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Why tower placement becomes a resource-allocation problem
In Bloons Tower Defense, balloons follow a track and the player places monkey towers to stop them before they reach its end. Each tower costs money, has its own range and attack properties, and occupies part of the map. Towers cannot overlap, and placement is restricted near the track. A choice that helps in one round can also use money or map area that would be valuable later.
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The modeled game begins with 40 lives and runs for 50 rounds. Those are parameters of the study’s game scenario, not statistics about players. As rounds progress, the challenges change, so a useful strategy must account not only for immediate coverage but also for what remains possible in later rounds.
How the researchers modeled the game
Delorme and Malinowski formulated the task as a temporal two-dimensional knapsack problem with irregular shapes and side constraints. A knapsack problem selects valuable items while respecting limits such as a budget. Here, the two dimensions represent the map and the need to fit towers without overlap; the temporal element represents how tower configurations and resources must work across successive rounds.
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They used integer-programming methods to choose configurations. The study compares two planning approaches:
| Approach | How it plans | Trade-off |
|---|---|---|
| Greedy framework | Determines a configuration round by round, keeping it compatible with the preceding configuration. | Focuses on the next round rather than jointly optimizing a longer sequence. |
| Multi-round framework | Optimizes configurations over several rounds together. | Can account for how current choices fit later rounds, but the more flexible computations are harder. |
The contrast is similar to choosing a move based on the immediate situation versus considering how a sequence of moves will fit together. Joint planning can preserve options for later, though it demands more computation.
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What the simulated results show
The paper reports that its generated solutions could win the modeled game and sometimes outperformed the human strategies used as baselines. Those human plans were derived from online wiki advice; the authors caution that they were not devised by Bloons experts.
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|---|---|---|
| MD1, lives remaining at the end of round 50 | 31 | 24 |
| MD2, lives remaining at the end of round 50 | 25 | 13 |
These are particular outcomes in the authors’ simulations, not expected results for players or a general measure of how much multi-round planning improves performance. They illustrate why anticipating later constraints may matter: an apparently strong placement now can consume money or space needed to survive subsequent rounds.
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Why these findings do not prove a universally optimal strategy
The more flexible MD2 multi-round computations were more difficult. Under the study’s time limits, not all solutions for later rounds were certified optimal, although the reported optimality gaps were below 8%. That qualification concerns optimization within the study’s model; it does not establish an optimal real-world deployment plan.
The experiments also simplify the game. They use the original 2007 Flash title, a single map and 50 rounds; consider only dart and super monkey towers; disallow selling towers; and prohibit building during a round. Some unavailable game details were estimated from measurements, and the authors note that the simulator may differ from the original game. The findings therefore apply to the modeled scenarios, not automatically to every version or strategy in the series. In particular, this was not a study of Bloons TD 6.
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What the game analogy can—and cannot—say about real decisions
The same broad mathematical ideas can be relevant when deciding where to locate facilities, aircraft or anti-drone detection systems: planners may need to balance coverage, limited space, costs and changing conditions. Maxence Delorme, quoted in the Phys.org report provided by Tilburg University, put the analogy this way: “Whether you’re placing towers in Bloons or deploying aircraft to protect an area, mathematics helps you find the location where they will have the greatest impact.”
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That is an analogy about optimization, not a claim that the game study validated those deployments. Real settings can involve threats that are not explicitly modeled and may not share the game’s evolving budgets and rounds. The study demonstrates how a constrained placement problem can be made tangible and analyzed; applying related methods outside the game requires models built for the real situation.
Paper, code and computational setup
The paper states that the simulator source code is public and links to the simulator repository and optimization algorithms. It reports Python for simulations and C++ for optimization, using Gurobi 11.0.2. The reported computational tests ran on a single-core virtual machine with an AMD EPYC-Rome 2.00 GHz processor and 64 GB RAM. These details describe the study’s implementation and test environment, rather than requirements for playing the game.
Read the Delorme and Malinowski paper in International Transactions in Operational Research, first published July 23, 2026, and the Phys.org report provided by Tilburg University, published October 5, 2026.
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