Report
University:
Whitman CollegeCourse:
MATH-125 | Calculus IAcademic year:
2012
Views:
314
Pages:
30
Author:
Josephine Costa
h means that g is more favorable to Left than h, g < h means that g is less favorable to Left than h, and g k h means that g is only more favorable to Left if Left is the first to move. 26 From our studies of numbers we can conclude that all of the usual order relations hold. For example, if we have three games g, h, and k such that g < h and h < k, then g < k. We know that 1 > 12 , so the game 1 is more favorable to Left than 12 . However, recall how in both games Left won regardless, so how could one be more favorable than the other? Observe what happens if we give Right an extra move in 21 . If Left goes first, then Right will surely win. If Right goes first, she can win by first deleting the red edge in 12 . Since adding -1 to 12 turned the game to Right’s favor, Left had less than a one-move advantage in 21 ; whereas Left clearly has a one-move advantage in 1. Thus 1 > 21 . 4.4 Sums of Games If we have two games, g and h, then the sum g + h is the compound game that we get by playing both g and h simultaneously side by side, as we did with 12 and 1 in the previous section. For example consider the game 12 + 12 . We see that Left can move to 0 + 12 or 21 + 0, and Right can move to 1 + 12 or 21 + 1. Thus 1 1 1 1 1 1 2 + 2 = { 2 |1 + 2 } ≡ 1. We can also verify that the compound game 2 + 2 is 1 by adding −1 to it. Although we leave it to the interested reader, it is not hard to deduce that in 12 + 12 + (−1), the second player to move will win, and thus 12 + 12 + (−1) = 0. Suppose we have the sum of any two games g = {GL |GR } and h = {HL |HR }. Then the legal moves for Left will be GL + h ∪ g + HL and the legal moves for Right will be GR + h ∪ g + HR . Thus g + h = {GL + h, g + HL |GR + h, g + HR }, and we see that addition for games is consistent with our definition of addition for numbers. This means that we can apply the properites of addition that we have proved for numbers to two-color Hackenbush games. For example it is not hard to see that the game below is really just 12 , since adding a zero game to any other game will not change its outcome. 27 If we have any Hackenbush game, then we get the negative of that game by interchanging red and blue edges. When we negate g = {GL |GR }, the possible moves in GR become moves for Left, and the moves in GL become moves for Right. Thus −g = {−GR | − GL }, and we see that our definition is consistent with our definition for numbers. Thus we can conclude that g − g is a zero game for any two-color Hackenbush game g. For example, the game 41 − 14 pictured below is a zero game. 4.5 Hackenbush Hotchpotch So far we have been looking at two-color Hackenbush games in which each player can only delete edges of a certain color. Consider the following simple game, in which either player could delete the single green edge. We see that both players can move to 0. Thus the game has value {0|0}. By Axiom 1, since 0 ≤ 0, {0|0} is not a number, but a pseudo-number. Hackenbush Hotchpotch is just like Hackenbush except that there can be green edges in a picture, which either player can delete. Since every green edge represents a move for both Left and Right, we see that in every Hackenbush Hotchpotch game, g = {GL |GR }, there will be some x corresponding to the resulting position from deleting a green edge such that x ∈ GL and x ∈ GR . Thus GL 6< GR , which means that G is not a number by Axiom 1. Thus every Hackenbush Hotchpotch game is a pseudo-number. Consider the game, g, below. 28 Since the first player to move can delete the green edge, causing endgame, we see that whoever moves first will win. Thus g k 0. Since there is no particular advantage to Left or Right, why are these games not equal to zero? Consider the sum g + g below. Whether Left or Right starts, Right has enough edges so that she can force Left to delete one of the green edges first. Right can then win by taking the second green edge. Thus the sum g + g is negative. Clearly, a zero game plus a zero game cannot be negative. In fact, fuzzy games are neither equal to, greater than, or less than zero, but rather “confused” with zero [3]. Consider what 1 happens when we add a small positive number to g, like 64 . 1 The game g + 64 is positive, since Left will win either by deleting the green edge or, if Right deletes 1 the green edge, by deleting the blue edge from the component 64 . If we add a small negative 1 number, like − 64 , to g then we get the game pictured below. 29 This game is negative since Right will always win by either taking the green edge or her red edge of 1 1 1 − 64 . Thus − 64 < g < 64 . It is not hard to see that this argument will hold for smaller and smaller fractions, thus g will be greater than all negative numbers and less than all positive numbers. In fact, any Hackenbush picture in which all of the edges directly connected to the ground are green has a value that lies strictly between all negative and positive numbers [3]. However, such a game can still have a value that is positive or negative. Consider the house below. The first person forced to delete one of the green edges will lose, since the other player can then end the game by deleting the second green edge. Since Right can force Left to delete the first green edge by deleting her red edges on her first one or two turns (depending on who goes first), Right 1 can always win. Thus the game has a negative value. However, as before, if we add 64 , Left will 1 1 win. Thus house + 64 > 0, which means that the house is greater than − 64 . Again, this will hold for any negative number, which means that the house has a negative value, but is greater than any negative number. Conclusion Although we have just scratched the surface of game theory, we get an idea of how surreal numbers can be used to analyze games. In addition to Hackenbush, they can be applied to all sorts of two-player games, examples of which can be found in On Numbers and Games and Winning Ways. As well as looking deeper into game theory, interested readers can look into more advanced topics involving surreal numbers, such as number theory, algebra, and analysis. For example, how would √ polynomials with surreal coefficients behave? Or what would the field S[i] look like, where i = −1? Conway explores all of these topics, and more, in On Numbers and Games. References [1] Donald E. Knuth. Surreal Numbers. Reading, MA: Addison-Wesley Pub., 1974. Print. [2] John H. Conway. On Numbers and Games. London: Academic, 1976. Print. [3] Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy. Winning Ways, for Your Mathematical Plays. London: Academic, 1982. Print. 30
An Introduction to Surreal Numbers - Report
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