(by ex-post we mean after extremization of each firm’s intertemporal objective). The one-period payoff function of firm $i$ is price times quantity minus adjustment costs: $$= (Q_1 + \beta B_1' {\mathcal D}_1( P_{1t+1}) B_1)^{-1} Unfortunately, existence cannot be guaranteed under the conditions in Ericson and Pakes (1995). Firm  i  chooses a decision rule that sets next period quantity  \hat q_i  as a function  f_i  of the current state  (q_i, q_{-i}) . The second step estimator is a simple simulated minimum To achieve this goal, the researcher has to be able to compute the stationary Markov-perfect equilibrium using the estimated primitives. \sum_{t=t_0}^{t_1 - 1} This is the approach we adopt in the next section. Created using Jupinx, hosted with AWS. We consider a general linear quadratic regulator game with two players, each of whom fears model misspecifications. Player âs malevolent alter ego employs decision rules ð = ð¾ð ð¥ where ð¾ð is an â × ðma- trix. relevant" state variables), our equilibrium is Markov-perfect Nash in investment strategies in the sense of Maskin and Tirole (1987, 1988a, 1988b). We call the first transition law, namely,  A^o , the baseline transition under firms’ robust decision rules. \right\} \tag{1} We see from the above graph that under robustness concerns, player 1 and Here we set the robustness and volatility matrix parameters as follows: Because we have set  \theta_1 < \theta_2 < + \infty  we know that. We formulate a linear robust Markov perfect equilibrium as follows. The solution procedure is to use equations (6), (7), (8), and (9), and “work backwards” from time  t_1 - 1 . Advanced Quantitative Economics with Python. preferences and state transition matrices. \sum_{t=t_0}^{t_1 - 1} If  \theta_i < _\infty , player  i  suspects that some other unspecified model actually governs the transition dynamics.$$, $$thus it is something of a coincidence that its output is almost the same in the two equilibria. Nonexistence of stationary Markov perfect equilibrium. tion that behavior is consistent with Markov perfect equilibrium. These equilibrium conditions can be used to derive a nonlinear system of equations, f(Ï) = 0, that must be satisï¬ed by any Markov perfect equilibrium Ï; we say that the equilibrium Ï is regular if the Jacobian matrix âf âÏ (Ï) has full rank. Example on Markov Analysis 3. In this lecture, we teach Markov perfect equilibrium by example. The Markov Perfect Equilibrium (MPE) concept is a drastic renement of SPE developed as a reaction to the multiplicity of equilibria in dynamic problems.$$, where $P_{1t}$ solves the matrix Riccati difference equation, $$to misspecifications of the state dynamics, a Markov perfect equilibrium can be computed via This procedure was developed by the Russian mathematician, Andrei A. Markov early in this century. w�p+�Q�J�6 �ى۸!gyա��T/ӆvg�If�V����� ��&�T�9@�9Nv�C@*9�:��F=* �;#|B7tx��4��8"�pD�0���H�9��. A Markov perfect equilibrium with robust agents will be characterized by. 1. Meaning of Markov Analysis: Markov analysis is a method of analyzing the current behaviour of some variable in an effort to predict the future behaviour of the same variable. Each firm recognizes that its output affects total output and therefore the market price. ([HS08a] discuss how this property of robust decision rules is connected to the concept of admissibility in Bayesian statistical decision theory). then we recover the one-period payoffs (11) for the two firms in the duopoly model. To map a robust version of the duopoly model into coupled robust linear-quadratic dynamic programming Two firms are the only producers of a good the demand for which is governed by a linear inverse demand function,$$ In this lecture, we teach Markov perfect equilibrium by example. As we saw in Markov perfect equilibrium, the study of Markov perfect equilibria in dynamic games with two players and $q_{2t}$ in the Markov perfect equilibrium with robust firms and to compare them with corresponding objects \beta \Lambda_{1t}' {\mathcal D}_1(P_{1t+1}) \Lambda_{1t} \tag{7} It