Daniel Graf, 1990 Jassen Game Theory Analysis of a Swiss Card Game

Tasks and Goals Methods • Is it possible to adapt game theoretical algorithms created for • Development of the game theoretical chance free games like chess and go for the „Schieber-Jass“? algorithms in C and Objective-C • Implementation of the minimax algorithm and alpha-beta- • Usage of the Apple frameworks Cocoa and pruning combined into a complete Jass-application Quartz Composer for the user interface design on Mac OS X • Calculation of the probabilities of special points („Weise“) • Simulation over thousands of points on 20 • Evaluate several algorithms against each other computers at my high school for the • Game with an intuitive user interface comparison of the different algorithms • Appealing representation of the cards in 3D space

Results Discussion • Weis-analysis: a selection of 9 out of 36 cards contains 9.48 • Optimisation of several parameters lead to a points in average whereof 7.89 can be written (calculated with a playing capability of a good amateur self-developed Monte-Carlo program) • Human - Computer comparison: • The program finds optimal strategies for small games with a ⊕ perfect memory (counting remaining cards) maximum of 24 cards. ⊖ intuitive decisions (psychology) • Adaption of the minimax algorithm and the alpha-beta- • The horizon effect has a negative influence pruning for the „Schieber-Jass“ when cards get valuable only later in the game. • Certainty theorem of Zermelo requires games with uncovered cards for finding optimal pure strategies. • Repeated and randomised distributions of the unknown cards Conclusion and a limited search depth allow the analysis of complete games with covered cards. • The programm allows significant statements about the influence of luck, logic and • A calculation time ratio of 20 to1 between the two parties (test strategy on the outcome of a jass game. series for 500‘000 points) results in: • Improving the strategic part of the • 53.37% of all points won algorithms would be possible by studying • 73% of all games won (victory with 2500 points) the many traditional conventions between two players in further detail. • Requirements concerning graphics and user interaction fulfilled

© 2010 Daniel Graf Literature Thanks • Bewersdorff, Jörg: Mit Glück, Logik und Bluff. Mathematik • Egg, Gottfried: Puur, Näll, As. Offizielles Schweizer im Spiel - Methoden, Ergebnisse und Grenzen. 4. Auflage, • Prof. Josef Vogt (supporting teacher) Sources Wiesbaden: Friedrich Vieweg und Sohn Verlag Januar 2007 Jassreglement. 4. Auflage, Neuhausen am Rheinfall: AGM • Prof. Peter Hammer (supervising expert SJf) • I created all graphics by myself. AGMÜLLER 1984 • Hirt, Martin / Matter, Daniel / Bänziger, Rolf / Hartmann, Werner: Gruppenunterricht zum Thema Spieltheorie. März • Family Graf KANTISARGANS 1999, EducETH • AGMüller Neuhausen www.kantisargans.ch Decision-Process for choosing a Card Player 3 1. Does an optimal card even exist?

Player 4 Certainty Theorem of Zermlo Example Situation Player 2 Requirements for optimal pure strategies: „Obenabe“ is played 2 rounds remaining my cards • game with two players • zero-sum-game (profit ≙ loss) remaining cards • finiteness 2. Card Distribution • perfect information Player 1 & 3 Assumption: The unknown cards • no random influences are distrbuted randomly among Player 1 ? ⇒ conversion in a game with exposed cards the other players. Therefore the Player 2 & 4 game tree analysis examines an arbitrary distribution. ? 3. Game Tree Analysis 28 Player 1 Problem: Game Tree Size of the Game Trees 0 28 Player 2 • Representation of all possible courses of the game as a tree Number of courses of the game • Circle: state of a game labelled with the 0 28 Player 3 • 10'000'000 1'521'933 maximum of achievable points of players 1'000'000 1 and 3 from this situation 100'000 11'594 0 28 18 Player 4 10'000 • Arrow: transition between two states, 1'000 231 therefore a played card 100 9 10 1 0 45 28 45 18 1 Minimax-Algorithm Number of 4 8 12 16 20 • A systematic simulation of all possible courses of the game that tries to find Improvement: Alpha-Beta-Pruning Limitation of the Search Depth the optimal game course. 3 Player A - MAX • The amount of points gets alternately maximized (player 1 and 3) and 3 -1 Player B - MIN search limited minimized (player 2 and 4). ✂ depth α-Cutoff search tree • Any aberration from this found strategy 3 10 -1 ? Player A - MAX has never a positive effect on the total ✂ β-Cutoff horizon amount of points. complete search tree 3 10 ? ? -1 -2 ? ? ? 4. Choice of a Card • Repeat the steps 2 and 3 several times. 5. Realisation in a complete user-friendly Application • Choose the card, which promises the highest benefit on Basic Structure of this Jass-Application Screenshot of the Jass-Program average. Please Calculate the choose a A best card! card!

B C 0 28 Game Admin Player Brain 45 45 Please animate 0 28 the new card distribution! Rosen Ass has 25 25 been clicked! 55 70 126 Graphics Interface

The programm consists of a total of 400 kilobytes of source code, which include the game-tree-algorithms, the organisation of the game and the calculation of the 3D- coordinates. The rendering of the game is done with OpenGL directly on the GPU.