Discrete-Time Cooperative Games Under Uncertainty

Abstract

In some economic processes in discrete-time, uncertainty may also arise. For instance, Smith and Zenou (Rev. Econ. Dyn. 6(1):54–79, 2003) considered a discrete-time stochastic job search model. Esteban-Bravo and Nogales (Comput. Oper. Res. 35:226–240, 2008) analyzed mathematical programming for stochastic discrete-time dynamics arising in economic systems, including examples in a stochastic national growth model and international growth model with uncertainty. The discrete-time counterpart of stochastic differential games is known as stochastic dynamic games. Basar and Ho (J. Econ. Theory 7:370–387, 1974) examined informational properties of the Nash solutions of stochastic nonzero-sum games. The elimination of the informational nonuniqueness in a Nash equilibrium through a stochastic formulation was first discussed in Basar (Int. J. Game Theory 5:65–90, 1976) and further examined in Basar (Automatica 11:547–551, 1975; In: New trends in dynamic system theory and economics, pp. 3–5, 1979; In: Dynamic policy games in economics, pp. 9–54, 1989). Basar and Mintz (In: Proceedings of the IEEE 11th conference on decision and control, pp. 188–192, 1972; Stochastics 1:25–69, 1973) and Basar (IEEE Trans. Autom. Control AC-23:233–243, 1978) developed an equilibrium solution of linear-quadratic stochastic dynamic games with noisy observation. Again, the SIAM Classics on Dynamic Noncooperative Game Theory by Basar and Olsder (Dynamic noncooperative game theory, 2nd edn. Academic Press, London, 1995) gave a comprehensive treatment of noncooperative stochastic dynamic games. Yeung and Petrosyan (J. Optim. Theory Appl. 145(3):579–596, 2010) provided the techniques in characterizing subgame consistent solutions for stochastic dynamic. Furthermore, they also presented a stochastic dynamic game in resource extraction. Analyses of noncooperative and cooperative discrete-time dynamic games with random game horizons were presented in Yeung and Petrosyan (J. Optim. Theory Appl. forthcoming, 2011). The recently emerging robust control techniques in discrete time along the lines of Hansen and Sargent (Robustness. Princeton University Press, Princeton, 2008) should prove to be fruitful in developing into stochastic dynamic interactive economic models.