=Paper=
{{Paper
|id=Vol-1964/PU1
|storemode=property
|title=Deep Barca: A Probabilistic Agent to Play the Game Battle Line
|pdfUrl=https://ceur-ws.org/Vol-1964/PU1.pdf
|volume=Vol-1964
|authors=Sean McCulloch,Daniel Bladow,Tom Dobrow,Haleigh Wright
|dblpUrl=https://dblp.org/rec/conf/maics/McCullochBDW17
}}
==Deep Barca: A Probabilistic Agent to Play the Game Battle Line==
https://ceur-ws.org/Vol-1964/PU1.pdf
Sean McCulloch et al. MAICS 2017 pp. 145–150
Deep Barca: A Probabilistic Agent to Play the Game Battle Line
S. McCulloch Daniel Bladow Tom Dobrow Haleigh Wright
Ohio Wesleyan University Gonzaga University Google, Inc University of California,
Santa Barbara
stmccull@owu.edu dbladow@zagmail.gonzaga.edu tomdobrow@gmail.com wright@umail.ucsb.edu
ABSTRACT based system derived from specific rules and strategies of a given
game. These rules can then either be applied directly to choose a
Recent years have seen an explosion of interest in “modern” board
move, or serve as the basis of an evaluation function in a search [7]
games. These differ from the “classic” games typically seen in
[8].
Artificial Intelligence research (e.g. Chess, Checkers, Go) in that
While these methods can produce competitive players, any
the modern games often have a large component of randomness or
approach that uses a variant of classical search will eventually
non-public information, making traditional game-tree methods
become confounded by the exponential growth of the game tree-
infeasible. Often, these modern games have an underlying
often sooner in modern games than in more classical games because
mathematical structure that can be exploited. In this paper, we
of the need to model randomness or hidden information. Efforts to
describe an intelligent agent to play the game Battle Line, which
reduce the game’s complexity may remove important information.
uses elements of theorem-proving and probability to play
Rule-based systems can only perform as well as the rules in which
intelligently without utilizing game trees. The agent is superior to
they are given, and since those rules are typically programmed in
the only other known computer player of the game and plays at a
by humans, they do not capture the autonomous nature that we
level competitive with top human players.
desire in our agents.
Our approach is to notice that these games often have
CCS CONCEPTS interesting underlying mathematical structure and to exploit that
• Computing methodologies ~ Probabilistic reasoning structure. If we can recast a game’s strategy as a mathematical
• Computing methodologies ~ Game tree search problem, then we have the ability to use theorems and approaches
from mathematics to design an agent that makes moves based on
ADDITIONAL KEYWORDS an underlying mathematical model. We feel that an agent made
Modern Board Games, Battle Line according to this paradigm is more autonomous than one that
follows a rule-based system or an evaluation of leaves of a game
tree because its strategy comes from the mathematical
1 INTRODUCTION underpinnings of the game itself.
Historically, the main work in developing intelligent agents to play We have used this approach in the past on a different game,
board games has focused on “classic” games such as Chess [5], called Football Strategy [8], where the rules of the game can be
Checkers[12], or Othello[4]. The agents that play these games viewed as a normal-form game, in the game-theoretic sense. By
often focus on variants and improvements to game-tree search. applying concepts of Nash Equilibria, our agent created a mixed
While modern agents have many clever approaches and strategy that made it competitive against top human players.
optimizations, there is still an exponentially-sized game tree to In this paper we focus on the two-player strategy card game
search. Even the recent “AlphaGo” program by Google [13] uses Battle Line, published by GMT Games in 2000 [1]. We have
neural networks, but does so in order to evaluate a game tree more developed an agent named “Deep Barca” that plays this game by
efficiently. exploiting the underlying probabilistic and logical nature of the
In recent years, an influx of new games have been created of card game.
