perfect bayesian equilibrium problem set

0000000496 00000 n Beforeplayingeach player puts a dollardown. We need to modify the idea of subgame perfection so that we are Sequential#rationality# # Receiver!best!responds!toLLby!playing!u(strictdomnt)since:! And so, there are equilibrium concepts that explicitly model player's beliefs about where they are in a tree for every information set. Problem set on repeated games and Bayesian games 1. sR�_ξ/��v�6pbEx&�. 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. An example of a Perfect Bayesian equilibrium in mixed strategy. The problem with this situation is that player 2’s beliefs are not 3. consistent with player 1’s strategy. i&KT2s8��t8$p�)�� flcˬbaEN�� De ne a Perfect Bayesian Equilibrium for this game. But assume that player 1 plays acompletely mixed strategy, playing L, M, and R with probabilities 1 , 3 4, and 4. A PBE consists of a pair of strategy proﬁle and belief system. A semisepa- rating equilibrium also arises when mixed strategies are played. Problem Set 3 - Solutions Due Wednesday, December 5 Important: hand in only the two-star problems. %%EOF 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. ex ante probability that a node in D will be reached under strategy profile a. First each of them names either himself or the other person as the one who will make the choice. Here, I will deﬁne sequential equilibrium and apply it to some important games. A simplificationof poker Consider the followingsimplificationof poker. (Market for Lemons) Here I ask that you work out some of the details in perhaps the most famous of all information economics models. ))Ce�:�;`A%c�~A��1P�P'�EG#�P`"RR��' Bayesian game. Now, we’ll de ﬁne a concept of consistency, which will be required in a perfect Bayesian Nash equilibrium. ��h�y2+�+80�00`��i�l�L@� ��L�7A� �K { � A seller is privately informed of the value vof the good that she sells to a buyer. By contrast to discussion in class, we give a complete formulation of the game. The theorem tells us at least one such equilibrium will exist. EK, Chapter 16. /Filter /FlateDecode a��e~�Y��8}��[T��I`V�7��j�7�q��t]ʙ��5��Y b) The beliefs are consistent with Bayes™rule, whenever possible. 4 0 obj << 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. Weak Perfect Bayesian Equilibrium Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign [email protected] June 16th, 2016 C. Hurtado (UIUC - Economics) Game Theory. By contrast to discussion in class, we give a complete formulation of the game. The notation a.b denotes problem number b from Chapter a in Watson. sets oﬀthe path of equilibrium. Introduction to social learning and herding. Obara (UCLA) Bayesian Nash Equilibrium February 1, 2012 6 / 28. sets to represent what each player knows at each stage of the game. In game theory, a Perfect Bayesian Equilibrium (PBE) is an equilibrium concept relevant for dynamic games with incomplete information (sequential Bayesian games). gW�ps��xT��2 Er��;cbߋ�o��K��xc��>� Xa��pC8�7��~. An example of a Perfect Bayesian equilibrium in mixed strategy. Because we can™t use Bayes™rule, WPB does not constrain beliefs! Suppose now that the game from part a is played twice. 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)). 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. 0000002379 00000 n Now look at Row. Consider the following game of complete but imperfect information. 136 0 obj <> endobj Anything goes 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? Player 1 observes her type and decides whether to choose L or R. If player 1 chooses R, the game ends. http://gametheory101.com/courses/game-theory-101/This lecture begins a new unit on sequential games of incomplete information. 3. 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. Networks: Lectures 20-22 Incomplete Information Incomplete Information In many game theoretic situations, one agent is unsure about the preferences or intentions of others. Get the latest machine learning methods with code. Problem Set 5 Due: November 21, 2006 Recall that what Osborne calls “Weak Sequential Equilibrium” is equivalent to our “Perfect Bayesian Equilibrium.” 1. The problem is that the set of actions available to agent 1 depends on the state of the world. (At the very least, this ensures information sets that can be reached with positive probability have beliefs assigned using Bayes’ rule.) ��β��䛻$�I��_�8\��9~8d�$��7$�i��'c��,��eR�� `@ these problems, we start by investigating a new set of solution concepts, then moev on to applications. Problem Set 10 1. xref 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)}. 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). (When constructing the normal form of each game, be … ��(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 Remember that the "weak" in "weak perfect Bayesian" refers to the lack of restrictions on oﬀ-the-equilibrium path beliefs. trailer (d) For what rangeof x is therea unique subgame perfect equilibrium outcome? 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. Formalizing the Game … If strategy sets and type sets are compact, payo functions are In The requirement that the equilibrium be perfect Bayesian, and not just Bayesian, \ equilibrium. Perfect Bayesian (Nash) Equilibria. Problem Set 5. There are 2 players: a professor and a student. Example 62 9.C.5 A WPBNE need not be subgame perfect. Rationality. Then a mixed strategy Bayesian Nash equilibrium exists. Also when I combine the matrices I find no Pure Strategy Bayesian Equilibrium. 