1. Game Theory

2. Your Task  Split into pairs and label one person A and the other B.  You are theoretically about to play a game involving money. Player A has two strategies of playing this game and player B has three strategies. The matrix on the next slide shows the possible winnings for A depending on the strategies chosen by each player. This is called a pay-off matrix.

3. Your Task Work in pairs to decide: 1) Which strategies give A the most amount of money? 2) Which strategies give B the most amount of money? 3) Which strategy allows for one person to lose the same amount of money that the other person wins? A B P Q XY Z 1 3 -2 2 2 4

4. Play-Safe Strategy  To find a play-safe strategy, we look for the worst possible outcome for each option, and then choose the option in which the worst possible outcome is least bad. A B P Q XY Z 1 3 -2 2 2 4 Minimum Outcome Maximum Outcome -2 2 324 Why?

5. Stable Solution  If neither player can improve their strategy if the other plays safe, the game has a stable solution.  Does our game have a stable solution? A B P Q XY Z 1 3 -2 2 2 4 Minimum Outcome Maximum Outcome -2 2 324 This is hence called the value of the game. The position of this point is called the saddle point.

6. Optimal Mixed Strategies  In many two-person zero-sum games, there is no stable solution. The optimal strategy is therefore found by using two or more options with a fixed probability of choosing each option. This is called a mixed strategy.  We will look at games with a 2x2 payoff matrix.

7. Your Task  Look at the example on your hand-out of how to work out the value of the game and optimal mixed strategy of both players for a 2x2 pay-off matrix.  Read through and understand the example, then apply your knowledge to the following question, in the same way, to find a solution! Your Question: A two-person zero-sum game has the following 2x2 pay-off matrix: Find the value of the game and the optimal mixed strategy of both players. A B P Q X Y -34 2 1

8. The Solution Let A choose option P with probability p and option Q with probability 1-p. If B chooses option X, expected pay-off for A = -3p + 2(1 - p) = 2 - 5p If B chooses option Y, expected pay-off for A = 4p + 1(1 – p) = 3p + 1 Equating the expected pay-offs we have 2 – 5p = 3p + 1 1 = 8p p = 1/8

9. The Solution A should choose option P with probability 1/8 and option Q with probability 7/8. Value of game = 2 – 5p = 2 – (5 x 1/8) = 11/8 Let B choose option X with probability q and option Y with probability 1 – q If A chooses option P, the expected pay-off for B = -3q + 4(1 – q) = 4 – 7q Equating this to the value of the game, we get: 4 – 7q = 11/8 q = 3/8 B should therefore choose option X with probability 3/8 and option Y with probability 5/8.

10. Game Theory in Real Life Bruce Mesquita (a political scientist) correctly predicted the fall of Egypt’s president, Hosni Mubarak and Pakistan’s president, Pervez Musharraf. He made these predictions by evaluating the motives of the players involved and using the idea of game theory in terms of a pay-off matrix. He then used a computer to predict how the “game” would be played. “Paul Migrom (a consultant) uses game-theory software to help companies win auctions cheaply. In one auction, his software recognized that large bundles of goods were being valued more than their component parts. He used this information to save two of his clients $1.2 billion. Auction software also has important applications on a smaller scale, to help users make extra money from online auction sites such as eBay.” “Mediation software is under development to help resolve conflict effectively. Each party provides the computer with their secret, bottom line information and the computer proposes an optimal agreement. One day, similar software may be used to resolve wars without fighting. Using secret information about the military capabilities and motives of both sides, the software may be able to predict the outcome a military conflict and broker an agreement without bloodshed.” Quotes from http://blogs.cornell.edu/info2040/2011/09/21/solving-real- life-problems-with-game-theory/http://blogs.cornell.edu/info2040/2011/09/21/solving-real- life-problems-with-game-theory/ Can you think of any more examples?

11. Evaluation 1) What do you feel are the advantages of Game Theory? 2) What is difficult about the idea of Game Theory? 3) How do you think the work we have done today could be made more difficult? 4) What have you learnt from today’s lesson?

12. A Well-Known Example Please click the following link to see the YouTube video: http://www.youtube.com/watch?v=iZKErrv VMaY&feature=player_detailpage