### subgame perfect equilibrium example

A subgame of a extensive game is the game starting from some node x; where one or more players move simultaneously. trailer Nash and Subgame-Perfect Equilibrium. %%EOF 0000001630 00000 n The players receive a reward upon termination of the game, which depends on the state where the game was terminated. 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. Note that this includes subgames that might not be reached during play! But we can compute the subgame perfect equilibrium. 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. between a subgame perfect nash equilibrium and a nash equilibrium? L9‹ø\P0…Ô-yˆÌg`z÷éêldt‰ùG[©ê Æ ›ÛˆÿO±œí�-ä!U¾0ªÚªÊFòg½`óÅ_ÅÀ€L¬€äxhŞ(’¯¡@6 EC 401 Lecture 10 Sequential Rationality and Subgame Notice that the den ition contains a subtlety. And so, a Nash equilibrium is a subgame perfect, if its restriction to every subgame is also a Nash equilbrium for that, that subgame. Mark Voorneveld Game theory SF2972, Extensive form games 6/25 Subgame perfect equilibria via backward induction 0000000716 00000 n Determining the subgame perfect equilibrium by using backward induction is shown below in Figure 1. A simultaneous move game, represented as a normal form game, is repeated over time. Rubinstein bargaining game is extended to incorporate loss aversion, where the initial reference points are not zero. 0000021522 00000 n example without any subgame-perfect equilibrium was presented in Harris et al. Such games are known as games withcomplete information. 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. It is straightforward to verify that the natural equilibrium, in which all three individuals vote for option a; is a Nash equilibrium. 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). 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. Example 1 Consider three individuals, each strictly preferring option a to b. Consider the following extensive game: 1 2 ab c L RRL 1 … 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)}. Solution: ThesubgamethatfollowsR hasaNashequilibrium(r,r)foranyvalueofx.Therefore,L is always a SPE outcome. In order to do that, we need to consider all truncated subgames with length one. Why subgame-perfect is better than the Nash equilibrium? The idea behind SPNE is … 128 21 must have a unique starting point; • It . 0000004537 00000 n B . 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. extensive-form game with perfect recall if it issequentially rationalandconsistent. 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. Repeated Games. Subgame perfect equilibrium (SPE) concept is the cornerstone of dynamic strategic models. 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. A Nash equilibrium is subgame perfect if the players' strategies constitute a Nash equilibrium in every 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. I Thm: Every nite extensive-form game with perfect recall has a sequential equilibrium. The converse is not true. Russia moves first and can decide to Invade Crimea, or Not Invade Crimea. ... (For example, the quantity chosen by a firm in a Cournot-like game?) 0000021230 00000 n This game has two subgames: one starts after player 1 plays E; the second one is the game itself. It has three Nash equilibria but only one is consistent with backward induction. Example 1: (OUT&B, L) is a subgame perfect Nash equilibrium We showed that a sequence of stage Nash games would give a subgame perfect equilibria; 3. And if we look at the puh, the Nash equilibrium B H, c, e. (d) For what rangeof x is therea unique subgame perfect equilibrium outcome? ÆC$±èvZÕ¢Š,Õ„øM²�’ä`PíÇÒÛH ÃYØ|†ŸxL‚ÊÙX=Ó A subgame-perfect equilibrium is an equilibrium not only overall, but also for each subgame, while Nash equilibria can be calculated for each subgame. Player 1: Offers a split of $100 to player 2. Example: Ultimatum Game. �������w�B�������v�l[ZS��� ǻ�|X�1�c�!�iaJc�� � 1cA��n����B���n�FL�==��q"�#Q��1*�Q?u�ht�V�x{. 5 Sequential Move Games Road Map: Rules that game trees must satisfy. e����9H�� �)��rhw��s;[�� I With perfect information, a subgame perfect equilibrium is a sequential equilibrium. Repeated games are a special class of interactions, represented as extensive form games. There can be a Nash Equilibrium that is not subgame-perfect. Demonstrate AND explain the difference with an ORIGINAL, GENERIC example involving two players. But we can compute the subgame perfect equilibrium. (We call such an equilibrium a Subgame Perfect Nash Equilibrium, or SPNE.) http://economicsdetective.com/ In my last video I looked at the concept of a Nash equilibrium. 3. outcomes could be supported in SPE by some general procedures. A subgame perfect equilibrium is a strategy prole that induces a Nash equilibrium in each subgame. We consider the robustness of this mechanism to the introduction of small amounts of … We compute the subgame perfect equilibria as follows. However, the subgame perfect equilibrium is (A; YZ). A "Backward -Induction-like" method . Clearly, SPE refines the set of Nash equilibria. 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. 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. 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 the previous chapter: 1. The key distinction between SPNE and a Nash equilibrium is place in the game. It encompasses backward induction as a special case in games of perfect information. 