Bibliographic record
Abstract
An important problem in single-player video game design is how to sequence game elements within a level (or “chunk”) of the game. Each element has two critical features: a reward (e.g., earning an item or being able to watch a cinematic) and a degree of difficulty (e.g., how much energy or focus is needed to interact with the game element). The latter property is a distinctive feature in video games. Unlike passive services (like a trip to the spa) or passive entertainment (like watching sports or movies), video games often require concerted effort to consume. We study how to sequence game elements to maximize overall experienced utility subject to the dynamics of adaptation to rewards and difficulty and memory decay. We find that the optimal design depends on the relationship between rewards and difficulty, leading to qualitatively different designs. For example, when the proportion of reward-to-difficulty is high, the optimal design mimics that of more passive experiences. By contrast, the optimal design of games with low reward-to-difficulty ratios resembles work-out routines with “warm-ups” and “cool-downs.” Intermediate cases may follow the classical “mini-boss, end-boss” design where difficulty has two peaks. Numerical results reveal optimal designs with “waves” of reward and difficulty with multiple peaks. Level designs with multiple peaks of difficulty are ubiquitous in video games. In summary, this paper provides practical guidance to game designers on how to match the design of single-player games to the relationship between reward and difficulty inherent in their game’s mechanics. Our model also has implications for other interactive services that share similarities with games, such as summer camps for children. This paper was accepted by Jeannette Song, operations management. Funding: This work was supported by the National Natural Science Foundation of China [Project 72201210] and Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2020-06488]. Supplemental Material: The data files and online appendices are available at https://doi.org/10.1287/mnsc.2022.4665 .
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".