Extensive Form Games and Backward Induction
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1 Recap Subgame Perfection ackward Induction Extensive Form ames and ackward Induction ISCI 330 Lecture 3 February 7, 007 Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide
2 Recap Subgame Perfection ackward Induction Lecture Overview Recap Subgame Perfection ackward Induction Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide
3 Recap Subgame Perfection ackward Induction Nash Equilibria iven our new definition of pure strategy, we are able to reuse our old definitions of: mixed strategies best response Nash equilibrium Theorem Every perfect information game in extensive form has a PSNE This is easy to see, since the players move sequentially. Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 3
4 0) yes Recap no yes no yes Subgame Perfection ackward Induction (,0) (0,0) (,) (0,0) (0,) Induced Normal Form Figure 5. The Sharing game. t the definition contains a subtlety. n agent s strategy requires a decision ce node, regardless of whether or not it is possible to reach that node given oice nodes. In the Sharing game above the situation is straightforward three pure strategies, and player has eight (why?). ut now consider the in Figure 5.. In fact, the connection to the normal form is even tighter we can convert an extensive-form game into normal form (3,8) (8,3) (5,5) (,0) (,0) Figure 5. perfect-information game in extensive form. define a complete strategy for this game, each of the players must choose each of his two choice nodes. Thus we can enumerate the pure strategies s as follows.,),(,),(,),(,)} Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 4
5 0) yes Recap no yes no yes Subgame Perfection ackward Induction (,0) (0,0) (,) (0,0) (0,) Induced Normal Form Figure 5. The Sharing game. t the definition contains a subtlety. n agent s strategy requires a decision ce node, regardless of whether or not it is possible to reach that node given oice nodes. In the Sharing game above the situation is straightforward three pure strategies, and player has eight (why?). ut now consider the in Figure 5.. In fact, the connection to the normal form is even tighter we can convert an extensive-form game into normal form (3,8) (8,3) (5,5) (,0) (,0) CE CF DE DF 3, 8 3, 8 8, 3 8, 3 3, 8 3, 8 8, 3 8, 3 5, 5, 0 5, 5, 0 5, 5, 0 5, 5, 0 Figure 5. perfect-information game in extensive form. define a complete strategy for this game, each of the players must choose each of his two choice nodes. Thus we can enumerate the pure strategies s as follows.,),(,),(,),(,)} Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 4
6 0) Recap Subgame Perfection ackward Induction (,0) (0,0) (,) (0,0) (0,) Figure 5. The Sharing game. Induced Normal Form t the definition contains a subtlety. n agent s strategy requires a decision ce node, regardless of whether or not it is possible to reach that node given oice nodes. In the Sharing game above the situation is straightforward three pure strategies, and player has eight (why?). ut now consider the in Figure 5.. In fact, the connection to the normal form is even tighter we can convert an extensive-form game into normal form (3,8) (8,3) (5,5) (,0) (,0) Figure 5. this perfect-information illustrates game the in extensive lack of form. compactness of the normal form define a complete strategy for this game, each of the players must choose each of his two choice nodes. Thus we can enumerate the pure strategies s as follows.,),(,),(,),(,)} CE CF DE DF 3, 8 3, 8 8, 3 8, 3 3, 8 3, 8 8, 3 8, 3 5, 5, 0 5, 5, 0 5, 5, 0 5, 5, 0 games aren t always this small even here we write down 6 payoff pairs instead of 5,E),(C,F),(D,E),(D,F)} Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 4
7 0) Recap (,0) (0,0) (,) Subgame (0,0) Perfection (0,) ackward Induction Figure 5. The Sharing game. Induced Normal Form t the definition contains a subtlety. n agent s strategy requires a decision ce node, regardless of whether or not it is possible to reach that node given oice nodes. In the Sharing game above the situation is straightforward three pure strategies, and player has eight (why?). ut now consider the in Figure 5.. In fact, the connection to the normal form is even tighter we can convert an extensive-form game into normal form (3,8) (8,3) (5,5) (,0) (,0) Figure 5. while perfect-information we can game write in extensive any extensive-form form. game as a NF, we can t do the reverse. e.g., matching pennies cannot be written as a perfect-information extensive form game define a complete strategy for this game, each of the players must choose each of his two choice nodes. Thus we can enumerate the pure strategies s as follows.,),(,),(,),(,)},e),(c,f),(d,e),(d,f)} CE CF DE DF 3, 8 3, 8 8, 3 8, 3 3, 8 3, 8 8, 3 8, 3 5, 5, 0 5, 5, 0 5, 5, 0 5, 5, 0 ntextensive to note that Formwe ames have to andinclude ackward theinduction strategies (,) and (,), even ISCI 330 Lecture 3, Slide 4