varying levels of complexity. These games often depend on non-
public information (such as a secret hand of cards or tiles) or 2 DESCRIPTION OF THE GAME
randomness, making the size of a typical game tree infeasibly large. In Battle Line there are two decks of cards: Troop cards and Tactic
There have been efforts to design agents to play these games. cards. Most of the game is played using the Troop cards, which is
Some of these efforts have involved designing variants of classical a deck of 60 cards consisting of the numbers 1-10 in six different
search methods [8] [10] [11], some have applied multiple agents to colors. On their turn, players play a Troop card from their hand on
the problem, and some have kept a simpler model of the game to one of nine flags. Each player will eventually place three cards on
reduce the complexity [14]. Another approach is to develop a rule- each of nine flags. The three card hand that is formed is called a
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Deep Barca: A Probabilistic Agent to Play the Game Battle Line pp. 145–150
MAICS 2017, April 2017, Fort Wayne, IN, USA McCulloch, Bladow, Dobrow, and Wright
formation. (See Figure 1 for an example of what flags and Figure 1 illustrates some of these situations. In the leftmost
formations look like). The player with the better formation will column, the top player has a straight flush, which beats the
claim the flag. Winning the game requires claiming 5 of the 9 flags opposing three of a kind. In the second column, both players have
or any 3 flags in a row. flushes, but the top one totals to 18 and will beat the bottom’s total
of 12.
2.1 Evaluating Formations The third column of Figure 1 shows a situation where the top
The values of the formations correspond to three-card poker hands. player has played the best possible hand-- the highest possible
The order in which cards are played does not matter; the cards in straight flush (10, 9, 8) that is finished first. Once the player plays
the final formation are evaluated collectively. The ranks of the the final card of this formation, it cannot be beaten. When this
different formations are as follows (the actual game uses military situation occurs, a player is allowed to claim a flag at the beginning
terminology for these ranks, but we have translated them into of their next turn, even before the opponent has competed their
terminology that is closer to that used in poker to aid three card hand on the flag. This prevents the opponent from
understanding). playing any more cards on the flag and is a powerful move, as it
x A straight flush is the highest possible hand. It consists reduces the available number of places for the opponent to play
of three cards, all the same color, in consecutive order cards.
(for example, the 7, 8, and 9 of Blue). It is also possible in other situations to use the state of the
x A three of a kind is the next highest hand, consisting of board to prove that a player will not be able to beat a completed
three cards of the same number (for example, the red, formation no matter what Troop cards are played. For example, in
blue, and green 8). the fourth column of Figure 1, the top player has completed a
straight totaling 27. The best formation the bottom player could
x A flush is three cards of the same color, but with no
possible complete is a straight totaling 9. In this case, the player is
relationship among the numbers (for example, the
also allowed to claim the flag at the start of their next turn.
Yellow 1, 4,and 8).
This situation can also arise when a needed card exists
x A straight is three cards in consecutive order, but with no
elsewhere on the board. In the rightmost formation of Figure 1, the
relationship among the colors (for example, a Red 2,
bottom player has two cards towards a straight flush, which if
Orange 3, and Purple 4).
completed would win the flag. However, the cards that can
x A nothing is the lowest formation, and is any collection
complete the straight flush have already been played (the 8 in the
of cards that does not fit one of the above rankings. (for
third column and the 5 in the same column by the top player). In
example, a Red 2, Orange 4, and Blue 6). Note that a
this case, the best formation that can be completed by the bottom
single pair (for example, a Green 2, Blue 2, and Orange
player is a flush, which will lose to the top player’s three of a kind,
7) counts as a nothing.
and so the top player can claim this flag at the start of their turn.
When both players have completed formations, if one player’s
formation is a higher rank, that player wins. If two players have
2.2 Tactics Cards
the same rank, then the highest total wins (so a three of a kind with
7’s beats a 3 of a kind with 4’s). If two players have the same rank The deck of 10 Tactics cards provides “special” cards that can bend
and the same total, whoever finishes their hand first wins the flag. the rules of the game in various ways. Some are wild cards (and
In practice among skilled players, most flags are won with a straight thus can take on a variety of colors or numbers depending on what
flush or three of a kind. makes the best possible formation), some change the resolving
rules of a flag (for example, making the hands consist of 4 cards
instead of 3, or turning all hands into “nothings” so only the highest
total wins), and some change the positions of the cards (allowing a
player to move their cards between flags, or steal a card from an
opponent’s formation). The possibility of an opponent responding
to a completed formation with a Tactic card play is the main reason
why flags are claimed at the start of the claimers turn.
