A seller is privately informed of the value vof the good that she sells to a buyer. I bidder i’s payo is u i(b;v) = 1(b i max j6=i b j)(v i b i). BNEs and Sequential rationality So far we have learned how to –nd BNEs in incomplete information games. ��(G��g~�4)��h̺�2�csRE�Y���q&��]�S����k��4�H+U�C�T��O��N�\�σ~/9���Mx��cÂXeQ�|ף��/PˠԬ�4N�_x�X�X� ��[��4�e�ᶽ���6�(�K�\��3{�[��j7�����&���:��F�sU_�è�a�^硓 If strategy sets and type sets are compact, payo functions are Networks: Lectures 20-22 Bayesian Games Existence of Bayesian Nash Equilibria Theorem Consider a nite incomplete information (Bayesian) game. De ne a Perfect Bayesian Equilibrium for this game. So now suppose 2 plays iat that last information set. 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)}. In a PBE, every agent’s strategy should be a best response under the belief system, and the belief system depends on agents’ strategy proﬁle when there is signaling among agents. xref %%EOF Also when I combine the matrices I find no Pure Strategy Bayesian Equilibrium. Recall from the answers to the last problem set that (af;dh) is subgame perfect; we see here that it is not weak perfect Bayesian. Beforeplayingeach player puts a dollardown. 145 0 obj <>stream Player 1 observes her type and decides whether to choose L or R. If player 1 chooses R, the game ends. A seller is privately informed of the value vof the good that she sells to a buyer. Suppose now that the game from part a is played twice. plausibility order on the set of histories is choice measurable, which is a necessary condition for a PBE to be a SE. 1.2 Perfect Bayesian Equilibrium Let G be an extensiev form game. !S�8{0ް��)���!kҿ�KVa��`%��Ŷn���*Ab�up�#�I���"� 136 0 obj <> endobj gW�ps��xT��2 Er��;cbߋ�o��K��xc���>� Xa���pC8�7���~. This problem addressed by sequential equilibrium, which explicitly requires that the players play a best reply at every information set (sequential rationality) and that the players’ beliefs are "consistent" with the other players’ strategies. Perfect Bayesian Equilibrium When players move sequentially and have private infor-mation, some of the Bayesian Nash equilibria may involve strategies that are not sequentially rational. Formalizing the Game … A PBE consists of a pair of strategy proﬁle and belief system. Because we can™t use Bayes™rule, WPB does not constrain beliefs! Solutions & Answers to Exercise Set 1 Giuseppe De Feo May 10, 2011 1 Equilibrium concepts Exercise 1 (Training and payment system, By Kim Swales) Two players: The employee (Raquel) and the employer (Vera). Problem Set 10 1. Usually, there will be two counterparts in the game, one in informed and the other not (informed workers and uninformed ﬁrms, informed normal incumbent and uninformed entrant). Now look at Row. 35. We are doing great! First note that if the opponent is strong, it is a dominant strategy for him to play F — ﬁght. %PDF-1.4 ��β������䛻$�I���_�8\��9~8d�$��7$�i��'c��,�����eR�� `@ Recall that: De nition 1 A ebhaviaolr sattrgey for player i is a function i: H i ( A i) such that for any h i H i, the suporpt of i ( h i) is ontacined in the set of actions available at h i. eW now augment a plyear s strategy to explicitly account for his beliefs. Player 1 observes her type and decides whether to choose L or R. If player 1 chooses R, the game ends. That means that all BNE are subgame perfect. • The professor draws a single card from a deck consisting of an equalnumber of kings and queens. 2. We need to modify the idea of subgame perfection so that we are Consider the NE (L, r) again. Now, we’ll de ﬁne a concept of consistency, which will be required in a perfect Bayesian Nash equilibrium. Now, if !0, it’s still well de ned. A seller is privately informed of the value vof the good that she sells to a buyer. 