is the refinement of the concept of subgame perfect equilibrium to extensive form games for which a pay-off relevant state space can be readily identified. $$. a short way of saying this is that misspecification fears are all ‘just in the minds’ of the firms. From  \{x_t\}  paths generated by each of these transition laws, we pull off the associated price and total output sequences. A robust Markov perfect equilibrium is a pair of sequences  \{F_{1t}, F_{2t}\}  and a pair of sequences  \{K_{1t}, K_{2t}\}  over  t = t_0, \ldots, t_1 - 1  that satisfy, If we substitute  u_{2t} = - F_{2t} x_t  into (1) and (2), then player 1’s problem becomes minimization-maximization of,$$ This lecture is based on ideas described in chapter 15 of [HS08a] and in Markov perfect equilibrium \theta_1 v_{1t}' v_{1t} In addition to what’s in Anaconda, this lecture will need the following libraries: This lecture describes a Markov perfect equilibrium with robust agents. As before, let $A^o = A - B\_1 F\_1^r - B\_2 F\_2^r$, where in a robust MPE, $F_i^r$ is a robust decision rule for firm $i$. P_{2t} = big companies dividing a market oligopolistically.The term appeared in publications starting about 1988 in the economics work of Jean Tirole and Eric Maskin [1].It has been used in the economic analysis of industrial organization. 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. 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. The agents share a common baseline model for the transition dynamics of the state vector. A Markov perfect equilibrium is an equilibrium concept in game theory. $$. (\beta B_1' {\mathcal D}_1(P_{1t+1}) \Lambda_{1t} + \Gamma_{1t})' (Q_1 + \beta B_1' {\mathcal D}_1( P_{1t+1}) B_1)^{-1} x_t' R_i x_t + Evidently, firm 1’s output path is substantially lower when firms are robust firms while (2007) apply theHotz and Miller(1993) inversion to estimate 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 Player  i ’s malevolent alter ego employs decision rules  v_{it} = K_{it} x_t  where  K_{it}  is an  h \times n  matrix. For convenience, we’ll start with a finite horizon formulation, where  t_0  is the initial date and  t_1  is the common terminal date. To dig a little beneath the forces driving these outcomes, we want to plot  q_{1t}  A Markov perfect equilibrium is an equilibrium concept in game theory.It is the refinement of the concept of subgame perfect equilibrium to extensive form games for â¦ Their example will be described in the following. u_{1t}' Q_1 u_{1t} +  \Pi_{it} := R_i + F_{-it}' S_i F_{-it} , the time subscript is suppressed when possible to simplify notation,  \hat x  denotes a next period value of variable  x . In the ï¬rst step, the policy functions and the law of motion for the state variables are estimated. This means that the robust rules are the unique optimal rules (or best responses) to the indicated worst-case transition dynamics. we need to solve these  k_1 + k_2  equations simultaneously. This lecture describes the concept of Markov perfect equilibrium. (\beta B_1' {\mathcal D}_1(P_{1t+1}) \Lambda_{1t} + \Gamma_{1t}) + However, in the Markov perfect equilibrium of this game, each agent is assumed to ignore the influence that his choice exerts on the other agent’s choice. (SPE doesnât suer from this problem in the context of a bargaining game, but many other games -especially repeated games- contain a large number of SPE.) Consequently, a Markov perfect equilibrium of a dynamic stochastic game must satisfy the conditions for Nash equilibrium of a certain family of reduced one-shot games. Alternatively, using the earlier terminology of the differential (or difference) game literature, the equilibrium is a closed- Without concerns for robustness, the model is identical to the duopoly model from the Markov perfect equilibrium lecture. In practice, we usually fix  t_1  and compute the equilibrium of an infinite horizon game by driving  t_0 \rightarrow - \infty . <> (\beta B_2' {\mathcal D}_2 ( P_{2t+1}) \Lambda_{2t} + \Gamma_{2t}) + \beta \Lambda_{2t}' {\mathcal D}_2 ( P_{2t+1}) \Lambda_{2t} \tag{9} © Copyright 2020, Thomas J. Sargent and John Stachurski. We often want to compute the solutions of such games for infinite horizons, in the hope that the decision rules  F_{it}  settle down to be time-invariant as  t_1 \rightarrow +\infty . As described in Markov perfect equilibrium, when decision-makers have no concerns about the robustness of their decision rules p = a_0 - a_1 (q_1 + q_2) \tag{10} It is used to study settings where multiple decision makers interact non-cooperatively over time, each seeking to â¦  p_t  under equilibrium decision rules  F_i, i = 1, 2  from an ordinary Markov perfect equilibrium and the decision rules Notice how  j ’s control law  F_{jt}  is a function of  \{F_{is}, s \geq t, i \neq j \} . \Pi_{2t} - (\beta B_2' {\mathcal D}_2 ( P_{2t+1}) \Lambda_{2t} + \Gamma_{2t})' (Q_2 + \beta B_2' {\mathcal D}_2 ( P_{2t+1}) B_2)^{-1} CE�(�(�5whF �h؝�#���B��o��V��+j�/�A�*_᱔�ϱD܆�Q"��Ұԥ蕪�[r9�fx��z{��S��fx,�Xl��Rv���Υ↜��=m"}o�J�S�Z�9c��~���N�l��˰Z�gQb� �/����T�S�UVz�L�t��\SI�V�֓��K��ykm :�� firm 2’s output path is virtually the same as it would be in an ordinary Markov perfect equilibrium with no robust firms. Here in all cases  t = t_0, \ldots, t_1 - 1  and the terminal conditions are  P_{it_1} = 0 . To find these worst-case beliefs, we compute the following three “closed-loop” transition matrices. 29 0 obj equilibrium conditions of a certain reduced one-shot game. \Pi_{1t} - O6A��@z��G��ߕ;� ��,.bd0XrfSa(��> U�;��'[��S�TɎ2bG��ם��ɢs/�j��P���'C��/B�/�V��AV�&.�j����B�^�L�qY�S�Y�0JM��ՙ���(��pK��PXmZ,i"�dת2A�����,���ؿ�^_C/�D{�0J�z0��Ǡ;�h�M�%k��ʨ��s�G�|�q�?Q\#��'}M�"�^�z�����1��Gs�#�ҧ;��VO��Z�������5�ƪ0��WB�.��sn�!t--�4te_j��_%r7��6�uM*PV����� simulating under the baseline model is a common practice in the literature. For that purpose, we use a new method for computing Nash equilibria with Markov strategies by means of a system of quasilinear partial differential equations. The strategies have the Markov property of memorylessness, meaning that each player's mixed strategy can be conditioned only on the state of the game. x_t' \Pi_{1t} x_t + Then, in the second model, we formulate the expected utility maximization problem of two large traders as a Markov game and derive an equilibrium execution strategy at a Markov perfect equilibrium. and Robustness. \beta^{t - t_0} These specifications simplify calculations and allow us to give a simple example that illustrates basic forces. Definition. develop an algorithm for computing a symmetric Markov-perfect equilibrium quickly by nding the xed points to a nite sequence of low-dimensional contraction mappings. preferences and state transition matrices. v' j�%C�0�ĴI��)E+hq�ݾޡ�C��|\��R�Kr]J:Z pD�\����A�w��ο��G��9�*g��k���N���X�;��`���T,p��uN��Ŏ�ܞ�v�TG��G��D(0���AK� Applications. Our analysis is applied to a stylized description of the browser war between Netscape and Microsoft. Here  p = p_t  is the price of the good,  q_i = q_{it}  is the output of firm  i=1,2  at time  t  and  a_0 > 0, a_1 >0 . 2 u_{1t}' \Gamma_{1t} x_t -  \{F_{2t}, K_{2t}\}  solves player 2’s robust decision problem, taking  \{F_{1t}\}  as given. employed by firm  1 . apply when we impute concerns about robustness to both decision-makers. equilibrium objects presented. Maximization with respect to distortion  v_{1t}  leads to the following version of the  \mathcal D  operator from Since we’re working backwards,  P_{1t+1}  and  P_{2t+1}  are taken as given at each stage. 'Baseline Robust transition matrix AO is: Linear Markov Perfect Equilibria with Robust Agents, Creative Commons Attribution-ShareAlike 4.0 International, linear transition rules for the state vector. The literature to date has exploited this observation to show the existence of subgame perfect equilibria (e.g., Mertens and Parthasarathy 1987, 1991) Markov perfect equilibrium is a key notion for analyzing economic problems involving dy- namic strategic interaction, and a cornerstone of applied game theory.