Introduction to Game Theory

Transcription

1 Introduction to Game Theory Part 2. Dynamic games of complete information Chapter 4. Dynamic games of complete but imperfect information Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

2 Topics covered 1 Extensive-form representation of games 2 Subgame-perfect Nash equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

3 Extensive-form representation It may seem that that static games must be represented in normal form and dynamic games in extensive form This is not the case Any game can be represented in either normal or extensive form Although for some games one of the two forms is more convenient to analyze We will discuss how static games can be represented using extensive form and how dynamic games can be represented using normal form V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

4 Normal-form representation The normal-form representation of a game specifies 1 the players in the game 2 the strategies available to each player 3 the payoff received by each player for each combination of strategies that could be chosen by the players V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

5 Extensive-form representation Definition The extensive-form representation of a game specifies 1 the players in the game 2 (a) when each player has the move (b) what each player can do at each of his or her opportunities to move (c) what each player knows at each of his or her opportunities to move 3 the payoff received by each player for each combination of moves that could be chosen by the players V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

6 Extensive-form representation We already analyzed several games represented in extensive form We propose to describe such games using game trees rather than words As an example, consider the following class of two-stage games of complete and perfect information 1 Player 1 chooses an action a 1 from the feasible set A 1 = {L, R} 2 Player 2 observes a 1 and then chooses an action a 2 from the set A 2 = {L, R } 3 Payoffs are u 1 (a 1, a 2 ) and u 2 (a 1, a 2 ), as shown in the following game tree V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

7 Extensive-form representation V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

8 Extensive-form representation We can extend in a straightforward manner the previous game tree to represent any dynamic game of complete and perfect information: players move in sequence all previous moves are common knowledge before the next move is chosen the players payoffs from each feasible combination of moves are common knowledge We propose to derive the normal-form representation of the previous dynamic game To represent a dynamic game in normal form, we need to translate the information in the extensive form into the description of each player s strategy V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

9 Normal-form representation of dynamic games Definition A strategy for a player 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 We could not apply the notion of Nash equilibrium to dynamic games of complete information if we allowed a player s strategy to leave the actions in some contingencies unspecified For player j to compute a best response to payer i s strategy, j may need to consider how i would act in every contingency, not just in the contingencies i thinks likely to arise In the previous game, player 2 has two actions but four strategies This is because there are two contingencies V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

10 Normal-form representation of dynamic games Strategies of player 2 Strategy 1 If player 1 plays L then play L, if player 1 plays R then play L { L if a f 2 (a 1 ) = 1 = L L if a 1 = R This strategy may be denoted by (L, L ) Strategy 2 If player 1 plays L then play L, if player 1 plays R then play R { L if a f 2 (a 1 ) = 1 = L R if a 1 = R This strategy may be denoted by (L, R ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

11 Normal-form representation of dynamic games Strategies of player 2 Strategy 3 If player 1 plays L then play R, if player 1 plays R then play L { R if a f 2 (a 1 ) = 1 = L L if a 1 = R This strategy may be denoted by (R, L ) Strategy 2 If player 1 plays L then play R, if player 1 plays R then play R { R if a f 2 (a 1 ) = 1 = L R if a 1 = R This strategy may be denoted by (R, R ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

12 Normal-form representation of dynamic games Player 1 has two actions but only two strategies: play L or R The reason is that player 1 has only one contingency in which he might be called upon to act Player 1 s strategy space is equivalent to the action space A 1 = {L, R} V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

13 Normal-form representation of dynamic games Recall the extensive-form representation We can now derive the normal-form representation of the game from its extensive-form representation Player 1 Player 2 (L,L ) (L,R ) (R,L ) (R,R ) L 3, 1 3, 1 1, 2 1, 2 R 2, 1 0, 0 2, 1 0, 0 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

14 Extensive-form of static games We turn to showing how a static (i.e., simultaneous-move) game can be represented in extensive form In a static game players do not need to act simultaneously It suffices that each choose a strategy without knowledge of the other s choice We can represent a simultaneous game between players 1 and 2 as follows 1 Player 1 chooses an action a 1 from the feasible set A 1 2 Player 2 does not observe player 1 s move but chooses an action a 2 from the feasible set A 2 3 Payoffs are u 1 (a 1, a 2 ) and u 2 (a 1, a 2 ) Alternatively, player 2 could move first and player 1 could then move without observing 2 s action V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

