Lecture 19: Infinitely Repeated Games Mauricio Romero Lecture 19: Infinitely Repeated Games Introduction to Infinitely Repeated Games Lecture 19: Infinitely Repeated Games Introduction to Infinitely Repeated Games I This happened because there was a last period from which we could induct backwards (and there was a domino effect!) I When the game is instead infinitely repeated, this argument no longer applies since there is no such thing as a last period I One of the features of finitely repeated games was that if the stage game had a unique Nash equilibrium, then the only subgame perfect Nash equilibrium was the repetition of that unique stage game Nash equilibrium I When the game is instead infinitely repeated, this argument no longer applies since there is no such thing as a last period I One of the features of finitely repeated games was that if the stage game had a unique Nash equilibrium, then the only subgame perfect Nash equilibrium was the repetition of that unique stage game Nash equilibrium I This happened because there was a last period from which we could induct backwards (and there was a domino effect!) I One of the features of finitely repeated games was that if the stage game had a unique Nash equilibrium, then the only subgame perfect Nash equilibrium was the repetition of that unique stage game Nash equilibrium I This happened because there was a last period from which we could induct backwards (and there was a domino effect!) I When the game is instead infinitely repeated, this argument no longer applies since there is no such thing as a last period I We start with a stage game whose utilities are given by u1; u2;:::; un I Each player i has an action set Ai I In each period t = 0; 1; 2;:::, players simultaneously choose an action ai 2 Ai and the chosen action profile (a1; a2;:::; an) is observed by all players I Then play moves to period t + 1 and the game continues in the same manner. I Lets first define what an infinitely repeated game is I Each player i has an action set Ai I In each period t = 0; 1; 2;:::, players simultaneously choose an action ai 2 Ai and the chosen action profile (a1; a2;:::; an) is observed by all players I Then play moves to period t + 1 and the game continues in the same manner. I Lets first define what an infinitely repeated game is I We start with a stage game whose utilities are given by u1; u2;:::; un I In each period t = 0; 1; 2;:::, players simultaneously choose an action ai 2 Ai and the chosen action profile (a1; a2;:::; an) is observed by all players I Then play moves to period t + 1 and the game continues in the same manner. I Lets first define what an infinitely repeated game is I We start with a stage game whose utilities are given by u1; u2;:::; un I Each player i has an action set Ai I Then play moves to period t + 1 and the game continues in the same manner. I Lets first define what an infinitely repeated game is I We start with a stage game whose utilities are given by u1; u2;:::; un I Each player i has an action set Ai I In each period t = 0; 1; 2;:::, players simultaneously choose an action ai 2 Ai and the chosen action profile (a1; a2;:::; an) is observed by all players I Lets first define what an infinitely repeated game is I We start with a stage game whose utilities are given by u1; u2;:::; un I Each player i has an action set Ai I In each period t = 0; 1; 2;:::, players simultaneously choose an action ai 2 Ai and the chosen action profile (a1; a2;:::; an) is observed by all players I Then play moves to period t + 1 and the game continues in the same manner. I Each information set of each player i associated with a finitely repeated game corresponded to a history of action profiles chosen in the past I We can represent each information set of player i by a history: 0 1 0 0 0 t 0 1 t−1 h = (;); h = (a := (a1;:::; an));:::; h = (a ; a ;:::; a ) I We denote the set of all histories at time t as Ht I It is impossible to draw the extensive form of this infinitely repeated game I We can represent each information set of player i by a history: 0 1 0 0 0 t 0 1 t−1 h = (;); h = (a := (a1;:::; an));:::; h = (a ; a ;:::; a ) I We denote the set of all histories at time t as Ht I It is impossible to draw the extensive form of this infinitely repeated game I Each information set of each player i associated with a finitely repeated game corresponded to a history of action profiles chosen in the past I We denote the set of all histories at time t as Ht I It is impossible to draw the extensive form of this infinitely repeated game I Each information set of each player i associated with a finitely repeated game corresponded to a history of action profiles chosen in the past I We can represent each information set of player i by a history: 0 1 0 0 0 t 0 1 t−1 h = (;); h = (a := (a1;:::; an));:::; h = (a ; a ;:::; a ) I It is impossible to draw the extensive form of this infinitely repeated game I Each information set of each player i associated with a finitely repeated game corresponded to a history of action profiles chosen in the past I We can represent each information set of player i by a history: 0 1 0 0 0 t 0 1 t−1 h = (;); h = (a := (a1;:::; an));:::; h = (a ; a ;:::; a ) I We denote the set of all histories at time t as Ht Prisoner's Dilemma C2 D2 C1 1; 1 −1; 2 D1 2; −1 0; 0 I For time t, Ht consists of 4t possible histories I This means that there is a one-to-one mapping between all possible histories and the information sets if we actually wrote out the whole extensive form game tree I As a result, we can think of each ht 2 Ht as representing a particular information set for each player i in each time t I For example, if the stage game is the prisoner's dilemma, at period 1, there are 4 possible histories: 1 f(C1; C2); (C1; D2); (D1; C2); (D1; D2)g = H : I This means that there is a one-to-one mapping between all possible histories and the information sets if we actually wrote out the whole extensive form game tree I As a result, we can think of each ht 2 Ht as representing a particular information set for each player i in each time t I For example, if the stage game is the prisoner's dilemma, at period 1, there are 4 possible histories: 1 f(C1; C2); (C1; D2); (D1; C2); (D1; D2)g = H : I For time t, Ht consists of 4t possible histories I As a result, we can think of each ht 2 Ht as representing a particular information set for each player i in each time t I For example, if the stage game is the prisoner's dilemma, at period 1, there are 4 possible histories: 1 f(C1; C2); (C1; D2); (D1; C2); (D1; D2)g = H : I For time t, Ht consists of 4t possible histories I This means that there is a one-to-one mapping between all possible histories and the information sets if we actually wrote out the whole extensive form game tree I For example, if the stage game is the prisoner's dilemma, at period 1, there are 4 possible histories: 1 f(C1; C2); (C1; D2); (D1; C2); (D1; D2)g = H : I For time t, Ht consists of 4t possible histories I This means that there is a one-to-one mapping between all possible histories and the information sets if we actually wrote out the whole extensive form game tree I As a result, we can think of each ht 2 Ht as representing a particular information set for each player i in each time t I It is simply a prescription of what player i would do at every information set or history I Therefore, it is a function that describes: [ t si : H ! Ai : t≥0 t I Intuitively, si describes exactly what player i would do at every possible history h , t t where si (h ) describes what player i would do at history h I What is a strategy in an infinitely repeated game? I Therefore, it is a function that describes: [ t si : H ! Ai : t≥0 t I Intuitively, si describes exactly what player i would do at every possible history h , t t where si (h ) describes what player i would do at history h I What is a strategy in an infinitely repeated game? I It is simply a prescription of what player i would do at every information set or history t I Intuitively, si describes exactly what player i would do at every possible history h , t t where si (h ) describes what player i would do at history h I What is a strategy in an infinitely repeated game? I It is simply a prescription of what player i would do at every information set or history I Therefore, it is a function that describes: [ t si : H ! Ai : t≥0 I What is a strategy in an infinitely repeated game? I It is simply a prescription of what player i would do at every information set or history I Therefore, it is a function that describes: [ t si : H ! Ai : t≥0 t I Intuitively, si describes exactly what player i would do at every possible history h , t t where si (h ) describes what player i would do at history h I There can be more complicated strategies such as the following: ( t t Ci if t = 0 or h = (C; C;:::; C); si (h ) = Di otherwise: I The above is called a grim trigger strategy t I For example in the infinitely repeated prisoner's dilemma, the strategy si (h ) = Ci t for all h and all t is the strategy in which player i always plays Ci regardless of the history I The above is called a grim trigger strategy t I For example in the infinitely repeated prisoner's dilemma, the strategy si (h ) = Ci t for all h and all t is the strategy in which player i always plays Ci regardless of the history I There can be more complicated strategies such as the following: ( t t Ci if t = 0 or h = (C; C;:::; C); si (h ) = Di otherwise: t I For example in the infinitely repeated prisoner's dilemma, the strategy si (h ) = Ci t for all h and all t is the strategy in which player i always plays Ci regardless