Games, Logic, and Math Kristy and Dan

GAMES

Game Theory Applies to social science Applies to social science Explains how people make all sorts of decisions Explains how people make all sorts of decisions Economic choices, animal life, and ethical behavior Economic choices, animal life, and ethical behavior Game continues based off of contestant’s risk- adversity Game continues based off of contestant’s risk- adversity Risk-averse Risk-averse Most people Most people This is what the game thinks people are This is what the game thinks people are Risk-neutral Risk-neutral Risk-loving Risk-loving

Deal or No Deal GAMES

How the Game Works Contestants are presented with 26 briefcases with amounts of money in them Contestants are presented with 26 briefcases with amounts of money in them They keep one briefcase that they are trying to figure out how much money there is in it They keep one briefcase that they are trying to figure out how much money there is in it They eventually sell or keep the money in the briefcase They eventually sell or keep the money in the briefcase Amounts start at one cent to a million dollars Amounts start at one cent to a million dollars Want to eliminate the lower numbers Want to eliminate the lower numbers Pick 7 briefcases Pick 7 briefcases Either take the banker’s offer or continue picking Either take the banker’s offer or continue picking Banker’s offer: bigger when they pick lower numbers and smaller when they get higher numbers Banker’s offer: bigger when they pick lower numbers and smaller when they get higher numbers

Example We have two briefcases left We have two briefcases left $5,000 $5,000 $75,000 $75,000 The offer from the bank is $50,000 The offer from the bank is $50,000

Answer Calculate the expected outcome Calculate the expected outcome is the sample probability space and X is the expected value is the sample probability space and X is the expected value This is much like the process of rolling a die and you want to find the expected answer:

Answer cont. The expected value for this example = $40,000 The expected value for this example = $40,000 The offer from the banker ($50,000) is $10,000 larger than the expected value The offer from the banker ($50,000) is $10,000 larger than the expected value SO YOU WOULD TAKE THE OFFER SO YOU WOULD TAKE THE OFFER But this contestant didn’t… But this contestant didn’t…

LOGIC

LOGIC Monty Hall Paradox

A player is shown three doors- behind two of them are goats, the third a car A player is shown three doors- behind two of them are goats, the third a car After selecting one door another door is revealed containing a goat After selecting one door another door is revealed containing a goat The player is then asked if he or she would like to switch doors or stay with the original selection The player is then asked if he or she would like to switch doors or stay with the original selection

The Math Behind the Paradox By switching doors the probability of selecting the car increases from one third to two thirds By switching doors the probability of selecting the car increases from one third to two thirds This is because the initial probability of choosing a goat is 2/3. Unless the player chooses the car (1/3 chance) switching will increase the chance of selecting the car. This is because the initial probability of choosing a goat is 2/3. Unless the player chooses the car (1/3 chance) switching will increase the chance of selecting the car.

Player initially picks Door 1 Car hidden behind Door 3 Car hidden behind Door 1 Car hidden behind Door 2 Host must open Door 2 Host opens either goat door Host must open Door 3 Probability 1/3 Probability 1/6 Probability 1/3 Switching wins Switching loses Switching wins In the problem as stated, these cases did not happen If the host has opened Door 3, switching wins twice as often as staying

LOGIC Birthday Problem

Go around and say birthdates and record numbers of doubles Go around and say birthdates and record numbers of doubles

Birthday problem logic You would think that once “n” number of birthdays are stated, that there would be 365-n/365 probability of having the same birthday You would think that once “n” number of birthdays are stated, that there would be 365-n/365 probability of having the same birthday With 23 people in a room, there is a 50% chance of two people having the same birthday With 23 people in a room, there is a 50% chance of two people having the same birthday P( at lease one birthday match)= 1-(364/365 x 363/365 x …) = > 50% P( at lease one birthday match)= 1-(364/365 x 363/365 x …) = > 50%