Game Theory

1. Game Theory Asia Burrill, Marc Relford, Bridgette Mallet

2. What is Game Theory? Game theory is a mathematical approach to capture behavior in strategic situations, primarily focused on an individual’s success against the choices of others.

3. Major Types of Game Theory • Ultimatum Game • Prisoner’s Dilemma • Dictator Game • Zero-sum Games • Also other lesser known games such as Volunteer’s Dilemma, Sequential game, and also Variable sum game.

4. Game theory is formally represented in a Prisoner’s Dilemma, as represented below: Prisoner’s Dilemma is the most basic and easily explained of game theory. It is a situation that involves two people, both being interrogated for crimes in separate rooms. Both have the option to plead guilty or not guilty, with each suffering different sentences. Both prisoners are affected by the other’s decision, yet both of their futures lie in each other’s decisions. The best choice for Dave in this diagram is to plead guilty if Henry pleads not guilty. The opposite is to not plead guilty and if Henry pleads guilty Dave is subject to five years. In each situation, both people are subject to longer sentences than each other. The only choice that would grant them the least amount of time each is for each of them to plead not guilty. As each person can benefit from pleading opposite, they both have the chance of spending more time in jail for pleading guilty together. Therefore, each prisoner is left to plead his own strategy, with no knowledge of the other’s decision. Prisoner’s Dilemma

5. Prisoner’s Dilemma: Real Life Applications • Prisoner’s dilemma is frequently used to determine advertising expenditures within a given market. For example, cigarette companies have the option of advertising during a given season. Cigarette company A has seen that by advertising in the Spring, they would see the biggest audiences, but they would have the most competition during that time. Cigarette company B is the in the same position, with their decision affecting the amount of people they can sell to during that particular time. There are other types of Game theory however, the next being discussed on the following slide.

6. Mixed Strategy Game Theory • This game is effective when two players are competing using a variety of pure strategies, especially when the opponent’s knowledge of your next move is crucial to the outcome. This game is most useful when the opposing players’ decisions are the primary influence for your move, usually coupled with a payout matrix.

7. Nash Equilibrium • Nash Equilibrium arises when players are pursuing their best strategy in response to the best-reply strategy of the other player. This is useful in games that involve dominant strategy equilibrium and iterated strategy equilibrium. The situations that illustrate Nash equilibrium in real life are often competitors in the same market the depend their strategy on what the other competitor chooses to do.

8. Continuous Strategy Game Theory • These are strategies that relate to a continuous variable rather than a discrete one. Examples of this with regards to economics could be the price and quantity of output. This can relate to the Cournot case, where moves are simultaneous.

9. Cournot Oligopoly • Cournot Oligopoly considers a market in which there are only two firms, A and B. • The Cournot model is based on the following assumptions: • There are few firms in the market and many buyers • The firms produce homogeneous products; therefore each firm has to charge the same market price. • Competition is in the form of output, meaning that each firm determines its level of output based on its estimate of the level of output of the other firm. • Barriers to entry exist. • Each firm aims to maximize profit, and assumes that the other firms do the same.

10. Math of Cournot Oligopoly • An example of Cournot oligopoly involves two firms. In order to get the proper strategy, both companies must know market demand. Next, the market demand must be transformed into a demand function that relates the outputs of each of the two firms. As that is finished, the next step is to derive the profit functions of the outputs of each company. The last two steps that derive the Cournot equilibrium are to derive the optimal output for Company A as a function of the output of Company B. Then, after solving the equations for the best response functions simultaneously, we have now reached Cournot equilibrium.

11. Conclusion • What we have found is that game theory is extremely effective in determining everything from advertising budgets and expenditures to the best strategy for conducting business within a market. The mathematical transitivity into practical applications makes this theory very important to businesses and mathematicians when prompted with these situations.