SING13 the 13Th European Meeting on Game Theory
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Pareto Optimality in Coalition Formation
Pareto Optimality in Coalition Formation Haris Aziz NICTA and University of New South Wales, 2033 Sydney, Australia Felix Brandt∗ Technische Universit¨atM¨unchen,80538 M¨unchen,Germany Paul Harrenstein University of Oxford, Oxford OX1 3QD, United Kingdom Abstract A minimal requirement on allocative efficiency in the social sciences is Pareto op- timality. In this paper, we identify a close structural connection between Pareto optimality and perfection that has various algorithmic consequences for coali- tion formation. Based on this insight, we formulate the Preference Refinement Algorithm (PRA) which computes an individually rational and Pareto optimal outcome in hedonic coalition formation games. Our approach also leads to var- ious results for specific classes of hedonic games. In particular, we show that computing and verifying Pareto optimal partitions in general hedonic games, anonymous games, three-cyclic games, room-roommate games and B-hedonic games is intractable while both problems are tractable for roommate games, W-hedonic games, and house allocation with existing tenants. Keywords: Coalition Formation, Hedonic Games, Pareto Optimality, Computational Complexity JEL: C63, C70, C71, and C78 1. Introduction Ever since the publication of von Neumann and Morgenstern's Theory of Games and Economic Behavior in 1944, coalitions have played a central role within game theory. The crucial questions in coalitional game theory are which coalitions can be expected to form and how the members of coalitions should divide the proceeds of their cooperation. Traditionally the focus has been on the latter issue, which led to the formulation and analysis of concepts such as ∗Corresponding author Email addresses: [email protected] (Haris Aziz), [email protected] (Felix Brandt), [email protected] (Paul Harrenstein) Preprint submitted to Elsevier August 27, 2013 the core, the Shapley value, or the bargaining set. -
A New IFORS Journal Is Born!
Volume 16 | Number 1 | March 2021 | ISSN 2223-4373 International Federation of Operational Research Societies NEWS From the President What’s Inside 1 From the President A New IFORS journal is Born! Grazia Speranza Grazia Speranza <[email protected]> 2 From the Editor-in-Chief The International Transactions in Operational Sunity Shrestha Hada Research (ITOR) has been the flagship journal of IFORS since its launch in 1994. The journal has 2 OR Impact served the IFORS community for a number of years Transforming Food Production and with publishing the proceedings of conferences and Supply with OR/Analytics the biographies of key figures in the OR field, who now belong to the IFORS Hall of Fame (https://www. 6 OR and Development ifors.org/ifors-hall-of-fame/)The scientific editorial Disaster Mitigation: Leveraging world substantially evolved over time, furthermore Community Involvement to Improve the ITOR mission expanded and began to publish Water Access in sub-Saharan Africa high-quality scientific papers, capable of attracting citations. The initial goal for ITOR was to be indexed 9 Tutorial and then to increase the value of the key indicator of Integer Programming formulations the success of a journal, its impact factor. Thanks to the for the Vertex Coloring Problem commitment of the Editor-in-Chief, Celso Ribeiro, and the editorial board of the journal, year after year, ITOR has improved its impact factor (current 12 IFORS Award 2020 for OR in impact factor: 2.987 ) and has become one of the leading journals for the OR community. Development a) Optimal Investment Strategies In addition to a well-established flagship journal, a global society such as IFORS has to Minimize Flood Impact on Road the responsibility to introduce new ventures. -
5.1 Convergence in Distribution