Tactic cards operate under two important restrictions. First, if
a player chooses to draw a Tactic card, they do so instead of
drawing a Troop card for that turn. This reduces the player’s hand
of playable Troop Cards. Second, a player is not allowed to have
played more than one Tactics card beyond what their opponent has
played. So, if Player A had played two Tactics cards, and Player B
had only played one, Player A would not be able to play another
Tactics card until Player B has played a second one.
Figure 1: Some examples of formations
2
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Deep Barca: A Probabilistic Agent to Play the Game Battle Line MAICS 17, April 2017, Fort Wayne, IN, USA
3 DESIGN OF THE AGENT winning three adjacent flags) we take the weighted average of the
probabilities of winning via each of these two conditions as a
Our goal in designing Deep Barca was to make the agent as function of the number of cards left in the Troop Deck. Thus, in the
autonomous as possible, giving it a set of underlying mathematical beginning of the game, the agent is far more concerned with simply
and probabilistic principles from which moves will be devised. The winning the majority of the flags since this goal in practice entails
agent models these principles in several different ways, and these doing as well as possible on as many flags as possible. Near the end
models generate rankings of the possible moves that the agent can of the game, the agent is more concerned with winning three
make. adjacent flags and will even willingly sacrifice a fringe flag if it
means winning three in a row somewhere else.
3.1 Enforcing the Rules
Before intelligently deciding on the best moves, it is necessary to
3.3 The MFA and BSFA Methods
The Multiple-Formation Approach (MFA) is designed to estimate
develop a program that enforces the rules of the game. Many of
the best hands that can be created once both players have committed
those rules are straightforward. The most complex involves when to a hand. It is computed for each of the agent’s top four formations
a flag can be claimed. computed above. For each potential final formation, the agent finds
As stated in the previous section, flags can be claimed by a the probability of winning via that formation (the probability of
player at the start of their turn if it can be proven that no possible making that formation multiplied by the probability of that
set of cards the opponent can play will beat the player’s formation. formation not being beaten by the opponent). The probability of the
To determine this, when a formation is finished, the program agent winning that flag is the union of the probabilities of winning
determines the set of superior formations: formations that can beat via each of its top formations.
the formation that was just completed. Then, the set of possible Table 1 shows what these probabilities would look like in the
situation where the agent has played a 9 on a flag, and the opponent
cards that can complete the formation are examined. These are
has played a 4 of a different color. For this table, we assume that
cards that are currently not on the board (though they may be in
no cards in any useful formation exists in the player’s hand or on
player’s hands). If any superior formation can be completed by any other flag. The top two formations the agent can make using
adding these cards to it, then that formation is feasible. A flag can the 9 are straight flushes (10-9-8 and 9-8-7). If these are
only be claimed if all superior formations are infeasible. successfully made, the opponent cannot possibly beat them using
the 4 that was committed to its flag. The next highest formation is
3.2 The Probabilistic Model for Decision Making three nines. This involves the agent drawing any two of the five
Deep Barca uses a general probabilistic model for decision making remaining nines in the deck, but can be beaten if the opponent
draws any of the three possible straight flushes using the 4. The
to play Battle Line. Each turn, for each card in its hand and for each
fourth highest formation is a flush totaling 26 (10-9-7). This loses
flag that card could be placed on, the agent evaluates the probability not only to any straight flush, but also to any three 4’s.
