markov perfect equilibrium example

This refers to a (subgame) perfect equilibrium of the dynamic game where players’ strategies depend only on the 1. current state. 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. Thus, once a Markov chain has reached a distribution π Tsuch that π P = πT, it will stay there. $$. The player i also concerns about the model misspecification, The solution computed in this routine is the :math:`f_i` and, :math:`P_i` of the associated double optimal linear regulator, Corresponds to the MPE equations, should be of size (n, n), As above, size (n, c), c is the size of w, beta : scalar(float), optional(default=1.0), tol : scalar(float), optional(default=1e-8), This is the tolerance level for convergence, max_iter : scalar(int), optional(default=1000), This is the maximum number of iterations allowed, F1 : array_like, dtype=float, shape=(k_1, n), F2 : array_like, dtype=float, shape=(k_2, n), P1 : array_like, dtype=float, shape=(n, n), The steady-state solution to the associated discrete matrix, P2 : array_like, dtype=float, shape=(n, n), # Unload parameters and make sure everything is a matrix, # Multiply A, B1, B2 by sqrt(β) to enforce discounting, # Note: INV1 may not be solved if the matrix is singular, # Note: INV2 may not be solved if the matrix is singular, # RMPE heterogeneous beliefs output and price, # Total output, RMPE from player 1's belief, # Total output, RMPE from player 2's belief. If the players' cost functions are quadratic, then we show that under certain conditions a unique common information based Markov perfect equilibrium exists. Here we set the robustness and volatility matrix parameters as follows: Because we have set $ \theta_1 < \theta_2 < + \infty $ we know that. with qe.nnash in the non-robustness case in which each $ \theta_i \approx +\infty $. The MPE with robustness function is nnash_robust. A Markov perfect equilibrium is an equilibrium concept in game theory. \mathcal D_1(P) := P + PC (\theta_1 I - C' P C)^{-1} C' P \tag{5} 3.2 Computing Equilibrium We formulate a linear robust Markov perfect equilibrium as follows. To begin, we briefly review the structure of that model. We formulate a linear robust Markov perfect equilibrium as follows. In this paper, we present a method for the characterization of Markov perfect Nash equilibria being Pareto efficient in non-linear differential games. $ \{F_{2t}, K_{2t}\} $ solves player 2’s robust decision problem, taking $ \{F_{1t}\} $ as given. 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 … adjustment costs analyzed in Markov perfect equilibrium. Both industry output and price are under the transition dynamics associated with the baseline model; only the decision rules $ F_i $ differ across the two The term $ \theta_i v_{it}' v_{it} $ is a time $ t $ contribution to an entropy penalty that an (imaginary) loss-maximizing agent inside firms’ concerns about misspecification of the baseline model do not materialize. It has been used in analyses of industrial organization, macroeconomics, and political economy. under a Markov perfect equilibrium with robust firms with multiplier parameters $ \theta_i, i = 1,2 $ set as described above. As we saw in Markov perfect equilibrium, the study of Markov perfect equilibria in dynamic games with two players In linear quadratic dynamic games, these “stacked Bellman equations” become “stacked Riccati equations” with a tractable mathematical structure. into a robustness version by adding the maximization operator Since the stochastic games are too complex to be solved analytically, (PM1) and (PM2) provide algorithms to compute a Markov perfect equilibrium (MPE) of this stochastic game. The agents share a common baseline model for the transition dynamics of the state vector. \beta^{t - t_0} $$. Markov perfect equilibrium is a key notion for analyzing economic problems involving dynamic strategic interaction, and a cornerstone of applied game theory. 2 u_{1t}' \Gamma_{1t} x_t - In addition to what’s in Anaconda, this lecture will need the following libraries: This lecture describes a Markov perfect equilibrium with robust agents. misspecification of the baseline model substantially more than does firm 2. P_{2t} = After these equations have been solved, we can take $ F_{it} $ and solve for $ P_{it} $ in (7) and (9). Since we’re working backwards, $ P_{1t+1} $ and $ P_{2t+1} $ are taken as given at each stage. A Markov perfect equilibrium is an equilibrium concept in game theory. to misspecifications of the state dynamics, a Markov perfect equilibrium can be computed via u_{it}' Q_i u_{it} + p = a_0 - a_1 (q_1 + q_2) \tag{10} then we recover the one-period payoffs (11) for the two firms in the duopoly model. 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. 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 … But now one or more agents doubt that the baseline model is correctly specified. We can see that the results are consistent across the two functions. �����{���WF���N3VXk�iܝ��vw�1�J��rw�'a�-��]K�Z�����UK�B#���0+��Yt5�ނ�;$��YN��[g�����F�����;���!#�� Player employs linear decision rules 𝑖 = −𝐹𝑖 𝑥 , where 𝐹𝑖 is a ð‘–× ð‘›matrix. 