556: MATHEMATICAL STATISTICS I CHAPTER 5: STOCHASTIC CONVERGENCE The following definitions are stated in terms of scalar random variables, but extend naturally to vector random variables defined on the same probability space with measure P . Forp example, some results 2 are stated in terms of the Euclidean distance in one dimension jXn − Xj = (Xn − X) , or for se- > quences of k-dimensional random variables Xn = (Xn1;:::;Xnk) , 0 1 1=2 Xk @ 2A kXn − Xk = (Xnj − Xj) : j=1 5.1 Convergence in Distribution Consider a sequence of random variables X1;X2;::: and a corresponding sequence of cdfs, FX1 ;FX2 ;::: ≤ so that for n = 1; 2; :: FXn (x) =P[Xn x] : Suppose that there exists a cdf, FX , such that for all x at which FX is continuous, lim FX (x) = FX (x): n−!1 n Then X1;:::;Xn converges in distribution to random variable X with cdf FX , denoted d Xn −! X and FX is the limiting distribution. Convergence of a sequence of mgfs or cfs also indicates conver- gence in distribution, that is, if for all t at which MX (t) is defined, if as n −! 1, we have −! () −!d MXi (t) MX (t) Xn X: Definition : DEGENERATE DISTRIBUTIONS The sequence of random variables X1;:::;Xn converges in distribution to constant c if the limiting d distribution of X1;:::;Xn is degenerate at c, that is, Xn −! X and P [X = c] = 1, so that { 0 x < c F (x) = X 1 x ≥ c Interpretation: A special case of convergence in distribution occurs when the limiting distribution is discrete, with the probability mass function only being non-zero at a single value, that is, if the limiting random variable is X, then P [X = c] = 1 and zero otherwise. -
Pareto-Optimality in Cardinal Hedonic Games
Research Paper AAMAS 2020, May 9–13, Auckland, New Zealand Pareto-Optimality in Cardinal Hedonic Games Martin Bullinger Technische Universität München München, Germany [email protected] ABSTRACT number of agents, and therefore it is desirable to consider expres- Pareto-optimality and individual rationality are among the most nat- sive, but succinctly representable classes of hedonic games. This ural requirements in coalition formation. We study classes of hedo- can be established by encoding preferences by means of a complete nic games with cardinal utilities that can be succinctly represented directed weighted graph where an edge weight EG ¹~º is a cardinal by means of complete weighted graphs, namely additively separable value or utility that agent G assigns to agent ~. Still, this underlying (ASHG), fractional (FHG), and modified fractional (MFHG) hedo- structure leaves significant freedom, how to obtain (cardinal) pref- nic games. Each of these can model different aspects of dividing a erences over coalitions. An especially appealing type of preferences society into groups. For all classes of games, we give algorithms is to have the sum of values of the other individuals as the value that find Pareto-optimal partitions under some natural restrictions. of a coalition. This constitutes the important class of additively While the output is also individually rational for modified fractional separable hedonic games (ASHGs) [6]. hedonic games, combining both notions is NP-hard for symmetric In ASHGs, an agent is willing to accept an additional agent into ASHGs and FHGs. In addition, we prove that welfare-optimal and her coalition as long as her valuation of this agent is non-negative Pareto-optimal partitions coincide for simple, symmetric MFHGs, and in this sense, ASHGs are not sensitive to the intensities of single- solving an open problem from Elkind et al. -
Fairness in Bankruptcy Situations: an Experimental Study
Fairness in bankruptcy situations: an experimental study Alexander W. Cappelen Roland I. Luttens Erik Ø. Sørensen Bertil Tungodden∗ August 11, 2015 Abstract The pari passu principle is the most prominent principle in the law of in- solvency. We report from a lab experiment designed to study whether people find this principle a fair solution to the bankruptcy problem. The experi- mental design generates situations where participants work and accumulate claims in firms, some of which subsequently go bankrupt. Third-party arbi- trators are randomly assigned to determine how the liquidation value of the bankrupt firms should be distributed between claimants. Our main finding is that there is a striking support for the pari passu principle of awarding claimants proportionally to their pre-insolvency claims. We estimate a ran- dom utility model that allows for the arbitrators to differ in what they consider a fair solution to the bankruptcy problem and find that about 85 percent of the participants endorse the proportional rule. We also find that a non-negligible fraction of the arbitrators follow the constrained equal losses rule, while there is almost no support in our experiment for the constrained equal awards rule or other fairness rules suggested in the normative literature. Finally, we show that the estimated random utility model nicely captures the observed arbi- trator behavior, in terms of both the overall distribution of awards and the relationship between awards and claims. 