of winning the game if the card being considered is placed on the
flag being considered. It then chooses the option that maximizes Formation Probability Probability Chance
this probability. Calculating the probability of winning the game of Making of Opponent to win
for each move is a three-part process. beating flag
First, for each flag the agent calculates the top four formations SF 10-9-8 .25 0 .25
that each player could make on the flag. These are the four strongest SF 9-8-7 .25 0 .25
formations that could possibly be made, given the Troop Cards 9-9-9 .82 .58 .35
remaining in the agent’s hand and the cards left in the deck of Troop Flush 26 .25 .94 .02
Cards. Table 1: A sample table used in the MFA for a 9 vs a 4 on an
Next, the agent calculates the probability of winning each flag empty board. For details of how the probabilities are computed,
given the move under consideration. We categorize each flag in one see section 3.4
of the following three ways:
a) If neither player has played a card towards a formation In our experiments, we found that the MFA approach did not
on the flag, we simply say the probability of winning work well on flags where one player had not played any cards. This
the flag is 50%. was because on an empty flag all of the top formations in the game
b) If both players have played at least one card towards a are still theoretically possible. Once a card is played on a flag, the
formation on the flag, we calculate the probability of space of potential formations is drastically reduced-- instead of
winning the flag via what we call the Multiple being able to create any feasible formation, now only formations
Formation Approach, or MFA, described in the next that include the played card can be considered. This had the effect
section. of making the agent undervalue playing cards on empty flags. The
c) If one player has played at least one card towards a reason for this is that the MFA does not take into account the need
formation on the flag, but the other player has yet to of a player to play a card on a flag (usually reducing the potential
play a card, we use a mix of the MFA from above and value of the formations that can be made on the flag) each turn.
the Best Single Formation Approach, or BSFA, also Instead, the probability is based on whether the formation can be
described in the next section. made by the end of the game.
Lastly, we calculate the probability of winning the game as a Thus, to make the agent less pessimistic, in situations where
function of the probabilities of winning each of the nine flags. Since one of the two flags had no cards played, we implemented the Best
we have two different win conditions (winning five out of nine, and
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MAICS 2017, April 2017, Fort Wayne, IN, USA McCulloch, Bladow, Dobrow, and Wright
Single Formation Approach (BSFA). In the BSFA, we find the The first situation arises in the turn before claiming a flag.
single best possible formation for the player with no cards and then Recall that a flag is claimed at the start of a turn, meaning that in
evaluate the chances of winning the flag if that formation was the normal course of events, the opponent has a turn to react
played vs the top four formations of the player who has played between the playing of a card that will win a flag and the actual
cards. This simulates the potential “goal” of the opponent who sees claiming of it. This turn exists to give the opponent a chance to
a card played and is trying to beat what our agent is trying to do.
play a Tactics card that may result in changing the resolution of a
The BSFA on its own yields very narrow decision making
because the agent will not even consider many decent or adequate flag. But even if the player cannot or does not wish to play a Tactics
formations (since they are not the single best formations that could card, they must take a turn. This turn often has rich strategic depth.
be made). Often even subpar formations are enough to yield a Since a flag that is claimed is ineligible to be played upon by
victory on a particular flag. The MFA on its own is weak as well either player, even if the formation is incomplete, it usually
since it is heavily biased in favor of the player who has yet to play behooves the player about to lose the flag to play a card on the flag
a card on the flag. This is due to the fact that all of the top that is about to be lost. This “discarding” action has several
formations generated for that player are simply the very strongest benefits:
formations in the game since the player has yet to commit any card x Since playing a card on a flag often decreases the
that would narrow his or her options. Using a mix of these two possible formations that can be formed, it is usually
approaches, our agent makes much better decisions than it would beneficial to delay committing to a flag as long as
using either approach by itself. possible. Playing a card on the flag that is about to be
claimed helps delay decisions about other flags.
