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Game Theory and Math Modeling Preparing Women for Math Modeling Marie Vanisko email@example.com@carroll.edu Montana Learning Center, 15 June 2010
Game Theory The Mathematics of Competition The players (two or more) can be people, organizations, or even countries. They choose from a list of available options called strategies. The strategies chosen by the players lead to outcomes with different preferences. Game theory analyzes the rational choice of strategies in order to achieve the desired outcome(s).
Early History of Game Theory John von Neumann In game theory von Neumann proved the minimax theorem. He gradually expanded his work in game theory, and with co-author Morgenstern, he wrote the classic text Theory of Games and Economic Behaviour (1944).
Two-Person Total-Conflict Games Total conflict games, often called zero- sum games, are ones in which one person’s gain is another person’s loss. Games with pure strategies have a clear “best” strategy for each player In some games, using the same strategy all the time is not good, so a system of mixed strategies with associated probabilities is used.
A Location Game with Pure Strategies Henry and Lisa plan to locate a new restaurant. Henry prefers a high elevation, while Lisa prefers locating at a low elevation. They have agreed to locate at the intersection of a highway and a route. Henry will select a route and simultaneously, Lisa will select a highway. Elevations are given → Hts. 000’s ft Hiwy 1 Hiwy 2 Hiwy 3 Rte 1104 6 Rte 265 9 Rte 323 7
Henry selects a row, Lisa a column Hts. 000’s ft Hiwy 1 Hiwy 2 Hiwy 3 Rte 1104 6 Rte 265 9 Rte 323 7 Suppose Henry selects whichever row has the highest minimum ht. (maximin strategy) Suppose Lisa selects the column with the lowest maximum ht. (minimax strategy) If row minimum and column maximum are the same, the result is the game’s value and called a saddlepoint.
A Practice Problem – Pure Strategy Charlie Brown will buy a toy for Snoopy. The costs are given. Charlie will select a row and simultaneously Snoopy will select a column. Is there a saddle point? Might Charlie use a different strategy? What if Snoopy selected a column and Charlie selected a row? Toy Cost Col. 1 Col. 2 Col. 3 Row 1372 Row 2 851 Row 3 694
Games Requiring Mixed Strategies In many sporting events, a team tries to surprise or mislead the opposition. A pitcher in baseball will vary the type of pitch he or she throws throughout the game to keep the batter off balance.
Duel Between Pitcher and Batter Assume that the baseball pitcher can throw two types of pitches, either a blazing fast ball (F) or a slow curve ball (C) into the strike zone. The pitcher faces a batter who attempts to guess whether F or C will be thrown at each pitch. Assume the batter has the following batting averages, depending on what the pitcher does - Batter anticipates F: 0.300 if pitch is F and 0.200 if pitch is C - Batter anticipates C: 0.100 if pitch is F and 0.500 if pitch is C
Enter the Linear Equations Let p be the probability that the pitcher will throw a fast (F) ball p: 0-----------------------------------------------1 Batting average of 0.100 Batting average of 0.500 Batting average of 0.200 Batting average of 0.300 Batting average of 0.400 Upper line is when batter anticipates C; lower line when batter anticipates F
Equations of the Lines for Pitcher Line “batter expects curve” passes through (0, 0.5) and (1, 0.1): A = -0.4 p + 0.5 Line “batter expects fast” passes through (0, 0.2) and (1, 0.3): A = 0.1 p + 0.2 The point of intersection of these lines will be the maximin for the pitcher, p = 0.6, meaning that the pitcher should pitch fast balls 60% of the time, yielding a 0.26 average for the batter.
Equations for the Batter How can the batter optimize his or her chances? Let q be the probability that the batter anticipates F. q:0-----------------------------------------------1 Batting average of 0.100 Batting average of 0.500 Batting average of 0.200 Batting average of 0.300 Batting average of 0.400 Upper line is when pitcher throws C; lower line when pitcher throws F
Equations of the Lines for Batter Line “pitcher throws curve” passes through (0, 0.5) and (1, 0.2): A = -0.3 q + 0.5 Line “pitcher throws fast” passes through (0, 0.1) and (1, 0.3): A = 0.2 q + 0.1 The point of intersection of these lines will be the maximin for the pitcher, q = 0.8, meaning that the batter should anticipate fast balls 80% of the time, yielding a 0.26 average for the batter – the value of the game.