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. Problem 4: Semiseparating perfect Bayesian equilibrium A semiseparating (or partially separating/pooling) equilibrium is an equilibrium in which some types of Sender send the same message, while some others send some other messages. Perfect Bayesian equilibrium implies that the signal generated by the information system has a sort of preeminence o the equilibrium path in the following sense. ��4���C�&)���L��di �5�9d/D�qp b��?���� H��8=�0�1v0;T7\bX����=��/Ki� ���.2�`r �7��A��E�u Now, if !0, it’s still well de ned. perfect Bayesian equilibrium ("pooling equilibrium"): the oﬀspring is always quiet and the parent always keep the food. Remember that the "weak" in "weak perfect Bayesian" refers to the lack of restrictions on oﬀ-the-equilibrium path beliefs. In contrast, in an equilibrium a player maximizes his expected payoﬀgiven the other players’ strategies. (Market for Lemons) Here I ask that you work out some of the details in perhaps the most famous of all information economics models. The relevant notion of equilibrium will be Perfect Bayesian Equilibria, or Perfect Bayesian Nash Equilibria. Perfect Bayesian equilibrium (PBE) was invented in order to refine Bayesian Nash equilibrium in a way that is similar to how subgame-perfect Nash equilibrium refines Nash equilibrium. Note that this equilibrium also satis–es requirement 4 because there are no o⁄-the-equilibrium path information sets, so it is also a SPBE. Exercise 319.3 in Osborne (Nash Equilibria of a Card Game). 0000001437 00000 n 2. A weak perfect Bayesian equilibrium for this game is that Player 1 chooses L, Player 2 believes that Player 1 chooses L with probability 1, and Player 2 chooses L™. If the entrant enters, then each ﬁrm simultaneously chooses F or A. First, player 1 chooses among three actions: L,M, and R. If player 1 chooses R then the game ends without a move by player 2. Raquel has to choose whether to pursue training that costs $1;000 to herself or not. Turning to the second subcase, suppose 2 plays iat his last information set, 1 plays h�|U�n�F��+xl,�Mq�c8�a r0rhY-����}�^���fw��^�E��L�˸��v߫JIP�wI�E�ϟ�"�Ld�"�YP��8���Q�CP=�V������D�p����=O����>4Q�l�s��R�������z�0Q�s��S7�1��s�]��������4����Su ��4N���c�l��j�������� ��J��uSm�����v�գ�`���/�I��N���;��9�q��)��XI�IHӓj�T��]��yBƐ!�~t�U�k��r�S���L]�=R� '=���+ϣ�bx�i��zFfL|�t�8��0�J�!9�����"#�[� �O �-_�'5NҾ�ndi �(�R*c��ܢ��x�q��M�%��5G�a�pP�� 8��S 9���.1>Cl\��XՈ��b����8���6+! <<8BE3CBBEA2A431468DEFE7D45530D756>]>> • The professor draws a single card from a deck consisting of an equalnumber of kings and queens. The theorem tells us at least one such equilibrium will exist. Example 62 9.C.5 A WPBNE need not be subgame perfect. Weak Perfect Bayesian Equilibrium Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu June 16th, 2016 C. Hurtado (UIUC - Economics) Game Theory. �\���q�'�� ��$fx3��0PȵghpH h�#��y�� We do not consider this to be a choice. Problem Set 3 - Solutions Due Wednesday, December 5 Important: hand in only the two-star problems. Formalizing the Game … Kreps and Wilson [7] give a series of examples to motivate the idea that further restrictions may be natural. From Bayesian Nash Equilibrium (BNE) to Perfect Bayesian Equilibrium (PBE) FØlix Muæoz-García School of Economic Sciences Washington State University. No later submissions will be accepted! 2 Perfect Bayesian equilibrium In this section we recall the notion of perfect Bayesian equilibrium introduced in (5); we employ the same notation, which makes use of the history-based deﬁnition of extensive-form game (see, for example, (18)). The problem set is shown below: Problem Set. U��0�dC㫮�������>?�c01��j��-������(� From Bayesian Nash Equilibrium (BNE) to Perfect Bayesian Equilibrium (PBE) FØlix Muæoz-García School of Economic Sciences Washington State University. As in (5), we restrict attention to ﬁnite extensive-form games with perfect recall. Bayesian Games Suggested Solutions by Tibor Heumann 1. We are doing great! Bayesian Games 3/4/14 This problem set is due on Tuesday, 3/25/14. On the Agenda 1 Formalizing the Game 2 Systems of Beliefs and Sequential Rationality 3 Weak Perfect Bayesian Equilibrium 4 Exercises C. Hurtado (UIUC - Economics) Game Theory. Solution: ThesubgamethatfollowsR hasaNashequilibrium(r,r)foranyvalueofx.Therefore,L is always a SPE outcome. (Again, comparing to the answers to the last problem set, we see that this weak PBE is not subgame perfect.) (Market for Lemons) Here I ask that you work out some of the details in perhaps the most famous of all information economics models. 