8 0) Recap (,0) (0,0) (,) Subgame (0,0) Perfection (0,) ackward Induction Figure 5. The Sharing game. Induced Normal Form t the definition contains a subtlety. n agent s strategy requires a decision ce node, regardless of whether or not it is possible to reach that node given oice nodes. In the Sharing game above the situation is straightforward three pure strategies, and player has eight (why?). ut now consider the in Figure 5.. In fact, the connection to the normal form is even tighter we can convert an extensive-form game into normal form (3,8) (8,3) (5,5) (,0) (,0) Figure 5. What perfect-information are the game (three) in extensive pure-strategy form. equilibria? define a complete strategy for this game, each of the players must choose each of his two choice nodes. Thus we can enumerate the pure strategies s as follows.,),(,),(,),(,)},e),(c,f),(d,e),(d,f)} CE CF DE DF 3, 8 3, 8 8, 3 8, 3 3, 8 3, 8 8, 3 8, 3 5, 5, 0 5, 5, 0 5, 5, 0 5, 5, 0 ntextensive to note that Formwe ames have to andinclude ackward theinduction strategies (,) and (,), even ISCI 330 Lecture 3, Slide 4
9 0) Recap (,0) (0,0) (,) Subgame (0,0) Perfection (0,) ackward Induction Figure 5. The Sharing game. Induced Normal Form t the definition contains a subtlety. n agent s strategy requires a decision ce node, regardless of whether or not it is possible to reach that node given oice nodes. In the Sharing game above the situation is straightforward three pure strategies, and player has eight (why?). ut now consider the in Figure 5.. In fact, the connection to the normal form is even tighter we can convert an extensive-form game into normal form (3,8) (8,3) (5,5) (,0) (,0) Figure 5. What perfect-information are the game (three) in extensive pure-strategy form. equilibria? define a complete strategy for this game, each of the players must choose each of his two choice nodes. (, ), Thus (C, we F can ) enumerate the pure strategies s as follows.,),(,),(,),(,)},e),(c,f),(d,e),(d,f)} (, ), (C, F ) (, ), (C, E) CE CF DE DF 3, 8 3, 8 8, 3 8, 3 3, 8 3, 8 8, 3 8, 3 5, 5, 0 5, 5, 0 5, 5, 0 5, 5, 0 ntextensive to note that Formwe ames have to andinclude ackward theinduction strategies (,) and (,), even ISCI 330 Lecture 3, Slide 4
10 0) Recap (,0) (0,0) (,) Subgame (0,0) Perfection (0,) ackward Induction Figure 5. The Sharing game. Induced Normal Form t the definition contains a subtlety. n agent s strategy requires a decision ce node, regardless of whether or not it is possible to reach that node given oice nodes. In the Sharing game above the situation is straightforward three pure strategies, and player has eight (why?). ut now consider the in Figure 5.. In fact, the connection to the normal form is even tighter we can convert an extensive-form game into normal form (3,8) (8,3) (5,5) (,0) (,0) Figure 5. What perfect-information are the game (three) in extensive pure-strategy form. equilibria? define a complete strategy for this game, each of the players must choose each of his two choice nodes. (, ), Thus (C, we F can ) enumerate the pure strategies s as follows.,),(,),(,),(,)},e),(c,f),(d,e),(d,f)} (, ), (C, F ) (, ), (C, E) CE CF DE DF 3, 8 3, 8 8, 3 8, 3 3, 8 3, 8 8, 3 8, 3 5, 5, 0 5, 5, 0 5, 5, 0 5, 5, 0 ntextensive to note that Formwe ames have to andinclude ackward theinduction strategies (,) and (,), even ISCI 330 Lecture 3, Slide 4
11 Recap Subgame Perfection ackward Induction Lecture Overview Recap Subgame Perfection ackward Induction Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 5
12 Recap Subgame Perfection ackward Induction Notice that the definition contains a subtlety. n agent s strategy requires a decision at each choice node, regardless of whether or not it is possible to reach that node given the other choice nodes. In the Sharing game above the situation is straightforward player has three pure strategies, and player has eight (why?). ut now consider the game shown in Figure 5.. Subgame Perfection (3,8) (8,3) (5,5) Figure 5. (,0) (,0) perfect-information game in extensive form. There s something intuitively wrong with the equilibrium (, ), (C, E) In order to define a complete strategy for this game, each of the players must choose an action at each of his two choice nodes. Thus we can enumerate the pure strategies of the players as follows. Why S = would {(,),(,),(,),(,)} player ever choose to play if he got to the second S = {(C,E),(C,F),(D,E),(D,F)} choice node? fter all, dominates for him It is important to note that we have to include the strategies (,) and (,), even though once is chosen the -versus- choice is moot. The definition of best response and Nash equilibria in this game are exactly as they are in for normal form games. Indeed, this example illustrates how every perfectinformation game can be converted to an equivalent normal form game. For example, the perfect-information game of Figure 5. can be converted into the normal form image of the game, shown in Figure 5.3. Clearly, the strategy spaces of the two games are Multi gent Systems, draft of September 9, 006 Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 6
13 Recap Subgame Perfection ackward Induction Notice that the definition contains a subtlety. n agent s strategy requires a decision at each choice node, regardless of whether or not it is possible to reach that node given the other choice nodes. In the Sharing game above the situation is straightforward player has three pure strategies, and player has eight (why?). ut now consider the game shown in Figure 5.. Subgame Perfection (3,8) (8,3) (5,5) (,0) (,0) Figure 5. perfect-information game in extensive form. There s something intuitively wrong with the equilibrium (, ), (C, E) In order to define a complete strategy for this game, each of the players must choose an action at each of his two choice nodes. Thus we can enumerate the pure strategies of the players as follows. Why S = would {(,),(,),(,),(,)} player ever choose to play if he got to the second S = {(C,E),(C,F),(D,E),(D,F)} choice