# Ceasar's Mind

## The Princess and Monster game

with one comment

The current assignment in my programming class (CIS 121) is to build two AIs for a variation on the classic Pursuit-Evasion game, the Princess and the Monster. I think it’s an extremely interesting problem and will present some thoughts on the game here, and would greatly appreciate any feedback or response to further guide me given how little I know about the subject.

For the uninitatiated, the Princess and Monster game goes like this: Take a connected graph of any size. Assign one node to be the location of the Monster and another to be the location of the Princess. Each turn, the Princess and Monster may move to any nodes adjacent to their respective locations. If the Monster and Princess move onto the same square, the Monster wins. Else, if the Princess can avoid the Monster for a certain amount of turns, the Princess wins.

The game in practice varies a lot. For example, in some variants, the game is played on a continuous and bounded plane rather than a graph. Other interesting variations include revealing the starting location of each player to all of the player at the start of the game, enabling the Monster to capture the Princess if the Monster can simply get within a certain range, allowing the two player to “hear” each other if they are within a certain range, and changing the permitted speeds of each of the players.

In particular, I’ll examine the case where the Monster wins if it moves onto a node adjacent to the Princess’ location, and where both players are notified of each other’s location if they are within three edges of each other. Furthermore, for simplicity, I’ll assume the game is played on a 10×10 grid, rather than a random undirected graph.

So some quick observations.

The Really Simple Case

If we simplify the situation so that the game is the pure game, with not knowledge of starting positions, no ability to hear, and no capture range, we can observe a few things. First off, so long as the monster’s strategy is non-deterministic, there is no way that the princess can guarantee she can evade capture. However, should the monster and princess choose a random-walk strategy, then given a 100 node graph, the probability that they meet is 1/100 * 1/100. So for now on, let’s just assume the princess adopts such a strategy.

Now we can make a far more interesting observation. With the princesses’ strategy in mind, we note that every time we explore a node and move on, the probability that the princess is at the node that we just explored is equal to the probability that the princess is at once of the adjacent nodes multiplied by the chance that the princess would move there. Furthermore, because the princess cannot reach any adjacent nodes from the location that we were just at, she in fact is less likely to be nearby. (Which is kind of weird, but it makes sense if you imagine us putting a dummy princess on each node, and each turn, splitting the dummies into multiple dummies, sending some out and keeping some around, so that, without the monster, each probability would remain 1.)

Given all that, the intuitive solution is to continuously move to the most probable area nearby. The problem with this approach is that it very easily leads to deterministic searches that cover only a small part of the graph. (See the video below.)

This is not say the approach is incorrect. It’s just that we aren’t looking far enough ahead.

The code used in the video above works like this. First we go through every node and score it equal to 1, representing it’s probability of 1%. Then we place the monster down, remove the 1% from where he’s standing, and distribute it to all of the other nodes evenly. Lastly, we simulate what would happen if the princess moved according to an arbtirary Markov process, which in this case is simple a 20% chance to move to any other neighbor. (For those unfamiliar, as I initially was, this preserves the chance that all nodes will equal 1% given the absence of the monster since the princess will leave the corner only 2*20% of the time when she is on a corner, and she will only enter a corner 20% of the time if she is on either of the adjacent nodes.) Specifically, this is implemented with two hashmaps. One keeps track of the current probability, and another is a copy of the first. Then we go through each node and tell it to distribute 20% of it’s current probability to each of neighbors in the copied hashmap (which is needed so as to not corrupt the flow). Finally we replace the hashmap we are using with the copy. (Note here that 20% is arbitrary. Ideally, we would use some machine learning techniques to figure out what exact matrix the princess is using. Note also that if the chance of the princess moving is 0, then the best strategy becomes to create a Eulerian path that explores each node exactly once.)

So basically, as mentioned on the Wikipedia page for Princess and Monster games, it becomes evident that simply standing still most of the time and only occasionally find a new spot to chill is a pretty viable strategy for the monster. This will kind of create a drain, with the monster being the gutter, simplifying the search from one that is completely blind to only that is only half blind. Unfortunately, I’m not at all on the specifics on how long to wait, but it seems to make some intuitive sense. (Note here that the more the princess moves, the better this strategy of waiting is. )