The Theory of Games By Tara Johnson, Lisa Craig and Amanda Parlin

What is Game Theory The discipline got its name with the 1944 book Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern and was worked with heavily in the 1950’s. In the 1970’s it was used in the field of Biology (1)Theory of Games and Economic BehaviorJohn von Neumann Oskar Morgenstern Game Theory is a type of math that has been used in economics, engineering, business and political science just to name a few. (1) Game theory uses math to collect data off of behaviors in strategy driven situations, in which one persons successful choice depends on the choice of another player. (1)

Games and Strategies The Question of Game Theory—How should I move to maximize my gain? Pure Strategy is characterized by 1 play used over and over Mixed strategy is characterized by making random moves based on select probabilities for each move. These games are determined in that we KNOW what the scenarios and payoff’s are before we make our choice. We take into account what my opponent may do, not fully knowing what they will do. Next we are going to show you an example of this theory…

Example of Game Strategy “The Shopping Ladies” Two ladies, Betty and Jo are shopping for cheap goods to sell on EBay for a potential profit There are 2 stores that sell 1 item a day The objective is to get to one of the stores before the other person. If they choose the store first, they are in a positive profit, and if they don’t choose the store first, they will be in a negative profit. Store 1 has a payoff of 5 or -5 if it has previously been picked Store 2 has a payoff of 3 or -3 if it has previously been picked

The Payoff Matrix BETTY S1S2 JS1 5-3 OS2-5 3 If Jo goes to store 1 and Betty also goes to store 1, Jo will be up 5, Betty will be down -5 If Betty goes to store 2 and Jo goes to store 2, Betty will be down -3 and Jo will be up 3 If Jo goes to store 2 and Betty goes to store 1, Jo will be down -5 and Betty will be up 5 If Jo goes to store 1 and Betty goes to store 2, Betty will be up 3 and Jo will be down -3

Pure Strategy The Payoff Matrix Betty 1 st 2 nd J1 st _5 -3 O2 nd -5 3 To Maximize her Payoff using Pure Strategy, Betty would identify the largest payoff in each row. In this case, row 1’s choice would be 5 and row two would be 3. Then choose the best choice. Knowing that Jo will want to maximize her gain—the best choice for Betty might be to pick Row 2 with a payoff of 3 rather than risk going for the 5 and losing 3. (3)

Mixed Strategies Mixed Strategies is opposite from Pure Strategy in that instead of one strategy used over and over, you “mix it up” and use various strategies to obtain a chance at better results The outcomes are not always known or determined You play these games over and over again with many different outcomes

Step oneThe Payoff Matrix [A] Betty 1 st 2 nd J1st 5-3 O2nd-5 3 Step two Jo’s mixed strategy[B]Betty’s mixed strategy [C] [.75.25][.5.5] [.75.25] are represented that Jo will pick Store #1 - 75% of the time and will pick store #2 - 25% of the time. [.5.5] are represented that Betty will pick either store 50% of the time. Step 3 is to multiply Jo’s Matrix probability to the Payoff Matrix and then multiply by Betty’s Matrix this will give you the Expected Value for the two of them. OutcomeWinnerProbability [B] * [A] * [C] = EVRow 1, Col 1 5(.75)(.5) = [.75.25] * [5 -3 ] * [.5 OR Row 1, Col 2-3(.75)(.5) = [-5 3].5] =.5 Row 2, Col 1-5(.25)(.5) =-.625 Row 2, Col 2 3(.25)(.5) =.375 Step 4 to show the EV for the pair by adding the probability column together for total EV Answer: This Expected value for this is based on one play and is valued at 50 cents The Expected Value Of Shopping Using Mixed Strategies

The Payoff Matrix Betty 1 st 2 nd J1 st 5 -3 O2 nd -5 3 For every choice Jo may make, there is a counterstrategy that Betty might make The Optimized Mixed Strategy for Jo is where Her Expected Value against Betty’s Counter strategy is the highest(3) The Optimized Mixed Strategy for Betty is where Her Expected Value against Jo’s counterstrategy is the smallest(3) We would use a linear programming program to define the “optimal” move to make Optimal Mixed Strategy

References 1.Game Theory 2. Linear Programming Applet 3.The Theory of Games Handout Chapter 9