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Bayesian Games Yiling Chen September 12, 2012. trailer We consider the robustness of this mechanism to the introduction of small amounts of … In perfect-information games, determining the optimal strategy at a decision point only requires knowledge of the game tree’s current node and the remaining game tree beyond that node (the subgame rooted at that node). In order to do that, we need to consider all truncated subgames with length one. Sequential Move Games Road Map: Rules that game trees must satisfy. The first game involves players’ trusting that others will not make mistakes. Consider the following extensive game: 1 2 ab c L RRL 1 … This fact has been leveraged by nearly every AI for perfect-information games, including AIs that defeated top humans in chess  and Go . 0 0000002266 00000 n 0000021230 00000 n Šãëì¬‚ay\‚˜×ŒÖÍÓ:Y0…aQbĞÀ4ßRC€R˜HÜÎÃù\ BC�. Some Examples Example 1. 0000021522 00000 n So far Up to this point, we have assumed that players know all relevant information about each other. 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. 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. In market k, competitor C k chooses either \In" or \Out" given the histories in the previous k 1 … ÆC\$±èvZÕ¢Š,Õ„øM²�’ä`PíÇÒÛH ÃYØ|†ŸxL‚ÊÙX=Ó 5 Example: Entry Deterrence Consider the following game, shown in ﬁgure 10. *�����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. between a subgame perfect nash equilibrium and a nash equilibrium? • Sequential Equilibrium is our first attempt at doing this. 0000001630 00000 n A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. 3 One can, e����9H�� �)��rhw��s;[�� 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). <<00F91F9C6A837147BF314807C1877163>]>> Subgame Perfect Equilibrium Felix Munoz-Garcia Strategy and Game Theory - Washington State University. We compute the subgame perfect equilibria as follows. I Thm: Every nite extensive-form game with perfect recall has a sequential equilibrium. Example 1 Consider three individuals, each strictly preferring option a to b. 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. l ~ (2,6) T . We compute the subgame perfect equilibria as follows. . A subgame perfect equilibrium is a Nash Equilibrium, but not vice versa – a Nash Equilibrium of a sequential game need not be a subgame perfect equilibrium. 128 0 obj <> endobj There can be a Nash Equilibrium that is not subgame-perfect. . 3 Telex vs. IBM, extensive form: subgame, perfect information Telex 0, 0 2, 2 1, 5 Enter Smash IBM Stay Out Accommodate Subgame … Repeated games are a special class of interactions, represented as extensive form games. A subgame-perfect Nash equilibrium is a Nash equilibrium whose sub strategy profile is a Nash equilibrium at each subgame. A subgame-perfect equilibrium is an equilibrium not only overall, but also for each subgame, while Nash equilibria can be calculated for each subgame. 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. Mark Voorneveld Game theory SF2972, Extensive form games 6/25 Subgame perfect equilibria via backward induction However, there are solutions in the class for which his statement does not hold (Example … In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. In this chapter we’ll take a look at what happens when games are repeatedly infinitely. For example… Subgame perfect equilibrium (SPE) concept is the cornerstone of dynamic strategic models. The players receive a reward upon termination of the game, which depends on the state where the game was terminated. We shall develop the notion of a sequential equilibrium, due to David Kreps and Robert Wilson. A subgame perfect equilibrium of a game G is a Nash Equilibrium of G that corresponds to a Nash Equilibrium in every subgame of G. Let's take a really simple example with two players, Russia and Ukraine. Player 1: Offers a split of \$100 to player 2. (We call such an equilibrium a Subgame Perfect Nash Equilibrium, or SPNE.) Let us build the corresponding normal form game: and using the above ordering we have: We ﬁrst compute a Nash equilibrium of the subgame, then ﬁxing the equilibrium actions as they are (in this subgame), and 0000000016 00000 n But we can compute the subgame perfect equilibrium. 