6 Extensive Form Games

Transcription

1 6 Extensive Form Games 6.1 Example: Representing a Simultaneous 22 Game Alice H HHHH O H HH o Q Bob H QQQ h o HHHH h Figure 1: The Battle of the Sexes in Extensive Form So far we have described games by using what is called the normal form representation, which is convenient for games where players move simultaneously (as in all examples so far). However, there is an alternative way to depict games using a game tree, which is referred to as the extensive form. Figure 1 depicts the battle of the sexes using this alternative (but equivalent) way. Every place where some player makes a decision is a node in the graph and as you see the lines fully describes all possibilities in the game. Observe that being a simultaneous game, Bob doesn t know what Alice played when making her decision and we have shown this in the graph by the ring around the nodes where Bob is choosing. This ring means that Bob must make the same decision no matter whether Alice picked opera or football. In the graph the Opera, Opera equilibrium is depicted. This is referred to as an information set. 6.2 Example: Battle of the Sexes with Sequential Moves Suppose now that everything is as in the battle of the sexes except that Alice makes his choice before Bob, where Bob observes the choice made by Alice before making his decision. Thinking about the available strategies, everything is like in the simultaneous move case for Alice, as her strategy set is S 1 = fo; Hg : 32

2 Alice H HHHH O H Bob HH Bob o Q H QQQ h o HHHH h Figure 2: The Battle of the Sexes with Alice Moving First However, as Bob now can condition is choice on what Alice did a strategy in an action for each possible move by Alice. We may write this as S 2 = foo ; oh ; ho ; hh g ; where the rst coordinate is the action following O by Alice and the second is the action following H by Alice. We can then write down the normal form corresponding to this sequential game as oo oh ho hh O 2; 1 2; 1 ; ; H ; 1; 2 ; 1; 2 ; where we note that (O; oo ) ; (O; oh ) and (H; hh ) : In particular the last of these seems pretty unintuitive as this is an equilibrium in which Bob would be behaving irrationally should Alice play O. We will need to re ne the set of Nash equilibria in order to avoid these type of non-credible equilibria. 6.3 The Extensive Form The extensive form (in a game in which preferences of all players are known) has the following components: A set of players, A Game Tree A player function that assigns each non-terminal node to a player. 33

3 An information function that represents what the player that moves knows about past play. Actions, which connect each non-terminal node with an immediate successor. Preferences over terminal nodes. De nition 1 An extensive form game is said to be a game of perfect information if every player knows the complete past history of play when taking an action Example: Twice Repeated Prisoner s Dilemma Suppose the rules are as follows. In period 1, the players play D C d ; 2; 1 : c 1; 2 1; 1 After the rst period, the players observe the outcome and then play the game a second time. This is not a game of perfect information because the players move simultaneously The Centipede Game Consider a short version of Rosenthals s centipede game where there are 4 non-terminal nodes ft 1 ; t 2 ; t 3 ; t 4 g and 5 terminal nodes fz 1 ; z 2 ; z 3 ; z 4 ; z 5 g where; 1. At node t 1 player 1 plays an action in A 1 (t 1 ) = fs; cg if s is played z 1 is reached and the payo s are (1; ) : If c is played t 2 is reached. 2. At node t 2 player 2 plays an action in A 2 (t 2 ) = fs; Cg if S is played z 2 is reached and the payo s are (; 1) : If C is played t 3 is reached. 3. At node t 3 player 1 plays an action in A 1 (t 3 ) = fs ; c g if s is played z 3 is reached and the payo s are (1; 1) : If c is played t 4 is reached. 34

4 4. At node t 4 player 2 plays an action in A 2 (t 4 ) = fs ; C g if S is played z 4 is reached and the payo s are (1; 1) : If C is played z 5 is reached and the payo s are (999; 999). DRAW EXTENSIVE FORM. This is a game of perfect information. 6.4 Strategies and Outcomes in Games of Perfect Information De nition 2 A (sub)history h in a perfect information game is a nite sequence of past actions (a 1 ; a 2 ; ::::; a k ). De nition 3 A pure strategy of player i in an extensive form game with perfect information is a function that assigns s i (h) in A (h) for every history h when it is is turn to play. Remark 1 At the root of the game when nothing has yet happened we can still de ne h as the trivial null history. Also, Given a pure strategy pro le, there is a unique path from the root to a single terminal node. This is the outcome path of the strategy pro le. The terminal node corresponding to the strategy pro le is called the outcome. Nash equilibria are de ned just like in the Normal form. Pro le s is Nash if and only if no player has a pro table deviation. 6.5 Backwards Induction (Subgame Perfection) Returning to the sequential battle of the sexes, where we derived the normal form oo oh ho hh O 2; 1 2; 1 ; ; ; H ; 1; 2 ; 1; 2 35