Mixed Strategies in a Football Game Suppose it is third down and short yardage to go for a first down. The offense can decide to run or pass. The defense can commit itself to defend more heavily against a run or a pass. The optimal strategies for each can be determined. Probability of a First Down Defense expects offense to Run Defense expects offense to Pass Offense decides to Run 0.50.8 Offense decides to Pass 0.70.2
Mixed Strategies in a Business Model Suppose you plan to manufacture a new product for sale next year. You can decide to make a small quantity, in anticipation of a poor economy and few sales, or a large quantity, hoping for brisk sales. If you want to avoid risk and believe that the economy is playing an optimal mixed strategy zero-sum game, what is your optimal strategy and the resulting expected value? Expected Monetary Gain Economy is PoorEconomy is Good Small Quantity Produced $500,000$300,000 Large Quantity Produced $100,000$900,000
Partial Conflict Games The Prisoner’s Dilemma Games of partial conflict are variable-sum games, in which the sum of payoffs to the players at the different outcomes varies. Consider the prisoner’s dilemma where two persons accused of a crime are being held incommunicado. Each has two choices: to maintain his or her innocence, or to sign a confession accusing the partner of committing the crime. The best outcome for both might be when neither confesses, whereas the worst would be when both confess. If only one confesses, the other is worse off.
The Arms Race as a Partial-Conflict Game Assume there are two nations, Red and Blue. Each can arm or disarm independently. Arming is done in preparation for possible war (non-cooperation). Disarming is done with negotiation or arms agreements (cooperation). Outcomes (Red, Blue) Blue Arms Blue Disarms Red Arms (2, 2)(4, 1) Red Disarms (1, 4)(3, 3)
The Nash Equilibrium When no player can benefit by departing unilaterally (by itself) from its strategy associated with an outcome, the strategies of the players, constitute a Nash equilibrium. Consider why the (Arm, Arm) position is a stable Nash equilibrium, while the (Disarm, Disarm) position, even though it is better for both, is an unstable equilibrium
John Nash and the Nash Equilibrium When in 1959, the 21-year old John Nash wrote his 27-page dissertation outlining his "Nash Equilibrium" for strategic non-cooperative games, the impact was enormous. It reflected his methodological call for the reduction of all cooperative games into a non- cooperative framework.
Interesting Websites on Game Theory http://mathforum.org/library/topics/game_theory/ http://www.math.ucla.edu/~tom/Game_Theory/Contents.html http://www.gametheory.net/ http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/tcfinitep. html http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/tcfinitep. html http://www.chass.utoronto.ca/~osborne/ http://kuznets.fas.harvard.edu/~aroth/alroth.html The applications we looked at are very simplified, but they can open the door to a fascinating journey into an exciting field of applied mathematics.
Interesting Websites for Math Modelers COMAP Hi MCM: http://www.comap.com/highschool/http://www.comap.com/highschool/ Census Bureau – general: http://www.census.gov/http://www.census.gov/ Wolfram Alpha to answer your questions: http://www.wolframalpha.com/http://www.wolframalpha.com/ Live traffic data from California: http://www.dot.ca.govhttp://www.dot.ca.gov (San Diego site works best.) Temperature Data: http://www.usatoday.com/weather/waverage.htmhttp://www.usatoday.com/weather/waverage.htm Drug Dosage Information: http://www.infomed.org/100drugshttp://www.infomed.org/100drugs Population Information: http://www.census.gov/ipc/www/idbnew.html http://www.census.gov/ipc/www/idbnew.html Population Pyramids: http://www.census.gov/ipc/www/idbpyr.htmlhttp://www.census.gov/ipc/www/idbpyr.html Center for Disease Control: http://www.cdc.govhttp://www.cdc.gov World Health Org Statistical Information System: http://www3.who.int/whosis/menu.cfm http://www3.who.int/whosis/menu.cfm U. S. Geological Survey: http://www.usgs.govhttp://www.usgs.gov NASA: http://www.nasa.govhttp://www.nasa.gov