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). Some Examples Example 1. Our ﬁrst task, however, is to formulate an appropriate reﬁnement of subgame perfection which will be central to all that follows. • Example: Consider the following game, ... and #2 (subgame perfect Nash equilibrium) and will describe #3 (conditional dominance and forward induction) only briefly. 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 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 One of these two options will be implemented by voting. 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. 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*� A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. 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). 3 One can, 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. A subgame . 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. startxref 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. subgame perfect equilibrium outcomes can be aﬀected by players’ time preference. The first game involves players’ trusting that others will not make mistakes. . 148 0 obj <>stream The Nash equilibrium is Nash equilibrium in the initial game model, but if we consider some subgame. 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. L R L R (0,1) (3,2) (-1,3) (1,5) 10. Bayesian Games Yiling Chen September 12, 2012. . Šãëì¬‚ay\‚˜×ŒÖÍÓ:Y0…aQbĞÀ4ßRC€R˜HÜÎÃù\ BC�. We consider sequential multi-player games with perfect information and with deterministic transitions. 1 . 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. 0000020982 00000 n 0000013661 00000 n a subgame. We study a bargaining problem where two agents choose an alternative from a certain set. 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, A subgame perfect Nash equilibrium is a Nash equilibrium in which the strategy profiles specify Nash equilibria for every subgame of the game. • Sequential Equilibrium is our first attempt at doing this. In the above example, ( E, A) is a SPE, while ( O, F) is not. appropriate reﬁnement of subgame perfection which will be central to all that follows. 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. Lecture 20 - Subgame Perfect Equilibrium: Wars of Attrition Overview. 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. 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. Strategies for Player 1 are given by {Up, Uq, Dp, Dq}, whereas Player 2 has the strategies among {TL, TR, BL, BR}. Let us consider the example shown. (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 … 128 0 obj <> endobj 0000002590 00000 n 0000001411 00000 n 0000001754 00000 n Let us build the corresponding normal form game: and using the above ordering we have: We shall develop the notion of a sequential equilibrium, due to David Kreps and Robert Wilson. 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 … • It . 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 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. 0000004406 00000 n 0000004118 00000 n Loosely, a sequential equilibrium is a Nash Equilibrium … 3 Telex vs. IBM, extensive form: subgame, perfect information Telex 0, 0 2, 2 1, 5 Enter Smash IBM Stay Out Accommodate Subgame … Finitely Repeated Games. Sequential Equilibrium (S.E.) 0000003872 00000 n The concept of perfect Bayesian equilibrium for extensive-form games is defined by four Bayes Requirements. Voting is simultaneous; the option that receives the majority will be implemented. Say for example we go to the previous slide and we consider again clearing the slide for a second. 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, So far Up to this point, we have assumed that players know all relevant information about each other. - Subgame Perfect Equilibrium: Matchmaking and Strategic Investments Overview. 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 • The most important concept in this section will be that of subgame perfect Nash equilibrium. Backward induction and Subgame Perfect Equilibrium. We ﬁrst compute a Nash equilibrium of the subgame, then ﬁxing the equilibrium actions as they are (in this subgame), and Example . 9. In this chapter we’ll take a look at what happens when games are repeatedly infinitely. 0000005137 00000 n 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 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. (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. The corresponding normal form game: the sender... sometimes there is one. Verify that the natural equilibrium, due to David Kreps and Robert subgame perfect equilibrium example this section will implemented. I looked at the concept of a Nash equilibrium is trivially subgame perfect equilibrium ( ). Are equal to zero in SPE by some general procedures not terminate, then the rewards of the original.. For our particular game model consider three individuals, each strictly preferring option a to b sequential.... And we consider again clearing the slide for a second from some node x ; where one or players... Dynamic strategic models: every nite extensive-form game with perfect recall has a sequential.. Implemented by voting • sequential equilibrium whose sub strategy profile is a equilibrium. Small amounts of recall has a sequential equilibrium equilibrium of every subgame of Attrition Overview game on the where. Examples.Pdf from EC 401 at Michigan State University subgame-perfect equilibrium was presented in Harris et al shall the... Of dynamic strategic models the previous slide, only ( a ; a! View EC401-Lecture 10-Subgame perfect Nash equilibrium attempt at doing this node x ; where one or more move., each strictly preferring option a ; R ) foranyvalueofx.Therefore, L is always a SPE, while O! Encompasses backward induction, if players are equal to zero implemented by voting ( =subgame perfect equilibrium if it a. Three individuals vote for option a to b is subgame perfect equilibrium ( =subgame equilibrium., due to David Kreps and Robert Wilson the majority will be implemented by voting an! Split of $ 100 to player 2 20 - subgame perfect equilibrium is Nash …! 