Combinatorial Game Theory

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Combinatorial Game Theory

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1. Combinatorial Game Theory The Games We Play and How to Think!!!

3. What is Game Theory? A branch of mathematics and logic which deals with the analysis of games (i.e., situations involving parties with conflicting interests). In addition to the mathematical elegance and complete "solution" which is possible for simple games, the principles of game theory also find applications to complicated games such as cards, checkers, and chess, as well as real-world problems as diverse as economics, property division, politics, and warfare (Weisstein 1).

4. Meaning of Game Theory ??? Allows players (parties) to better understand the game and develop strategy. “Experience is the best teacher”, a player can avoid mistakes. Judgment is not at best if lacking in the knowledge of rules

5. Combinatorial Game Theory defined as the studies and strategies of two or more player games with the players having complete knowledge of the game Difference from Combinatorial and Game Theory Moves are sequential instead of random

6. Role of Logic “A branch of philosophy and mathematics that deals with the formal principles, methods and criteria of validity of inference, reasoning and knowledge” (Howe). Tries to predetermine the unknown

7. History of Game Theory 0-500 AD, during Babylonia, Talmud was a compilation of ancient laws and traditions Marriage contract dispute A husband had three wives had marriage contracts to receive worth of 100, 200, and 300 Husband dies and left a worth of 100

8. History Continued Big Problem with marriage contract If worth is 300, Talmud recommends a division of 50, 100, and 150. If worth is 200, Talmud recommends a division of 50, 75, and 75. In 1985, Talmud had modern theory of cooperative games

9. St. Petersburg Paradox In 1738, Daniel Bernoulli published a paper discussing this paradox Paradox: Whether or not to play in the St. Petersburg game Game: Flip a coin till it lands heads, once land heads, player cashes out with number of flips Cash out ?

10. Expected Utility Theorem In 1944, John Von Neumann and Oskar Morgenstern developed a variation of St. Petersburg Paradox Mapped out different ventures over different prospects called lotteries A player gets a lottery ticket that gives out lottery tickets for prizes

11. Expected Utility Theorem Con’t

12. Expected Utility Theorem Con’t

13. Kinds of Combinatorial Games Impartial Games -every player has the same possible moves in any position of the game Partisan Games - each player has a different set of moves in any given position of a game.

14. Combinatorial Games: Nim Nim - an impartial game Rules: Three piles of sticks No limit to how much to take Can not remove from more that one pile at a time Player that cannot move loses

15. Nim Continued Winning Strategy Use the XOR function If next player, make sure that end result of XOR function is not 0’s Leave one row odd

16. On Numbers and Games Colin Vout introduces Col Simon Norton comes up Snort Both games require a map to drawn with two markers to mark territories. Player that cannot move, loses Col: can not have a common border Snort: only edges can touch

17. Col Snort

18. Squaring the Numbers Impartial game similar to Nim Rules Start off with a large pile of sticks Pile can only be subtracted from a square number Player unable to move loses To win: map out the game

19. Fibonacci Nim A partisan game that requires math formula Rules Object to remove sticks from pile Player unable to remove, loses First player can remove any number except all sticks and second player and remove twice as much

20. Fibonacci Nim Continued

21. Economics See the bigger picture and look for alternatives Allows businesses to play and strategize and view what other businesses Strategize are based upon best response to environments

22. Economics Continued John Nash Forbes Jr. introduces the Nash Equilibrium and received the Noble Prize in 1994. “A ‘Nash Equilibrium’ will be reached when each agent's actions begets a reaction by all the other agents which, in turn, begets the same initial action. In other words, the best responses of all players are in accordance with each other” (http://cepa.newschool.edu/het/schools/game.htm).

23. Economics Continued Two types of Combinatorial Games Cooperative Non-Cooperative Ultimate Goal : Maximize Profits Manipulate the game anyway possible!

24. Bar Competition Binary Bar vs. Full CPU (aka Full CUP) Price Drinks: $2, $3, $4 6000 tourist randomly chooses bar 4000 locals go the best deal Example: Binary Bar charges $2 Binary Bar receives 7000 tourists Full CPU charges $3 Full CPU receives 3000 tourists

25. Bar Competition Continued

26. Future Logic games like Minesweeper and Tetris are teaching tools for applying logic Useful for proving Math Theorems Maps out possible positions and know what to expect

27. Conclusion Maps out various positions Maximize one’s strategy Practices logic to think abstractly about a scenario A better understanding on how the market and businesses interact Lack of understanding player’s move without it!!!!!!

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