0000001303 00000 n Show that there is a unique separating perfect Bayesian equilibrium. 0000002055 00000 n Because we can™t use Bayes™rule, WPB does not constrain beliefs! In the following two extensive games, derive the strategic games and find all the pure-strategy Nash, Subgame-perfect, and Perfect Bayesian Equilibria. Theorem Consider a Bayesian game with continuous strategy spaces and continuous types. sR�_ξ/��v�6pbEx&�. So (af;di) is weak perfect Bayesian. these problems, we start by investigating a new set of solution concepts, then moev on to applications. First note that if the opponent is strong, it is a dominant strategy for him to play F — ﬁght. Perfect Bayesian Equilibrium When players move sequentially and have private infor-mation, some of the Bayesian Nash equilibria may involve strategies that are not sequentially rational. We’re headed toward restricting these beliefs in a suitable way. Sequential#rationality# # Receiver!best!responds!toLLby!playing!u(strictdomnt)since:! Networks: Lectures 20-22 Incomplete Information Incomplete Information In many game theoretic situations, one agent is unsure about the preferences or intentions of others. Consider the following game in the normal form: Player 2 C N P Player 1 C 6, 6 0, 7 0, 0 N 7, 0 3, 3 0, 0 P 0, 0 0, 0 1, 1 a) Find all the pure strategy Nash equilibria. 3. The problem is that there are usually no proper subgames. Problems with Weak Perfect Bayesian Equilibrium Example Beliefs are generated by Bayes rule wherever possible 1(S) = 1(S 2) = 0:5 But, notice that P2™s information set is never reached, so we can use Bayes™rule 2(S 1jd) = 2(S 1 \d) 2(d) 2(d) = 0! 0000003439 00000 n %PDF-1.4 %���� Each type is chosen with equal probability. 0000000496 00000 n An example of a Perfect Bayesian equilibrium in mixed strategy. /Filter /FlateDecode Problem Set 2 Spring 2016 Luca Merlino T.A.s Stefan Bergheimer and Luca Livio Due Date: March 22, 2015, 8 a.m. 1 Game Theory 1.1 Trembling Hand Perfection Two people are engaged in the following game to select either a good or a bad outcome. 136 10 (Is there a pooling equilibrium?) (d) For what rangeof x is therea unique subgame perfect equilibrium outcome? The problem with this situation is that player 2’s beliefs are not 3. consistent with player 1’s strategy. Das perfekt bayessche Gleichgewicht ist ein Lösungskonzept in der Spieltheorie. Here, I will deﬁne sequential equilibrium and apply it to some important games. First, it constrains only how individual players update beliefs on consecutive information sets—that is, from one informa-tion set to the next one that arises for the same player—thus lending itself to straightforward application in a way familiar to practitioners. Game Theory: Lecture 18 Perfect Bayesian … Each type is chosen with equal probability. Bayesian Games Suggested Solutions by Tibor Heumann 1. EK, Chapter 16. Turn in a single problem set for each pair. It is a refinement of Bayesian Nash equilibrium (BNE). (At the very least, this ensures information sets that can be reached with positive probability have beliefs assigned using Bayes’ rule.) 0 Problems with Weak Perfect Bayesian Equilibrium Example Beliefs are generated by Bayes rule wherever possible 1(S) = 1(S 2) = 0:5 But, notice that P2™s information set is never reached, so we can use Bayes™rule 2(S 1jd) = 2(S 1 \d) 2(d) 2(d) = 0! This problem addressed by sequential equilibrium, which explicitly requires that the players play a best reply at every information set (sequential rationality) and that the players’ beliefs are "consistent" with the other