Game Theory Slides (EC-2021 Week 1) PDF

Topic 1: Game Theory Static Games of Complete Information Players decide upon their actions at the same time (simultaneously) the benefits from choosing certain actions are known to all players Prisoner’s Dilemma Two suspects are charged with a crime and have the choice to either...

Topic 1: Game Theory Static Games of Complete Information Players decide upon their actions at the same time (simultaneously) the benefits from choosing certain actions are known to all players Prisoner’s Dilemma Two suspects are charged with a crime and have the choice to either Confess or Deny if neither Confesses they are both sentenced to 1 month in jail if both Confess they are both sentenced to 6 months in jail if one suspect Confesses but the other Denies: the suspect who Confesses is released Prisoner’s Dilemma in Normal Form Representation The setup of a game: the players involved the strategies available to each player the payoff received by each player Prisoner’s Dilemma in Normal Form Representation What would you choose to do? Equilibrium and Dominant Strategies A rational player will not play a strictly dominated strategy regardless of the strategy chosen by Suspect 2, Suspect 1 will always find it optimal to Confess Equilibrium and Dominant Strategies The strategy Deny is strictly dominated by the strategy Confess The flipside of this: Confess is the dominant strategy (the strategy that is the best response to all possible strategies of the other player) Equilibrium and Dominant Strategies We end up with the situation where both suspects choose to Confess note that this outcome is inferior to the one in which both Deny Equilibrium and Dominant Strategies There are no strictly dominated strategies to be eliminated and hence all outcomes are possible equilibria Best responses In game theory… a rational player will want to choose her best response given her beliefs of what the other players will choose Best responses What is the best response of a player for each possible strategy of the other player? Nash Equilibrium A Nash equilibrium is a situation where the chosen strategy of all players is a best response to the best responses of all the other players Strategy pairs that do not constitute a Nash equilibrium are not plausible outcomes Nash Equilibrium The Nash equilibrium is here A Cooperation Game Two students are working on a joint project each student may either Work Hard or Shirk notice the similarity of the (Shirk,Skirk) NE with the NE of the Battle of the Sexes / Bach or Stravinsky A Man and a Woman have a date problem: they forgot where they planned to go to (even worse: they also forgot their mobiles) they remember that it was either the Opera or a Prize Fight Battle of the Sexes / Bach or Stravinsky Both the Man and the Woman prefer to spend the evening together: the Man prefers the evening together at a Prize Fight the Woman prefers the evening together at the Opera Battle of the Sexes / Bach or Stravinsky Two Nash equilibria in this game: (Opera,Opera) is better for the Woman and (Fight,Fight) is better for the Man Players agree that it is better to co- operate, but they disagree about the best outcome Stag Hunt A group of hunters are involved in the pursuit of a stag if all hunters remain attentive, they will successfully catch the stag each hunter is tempted to devote his energy towards catching a rabbit each hunter would prefer his share of the stag as opposed to a rabbit the worst possible outcome is where a hunter is left empty-handed Stag Hunt We have two Nash equilibria: (Stag, Stag) - one where both hunters remain attentive (Rabbit, Rabbit) - the other where both hunters pursue rabbits Stag Hunt (Stag,Stag) suggests itself as a more plausible outcome: it is a focal point Focal Points Focal points are strategy combinations that stand out in some way around which players can co- ordinate More on Focal Points think back to the Battle of the Sexes – what would happen if the Woman was the dominant personality? isn’t it more likely that we would end up at her most preferred outcome? Strict and Non-Strict Equilibria All the equilibria we have identified thus far have been strict Nash equilibria – a deviation has meant a player will be worse off but the notion of a Nash equilibrium only requires that: “the outcome of a deviation be no better for the deviating player than the equilibrium outcome” Strict and Non-Strict Equilibria there is a unique non-strict Nash equilibrium of (Up, Left)