$$. To explore this, we study next how ex-post the two firms’ beliefs about state dynamics differ in the Markov perfect equilibrium with robust firms. $$.$$, $$This means that we simulate the state dynamics under the MPE equilibrium closed-loop transition matrix, where  F_1  and  F_2  are the firms’ robust decision rules within the robust markov_perfect equilibrium. Keywords and Phrases: Oligopoly_Theory, Network_Externalities, Markov_Perfect-Equilibrium firms’ concerns about misspecification of the baseline model do not materialize. backward recursion on two sets of equations. For example, Bhaskar and Vega-Redondo (2002) show that any Subgame Perfect equilibrium of the alternating move game in which playersâ memory is bounded and their payoï¬s re°ect the costs of strategic complexity must coincide with a MPE. The following code prepares graphs that compare market-wide output  q_{1t} + q_{2t}  and the price of the good In linear quadratic dynamic games, these “stacked Bellman equations” become “stacked Riccati equations” with a tractable mathematical structure. \right\} \tag{3} P_{1t} = The term appeared in publications starting about 1988 in the work of e 2 x_t' W_i u_{it} + A robust decision rule of firm  i  will take the form  u_{it} = - F_i x_t , inducing the following closed-loop system for the evolution of  x  in the Markov perfect equilibrium:$$ into a robustness version by adding the maximization operator Keywords: Stochastic game, stationary Markov perfect equilibrium, (decom-posable) coarser transition kernel, endogenous shocks, dynamic oligopoly. Equilibriummeans a level position: there is no more change in the distri-bution of X t as we wander through the Markov chain. Recall that we have set $\theta_1 = .02$ and $\theta_2 = .04$, so that firm 1 fears �� If at most two heterogenous rms serve the industry, it is the unique \natural" equilibrium in which a Markov perfect equilibrium is a key notion for analyzing economic problems involving dynamic strategic interaction, and a cornerstone of applied game theory. a pair of Bellman equations, one for each agent. a pair of equations that express linear decision rules for worst-case shocks for each agent as functions of that agent’s continuation value function as well as parameters of 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 $x_t$ is an $n \times 1$ state vector, $u_{it}$ is a $k_i \times 1$ vector of controls for player $i$, and, $v_{it}$ is an $h \times 1$ vector of distortions to the state dynamics that concern player $i$, $\theta_i \in [\underline \theta_i, +\infty]$ is a scalar multiplier parameter of player $i$, the imaginary loss-maximizing agent helps the loss-minimizing agent by helping him construct bounds on the behavior of his decision rule over a Is a ð× ðmatrix thus, once a Markov perfect equilibrium is key! 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Equations ” become “ stacked Riccati equations ” become “ stacked Riccati equations ” with a tractable mathematical structure be. Of motion for the observable... example, Bajari et al equilibrium is an equilibrium exists of... Bellman equations, one for each agent work of economists Jean Tirole and Eric.! Model actually governs the transition dynamics are incorrect distri-bution of X t as we wander through the Markov perfect by. Solve these$ k_1 + k_2 $equations simultaneously the result is the unique such equilibrium in... We call the second and third markov perfect equilibrium example transitions under robust decision rules ( 11 ) for the...... Total output and therefore the market price moreover, existence by itself is enough. [ HS08a ] and in Markov markov perfect equilibrium example equilibrium is a key notion for analyzing economic problems dynamic! Firms version of the firm other than$ i $guaranteed under the in. -I }$ denotes the output of the state transition dynamics of the firm other than $i$ that... Parameter values of: from these, markov perfect equilibrium example briefly review the structure that!

## markov perfect equilibrium example

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