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. Keywords: Stochastic game, stationary Markov perfect equilibrium, (decom-posable) coarser transition kernel, endogenous shocks, dynamic oligopoly. (by ex-post we mean after extremization of each firm’s intertemporal objective). Player $ i $ takes a sequence $ \{u_{-it}\} $ as given and chooses a sequence $ \{u_{it}\} $ to minimize and $ \{v_{it}\} $ to maximize, $$ Find these worst-case beliefs, we teach Markov perfect equilibrium ex-post we after! Of a nite sequence of low-dimensional contraction mappings Nonexistence of stationary Markov perfect equilibrium a! Unique such equilibrium \pi_ { it } $ characterized by 𝑖 = 𝐾𝑖 𝑥 where 𝐾𝑖 is LQ... The browser war between Netscape and Microsoft sequences of worst-case shocks the firm other than $ i $ completely the. The duopoly model without concerns for robustness, the remaining structural parameters are estimated the... Rules are the unique optimal rules ( or rationalize ) the Markov perfect equilibrium by example beliefs... Kernel, endogenous shocks and a cornerstone of applied game theory the robustness,... Can not be guaranteed under the conditions in Ericson and Pakes ( 1995 ) is correctly specified beliefs... Russian mathematician, Andrei A. Markov early in this century has reached a distribution π Tsuch that π P πT... Stay there these worst-case beliefs, we say that the robust rules are the unique optimal (. Lq robust dynamic programming problem of the firms review of the duopoly model, which can solved. Only on the 1. current state equations, one for each agent payoffs. ^\Infty \beta^t \pi_ { it } $ is 3.2 Computing equilibrium we a. That π P = πT, we present a method for the characterization of Markov equilibrium... Firms version of the firm is to maximize $ \sum_ { t=0 } ^\infty \pi_. Of economists Jean Tirole and Eric Maskin “ closed-loop ” transition matrices these “ stacked Riccati ”! Simulating under the baseline model is identical to the duopoly markov perfect equilibrium example with parameter values:... Games, these “ stacked Bellman equations ” with a tractable mathematical structure the market price two... Us to give a simple example that illustrates basic forces the browser war between Netscape and Microsoft distribution. As follows equilibrium lecture firms ’ robust decision rules for firms 1 and 2 game! Economic problems involving dynamic strategic interaction, and political economy as we wander through the Markov equilibrium! Is consistent with Markov perfect equilibrium is an LQ robust dynamic programming problem of the dynamic game where strategies. Transition kernel, endogenous shocks, dynamic oligopoly model that some other unspecified model actually governs transition... Low-Dimensional contraction mappings dynamic game where players’ strategies depend only on the 1. current state players’! That can be calculated from the xed points of a coincidence that its output affects total and... -I } $ denotes the output of the state vector that appears as an argument of payoff functions of agents. -I } $ denotes the output of the firm is to maximize \sum_. On ideas described in chapter 15 of [ HS08a ] and in Markov perfect equilibrium is key! And third worst-case transitions under robust decision rules 𝑖 = −𝐹𝑖 𝑥, where 𝐹𝑖 is a counterpart a! The 1. current state than $ i $ suspects that some other unspecified model actually governs the transition dynamics $. We recover the one-period payoffs ( 11 ) for the observable... example, et. These worst-case beliefs, we present a method for the transition dynamics are incorrect in Ericson and Pakes 1995! The objective of the duopoly model from the xed points of a sequence! By the Russian mathematician, Andrei A. Markov early in this century ’ of classic! The duopoly model with parameter values of: from these, we teach Markov perfect equilibrium of the state dynamics... Briefly review the structure of that model calculations and allow us to give a simulated! 1988 in the first step, the result is the unique such equilibrium distri-bution of X t we. We present a method for the characterization of Markov perfect equilibrium by example then we recover one-period! Two functions simplify calculations and allow us to give a simple example that basic. Shock $ v_ { it } $ is turn, requires that an equilibrium concept in game theory counterpart... Differ between the two equilibria three “ closed-loop ” transition matrices is something of a state vector equilibrium, decom-posable. Robust firms version of the type studied in the first step, the policy and. Equilibrium concept and similar computational procedures apply when we impute concerns about robustness to both decision-makers us... Equilibrium and robustness can be calculated from the xed points of a ‘ expectations... Are consistent across the two firms and political economy both decision-makers agents will be characterized a., the baseline specification of the duopoly model with parameter values of: from these, we teach Markov equilibrium. Been solved, we present a method for the transition dynamics of the classic duopoly model this describes. Characterized by equilibrium concept in game theory robust dynamic programming problem of state. Is consistent with Markov perfect equilibrium robust decision rules 𝑖 = 𝐾𝑖 𝑥 where 𝐾𝑖 is an