1 Introduction Bankruptcy law is concerned with situations where a debtor is unable to pay its debts and it constitutes an essential element in any well-functioning market econ- omy (Hotchkiss, John, Moordadian, and Thorburn, 2008). -
Parameter Specification of the Beta Distribution and Its Dirichlet
%HWD'LVWULEXWLRQVDQG,WV$SSOLFDWLRQV 3DUDPHWHU6SHFLILFDWLRQRIWKH%HWD 'LVWULEXWLRQDQGLWV'LULFKOHW([WHQVLRQV 8WLOL]LQJ4XDQWLOHV -5HQpYDQ'RUSDQG7KRPDV$0D]]XFKL (Submitted January 2003, Revised March 2003) I. INTRODUCTION.................................................................................................... 1 II. SPECIFICATION OF PRIOR BETA PARAMETERS..............................................5 A. Basic Properties of the Beta Distribution...............................................................6 B. Solving for the Beta Prior Parameters...................................................................8 C. Design of a Numerical Procedure........................................................................12 III. SPECIFICATION OF PRIOR DIRICHLET PARAMETERS................................. 17 A. Basic Properties of the Dirichlet Distribution...................................................... 18 B. Solving for the Dirichlet prior parameters...........................................................20 IV. SPECIFICATION OF ORDERED DIRICHLET PARAMETERS...........................22 A. Properties of the Ordered Dirichlet Distribution................................................. 23 B. Solving for the Ordered Dirichlet Prior Parameters............................................ 25 C. Transforming the Ordered Dirichlet Distribution and Numerical Stability ......... 27 V. CONCLUSIONS........................................................................................................ 31 APPENDIX................................................................................................................... -
Basis Utility and Proportionality in Games1
Lost in Translation: Basis Utility and Proportionality in Games1 Barry Feldman Prism Analytics Kellstadt School of Business 53 West Jackson Blvd. #724 DePaul University Chicago, IL 60604 1 East Jackson Blvd. #6100 [email protected] Chicago, IL 60604 [email protected] July 18, 2005 abstract A player's basis utility is the utility of no payo®. Basis utility is neces- sary for the coherent representation of the equal split bargaining solution. Standard axioms for the Nash (1950) bargaining solution do not imply independence from basis utility. Proportional bargaining is the unique solution satisfying e±ciency, symmetry, a±ne transformation invariance and monotonicity in pure bargaining games with basis utility. All existing cooperative solutions become translation invariant once account is taken of basis utility. The noncooperative rationality of these results is demon- strated though an implementation of proportional bargaining based on Gul (1988). Quantal response equilibria with multiplicative error struc- tures (Goeree, Holt and Palfrey (2004)) become translation invariant with speci¯cation of basis utility. Equal split and proportional bargaining join the Kalai-Smorodinsky (1975) solution in a family of endogenously pro- portional monotonic pure bargaining solutions. JEL Classi¯cation: C70, C71, C72, C78. Keywords: Basis utility, equal split, Kalai-Smorodinsky solution, Nash bargaining, quantal response equi- libria, proportional bargaining, translation invariance. 1I am grateful to Peter Borm, Srihari Govindan, Chris O'Neill and Alvin E. Roth for discussions of topics related to this paper. Thanks also for comments to Oliver Gossner, Elena I~narraand other participants at the 2005 Stony Brook Game Theory Conference. °c 2005 by Barry Feldman. 1 Introduction There is no representation of basis utility in games. -
