Full Report for Las Médulas(size=5) by Drew Edwards

Las Médulas is an ancient Roman gold mine in Spain. Think of your stones as miners digging corridors through the rock. If they dig too closely together, the mine will collapse. To win, trap your opponent so that you have space to mine safely and they do not.

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!

Play

On your turn, Mine and then Move one of your miners (i.e. stones).

Mining:

Place one tile on any space connected to one of your miners by other tiles. Your opponents miners block your connections.
No tile on the board may touch more than 3 other tiles.

Movement:

After placing a tile, move one of your miners to any tile connected to that miner. You may move through your own miners, and you may choose not to move a miner.

Removal:

At the end of your turn, remove any enemy miner that is isolated (not connected to a same colored miner) and has a clear path to 2 or more tiles occupied by your own miners. If your opponent removes your miners, return one miner to the board each turn on the tile you place during that turn (the tile must be connected to one of your miners that is already on the board, as usual).

Winning

The first player who cannot Mine (place a tile) loses.

Miscellaneous

General comments:

Play: Combinatorial

Family: Combinatorial 2020

Mechanism(s): Connection,Movement,Capture,Stalemate

Components: Board

BGG Stats

BGG Entry	Las Médulas(size=5)
BGG Rating	8.5
#Voters	2
SD	0.5
BGG Weight	3
#Voters	1
Year	2020

BGG Ratings and Comments

User	Rating	Comment
PSchulman	8
rsb762gm	9	Fun, moderately strategic, fast moving game with simple rules.

Kolomogorov Complexity Analysis

Size (bytes)	27431
Reference Size	10293
Ratio	2.67

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	4647.50 (215.17µs/playout)
Reference Size	309281.54 (3.23µs/playout)
Ratio (low is good)	66.55

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

Label	Its/s	SD	Nodes/s	SD	Game length	SD
Random playout	13,723	498	797,921	28,961	58	5
search.UCB	14,029	798			56	6
search.UCT	14,660	754			56	6

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)

1: White win %	48.90±3.09	Includes draws = 50%
2: Black win %	51.10±3.10	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	331	962	0.6253 <= 0.6559 <= 0.6853	47.19	0.00	52.81	58.02
2	UCT (its=3)	542	0	458	1000	0.5110 <= 0.5420 <= 0.5727	49.60	0.00	50.40	57.61
3	UCT (its=4)	600	0	400	1000	0.5693 <= 0.6000 <= 0.6299	52.80	0.00	47.20	57.84
4	UCT (its=5)	622	0	378	1000	0.5915 <= 0.6220 <= 0.6515	53.00	0.00	47.00	57.77
5	UCT (its=6)	631	0	327	958	0.6281 <= 0.6587 <= 0.6880	49.90	0.00	50.10	57.76
6	UCT (its=7)	513	0	487	1000	0.4820 <= 0.5130 <= 0.5439	53.10	0.00	46.90	57.26
7	UCT (its=8)	527	0	473	1000	0.4960 <= 0.5270 <= 0.5578	52.30	0.00	47.70	57.15
8	UCT (its=9)	515	0	485	1000	0.4840 <= 0.5150 <= 0.5459	48.90	0.00	51.10	57.37
9	UCT (its=10)	546	0	454	1000	0.5150 <= 0.5460 <= 0.5766	49.60	0.00	50.40	57.20
10	UCT (its=11)	543	0	457	1000	0.5120 <= 0.5430 <= 0.5737	49.10	0.00	50.90	57.36
11	UCT (its=12)	582	0	418	1000	0.5512 <= 0.5820 <= 0.6122	52.00	0.00	48.00	57.06
12	UCT (its=13)	554	0	446	1000	0.5230 <= 0.5540 <= 0.5845	49.40	0.00	50.60	57.32
13	UCT (its=14)	592	0	408	1000	0.5612 <= 0.5920 <= 0.6221	49.40	0.00	50.60	57.46
14	UCT (its=15)	594	0	406	1000	0.5633 <= 0.5940 <= 0.6240	50.40	0.00	49.60	57.35
15	UCT (its=16)	592	0	408	1000	0.5612 <= 0.5920 <= 0.6221	51.80	0.00	48.20	57.48
16	UCT (its=17)	614	0	386	1000	0.5834 <= 0.6140 <= 0.6437	50.20	0.00	49.80	57.09
17	UCT (its=18)	590	0	410	1000	0.5592 <= 0.5900 <= 0.6201	48.20	0.00	51.80	57.53
18	UCT (its=49)	631	0	296	927	0.6500 <= 0.6807 <= 0.7099	51.24	0.00	48.76	56.96
19	UCT (its=133)	631	0	342	973	0.6180 <= 0.6485 <= 0.6779	47.89	0.00	52.11	55.96
20	UCT (its=362)	630	0	370	1000	0.5996 <= 0.6300 <= 0.6594	51.40	0.00	48.60	55.08
21	UCT (its=983)	631	0	145	776	0.7842 <= 0.8131 <= 0.8390	48.58	0.00	51.42	53.82
22	UCT (its=2671)	305	0	150	455	0.6259 <= 0.6703 <= 0.7119	50.77	0.00	49.23	53.70
23	UCT (its=2671)	513	0	487	1000	0.4820 <= 0.5130 <= 0.5439	53.30	0.00	46.70	53.86

