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Game Theory

Game Theory. What is game theory (GT)?. GT is the study of multi-agent decision problems GT is used to predict what will happen (equilibrium) when: There are more than one agent but not too many for each of them to be negligible

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Game Theory

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  1. Game Theory

  2. What is game theory (GT)? • GT is the study of multi-agent decision problems • GT is used to predict what will happen (equilibrium) when: • There are more than one agent but not too many for each of them to be negligible • Each agent’s payoff depend on what they decide to do and what others decide to do • Examples: members of a team, oligopolies, international negotiations,

  3. Basic Textbook: • “Game Theory for Applied Economists” by Robert Gibbons • The slides are also based in this textbook

  4. Elements of a game… • Players: • agents that take actions • “nature” is also a player • Information sets: What is known by each agent at each moment that a decision must be taken • Actions: • In each moment of a game, what can each agent choose. Examples: q=20,30,40; High, medium or low taxes • Outcomes: what is the result for each set of possible actions • Payoffs: Depending on the outcome, what each agent gets( could be in terms of utility or directly money)

  5. Static games of complete information

  6. What is a static game of complete information? • Static (or simulateneous-move) game: • Players choose actions • Each player chooses an action without knowing what the other players have choosen • Complete information: • Every player will know the payoff that each agent will obtain depending on what actions have been taken

  7. The normal-form representation of a game • Basic tool to analyze games • Very useful for static games under complete information. • It consists of: • The player in the game • The strategies available to each player • In this type of games, strategies are actions, but they will be very different in more complicated dynamic games !!! • The payoff received by each player for each combination of strategies the could be chosen by the players • Usually represented by a table

  8. The normal-form representation of a game The Prisioners’ Dilemma Two prisoners. They are being question by the police in different rooms. Each can confess or not Prisoner B Prisoner A Prisoner A Be sure that you recognize the elements of the game !!!!

  9. Solutions concepts for static games with complete information • Given a game, we will apply a solution concept to predict which will be the outcome (equilibrium) that will prevail in the game • Elimination of strictly dominated strategies • Nash equilibrium • These solution concepts can also be applied to more complicated games… but they are terribly useful for static games with complete information

  10. Elimination of strictly dominated strategies • Intuitive solution concept • Based on the idea that a player will never play a strategy that is strictly dominated by another strategy • An strategy si’of player i is strictly dominated by si’’if player i’s payoff is larger for si’’than for si’independently of what the other players play! • In the prisoners' dilemma, the strategy “not confess” is strictly dominated by “confess”

  11. Iterative Elimination of strictly dominated strategies • In some games, the iterative process of eliminating strictly dominated strategies lead us to a unique prediction about the result of the game (solution) • In this case, we say that the game is solvable by iterative elimination of strictly dominated strategies • Let’s see an example…

  12. Iterative Elimination of strictly dominated strategies If the red player is rational, it will never play “Right” because it is strictly dominated by “Middle”. If the blue player knows that red is rational, then he will play as if the game were…

  13. Iterative Elimination of strictly dominated strategies In this case if the blue player is rational and he knows that red is rational, then the blue player will never play Down. So, the red player will play as if the game were:

  14. Iterative Elimination of strictly dominated strategies So, if the red player is rational, and he know that blue is rational, and he knows that blue knows that he is rational, then the red player will play “Middle” The solution of the game is (UP, Middle)

  15. Problems with this solution concept… We need to assume that rationality is common knowledge “Decisions are tough”…. In many games there are no strategies that are strictly dominated… or there are just a few and the process of deletion does not take us to a solution but only to a smaller game…

  16. Example of a game were there are no dominated strategies In this game, no strategies are dominated, so the concept of iterated elimination of dominated strategies is not very useful… Let’s study the other solution concept…

  17. Some notation, before defining the solution concept of Nash Equilibrium, • SA = strategies available for player A (a SA) • SB = strategies available for player B (b SB) • UA = utility obtained by player A when particular strategies are chosen • UB = utility obtained by player B when particular strategies are chosen

  18. Nash Equilibrium • In games, a pair of strategies (a*,b*) is defined to be a Nash equilibrium if a* is player A’s best strategy when player B plays b*, and b* is player B’s best strategy when player A plays a* • It has some resemblance with the market equilibrium where each consumer and producer were taking optimal decisions…

  19. Nash Equilibrium in Games • A pair of strategies (a*,b*) is defined to be a Nash equilibrium if UA(a*,b*)  UA(a’,b*) for all a’SA UB(a*,b*)  UB(a*,b’) for all b’SB

  20. Intuition behind Nash Eq. • If a fortune teller told each player of a game that a certain (a*,b*) would be the predicted outcome of a game • The minimum criterion that this predicted outcome would have to verify is that the prediction is such that each player is doing their best response to the predicted strategies of the other players… that is the NE • Otherwise, the prediction would not be internally consistent, would be unstable…

  21. Intuition behind Nash Eq. • If a prediction was not a NE, it would mean that at least one individual will have an incentive to deviate from the prediction • …The idea of convention… if a convention is to develop about how to play a given game, then the strategies prescribed by the convention must be a NE, else at least one player will not abide by the convention

  22. Checking whether or not a pair of strategies is a NE • In the Prisioners Dilema: • (No Confess, No Confess) • Notice that it is not optimal from the “society-of-prisoners” point of view • In the previous 3x3 game: BxR • Notice that these NE also survived the iterated process of elimination of dominated strategies… This is a general result…

  23. Relation between NE and iterated… • If the process of iterative deletion of dominates strategies lead to a single solution… this solution is a NE • The strategies that are part of a NE will survive the iterated elimination of strictly dominated strategies • The strategies that survive the iterated elimination of strictly dominated strategies are NOT necessarily part of a NE

