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

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$\times$8 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 $\sim$90 to $\sim$6 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.