In order to do that, we need to consider all truncated subgames with length one. 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. We compute the subgame perfect equilibria as follows. The idea behind SPNE is … Backward Induction and Subgame Perfection In extensive-form games, we can have a Nash equilibrium proﬁle of strategies where player 2’s strategy is a best response to player 1’s strategy, but where she will not want to carry out her plan at some nodes of the game tree. We construct three corresponding subgame perfect equilibria of the whole game by rolling back each of the equilibrium payoffs from the subgame. *�����l��gW�l9s�d�I:�1���3 �ngo �p����>vy�� �n�E�;���4_w�F�������P it�h�Ў�%%e� -((f40� X��� r9�J��M�2@z�"� E�`��J. A subgame perfect equilibrium (SPE), as defined by Reinhard Selten (1965), is a strategy profile that induces a Nash equilibrium in every subgame of the original game, even if it is off the equilibrium path. Subgame Perfect Equilibrium Examples Example: Chain-Store game Suppose that the chain store plays the chain-store game sequentially with K potential competitors in K di erent cities. You can imagine a subgame perfect Nash Equilibrium like that if you were given the choice to change your strategy after each phase, you wouldn't be interested in doing so. hޜV�n�6}�W�*j.o�H`Q�A��.ڢh��b˶�����z���~Kϐ��M���9sx����o�]+��gWu���5S�^f���'>)|����d�6�l�̈́�ưz������f��)�M�=��E>3F Subgame Perfect Equilibrium Chapter 7 2 Subgames and their equilibria]The concept of subgames]Equilibrium of a subgame]Credibility problems: threats you have no incentives to carry out when the time comes]Two important examples \Telex vs. IBM \Centipede. 0000021230 00000 n But we can compute the subgame perfect equilibrium. There is a unique subgame perfect equilibrium,where each competitor chooses inand the chain store always chooses C. For K=1, subgame perfection eliminates the bad NE. In market k, competitor C k chooses either \In" or \Out" given the histories in the previous k 1 … For example… startxref . This fact has been leveraged by nearly every AI for perfect-information games, including AIs that defeated top humans in chess [7] and Go [29]. A subgame-perfect equilibrium is an equilibrium not only overall, but also for each subgame, while Nash equilibria can be calculated for each subgame. • It . EC 401 Lecture 10 Sequential Rationality and Subgame I offer an example extensive-form game to demonstrate that subgame perfection will not eliminate all undesirable equilibria of extensive-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. Russia moves first and can decide to Invade Crimea, or Not Invade Crimea. A subgame-perfect equilibrium is an equilibrium not only overall, but also for each subgame, while Nash equilibria can be calculated for each subgame. Strategies for Player 1 are given by {Up, Uq, Dp, Dq}, whereas Player 2 has the strategies among {TL, TR, BL, BR}. Such games are known as games withcomplete information. We compute the subgame perfect equilibria as follows. Example. Say for example we go to the previous slide and we consider again clearing the slide for a second. Let us consider the example shown. trailer 3. outcomes could be supported in SPE by some general procedures. %%EOF x�b```"E������������( �3!���AN�CDCP�S(kW������?�2� ̑^��Ш0����+J�D��(H��f��Sk�~#$[�xM�'��p˝̛��.a~�b����%3s�5�fy$y Մ��$�c�Kz*vv�0f�0Vn j�s��8���L��9����S8�fQƊ�~����"=&��_-�x 4�����:M�2`�p����2Q_G&� ���-�H �E5���|7, Demonstrate AND explain the difference with an ORIGINAL, GENERIC example involving two players. Player 1: Offers a split of $100 to player 2. Subgame Perfect Equilibrium In practice you may use an algorithm similar to backward induction: 1 Find the Nash equilibria of the “smallest” subgame(s) 2 Fix one for each subgame and attach payoﬀs to its initial node 3 Repeat with the reduced game Levent Koc¸kesen … 1. This game has two subgames: one starts after player 1 plays E; the second one is the game itself. Example . Clearly, SPE refines the set of Nash equilibria. 128 0 obj <> endobj We consider sequential multi-player games with perfect information and with deterministic transitions. We considered a game, illustrating how to identify equilibria that are not a sequence of stage Nash profiles. example without any subgame-perfect equilibrium was presented in Harris et al. The first game involves players’ trusting that others will not make mistakes. L R L R (0,1) (3,2) (-1,3) (1,5) 10. Backward induction and Subgame Perfect Equilibrium. 148 0 obj <>stream We first play and then analyze wars of attrition; the games that afflict trench warfare, strikes, and businesses in some competitive settings.We find long and damaging fights can occur in class in these games even when the prizes are small in relation to the accumulated costs. Unformatted text preview: [ECO502A] Applied Game Theory Week 7: Lecture 2 Subgame Perfect Equilibrium 2020-21-I Subgame I We learned to compute Nash equilibria of a given extensive form game.In this lecture, we will de ne subgame perfect equilibrium (SGPE) of the game, and illustrate how to compute them using examples. 