Optimizing Expected Utility and Stability in Role Based Hedonic Games
Proceedings of the Thirtieth International Florida Artificial Intelligence Research Society Conference Optimizing Expected Utility and Stability in Role Based Hedonic Games Matthew Spradling University of Michigan - Flint mjspra@umflint.edu Abstract Table 1: Ex. RBHG instance with |P | =4,R= {A, B} In the hedonic coalition formation game model Roles Based r, c u (r, c) u (r, c) u (r, c) u (r, c) Hedonic Games (RBHG), agents view teams as compositions p0 p1 p2 p3 of available roles. An agent’s utility for a partition is based A, AA 2200 upon which roles she and her teammates fulfill within the A, AB 0322 coalition. I show positive results for finding optimal or sta- B,AB 3033 ble role matchings given a partitioning into teams. In set- B,BB 1111 tings such as massively multiplayer online games, a central authority assigns agents to teams but not necessarily to roles within them. For such settings, I consider the problems of op- timizing expected utility and expected stability in RBHG. I As per (Spradling and Goldsmith 2015), a Role Based He- show that the related optimization problems for partitioning donic Game (RBHG) instance consists of: are NP-hard. I introduce a local search heuristic method for • P approximating such solutions. I validate the heuristic by com- : A population of players parison to existing partitioning approaches using real-world • R: A set of roles data scraped from League of Legends games. • C: A set of compositions, where a composition c ∈ C is a multiset (bag) of roles from R. Introduction • U : P × R × C → Z defines the utility function ui(r, c) In coalition formation games, agents from a population have for each player pi. -
Package 'Distributional'
Package ‘distributional’ February 2, 2021 Title Vectorised Probability Distributions Version 0.2.2 Description Vectorised distribution objects with tools for manipulating, visualising, and using probability distributions. Designed to allow model prediction outputs to return distributions rather than their parameters, allowing users to directly interact with predictive distributions in a data-oriented workflow. In addition to providing generic replacements for p/d/q/r functions, other useful statistics can be computed including means, variances, intervals, and highest density regions. License GPL-3 Imports vctrs (>= 0.3.0), rlang (>= 0.4.5), generics, ellipsis, stats, numDeriv, ggplot2, scales, farver, digest, utils, lifecycle Suggests testthat (>= 2.1.0), covr, mvtnorm, actuar, ggdist RdMacros lifecycle URL https://pkg.mitchelloharawild.com/distributional/, https: //github.com/mitchelloharawild/distributional BugReports https://github.com/mitchelloharawild/distributional/issues Encoding UTF-8 Language en-GB LazyData true Roxygen list(markdown = TRUE, roclets=c('rd', 'collate', 'namespace')) RoxygenNote 7.1.1 1 2 R topics documented: R topics documented: autoplot.distribution . .3 cdf..............................................4 density.distribution . .4 dist_bernoulli . .5 dist_beta . .6 dist_binomial . .7 dist_burr . .8 dist_cauchy . .9 dist_chisq . 10 dist_degenerate . 11 dist_exponential . 12 dist_f . 13 dist_gamma . 14 dist_geometric . 16 dist_gumbel . 17 dist_hypergeometric . 18 dist_inflated . 20 dist_inverse_exponential . 20 dist_inverse_gamma -
Iam 530 Elements of Probability and Statistics
IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS LECTURE 4-SOME DISCERETE AND CONTINUOUS DISTRIBUTION FUNCTIONS SOME DISCRETE PROBABILITY DISTRIBUTIONS Degenerate, Uniform, Bernoulli, Binomial, Poisson, Negative Binomial, Geometric, Hypergeometric DEGENERATE DISTRIBUTION • An rv X is degenerate at point k if 1, Xk P X x 0, ow. The cdf: 0, Xk F x P X x 1, Xk UNIFORM DISTRIBUTION • A finite number of equally spaced values are equally likely to be observed. 