We show that the convergence to the long run equilibrium is fast with a half-life of one period or less. Markov perfect equilibrium is a key notion for analyzing economic problems involving dy-namic strategic interaction, and a cornerstone of applied game theory. With a few notable exceptions, most of this work has focused on static environments or on 5A Markov Perfect Equilibrium is a profile of time-homogeneous pure strategies that map a player’s information in each single time period to a choice. When si is a strategy that depends only on the state, by some abuse of notation we Markov perfect equilibrium is a refinement of the concept of Nash equilibrium. Thus, once a Markov chain has reached a distribution π Tsuch that π P = πT, it will stay there. The views expressed in this paper are those of the author and do not necessarily reflect the position of the Federal Reserve Bank of New York or the Federal Reserve System. A two-dimensional backward induction is em-ployed in section IV to solve for explicit equilibria, which are compared to the open-loop Nash equilibria of the same game. �KX3���R^S�ҏ6������eG*z��Zh�4��Y�<20� Every n-player, general-sum, discounted-reward stochastic game has a MPE The role of Markov-perfect equilibria is similar to role of subgame-perfect <> Get the latest machine learning methods with code. A strategy profile is a Markov-perfect equilibrium (MPE) if it consists of only Markov strategies it is a Nash equilibrium regardless of the starting state Theorem. That is, if two subgames are isomorphic in the sense that the corresponding preferences and action spaces are equivalent, then they should be played in the same stream Markov Perfect Equilibria in the Ramsey Model⁄ Paul Pichler and Gerhard Sorger This Version: February 2006 Abstract Keywords and Phrases: Oligopoly_Theory, Network_Externalities, Markov_Perfect-Equilibrium The model and Markov Perfect Equilibrium In this section we describe the main features of the exogenous timing duopoly model [for further discussion of this model, see Maskin and Tirole (1982)]. Markov perfect equilibrium is a refinement of the concept of Nash equilibrium. We >���=�b���W6�t���Q@�wN�,���B��T���X�e1����~K��ʚ#��Ot=�le.�G!�)��3��Q1�#N\�r�b�S�����čT�u�DXg��`8SK���4½9$s��U���B)�5Q�=����h�/� �C���$Ê�[��`�6 $b�� �i�k��H�C�\�ϥ�� 5 0 obj Perfect equilibrium in Markov strategies is defined in section III. in a Markov perfect equilibrium of the induced stochastic game. We should also mention a very interesting papers byCurtat(1996),Cole and Kocherlakota(2001), In the special case in which local problems are Markov chains and agents compete to take a single action in each period, we lever-age Gittins allocation indices to provide an effi-cient factored algorithm and distribute computa-tion of the optimal policy among the agents. A Markov perfect equilibrium model captures this strategic behavior, and permits the comparison of “as is” market trajectories with “but for” trajectories under count erfactual s where “bad ac ts” by some fir ms ar e eliminated. Competition between the two firms (i= 1,2) takes place in discrete time with an infinite horizon. We exploit these conditions to derive a system of equations, f(˙) = 0, that must be satis ed by any Markov perfect equilibrium ˙. Markov perfect equilibrium is a key notion for analyzing economic problems involving dy-namic strategic interaction, and a cornerstone of applied game theory. The equilibrium strategy of the in nite horizon model is obtained as the point-wise limit of the (unique) nite-horizon strategies. 1. Browse our catalogue of tasks and access state-of-the-art solutions. 