Differential games with hierarchical play
Bibliographic record
Abstract
The preceding chapter dealt with differential games in which all players make their moves simultaneously. We now turn to a class of differential games in which some players have priority of moves over other players. To simplify matters, we focus mostly on the case where there are only two players. The player who has the right to move first is called the leader and the other player is called the follower. A well-known example of this type of hierarchical-moves games is the Stackelberg model of duopoly, which is often contrasted with the Cournot model of duopoly. The plan of this chapter is as follows. In section 5.1 we review the one-shot Cournot duopoly game and the corresponding one-shot Stackelberg game. We also present a modified version of the one-shot Stackelberg game as a quick means of raising the issue of time inconsistency (sometimes referred to as dynamic inconsistency) in Stackelberg games, which we further expound in the rest of the chapter. In sections 5.2 and 5.3 we define the concepts of open-loop Stackelberg equilibrium and (nondegenerate) Markovian Stackelberg equilibrium for differential games. We show in section 5.2 that open-loop Stackelberg equilibria are, in general, not time consistent. There are, of course, exceptions to this rule which we also consider. In section 5.3 we turn to the analysis of nondegenerate Markovian Stackelberg equilibria. In general, it is difficult to find such equilibria. However, we are able to provide some rules of thumb which work in a number of situations.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.016 | 0.002 |
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".