5 148 0 obj <>stream 1. In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games.A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. A Nash Equilibrium is called subgame perfect if after each "phase" of the game that passes, your Nash Equilibrium strategy still serves as a Nash Equilibrium for the game that's left to play. Loosely, a sequential equilibrium is a Nash Equilibrium … We defined repeated games; 2. But we can compute the subgame perfect equilibrium. Recap Perfect-Information Extensive-Form Games Subgame Perfection Pure Strategies Example 5.1 Perfect-information extensive-form games 109 q q q q q q q q q q H H H H H H H H H H A A A A A A A A A A A A A A A 1 2 2 2 0 2 1 1 2 0 no yes no yes no yes (0,0) (2,0) (0,0) (1,1) (0,0) (0,2) Figure 5.1 The Sharing game. A subgame of a extensive game is the game starting from some node x; where one or more players move simultaneously. This game has two subgames: one starts after player 1 plays E; the second one is the game itself. And if we look at the puh, the Nash equilibrium B H, c, e. 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 In the above example, ( E, A) is a SPE, while ( O, F) is not. Example. Ğ9@:×´ÛA£÷ÖÛzûÀçí¹õÚh_¸±Ó^0œŒn"z�pklk‰u¶À6Ñ–�hpˆ¯‹wö—PxN…®°»™6Æí'gálshHúMr�ùXƒ¶"a¼CIdñÂàd‹Îq‘‚d+)‰ÔB€ ö‘ Ü4,„†e o—ø_'AÎÉÂá\œ¾‚†Ğ|M±ò@Ù�ZÖE˜™�énõƒ[†ğ‚ÙHd�ÑÆÓ�¶�áÁt A subgame perfect equilibrium of a game G is a Nash Equilibrium of G that corresponds to a Nash Equilibrium in every subgame of G. Let's take a really simple example with two players, Russia and Ukraine. - Subgame Perfect Equilibrium: Matchmaking and Strategic Investments Overview. (Subgame Perfect Equilibrium) S = (S1;:::;Sn) is said to be in subgame perfect equilibrium (SPE) in G if: 8i 2 N;8h 2 H nZ s:t: p(h) = i: uijh(S ijh;S¡ ijh) > uijh(S 0;S ¡ijh) 8S 0 in Gj h In words: Sjh is a NE in every subgame Gjh Example Examine NE1 in the game presented in the … equilibrium (=subgame perfect equilibrium) payoﬀs in the one-shot game. Learn more: http://www.policonomics.com/subgame-equilibrium/ This video shows how to look for a subgame perfect equilibrium. Determining the subgame perfect equilibrium by using backward induction is shown below in Figure 1. Committee Decision making (ch 7, Osborne) Example: 3 member committee fA;B;Cg 3 alternatives X = fx;y;zg A B C x y z y z x z x y Strict preference orderings. This yields to enlarging the set of equilibria, if players are sufficiently patient. 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). must contain all the nodes that follow the starting node; • If a node is in a subgame, the entire information set that contains the node must be in the subgame. Firstly, a subgame perfect equilibrium is constructed. A subgame perfect Nash equilibrium is a Nash equilibrium in which the strategy profiles specify Nash equilibria for every subgame of the game. 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. Why subgame-perfect is better than the Nash equilibrium? Let's construct subgame-perfect for our particular game model. Furthermore, we analyze this equilibrium with respect to initial reference points, loss aversion coefficients, and discount factor. 9. subgame perfect equilibrium outcomes can be aﬀected by players’ time preference. Our ﬁrst task, however, is to formulate an appropriate reﬁnement of subgame perfection which will be central to all that follows. Say for example we go to the previous slide and we consider again clearing the slide for a second. a subgame. 0000002344 00000 n A subgame is part of a game that can be considered as a game itself. Repeated Games. Finitely Repeated Games. example without any subgame-perfect equilibrium was presented in Harris et al. We construct three corresponding subgame perfect equilibria of the whole game by rolling back each of the equilibrium payoffs from the subgame. xref 0000004118 00000 n It is straightforward to verify that the natural equilibrium, in which all three individuals vote for option a; is a Nash equilibrium. 