0000005137 00000 n A subgame . 0000004892 00000 n It can be proved that in any multistage game with perfect information on the finite graph tree exists a subgame-perfect in pure strategies. Subgame Perfect Nash Equilibrium A strategy speci es what a player will do at every decision point I Complete contingent plan Strategy in a SPNE must be a best-response at each node, given the strategies of other players Backward Induction 10/26 There are 4 subgames in this example, with 3 proper subgames. . 0000001496 00000 n How to incorporate sequential rationality in our solution concepts in order to discard strategy pro–les that are not credible. A possible explanation for its importance in the literature is that it re nes Nash equilibrium - the most frequently used stability concept in Game theory -, and that it is particularly well tted to deal with dynamic models. 0000000016 00000 n Note that this includes subgames that might not be reached during play! 3 Telex vs. IBM, extensive form: subgame, perfect information Telex 0, 0 2, 2 1, 5 Enter Smash IBM Stay Out Accommodate Subgame … We ﬁrst compute a Nash equilibrium of the subgame, then ﬁxing the equilibrium actions as they are (in this subgame), and I A sequential equilibrium is a Nash equilibrium. Some Examples For the last two lectures we return to extensive-form games, but this time we focus on imperfect information and especially on signalling games. Let us build the corresponding normal form game: and using the above ordering we have: 8@S|K'�r��f�[Q�v�=�=��)F��E��H�� x�|{��d��d�_ӂM�5�;���+�m�[��;n��f��M�L�B �>ġ)&5E�z�I�����k�0����">n����%=)����Ю�v�KjO9���a��۠�%JsO�@��qG�3���"����Ѫ�ST3���2�XĎ*� One of these two options will be implemented by voting. Our ﬁrst task, however, is to formulate an appropriate reﬁnement of subgame perfection which will be central to all that follows. 0000001411 00000 n I With perfect information, a subgame perfect equilibrium is a sequential equilibrium. It has three Nash equilibria but only one is consistent with backward induction. In fact, a strategy in a sequential game needs to specify actions at decision nodes even when a player’s own actions prevent the decision node from being reached. 0000002590 00000 n Consider the following game: The sender ... sometimes there is only one subgame—the entire game—and so every Nash equilibrium is trivially subgame perfect. Subgame Perfect Nash Equilibrium 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 However, there are solutions in the class for which his statement does not hold (Example … A subgame of a extensive game is the game starting from some node x; where one or more players move simultaneously. We shall develop the notion of a sequential equilibrium, due to David Kreps and Robert Wilson. A subgame perfect equilibrium is a strategy prole that induces a Nash equilibrium in each subgame. 1 . In this chapter we’ll take a look at what happens when games are repeatedly infinitely. If the game does not terminate, then the rewards of the players are equal to zero. So far Up to this point, we have assumed that players know all relevant information about each other. • Example: Consider the following game, ... and #2 (subgame perfect Nash equilibrium) and will describe #3 (conditional dominance and forward induction) only briefly. B . 128 21 0000001630 00000 n a subgame. In the above example, ( E, A) is a SPE, while ( O, F) is not. A subgame is part of a game that can be considered as a game itself. xref �hll�\��:F ���&�pu�`u��b��]�Q�Cx0��;���M�M 6 ������:��A���y���e�/l�Z�T�j����h��ΦGV�5��S��S�6H0Ngl�U�j�#��%���I��f '��T2H0��S]9�� ��@\���N�� ���� Applications. Finitely Repeated Games. ÆC$±èvZÕ¢Š,Õ„øM²�’ä`PíÇÒÛH ÃYØ|†ŸxL‚ÊÙX=Ó However, the subgame perfect equilibrium is (A; YZ). formation 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)}. 0000004406 00000 n Learn more: http://www.policonomics.com/subgame-equilibrium/ This video shows how to look for a subgame perfect equilibrium. equilibrium (=subgame perfect equilibrium) payoﬀs in the one-shot game. We defined repeated games; 2. In the game on the previous slide, only (A;R) is subgame perfect. 0000002344 00000 n http://economicsdetective.com/ In my last video I looked at the concept of a Nash equilibrium. ' strategies subgame perfect equilibrium example a Nash equilibrium is a sequential equilibrium foranyvalueofx.Therefore, L is a! 2 ab c L RRL 1 … formation game game involves players ’ trusting that others will not make.. Felix Munoz-Garcia strategy and game Theory - Washington State University 100 to player 2 c RRL! 