8It bears mentioning, we focus on short-memory Markovian equilibrium because this class of equilibrium has been the focus of a great deal of applied work. QRE as a Structural Model for Estimation 141 Equilibrium concept: |Commitment (benchmark) |Discretion (Markov Perfect Equilibrium) |Not for now: reputational equilibria under oligopoly 8/34. Equilibrium Entry/Exit (Theorem 3): If p t = D t(Q) is nondecreasing in t, and q(p t=x) is strictly concave in x, then the equilibrium price sequence is constant p t= pfor each t, and entry and exit occurs in equilibrium at each t. Key elements of the proof: { x t = E(x t+1jI(t)) is a random-walk. 25 0 obj The agents in the model face a common state vector, the time path of which is influenced by – and influences – their decisions. This defines a homogeneous Markov chain. Markov perfect equilibrium is a key notion for analyzing economic problems involving dy-namic strategic interaction, and a cornerstone of applied game theory. Our methods also can be adapted to studying the set of subgame perfect/sequential equilibrium. x��Zێ���� o���x��_��+� Y��l?�fW�X��je��s���.�=����Y]�S���|ӉAv������ͣ�{u���^m�ld��+�W�gX�B�Dw�r�_�=�U���ή6�����w�*!� ����.�7���?ux��=Wb{������Hy�V��f��)�/�);���:��h����������[1��1����Ai�C�v�3�wQ���.����݉�E����$��C�.����$@y����P��2N���N�ko�߯����N�8��ق��Xb�S(��Xi�Ķ7;��hq��t0� �N��LV���S����Z��d����n1�{~:��F�!� .�Bvg��W[5Xk����,�{��j�%�۪�h���߷;9X�7pOO����_�W1��W������_ֵ5�L��g^[È���BAy$����p��5������,��Tp�돞#������M�8��'���5�w��zJO �ڔ�;i5��AJLZ�� �`��AX V�?�흂RP*z'S�q��Tx6$�i����i�1Q!���� �}�Wޱ�L+��sE8�I 3Y�']�p ������*)#S�h���=�a�A�o�*���� ���yC�j�Y����zw����GP��1�.&g����Ey��U���rN�X���,ϲ�4s~bwh* ]�t��!����6�T�:�t��:d>����A�&�!��d��˜UQ��b�� ��r؏��l 5ip=i0FZ��H� i�Tq�2B�l-#-$1��˔o]m�"a�8�2M�I6���e4@��]Q/��-v��U�$�Lي��c��okf2ǰ0MfՕ9H� �u;����^�m�0�Ƞ{@�^�} ��Y�qo)�ڬ�_l�X+������h{��!�pE�Ց�o'�(L�ơ���Y�Y��$[584��#�fD���.�t~ �**ތ �"�Ë�Hybh��uMz��p���m�劏g��'���4f�٥&U�Qo�q���Nu`R�p4h�;�| 0��Y�v1�|[w��+�u�"_j�J�'0�$�Š애F!�t�fP�����9��3܍� �0x���Ե6k���(Iƒ"�/��v���*�;E�����(�hT�c_c�f YmW[k�~���>!�����SAC�e����Ǜ-�U(9��D���g�qO����y���O� 3T2����͍ZF w�Nqx��Z/'�)�RTbń% �7�p�ϖZṴ��l�`d\g�qJ�5��F��6�M�3Z1�b� | ̃"D��O$̾P20�`jԔkP>! { When the supply function is concave in x, Jensen inequality holds: Moreover, we show that, as the market becomes large, if the equilibrium distribution of firm states obeys a certain “light-tail” condition, then oblivious equilibria closely approximate Markov perfect equilibria. %PDF-1.5 More recent work has used stochastic games to model a wide range of topics in industrial organization, including advertising (Doraszelski, 2003) capacity accumulation (Besanko and It is used to study settings where multiple decision makers interact non-cooperatively over time, each seeking to pursue its own objective. the market. %���� 4. Tip: you can also follow us on Twitter In this lecture, we teach Markov perfect equilibrium by example. A Markov perfect equilibrium with robust agents will be characterized by • a pair of Bellman equations, one for each agent. We will focus on settings with • two players Basic Setup¶. Markov perfect equilibrium is a key notion for analyzing economic problems involving dy-namic strategic interaction, and a … • a pair of equations that express linear decision rules for each agent as functions of that agent’s continuation value function as well as parameters of preferences and state tran- Equilibrium exists and is unique (refer to the paper) Ann Oper Res (2020) 287:573–591 https://doi.org/10.1007/s10479-018-2778-2 S.I. We define Markov strategy and Markov perfect equilibrium (MPE) for games with observable actions. We will focus on settings with • two players 5.2.1 Markov Perfect Equilibrium 122 5.2.2 Logit Markov QRE 125 5.3 Evolutionary Dynamics and Logit QRE 127 5.4 Stochastic Learning Equilibrium 133 5.4.1 Some Alternative Learning Rules 134 5.4.2 Beliefs and Probabilistic Choice 137 5.4.3 History Formation 138 5.4.4 Stochastic Learning Equilibrium 138 6. In this lecture, we teach Markov perfect equilibrium by example. 7For this set up, one can guess the unique subgame perfect Nash equilibrium strategies of the nite horizon model. The equilibrium concept used is Markov perfect equilibrium (MPE), where the set of states are all possible coalition structures. Tip: you can also follow us on Twitter Equilibriummeans a level position: there is no more change in the distri-bution of X t as we wander through the Markov chain. 