players’ strategies. Since these are dynamic games, we will also need to strengthen our Bayesian Nash equilibria to include the notion of perfection—as in subgame perfection. (For other parameter values, the game has a pooling equilibrium in which the oﬀspring is always quiet and the parent always gives the food.) ))Ce�:�;`A%c�~A��1P�P'�EG#�P`"RR���' A perfect Bayesian equilibrium has two components -- strategies and beliefs : By contrast to discussion in class, we give a complete formulation of the game. That is for any information set … Consider the following game in the normal form: Player 2 C N P Player 1 C 6, 6 0, 7 0, 0 N 7, 0 3, 3 0, 0 P 0, 0 0, 0 1, 1 a) Find all the pure strategy Nash equilibria. endstream endobj 137 0 obj <> endobj 138 0 obj <> endobj 139 0 obj <>/ProcSet[/PDF/Text]/ExtGState<>>> endobj 140 0 obj <> endobj 141 0 obj <> endobj 142 0 obj <> endobj 143 0 obj <>stream PERFECT BAYESIAN AND SEQUENTIAL EQUILIBRIUM 241 similar to the no-signaling condition defined below corresponds to the definition of perfect Bayesian equilibrium given in our [4] paper.] By contrast to discussion in class, we give a complete formulation of the game. Perfect Bayesian (Nash) Equilibria. In the following game, nature –rst chooses one of two types of player 1 (in the –gure, the two types are denoted t 1 and t 2). Then, the belief on player 2’s information set is well de ned. 0000000016 00000 n Problem set on repeated games and Bayesian games 1. Receiver's#beliefs#for#theinfo#set#on#theequilibrium#path:#p=½=1Rp# 2. Problem Set 5. There are no one-star problems on this problem set. Problem Set 5 Due: November 21, 2006 Recall that what Osborne calls “Weak Sequential Equilibrium” is equivalent to our “Perfect Bayesian Equilibrium.” 1. In this equilibrium, every strategy is rational given the beliefs held and every belief is consistent with the strategies played. I bidder i’s payo is u i(b;v) = 1(b i max j6=i b j)(v i b i). A simplificationof poker Consider the followingsimplificationof poker. Weak Perfect Bayesian Equilibrium • Deﬁnition: (δ∗,μ∗) is a Weak Perfect Bayesian equilibrium iﬀ a) the behaviour strategy proﬁle δ∗is sequentially rational given μ∗,and b) wherever possible μ∗is computed from δ∗using Bayes rule. The problem is that there are usually no proper subgames. 0000002301 00000 n Problem Set 5. Consider the following game of complete but imperfect information. Suppose now that the game from part a is played twice. Problem set on repeated games and Bayesian games 1. If Row ﬁghts, he gets 1 if the opponent is weak and — by the dominance argument just made — he gets -1 if the opponent is strong. Show that in period 2, a worker will be paid w 2 (Y 1) = ˇ(Y 1)q H;0 Yh + (1 ˇ(Y 1))q L;0 Yh; where ˇ(Y 1) is the probability that the market assigns to the worker being high ability after observing his output level Y 1 2 Yh;Yl = 0 in the rst period. startxref First each of them names either himself or the other person as the one who will make the choice. (When constructing the normal form of each game, be … A simplificationof poker Consider the followingsimplificationof poker. Tip: you can also follow us on Twitter In The requirement that the equilibrium be perfect Bayesian, and not just Bayesian, \ equilibrium. Weak Perfect Bayesian Equilibrium Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu June 16th, 2016 C. Hurtado (UIUC - Economics) Game Theory. sets oﬀthe path of equilibrium. The problem is that the set of actions available to agent 1 depends on the state of the world. Obara (UCLA) Bayesian Nash Equilibrium February 1, 2012 6 / 28. Then a mixed strategy Bayesian Nash equilibrium exists. http://gametheory101.com/courses/game-theory-101/This lecture begins a new unit on sequential games of incomplete information. Since this equilibrium reaches every information set, it must be weak perfect Bayesian. An example of a Perfect Bayesian equilibrium in mixed strategy. >> If Row ﬁghts, he gets 1 if the opponent is weak and — by