<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Archiving and Interchange DTD v1.0 20120330//EN" "JATS-archivearticle1.dtd">
<article xmlns:xlink="http://www.w3.org/1999/xlink">
  <front>
    <journal-meta />
    <article-meta>
      <title-group>
        <article-title>TAG: A Tabletop Games Framework</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Raluca D. Gaina</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Martin Balla</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Alexander Dockhorn</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Rau´ l Montoliuy</string-name>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Diego Perez-Liebana</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
        </contrib>
        <aff id="aff0">
          <label>0</label>
          <institution>School of Electronic Engineering and Computer Science, Queen Mary University of London</institution>
          ,
          <country country="UK">UK</country>
        </aff>
      </contrib-group>
      <abstract>
        <p>Tabletop games come in a variety of forms, including board games, card games, and dice games. In recent years, their complexity has considerably increased, with many components, rules that change dynamically through the game, diverse player roles, and a series of control parameters that influence a game's balance. As such, they also encompass novel and intricate challenges for Artificial Intelligence methods, yet research largely focuses on classical board games such as chess and Go. We introduce in this work the Tabletop Games (TAG) framework, which promotes research into general AI in modern tabletop games, facilitating the implementation of new games and AI players, while providing analytics to capture the complexities of the challenges proposed. We include preliminary results with sample AI players, showing some moderate success, with plenty of room for improvement, and discuss further developments and new research directions.</p>
      </abstract>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>1 Introduction</title>
      <p>
        In the last few decades, tabletop games have gone through a
‘Renaissance‘, gaining more popularity than ever: thousands
of them are published each year and welcomed by an
expanding audience of gamers
        <xref ref-type="bibr" rid="ref10 ref11 ref6 ref7">(Engelstein and Shalev 2019)</xref>
        .
Many of these games are typically designed with richer
mechanics and rules, and less focus on chance-based elements.
Modern tabletop games are very diverse and complex in
general, which can provide various challenges, such as unique
game state representations, partial observability on various
levels, actions outside of a player’s turn, cooperation, and
competition in the same game, etc. Therefore, they have a
very different and complex set of mechanisms that would
require a lot of effort to develop using previous approaches.
      </p>
      <p>
        AI Research in board games has mostly been, with some
exceptions, focused on traditional board games, either in
isolation (chess, Othello, Go, etc.) or as part of general
game playing (GGP) frameworks, such as GGP
        <xref ref-type="bibr" rid="ref15">(Genesereth,
Love, and Pell 2005)</xref>
        , Ludii
        <xref ref-type="bibr" rid="ref25 ref30">(Piette et al. 2019)</xref>
        and
OpenSpiel
        <xref ref-type="bibr" rid="ref10 ref11 ref25 ref6 ref7">(Lanctot and others 2019)</xref>
        . While these frameworks also
allow the definition of additional games, they are limited to
common mechanics or require extensive development effort.
      </p>
      <p>
        In this work, we introduce the Tabletop Games (TAG)
framework, which is a collection of tabletop games, agents,
and development tools meant to provide a common platform
for AI research. This work is mainly motivated by three
factors: i) the characteristics of modern tabletop games
(multiplayer, partially observable, large action spaces, competition,
and collaboration, etc.) provide an interesting challenge to
AI research. These modern games provide many
characteristics not implemented in existing frameworks, e.g. changing
player roles or varying forms of cooperative and competitive
game-play; ii) TAG presents tabletop games from a GGP
perspective, by providing a common API for games and playing
agents; and iii) we aim to provide a framework that can
incorporate different games under a common platform, making
it possible for the research community to implement their
own games and AI players to expand TAG’s collection.
        <xref ref-type="bibr" rid="ref8">De
Araujo et al. (2019</xref>
        ) highlighted the need for such a
framework in a survey that described the different schemes and
data structures used in the literature of digital board games.
      </p>
      <p>In order to provide the necessary flexibility to support the
great variety of existing tabletop games, TAG requires the
user to implement games via a programming language (Java),
instead of using a game description language as other general
frameworks do. TAG provides many customisable
components to simplify the development of additional games not
currently in the framework. It is able to handle partial
observability, providing simple means of adding custom observation
schemes, game-state heuristics, and agent statistics, as well
as supporting the development of graphical user interfaces
for human play with computer players.</p>
      <p>While we are working on extending the framework with
additional game mechanics, games and agents, the current
version is publicly available1. The contributions of this paper
are two-fold: firstly, we present the framework, its structure,
games and AI players implemented (see Section 3). Secondly,
we provide a discussion on a baseline experimentation
(Section 4), aimed at illustrating insights into the games
implemented, their features, and performance of vanilla AI players.
