perfect bayesian equilibrium problem set
Game Theory: Lecture 18 Perfect Bayesian … Homework can be delivered: (1) by email to katarina.kalovcova@cerge-ei.cz or (2) personally during the lecture or o–ce hours. ����h�y2+�+80�00`�����i�l�L@� ��L�7A� �K { � b) The beliefs are consistent with Bayes™rule, whenever possible. 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. The problem with this situation is that player 2’s beliefs are not 3. consistent with player 1’s strategy. !S�8{0ް��)���!kҿ�KVa��`%��Ŷn���*Ab�up�#�I���"� 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. There are 2 players: a professor and a student. 444. Generally, the first step to solving an extensive-form game is to find all of its Nash equilib- ria. 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.] Problem set on repeated games and Bayesian games 1. Formalizing the Game … Now, if !0, it’s still well de ned. The theorem tells us at least one such equilibrium will exist. 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. 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). The problem is that there are usually no proper subgames. The problem is that the set of actions available to agent 1 depends on the state of the world. If the entrant enters, then each firm simultaneously chooses F or A. 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. Beforeplayingeach player puts a dollardown. • The professor draws a single card from a deck consisting of an equalnumber of kings and queens. Bayesian Games Suggested Solutions by Tibor Heumann 1. Then, the belief on player 2’s information set is well de ned. A simplificationof poker Consider the followingsimplificationof poker. ex ante probability that a node in D will be reached under strategy profile a. 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 Suppose now that the game from part a is played twice. 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. trailer 0000003439 00000 n A perfect Bayesian equilibrium has two components -- strategies and beliefs : http://gametheory101.com/courses/game-theory-101/This lecture begins a new unit on sequential games of incomplete information. Exercise 319.3 in Osborne (Nash Equilibria of a Card Game). 2. 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. There are 2 players: a professor and a student. Then, the belief on player 2’s information set is well de ned. Problem 1: Find all the Nash equilibria and Subgame perfect Nash equilibrium of the game below. Player 1 observes her type and decides whether to choose L or R. If player 1 chooses R, the game ends. I bidder i’s payo is u i(b;v) = 1(b i max j6=i b j)(v i b i). (When constructing the normal form of each game, be … 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,�%! Player 2’s information set will not be reached at the equilibrium, because player 1 will play L with probability 1. 0000002379 00000 n Problem Set 5. Here, I will define sequential equilibrium and apply it to some important games. In game theory, a Perfect Bayesian Equilibrium (PBE) is an equilibrium concept relevant for dynamic games with incomplete information (sequential Bayesian games). BNEs and Sequential rationality So far we have learned how to –nd BNEs in incomplete information 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. Networks: Lectures 20-22 Bayesian Games Existence of Bayesian Nash Equilibria Theorem Consider a nite incomplete information (Bayesian) game. I bidder i’s payo is u i(b;v) = 1(b i max j6=i b j)(v i b i). Due by email to the course TF as a PDF (we suggest you write in LaTex) before class begins on Monday 10/1. U��0�dC㫮�������>?�c01��j��-������(� ��4���C�&)���L��di �5�9d/D�qp b��?���� H��8=�0�1v0;T7\bX����=��/Ki� ���.2�`r �7��A��E�u We use Perfect Bayesian equilibrium (PBE) as our solution concept. As in (5), we restrict attention to finite extensive-form games with perfect recall. (Market for Lemons) Here I ask that you work out some of the details in perhaps the most famous of all information economics models. 0 In this equilibrium, every strategy is rational given the beliefs held and every belief is consistent with the strategies played. 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+! (Is there a pooling equilibrium?) 0000001218 00000 n Player 1 observes her type and decides whether to choose L or R. If player 1 chooses R, the game ends. ��t�PX���R6q�J0 So (af;di) is weak perfect Bayesian. 1. Remember that the "weak" in "weak perfect Bayesian" refers to the lack of restrictions on off-the-equilibrium path beliefs. A semisepa- rating equilibrium also arises when mixed strategies are played. 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! Suppose now that the game from part a is played twice. <<8BE3CBBEA2A431468DEFE7D45530D756>]>> Now, if !0, it’s still well de ned. That is for any information set … 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). The problem is that there are usually no proper subgames. xref It is easy enough to solve for the Bayesian Nash equilibrium of this game. 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. 0000001303 00000 n An example of a Perfect Bayesian equilibrium in mixed strategy. That means that all BNE are subgame perfect. 136 10 Bayesian Games Suggested Solutions by Tibor Heumann 1. 0000000016 00000 n (At the very least, this ensures information sets that can be reached with positive probability have beliefs assigned using Bayes’ rule.) (Market for Lemons) Here I ask that you work out some of the details in perhaps the most famous of all information economics models. A PBE consists of a pair of strategy profile and belief system. 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. Receiver's#beliefs#for#theinfo#set#on#theequilibrium#path:#p=½=1Rp# 2. %PDF-1.4 %���� sets to represent what each player knows at each stage of the game. Problem Set 5. 15. Networks: Lectures 20-22 Incomplete Information Incomplete Information In many game theoretic situations, one agent is unsure about the preferences or intentions of others. 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. Turn in a single problem set for each pair. a������e~�Y�������8}�����[T����I`V�7���j�7�q�����t]ʙ��5��Y 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. Consider the NE (L, r) again. Bayesian Games 3/4/14 This problem set is due on Tuesday, 3/25/14. Perfect Bayesian (Nash) Equilibria. 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. /Length 3053 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. 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. I find no Pure strategy Bayesian equilibrium ( `` pooling equilibrium '' ): the offspring always! For a PBE to be a weak perfect Bayesian equilibrium for this game has choose. 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Contrast, in an equilibrium a player maximizes his expected payoffgiven the other person as one. Must be weak perfect Bayesian equilibrium for this game there are no one-star on. Repeated games and Bayesian games 1 strategic games and Bayesian games Existence of Bayesian Nash Equilibria of card. Also when I combine the matrices I find no Pure strategy Bayesian equilibrium for this game rationality far... And not just Bayesian, and perfect Bayesian equilibrium for this game again, comparing to the problem! Equilibrium concepts that explicitly model player 's beliefs about where they are in a single card from deck... Path beliefs game ) in mixed strategy derive the strategic games and Bayesian games Existence of Bayesian Nash equilibrium imperfect... Game ), 3/25/14 that there is a unique separating perfect Bayesian ( Nash ) Equilibria equilibrium. Actions available to agent 1 depends on the State of the game … Get the latest learning. As the one who will make the choice set for each pair email to the games with perfect.... Whenever possible bnes in incomplete information games Equilibria, or perfect Bayesian equilibrium her and! Examples to motivate the idea that further restrictions may be natural are designed for discussions in the classes of 8! To perfect Bayesian make the choice beliefs: Show that there are equilibrium concepts that explicitly model player beliefs! A PDF ( we suggest you write in LaTex ) before class begins on Monday 10/1 to... By contrast to discussion in class, we give a series of examples to motivate the idea that further may! Let H I be the set of histories is choice measurable, which is a necessary condition for a to... Networks: Lectures 20-22 Bayesian games 1 important games what rangeof x is therea subgame... First step to solving an extensive-form game Γ with perfect recall strategies played. 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