15 Extensive-form of static games To represent that some player ignores the previous moves, we introduce the notion of a player s information set Definition An information set for a player is a collection of decision nodes satisfying (i) the player has the move at every node in the information set (ii) when the play of the game reaches a node in the information set, the player does not know which node in the information set has (or has not) been reached (iii) it is the largest set satisfying (i) and (ii) Part (ii) implies that the player must have the same set of feasible actions at each decision in an information set In an extensive-form game, we will indicate that a collection of decision nodes constitutes an information set by connecting the nodes by a dotted line V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

16 Extensive-form of the Prisonners Dilemma Fink = confess V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

17 Information set: an example We propose a second example of the use of an information set in representing ignorance of a previous play Consider the following dynamic game of complete but imperfect information 1 Player 1 chooses an action a 1 from the feasible set A 1 = {L, R} 2 Player 2 observes a 1 and then chooses an action a 2 from the feasible set A 2 = {L, R } 3 Player 3 observes whether or not (a 1, a 2 ) = (R, R ) and then chooses an action a 3 from the feasible set A 3 = {L, R } V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

18 Information set: an example Player 3 has two information sets 1 a singleton information set following R by player 1 and R by player 2 2 a non-singleton information set that includes every other node at which player 3 has the move V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

19 Perfect and imperfect information We previously defined perfect information to mean that at each move in the game the player with the move knows the full history of the play of the game thus far An equivalent definition is that every information set is a singleton Imperfect information means that there is at least one non-singleton information set The extensive-form representation of a simultaneous-move game (such as the Prisoners Dilemma) is a game of imperfect information V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

20 Subgames We gave a formal definition of a subgame for repeated games We extend this definition to general dynamic games of complete information in terms of the game s extensive-form representation Definition A subgame in an extensive-form game is a game that (a) begins at a decision node n that is a singleton information set but is not the game s first decision node (b) includes all the decision and terminal nodes following n in the game tree but no nodes that do not follow n (c) does not cut any information sets, i.e., if a decision node n follows n in the game tree, then all other nodes in the information set containing n must also follow n, and so must be included in the subgame V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

21 Subgames: example There are two subgames, one beginning at each of player 2 s decision nodes V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

22 Subgames: example There are no subgames V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

23 Subgames: example There is only one subgame: it begins at player 3 s decision node following R by player 1 and R by player 2 Because of part (c), a subgame does not begin at either of player 2 s decision nodes, even though both of these nodes are singleton information sets V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

24 Subgame perfect Nash equilibrium Definition A profile of strategies of a dynamic game with complete information is a subgame perfect Nash equilibrium if it is a Nash equilibrium of the initial game and the players strategies restricted to every subgame constitute a Nash equilibrium of the subgame We already encountered two game solutions for dynamic games: backwards induction outcome and subgame perfect outcome The difference is that a subgame perfect Nash equilibrium is a collection of strategies and a strategy is a complete plan of actions Whereas an outcome describes what will happen only in the contingencies that are expected to arise, not in every contingency that might arise V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

25 Equilibrium vs outcome Consider the standard two-stage game of complete and perfect information defined as follows 1 Player 1 chooses an action a 1 from a feasible set A 1 2 Player 2 observes a 1 and then chooses an action a 2 from a feasible set A 2 3 Payoffs are u 1 (a 1, a 2 ) and u 2 (a 1, a 2 ) Assume that for each a 1 in A 1, player 2 s optimization problem argmax{u 2 (a 1, a 2 ) : a 2 A 2 } has a unique solution, denoted by R 2 (a 1 ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

26 Equilibrium vs outcome Player 1 s problem at the first stage amounts to argmax{u 1 (a 1, R 2 (a 1 )) : a 1 A 1 } Assume that the previous optimization problem for player 1 also has a unique solution, denoted by a 1 The pair of actions (a 1, R 2(a 1 )) is the backwards induction outcome of this game V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

