Week 8: Basic concepts in game theory Part 1: Examples of games We introduce here the basic objects involved in game theory. To specify a game ones gives The players. The set of all possible...

Week 8: Basic concepts in game theory Part 1: Examples of games We introduce here the basic objects involved in game theory. To specify a game ones gives The players. The set of all possible strategies for each player. The payos: if each player Week 8: Basic concepts in game theoryPart 1: Examples of gamesWe introduce here the basic objects involved in game theory. To specify a game onesgives•Theplayers.•The set of all possiblestrategiesfor each player.•Thepayoffs: if each player picks a certain strategy then each player receive a payoffrepresented by a number. The payoff for every player depends, in general, fromthe strategies of all other players. Payoffs have many different meanings, e.g., anamount of money, a number of years of happiness, the fitness in biology, etc... Ourconvention is that the highest payoff is deemed the most desirable one.In the rest of this section we introduce some of the standard games in game theory,together with the ”story behind them”The prisoner’s dilemma:. This is one of the most famous game in game theory. Onepossible story associated to it goes as follows. A prosecutor seeks two arrest two bankrobbers. He lacks proofs of their involvement in the heist but has managed to have themarrested for a minor fraud charge. He offers to each them the following choices. If youconfess but your accomplice does not then you will go free and your accomplice will bepunished 8 years in jail. If you both confess then you will each get 6 years in jail. Finally,if none confess then they both will convicted for the minor fraud charge, say for 1 year inprison.We will represent payoffs and strategies using a table. We will call the player Robert(R is for the ”row” player) and Collin (C is for ”column” player). For both Robert andCollin the strategies are confess or not confess.RobertCollinconfessnot confessconfess-6-6-80not confess-6-8-1-11 The rows represents the strategies of Robert, the columns the strategies of Collin. Thenumerical entries are understood as follows: in each box the lower left entry is the payofffor Robert while the the upper right entry is the payoff for Collin.Solution of the prisoner’s dilemma:To see what is the best option for the prisoner’sdilemma one observes that if your accomplice confess then it is better for you to confess(8 years in jails) rather than not confessing (10 years in jail). On the other hand if youraccomplice does not confess it is better for you to confess (0 years in jails) rather than notconfessing (1 years in jail). Therefore the outcome of the game is that rational playerswill both confess and end in jail for 8 years. We will see later that the pairs of strategies”confess” and ”confess” is an equilibrium for this game. The ”dilemma” here is that ifboth do not confess they would get both 1 year in jail instead of 8 years, a much morepreferable outcome for both! However without communication between the players, thereis no mechanism to enforce this.Duopoly:A strategic situation similar to the prisoner’s dilemma occurs in many differentcontexts. Imagine for example two countries who produce oil and each can choose betweenproducing 2 millions barrels/day or 4 millions barrels/day. The total output will be then4, 6, or 8 millions barrels/day and the corresponding price will be $20, $12, $7 per barrel( a decreasing price reflects basic law of supply and demand). The payoff table is givenbyCountry ACountry B2 million4 million2 million404048244 million24482828Analyzing the game as in the prisoner’s dilemma one finds that the best strategy is toproduce $4 million per day because it leads to a higher revenue no matter what the othercountry does. This is of course in many ways a socially bad outcome since it leads towasting a precious resource and lower revenue for the oil producing countries.Battle of the sexes:Robert and Chelsea are planning an event of entertainment. Aboveall they value spending time together but Robert likes to go to the game while Chelseaprefers to go to the ballet. They both need to decide what to do tonight without commu-nicating with each other. To produce a numerical outcome we assume that the value 1 isgiven to having your favorite entertainment while a value 2 is assigned to being together.This leads to2