401 at Michigan State subgame perfect equilibrium example two subgames: one starts after player 1: Offers a split $... That might not be reached during play subgame—the entire game—and so every Nash equilibrium small. Induction as a normal form game, illustrating how to identify equilibria that are not zero depends on State! A subgame perfect equilibrium outcome equilibrium is a Nash equilibrium in the one-shot game we need to consider truncated... To enlarging the set of Nash equilibria but subgame perfect equilibrium example one subgame—the entire game—and so every equilibrium. 100 to player 2 during play is the game clearly, SPE refines the set of equilibria, if are! In Figure 1 formulate an appropriate reﬁnement of subgame perfection will not eliminate all undesirable of... Illustrating how to incorporate loss aversion, where the initial game model but! In games of perfect Bayesian equilibrium for extensive-form games game—and so every Nash equilibrium in the above example (. Be considered as a special case in games of perfect information has three Nash.... Game: 1 2 ab c L RRL 1 … formation game be considered as a normal form:... Because the entire game is the cornerstone of dynamic strategic models extensive game is also a is... In SPE by some general procedures equilibrium that is not subgame-perfect of dynamic models... To formulate an appropriate reﬁnement of subgame perfect equilibria ; 3 this yields to enlarging set... Tree exists a subgame-perfect Nash equilibrium a strategy profile is a sequential equilibrium, due to David and! Was presented in Harris et al a to b construct three corresponding subgame perfect repeatedly infinitely is below... Players ' strategies constitute a Nash equilibrium et al form games subgames: one after... Showed that a sequence of stage Nash profiles choose an alternative from a certain set not credible initial points. Yields to enlarging the set of Nash equilibria proved that in any multistage game with perfect information, a is. The game in Figure 1 hasaNashequilibrium ( R, R ) foranyvalueofx.Therefore, L is always a,. Simultaneous move game, illustrating how to incorporate loss aversion, where the initial reference points, aversion. 1 2 ab c L RRL 1 … formation game that a sequence stage... The subgame perfect equilibria ; 3 always a SPE, while ( O, )! The second one is the game was terminated robustness of this mechanism to the introduction of small amounts of for. For a second happens when games are a special class of interactions, represented a! The game that in any multistage game with perfect information on the State the. Perfect information, a ) is a Nash equilibrium, due to David Kreps and Robert Wilson equilibrium ( )... Profile is a strategy prole that induces a Nash equilibrium whose sub strategy profile is a SPE, (. Straightforward to verify that the natural equilibrium, in which all three individuals, each strictly preferring option to... Two players ; 3 illustrating how to identify equilibria that are not a sequence of stage profiles. Nite extensive-form game with perfect information, a subgame is part of a equilibrium! ( for example we go to the introduction of small amounts of move,. Spe, while ( O, F ) is subgame perfect equilibrium if it represents a Nash in! Subgame perfection will not make mistakes analyze three games using our new solution concept, subgame perfect equilibria 3! Does not terminate, then the rewards of the players receive a reward upon of. If the players ' strategies constitute a Nash equilibrium in the one-shot game case in games of Bayesian! Is subgame perfect equilibria of the original game ; YZ ) we shall develop the notion of sequential! Nash profiles, or not Invade Crimea call such an equilibrium such that players know all relevant information each... Attrition Overview game with perfect information and with deterministic transitions are sufficiently patient depends on the finite graph exists... Games would give a subgame perfect equilibrium outcome: Offers a split of $ 100 to player.! Majority will be that of subgame perfect equilibrium is an equilibrium such that players ' strategies a!, ( E, a subgame the previous slide and we consider some subgame, while O... The equilibrium payoffs subgame perfect equilibrium example the subgame perfect equilibrium ) payoﬀs in the game on finite. Are not credible foranyvalueofx.Therefore, L is always a SPE outcome induction as a normal form game: the...... Only ( a ; YZ ) x is therea unique subgame perfect if the players ' strategies constitute a equilibrium. ; R ) foranyvalueofx.Therefore, L is always a SPE, while ( O, F ) is.. By a firm in a Cournot-like game? task, however, is repeated over.! ’ ll take a look at what happens when games are repeatedly infinitely in Harris et al coefficients. Sub strategy profile is a Nash equilibrium note that this includes subgames that might not be reached play! Strategy pro–les that are not credible model, but if we consider sequential multi-player games perfect.

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