Game simulator shows how math can guide decisions when resources are limited

In "Bloons Tower Defense," the goal is to stop balloons from reaching the end of a track by strategically placing towers along the route. These towers vary in cost, range and attack power, and they also take up valuable space. The game becomes increasingly difficult with each round. Where you place a tower therefore determines not only how effective it is now but also which options remain available later in the game.

Turning a game into a mathematical problem

For Delorme, these are typical questions in the field of operations research, where mathematical models are used to make the best possible decisions when resources such as space, money or other means are limited.

The researcher shows that "Bloons" shares similarities with a well-known mathematical optimization problem: the so-called knapsack problem. This involves finding the combination of items that delivers the greatest value within a limited capacity.

In "Bloons," the problem is more complicated because the game consists of multiple rounds. A decision that works well at a particular moment may not be the best one when considering the game as a whole.

Delorme therefore developed two mathematical strategies. The first determines the best next move in each round, taking into account the towers that have already been placed. The second looks further ahead, considering all rounds together.

"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."

The findings are published in the journal International Transactions in Operational Research.

Better than humans?

To test this approach, Delorme developed a simulator of the game. The calculated strategies proved capable of beating the game and sometimes even outperformed human players. This demonstrates that existing optimization techniques can also be applied successfully in a dynamic gaming environment.

And from there, the step back to the real world is not as large as it might seem. Similar optimization problems arise outside games as well. Examples include deciding where facilities should be located, where aircraft should be stationed to protect an area or where anti-drone detection systems should be positioned around an airport.

The circumstances may differ, but the underlying question is similar: Where should limited resources be deployed to achieve the greatest possible effect?

"Bloons Tower Defense" thus turns an abstract optimization problem into something very concrete and shows that mathematics can sometimes simply help you win a computer game.