  24. Finding out the NE of a game… • The underlining trick… let’s see it with previous games… • One cannot use this if the strategies are continuous (ie. Production level). We will see afterwards…

  25. Multiple Nash Equilibrium Some games can have more than one NE… In this case, the concept of NE is not so useful because it does not give a clear prediction…as in this game called The Battle of the Sexes SEX B SEX A Prisoner A

  26. Nash Equilibrium with continuous strategies Example: Duopoly. Firm i & Firm j Strategies: Output level, that is continuous Prisoner A

  27. Prisoner A

  28. One can draw the best response functions. The intersection point is the NE Prisoner A

  29. qj a-c Ri(qj) (a-c)/2 (a-c)/3 Rj(qi) a-c qi Prisoner A (a-c)/2 (a-c)/3

  30. Nash Equilibrium in Mixed Strategies • So far, we have used the word strategy. To be more explicit, we were referring to pure strategies • We will also use the concept of mixed strategy • In a static game with complete information, a mixed strategy is a vector that tell us with what probability the player will play each action that is available to him

  31. Nash Equilibrium in Mixed Strategies Consider the following game: matching pennies Player 2 Player 1 Prisoner A An example of a mixed strategy for player 1 would be: (1/3,2/3) meaning that player 1 will play Heads with probability 1/3 and Tails with probability 2/3 We can obviously say that a mixed strategy is (q,1-q) where q is the probability of Heads

  32. Nash Equilibrium in Mixed Strategies • Why are Mixed Strategies useful? • Because in certain games, players might find optimal to have a random component in their behavior • For instance, if it was the case that the Inland Revenue would never inspect individuals taller than 190 cms, these individuals will have lots of incentives not to declare their income truthfully!

  33. Nash Equilibrium in Mixed Strategies Notice that the matching pennies game does not have an equilibrium in pure strategies Player 2 Player 1 Prisoner A Does this game have an equilibrium in mixed strategies?

  34. Player 2 Player 1 Prisoner A

  35. Prisoner A

  36. Prisoner A

  37. Drawing the best responses Notice the vertical and horizontal lines are because of the “any between [0,1]” r r*(q); player 1 br 1 1 q*(r); player 2 br 1/2 The Nash eq. in mixed strategies is the intersection of the best response…in this case (1/2,1/2) for player 1 and (1/2,1/2) for player 2 0 1/2 1 q

  38. Existence of Nash Equilibrium • A game could have more than one Nash Equilibrium • The same game could have equilibria in both pure and mixed strategies or only pure or only mixed • Notice that this is a bit artificial… any pure strategy is a mixed strategy where one action has probability 1 • Any game has at least one NE, but this one could be in mixed strategies

  39. Dynamic games of complete information

  40. Extensive-form representation -In dynamic games, the normal form representation is not that useful. The extensive-form representation will be a very useful tool in this setting. It consists of: -players -when each player has the move -what each player can do at each of his opportunities to move -what each player knows at each of his opportunities to move -the payoff received by each player for each combination of moves that could be chosen by the players -Usually represented by a tree…

  41. The dormitory game: A chooses loud (L) or soft (S). B hears the noise and chooses (L) Or (S). 7,5 L B L S 5,4 A L 6,4 S B S 6,3 Example of Extensive form representation

  42. Strategies for dynamic games -In dynamic games, we have to be much more careful defining strategies. A strategy is a complete plan of action – it specifies a feasible action for the player in every contingency in which the player might be called on to act In the previous example, a strategy for player A is (L). Another possible strategy for player A is (H) An example of a strategy for player B is (L,S) that means that player B will play L if he gets to his first node and will play S if he get to the second node. Other strategies would be (L,L); (S,S); and (S,L)

  43. Extensive-form representation -It could be that when a player moves, he cannot distinguish between several nodes… he does not know in what node he is! -For instance, it could be that when player B moves, he has not heard the noise from player A -We reflect this ignorance by putting these two nodes together in the same circle… as in the following slide…

  44. A chooses loud (L) or soft (S). B makes a similar choice without knowing A’s choice 7,5 L B L S 5,4 A L 6,4 S B S 6,3 A Dormitory Game Notice that this game will actually be static !!!!!!

  45. Information set • -This take us to the notion of an information set!! • An information set for a player is a collectionof decision nodes satisfying: • The player has the move at every node in the information set • When the play of the game reaches a node in the information set, the player with the move does not know which node in the information set has (or has not) been reached • As a consequence, the nodes surrounded by the same circle are part of the same information set

  46. Strategies -We can be more precise defining what a strategy is. A player’s strategy is an action for each information set that the player has !!!

  47. Dynamic games with complete information Can be divided in: Perfect information: at each move in the game, the player with the move knows the full history of the play of the game so far… Imperfect information: at some move the player with the move does not know the history of the game Notice, in complete games with perfect information each information set must have one and only one node (the information set is singleton). If info is complete but there is an information set with more than one node, it must be an imperfect information game.

  48. Dynamic games with complete information Another classification: Non-repeated games: The game is just played once Finitely repeated games: The game is repeated a finite number of times Infinitely repeated games: The game is repeated an infinite amount of times

  49. Non-repeated dynamic games with perfect information Two main issues: -Is the Nash Equilibrium an appropriate solution concept? -If not… Define a better solution concept…

  50. A chooses loud (L) or soft (S) B makes a similar choice knowing A’s choice 7,5 L B L S 5,4 A Thus, we should put B’s strategies in a form that takes the information on A’s choice into account L 6,4 S B S 6,3 A Two-Period Dormitory Game

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