5 we already saw that (H; hh ) seemed implausible: It seems that a more reasonable approach would be to ask: 1. What would Bob do after O? Clearly o is the reasonable answer. 2. What would Bob do after H? Clearly h is the reasonable answer. 3. Hence, Alice should reason that if Bob is rational, then he should play oh : 4. Clearly, the unique best response to oh is O: 5. (O; oh ) is the unique backwards induction (subgame perfect) equilibrium in the example. Notice that oh is also weakly dominant, so that the backwards induction equilibrium corresponds to iterated elimination of weakly dominated strategies. This is a general property. De nition 4 A subgame in an extensive form game of perfect information is the extensive form that is obtained by picking a node t (which corresponds with a unique history h) and rede ning strategy sets and payo s so that a strategy only takes on actions from t (h) on and the payo s are over the terminal nodes that can be reached from t (h). That is, a subgame is any part of the game that can be analyzed as a stand alone game. This is true more generally than for games of perfect information. De nition 5 A subgame perfect Nash equilibrium s is a strategy pro le that induces Nash equilibrium play in any subgame Example: An Entry Game Suppose that: First, a challenger decides whether to enter or no. 36

6 Then, if the challenger enters, an incumbent plays either ght or accommodate. Suppose that incumbent ranks outcomes as follows U I (No entry) > U I (Enter, Accommodate) > U I (Enter, Fight) and that the entrant has the following preferences U E (Enter, Accommodate) > U E (No entry) > U I (Enter, Fight) One way to make payo s cardinal and write down the normal form is Fight Accommodate Entry 1; 1 1; 1 ; No entry ; 2 ; 2 which has two Nash equilibria: (No entry, Fight) and (Entry, Accommodate) : However, to nd the subgame perfect equilibrium we can backwards induct. In the subgame that follows entry (which has a single player), the unique Nash equilibrium is to Accommodate. Hence, the entrant will count on the incumbent to play Accommodate, and will therefore enter Example: The Centipede Game Draw game: Let s nd the subgame perfect equilibrium: 1. At node 4, player 2 gets 1 if playing stop and 999 if playing continue. Hence, player 2 stops. 2. At node 3, player 1 gets 1 if playing stop. If playing continue she knows that player 2 will stop, which gives a payo of 1: Hence, stop is better given that player 2 plays rationally in the nal subgame. 37

7 3. At node 2, player 2 gets 1 if playing stop and he knows that player 1 will stop in the next round, so stop is optimal given subgame perfect continuation play. 4. Hence, at node 1 player 1 must stop. We conclude that the unique subgame perfect equilibrium is as follows: The subgame perfect equilibrium strategy is to always stop. That is, (ss ; SS ) The subgame perfect equilibrium outcome is that player 1 stops at node 1. Notice that the normal form payo matrix is SS SC CS CC ss 1; 1; 1; 1; sc 1; 1; 1; 1; : cs ; 1 ; 1 1; 1 1; 1 cc ; 1 ; 1 1; 1 999; 999 We observe that: Regardless of what player 2 is doing, ss and sc result in the same payo s for both players (because the action s makes all later actions irrelevant) Regardless of what player 2 is doing, SS and SC result in the same payo s for both players (because the action S makes all later actions irrelevant) Hence, we may replace ss and sc by s and SS and SC by S which gives us the reduced normal form representation of the game, S CS CC s 1; 1; 1; cs ; 1 1; 1 1; 1 : cc ; 1 1; 1 999;

8 Notice that S CS CC s 1; 1; 1; cs ; 1 1; 1 1; 1 ; cc ; 1 1; 1 999; 999 so that the unique Nash equilibrium in the reduced normal form is (s; S), which gives a payo of (1; ) despite the fact that (cc ; CC ) gives a payo of (999; 999) : Hence, this is an example where it is irrelevant whether we impose optimizing behavior of the equilibrium path or not. 6.6 The Zermelo-Kuhn Theorem I will now argue that: Theorem 1 Every nite extensive form game with perfect information has at least one pure strategy subgame perfect equilibrium. To prove the result we rst let Z be the set of all terminal nodes T 1 the set of all nodes immediately before a terminal node, T 2 the set of nodes immediately before a node in t 1 ; etc. By niteness we have that there is some maximal k such that the nodes in T k are k steps removed from the terminal node. Moreover, an extensive form game must have a root, so T k = ft k g, a single node. Now: Let i be a player at a node in T 1 : Then, the player simply picks a terminal node. If there is a unique optimal node, the player picks that. If there is no unique terminal node, then the player picks one of the optimal actions. Hence, given a node t in T 1 optimality in the last stage subgames (which are non-strategic decisions) creates a mapping from T 1 onto Z: We let z 1 (t) denote the node picked in the equilibrium of the subgame for every t in T 1 : Let i be a player at a node in T 2 : Then, i picks a node in T 1. However, for each node in T 1 that the player can reached there is a nal outcome in Z, so the player essentially 39

9 picks a nal outcome in Z: Clearly, to be a Nash equilibrium also in the game starting at the node in T 2 the player must pick the best outcome if unique and one of the optimal outcomes if non-unique. Continuing inductively, we eventually reach the root of the game, which, by plugging in the subgame perfect continuation strategies collapses to another decision problem with at least one solution. 4