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	54.12
Branching factor	16.63
Complexity	10^55.41	Based on game length and branching factor
Computational Complexity	10^6.87	Sample quality (100 best): 12.28
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

Distinct actions	3716	Number of distinct moves (e.g. "e4") regardless of position in game tree
Good moves	454	A good move is selected by the AI more than the average
Bad moves	3262	A bad move is selected by the AI less than the average
Terrible moves	647	A terrible move is never selected by the AI Too many terrible moves to list.
Response distance	3.58	Mean distance between move and response; a low value relative to the board size may mean a game is tactical rather than strategic.
Samples	1000	Quantity of logged games played

Board Coverage

A mean of 52.72% of board locations were used per game.

Colour and size show the frequency of visits.

Game Length

Game length frequencies.

Mean	54.12
Mode	[57]
Median	55.0

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 25% of the game turns. Ai Ai found 3 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	74.43	67.23	81.91
Mean no. of effective moves	8.40	7.89	8.93
Effective game space	10^41.05	10^20.10	10^20.96
Mean % of good moves	29.50	29.59	29.41
Mean no. of good moves	3.98	4.32	3.63
Good move game space	10^23.69	10^12.62	10^11.07

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	80.00%	A hot turn is one where making a move is better than doing nothing.
Momentum	27.27%	% of turns where a player improved their score.
Correction	38.18%	% of turns where the score headed back towards equality.
Depth	3.01%	Difference in evaluation between a short and long search.
Drama	0.57%	How much the winner was behind before their final victory.
Foulup Factor	50.91%	Moves that looked better than the best move after a short search.
Surprising turns	5.45%	Turns that looked bad after a short search, but good after a long one.
Last lead change	76.36%	Distance through game when the lead changed for the last time.
Decisiveness	10.91%	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
f1,Pass,i3,h3-i3,g1,Pass,c9,Pass
g1,f2-g1,i3,h3-i3,f1,g1-f2,c9,Pass
e3,Pass,c5,Pass,f6,Pass,g4,Pass
e3,Pass,c5,Pass,h6,Pass,b5,Pass
f3,Pass,g4,Pass,g5,g6-g5,d3,Pass
g5,g6-g5,g4,Pass,f3,Pass,d3,Pass
g5,Pass,g4,Pass,f3,g6-g5,d3,Pass
h5,Pass,g4,h3-g4,g7,g6-g7,c8,Pass
h5,Pass,d9,Pass,f3,g6-h5,e7,d8-d9
h5,Pass,d9,Pass,f3,g6-h5,e7,Pass
f6,g6-f6,d4,Pass,c7,Pass,d7,Pass
f6,Pass,d4,Pass,c7,g6-f6,d7,Pass

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	9	10
1	18	36	360	684	6930	13194	147880	310469	2825574	5483659

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.

Puzzle	Solution
White to win in 3 moves
White to win in 3 moves
Black to win in 3 moves
White to win in 3 moves
Black to win in 3 moves
White to win in 3 moves
White to win in 3 moves
White to win in 3 moves