the dominance argument just made — he gets -1 if the opponent is strong. That means that all BNE are subgame perfect. In general, the Perfect Bayesian Equilibrium (PBE) is the concept we are using when solving dynamic games with incomplete information (such as signaling game and repu-tation game). For any extensive-form game Γ with perfect recall, a Nash equilibrium in behav-ior strategies exists. In those games we ignored these equilibria by focusing on subgame perfect equilibria; in the latter equilibria each agent’s action is sequentially rational at each information set. Es dient dem Lösen von dynamischen Spielen mit unvollständiger Information. trailer This is a simple Bayesian game where I the set of players (bidders) is N I the set of states is V 1:::; V n I the set of actions for bidder i is A i = < + I the set of types for bidder i is V i I bidder i’s interim belief is p i(v ijv i). stream 0000001218 00000 n Generally, the ﬁrst step to solving an extensive-form game is to ﬁnd all of its Nash equilib- ria. Obara (UCLA) Bayesian Nash Equilibrium February 1, 2012 6 / 28. 15. tion of perfect Bayesian equilibrium that meets several goals. Suppose for example that an o path information system ˇ0 is chosen 3This description includes any pair of distributions on a nite set as inKamenica and Gentzkow(2011). 1. And so, there are equilibrium concepts that explicitly model player's beliefs about where they are in a tree for every information set. In the following game, nature –rst chooses one of two types of player 1 (in the –gure, the two types are denoted t 1 and t 2). Due by email to the course TF as a PDF (we suggest you write in LaTex) before class begins on Monday 10/1. Get the latest machine learning methods with code. a������e~�Y�������8}�����[T����I`V�7���j�7�q�����t]ʙ��5��Y We do not consider this to be a choice. (Market for Lemons) Here I ask that you work out some of the details in perhaps the most famous of all information economics models. sets to represent what each player knows at each stage of the game. Now look at Row. On the Agenda 1 Formalizing the Game 2 Systems of Beliefs and Sequential Rationality 3 Weak Perfect Bayesian Equilibrium 4 Exercises C. Hurtado (UIUC - Economics) Game Theory. i&KT2s8��t8$p�)�� �flcˬbaEN����� BNEs and Sequential rationality So far we have learned how to –nd BNEs in incomplete information games. Then, the belief on player 2’s information set is well de ned. Introduction to social learning and herding. Rationality. Problem Set 5. b) The beliefs are consistent with Bayes™rule, whenever possible. xڍZK�� �ϯ�\5툢Vn�ͤR���T����A��jd�G�������%�;{iK$�x| �~z��%���k��χ�"y(�r����y��Ȭ�1I�y��Q�2i���j�o6ڭ���գͳ�ieʨZ�6z_������f��8Q���D�V��~���i�U�D¿[�"�E2}�EY}����}�Ų���a����?��C�.s˧��ޘR�|����Fߒ8[�$��U�# ��l����c���ߗ�#������ޚve�/�f�]HW�0`����|Ť�e:��%��~����TP9l���r���ǥ>��"��7��u��U2>�a5:Y_��ŭ�z 7.- (Revisiting the War of Attrition, exercise 6 Problem set 1). Problem 1: Find all the Nash equilibria and Subgame perfect Nash equilibrium of the game below. 2. The notation a.b denotes problem number b from Chapter a in Watson. 2 Perfect Bayesian Equilibrium - De–nition A strategy pro–le for N players (s 1;s 2;:::;s N) and a system of beliefs over the nodes at all infor-mation sets are a PBE if: a) Each player™s strategies specify optimal actions, given the strategies of the other players, and given his beliefs. Beforeplayingeach player puts a dollardown. Menon Business Economics 2 PROBLEM SET Solution (b): Let be the probability game 1 given or , and be the probability game 1 given or . In game theory, a Perfect Bayesian Equilibrium (PBE) is an equilibrium concept relevant for dynamic games with incomplete information (sequential Bayesian games). And there's two, two solution concepts in particular known as sequential equilibrium and perfect Bayesian equilibrium that have key features where they have players, as part of the equilibrium you specify what the beliefs of the players are. x�b```f``r�,����������������� ,6Sp�}Nj�=�z�u�3L���~B���ً����*���,�\���YM�g++S)Y�P�v��@�xE#�\��IOx4���0�h�m�lC��elK&��Q 8r>t����>M���t9ME{|�FgN�!