A subgame-perfect equilibrium is an equilibrium not only overall, but also for each subgame, while Nash equilibria can be calculated for each subgame. oT begin, consdier a date where erypla two maesk an eor. Motivate each answer; answers without motivation will not be awarded any points. 1 . There are three Nash equilibria in the dating subgame. Solution. 1 Nash equilibrium 2 Subgame perfect equilibrium 3 Relation between strategies and beliefs: assessments. involve (subgame perfect) Nash equilibria also o⁄the equilibrium path. This lecture shows how games can sometimes have multiple subgame perfect equilibria. 1 Extensive form games with perfect information 1.1 Tic-Tac-Toe Two people play the following game. Proposition 66 A sequential eq. Subgame Perfect Nash Equilibrium: a pro le of strategies s = (s1;s2;:::;sn) is a subgame perfect Nash equilibrium if a Nash equilibrium is played in every subgame. 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. Question Three houses share exclusive access to a beach, but it is dirty due to trash washed ashore. 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. Note that since the entire game is alwysa a subgame, ayn SPE ustm also be a NE. Solve for all Nash equilibria and all subgame-perfect Nash equilibria of the two-player game. 2 Subgame Perfect Equilibria In previous lectures, we studied Nash Equilibria in normal form games. A subgame perfect Nash equilibrium is a Nash equilibrium in which the strategy profiles specify Nash equilibria for every subgame of the game. In other words, (Not Start ET, ET) is indeed a sub-game perfect equilibrium. There are many other Nash equilibria. Solution ( )( ) s.t. (binding) and (( ) ). Not a valid game (node d and e are in same information set but have different action sets). ¥ (b) Can you find a Nash equilibrium of the game that results in an outcome erPyal one will certainly accept ayn oer greater than 1 v 1 and reject ayn eor less than 1 v 1. This follows directly from Nash’s Theorem. (d) For what rangeof x is therea unique subgame perfect equilibrium outcome? Exercise 3 a) Nash Bargaining Problem - X (outcomes): Any offer ( ), where . I With perfect information, a subgame perfect equilibrium is a sequential equilibrium. Let us consider the example shown. In other words, you can suggest any integer between 0 and 10. Hence imagine that we are in a subgame where at least one player did not choose C in period 1. (b)Find the set of pure strategy Nash equilibria 2.Now suppose that Bonnie submits her suggestion before Clyde and suppose that Clyde hears about Bonnie’s suggestion before she makes hers. 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)}. References: Watson, Ch. If a strategy profile results in (ET, ET) being played in the smalles subgame, the a subgame perfect equilibrium must entail Stephen playing Not Start. L R L R (0,1) (3,2) (-1,3) (1,5) 10. Identify which Nash equilibrium are also subgame perfect Nash equilibrium. When players receive the same payoff for two different strategies, they are indifferent and therefore may select either. • The most important concept in this section will be that of subgame perfect Nash equilibrium. [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. Hand-In Exercises Game Theory Economic Theory, EC4010 Jacco Thijssen All questions carry equal weight. Extensive form of a sequential game carries more information than normal form, specifically which moves do not exist within the sequence. Player 1 makes a proposal to divide the money in integers. games where players are not moving simultaneously, but rather in a sequence over time. This video explains how to find the sublime perfect Nash equilibrium of a sequential strategic game. Terminology: J orgen used ‘normal-form game’, the book of Peters uses ‘strategic game’. And its uniqueness is shown. Please write clearly (or type) on A4-size paper, stapled together. 2) Every sequential game has a Nash equilibrium. 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. Extensive Form Refinements of Nash Equilibrium Extensive Form Refinements of Nash Equilibrium I We now turn to a study of sequential or dynamic games, i.e. Determine every subgame of this game 3. A subgame perfect Nash equilibrium is an equilibrium such that players' strategies constitute a Nash equilibrium in every subgame of the original game. Two players must share 10 euros. 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) . Hence, the unique subgame perfect equilibrium has player 2 first choose the low level of investment, then player 1 o ffering to keep all the value to himself, and finally player 2 accepting the offer and getting − . Problem 7. 3) Every sequential game has a subgame per- fect equilibrium. ( ) F.O.C. What is the subgame perfect equilibrium? Nash equilibrium We can compute, for each pro le of pure strategies, the corresponding (expected) payo s: every extensive form game has a corresponding strategic/normal-form game. A beach clean-up exercise costs $100$, but has a value of $200$ to each household. Theorem), we can conclude that a Nash equilibrium in behavior strategies must always exist in these games. l ~ (2,6) T . Bayesian Games Yiling Chen September 12, 2012. 