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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!!!!!!