Differential games with special structures
Bibliographic record
Abstract
In this chapter we are concerned with the analysis of tractable game structures. More specifically, we investigate various classes of differential games for which one can derive analytical characterizations of open-loop and Markov perfect Nash equilibria. We stress analytical tractability since we believe that analytical solutions have the advantage of shedding light on the qualitative properties of equilibria in a general way. We identify three different classes of games for which we discuss the derivation of both open-loop and Markov perfect Nash equilibria. The first class is that of linear quadratic games. Such a game is characterized by a linear system of state equations and quadratic objective functions. The class of linear quadratic differential games has not only gained popularity among dynamic game theorists but also among macroeconomists interested in issues such as policy coordination, optimal stabilization policies, and the like. For this reason we present some macroeconomic interpretations of linear quadratic games. The second class of solvable differential games we consider consists of games in which the state variables enter both the state equations and the objective functions linearly. We refer to this class as linear state games. We show that these games have the property that an open-loop Nash equilibrium is Markov perfect and that the optimal value functions are linear with respect to the state variables. The third class of games discussed in this chapter is exponential games. In this class the state variables enter the objective functions via an exponential term while the state equation is independent of the state variables.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".