4.2 Markov Chains at Equilibrium Assume a Markov chain in which the transition probabilities are not a function of time t or n,for the continuous-time or discrete-time cases, respectively. Markov perfect equilibrium Eggertsson: Federal Reserve Bank of New York (e-mail: gauti.eggertsson@ny.frb.org). The agents in the model face a common state vector, the time path of which is influenced by – and influences – their decisions. INTRODUCTION IN MANY BRANCHES OF APPLIED ECONOMICS, it has become common practice to estimate structural models of decision-making and equilibrium. a Markov perfect equilibrium of a dynamic stochastic game must satisfy the conditions for a Nash equilibrium of a certain reduced one-shot game. Get the latest machine learning methods with code. Following convention in the literature, we maintain that players do not switch between equilibria within the process of a dynamic game. %PDF-1.4 The model and Markov Perfect Equilibrium In this section we describe the main features of the exogenous timing duopoly model [for further discussion of this model, see Maskin and Tirole (1982)]. VP�*y� We will focus on settings with • two players Browse our catalogue of tasks and access state-of-the-art solutions. Higher network effects increase the inequality of the market structure. The MPE solutions determine, jointly, both the expected equilibrium value of coalitions and the Markov state transition probability that describes the path of coalition formation. (SPE doesn’t su er from this problem in the context of a bargaining game, but many other games -especially repeated games- contain a large number of SPE.) Markov perfection implies that outcomes in a subgame depend only on the relevant strategic elements of that subgame. Our analysis is applied to a stylized description of the browser war between Netscape and Microsoft. Markov-perfect equilibrium where the equilibrium path market share difference is linear in the price differences between the firms in the preceding period. 1For example, while Markov perfect equilibria in standard in nitely repeated games are simply in nite repetitions of Nash equilibria of the stage game, there can be nontrivial strategic dynamics in Markov perfect equilibria of asynchronous-move games - see Maskin … perfect equilibrium payoffs for the seller range from capturing the full social surplus all the way down to capturing only the current flow value of each good and that each of these payoffs is realized in a Markov perfect equilibrium that follows the socially efficient allocation path. ޮ)[y[��V�٦~�g�W7��~�t�)5:k��95l\��8�]�S�+�:8�{#�������tXC�$. Keywords and Phrases: Oligopoly_Theory, Network_Externalities, Markov_Perfect-Equilibrium perfect equilibrium. Competition between the two firms (i= 1,2) takes place in discrete time with an infinite horizon. %�쏢 Markov perfect equilibrium is a key notion for analyzing economic problems involving dy-namic strategic interaction, and a cornerstone of applied game theory. Decisions of two agents affect the motion of a state vector that appears as an argument of payoff functions of both agents. Markov perfect equilibrium model from observations on partial trajectories, and discuss estimation of the impacts of firm conduct on consumers and rival firms. $瀁E�eə��Ȇr r��������^X�:ɑ�a�����(m-� We will focus on settings with • two players Product innovation 1 Introduction Since the seminal contributions of Sethi [19], Skiba [20] and Dechert and Nishimura [9], it has been shown that rational planning over an infinite planning horizon can go along ��D�w0�9��7�+�^?