In Section 5 we discuss challenges and opportunities included
for this framework and we conclude with final remarks on
future developments in Section 6.</p>
      <p>
        1https://github.com/GAIGResearch/TabletopGames
AI research and board games have been closely related since
the beginnings of the field, with game-playing agents for
TicTac-Toe, Checkers and chess
        <xref ref-type="bibr" rid="ref5">(Campbell and others 2002)</xref>
        , and
the more recent breakthroughs in Go
        <xref ref-type="bibr" rid="ref28 ref35">(Silver et al. 2017)</xref>
        . One
of the most well-known contests, the General Game Playing
(GGP) competition
        <xref ref-type="bibr" rid="ref15">(Genesereth, Love, and Pell 2005)</xref>
        ,
featured classical board games written in the Game Description
Language and promoted research into generic game-players.
        <xref ref-type="bibr" rid="ref25 ref30">(Piette et al. 2019)</xref>
        later introduced the “ludemic” general
game system Ludii, which builds upon ideas from GGP. Ludii
defines games as structures of ludemes, high-level, easily
understandable game concepts, which allows for concise and
human-understandable game descriptions, as well as easy
implementation of new games. The current main focus of
the Ludii project is on classical and ancient board games.
Kowalski et al. presented in
        <xref ref-type="bibr" rid="ref23 ref25">(Kowalski et al. 2019)</xref>
        a new
GGP language, called Regular Boardgames (RBG), with a
similar focus as Ludii, but which describes games as regular
expressions instead, for increased efficiency.
      </p>
      <p>
        We consider direct code implementations to be more
accessible, faster to execute and easier to work with in many
cases. More similarly to our framework in this regard,
OpenSpiel
        <xref ref-type="bibr" rid="ref10 ref11 ref25 ref6 ref7">(Lanctot and others 2019)</xref>
        provides a collection of
games and algorithms written in C++ and exposed to Python.
Most of their games are still traditional board games, with
some exceptions, such as the inclusion of Hanabi.
Differently, TAG shifts the development effort onto the framework,
rather than the games, by making a wide range of
components, rules, actions, etc. available to users. Our system allows
for fast prototyping of new games and immediate extraction
of insights and features of interest in each game through
readily-available game and AI-facilitated analysis. We
further support many research directions, from simulation-based
game-playing to parameter optimisation of games and
artificial players.
      </p>
      <p>
        Here, we focus on more types of games, including board,
card, dice and role-playing games, among others, often
grouped under the tabletop games umbrella term. Tabletop
Simulator
        <xref ref-type="bibr" rid="ref19">(Henry 2015)</xref>
        is an example of software
facilitating implementation of tabletop games components in a
physics-based simulated environment; this allows players
to interact with the games as they would in the real world,
but the many games implemented lack support for automatic
rule execution, nor does the software facilitate AI research
as targeted with TAG. However, tabletop games research has
been gaining popularity in recent years. Research in
gameplaying agents for card games is common in competitive
(Poker (Moravcˇ´ık and others 2017) and Bridge
        <xref ref-type="bibr" rid="ref10 ref11 ref6 ref7">(Cazenave
and Ventos 2019)</xref>
        ) and cooperative (Hannabi
        <xref ref-type="bibr" rid="ref1 ref34">(Bard and
others 2020)</xref>
        ) games.
      </p>
      <p>
        Asymmetric player roles is one feature often
encountered in modern tabletop games, and these have been
studied in games such as The Resistance
        <xref ref-type="bibr" rid="ref25 ref33">(Serrino et al. 2019)</xref>
        and Ultimate Werewolf
        <xref ref-type="bibr" rid="ref10 ref11 ref6 ref7">(Eger and Martens 2019)</xref>
        . This is
just another complexity added in modern tabletop games,
yet more lead to the need for intricate strategic planning.
To this extent, Monte Carlo Tree Search (MCTS) methods
have been tried in Settlers of Catan
        <xref ref-type="bibr" rid="ref37">(Szita, Chaslot, and
Spronck 2009)</xref>
        and Risk
        <xref ref-type="bibr" rid="ref16">(Gibson, Desai, and Zhao 2010)</xref>
        ,
and Rolling Horizon Evolutionary Algorithms (RHEA) in
Splendor
        <xref ref-type="bibr" rid="ref25">(Bravi et al. 2019)</xref>
        , all showing a great
improvement in performance. Other games have been more recently
highlighted as important challenges for AI players due to
their strategic complexity (Pandemic
        <xref ref-type="bibr" rid="ref10 ref11 ref21 ref34 ref6 ref7">(Chaco´n and Eger 2019;
Sfikas and Liapis 2020)</xref>
        ) or very large action spaces (Blood
bowl
        <xref ref-type="bibr" rid="ref21 ref25">(Justesen et al. 2019)</xref>
        ).
