emph{competition graph} $C(D)$ of a digraph $D$ is defined to be a graph whose vertex set is the same as $D$ and which has an edge joining two distinct vertices $x$ and $y$ if and only if there are arcs $(x,z)$ and $(y,z)$ for some vertex $z$ in $D$. Competition graphs have been extensively studied for more than four decades. Cohen~\cite{cohen1968interval, cohen1977food, cohen1978food} empirically observed that most competition graphs of acyclic digraphs representing food webs are interval graphs. Roberts~\cite{roberts1978food} asked whether or not Cohen's observation was just an artifact of the construction, and then concluded that it was not by showing that if $G$ is an arbitrary graph, then $G$ together with additional isolated vertices as many as the number of edges of $G$ is the competition graph of some acyclic digraph. Then he asked for a characterization of acyclic digraphs whose competition graphs are interval graphs. Since then, the problem has remained elusive and it has been one of the basic open problems in the study of competition graphs. There have been a lot of efforts to settle the problem and some progress has been made. While Cho and Kim~\cite{cho2005class} tried to answer his question, they could show that the competition graphs of doubly partial orders are interval graphs. They also showed that an interval graph together with sufficiently many isolated vertices is the competition graph of a doubly partial order. In this thesis, we study the competition graphs of $d$-partial orders some of which generalize the results on the competition graphs of doubly partial orders. For a positive integer $d$, a digraph $D$ is called a \emph{$d$-partial order} if V(D) \subset \RR^d and there is an arc from a vertex $\mathbf{x}$ to a vertex $\mathbf{y}$ if and only if $\mathbf{x}$ is componentwise greater than $\mathbf{y}$. A doubly partial order is a $2$-partial order. We show that every graph $G$ is the competition graph of a $d$-partial order for some nonnegative integer $d$, call the smallest such $d$ the \emph{partial order competition dimension} of $G$, and denote it by $\dim_\text{poc}(G)$. This notion extends the statement that the competition graph of a doubly partial order is interval and the statement that any interval graph can be the competition graph of a doubly partial order as long as sufficiently many isolated vertices are added, which were proven by Cho and Kim~\cite{cho2005class}. Then we study the partial order competition dimensions of some interesting families of graphs. We also study the $m$-step competition graphs and the competition hypergraph of $d$-partial orders.1 Introduction 1 1.1 Basic notions in graph theory 1 1.2 Competition graphs 6 1.2.1 A brief history of competition graphs 6 1.2.2 Competition numbers 7 1.2.3 Interval competition graphs 10 1.3 Variants of competition graphs 14 1.3.1 m-step competition graphs 15 1.3.2 Competition hypergraphs 16 1.4 A preview of the thesis 18 2 On the competition graphs of d-partial orders 1 20 2.1 The notion of d-partial order 20 2.2 The competition graphs of d-partial orders 21 2.2.1 The regular (d − 1)-dimensional simplex △ d−1 (p) 22 2.2.2 A bijection from H d + to a set of regular (d − 1)-simplices 23 2.2.3 A characterization of the competition graphs of d-partial orders 25 2.2.4 Intersection graphs and competition graphs of d-partial orders 27 2.3 The partial order competition dimension of a graph 29 3 On the partial order competition dimensions of chordal graphs 2 38 3.1 Basic properties on the competition graphs of 3-partial orders 39 3.2 The partial order competition dimensions of diamond-free chordal graphs 42 3.3 Chordal graphs having partial order competition dimension greater than three 46 4 The partial order competition dimensions of bipartite graphs 3 53 4.1 Order types of two points in R 3 53 4.2 An upper bound for the the partial order competition dimension of a graph 57 4.3 Partial order competition dimensions of bipartite graphs 64 5 On the m-step competition graphs of d-partial orders 4 69 5.1 A characterization of the m-step competition graphs of dpartial orders 69 5.2 Partial order m-step competition dimensions of graphs 71 5.3 dim poc (Gm) in the aspect of dim poc (G) 76 5.4 Partial order competition exponents of graphs 79 6 On the competition hypergraphs of d-partial orders 5 81 6.1 A characterization of the competition hypergraphs of d-partial orders 81 6.2 The partial order competition hyper-dimension of a hypergraph 82 6.3 Interval competition hypergraphs 88 Abstract (in Korean) 99Docto