Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Discrete bidding games. | Discrete bidding games Mike Develin Sam Payne American Institute of Mathematics 360 Portage Ave. Palo Alto CA 94306 develin@ Stanford University Dept. of Mathematics 450 Serra Mall Stanford CA 94305 spayne@ Submitted Feb 28 2008 Accepted May 26 2010 Published Jun 7 2010 Mathematics Subject Classification 91A46 91B26 91A60 Abstract We study variations on combinatorial games in which instead of alternating moves the players bid with discrete bidding chips for the right to determine who moves next. We consider both symmetric and partisan games and explore differences between discrete bidding games and Richman games which allow real-valued bidding. Unlike Richman games discrete bidding game variations of many familiar games such as chess Connect Four and even Tic-Tac-Toe are suitable for recreational play. We also present an analysis of Tic-Tac-Toe for both discrete and real-valued bidding. Contents 1 Introduction 2 A game of bidding Tic-Tac-Toe. 4 A game of bidding chess. 5 2 Preliminaries 7 Game model. 7 Bidding. 7 Tie-breaking. 7 3 General theory 8 Value of the tie-breaking advantage. 8 Using the tie-breaking advantage . 9 Classical Richman calculus. 10 Discrete Richman calculus. 11 Supported by the Clay Mathematics Institute. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R85 1 Discrete bidding with large numbers of chips. 14 Periodicity. 16 4 Examples 17 Tug o War . 18 Ultimatum. 19 5 A partial order on games 21 6 Bidding Tic-Tac-Toe 23 Optimal moves . 23 Chip tables . 32 7 Appendix other tie-breaking methods 38 1 Introduction Imagine playing your favorite two-player game such as Tic-Tac-Toe Connect Four or chess but instead of alternating moves you bid against your opponent for the right to decide who moves next. For instance you might play a game of bidding chess in which you and your opponent each start with one hundred bidding chips. If you bid twelve for the first move .
