To find SPE 1. Game Theory: Lecture 13 Extensive Form Games Introduction We have studied extensive form games which model sequential decision making. I A sequential equilibrium is a Nash equilibrium. Title: Game Theory 2: Extensive-Form Games and Subgame Perfection Created Date: Identify which Nash equilibrium are also subgame perfect Nash equilibrium. A Nash equilibrium of a finite extensive-form game Γ is a Nash equilibrium 1 Subgame perfection in perfect information games The centipede game is an example of a game of perfect … Solution. I With perfect information, a subgame perfect equilibrium is a sequential equilibrium. Extensive Form Games • Strategic (or normal) Form G ames – Time is absent • Extensive Form Games – Capture time – With the introduction of time, players can adopt strategies contingent ... • Subgame Perfect Equilibrium requires that players play a Nash Equlibrium in every subgame of the game. This yields the unique subgame perfect equilibrium in which each player uses the strategy l,l. Subgame perfect equilibrium refines the concept of Nash equilibrium accordingly. extensive form to strategic form as well. I player 1: 3; player 2: 8 I Overall, a pure strategy for a player in a perfect-information game is a complete specification of which deterministic action The solution concept we now define ignores the sequential nature of the extensive form and treats strategies as choices to be made by players before all play begins (i.e. It requires each player’s strategy to be “optimal” not only at the start of the game, but also after every history. Solution Subgame Perfect Nash Equilibrium is a re nement of Nash Equilibrium It rules out equilibria that rely on incredible threats in a dynamic environment All SPNE are identi ed by backward induction 26/26. I Thm: Every nite extensive-form game with perfect recall has a sequential equilibrium. Not a valid game (node d and e are in same information set but have different action sets). • A proper subgame is a subset of the nodes of the game starting with an initial node and including all its successors that preserves all information sets of the game and over which Definition: A strategy profile for an extensive-form game is a subgame perfect Nash equilibrium (SPNE) if it spec-ifies a Nash equilibrium in each of its subgames. extensive-form game with perfect recall if it issequentially rationalandconsistent. Levent Koc¸kesen (Koc¸ University) Extensive Form Games II 10 / 51 Subgame Perfect Equilibrium Proposition Let Γ be an extensive form game with perfect information and s∗ be a subgame perfect equilibrium of Γ. Subgame Perfect Equilibrium Extensive form game strategies A pure strategy of a player specifies an action choice at each information set of that player Definition A strategy profile in an extensive form game is a subgame perfect equilibrium (SPE) if it induces a Nash equilibrium in every subgame of the game. We will focus on it in this unit. Identify the corresponding normal form representations and hence obtain all Nash equilibrium. The idea behind SPNE is that even if a NE strategy pro-file dictates that certain subgames are not reached, we require that what the players would do conditional on just like in strategic games). Then s∗ is a backward induction equilibrium of Γ. Clearly every SPE is a NE but not conversely. Subgames • A subgame is a part of an extensive form game that constitutes a valid extensive form game on its own Definition A node x initiates a subgame if all the information sets that contain either x or a successor of x contain only nodes that are successors of x. Definition 1. As such, not all Nash equilibria are sensible in extensive form games. For each of the following games: Identify all subgames. In that sense we say that Subgame-Perfect Nash Equilibrium • Subgame perfect Nash equilibrium can be seen as an extension of the backwards induction method to deal with extensive form games. It is much easier to do this in the extensive form than it is in the normal form of the game. Equilibrium notion for extensive form games: Subgame Perfect (Nash) Equilibrium. A set of strategies is a subgame perfect equilibrium if the strategies within it form Nash equilibria in all subgames of the overall game. Recap Perfect-Information Extensive-Form Games Subgame Perfection Pure Strategies I In the sharing game (splitting 2 coins) how many pure strategies does each player have? Obtain all Nash equilibrium if the strategies within it form Nash equilibria are sensible in extensive form than is... 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