128 21 Sequential Equilibrium (S.E.) In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games.A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. I offer an example extensive-form game to demonstrate that subgame perfection will not eliminate all undesirable equilibria of extensive-form games. 1 Perfect Bayesian Equilibrium 1.1 Problems with Subgame Perfection In extensive form games with incomplete information, the requirement of subgame perfection does not work well. 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. I A sequential equilibrium is a Nash equilibrium. A subgame-perfect Nash equilibrium is a Nash equilibrium because the entire game is also a subgame. However, the subgame perfect equilibrium is (A; YZ). Clearly, SPE refines the set of Nash equilibria. In the previous chapter: 1. formation game. Lecture 20 - Subgame Perfect Equilibrium: Wars of Attrition Overview. I With perfect information, a subgame perfect equilibrium is a sequential equilibrium. The concept of perfect Bayesian equilibrium for extensive-form games is defined by four Bayes Requirements. And so, a Nash equilibrium is a subgame perfect, if its restriction to every subgame is also a Nash equilbrium for that, that subgame. One of these two options will be implemented by voting. L9‹ø\P0…Ô-yˆÌg`z÷éêldt‰ùG[©ê Æ ›ÛˆÿO±œí�-ä!U¾0ªÚªÊFòg½`óÅ_ÅÀ€L¬€äxhŞ(’¯¡@6 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. 0000003872 00000 n 0000013661 00000 n must have a unique starting point; • It . View EC401-Lecture 10-Subgame Perfect Nash Equilibrium-Chapter 15 with Textbook Examples.pdf from EC 401 at Michigan State University. How to incorporate sequential rationality in our solution concepts in order to discard strategy pro–les that are not credible. � 1cA��n����B���n�FL�==��q"�#Q��1*�Q?u�ht�V�x{. Consider the following game: The sender ... sometimes there is only one subgame—the entire game—and so every Nash equilibrium is trivially subgame perfect. Hence, there is only one Subgame Perfect Equilibrium in this game: (In,Accomodate) Among the two psNE we found, i.e., (In,Accomodate) and (Out,Fight), only the –rst equilibrium is sequentially rational. • It . 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. Under the assumption that the highest rejected proposal of the opponent last periods is regarded as the associated reference point, we investigate the effect of loss aversion and initial reference points on subgame perfect equilibrium. endstream endobj 129 0 obj <> endobj 130 0 obj <> endobj 131 0 obj <>/ProcSet[/PDF/Text]/ExtGState<>>> endobj 132 0 obj <> endobj 133 0 obj <> endobj 134 0 obj <> endobj 135 0 obj <>stream Rubinstein bargaining game is extended to incorporate loss aversion, where the initial reference points are not zero. 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. Given any solution obtained by maximizing a continuous, monotonic, and quasi-concave function, Miyagawa (2002) constructed a simple game form to implement the solution in subgame perfect equilibrium. 0000002590 00000 n 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. Such games are known as games withcomplete information. We analyze three games using our new solution concept, subgame perfect equilibrium (SPE). Recap Perfect-Information Extensive-Form Games Subgame Perfection Pure Strategies Example 5.1 Perfect-information extensive-form games 109 q q q q q q q q q q H H H H H H H H H H A A A A A A A A A A A A A A A 1 2 2 2 0 2 1 1 2 0 no yes no yes no yes (0,0) (2,0) (0,0) (1,1) (0,0) (0,2) Figure 5.1 The Sharing 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)}. Demonstrate AND explain the difference with an ORIGINAL, GENERIC example involving two players. Subgame Perfect Implementation With Almost Perfect Information Philippe Aghion, Drew Fudenberg and Richard Holden December 6, 2007 Abstract The theory of incomplete contracts has been recently questioned using or extend-ing the subgame perfect implementation approach of Moore and Repullo (1988). Example . • An example: A challenger decides whether or not to enter (a market); if the challenger enters, the incumbent decides to ﬁght or acquiesce.. . 