Despirala-Engine

Finds the optimal strategy for the dice game Despirala. Can also evaluate luck and mistakes made during play.

Rules of the game

At the start of each turn you roll all six dice for free and then gain five goods.
After that you choose to either reroll the dice (for the cost of one good) or choose some combination that you have not attempted before.
After picking a combination you have to complete it.
To that end, you repeatedly take the dice not found in the combination and reroll only them (for the cost of one good).
If you successfully complete the combination, you win the amount of points it is worth and your turn ends.
If you run out of goods before completing the chosen combination, your turn ends and you do not get any points for that combination.
The six single number combinations (ones, twos, ...) are an exception: you choose when you are done rolling and get points proportional to the number of correct dice.
At the end of the game you get one bonus point per good you have left over.

Under normal play, the goal is to maximize your score. Under misère play, the goal is to minimize your score, but rerolling is not allowed.

List of combinations

Name	Contents	Points
Collect ones	N ones	Sum of dice in the combination
Collect twos	N twos	Sum of dice in the combination
Collect threes	N threes	Sum of dice in the combination
Collect fours	N fours	Sum of dice in the combination
Collect fives	N fives	Sum of dice in the combination
Collect sixes	N sixes	Sum of dice in the combination
Three pairs	X, X, Y, Y, Z, Z	Sum of dice in the combination
Two triples	X, X, X, Y, Y, Y	Sum of dice in the combination
Four of a kind	X, X, X, X	40
Kamerun	4, 5, 5, 6, 6, 6	45
Straight	1, 2, 3, 4, 5, 6	50
Six of a kind	X, X, X, X, X, X	60
General	6, 6, 6, 6, 6, 6	70
Despirala	1, 1, 1, 1, 1, 6	80

Note: for combinations such as pairs/triples/X of a kind, you have to pick which pairs/triples/X you are doing. E.g. Three pairs: threes, fives and sixes.

User interface

The UI is entirely in the console. It should be fairly intuitive. Moves are written in the format: name of the move, followed by its arguments given as numbers. E.g. "Collect 3", "Two triples 3 5", "Despirala", "Reroll". While collecting the two possible moves are "Continue collecting" and "Stop collecting" (or just "Continue" and "Stop"). Move names are case insensitive. To list all currently possible moves use "List options" (or just "List" or "Options").

Engine

The engine precomputes the expected scores of reachable positions. If wanted, it can do this with Monte Carlo sampling in each state. However, exactly computing the value of each state is tractable. Thus, the engine can optimally solve the game. Using these precomputed values it can evaluate the expected score from any states and the delta in any transition between states (i.e. luck when rolling and mistakes when choosing a move). Note that the engine solves for maximizing the expected score, without caring about other players or winning chances.

Competitive play

There is a competitive strategy. It tries to optimize the winning chances/expected rank. It explores the possible moves of the current player and assume the game is played according to the EV maximizing strategy from there on. It then uses the distribution of the scores of the other players and the distributions of all the moves to choose the move which results in the best expected rank. It works through Monte Carlo simulations of the games (instead of exact computations).

Its win rate, if we count draws as half-wins, against the EV maximizing strategy is around 55% as the second player (its average rank from over 30k games is 1.449). Simlarly, as the first player it has a win rate around 54% (its average rank from over 30k games is 1.462). The strategy's win rate against itself is about 51% as the second player (its average rank from over 30k games is 1.492).

Performance stats

All stats are obtained with a million tests. Note that for an exact model, "expected score" will actually be equal to the expected value of the score, i.e. empirical mean will approach the calculated "expected score".

Stats for normal_exact.model

Expected score: 443.616

Mean: 443.754
Stdev: 61.373
5th percentile: 310
25th percentile: 420
50th percentile: 468
75th percentile: 484
95th percentile: 501
Mode: 478

Stats for normal_1000.model

Expected score: 443.665

Mean: 443.667
Stdev: 61.388
5th percentile: 311
25th percentile: 421
50th percentile: 468
75th percentile: 484
95th percentile: 501
Mode: 479

Stats for normal_50.model

Expected score: 445.344

Mean: 442.396
Stdev: 62.388
5th percentile: 307
25th percentile: 417
50th percentile: 467
75th percentile: 483
95th percentile: 501
Mode: 476

Stats for normal_5.model

Expected score: 454.471

Mean: 429.211
Stdev: 70.139
5th percentile: 282
25th percentile: 392
50th percentile: 459
75th percentile: 479
95th percentile: 498
Mode: 473

Stats for normal_1.model

Expected score: 497.152

Mean: 329.579
Stdev: 92.806
5th percentile: 179
25th percentile: 261
50th percentile: 326
75th percentile: 398
95th percentile: 477
Mode: 468

Stats for misere_exact.model

Expected score: 105.973

Mean: 105.917
Stdev: 52.408
5th percentile: 28
25th percentile: 71
50th percentile: 99
75th percentile: 137
95th percentile: 201
Mode: 80

Stats for misere_50.model

Expected score: 105.956

Mean: 105.889
Stdev: 52.443
5th percentile: 28
25th percentile: 71
50th percentile: 99
75th percentile: 137
95th percentile: 201
Mode: 80

Stats for misere_5.model

Expected score: 103.943

Mean: 107.710
Stdev: 52.700
5th percentile: 29
25th percentile: 72
50th percentile: 101
75th percentile: 138
95th percentile: 204
Mode: 80

Stats for misere_1.model

Expected score: 100.375

Mean: 113.820
Stdev: 53.628
5th percentile: 33
25th percentile: 77
50th percentile: 109
75th percentile: 145
95th percentile: 211
Mode: 83

indjev99 / Despirala-Engine