Full Report for Throngs by Dale Walton

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!

Definitions:

• The vicinity of an intersection is that intersection together with all the intersections immediately adjacent to it.
• The action-potential of an intersection is the number of the moving player’s stones in the intersection’s vicinity minus the number of the opponent’s stones there. For example, the action-potential of an intersection on Black's move is 3 if its vicinity contains either: 5 Black and 2 White, 4 Black and 1 White, or 3 Black and no White.

The structure of the game

Before play begins, one player places a Black stone and two White stones on three different intersections. Then the other player decides to play either as Black or as White. After this the players alternate turns taking two moves each, beginning with Black.

A move begins on an intersection that has sufficient action-potential (calculated as described above). Before movement, the following deductions must be made from the potential, based on the intersection's contents:

• If the intersection is occupied by the enemy, you MUST deduct 2 actions to remove it and replace it with your piece.
• If the intersection is empty, you MUST deduct one action to add one of your own pieces, with the following exception:
  • This empty-intersection deduction is waived if there is no other empty intersection next to the chosen intersection and the action-potential at that intersection is zero.
• No deduction is taken if the intersection is already occupied by your own piece.

If you have any leftover actions, you MAY now move the piece in the intersection; in a sequence of steps and jumps, spending one action for each space moved during the sequence.

• A step is a move to an adjacent empty intersection.
• A jump is a movement in a straight line over occupied spaces.

The distance along the path traveled may not exceed the number of actions that remain.

Voluntary passing and partial passing are allowed, with the following exception:

• If possible, a piece must be added during at least one move of any turn that follows a fully-passed turn.

Ending the game:

Play continues until no moves are left, and the player with the most pieces on the board wins.

Resigning in advance is a courtesy.

Miscellaneous

General comments:

Play: Combinatorial

Family: Combinatorial 2020

Mechanism(s): Connection,Movement,Scoring

Components: Board

BGG Stats

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.

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 98.11% 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 2% of the game turns. Ai Ai found 7 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
Add g7,Add f10,Add k6,Play white,Add f8,Add h6
Add g7,Add f10,Add k6,Play white,Add f8
Add h1,Add a11,Add c11,Play black
Add k1,Add f3,Add l7,Play black
Add l2,Add a12,Add i4,Play black
Add k3,Add j6,Add j7,Play black
Add i4,Add h6,Add g4,Play black
Add i4,Add h6,Add d11,Play black
Add g5,Add e8,Add e10,Play black
Add b6,Add f5,Add k4,Play black
Add l6,Add b9,Add g11,Play black
Add c7,Add b10,Add h1,Play black

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)

No solutions found to depth 6.