���%� ��ȁ�{ A Markov perfect equilibrium is an equilibrium concept in game theory.It has been used in analyses of industrial organization, macroeconomics, and political economy.It is a refinement of the concept of subgame perfect equilibrium to extensive form games for which a pay-off relevant state space can be identified. Following convention in the literature, we maintain that players do not switch between equilibria within the process of a dynamic game. stream Our analysis is applied to a stylized description of the browser war between Netscape and Microsoft. This is … Dis- It is used to study settings where multiple decision-makers interact non-cooperatively over time, each pursuing its own objective. The Markov Perfect Equilibrium (MPE) concept is a drastic re nement of SPE developed as a reaction to the multiplicity of equilibria in dynamic problems. (PM1) and (PM2) provide algorithms to compute a Markov perfect equilibrium (MPE) of this stochastic game. If πTP = πT, we say that the distribution πT is an equilibrium distribution. • Linear Markov perfect equilibria 4 • Application 5 • Exercises 6 • Solutions 7 2 Overview This lecture describes the concept of Markov perfect equilibrium. xڽYɎ#7��+��H�B � �[��9���rH��R�TU{��3A�\��S�d���bMfAX0$�r�{��e���T`�[^��O��;�_^n~�ڽjm�ZM����ys�@g��Ն����BYi�\Å��V��.�3��������H?X�Q�{`�NA�����ӹ�����JLVU��q��G�cu���KQ�݊)Y�L�+)w��\L�C����d�B����π��!,)����e�|T�x�Z�Vx-�*���O�Y�g�J�2Е�%0������_J|�b��.d����Uj���'��^�og��Q�=V�0�v�!f:��;VYkH �1�oS %�5;�� Durable Good Monopoly Commitment Optimal pricing with commitment VC(D 1) = max fP t;Xt;Dtg 0 E 0 X1 t=0 t(P t W t)X t subject to durable stock dynamics D The overwhelming focus in stochastic games is on Markov perfect equilibrium. Markov perfect equilibrium model from observations on partial trajectories, and discuss estimation of the impacts of firm conduct on consumers and rival firms. A strategy profile is a Markov-perfect equilibrium (MPE) if it consists of only Markov strategies it is a Nash equilibrium regardless of the starting state Theorem. 4. In this lecture, we teach Markov perfect equilibrium by example. • a pair of equations that express linear decision rules for each agent as functions of that agent’s continuation value function as well as parameters of preferences and state tran- Generally, Markov Perfect equilibria in games with alternating moves are difierent than in games with simultaneous moves. <> any Subgame Perfect equilibrium of the alternating move game in which players’ memory is bounded and their payofis re°ect the costs of strategic complexity must coincide with a MPE. In this lecture, we teach Markov perfect equilibrium by example. MPE equilibrium cannot be taken for granted. Every n-player, general-sum, discounted-reward stochastic game has a MPE The role of Markov-perfect equilibria is similar to role of subgame-perfect This refers to a (subgame) perfect equilibrium of the dynamic game where players’ strategies depend only on the 1. current state. (PM1) and (PM2) provide algorithms to compute a Markov perfect equilibrium (MPE) of this stochastic game. KEYWORDS: Markov perfect equilibrium, dynamic games, incomplete models, bounds estimation. 5A Markov Perfect Equilibrium is a profile of time-homogeneous pure strategies that map a player’s information in each single time period to a choice. * QV.�DZN �2�i}��y���T:���������y��]G��s-�����0��Fn�ۺ�2#YѴ3"9�7�����G;L4w(p��u�ʧ��{�S���F�#K퉂QKG�{机��X-rVdc�O��ԣ@. Informally, a Markov strategy depends only on payoff-relevant past events. A Markov Perfect Equilibrium is a set of functions such that: The policy functions solve the incumbent and entrants problems given beliefs The perceived aggregate transition probabilities are consistent with the optimal response of all agents. In section V we consider the limit of equilibrium behav- More recent work has used stochastic games to model a wide range of topics in industrial organization, including advertising (Doraszelski, 2003) capacity accumulation (Besanko and In game theory, a Perfect Bayesian Equilibrium (PBE) is an equilibrium concept relevant for dynamic games with incomplete information (sequential Bayesian games).It is a refinement of Bayesian Nash equilibrium (BNE). A Markov perfect equilibrium with robust agents will be characterized by • a pair of Bellman equations, one for each agent. In the latter case, MPE are trivial. D�hi-5���+��P� ( PM1 ) and ( PM2 ) provide algorithms to compute a Markov perfect by! 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