Full Report for Lava by Michael Amundsen,Alek Erickson

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!

INTRODUCTION

On each turn, take N checkers (where N is any number from 1 to stack size) from the top of a stack you control and move them N spaces in a straight line to a destination containing N or fewer checkers. Then, add a new checker of your colour to the destination. Stacks may not jump over enemies, but may land on enemy or friendly stacks, thereby burying them.

When the board is full, the player controlling more stacks wins the game.

Miscellaneous

General comments:

Requires αβ search to play well.

Play: Combinatorial

Family: Combinatorial 2021

Mechanism(s): Movement,Stacking,Territory

Components: Board

Level: Standard

BGG Stats

BGG Entry	Lava
BGG Rating	6.33333
#Voters	3
SD	1.88562
BGG Weight	4
#Voters	1
Year	2021

BGG Ratings and Comments

User	Rating	Comment
evanvariano	9	Really nice 3D-thinking; beautifully simple ruleset.
rayzg	5	The rules are ingenious, but I'm not a fan of territory games (because they tend to move too slowly for my tastes) nor games with stacks of pieces that dictate how far they can move (because movement is too limited). Unfortunately, the game met my expectations for a territory and stacking game. Worse, though, is that games last way too long! We didn't bother finishing the only game that we played because the total number of occupied spaces on the board increased way too slowly.
pezpimp	5	Based on one play: It sounds very interesting but once you start playing you notice that the movement is very restrictive and thus can get quite frustrating. It is also really long, the game can go on and on, thus the end game just drags on.

Kolomogorov Complexity Analysis

Size (bytes)	27839
Reference Size	10673
Ratio	2.61

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

Playouts per second	5630.87 (177.59µs/playout)
Reference Size	382292.22 (2.62µs/playout)
Ratio (low is good)	67.89

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.

State Space Complexity

% new positions/bucket

State Space Complexity	71579204
State Space Complexity bounds	64717257 < 71579204 < ∞
State Space Complexity (log 10)	7.85
State Space Complexity bounds (log 10)	7.81 <= 7.85 <= ∞
Samples	498526
Confidence	0.00	0: totally unreliable, 100: perfect

State space complexity (where present) is an estimate of the number of distinct game tree reachable through actual play. Over a series of random games, Ai Ai checks each position to see if it is new, or a repeat of a previous position and keeps a total for each game. As the number of games increase, the quantity of new positions seen per game decreases. These games are then partitioned into a number of buckets, and if certain conditions are met, Ai Ai treats the number in each bucket as the start of a strictly decreasing geometric sequence and sums it to estimate the total state space. The accuracy is calculated as 1-[end bucket count]/[starting bucklet count]

Playout/Search Speed

Label	Its/s	SD	Nodes/s	SD	Game length	SD
Random playout	6,090	78	829,256	10,575	136	15
search.UCT	6,079	144			154	11
search.AlphaBeta	405,314	50,631			129	11

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)

1: Black win %	50.90±3.10	Includes draws = 50%
2: Red win %	49.10±3.09	Includes draws = 50%
Draw %	0.00	Percentage of games where all players draw.
Decisive %	100.00	Percentage of games with a single winner.
Samples	1000	Quantity of logged games played

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

Match	AI	Strong Wins	Draws	Strong Losses	#Games	Strong Score	p1 Win%	Draw%	p2 Win%	Game Length
0	Random
1	UCT (its=2)	631	0	312	943	0.6385 <= 0.6691 <= 0.6984	48.46	0.00	51.54	135.83
3	UCT (its=8)	631	0	311	942	0.6392 <= 0.6699 <= 0.6991	51.80	0.00	48.20	136.25
5	UCT (its=60)	631	0	83	714	0.8582 <= 0.8838 <= 0.9052	49.86	0.00	50.14	136.74
6	UCT (its=164)	631	0	120	751	0.8123 <= 0.8402 <= 0.8647	48.60	0.00	51.40	136.15
7	UCT (its=445)	381	0	58	439	0.8330 <= 0.8679 <= 0.8964	52.16	0.00	47.84	138.37
8	UCT (its=445)	480	0	520	1000	0.4492 <= 0.4800 <= 0.5110	47.60	0.00	52.40	144.79

