# Smitsimax

MSmits
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## Smitsimax

I first thought of this search during the Code of Kutulu contest, where I placed 4th. Kutulu is a 4-player simultaneous game in which it is very hard to write a good search, because of the exploding searchspace. Smitsimax helped me solve this problem. I was inspired by my work to code an ultimate tic tac toe (UTTT) bot using MCTS. This is where my use of the UCB formula comes from. Disclaimer: I am not completely sure if my search doesn't already exist in some form, but if it doesn't, I have coined the name "Smitsimax" , more as joke than anything else. If anyone is more versed in search algorithms and knows its true name, do let me know.

EDIT: It has been pointed out to me that a similar algorithm is described here: https://www.researchgate.net/publication/224396568_CadiaPlayer_A_Simulation-Based_General_Game_Player You're free to keep calling it Smitsimax, because it is a much used name for it on CG. I just can't claim credit for something someone else invented earlier (even though I did not know).

CSB is a two player zero sum game where both players have two racing pods. The zero-sum nature makes minimax a natural choice as a search algorithm. The main problem is the simultaneous nature of the game. Both players decide what to do simultaneously and after this, the turn is calculated. The basic version of minimax requires players to move one after the other, knowing what the other player did the turn (ply) before. To solve the problem of simultaneous gameplay, you can use the "paranoid" option. This means you assume the other player is going to know what you did and chooses the best possible counter. This is an assumption that can be good or bad, depending on the state of the game and the possible strategies you can use. A possible way to deal with this is to calculate all possible combinations of moves. If one player has 6 possible moves and the other player does as well, then there are a total of 36 possible combinations. You can then average the result, maximize your worst result or minimize your opponents best result for each of your moves. What ends up being the most succesful approach is a matter of trial and error. The search space can quickly get out of hand if you don't prune enough.

Smitsimax has a few things in common with minimax, but is also different. Both search algorithms have a tree of possible moves. Each move has a node and each node has children that correspond with the moves on the next turn. In minimax there is only one tree. The tree has moves by both players in it and each node corresponds to a single absolute gamestate. When you get to this specific node, you know exactly what the game looks like.

If we consider CSB specifically, Smitsimax has a separate tree for each pod (meaning two trees per player). Each node on tree does not directly correspond to a gamestate. The trees are completely independent at first glance. During each search iteration, moves are selected on each tree, randomly at first and by Upper Bound Confidence formula later. Each turn is simulated with the selected moves. When a predetermined depth level is reached, an evaluation is done for each player (pod) in the game and the result is backpropagated along the respective trees of each pod. The first time this is done, every node does correspond to a single gamestate. However, the next time the same node is selected by a pod, the other pods may select different nodes, leading to a different gamestate. The differences in possible gamestates will be larger for greater depth levels. If the search is able to converge, this is not a problem. The (best) branches to which each tree converges should, together, correspond to a single gamestate per node and lead to good opponent prediction and a good choice of move for the players pods.

Let's look at this step by step in pseudo code, written for CSB:

At the start of a turn you get all the information you need as input. You have your Pod class with information such as:

class Pod
{
position
velocity
shield
...
}


You need two instances of each pod. One is the base instance you get as you update using the turn-input. You need to keep this information so that you can reset to it after every search-run. The other is the evolving pod that changes during the search.

There are various ways to set up your simulation and Smitsimax-search. I currently use a static Sim class that looks somewhat like this:

static class Sim
{
Pods[4]  // all 4 pods that will be simulated
Node[4] current  // 4 current nodes, one for each pod. Current nodes start as a reference to the root nodes of each tree
float[4] lowestscore // the lowest score earned by the tree, one for each pod.
float[4] highestscore // the highest score earned by the tree, one for each pod.
float[4] scaleparameter  // the scaleparameter calculated by subtracting the lowest score from the highest, needed for the UCB formula.

void Reset() // To reset the sim after search

void Search() // The important bit, the actual search

void Play()  // a CSB specific algorithm that handles movement and collisions (see Magus CSB postmortem)

void BackPropagate()  // each tree has the score result backpropagated along the branch of the tree.
}


NOTE: I have changed my CSB bot to no longer use the lowestscore/highestscore normalization method as it was kind of heavy. It makes sense to start with something like this though.

The tree consists of nodes and each node in the tree needs the following things:

class Node
{
Node parent // each node has a parent except the root node
float score // the total score obtained by this node. To get the average, you can divide by the number of visits
int firstChildIndex // I use a preallocated array that I use this index for
int childCount // the number of children
int visits // the number of times this node has been visited
int angle // this is part of the move, a CSB bot needs an angle
int thrust // a CSB bot needs a thrust
....
}


The Search method looks like this:

void Search()
{
CreateRootNodes() // create 4 nodes, one for each tree. These are also the "current" nodes of the sim.

while (we still have calculation time left)
{
Reset() // reset the sim using the pod instances you obtained during the update
int depth = 0;

while (depth < maximum_depth)
{
for (each pod, 4 times total)
{
Node node = current[i];

if (node.visits == 1)
node.MakeChildren() // give the node children if it doesnt have them yet, each with a possible move

Node child = node.Select() // select a node that you want to use for the sim.
//At first it is best to select randomly a few times.
//I currently random 10 times. After this I use the UCB formula to select a child.

child.visits++; // increment the child visitcount

current[i] = child; // the child becomes the current node

pod.ApplyMove // do the stuff the child node tells you to do (rotate, accelerate, shield etc.)
}

Play(); // Simulate what actually happens, including movement and collisions
depth++;
}

float[4] score = GetScore();
// get a score for each pod when the simulation depth is reached.
//The way the score is calculated is not that different from what it would be in minimax or GA or any other search method.

for (each pod, 4 times total)
{
// update the lowest score, highest score and scaleparameter for this pod.
// the scale parameter is the difference between the highest and lowest score
}

Backpropagate()
// We go up the tree back to the root node, adding the score to each node we pass. We do this for each pod (4x)
}
}


The UCB formula that is used on child selection looks like this:

float ucb = (score / (visits_of_child * scale_param)) + exploration_param * Sqrt(Log(visits of parent)) * (1/Sqrt(visits_of_child);


Score divided by visits is the average score of the node. The scale parameter normalizes the first term of this score in a range between 0 and 1. The other term in the formula is identical to UCB as it would be used in MCTS (for example). The exploration parameter controls how often the search will try moves with low score and low number of visits.