2. (Again, comparing to the answers to the last problem set, we see that this weak PBE is not subgame perfect.) It is a refinement of Bayesian Nash equilibrium (BNE). Private Provision of Public Good. That means that all BNE are subgame perfect. 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. Weak Perfect Bayesian Equilibrium Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign [email protected] June 16th, 2016 C. Hurtado (UIUC - Economics) Game Theory. 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 . 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. Now look at Row. 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). 0000002301 00000 n A simplificationof poker Consider the followingsimplificationof poker. /Length 3053 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). 145 0 obj <>stream 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,�%! 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. perfect Bayesian equilibrium ("pooling equilibrium"): the oﬀspring is always quiet and the parent always keep the food. 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). 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. !S�8{0ް��)��!kҿ�KVa��`%��Ŷn��*Ab�up�#�I��"� 0000001437 00000 n Problem 1: Find all the Nash equilibria and Subgame perfect Nash equilibrium of the game below. Turn in a single problem set for each pair. U��0�dC㫮��>?�c01��j��-��(� 0000001303 00000 n 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. Problem Set 10 1. We are doing great! Turning to the second subcase, suppose 2 plays iat his last information set, 1 plays 35. BNEs and Sequential rationality So far we have learned how to –nd BNEs in incomplete information games. 0000001218 00000 n 136 10 Theorem Consider a Bayesian game with continuous strategy spaces and continuous types. Bayesian Games 3/4/14 This problem set is due on Tuesday, 3/25/14. Es dient dem Lösen von dynamischen Spielen mit unvollständiger Information. Formalizing the Game … Homework can be delivered: (1) by email to [email protected] or (2) personally during the lecture or o–ce hours. 0000003439 00000 n Player 2’s information set will not be reached at the equilibrium, because player 1 will play L with probability 1. Solution: ThesubgamethatfollowsR hasaNashequilibrium(r,r)foranyvalueofx.Therefore,L is always a SPE outcome. 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! Generally, the ﬁrst step to solving an extensive-form game is to ﬁnd all of its Nash equilib- ria. Each group submits one copy of problem set with the names of all members. Since this equilibrium reaches every information set, it must be weak perfect Bayesian. 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™. 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. Problem Set 5. Obara (UCLA) Bayesian Nash Equilibrium February 1, 2012 6 / 28. Player 1 observes her type and decides whether to choose L or R. If player 1 chooses R, the game ends. %PDF-1.4 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. In this equilibrium, every strategy is rational given the beliefs held and every belief is consistent with the strategies played. 1. (For other parameter values, the game has a pooling equilibrium in which the oﬀspring is always quiet and the parent always gives the food.) 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). There are 2 players: a professor and a student. tion of perfect Bayesian equilibrium that meets several goals. As in (5), we restrict attention to ﬁnite extensive-form games with perfect recall. 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). Player 2’s information set will not be reached at the equilibrium, because player 1 will play L with probability 1. I bidder i’s payo is u i(b;v) = 1(b i max j6=i b j)(v i b i). 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. Game Theory: Lecture 18 Perfect Bayesian … 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. %PDF-1.4 %�� In a perfect Bayesian equilibrium, “wherever possible”, beliefs must be computed using Bayes’ rule and the strategies of the players. By contrast to discussion in class, we give a complete formulation of the game. 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! 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. 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. Consider the NE (L, r) again. I bidder i’s payo is u i(b;v) = 1(b i max j6=i b j)(v i b i). There are no one-star problems on this problem set. 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. Browse our catalogue of tasks and access state-of-the-art solutions. 0000002055 00000 n 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. 15. Now, we e xtend this notion to the games with incom-plete information. Suppose now that the game from part a is played twice. Problem Set 5. We do not consider this to be a choice. 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 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. So (af;di) is weak perfect Bayesian. Tip: you can also follow us on Twitter 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. From Bayesian Nash Equilibrium (BNE) to Perfect Bayesian Equilibrium (PBE) FØlix Muæoz-García School of Economic Sciences Washington State University. Bayesian Games Suggested Solutions by Tibor Heumann 1. Das perfekt bayessche Gleichgewicht ist ein Lösungskonzept in der Spieltheorie. For any extensive-form game Γ with perfect recall, a Nash equilibrium in behav-ior strategies exists. A seller is privately informed of the value vof the good that she sells to a buyer. 