What is the Royal Game of Ur?

The Royal Game of Ur is one of the oldest board games in the world, with evidence of it being played over 5000 years ago! It is a two-player racing game where each player moves their 7 pieces around the board, competing to be the first to advance all their pieces to the end. The roll of 4D2 dice is used to decide how far each piece can be moved, and players can capture each other's pieces by landing on them. There has been a lot of academic debate on the rules used for the game throughout history, but this dataset uses the simple rules proposed by Irving Finkel, which uses Bell's path around the board. You can learn more about these rules at https://royalur.net/rules/.

What is the purpose of this dataset?

This dataset contains evaluations of moves from a world-leading AI agent that plays the Royal Game of Ur, called Expectimax Depth 8. We hope that people can use these evaluations for two purposes:

1. Build up a greater understanding of the game. Currently, our understanding of the best moves to play is limited and is subject to our human biases. We hope that we can build a greater understanding of strategy in the Royal Game of Ur by taking a statistical approach to visualize the best moves under different conditions.

2. Build a machine-learning agent to approximate the evaluations using less computational power. Each evaluation of Expectimax Depth 8 can take up to 10 seconds on an M1 chip, which makes it infeasible for use in real-time play against human players. Additionally, if we are able to approximate the evaluations of Expectimax Depth 8 more cheaply, then we may be able to build even stronger AI agents by using the approximate models as the utility function for expectimax, increasing the effective depth that the agent can see. This approach would be similar to Stockfish NNUE, the strongest chess AI in the world.

What is Expectimax Depth 8?

Expectimax Depth 8 is an AI agent that looks 8 moves into the future to score each possible move from a position. This is achieved by scoring each possible position after 8 moves using a simple utility function, and then combining all these scores into a single, more accurate, score value for each move. The utility function used counts the advancement of all pieces of the light player, minus the advancement of all pieces of the dark player. The advancement of pieces is defined as the number of tiles that the piece has been moved without being captured. Therefore, a piece at the end of the board is valued more highly than a piece that has only just been moved onto the board. These scores are then combined by assuming each player will select the move with the best score value for them, similarly to minimax. However, when a dice roll is involved, the scores of each possible dice outcome are combined using a sum of each score after the given dice roll multiplied by the probability of that dice roll. You can learn more about the Expectimax algorithm on Wikipedia.

Technically, this is the implementation of Expectimax Depth 8 that was used for this dataset with this utility function.

How is the dataset made?

The A1, A2, A3, ... C5, C6, C7, C8 columns denote the individual cells of the game board. They are a columnar "explosion" of the game column, removing the empty cells at A5, A6, C5 and C6.

The dataset was made as follows:

1. Many random playing agents are created, and start playing the game.
2. For every move they make, every possible moves for every possible dice throw (but 0) are analyzed through Expectimax Depth 8 and noted in the dataset.
3. The moves are ordered by their utility value after Expectimax has analyzed them and ranked from 1 to N, and noted in the rank column.
4. The board state is also noted.

We hope you find it fun to work on this dataset!

Discussion on Discord: https://discord.gg/fyNjxBPCSz Learn more about the game and analysis: https://royalur.net/learn/ Learn even more about in depth analysis: https://github.com/qwertyuu/go-ur

This work's Github repo (also has bonus training and model inference): https://github.com/qwertyuu/python-ur-machine-learning

Videos of Raphaël showcasing the process of analyzing, generating and training models around the data in this dataset:

• Part 1: https://www.youtube.com/watch?v=rXu9zFYt6HE
• Part 2: https://www.youtube.com/watch?v=42X-p-wSEgQ