3.4 Evaluating Formations Probabilistically x Since proving a flag is unbeatable often involves using
In both the MFA and BSFA models described above, the agent cards already played on the board as a reference,
needs to determine the likelihood of whether one formation will discarding is a good chance to play a card that will help
defeat another, especially in a situation where the formations are a later proving effort.
currently incomplete, and the cards needed to complete the x Since once the flag is claimed the flag will become
formation will need to be drawn from the deck. We use the unplayable, not playing on the flag before it is claimed
will mean the owner of the losing flag will have less
following probabilities to aid our calculations:
total spaces to play on in the game (and thus have less
x If a card is present on the board, and is in the formation, total options in the game). This loss of tempo is
it has a 100% probability of being part of the final especially noticeable at the end of the game, since it is
formation. possible for one player to have no legal places to play a
x If a card is present on the board but is part of another card and be forced to pass.
flag, it has a 0% probability of bring part of the final Deep Barca handles the situation in which it is about to lose a
formation. flag in a different way than it handles the normal course of the
x If a card is not present on the board but is in the agent’s game. If it recognizes (via its internal proving mechanism) that it
hand, then the agent gives a 100% chance of that card
is about to lose a flag, it attempts to find a card to discard. The
being a part of the formation if the desired formation
belongs to the agent and a 0% chance of the card being decision of which card to dump is made by creating a set of
part of the formation if the desired formation belongs to “wishlists”-- cards that each possible formation (for each player)
the opponent. are hoping to draw or to play on a flag. The agent will choose a
x If the card is not present in the board or in the agent’s card that appears most frequently on the opponent’s wishlist and
hand, the card must be drawn to be useful. The cards in least frequently on its own. The hope is that this discarding action
the Troop deck are assumed to be split evenly among will thus not adversely affect any of the top formations the agent is
both players. The probability of the agent drawing a trying to make but possibly will aid in proving that a formation the
needed card is their share of the Troop deck divided by opponent is trying to make is impossible.
the total number of unseen cards (both in the Troop These wishlists are also used by the agent to resolve conflicts
deck and the opponent’s hand). The complement of that
where a single card is one of the top formations on multiple flags.
probability is the probability that the opponent will draw
By noting what cards are desired by the agent in other places, we
(or has drawn) that card. Notice that the agent will do
this calculation even if the card is actually in the reduce the probability that such a card will be available on the
opponent’s hand, since the agent has no legitimate way conflicted flag.
of knowing this fact.
3.6 Drawing and Playing Tactics Cards
3.5 Discarding on a Lost Flag In addition to the Troop deck, Battle Line has a deck of Tactics
The above probabilistic behavior forms the bulk of our agent, and Cards. These cards do many different things, and if played at the
thus it chooses moves largely autonomously within the models it is right time, can swing the ownership of a flag from one player to
given. However, our models do not cover some areas. Thus, there another, for example by playing a wild card to complete a formation
is a need to develop approaches for these specific situations. Even that would be proven impossible using played Troop cards. The
here, we attempted to make the decisions as autonomous as agent has two decisions regarding Tactics cards. The first decision
possible, avoiding low-level pre-programmed rules. is deciding when to draw one. The agent will be drawing a Tactics
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Deep Barca: A Probabilistic Agent to Play the Game Battle Line MAICS 17, April 2017, Fort Wayne, IN, USA
card instead of a Troop card, and since there are limitations on how we found some emergent behaviors that were particularly
many Tactics cards can be played, it is not a good idea to have an noteworthy.
excess. The second decision is when to play one. Since there are Eight > Ten: We noticed that early in the game, on an empty
limitations on the frequency of playing these cards, the timing of flag, the agent often chose to play an 8 on the flag when it could
the play should be carefully considered. also play a 10. Most high level human players would prefer a Troop
Since the different Tactics cards are so different and the Card with value 10 to a Troop Card with value 8, all else equal. The
situations in which they can be played are so different, we had to 10 is a higher value, and thus potentially can make hands with
resort to several ad-hoc rules: higher totals than the 8. However, the 8 is eligible for more straight
x If playing a Tactic card could win the agent the game, flushes (10-9-8; 9-8-7; 8-7-6 as opposed to just 10-9-8), which are
play it. the strongest formations. Because of this, our agent’s preference of
x If the agent is about to lose a flag, and the loss can be an 8 over a 10 is not necessarily bad, and many human players even
prevented by the play of a Tactics card, play it if it have this preference. We were pleased to see this behavior emerge,
raises the agent’s chance of winning the flag to over as it was unexpected and not caused by direct human intervention.
