Full Report for Tixel by Martijn Althuizen

Rules

Representative game (in the sense of being of mean length). Wherever you see the 'representative game' referred to in later sections, this is it!

Disclaimer

The full rules are long and complex. I suggest reading the rules at the Nestorgames web site if you are not fully familiar with them.

Definitions

Active

• An active piece can move.
• If you have at least one active piece on the board, you can add pieces.
• After your first move, if you have no active pieces at the start of your turn, you lose.
• Visually, an active piece is rotated diagonally so its corners are outside the board squares.

Inactive

• An inactive piece cannot move.
• Visually, an inactive piece is aligned with the board squares.

Play

The following actions are available at the start of a turn:

Add

Add a piece to the board; after your first move you can only do this if you already have at least one active piece. The piece must physically fit on the board without overlapping another piece.

Slide

Slide an active piece orthogonally; it may inactivate pieces that it passes, and may end in an inactive state. If the sliding piece ends in an inactive state, you get a bonus move.

Bonus moves:

Add

Add a piece to the board.

Remove

Remove a piece from the board.

Slide

As above. This can trigger another bonus action, and so on.

Activate

Rotate an inactive piece so it is active.

Pivot

Re-orient a Tixel or Regatta piece so it is visually different.

Variants

Bounded/Unbounded board

In Tix and Tixel (but not Regatta), the edge of the board is unbounded by default, meaning that pieces can protrude beyond the edge of the board. There is an option to play with the board bounded, making the edgesmore deadly; this is recommended only for board sizes 8 or larger.

Promotions

In Poka Yoke, the first two pieces you remove from the board are promoted from Tixel pieces to Regatta pieces.

Tix

Played only with Tix pieces (simple squares).

Tixel

Played only with Tixel pieces (one hollow edge).

Regatta

Played only with Regatta pieces (one hollow edge, one rounded corner). The board edge is bounded.

Poka Yoke

Played only with Tixel pieces (one hollow edge). Promotions are allowed.

Game End

If you cannot move on your turn, you lose.

Miscellaneous

General comments:

• Requires αβ search to play well.

Play: Combinatorial

Family: Combinatorial 2014,Tix

Mechanism(s): Stalemate

BGG Stats

BGG Ratings and Comments

Kolomogorov Complexity Analysis

Ai Ai calculates the size of the implementation, and compares it to the Ai Ai implementation of the simplest possible game (which just fills the board). Note that this estimate may include some graphics and heuristics code as well as the game logic. See the wikipedia entry for more details.

Playout Complexity Estimate

Tavener complexity: the heat generated by playing every possible instance of a game with a perfectly efficient programme. Since this is not possible to calculate, Ai Ai calculates the number of random playouts per second and compares it to the fastest non-trivial Ai Ai game (Connect 4). This ratio gives a practical indication of how complex the game is. Combine this with the computational state space, and you can get an idea of how strong the default (MCTS-based) AI will be.

Playout/Search Speed

Random: 10 second warmup for the hotspot compiler. 100 trials of 1000ms each.

Other: 100 playouts, means calculated over the first 5 moves only to avoid distortion due to speedup at end of game.

Mirroring Strategies

Rotation (Half turn) lost each game as expected.
Reflection (X axis) lost each game as expected.
Reflection (Y axis) lost each game as expected.
Copy last move lost each game as expected.

Mirroring strategies attempt to copy the previous move. On first move, they will attempt to play in the centre. If neither of these are possible, they will pick a random move. Each entry represents a different form of copying; direct copy, reflection in either the X or Y axis, half-turn rotation.

Heuristic Values

This chart shows the heuristic values thoughout a single representative* game. The orange line shows the difference between player scores. (* Representative, in the sense that it is close to the mean game length.)

Win % By Player (Bias)

Note: that win/loss statistics may vary depending on thinking time (horizon effect, etc.), bad heuristics, bugs, and other factors, so should be taken with a pinch of salt. (Given perfect play, any game of pure skill will always end in the same result.)

Note: Ai Ai differentiates between states where all players draw or win or lose; this is mostly to support cooperative games.

UCT Skill Chains

Search for levels ended: time limit reached.

Level of Play: Strong beats Weak 60% of the time (lower bound with 95% confidence).

Draw%, p1 win% and game length may give some indication of trends as AI strength increases.

1st Player Win Ratios by Playing Strength

This chart shows the win(green)/draw(black)/loss(red) percentages, as UCT play strength increases. Note that for most games, the top playing strength show here will be distinctly below human standard.

Complexity

Computational complexity (where present) is an estimate of the game tree reachable through actual play. For each game in turn, Ai Ai marks the positions reached in a hashtable, then counts the number of new moves added to the table. Once all moves are applied, it treats this sequence as a geometric progression and calculates the sum as n-> infinity.

Move Classification

Board Coverage

A mean of 68.52% of board locations were used per game.

Colour and size show the frequency of visits.

Game Length

Game length frequencies.

Change in Material Per Turn

This chart is based on a single representative* playout, and gives a feel for the change in material over the course of a game. (* Representative in the sense that it is close to the mean length.)

Actions/turn

Table: branching factor per turn, based on a single representative* game. (* Representative in the sense that it is close to the mean game length.)

Action Types per Turn

This chart is based on a single representative* game, and gives a feel for the types of moves available throughout that game. (* Representative in the sense that it is close to the mean game length.)

Red: removal, Black: move, Blue: Add, Grey: pass, Purple: swap sides, Brown: other.

Trajectory

This chart shows the best move value with respect to the active player; the orange line represents the value of doing nothing (null move).

The lead changed on 5% of the game turns. Ai Ai found 1 critical turn (turns with only one good option).

Position Heatmap

This chart shows the relative temperature of all moves each turn. Colour range: black (worst), red, orange(even), yellow, white(best).

Good/Effective moves

These figures were calculated over a single game.

An effective move is one with score 0.1 of the best move (including the best move). -1 (loss) <= score <= 1 (win)

A good move has a score > 0. Note that when there are no good moves, an multiplier of 1 is used for the game space calculation.

Quality Measures

These figures were calculated over a single representative* game, and based on the measures of quality described in "Automatic Generation and Evaluation of Recombination Games" (Cameron Browne, 2007). (* Representative, in the sense that it is close to the mean game length.)

Opening Heatmap

Colour shows the success ratio of this play over the first 10moves; black < red < yellow < white.

Size shows the frequency this move is played.

Unique Positions Reachable at Depth

Note: most games do not take board rotation and reflection into consideration.
Multi-part turns could be treated as the same or different depth depending on the implementation.
Counts to depth N include all moves reachable at lower depths.
Inaccuracies may also exist due to hash collisions, but Ai Ai uses 64-bit hashes so these will be a very small fraction of a percentage point.

Shortest Game(s)

69216 solutions found at depth 3.

Puzzles

Puzzle	Solution
Black to win in 8 moves	Show Solution
White to win in 5 moves	Show Solution

Weak puzzle selection criteria are in place; the first move may not be unique.