      </p>
      <p>
        Research has not only focused on game-playing AI,
however. Ticket to Ride
        <xref ref-type="bibr" rid="ref28 ref9">(de Mesentier Silva et al. 2017)</xref>
        was
used as an example for employing AI players for play-testing
games, characterising their features based on different
playstyles and finding possible bugs or gaps in the ruleset. Further,
the use of Procedural Content Generation for such games is
highlighted by
        <xref ref-type="bibr" rid="ref1 ref18">(Guzdial et al. 2020)</xref>
        ; given the rule
complexities and the multitude of components in modern
tabletop games, AI methods can provide a more efficient way of
searching the possibility space for interesting variations.
      </p>
      <p>The Tabletop Games (TAG) framework introduced in this
paper brings together all of these different research directions
and provides a common ground for the use of AI algorithms
in a variety of tabletop games, removing the effort of creating
different frameworks for different purposes and simplifying
the overall development process. As far as we know, TAG is
the first framework that allows the development of multiple
games and AI players under a common API for complex
modern tabletop games.</p>
      <p>3</p>
    </sec>
    <sec id="sec-2">
      <title>The Framework</title>
      <p>TAG was designed to capture most of the complexity that
modern tabletop games provide, with a few games
implemented already and more in progress.
3.1</p>
      <sec id="sec-2-1">
        <title>Concepts</title>
        <p>
          Our framework includes handy definitions for various
concepts and components common across many tabletop
games
          <xref ref-type="bibr" rid="ref10 ref11 ref6 ref7">(Engelstein and Shalev 2019)</xref>
          .
        </p>
        <p>We define an action as an independent unit of game logic
that modifies a given game state towards a specific effect (e.g.
player draws a card; player moves their pawn). These actions
are executed by the game players and are subject to certain
rules: units of game logic, part of a hierarchical structure (a
game flow graph). Rules dictate how a given game state is
modified and control the flow through the game graph (for
instance, checking the end of game conditions and the turn
order). This turn order defines which player is due to play
at each time, possibly handling player reactions forced by
actions or rules. At a higher level, games can be structured
in phases, which are time frames where specific rules apply
and/or different actions are available for the players.</p>
        <p>All tabletop games use components (game objects
sharing certain properties), whose state is modified by actions
and rules during the game. TAG includes several predefined
components to ease the development of new games, such
as tokens (a game piece of a particular type), dice (with N
sides), cards (with text, images or numbers), counters (with
a numerical value), grid and graph boards. Components can
also be grouped into collections: an area groups components
in a map structure in order to provide access to them using
their unique IDs, while a deck is an ordered collection with
specific interactions available (e.g. shuffle, draw, etc.). Both
areas and decks are considered components themselves.
3.2</p>
      </sec>
      <sec id="sec-2-2">
        <title>Structure</title>
        <p>The TAG framework brings together all of the concepts and
components described previously and allows quick
implementation and prototyping of new games. To this end, a
flexible API is provided for all functionality needed to define
a game, with multiple abstract classes that can be extended
for specific implementations. The framework provides some
generic functionality, from ready-made components, rules,
actions, turn orders and game phases, to a fully functional
game loop and a prototyping GUI. The GUI allows users to
start interacting with the games as soon as they have the two
main classes required set up: a Game State (GS) class, and a
Forward Model (FM) class.</p>
        <p>GS is a container class, including all variables and game
components which would allow one to describe one specific
moment in time. It defines access methods in the game state
to retrieve existing game components, make custom and
partially observable copies of the state, and define an evaluation
function that can be used by the playing agents. The FM
encompasses the logic of the game: performs the game setup,
defines what actions players can take in a particular game
state, applies the effect of player actions and any other game
rules applicable, uses a turn order to decide which player is
due to play next (or may wait for all players to return an
action before processing for simultaneous-actions games), and
checks for any end of game conditions. The FM is available
to AI players for game simulations.</p>
        <p>
          For each game, users can further implement specific
actions, rules, turn orders, game parameters (for easy
modification of game mechanics), a GUI and provision of game data.