50%, or if losing the flag would lose the game. Playing low-value viable formations: If the agent has two
cards toward a straight flush (for example, the 5 and 6 of red), and
Other than that, except for very specific situations relating to
the opponent has two cards towards a three of a kind (for example,
specific cards, the agent will not play a Tactics card. This has the
two 7’s), the agent would often give up on the straight flush, going
result of the agent being rather conservative with their Tactics
all the way down to a straight (for example, playing a blue 4). This
cards, which mirrors the way they tend to be used by expert-level
was especially common in situations where many of the cards of
humans.
the opposing three of a kind were already played (and thus making
Similarly, the agent will draw Tactics cards defensively. If it
the three of a kind less likely) or in situations where there was just
determines that the opponent is one card away from winning a flag
one possible card that would complete the straight flush.
that can only be beaten by a Tactics card, it will draw one. It will
At first glance, it appeared that the agent was making a mistake
draw a second if the opponent has two or more cards or if the one
giving up a flag so easily. But in reality, once two cards of the same
tactic card the agent possesses is one of the “less useful” ones. In
value are played on a flag, the only possible outcomes are a three
either case, the agent will not draw a Tactics card that it cannot
of a kind or a nothing. The agent has recognized this fact and
play.
decided to create a formation that can only be beaten by the
opponent drawing the card to complete their three of a kind.
4 RESULTS While the actual usefulness and timings of these moves against
top players can be debated, the ability to find these kinds of moves
4.1 Evaluating the Quality of Deep Barca without having the strategies directly programmed in validates our
We evaluated the quality of our agent against computer and approach to make the agent largely autonomous and based on
human players. To the best of our knowledge, there is only one underlying mathematical principles.
other computer agent, which is the IOS App Reiner Knizia’s
Battleline, by Gourmet Gaming [6] We played a best-of-three 5 FUTURE WORK
match, in which our agent handily won 2-0. In one of the games, There are many possible improvements we could make to the agent
our agent won without losing a flag. Against human players, we to strengthen quality of play and reduce ad hoc decision making
tested against a group of 5 experienced human players, and the even further.
agent won about 50% of the time. One of the authors is a former We would like to improve the way the agent draws and plays
three-time world champion at Battle Line [2] at the World Tactics cards to be less ad-hoc.
Boardgaming Championships [3]. We played a 10 game series, in We would like to improve the agent’s ability to make longer-
which Deep Barca won three of the games. term decisions. Since we evaluate the quality of each move based
With these results we can confidently say that our agent can only on the game state one turn in the future, the agent can often
competitively play against even the very best human Battle Line miss moves involving multiple Tactic Cards used together, since
players. Also, our agent was able to make its decision for each turn the play of any of the cards by itself is weak.
in between a tenth of a second and one second, which is far faster The agent’s ability to handle the scenario in which multiple
than a human player could play. This is encouraging, since it leaves flags are being claimed at the same time is very weak. We would
the option for more complex decision-making without those like to be able to evaluate which of the claimed flags are worth
decisions taking an undue amount of time. fighting for and which of the flags might be too detrimental to lose,
but currently we do not.
4.2 Emergent Behavior Our agent keeps the top four possible formations in its MFA
Since our agent generally followed a general mathematical model model of rating a flag, mainly for efficiency reasons. We would
for decision making, we often found that it would make moves that like to examine what would happen if that number was increased.
went against our immediate human intuition. In our experiments,
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MAICS 2017, April 2017, Fort Wayne, IN, USA McCulloch, Bladow, Dobrow, and Wright
Since our agent evaluates its moves so quickly, it would be
interesting to see if it would be possible to incorporate some
lookahead search into the agent, even if just for one or two moves.
ACKNOWLEDGMENTS
This work was partially sponsored by the National Science
Foundation’s Research Experience for Undergraduates Program,
Grant # 1262850
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