Full Report for Wizard's Garden by Tim Schutz

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!

Setup

Players take turns to place two pieces up in the playing area, so no two pieces touch orthogonally.

Definitions

Seed

An offboard piece is a seed.

Flower

An onboard piece is a flower.

Play

Each turn, a player performs the following steps:

• Place a flower of either colour orthogonally adjacent to another piece.
• Flip all adjacent flowers (light becomes dark, and vice versa).
• Remove all lines of four identical flowers formed (diagonals included), scoring one point for each.

Points are scored using the flowers themselves, so the pool gradually depeles as the game continues.

Black flowers are deemed more valuable than white flowers, so the last player to complete a black row will win in the case of a tie.

Goal

The game ends when no further moves can be made. The three possible reasons a player cannot make a move are that there areno seeds left to play, there are no seeds left on the board to plant next to (the board is empty), or there are no spaces lefton the board to plant on (the board is full).

The player who has harvested the most flowers wins

Miscellaneous

General comments:

Play: Combinatorial

Family: Combinatorial 2019,Minimal

Mechanism(s): Scoring,Shared Pieces

Components: Board

Level: Standard

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.

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.

Levels of Play

Level of Play: Strong beats Weak 60% of the time (lower bound with 90% confidence).

Draw%, p1 win% and game length may give some indication of trends as AI strength increases; but be aware that the AI can introduce bias due to horizon effects, poor heuristics, etc.

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 99.73% 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 10% of the game turns. Ai Ai found 5 critical turns (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.)

Openings

Moves	Animation
Wc1,Wb3,Ba2,Ba4,Bb2
Wc1,Ba2,Wb3,Ba4,Bb2
Wd1,Wb3,Wc4,Wb1,Wa1
Wd1,Wc4,Wb3,Wb1,Wa1
Wb2,Wa3,Wd3,Bc1,Bc3
Wb2,Wd3,Wa3,Bc1,Bc3
Wd2,Ba2,Bd4,Ba4,Ba1
Wa3,Wb2,Wd3,Bc1,Bc3
Wb3,Wc1,Ba2,Ba4,Bb2
Wb3,Wd1,Wc4,Wb1,Wa1
Wb3,Wc4,Wd1,Wb1,Wa1
Wd3,Wb2,Wa3,Bc1,Bc3

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)

4 solutions found at depth 5.

Puzzles

Puzzle	Solution
Player 1 to win in 2 moves	Show Solution

Selection criteria: first move must be unique, and not forced to avoid losing. Beyond that, Puzzles will be rated by the product of [total move]/[best moves] at each step, and the best puzzles selected.