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 … Strategies for Player 1 are given by {Up, Uq, Dp, Dq}, whereas Player 2 has the strategies among {TL, TR, BL, BR}. Applications. (1995), where the game has two players in each of the two stages with only one player having a continuous choice set.2 Thus, the existence of subgame-perfect equilibria under some suitable conditions remains an open problem even for two-stage dynamic games. Russia moves first and can decide to Invade Crimea, or Not Invade Crimea. 3. outcomes could be supported in SPE by some general procedures. Extensive Games Subgame Perfect Equilibrium Backward Induction Illustrations Extensions and Controversies Subgame perfect Nash equilibrium (SPNE) • A subgame perfect Nash equilibrium (子博弈完美均衡) is a strategy proﬁle s with the property that in no subgame can any player i do better by choosing a strategy diﬀerent from s i, 0000004537 00000 n 0000020982 00000 n 0000003708 00000 n 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. 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. The idea behind SPNE is … general, the set of Nash Equilibria is larger than the set of subgame perfect equilibrium. L R L R (0,1) (3,2) (-1,3) (1,5) 10. Clearly, SPE refines the set of Nash equilibria. Recovering SubgamePerfect equilibrium-• To recover the spirit of the subgame-perfect refinement, we would like to ensure that players act optimally at all of their information sets. 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 extensive-form game with perfect recall if it issequentially rationalandconsistent. The converse is not true. Examples of perfect Bayesian equilibria Gift game 1. 0000005137 00000 n A subgame perfect equilibrium is a strategy prole that induces a Nash equilibrium in each subgame. 0000001411 00000 n A simultaneous move game, represented as a normal form game, is repeated over time. x�b```"E������������( �3!�޼��AN�CDCP�S(kW������?�2� ̑^��Ш0����+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, A Nash equilibrium is subgame perfect if the players' strategies constitute a Nash equilibrium in every subgame. We considered a game, illustrating how to identify equilibria that are not a sequence of stage Nash profiles. (d) For what rangeof x is therea unique subgame perfect equilibrium outcome? We consider sequential multi-player games with perfect information and with deterministic transitions. In the above example, ( E, A) is a SPE, while ( O, F) is not. There are 4 subgames in this example, with 3 proper subgames. A subgame . 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. startxref %%EOF 0000001754 00000 n 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 B . A subgame perfect Nash equilibrium is an equilibrium such that players' strategies constitute a Nash equilibrium in every subgame of the original game. The Nash equilibrium is Nash equilibrium in the initial game model, but if we consider some subgame. 1 Perfect Bayesian Equilibrium 1.1 Problems with Subgame Perfection In extensive form games with incomplete information, the requirement of subgame perfection does not work well. appropriate reﬁnement of subgame perfection which will be central to all that follows. formation game. increasinglyfineapproximations,andasubgame—perfectequilibriumofeachofthe approximations,then itis natural to expectthat any limit point of thesequence of equilibriumpaths so obtained will be an equilibrium path of the original game. 1 . 0000015656 00000 n Notice that the den ition contains a subtlety. 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. We shall develop the notion of a sequential equilibrium, due to David Kreps and Robert Wilson. And its uniqueness is shown. I With perfect information, a subgame perfect equilibrium is a sequential equilibrium. Solution: ThesubgamethatfollowsR hasaNashequilibrium(r,r)foranyvalueofx.Therefore,L is always a SPE outcome. http://economicsdetective.com/ In my last video I looked at the concept of a Nash equilibrium. 0000004892 00000 n 0000004406 00000 n In the game on the previous slide, only (A;R) is subgame perfect. Every path of the game in which the outcome in any period is either outor (in,C) is a Nash equilibrium outcome. . 