Combinatorial Game Theory: An Introduction to Tree Topplers

The purpose of this thesis is to introduce a new game, Tree Topplers, into the field of Combinatorial Game Theory. Before covering the actual material, a brief background of Combinatorial Game Theory is presented, including how to assign advantage values to combinatorial games, as well as information on another, related game known as Domineering. Please note that this document contains color images so please keep that in mind when printing

Re-establishing an Ecological Discourse in the Debate over the Value of Ecosystems and Biodiversity

The approach of conceptualizing biodiversity and ecosystems as goods and services to be represented by monetary values in policy is being championed not just by economists, but also by ecologists and conservation biologists. This new environmental pragmatism is now being pushed forward internationally under the guise of hardwiring biodiversity and ecosystems services into finance. This conflicts with the realisation that biodiversity and ecosystems have multiple incommensurable values. The current trend is to narrowly define a set of instrumental aspects of ecosystems and biodiversity to be associated with ad hoc money numbers. We argue that ecosystem science has more to offer the policy debate than pseudo-economic numbers based on assumptions that do not reflect ecological or social complexity. Re-establishing the ecological discourse in biodiversity policy implies a crucial role for biophysical indicators as policy targets e.g., the Nature Index for Norway. Yet there is a recognisable need to go beyond the traditional ecological approach to create a social ecological economic discourse. This requires reviving and relating to a range of alternative ecologically informed discourses (e.g. intrinsic values, deep ecology, ecofeminism) in order to transform the increasingly dominant and destructive relationship of humans separated from and domineering over Nature. (author's abstract)Series: SRE - Discussion Paper

New Results for Domineering from Combinatorial Game Theory Endgame Databases

We have constructed endgame databases for all single-component positions up to 15 squares for Domineering, filled with exact Combinatorial Game Theory (CGT) values in canonical form. The most important findings are as follows. First, as an extension of Conway's [8] famous Bridge Splitting Theorem for Domineering, we state and prove another theorem, dubbed the Bridge Destroying Theorem for Domineering. Together these two theorems prove very powerful in determining the CGT values of large positions as the sum of the values of smaller fragments, but also to compose larger positions with specified values from smaller fragments. Using the theorems, we then prove that for any dyadic rational number there exist Domineering positions with that value. Second, we investigate Domineering positions with infinitesimal CGT values, in particular ups and downs, tinies and minies, and nimbers. In the databases we find many positions with single or double up and down values, but no ups and downs with higher multitudes. However, we prove that such single-component ups and downs easily can be constructed. Further, we find Domineering positions with 11 different tinies and minies values. For each we give an example. Next, for nimbers we find many Domineering positions with values up to *3. This is surprising, since Drummond-Cole [10] suspected that no *2 and *3 positions in standard Domineering would exist. We show and characterize many *2 and *3 positions. Finally, we give some Domineering positions with values interesting for other reasons. Third, we have investigated the temperature of all positions in our databases. There appears to be exactly one position with temperature 2 (as already found before) and no positions with temperature larger than 2. This supports Berlekamp's conjecture that 2 is the highest possible temperature in Domineering

11 x 11 Domineering is Solved: The first player wins

We have developed a program called MUDoS (Maastricht University Domineering Solver) that solves Domineering positions in a very efficient way. This enables the solution of known positions so far (up to the 10 x 10 board) much quicker (measured in number of investigated nodes). More importantly, it enables the solution of the 11 x 11 Domineering board, a board up till now far out of reach of previous Domineering solvers. The solution needed the investigation of 259,689,994,008 nodes, using almost half a year of computation time on a single simple desktop computer. The results show that under optimal play the first player wins the 11 x 11 Domineering game, irrespective if Vertical or Horizontal starts the game. In addition, several other boards hitherto unsolved were solved. Using the convention that Vertical starts, the 8 x 15, 11 x 9, 12 x 8, 12 x 15, 14 x 8, and 17 x 6 boards are all won by Vertical, whereas the 6 x 17, 8 x 12, 9 x 11, and 11 x 10 boards are all won by Horizontal

An update on domineering on rectangular boards

Domineering is a combinatorial game played on a subset of a rectangular grid between two players. Each board position can be put into one of four outcome classes based on who the winner will be if both players play optimally. In this note, we review previous work, establish the outcome classes for several dimensions of rectangular board, and restrict the outcome class in several more.Comment: 9 pages. References fixe

Courtship in Drosophila Mosaics: Sex-Specific Foci for Sequential Action Patterns

Mosaic fate mapping is used to locate the foci determining sex-specific steps in the mating behavior of Drosophila. Male performance of following females and displaying wing vibration toward them requires that a focus inside the head be constituted of male tissue, regardless of the sex of the head sense organs, the legs, the wings, or the thoracic ganglion. For attempted copulation to occur, a second focus in the thoracic region must also be male. Courtship by males is induced by a posteriorly located focus in the female, but an anterior female focus determines receptivity to attempted copulation. The interplay of male and female foci in the complex behavioral sequence is delineated

Bridging the gap between critical theory and critique of power? Honneth’s approach to ‘social negativity’

In this paper, I will analyze Axel Honneth’s theory against the background of some of the criticisms that Amy Allen levelled against it. His endeavor seems to partially compromise his ability to identify the domineering forms of power that the subject does not acknowledge consciously and affectively. I will argue that, despite some significant limitations, Honneth’s theory has become increasingly able to analyze social negativity since The struggle for recognition. Also, in both defending Honneth’s methodology and delimiting its scope, I aim to contribute to the debate between two understandings of power: power as ‘domination’ and power as ‘constitution’

The switch operators and push-the-button games: a sequential compound over rulesets

We study operators that combine combinatorial games. This field was initiated by Sprague-Grundy (1930s), Milnor (1950s) and Berlekamp-Conway-Guy (1970-80s) via the now classical disjunctive sum operator on (abstract) games. The new class consists in operators for rulesets, dubbed the switch-operators. The ordered pair of rulesets (R 1 , R 2) is compatible if, given any position in R 1 , there is a description of how to move in R 2. Given compatible (R 1 , R 2), we build the push-the-button game R 1 R 2 , where players start by playing according to the rules R 1 , but at some point during play, one of the players must switch the rules to R 2 , by pushing the button ". Thus, the game ends according to the terminal condition of ruleset R 2. We study the pairwise combinations of the classical rulesets Nim, Wythoff and Euclid. In addition, we prove that standard periodicity results for Subtraction games transfer to this setting, and we give partial results for a variation of Domineering, where R 1 is the game where the players put the domino tiles horizontally and R 2 the game where they play vertically (thus generalizing the octal game 0.07).Comment: Journal of Theoretical Computer Science (TCS), Elsevier, A Para{\^i}tr