2,0 1,2 4,1 3,4 6,3 8,6 1 12 2 U U U U U D D DD D Obara (UCLA) SPE February 20, 2012 18 / 29. While I am not sure why you would want to find non-subgame perfect Nash equilibria in an extensive form game, I am sure you would need to convert it to normal form to do it. Player 2 can accept or reject the proposal. 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. † Subgame Perfect Equilibria (SPE). Back to Game Theory 101 Such games are known as games withcomplete information. Subgame Perfect Equilibrium Felix Munoz-Garcia Strategy and Game Theory - Washington State University. Consider the following game: player 1 has to decide between going up or down (U/D), while player 2 has to decide between going left or right (L/R). extensive and normal form, and nd all Nash equilibria (pure strategy)and subgame perfect Nash equilibria (in pure and mixed). 9. Recursively, if VS is the set of subgame-perfect payoffs for an S-period game, it is easy to see that the corresponding set for S +1 is given by VS+1 = φ(VS), and this way we can “recurse backwards” to find the set of all subgame perfect payoffs at the start of the full repeated game. 4 Subgame Perfect Nash Equilibrium 5 Exercises C. Hurtado (UIUC - Economics) Game Theory. De nition 5 Astrategypro le s is a subgame eprfect qeuilibrium of G if it induecs a Nash qeuilibrium in every subgame of G . Determine the subgame perfect equilibrium correction 1. and #2 (subgame perfect Nash equilibrium) and will describe #3 (conditional dominance and forward induction) only briefly. a subgame. Exercise 5. Note that this includes subgames that might not be reached during play! It encompasses backward induction as a special case in games of perfect information. equilibrium (=subgame perfect equilibrium) payoffs in the one-shot game. - D (failure to agree) - Utility: ( ) , ( ) - U (payoff set): ( ) ( ), for some lottery L over . A subgame-perfect Nash equilibrium is a Nash equilibrium whose sub strategy profile is a Nash equilibrium at each subgame. Applications. Most games have only one subgame perfect equilibrium, but not all. Hence, we have the following important result: Theorem 1. : =0, , Comparison to SPNE (from exercise 2): () () For agents who are not impatient the subgame perfect Nash equilibrium is the same as the Nash bargaining solution. Example . Using a lemma on topological sorting, this paper proves that the following three propositions are equivalent: 1) Preferences over the outcomes are acyclic. Indeed, this example illustrates how every perfect- information game can be converted to an equivalent normal form game. A strategy is in NE if no single player can gain by deviating from the strategy. Solution: ThesubgamethatfollowsR hasaNashequilibrium(r,r)foranyvalueofx.Therefore,L is always a SPE outcome. Takeaway Points. How to incorporate sequential rationality in our solution concepts in order to discard strategy pro–les that are not credible. Furthermore, we analyze this equilibrium with respect to initial reference points, loss aversion coefficients, and discount factor. Thus, the subgame perfect equilibrium through backwards induction is (UA, X) with the payoff (3, 4). There are cards on the table numbered one through In the etnry game, only ( A,In ) is subgame perfect… In finitely repeated games. Explain how the subgame perfection proceeds 2. (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 I A sequential equilibrium is a Nash equilibrium. For finitely repeated games, if a stage game has only one unique Nash equilibrium, the subgame perfect equilibrium is to play without considering past actions, treating the current subgame as a one-shot game. We construct three corresponding subgame perfect equilibria of the whole game by rolling back each of the equilibrium payoffs from the subgame. For example, the perfect-information game of Figure 5.2 can be converted into the normal form im-age of the game, shown in Figure 5.3. 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. So (Start GS, GS) is a subgame perfect Nash Equilibrium and the only SPE involving (GS, GS) being played. 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? Obara (UCLA) SPE February 20, 2012 17 / 29. 9.D. Firstly, a subgame perfect equilibrium is constructed. The den ition of best response and Nash equilibria in this ga me are exactly as they are in for normal form games. So far Up to this point, we have assumed that players know all relevant information about each other. Backward induction and Subgame Perfect Equilibrium. Entry Game, cont. (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). Nash equilibrium, subgame perfect equilibrium, and \backward induc-tion" can still be de ned. Sequential Move Games Road Map: Rules that game trees must satisfy. This causes multiple SPE. Subgame Perfect Equilibrium Examples Example: Centipede game Consider the following game with two players. Solution. 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. † Games with imperfect information. For any extensive-form game Γ with perfect recall, a Nash equilibrium in … subgame perfect equilibrium starting at a date where he gets to make an oer. B . A "Backward -Induction-like" method . 1. It seems to work, but why is it the right way to refine WPBNE? , 4 ) at a date where erypla two maesk an eor a NE: Centipede game the. Least one player did not choose C in period 1 in integers in normal form game special. 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