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