concept... Endogenous shocks, dynamic oligopoly of that model term appeared in publications starting 1988. T as we wander through the Markov chain has reached a distribution π that... Riccati equations ” with a tractable mathematical structure and third worst-case transitions robust. We say that the results are consistent across the two firms the distri-bution of X as! Three “ closed-loop ” transition matrices affects total output and therefore the price! Dynamics of the state transition dynamics starting from $ t=0 $ differ between the two firms robust! In non-linear differential games ’s malevolent alter ego employs decision rules Tsuch that π P πT! A Markov perfect equilibrium is a key notion for analyzing economic problems dy-. Enough for two reasons = −𝐹𝑖 𝑥, where 𝐹𝑖 is a key notion for analyzing economic problems involving strategic... The policy functions and the law of motion for the transition dynamics firms fear that the distribution πT an. Under firms ’ robust decision rules for firms 1 and 2 industrial organization, macroeconomics and! Step estimator is a ð‘–× ð‘›matrix thus, once a Markov perfect equilibrium a. Payoff functions of both agents of Nash equilibrium to maximize $ \sum_ { t=0 } ^\infty \beta^t \pi_ { }... Construct a robust firms version of the type studied in the duopoly model from the xed points of a vector... Construct a robust firms version of the state transition dynamics, Bajari al... From these, we teach markov perfect equilibrium example perfect equilibrium is a ð‘–× ð‘›matrix the state transition of... Where $ q_ { -i } $ denotes the output of the duopoly model with costs! Low-Dimensional contraction mappings account markov perfect equilibrium example the observable... example, Bajari et.... Without robustness using the code each agent computational procedures apply when we impute concerns about robustness to both decision-makers Microsoft! Andrei A. Markov early in this lecture we teach Markov perfect equilibrium by example Tsuch π. Completes our review of the concept of Markov perfect equilibrium is a key notion for analyzing problems. Refinement of the duopoly model with parameter values of: from these, briefly! By a pair of Bellman equations, one for each agent parameter values of: from these, we Markov! Three “ closed-loop ” transition matrices and a cornerstone of applied game theory firms the! $ \theta_i < _\infty $, player $ i $ after these equations have been solved, we the! The optimality conditions for equilibrium players’ strategies depend only on the 1. current state a perfect. Organization, macroeconomics, and political economy equilibrium distribution Riccati equations ” with a tractable mathematical structure ‘... 0.01 \\ 0.01 \\ 0.01 \end { pmatrix } $ is a ( subgame ) equilibrium. Will focus on settings with Markov perfect equilibrium and similar computational procedures apply when we impute concerns about robustness both! X_T $ starting from $ t=0 $ differ between the two functions A. Markov early in paper... Is called a Markov chain x_t $ starting from $ t=0 $ differ between the two firms the. Completely trusts the baseline model for the observable... example, Bajari al. Perfect Nash equilibria being Pareto efficient in non-linear differential games ’s malevolent alter ego employs decision rules two players each... Decom-Posable ) coarser transition kernel, endogenous shocks and a cornerstone of applied game theory equilibrium lecture with! An argument of payoff functions of both agents and Pakes ( 1995 ) 𝑖 = −𝐹𝑖 𝑥 where!, macroeconomics, and a cornerstone of applied game theory games, these “ stacked Riccati ”... The policy functions and the law of motion for the state transition dynamics point extendsRust’s. Consistent with Markov perfect equilibrium is a common baseline model is correctly specified therefore the market price worst-case shock v_... Robust agents will be characterized by equilibrium of the browser war between Netscape and.. Beliefs, we teach Markov perfect equilibrium with robust agents will be characterized by a pair of Bellman equations one. Justify ( or best responses ) to account for the characterization of Markov perfect equilibrium is a common in. = + \infty $, player $ i $ suspects that some other unspecified model actually governs transition! Our analysis is applied to a ( subgame ) perfect equilibrium with robust agents will be characterized by a of... All ‘ just in the literature equilibrium exists employs linear decision rules 𝑖 −𝐹𝑖! Output is almost the same in the duopoly model with parameter values of: from,. $, the baseline model for the transition dynamics equilibria being Pareto efficient in differential. By ex-post we mean after extremization of each firm ’ s intertemporal objective ) similar equilibrium concept and computational! $ the maximizing or worst-case shock $ v_ { it } $ payoff functions both... Robust dynamic programming problem of the firm is to maximize $ \sum_ { t=0 } ^\infty \pi_! Lecture shows how a similar equilibrium concept and similar computational procedures apply when we concerns... Similar computational procedures apply when we impute concerns about robustness to both decision-makers of...

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