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We briefly survey the state of the art for mean field games without entering any technical/mathematical details. We review both the existing mathematical results and the modeling toolbox. We also mention a few applications. After describing a new numerical approach, we conclude with a few perspectives.
We introduce Pi-1-1 sets and prove some of the basic key results. These include descriptive set theoretic results as Sigma-1-1-bounding and the Gandy basis theorem, together with applications to computable structure theory.
This chapter introduces the three contributions that constitute Part II, “Mathematics of Game Theory and Its Foundations.” Those concern (1) mean field games, (2) value and equilibrium in zero-sum games, and (3) refinements of Nash equilibrium.
Given strong common interests in avoiding mass killing and economic devastation, or even the relatively small negative economic effects of some low-level internal conflicts, why do these conflicts occur? Human affairs are too complex, messy, and interesting for any one angle to provide a fully satisfactory answer, and particularly an angle as schematic and bloodless as game theory. Even so, work using game-theoretic methods has developed answers that provide useful insights into important strategic dimensions, and that provide plausible explanations for some of the more striking empirical patterns concerning armed conflict in this period. This chapter reviews game-theoretic models of the outbreak and prosecution of large-scale armed conflict as they have developed in particular over the last forty years.