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

Game length	121.65
Branching factor	35.86
Complexity	10^174.96	Based on game length and branching factor
Computational Complexity	10^7.65	Sample quality (100 best): 0.55
Samples	1000	Quantity of logged games played

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 Size	37	Quantity of distinct board cells
Distinct actions	499	Quantity of distinct moves (e.g. "e4") regardless of position in game tree
Good moves	241	A good move is selected by the AI more than the average
Bad moves	257	A bad move is selected by the AI less than the average
Response distance%	47.87%	Distance from move to response / maximum board distance; a low value suggests a game is tactical rather than strategic.
Samples	1000	Quantity of logged games played

Board Coverage

A mean of 100.00% of board locations were used per game.

Colour and size show the frequency of visits.

Game Length

Game length frequencies.

Mean	121.65
Mode	[124]
Median	121.0

Change in Material Per Turn

Mean change in material/round	0.28	Complete round of play (all players)

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 23% of the game turns. Ai Ai found 10 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

Measure	All players	Player 1	Player 2
Mean % of effective moves	77.48	80.10	74.85
Mean no. of effective moves	25.58	20.13	31.03
Effective game space	10^156.36	10^72.74	10^83.62
Mean % of good moves	38.27	12.81	63.73
Mean no. of good moves	15.73	2.39	29.07
Good move game space	10^91.43	10^14.81	10^76.62

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

Measure	Value	Description
Hot turns	87.70%	A hot turn is one where making a move is better than doing nothing.
Momentum	22.13%	% of turns where a player improved their score.
Correction	40.98%	% of turns where the score headed back towards equality.
Depth	8.16%	Difference in evaluation between a short and long search.
Drama	2.16%	How much the winner was behind before their final victory.
Foulup Factor	76.23%	Moves that looked better than the best move after a short search.
Surprising turns	18.85%	Turns that looked bad after a short search, but good after a long one.
Last lead change	40.98%	Distance through game when the lead changed for the last time.
Decisiveness	4.92%	Distance from the result being known to the end of the game.

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
d1-c2,d7-e6,c2-b3,e6-e4,b3-d3,e4-f4,d3-d6,f4-f2,d6-d5,f2-c2
d1-c2,d7-e6,c2-a4,e6-f5,a4-d4,e6-e5,d4-c4,f5-d5,c4-c3,e5-c5
d1-c2,d7-e6,c2-c3,e6-g4,c3-c5,g4-d7,c5-f5,d7-d6,f5-e5,d7-d4
d1-c2,d7-e6,c2-c3,e6-d6,c3-c4,d6-d4,c3-b4,e6-e5,b4-c3,e5-d5
d1-c2,d7-e6,c2-d2,e6-e4,d2-d4,e4-f4,c2-c3,f4-f3,c3-c4,e4-e2
d1-c2,d7-e6,c2-d2,e6-d6,d2-d4,d6-d5,d4-e3,d6-c6,e3-e5,e6-d6
d1-c2,d7-e6,c2-d1,e6-d7,d1-d3,d7-d5,c2-d2,e6-d7,d2-d4,d7-d6
d1-c2,d7-e6,c2-d1,e6-e4,d1-d3,e4-f4,d3-d6,f4-f2,d6-d5,f2-c2
d1-c2,d7-d6,c2-b3,d6-c7,b3-d3,d6-e5,d3-c4,e5-e3,c2-c3,c7-c5
d1-c2,d7-d6,c2-b3,d6-f4,b3-d3,f4-e4,d3-d6,f4-f2,d6-d5,f2-c2
d1-c2,d7-d6,c2-b3,d6-c6,b3-d3,d6-e5,d3-d2,c6-c4,c2-c3,c4-b5
d1-c2,d7-d6,c2-b3,d6-c6,b3-d3,d6-e5,d3-d2,c6-c4,d2-e1,e5-e4

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

0	1	2	3	4	5	6	7	8
1	3	12	81	609	5573	51233	490147	4642168

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 8.

Puzzles

Puzzle	Solution
Black to win in 6 moves
Black to win in 5 moves
Red to win in 6 moves
Red to win in 6 moves
Red to win in 4 moves
Red to win in 4 moves
Black to win in 4 moves
Black to win in 4 moves
Black to win in 4 moves
Black to win in 4 moves

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