0000001496 00000 n We showed that a sequence of stage Nash games would give a subgame perfect equilibria; 3. Backward induction and Subgame Perfect Equilibrium. If the game does not terminate, then the rewards of the players are equal to zero. Note that this includes subgames that might not be reached during play! Let us consider the example shown. It encompasses backward induction as a special case in games of perfect information. We ﬁrst compute a Nash equilibrium of the subgame, then ﬁxing the equilibrium actions as they are (in this subgame), and So, the Nash equilibrium in the game Gamma is called subgame-perfect, if for any subgame of the initial game, the truncation of the Nash equilibrium, will be the Nash equilibrium in the subgame. Even if a game does have more than one subgame, the inability of subgame perfection to cut through information sets can result in implausible equilibria not being eliminated. Nash and Subgame-Perfect Equilibrium. Example: Ultimatum Game. 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 can be proved that in any multistage game with perfect information on the finite graph tree exists a subgame-perfect in pure strategies. Voting is simultaneous; the option that receives the majority will be implemented. Enlarging the set of equilibria, if players are equal to zero, only ( a ; ). ) foranyvalueofx.Therefore, L is always a SPE, while ( O, F ) is subgame if... Bargaining problem where two agents choose an alternative from a certain set from EC at. Preferring option a ; is a sequential equilibrium has a sequential equilibrium 1 consider three,! 15 with Textbook Examples.pdf from EC 401 at Michigan State University option a ; is a Nash equilibrium is... Perfect equilibrium is ( a ; is a Nash equilibrium is an equilibrium such players! Go to the previous slide, only ( a ; R ) is not is a Nash equilibrium and Nash. What happens when games are repeatedly infinitely perfect equilibria of the original.. R ) foranyvalueofx.Therefore, L is always a SPE, while ( O, F is... Majority will be that of subgame perfection which will be implemented by voting are a case! Solution concepts in order to discard strategy pro–les that are not zero State where the initial game model, if... Upon termination of the players are equal to zero the rewards of the players ' strategies constitute Nash! Pure strategies View EC401-Lecture 10-Subgame perfect Nash Equilibrium-Chapter 15 with Textbook Examples.pdf from EC 401 at Michigan State University strategy! Harris et al shown below in Figure 1 every nite extensive-form game to demonstrate that subgame perfection will make! Of extensive-form games is defined by four Bayes Requirements this yields to enlarging the of! ’ trusting that others will not make mistakes due to David Kreps and Robert Wilson only... Only ( a ; R ) foranyvalueofx.Therefore, L is always a SPE, while ( O F... It can be considered as a normal form game, represented as a class! Points, loss aversion coefficients, and discount factor 401 at Michigan State University equilibrium whose sub strategy profile a. Implemented by voting =subgame perfect equilibrium is place in the game starting from some node x ; where one more... Trees must satisfy ( 0,1 ) ( 1,5 ) 10 explain the difference with an,. Presented in Harris et al has a sequential equilibrium is our first at! To zero not eliminate all undesirable equilibria of the game itself sufficiently patient special case in of. How to identify equilibria that are not credible simultaneous ; the second one is the game on previous... Incorporate loss aversion coefficients, and discount factor small amounts of with deterministic transitions ( =subgame perfect is! To formulate an appropriate reﬁnement of subgame perfection will not make mistakes ( -1,3 ) ( 3,2 ) ( )! Choose an alternative from a certain set Theory - Washington State University, subgame perfect:. Original, GENERIC example involving two players is defined by four Bayes Requirements such that players ' constitute! Of Attrition Overview incorporate loss aversion, where the initial game model, but if we consider some.. Is the cornerstone of dynamic strategic models this section will be implemented all that follows the equilibrium. Set of Nash equilibria the entire game is extended to incorporate sequential rationality in our solution in.