### Why does this work at all?

The first few search iterations every pod will select nodes randomly, or just very badly. That means all evaluations are bad and untrustworthy. At some point, pods will start "discovering" better moves. This will lead to several things happening.

• Your better moves will be evaluated as better moves, meaning they are more likely to be picked next time (by UCB formula).
• The opponents pods will also be more likely to pick good moves during the search.
• The parts of the evaluation that contain opponent-state information (travelled distance and such) will be scored more accurately.
• Because of better opponent prediction, your own pods moves are scored more accurately.

This has great potential for convergence to an optimal series of moves for all pods. If player 1's runner pod has found a good path, player 2's blocker pod will converge to moves that best counter this path, which will lead to player 1's runner adjusting its path, which means the blocker will adjust again... etcetera.

Since we use exploration and expansion by UCB, all moves are evaluated statistically. Your opponent is more likely to pick its best moves and you are more likely to pick yours. However, if some unlikely move is a great counter to the current best opponent path, the exploration part of your search will soon (if the exploration parameter is not too small) pick it up and multiple succesful iterations will cause the opponent to "rethink" its best path. As with every search, the more calculation time you have, the better this search will converge. With Smitsimax, you can freely select your simulation depth. If you set the depth too high, your search can never converge. You will end up using random moves, or too few moves in the last few depth levels. However, because of UCB, you will not need to explore the entire searchspace, like you would with (non-pruned) minimax. The exploration and expansion will focus on interesting parts of the tree and you can get much deeper than you would expect. My CSB bot currently uses roughly 60k simulations per turn where my rivals in the top 10 use over a million for the same depth of search. I can only imagine how well this bot will perform once it is properly optimized.

### Similarities with Monte Carlo Tree Search (MCTS)

Smitsimax shares some features with MCTS. One of them is the use of statistics to guide the search. The other is the use of a selection and backpropagation phase. There are also some clear differences:

1. MCTS has a single tree and a single gamestate per node. It shares this feature with minimax. Smitsimax has a tree for every agent. A node on this tree can correspond to many possible gamestates with different likelyhoods.

2. Smitsimax only uses random moves to avoid "resonance" of bad moves on the first few visits to a node. It is possible all pods start doing bad moves at the beginning of the search. Because bad moves can seem good when compared to other bad moves, the tendency to explore disappears and in the worst case, the search will converge on bad moves for all pods. Random selection at the start helps avoid this problem. Smitsimax does not use a random rollout.

3. MCTS expands one node on every iteration of the search. Smitsimax expands all the way to the maximum depth on every iteration. This is necessary because there is no random rollout apart from the first few times a node is visited.

NOTE: I have since changed to a more MCTS like implementation after a suggestion by RoboStac. He tried to replace my full rollout with a random rollout and it worked a bit better. I think it did for me as well, though it is not a big change. There is still a large difference with MCTS because you use a separate tree for every agent and the nodes in the tree do not have a single corresponding gamestate.

### What are the advantages and limitations of Smitsimax?

• Maximum quality opponent prediction. The opponent prediction has the same quality as your own search, since everything is symmetric between you and your opponent. Genetic algorithms don't share this feature and minimax might be less effective at this because of (for example) the paranoid prediction option.

• Quick convergence: Relatively few sims needed for a reliable result, or greater achievable depth levels.

• Potential for emergent behavior. An example of this is seen in my CSB blocker-pod. I currently have it set to try to reduce the opponent travelled distance, but it also gets more score if my runner pod travels farther. This sometimes leads to my blocker using its shield to boost my runner ahead. My runner and blocker each have their own tree, but they share goals in the evaluation, which tends to make them cooperate. This is a strong feature of genetic algorithms that is also present in Smitsimax.

• Few heuristics needed. In the evaluation you merely have to give "score" for what you want to achieve and you dont have to specify how to achieve it. The possible moves are decided when creating children on the nodes and if done right, will be selected to achieve the highest score. If you have some experience using this type of search, you can quickly code a usable bot.

#### Limitations

• Because gamestates that correspond to a node are not uniquely determined (the opponent may do different things on each iteration) this search might not be usable for many games. For example, if a game allows 4 types of moves: A, B, C and D and if in some situations caused by the opponent, move C is illegal, then this search will not work. The allowed (legal) moves have to be independent of opponents choices. This is the case in CSB, as you can always thrust, steer and shield, no matter what the opponent does. The same is true for Code of Kutulu. In that game you always know which moves are legal, no matter what the opponent does.

• It is hard to use heuristics as part of the search, because you really dont know what is going to happen until after you finish your sim and even then, things will be different on the next iteration. You lean heavily on the evaluation score telling your pods what is good and what is bad. For example, you can't have a runner-node select its children to "steer away to the right if it sees a blocker to the left". In one particular iteration of the search, there could be a blocker to the left, but next time when you get to this node, there might not be.