Human-like Strategies

Even a board games novice is likely to notice some useful principles (or strategies) after playing a few games of ROGOUR. For example:

• Back you go, little fella. AKA hit.

  Hit an opponent piece if you can.

• Hit them hard, AKA Chuck Norris.

  Specifically, hit the most advanced opponent piece.

• Let’s do that again, AKA Donkey in Duloc.

  Pick moves with give you an extra turn.

• Play it safe.

  Prefer positions with a piece on square 4, which is protected from hits.

• Play like a Bear

  Bear off a piece if you can.

• The Homestretch

  Play pieces into the homestretch (y or z), where they can never be hit again.

• Catch me if you can, AKA Frank Abagnale.

  Minimize the probability of getting hit on the opponent next turn. This is slightly less straightforward than the others, but takes only a little practice to master.

• Extra hitter.

  Play the move which gives an extra move and has the highest probability of hitting an opponent piece in that second move.

Simple-minded Humans

Lets start with the most simple minded humans, so simple they can manage only one of those principles. They look at all possible moves, discard non-compatible ones and choose a move at random from those remaining. How well do they fare against the most simple player possible, Mister Random himself?

It is only fitting to let the computer evaluate those human-like strategies. For each principle, the computer plays 10,000 games against Mr Random (5000 as green, 5000 as red), and counts wins and losses. Below is a table listing the win percentages of each strategy, from highest to lowest:

Principle win rate (%) against random player

Frank	Chuck	Hit	Donkey	Extra	H-stretch	bear	safe
93.4-94.3	85.9-87.2	85.4-86.8	69.2-71.0	68.6-70.4	62.4-64.3	58.4-60.4	58.4-60.3

Note that the win rate is given as a 95% credible interval [1], mainly to demonstrate that 10,000 trial are sufficient to separate between principles.

This simple table is already quite interesting. Even the weakest principle, safe, wins on average 0.2 points per game, a rating difference of about 65 ELO points. The strongest, Frank, is 470 ELO points stronger. Also, Frank is faring considerably better than Chuck: not getting hit seems to be more important than hitting, at least against a random opponent. Or maybe it is that the Frank principle applies to all positions, while Chuck only to positions with possible hits. That said, those numbers do not necessarily reflect the relative strength of the various principle when pitted one against the other. When we let everyone play everyone else (28 matches, again 10,000 games each) and convert the pairwise wins/loses into ELO we get the ratings below:

Principle ELO rating from pairwise matches

Frank	Chuck	Hit	Donkey	Extra	H-stretch	Safe	Bear
1759	1590	1585	1444	1444	1406	1405	1367

In fact, the agreement between the two lists is very good. The order is almost identical, only in this pool safe is doing better than bear, whereas against random they were equal.

Advanced Humans

But surly this is taking a very dim view of humanity? Real humans can easily deal with multiple criteria, so why not our computer masquerading as a human? It is quite easy to combine principles: each one is a filter, and we can create a compound-filter by stringing together several principles and applying them in succession. For example “hit;Frank” plays by removing all non-hitting moves and selecting the safest among the remaining moves, while “Frank;hit” discard moves which are not as safe as the maximum and picks a hitting move. However, there are 109600 possible compound-filters, so it is not feasible to find the best by trying out all pairs.

But even if finding the best is hard, we can certainly evolve good ones. If we view those principles as “genes” and the compound-filter as a “genome” we can start with a pool of random genomes and evolve it over time. This extremely simplified evolutionary process proceeds generation by generation. In each generation we take an existing genome and create a new one by adding, removing or changing a gene. If the new genome outperforms the weakest in the pool it replaces it, otherwise it is discarded. If we keep track of the genomes coming out from the pool we get a list of progressively stronger players. This competitive process, like most evolutionary ones, is very effective. After 300 generations the pool already contains very similar players of roughly the same strength. A sample of progressively stronger players from this process, together with random and the strongest looks like this:

• Random
• Frank

1. safe;homestretch;hit;Extra
2. homestretch;Frank;bear;safe;Donkey;hit;Chuck;Extra
3. safe;homestretch;hit;Extra;Chuck;Frank;bear;Donkey
4. Extra;Chuck;hit;bear;homestretch;Frank;Donkey;safe
5. hit;Donkey;safe;homestretch;Extra;bear;Frank
6. hit;Donkey;safe;homestretch;Extra;bear;Frank;Chuck
7. hit;safe;Extra;bear;Frank;Donkey;homestretch;Chuck
8. hit;safe;Extra;bear;Frank;Donkey
9. hit;safe;Donkey;bear;Frank;Extra;homestretch;Chuck

Like the products of most evolutionary systems, it is almost impossible to understand why each outperforms the previous ones, but when we play all pairwise matches and assign ELO’s we get the following picture:

ELO rating from pairwise matches

Random	Frank	1	2	3	4	5	6	7	9	8
760	1323	1363	1413	1565	1638	1683	1686	1687	1690	1693

This relatively simple process generates quite strong players, yet ones which make human-like mistakes. Just play (8) and you will see what I mean. Having human-like opponents of various strengths is actually quite nice, because usually it is hard to “degrade” a strong computer players in natural ways.

[1]	The exact details of how to compute those credible intervals are in the source.