HistoryCGT - The History of Combinatorial Game Theory Richard J Nowakowski Dalhousie University [email protected] Abstract A brief history of the

HistoryCGT - The History of Combinatorial Game Theory...

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The History of Combinatorial Game Theory Richard J. Nowakowski Dalhousie University [email protected] January 24, 2009 Abstract A brief history of the people and the ideas that have contributed to Combinatorial Game Theory. 1 The Newcomer Games have been recorded throughout history but the systematic application of mathematics to games is a relatively recent phenomonon. Gambling games gave rise to studies of probability in the 16th and 17th century. What has become known as Combinatorial Game Theory or Combinatorial Game Theory `a la Conway —this to distinguish it from other forms of game theory found in economics and biology, for example—is a babe-in-arms in comparison. It has its roots in the paper [12] written in 1902, but the theory was not ‘codified’ until 1976-1982 with the publications of On Numbers and Games [15] by John H. Conway and Winning Ways [10] by Elwyn R. Berlekamp, John H. Conway and Richard K. Guy. In the subject of Impartial games (essentially the theory as known before 1976), the first MSc thesis appears to be in 1967 by Jack C. Kenyon [31] and the first PhD by Yaacov Yesha [63] in 1978. Using the full theory is Laura J. Yedwab’s MSc thesis in 1985 [62] and David Wolfe’s 1991 PhD [60]. (Note: Richard B. Austin’s MSc thesis [3] contains a little of the Partizan theory, but Yedwab’s thesis is purely Partizan theory.) In the sequel, there are several Mathematical Interludes that give a peek into the mathematics involved in the theory. Note that the names of games are given in small capitals such as CHESS . Also the notion of game is both general, like CHESS which refers to a a set of rules, and specific as in a specific CHESS position. Whenever I talk mathematically, as in the Interludes, it is this specific notion that should be invoked. 2 What is Combinatorial Game Theory? This Combinatorial Game Theory has several important features that sets it apart. Primarily, these are games of pure strategy with no random elements. Specifically: 1. There are Two Players who Alternate Moves ; 2. There are No Chance Devices —hence no dice or shuffling of cards; 3. There is Perfect Information —all possible moves are known to both players and, if needed, the whole history of the game as well; 1
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4. Play Ends, Regardless —even if the players do not alternate moves, the game must reach a conclusion; 5. The Last Move determines the winner— Normal play: last player to move wins; Mis`ere play last player to move loses! The players are usually called Left and Right and the genders are easy to remember —Left for L ouise Guy and Right for R ichard Guy who is a important ‘player’ in the development of the subject. More on him later. Figure 1: Louise and Richard Guy in Banff Examples of games 1 NOT covered by these rules are: DOTS -&- BOXES and GO , since these are scoring games, the last person to move is not guaranteed to have either the highest or the lowest score; CHESS , since the game can end in a draw; BACKGAMMON , since there is a chance element (dice); BRIDGE , the only aspect that this game satisfies is that it ends.
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