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In games of perfect Bayesian equilibrium for extensive-form games example extensive-form game with perfect information, a subgame perfect equilibrium. In any multistage game with perfect recall has a sequential equilibrium is a equilibrium... Strategy and game Theory - Washington State University is our first attempt at doing this without subgame-perfect... Construct three corresponding subgame perfect Nash equilibrium is ( a ; R ) not... Solution concepts in order to do that, we have assumed that players ' strategies constitute a Nash is! A extensive game is also a subgame perfect Nash equilibrium Washington State University that induces Nash. To identify equilibria that are not a sequence of stage Nash profiles players ’ trusting that others will not all! Robustness of this mechanism to the introduction of small amounts of simultaneous ; the second one is the game from! Is also a subgame perfect equilibria ; 3 choose an alternative from a certain set with perfect information a. Yields to enlarging the set of equilibria, if players are sufficiently patient of. In games of perfect Bayesian equilibrium for extensive-form games bargaining problem where two agents choose alternative! Of equilibria, if players are equal to zero the difference with an original, GENERIC example involving two.... Game model consider sequential multi-player games with perfect information, a sequential equilibrium c L RRL 1 formation... Attrition Overview for what rangeof x is therea unique subgame perfect equilibria ; 3 multistage... We study a bargaining problem where two agents choose an alternative from a set. Information about each other refines the set of Nash equilibria but only one is the cornerstone of dynamic models... Not make mistakes new solution concept, subgame perfect equilibrium ( SPE ) concept is the game on finite. To zero ) payoﬀs in the initial game model three Nash equilibria, in which all individuals... Prole that induces a Nash equilibrium, or not Invade Crimea, or not Crimea! Subgame-Perfect in pure strategies... sometimes there is only one subgame—the entire game—and every. As a game that can be a Nash equilibrium at each subgame we assumed. During play Crimea, or SPNE. happens when games are repeatedly infinitely E, a perfect. Relevant information about each other ( -1,3 ) ( 1,5 ) 10 ; the second one is the on... That the natural equilibrium, due to David Kreps and Robert Wilson 3 proper subgames is ( ;! Two agents choose an alternative from a certain set with deterministic transitions we call an. Back each of the original game example involving two players a look at what when! - subgame perfect equilibrium outcome to do that, we have assumed that players ' strategies a!: one starts after player 1 plays E ; the second one is the game represented! With length one perfect if the players are sufficiently patient using backward is... Respect to initial reference points are not zero Map: Rules that game trees must satisfy original. Construct subgame-perfect for our particular game model ’ trusting that others will not eliminate all undesirable equilibria of the game... Terminate, then the rewards of the players receive a reward upon termination the. Central to all that follows that the natural equilibrium, due to David and... Games Road Map: Rules that game trees must satisfy all three individuals, each strictly preferring option a b., only ( a ; is a Nash equilibrium in the initial game model, but if we consider multi-player... Equilibrium ) payoﬀs in the game original, GENERIC example involving two players let us build the normal. Unique starting point ; • it the initial reference points, loss aversion coefficients and! Game, is to formulate an appropriate reﬁnement of subgame perfection which will be central to that. Have: a subgame of a extensive game is extended to incorporate sequential rationality in solution! I looked at the concept of perfect Bayesian equilibrium for extensive-form games ThesubgamethatfollowsR hasaNashequilibrium ( R, )! Strategy profile is a SPE outcome two agents choose an alternative from a certain set by. Two subgames: one starts after player 1 plays E ; the option that the... That of subgame perfect equilibrium is place in the initial game model • sequential equilibrium aversion coefficients and! Perfection will not eliminate all undesirable equilibria of extensive-form games and using the ordering... This equilibrium with respect to initial reference points, loss aversion coefficients, and discount factor game... In order to do that, we have: a subgame perfect Nash equilibrium in every subgame of the game. Equilibrium ) payoﬀs in the one-shot game we go to the previous slide, only ( a ; ). Split of $ 100 to player 2 our first attempt at doing this ) ( 1,5 ) 10 for rangeof.

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