Quantum Frog: Emergent Cooperation and Difficulty Scaling in a Quantized-Time Cooperative Game

Computer Science > Artificial Intelligence [Submitted on 22 Apr 2026] Quantum Frog: Emergent Cooperation and Difficulty Scaling in a Quantized-Time Cooperative Game Saad Mankarious We introduce \emph{Quantum Frog}, a two-player cooperative game built on a novel \emph{quantized-time} mechanic in which the environment advances only when a player acts. Inspired by the classic arcade game Frogger, Quantum Frog requires two frogs to cross an 8\times8 grid of traffic and reach the far side together. We use reinforcement learning (RL) as an analytical lens to answer four design questions: (1) how does game difficulty scale with traffic density, (2) what is the optimal single-agent policy and why, (3) how large is the cooperation gap between independent and cooperative two-agent play, and (4) what joint strategy emerges when agents are incentivised to cooperate? We train agents through five escalating stages, Tabular Q-Learning, Deep Q-Network (\DQN), Independent \DQN~(\IDQN), and Multi-Agent Proximal Policy Optimisation (\MAPPO\ with a centralised critic), evaluating each against traffic densities of one to six cars. Our key findings are: (i) the quantized-time mechanic makes a \emph{rush strategy} (moving directly upward at every step) universally optimal, as time exposure to traffic is minimised; (ii) adding an uncoordinated second player is harder than sextupling the traffic for a single expert player; (iii) cooperative training recovers +32--34 percentage points of joint success rate relative to independent agents and reduces episode length from \sim90 to \sim6 steps; and (iv) the emergent cooperative strategy is synchronised rushing, not complex positional coordination, illustrating that shared incentives alone suffice to align agents in time-critical cooperative tasks. These findings provide concrete, empirically grounded guidance for the commercial design of Quantum Frog and offer broader insights into the role of environment mechanics in shaping multi-agent learning dynamics. Subjects: Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Multiagent Systems (cs.MA) Cite as: arXiv:2605.23930 [cs.AI]   (or arXiv:2605.23930v1 [cs.AI] for this version)   https://doi.org/10.48550/arXiv.2605.23930 Focus to learn more Submission history From: Saad Mankarious [view email] [v1] Wed, 22 Apr 2026 00:55:08 UTC (118 KB) Access Paper: HTML (experimental) view license Current browse context: cs.AI < prev   |   next > new | recent | 2026-05 Change to browse by: cs cs.LG cs.MA References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)