The last is useful when the game requires large amounts of
data such as tile patterns, cards and board node connections,
and it is provided via JSON files. A full guide on using the
framework and implementing new games is available in the
wiki provided with the code and in
          <xref ref-type="bibr" rid="ref1 ref13">(Gaina et al. 2020)</xref>
          .
3.3
        </p>
      </sec>
      <sec id="sec-2-3">
        <title>Games</title>
        <p>
          There are currently 7 games implemented in the framework,
varying from very simple test games (Tic-Tac-Toe) to strategy
games (Pandemic
          <xref ref-type="bibr" rid="ref26">(Leacock 2008)</xref>
          ), as well as diverse
challenges for AI players. A few games are currently in active
development (Descent
          <xref ref-type="bibr" rid="ref12">(Fantasy Flight Publishing, Inc. 2012)</xref>
          ,
Carcassonne
          <xref ref-type="bibr" rid="ref39">(Wrede 2000)</xref>
          and Settlers of Catan
          <xref ref-type="bibr" rid="ref38">(Teuber
1995)</xref>
          ), and many more are in the project’s backlog, including
games from other frameworks to allow for easy comparison
(see Section 2).
        </p>
        <p>All games implemented can be found in the games
package, each registered in the games.GameType class; this
class allows specifying properties for each game, to allow for
automatic listing for experiments (e.g. a list of all games with
the “cooperative” tag). We highlight next some particularities
of the games currently implemented in the framework.
Tic-Tac-Toe 2 players alternate placing their symbol in a
N N grid until one player completes a line, column or
diagonal and wins the game; if all cells in the grid get filled
up without a winner, the game is a draw.</p>
        <p>This is the simplest game included in the framework, meant
to be used as a quick reference for the minimum requirements
to get a game up and running. Its implementation makes use
of mostly default concepts and components, but it implements
a scoring heuristic and a custom GUI for an easier interaction
given the specific game mechanics.</p>
        <p>
          Love Letter
          <xref ref-type="bibr" rid="ref22">(Kanai 2012)</xref>
          2 to 4 players start the game
with one card each, representing a character, a value and a
unique effect. A second card is drawn at the start of each
turn, one of which must be played afterwards. After the last
card of the deck is drawn, the player with the highest valued
card wins the current round. A player wins the game after
winning 5 rounds. Love Letter features partial observability,
asymmetric and changing player roles and a point system
over several rounds. Figure 1 shows an example game state.
Uno
          <xref ref-type="bibr" rid="ref32">(Robbins 1971)</xref>
          The game consists of coloured cards
with actions or numbers. Numbered cards can only be played
in case either the colour or the number matches the newest
card on the discard pile. Action cards let 2 to 10 players
draw additional cards, choose the next colour to be played or
reverse the turn order. A player wins after gaining a number
of points over several rounds (computed as the sum of all
other players’ card values). Uno features stochasticity, partial
observability and a dynamically changing turn order.
Virus!
          <xref ref-type="bibr" rid="ref4">(Cabrero and others 2015)</xref>
          2 to 6 players have a
body each that consists of four organs, which can be:
infected (by an opponent playing a virus card), vaccinated (by
a medicine card), immunised (by 2 medicine cards) or
destroyed (by opponents playing 2 consecutive virus cards). The
winner is the first player who forms a healthy and complete
body. Virus! features stochasticity and partial observability,
with the draw pile and opponents’ cards being hidden.
Exploding Kittens
          <xref ref-type="bibr" rid="ref20">(Inman and others 2015)</xref>
          2 to 5
players try to avoid drawing an exploding kitten card while
collecting other useful cards. Each card gives a player access to
unique actions to modify the game-state, e.g. selecting the
player taking a turn next and shuffling the deck. This game
features stochasticity, partial observability and a dynamic turn
order with out-of-turn actions: in contrast to previous games,
Exploding Kittens keeps an action stack so that players have
the chance to react to cards played.
        </p>
        <p>
          Colt Express
          <xref ref-type="bibr" rid="ref31">(Raimbault 2014)</xref>
          2 to 6 players control a
bandit each, with a unique special ability. Their goal is to
collect the most money while traversing the two-level
compartments in a train and avoiding the sheriff (a non-playing
character moved by players and round card events). The game
consists of several rounds, each with a planning (players play
action cards) and an execution (cards are executed in the
same order) phase. This processing scheme forces players
to adapt their strategy according to all the moves already
played, in an interesting case of partial observability and
nondeterminism: the opponents’ type of action may be known
(sometimes completely hidden in a round), but not how it
will be executed. Additionally, the overall strategy should be
adapted to a bandit’s unique abilities.
