Note that since the entire game is alwysa a subgame, ayn SPE ustm also be a NE. and #2 (subgame perfect Nash equilibrium) and will describe #3 (conditional dominance and forward induction) only briefly. oT begin, consdier a date where erypla two maesk an eor. Theorem), we can conclude that a Nash equilibrium in behavior strategies must always exist in these games. And its uniqueness is shown. Example . Most games have only one subgame perfect equilibrium, but not all. Solution. ( ) F.O.C. Tutorial 3 Subgame perfect equilibrium Exercise 1 Subgame perfection We consider the following two-player sequential game in which player 1 has to choose between G and D and Player 2 has to choose between A and B. Motivate each answer; answers without motivation will not be awarded any points. 9. It may be found by backward induction, an iterative process for solving finite extensive form or sequential games.First, one determines the optimal strategy of the player who makes the last move of the game. Using a lemma on topological sorting, this paper proves that the following three propositions are equivalent: 1) Preferences over the outcomes are acyclic. games where players are not moving simultaneously, but rather in a sequence over time. Indeed, this example illustrates how every perfect- information game can be converted to an equivalent normal form game. (binding) and (( ) ). (a)De ne this as a game (b)Find the set of pure strategy subgame-perfect equilibria (c)Give an example of a Nash equilibrium that is not subgame-perfect 7 This lecture shows how games can sometimes have multiple subgame perfect equilibria. a subgame. It encompasses backward induction as a special case in games of perfect information. How to incorporate sequential rationality in our solution concepts in order to discard strategy pro–les that are not credible. Players 1 and 2 are bargaining over one dollar in two periods: In the first period, Player 1 proposes s 1 for himself and 1 − s 1 for player 2. Thus, the subgame perfect equilibrium through backwards induction is (UA, X) with the payoff (3, 4). The den ition of best response and Nash equilibria in this ga me are exactly as they are in for normal form games. References: Watson, Ch. Explain how the subgame perfection proceeds 2. Player 1 makes a proposal to divide the money in integers. Finally, we analyze a game in which a firm has to decide whether to invest in a machine that will reduce its costs of production. There are many other Nash equilibria. In finitely repeated games. Exercise 5. Furthermore, we analyze this equilibrium with respect to initial reference points, loss aversion coefficients, and discount factor. Example 1: (OUT&B, L) is a subgame perfect Nash equilibrium Example 2: (IN;H;d) is one SPE (OUT;d;H) is another SPE. Solution ( )( ) s.t. Entry Game, cont. [Answer:] For Nash equilibria, consider the following subgame 1 2 L 0 2 R 0 2 L 0 1 0, 3 1, 3 R 0 1 2, 0, 2, 1 and it is easy to see that (R 0 1, R 0 2) is Nash equilibrium. Determine every subgame of this game 3. involve (subgame perfect) Nash equilibria also o⁄the equilibrium path. This causes multiple SPE. In this case case this simply involves checking that the actions choosen, according to the strategies, at some –rst period actions other than (C;C) for a Nash equilibrium. ECON 504 Sample Questions for Final Exam Levent Koçkesen Therefore,the set of subgame perfectequilibria is {(Rl,l),(Lr,r),(L3 4 l ⊕ 1 4 r, 1 4 l ⊕ 2 4 r)}. Tuhs, starting from a period in which she eors, player wot can secure at astel 1 1 v 1 yb proposing a split (1 v 1, 1 1 v 1) . What is the subgame perfect equilibrium? - D (failure to agree) - Utility: ( ) , ( ) - U (payoff set): ( ) ( ), for some lottery L over . Not a valid game (node d and e are in same information set but have different action sets). Note that this includes subgames that might not be reached during play! (iv) Since it is a dynamic game with complete and perfect information, based on Figure 16, we have the unique subgame-perfect Nash equilibrium: (AG, DE). Solution. Hence, we have the following important result: Theorem 1. is necessarily a subgame perfect Nash equilibrium Thus, sequential equilibrium strengthens both subgame perfection and weak perfect Bayesian Nash equi-librium Behavioral motivation for sequential equilibrium? 3) Every sequential game has a subgame per- fect equilibrium. A subgame perfect Nash equilibrium is an equilibrium such that players' strategies constitute a Nash equilibrium in every subgame of the original game. Player 2 can accept or reject the proposal. Hence imagine that we are in a subgame where at least one player did not choose C in period 1. So (Start GS, GS) is a subgame perfect Nash Equilibrium and the only SPE involving (GS, GS) being played. 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