27 Equilibrium vs outcome To define a subgame perfect Nash equilibrium we need to construct strategies For player 1 a strategy coincides with an action since there is only one contingency in which player 1 can be called upon to act the beginning of the game A strategy for player 2 is a function a 1 f 2 (a 1 ) from A 1 to A 2 R2 (a 1) is an action but not a strategy the best response function R2 is a possible strategy for player 2 In this game, the subgames begin with player 2 s move in the second stage There is one subgame for each player 1 s feasible action a 1 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

28 Equilibrium vs outcome The profile of strategies f (a 1, R 2) is a subgame perfect Nash equilibrium We have to show that f = (a 1, R 2) is a Nash equilibrium and that the restriction to each subgame is also a Nash equilibrium Subgames are simply single-person decision problems Being a Nash equilibrium reduces to requiring that player 2 s action be optimal in every subgame This is exactly the problem that the best-response function R2 solves Now we have to prove that f is a Nash equilibrium Recall that a 1 satifies u 1 (a 1, R 2 (a 1)) u 1 (a 1, R 2 (a 1 )) a 1 A 1 implying that a 1 is a best response to R 2 R2 is a best response to a 1 since for every strategy f 2 : A 1 A 2 u 2 (a 1, R 2 (a 1)) u 2 (a 1, f 2 (a 1)) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

29 Equilibrium vs outcome Consider the standard two-stage game of complete but imperfect information defined as follows: Players i 1 and i 2 simultaneously choose actions a i1 and a i2 from feasible sets A i1 and A i2, respectively Players i 3 and i 4 observe the outcome of the first stage, (a i1, a i2 ), and then simultaneously choose actions a i3 and a i4 from feasible sets A i3 and A i4, respectively Payoffs are u i (a i1,..., a i4 ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

30 Equilibrium vs outcome We will assume that for each feasible outcome (a i1, a i2 ) of the first game, the second-stage game that remains between players i 3 and i 4 has a unique Nash equilibrium denoted by (â i3 (a i1, a i2 ), â i4 (a i1, a i2 )) Assume that (a i 1, a i 2 ) is the unique Nash equilibrium of the first-stage interaction between i 1 and i 2 defined by the following simultaneous-move game 1 Players i 1 and i 2 simultaneously choose actions a i1 and a i2 from feasible sets A i1 and A i2, respectively 2 Payoffs are u i (a i1, a i2, â i3 (a i1, a i2 ), â i4 (a i1, a i2 )) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

31 Equilibrium vs outcome Proposition In the two-stage game of complete but imperfect information defined above, the subgame perfect outcome is (a i 1, a i 2, â i3 (a i 1, a i 2 ), â i4 (a i 1, a i 2 )) but the subgame perfect Nash equilibrium is (a i 1, a i 2, â i3, â i4 ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

32 Subgame perfect Nash equilibrium and credible threats Consider the following dynamic game with complete and perfect information The backwards induction outcome of the game is (R, L ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

33 Subgame perfect Nash equilibrium and credible threats The subgame perfect Nash equilibrium is the profile (R, f 2 ) where f 2 : {L, R} {L, R } is defined by f 2 (L) = R and f 2 (R) = L V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

34 Subgame perfect Nash equilibrium and credible threats Recall that the normal-form representation of this game is given by Player 1 Player 2 (L,L ) (L,R ) (R,L ) (R,R ) L 3, 1 3, 1 1, 2 1, 2 R 2, 1 0, 0 2, 1 0, 0 There are two Nash equilibria: (R, (R, L )) and (L, (R, R )) The first one corresponds to the subgame perfect Nash equilibrium (R, f 2 ) The second one corresponds to a non-credible threat of player 2 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35

35 Subgame perfect Nash equilibrium and credible threats Player 2 is threatening to play R if player 1 plays R If the threat works then 2 is not given the opportunity to carry out the threat The threat should not work because it is not credible: if player 2 were given the opportunity to carry it out, then player 2 would decide to play L rather than R Observe that players strategies do not constitute a Nash equilibrium in one of the subgames V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory September, / 35