lecture11 - Lecture XI Subgame Perfect Equilibrium Markus M...

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Unformatted text preview: Lecture XI: Subgame Perfect Equilibrium Markus M. M¨obius April 3, 2004 • Gibbons, chapter 2.1.A,2.1.B,2.2.A • Osborne, sections 5.4, 5.5 1 Introduction Last time we discussed extensive form representation and showed that there are typically lots of Nash equilibria. Many of them look unreasonable because they are based on out of equilibrium threats. For example, in the entry game the incumbent can deter entry by threatening to flood the market. In equilibrium this threat is never carried out. However, it seems unreasonable because the incumbent would do better accommodating the entrant if entry in fact occurs. In other words, the entrant can call the incumbent’s bluff by entering anyway. Subgame perfection is a refinement of Nash equilibrium. It rules out non-credible threats. 2 Subgames Definition 1 A subgame G of an extensive form game G consists of 1. A subset T of the nodes of G consisting of a single node x and all of its successors which has the property that t ∈ T , t ∈ h ( t ) then t ∈ T . 2. Information sets, feasible moves and payoffs at terminal nodes as in G . 1 2.1 Example I: Entry Game 2 In Out 1 F A 2-1-1 1 1 This game has two subgames. The entire game (which is always a sub- game) and the subgame which is played after player 2 has entered the market: 1 F A-1-1 1 1 2.2 Example II 1 L R 2 L R 2 L R 1 a b 2 a b 2 a b 2 This game has also two subgames. The entire game and the subgame (a simultaneous move game) played after round 2: 1 a b 2 a b 2 a b This subgame has no further subgames: otherwise the information set of player 2 would be separated which is not allowed under our definition. 3 Subgame Perfect Equilibrium Definition 2 A strategy profile s * is a subgame perfect equilibrium of G if it is a Nash equilibrium of every subgame of G. Note, that a SPE is also a NE because the game itself is a (degenerate) subgame of the entire game. Look at the entry game again. We can show that s 1 = A and s 2 = Entry is the unique SPE. Accomodation is the unique best response in the subgame after entry has occurred. Knowing that, firm 2’s best response is to enter. 3.1 Example: Stackelberg We next continue the Stackelberg example from the last lecture. We claim that the unique SPE is q * 2 = 1 2 and q * 1 ( q 2 ) = 1- q 2 2 . The proof is as follows. A SPE must be a NE in the subgame after firm 1 has chosen q 1 . This is a one player game so NE is equivalent to firm 1 maximizing its payoff, i.e. q * 1 ( q 1 ) ∈ arg max q 1 [1- ( q 1 + q 2 )]. This implies that q * 1 ( q 2 ) = 1- q 2 2 . Equivalently, firm 1 plays on its BR curve. A SPE must also be a NE in the whole game, so q * 2 is a BR to q * 1 : u 2 ( q 1 ,q * 2 ) = q 2 (1- ( q 2 + q * 1 ( q 2 ))) = q 1 1- q 1 2 3 The FOC for maximizing u 2 is q * 2 = 1 2 ....
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lecture11 - Lecture XI Subgame Perfect Equilibrium Markus M...

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