(PM1) and (PM2) provide algorithms to compute a Markov perfect equilibrium (MPE) of this stochastic game. This chapter was originally published in The New Palgrave Dictionary of Economics, 2nd edition, 2008. © Copyright 2020, Thomas J. Sargent and John Stachurski. \Gamma_{1t})' (Q_1 + \beta B_1' P_{1t+1} B_1)^{-1} The concept of Markov perfect equilibrium was rst introduced by Maskin and Tirole, 1988. For each player i and state x, a set Ai(x) of actions available to player i in state x. We review the recent literature in macroeconomics that analyses Markov equilibria in dynamic general equilibrium model. u_{-it}' S_i u_{-it} + v_i^{j+1}(q_i, q_{-i}) = \max_{\hat q_i} A state space X (which we assume to be finite for the moment). Player ’s malevolent alter ego employs decision rules 𝑖 = 𝐾𝑖 𝑥 where 𝐾𝑖 is an ℎ × ð‘›ma- trix. 2002. Markov perfect equilibrium is a refinement of the concept of Nash equilibrium. Under mild regularity conditions, for economies with either bounded or unbounded state spaces, continuous monotone Markov perfect Nash equilibrium (henceforth MPNE) are shown to exist, and form an antichain. Spear, S., and S. Srivastava. Replicate the pair of figures showing the comparison of output and prices for the monopolist and duopoly under MPE. Not affiliated $ x_t $ is an $ n \times 1 $ state vector and $ u_{it} $ is a $ k_i \times 1 $ vector of controls for player $ i $, $ \{F_{1t}\} $ solves player 1’s problem, taking $ \{F_{2t}\} $ as given, and, $ \{F_{2t}\} $ solves player 2’s problem, taking $ \{F_{1t}\} $ as given, $ \Pi_{it} := R_i + F_{-it}' S_i F_{-it} $. 1992. 1988. Player $ i $ takes $ \{u_{-it}\} $ as given and minimizes, $$ After defining the Markov equilibrium concept we first summarize what is known about the existence and uniqueness of such equilibria in models where sequential equilibria can be obtained by solving a suitable social planner problem. Atkeson, A., and R. Lucas. Now we evaluate the time path of industry output and prices given Markov perfect equilibrium is a refinement of the concept of Nash equilibrium. The first figure shows the dynamics of inventories for each firm when the parameters are. \beta \Lambda_{1t}' P_{1t+1} \Lambda_{1t} \tag{11} Two firms are the only producers of a good, the demand for which is governed by a linear inverse demand function. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International. The second panel shows analogous curves for price. x_t' \Pi_{1t} x_t + 1998. 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 monopolist initial condition is $ q_0 = 2.0 $ to mimic the industry initial condition $ q_{10} = q_{20} = 1.0 $ in the MPE case. \sum_{t=t_0}^{t_1 - 1} Equilibria based on such strategies are called stationary Markov perfect equilibria. Asset prices in an exchange economy. Markov perfect equilibrium is a key notion for analyzing economic problems involving dy-namic strategic interaction, and a cornerstone of applied game theory. Is an e cient allocation of the processes achievable in equilibrium? This is close enough for rock and roll, as they say in the trade. $ I_{it} $ = inventories of firm $ i $ at beginning of $ t $, $ q_{it} $ = production of firm $ i $ during period $ t $, $ p_{it} $ = price charged by firm $ i $ during period $ t $, $ S_{it} $ = sales made by firm $ i $ during period $ t $, $ E_{it} $ = costs of production of firm $ i $ during period $ t $, $ C_{it} $ = costs of carrying inventories for firm $ i $ during $ t $, $ C_{it} = c_{i1} + c_{i2} I_{it} + 0.5 c_{i3} I_{it}^2 $, $ E_{it} = e_{i1} + e_{i2}q_{it} + 0.5 e_{i3} q_{it}^2 $ where $ e_{ij}, c_{ij} $ are positive scalars, $ S_t = \begin{bmatrix} S_{1t} & S_{2t} \end{bmatrix}' $, $ D $ is a $ 2\times 2 $ negative definite matrix and. © 2020 Springer Nature Switzerland AG. Let’s have a look at the different time paths, We can now compute the equilibrium using qe.nnash, Now let’s look at the dynamics of inventories, and reproduce the graph Klein, P., and V. Rios-Rull. 2. Blackwell, D. 1965. \pi_i(q_i, q_{-i}, \hat q_i) = a_0 q_i - a_1 q_i^2 - a_1 q_i q_{-i} - \gamma (\hat q_i - q_i)^2 , \tag{3} Markov perfect equilibrium. x_{t+1} = (A - B_1 F_1 -B_1 F_2 ) x_t \tag{14} Kydland, F., and E. Prescott. These iterations can be challenging to implement computationally. As we saw in the duopoly example, the study of Markov perfect equilibria in games with two players leads us to an interrelated pair of Bellman equations. Individual payoff maximization requires that each agent solve a dynamic programming problem that includes this transition law. Kubler, F., and K. Schmedders. These equilibrium conditions can be used to derive a nonlinear system of equations, f(σ) = 0, that must be satisfied by any Markov perfect equilibrium σ; we say that the equilibrium σ is regular if the Jacobian matrix ∂f ∂σ (σ) has full rank. F_{1t} 3.2 Computing Equilibrium We formulate a linear robust Markov perfect equilibrium as follows. $$, $$ In addition, we provide ... prove existence of subgame perfect Nash equilibrium in a class of such games. 2 u_{1t}' \Gamma_{1t} x_t Secondly, making use of the specific structure of the tran-sition probability and applying the theorem of Dvoretzky, Wald and Wolfowitz [27] we obtain a desired pure stationary Markov perfect equilibrium. Recursive methods 1 ( discounted ) stochastic game with N players consists of the following elements we address these in! Rules for price and output in this lecture, we teach Markov perfect equilibrium as follows out... Firms are the same way employs decision rules for price and quantity take the form $ u_ it. These, we teach Markov perfect equilibrium is a 𝑖×𝑛matrix decision rules for price and in! Initial condition has been set to $ q_ { 20 } = -F_i $! Н‘¥, where 𝐹𝑖 is a Markov perfect equilibria ( MPEs ) economists Jean Tirole and Maskin. 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