7.- (Revisiting the War of Attrition, exercise 6 Problem set 1). 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). 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 M.Phil. 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. Receiver's#beliefs#for#theinfo#set#on#theequilibrium#path:#p=½=1Rp# 2. 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+! 0 (Is there a pooling equilibrium?) sets are reached, this must be a weak perfect Bayesian equilibrium. 1.2 Perfect Bayesian Equilibrium Let G be an extensiev form game. 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. First note that if the opponent is strong, it is a dominant strategy for him to play F — ﬁght. 2. That means that all BNE are subgame perfect. Consider the NE (L, r) again. But assume that player 1 plays acompletely mixed strategy, playing L, M, and R with probabilities 1 , 3 4, and 4. Show that there is a unique separating perfect Bayesian equilibrium. Each type is chosen with equal probability. �\��q�'�� $fx3��0PȵghpH h�#��y�� Now, if !0, it’s still well de ned. 2. BNEs and Sequential rationality So far we have learned how to –nd BNEs in incomplete information games. 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. It is easy enough to solve for the Bayesian Nash equilibrium of this game. Problem set on repeated games and Bayesian games 1. A perfect Bayesian equilibrium has two components -- strategies and beliefs : We do not consider this to be a choice. Each type is chosen with equal probability. 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. Game theory: Problem set II These problems are designed for discussions in the classes of Week 8 of Michaelmas term.1 1. In contrast, in an equilibrium a player maximizes his expected payoﬀgiven the other players’ strategies. Now, if !0, it’s still well de ned. It is easy enough to solve for the Bayesian Nash equilibrium of this game. Then, the belief on player 2’s information set is well de ned. So (cf;eh) is weak perfect Bayesian. (Market for Lemons) Here I ask that you work out some of the details in perhaps the most famous of all information economics models. stream plausibility order on the set of histories is choice measurable, which is a necessary condition for a PBE to be a SE. 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. We’re headed toward restricting these beliefs in a suitable way. Raquel has to choose whether to pursue training that costs $1;000 to herself or not. We are doing great! From Bayesian Nash Equilibrium (BNE) to Perfect Bayesian Equilibrium (PBE) FØlix Muæoz-García School of Economic Sciences Washington State University. So now suppose 2 plays iat that last information set. 0000001525 00000 n 444. 0000000016 00000 n Reading: Osborne, Chapter 9. 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). In the following two extensive games, derive the strategic games and find all the pure-strategy Nash, Subgame-perfect, and Perfect Bayesian Equilibria. Due by email to the course TF as a PDF (we suggest you write in LaTex) before class begins on Monday 10/1. 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. First each of them names either himself or the other person as the one who will make the choice. Because we can™t use Bayes™rule, WPB does not constrain beliefs! Networks: Lectures 20-22 Bayesian Games Existence of Bayesian Nash Equilibria Theorem Consider a nite incomplete information (Bayesian) game. Kreps and Wilson [7] give a series of examples to motivate the idea that further restrictions may be natural. ��t�PX��R6q�J0 2. We use Perfect Bayesian equilibrium (PBE) as our solution concept. The problem is that there are usually no proper subgames. startxref Bayesian Games Suggested Solutions by Tibor Heumann 1. That is for any information set … 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.] Then, the belief on player 2’s information set is well de ned. >> Here, I will deﬁne sequential equilibrium and apply it to some important games. (Market for Lemons) Here I ask that you work out some of the details in perhaps the most famous of all information economics models. 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). If the entrant enters, then each ﬁrm simultaneously chooses F or A. ��4��C�&)��L��di �5�9d/D�qp b��?�� H��8=�0�1v0;T7\bX��=��/Ki� ��.2�`r �7��A��E�u The problem set is shown below: Problem Set. • The professor draws a single card from a deck consisting of an equalnumber of kings and queens. 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. • The professor draws a single card from a deck consisting of an equalnumber of kings and queens. <<8BE3CBBEA2A431468DEFE7D45530D756>]>> First note that if the opponent is strong, it is a dominant strategy for him to play F — ﬁght. 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. 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. No later submissions will be accepted! The relevant notion of equilibrium will be Perfect Bayesian Equilibria, or Perfect Bayesian Nash Equilibria. Let H i be the set of information sets at which player i moves. Exercise 319.3 in Osborne (Nash Equilibria of a Card Game). A seller is privately informed of the value vof the good that she sells to a buyer. The problem is that there are usually no proper subgames.

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