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A Nash Equilibrium for Differential Games with Moving-Horizon Strategies

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Abstract

Our paper aims at introducing a moving-horizon interaction in a strategic context. We assume that, in each instant of time, players can predict the effects of their actions and those of their opponents on a finite moving horizon. We define an equilibrium concept, which is consistent in this setting, and develop an appropriate algorithm to compute it by using nonlinear model predictive control techniques. Focusing on the length of forecasting horizon, we propose two economic interpretations for our equilibrium, based on the limited rationality and political economy literature: a simple 2 players’ nonlinear policy game, and what happens to debt stabilization when policymakers have different values of the forecasting horizon. To provide some practical insights of our approach, we consider a debt stabilization game in a monetary union. We consider three players; two nonlinear differential constraints; and assume that one player controls one instrument which is not additive but has some multiplicative effects on the state variables.

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Notes

  1. The NMPC is common in many fields as an approximation tool for global solutions. It is often used to solve dynamic decision models without having to resort to local approximations by linearization techniques. E.g., among others, Abedinia et al. (2019), Gao et al. (2019), Ghadimi et al. (2018), Saeedi et al. (2019).

  2. The former assumes hyperbolic discounting and imperfect commitment technology. The latter builds on the notion that agents’ inferences are based on a selected subset of possible events, not on the entire state space.

  3. If the interval becomes infinitesimal, we replicate van den Broek (2002), who only focuses on a continuous LQ framework. The approach can be also generalized to the case where state and control variables both evolve in a discrete manner.

  4. See Grüne & Semmler (2004).

  5. Another issue is whether there could be multiple equilibria which need to be detected by the algorithm. Grüne et al. (2015) give a model example with multiple equilibria where, for a sufficiently long horizon, multiple equilibria can be detected.

  6. We would like to note that the existence of the NMPC solution in the sense of a numerical convergence to a turnpike solution is studied in Grüne et al. (2015, Sect. 3). As to the existence of the NMPC Nash equilibrium, as a recent study shows, there do not seem to be generic results obtained yet: Stieler (2018, p.5) writes: “In noncooperative MPC we show that the mechanism developed in MO [multiobjective] MPC – i.e. choosing the proper solution by means of constraints on the objective functions – does generally not work for NE. For the special case of affine-quadratic games sufficient conditions for the MPC closed-loop trajectory to converge are presented.”

  7. For some extensions, see, e.g., van Aarle et al. (1995, 1997), Tabellini & La Via (1989), Engwerda et al. (2002), Di Bartolomeo & Di Gioacchino (2008), Engwerda et al. (2015), Neck & Blueschke (2014), Aarle et al. (2017), Anevlavis et al. (2018).

  8. The aim of our examples is illustrative. Qualitative results are robust for a large set of realistic parameters. Details are available upon request.

  9. This assumption is common in the literature of fiscal-monetary coordination games. Following this literature, we also assume that players’ threat points are zero. Both assumptions can be easily removed to study other heterogeneity dimensions associated with this game.

  10. See Petit (1990) for a discussion about the different cooperative solution concepts in the differential games.

  11. Since our set up of the differential game contains two non-linearities, a nonlinear state equation and the Nash product the results cannot be easily compared to an LQ game.

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Authors and Affiliations

  1. Universita degli Studi di Roma La Sapienza, Rome, Italy

    Enrico Saltari & Giovanni Di Bartolomeo

  2. The New School, New York, USA

    Willi Semmler

  3. University of Bielefeld, Bielefeld, Germany

    Willi Semmler

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  1. Enrico Saltari
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Correspondence to Enrico Saltari.

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The authors are grateful for comments from James Bullard, Jacob Engwerda, Lars Grüne, Paul Levine, Joseph Plasmans, Davide Ticchi, and participants from the 14th Viennese Conference on Optimal Control and Dynamic Games (3-6 July, 2018, Vienna, Austria), 24th International Conference of the Society for Computational Economics (June 19-21, 2018, Milan, Italy), and VII MPT Workshop (December, 14-15, 2017, Rome, Italy). Willi Semmler gratefully acknowledges a Visiting Professorship at Sapienza University in Fall 2016. The authors also acknowledge financial support by Sapienza University of Rome.

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Saltari, E., Semmler, W. & Di Bartolomeo, G. A Nash Equilibrium for Differential Games with Moving-Horizon Strategies. Comput Econ 60, 1041–1054 (2022). https://doi.org/10.1007/s10614-021-10177-8

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