�h�C)HP,�%! Here, I will deﬁne sequential equilibrium and apply it to some important games. /Length 3053 But assume that player 1 plays acompletely mixed strategy, playing L, M, and R with probabilities 1 , 3 4, and 4. 1. Player 2’s information set will not be reached at the equilibrium, because player 1 will play L with probability 1. Let H i be the set of information sets at which player i moves. This is a simple Bayesian game where I the set of players (bidders) is N I the set of states is V 1:::; V n I the set of actions for bidder i is A i = < + I the set of types for bidder i is V i I bidder i’s interim belief is p i(v ijv i). Private Provision of Public Good. A semisepa- rating equilibrium also arises when mixed strategies are played. Problem Set 10 1. Problem Set 1 CS 286r beginning of class, Monday 10/1 Preamble You may work in pairs and not discuss this problem set with anyone other than your (optional) partner. First each of them names either himself or the other person as the one who will make the choice. Perfect Bayesian Equilibrium 1 An Example Player 1 L M R’ 2 1 0 0 0 2 0 1 R 1 L’ R’L’ 3 Player 2 Each player has one information set Player 1 ’ strategies: = {,, } Player2’ strategies: = {’, ’} One sub-game (the whole game) : it implies that all NE are SPNE 2. 0000001525 00000 n But assume that player 1 plays acompletely mixed strategy, playing L, M, and R with probabilities 1 , 3 4, and 4. Problem Set 2 Spring 2016 Luca Merlino T.A.s Stefan Bergheimer and Luca Livio Due Date: March 22, 2015, 8 a.m. 1 Game Theory 1.1 Trembling Hand Perfection Two people are engaged in the following game to select either a good or a bad outcome. information set. sets are reached, this must be a weak perfect Bayesian equilibrium. For question 3, I initially tried to solve the first problem using Mixed Bayesian Nash Equilibrium but that doesn't make sense since both Player 1 and Player 2 have weakly dominated strategies, so why would they mix? ��t�PX���R6q�J0 2. It is easy enough to solve for the Bayesian Nash equilibrium of this game. ex ante probability that a node in D will be reached under strategy profile a. Game theory: Problem set II These problems are designed for discussions in the classes of Week 8 of Michaelmas term.1 1. M.Phil. ����h�y2+�+80�00`�����i�l�L@� ��L�7A� �K { � Player 2’s information set will not be reached at the equilibrium, because player 1 will play L with probability 1. In fact, there is a perfect Bayesian equilibrium where player 1 plays D and player 2 plays U' and player 2 holds the belief that player 1 will definitely play D (i.e player 2 places a probability of 1 on the node reached if player 1 plays D). Consider the NE (L, r) again. In a perfect Bayesian equilibrium, “wherever possible”, beliefs must be computed using Bayes’ rule and the strategies of the players. 4 0 obj << Reading: Osborne, Chapter 9. Anything goes Homework can be delivered: (1) by email to katarina.kalovcova@cerge-ei.cz or (2) personally during the lecture or o–ce hours. There are 2 players: a professor and a student. We use Perfect Bayesian equilibrium (PBE) as our solution concept. Now, we e xtend this notion to the games with incom-plete information. Bayesian game. 444. Each group submits one copy of problem set with the names of all members. Browse our catalogue of tasks and access state-of-the-art solutions. 0000002379 00000 n It is easy enough to solve for the Bayesian Nash equilibrium of this game. By contrast to discussion in class, we give a complete formulation of the game. So (cf;eh) is weak perfect Bayesian. There are 2 players: a professor and a student. Equilibrium also satis–es requirement 4 because there are no o⁄-the-equilibrium path information,. 1 ) designed for discussions in the following two extensive games, derive the strategic games find. 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