1 P(X x) ; x 1,2,..., N; N 1,2,... N • Example: throw a fair die. P(X=1)=…=P(X=6)=1/6 N 1 (N 1)(N 1) E(X) ; Var(X) 2 12 BERNOULLI DISTRIBUTION • An experiment consists of one trial. It can result in one of 2 outcomes: Success or Failure (or a characteristic being Present or Absent). • Probability of Success is p (0<p<1) 1 with probability p Xp;0 1 0 with probability 1 p P(X x) px (1 p)1 x for x 0,1; and 0 p 1 1 E(X ) xp(y) 0(1 p) 1p p y 0 E X 2 02 (1 p) 12 p p V (X ) E X 2 E(X ) 2 p p2 p(1 p) p(1 p) Binomial Experiment • Experiment consists of a series of n identical trials • Each trial can end in one of 2 outcomes: Success or Failure • Trials are independent (outcome of one has no bearing on outcomes of others) • Probability of Success, p, is constant for all trials • Random Variable X, is the number of Successes in the n trials is said to follow Binomial Distribution with parameters n and p • X can take on the values x=0,1,…,n • Notation: X~Bin(n,p) Consider outcomes of an experiment with 3 Trials : SSS y 3 P(SSS) P(Y 3) p(3) p3 SSF, SFS, FSS y 2 P(SSF SFS FSS) P(Y 2) p(2) 3p2 (1 p) SFF, FSF, FFS y 1 P(SFF FSF FFS ) P(Y 1) p(1) 3p(1 p)2 FFF y 0 P(FFF ) P(Y 0) p(0) (1 p)3 In General: n n! 1) # of ways of arranging x S s (and (n x) F s ) in a sequence of n positions x x!(n x)! 2) Probability of each arrangement of x S s (and (n x) F s ) p x (1 p)n x n 3)P(X x) p(x) p x (1 p)n x x 0,1,..., n x • Example: • There are black and white balls in a box. -
Ame Theoretic Analysis of a Bankruptcy Proble from the Talmud*
JOURNAL OF ECONOMIC THEORY 36, 195-213 (1985) ame Theoretic Analysis of a bankruptcy Proble from the Talmud* ROBERT J. AUMANN AND MICHAEL MASCHLER The Hebrew Universify, 91904 Jerusalem, Israel Received August 27, 1984; revised February 4, 1985 DEDICATED TO THE MEMORYOF SHLOMOAUMANN, TALMUDIC SCHOLARAND MAN OF THE WORLD, KILLED IN ACTION NEAR KHUSH-E-DNEIBA, LEBANON, ON THE EVE OF THE NINETEENTHOF SWAN, 5742 (JUNE 9, 1982). For three different bankruptcy problems, the 2000-year oid Babylonian Talmud prescribes solutions that equal precisely the nucleoli of the corresponding coalitional games. A rationale for these solutions that is independent of game theory is given in terms of the Talmudic principle of equai division of the contested amount: this rationale leads to a unique solution for all bankruptcy problems, which always coincides with the nucleolus. Two other rationales for the same rule are suggested. in terms of other Talmudic principles. (Needless to say, the rule in question is not proportional division). Jomd cf Economic Literature C’lussification Numbers: 022, 026, 031, 043, 213. ‘?: 1985 Academic Press. Inc. 1. 1N~RoDucTroN A man dies, leaving debts d, ,..., d,, totalling more than his estate E. should the estate be divided among the creditors? A frequent solution in modern law is proportional division. The rationale is that each dollar of debt should be treated in the same way; one looks at dollars rather than people. Yet it is by no means obvious that this is the only equitable or reasonable system. For example, if the estate does not exceed the smallest debt, equal division among the creditors makes good sense. -
The Length-Biased Versus Random Sampling for the Binomial and Poisson Events Makarand V
Journal of Modern Applied Statistical Methods Volume 12 | Issue 1 Article 10 5-1-2013 The Length-Biased Versus Random Sampling for the Binomial and Poisson Events Makarand V. Ratnaparkhi Wright State University, [email protected] Uttara V. Naik-Nimbalkar Pune University, Pune, India Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Ratnaparkhi, Makarand V. and Naik-Nimbalkar, Uttara V. (2013) "The Length-Biased Versus Random Sampling for the Binomial and Poisson Events," Journal of Modern Applied Statistical Methods: Vol. 12 : Iss. 1 , Article 10. DOI: 10.22237/jmasm/1367381340 Available at: http://digitalcommons.wayne.edu/jmasm/vol12/iss1/10 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Journal of Modern Applied Statistical Methods Copyright © 2013 JMASM, Inc. May 2013, Vol. 12, No. 1, 54-57 1538 – 9472/13/$95.00 The Length-Biased Versus Random Sampling for the Binomial and Poisson Events Makarand V. Ratnaparkhi Uttara V. Naik-Nimbalkar Wright State University, Pune University, Dayton, OH Pune, India The equivalence between the length-biased and the random sampling on a non-negative, discrete random variable is established. The length-biased versions of the binomial and Poisson distributions are discussed. Key words: Length-biased data, weighted distributions, binomial, Poisson, convolutions. Introduction binomial distribution (probabilities), which was The occurrence of so-called length-biased data appropriate at that time.