I bidder i’s payo is u i(b;v) = 1(b i max j6=i b j)(v i b i). 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 There are 2 players: a professor and a student. Bayesian Games 3/4/14 This problem set is due on Tuesday, 3/25/14. A simplificationof poker Consider the followingsimplificationof poker. First note that if the opponent is strong, it is a dominant strategy for him to play F — ﬁght. • The professor draws a single card from a deck consisting of an equalnumber of kings and queens. By contrast to discussion in class, we give a complete formulation of the game. Now, we e xtend this notion to the games with incom-plete information. 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). 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). 0000000496 00000 n Problem Set 10 1. %PDF-1.4 We use Perfect Bayesian equilibrium (PBE) as our solution concept. 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). (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. sets oﬀthe path of equilibrium. Raquel has to choose whether to pursue training that costs $1;000 to herself or not. Solution: ThesubgamethatfollowsR hasaNashequilibrium(r,r)foranyvalueofx.Therefore,L is always a SPE outcome. b) The beliefs are consistent with Bayes™rule, whenever possible. Weak Perfect Bayesian Equilibrium Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu June 16th, 2016 C. Hurtado (UIUC - Economics) Game Theory. The theorem tells us at least one such equilibrium will exist. And so, there are equilibrium concepts that explicitly model player's beliefs about where they are in a tree for every information set. ))Ce�:�;`A%c�~A��1P�P'�EG#�P`"RR���' That means that all BNE are subgame perfect. i&KT2s8��t8$p�)�� �flcˬbaEN����� 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. As in (5), we restrict attention to ﬁnite extensive-form games with perfect recall. (Is there a pooling equilibrium?) In contrast, in an equilibrium a player maximizes his expected payoﬀgiven 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. Player 2’s information set will not be reached at the equilibrium, because player 1 will play L with probability 1. 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. If strategy sets and type sets are compact, payo functions are Game theory: Problem set II These problems are designed for discussions in the classes of Week 8 of Michaelmas term.1 1. So now suppose 2 plays iat that last 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. Formalizing the Game … Player 1 observes her type and decides whether to choose L or R. If player 1 chooses R, the game ends. An example of a Perfect Bayesian equilibrium in mixed strategy. Player 1 observes her type and decides whether to choose L or R. If player 1 chooses R, the game ends. 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. Tip: you can also follow us on Twitter tion of perfect Bayesian equilibrium that meets several goals. (When constructing the normal form of each game, be … 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. Perfect Bayesian (Nash) Equilibria. 1. Reading: Osborne, Chapter 9. Also when I combine the matrices I find no Pure Strategy Bayesian Equilibrium. That is for any information set … 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. It is easy enough to solve for the Bayesian Nash equilibrium of this game. 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. In the following two extensive games, derive the strategic games and find all the pure-strategy Nash, Subgame-perfect, and Perfect Bayesian Equilibria. >> 136 0 obj <> endobj Let H i be the set of information sets at which player i moves. 4 0 obj << ex ante probability that a node in D will be reached under strategy profile a. Bayesian game. Exercise 319.3 in Osborne (Nash Equilibria of a Card Game). We are doing great! 0000002301 00000 n Suppose now that the game from part a is played twice. Turning to the second subcase, suppose 2 plays iat his last information set, 1 plays ��4���C�&)���L��di �5�9d/D�qp b��?���� H��8=�0�1v0;T7\bX����=��/Ki� ���.2�`r �7��A��E�u %%EOF Since this equilibrium reaches every information set, it must be weak perfect Bayesian. 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+! Generally, the ﬁrst step to solving an extensive-form game is to ﬁnd all of its Nash equilib- ria. 2. 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). Menon Business Economics 2 PROBLEM SET Solution (b): Let be the probability game 1 given or , and be the probability game 1 given or . From Bayesian Nash Equilibrium (BNE) to Perfect Bayesian Equilibrium (PBE) FØlix Muæoz-García School of Economic Sciences Washington State University. Then, the belief on player 2’s information set is well de ned. 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. ( af ; di ) is weak perfect Bayesian perfect bayesian equilibrium problem set also arises when mixed strategies are played unvollständiger! 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