node? fter all, dominates for him e does it to threaten player, to prevent him from choosing F, and so gets 5 It is important to note that we have to include the strategies (,) and (,), even though once is chosen the -versus- choice is moot. The definition of best response and Nash equilibria in this game are exactly as they are in for normal form games. Indeed, this example illustrates how every perfectinformation game can be converted to an equivalent normal form game. For example, the perfect-information owever, this game of seems Figure 5. like canabe non-credible converted into the normal threat form image of the game, shown in Figure 5.3. Clearly, the strategy spaces of the two games are If player reached his second decision node, would he really follow through and play? Multi gent Systems, draft of September 9, 006 Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 6
14 Recap Subgame Perfection ackward Induction Formal Definition Define subgame of rooted at h: the restriction of to the descendents of. Define set of subgames of : subgames of rooted at nodes in s is a subgame perfect equilibrium of iff for any subgame of, the restriction of s to is a Nash equilibrium of Notes: since is its own subgame, every SPE is a NE. this definition rules out non-credible threats Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 7
15 (0,0) (,0) (0,0) (,) (0,0) (0,) Recap Subgame Perfection ackward Induction ack to the Example Figure 5. The Sharing game. Notice that the definition contains a subtlety. n agent s strategy requires a decision at each choice node, regardless of whether or not it is possible to reach that node given the other choice nodes. In the Sharing game above the situation is straightforward player has three pure strategies, and player has eight (why?). ut now consider the game shown in Figure 5.. (3,8) (8,3) (5,5) (,0) (,0) Figure 5. perfect-information game in extensive form. In order to define a complete strategy for this game, each of the players must choose an action at each of his two choice nodes. Thus we can enumerate the pure strategies of the players as follows. Which equilibria from the example are subgame perfect? S = {(,),(,),(,),(,)} S = {(C,E),(C,F),(D,E),(D,F)} It is important to note that we have to include the strategies (,) and (,), even though once is chosen the -versus- choice is moot. The definition of best response and Nash equilibria in this game are exactly as they are in for normal form games. Indeed, this example illustrates how every perfectinformation game can be converted to an equivalent normal form game. For example, the perfect-information game of Figure 5. can be converted into the normal form image of the game, shown in Figure 5.3. Clearly, the strategy spaces of the two games are Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 8
16 (0,0) (,0) (0,0) (,) (0,0) (0,) Recap Subgame Perfection ackward Induction ack to the Example Figure 5. The Sharing game. Notice that the definition contains a subtlety. n agent s strategy requires a decision at each choice node, regardless of whether or not it is possible to reach that node given the other choice nodes. In the Sharing game above the situation is straightforward player has three pure strategies, and player has eight (why?). ut now consider the game shown in Figure 5.. (3,8) (8,3) (5,5) (,0) (,0) Figure 5. perfect-information game in extensive form. In order to define a complete strategy for this game, each of the players must choose an action at each of his two choice nodes. Thus we can enumerate the pure strategies of the players as follows. Which equilibria from the example are subgame perfect? (, ), (C, F ) is subgame perfect S = {(,),(,),(,),(,)} (, S ) is an non-credible threat, so (, ), (C, E) is not = {(C,E),(C,F),(D,E),(D,F)} subgame perfect (, ) is also non-credible, even though is off-path It is important to note that we have to include the strategies (,) and (,), even though once is chosen the -versus- choice is moot. The definition of best response and Nash equilibria in this game are exactly as they are in for normal form games. Indeed, this example illustrates how every perfectinformation game can be converted to an equivalent normal form game. For example, the perfect-information game of Figure 5. can be converted into the normal form image of the game, shown in Figure 5.3. Clearly, the strategy spaces of the two games are Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 8
17 Recap Subgame Perfection ackward Induction Lecture Overview Recap Subgame Perfection ackward Induction Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 9
18 Recap Subgame Perfection ackward Induction Centipede ame 5 Reasoning and Computing with the Extensive Form (3,5) D D D D D (,0) (0,) (3,) (,4) (4,3) Figure 5.9 Play this as a fun game... The centipede game place. In other words, you have reached a state to which your analysis has given a probability of zero. ow should you amend your beliefs and course of action based on this measure-zero event? It turns out this seemingly small inconvenience actually raises a fundamental problem in game theory. We will not develop the subject further here, but let us only mention that there exist different accounts of this situation, and they depend on the probabilistic assumptions made, on what is common knowledge (in Extensive Form ames and ackward Induction ISCI 330 Lecture 3, Slide 0
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