        </p>
      </sec>
      <sec id="sec-2-4">
        <title>Pandemic (Leacock 2008) Pandemic is a cooperative</title>
        <p>board game for 2 to 4 players. The board represents a world
map, with major cities connected by a graph. Four diseases
break out and the objective of the players is to cure them all.
Diseases keep spreading after each player’s turn, sometimes
leading to outbreaks. Each player is assigned a unique role
with special abilities and is given cards that can be used for
travelling between cities, building research stations or
curing diseases. Additionally, they have access to special event
cards, which can be played anytime (also out-of-turn). All
players lose if they run out of cards in the draw deck, if too
many outbreaks occur or if the diseases spread too much.
Pandemic features partial observability with face-down decks
of cards and asymmetric player roles. It employs a reaction
system to handle event cards and is the only game currently
using the graph-based rule system.
3.4</p>
      </sec>
      <sec id="sec-2-5">
        <title>AI Players</title>
        <p>All implemented players follow a simple interface, only
requiring one method to be implemented: getAction. This
receives a game state object reduced to the specific player’s
observation of the current state of the game. How this
reduced game state is built is game-dependent, usually
randomising unknown information. This method expects an
action to be returned out of those available and is called
whenever it is the player’s turn and they have more than 1
action available (i.e. the player actually has a decision to
make). If no decision is required, the agent can choose to
still receive and process the information on the game state
(in the registerUpdatedObservation function) but
an action is not requested. They may also choose to
implement the initializePlayer and finalizePlayer
functions which are called at the beginning and end of the
game, respectively. Each player has a player ID assigned by
the game engine, and they receive the forward model of the
game currently being played. The FM can then be used to
advance game states given actions, compute actions available,
or reset a game to its initial state. The rest of this section
defines the sample players implemented in the framework.
These agents use the game’s score to evaluate game states (as
implemented on the game side), but their heuristic functions
may be swapped with a different object implementing the
IStateHeuristic interface. Custom heuristics take the
current state as input and return a scalar number representing
the value of that state without any other restrictions. Agents
can be given a heuristic function on initialisation and then
instead of using the reward directly from the game they process
every state they visit using the provided heuristic.
Human Players Two types of human interaction are
available, both of which interrupt the game loop to wait for
human actions on their turn. Console allows human play using
the console. It outputs the game state and available actions
in the console and the player inputs the index of the
action they choose to play. GUI allows human play with a
Graphical User Interface, which is game-specific. It uses
an ActionController object to register player action
requests, which are then executed in the game.</p>
        <p>Random The simplest automatic player chooses random
actions out of those available on its turn.</p>
      </sec>
      <sec id="sec-2-6">
        <title>One Step Look Ahead (OSLA) A greedy exhaustive</title>
        <p>search algorithm, it evaluates all actions from a given game
state and picks that which leads to the highest valued state.</p>
      </sec>
      <sec id="sec-2-7">
        <title>Rolling Horizon Evolutionary Algorithm (RHEA)</title>
        <p>
          RHEA
          <xref ref-type="bibr" rid="ref29">(Perez-Liebana et al. 2013)</xref>
          evolves a sequence of
L = 10 actions over several generations, choosing the first
action of the best sequence found to play in the game. The
algorithm is randomly initialised with a sequence of actions.
At each generation it creates a mutation of the current best
solution, keeping the best solution of the two. This process
repeats until the given budget is exhausted.
        </p>
        <p>Given the variable action spaces and that actions available
are highly dependent on the current game state, the mutation
operator chooses a gene in the individual (i.e. position in the
action sequence) and changes all actions from that point until
the end of the individual to new random valid actions. The
game’s forward model is therefore used in both mutation (to
advance the game state given the last action, in order to find
the available actions for the given position in the sequence)
and evaluation (game states reached through the sequence of
actions are evaluated using the game’s heuristic, added up
for a discounted total with discount factor = 0:9, and this
total becomes the fitness of the individual). It is important to
note that RHEA evolves only its own actions and opponents
are given a random model (with intermediate states after
opponent actions ignored in fitness evaluations).</p>
        <p>
          Monte Carlo Tree Search (MCTS) MCTS
          <xref ref-type="bibr" rid="ref3">(Browne and
others 2012)</xref>
          incrementally builds an asymmetric game tree
balanced towards to most promising parts of the game state
space. It uses multiple iterations of four steps: first, it
navigates through the tree, using a tree policy, until reaching a
node which is not yet fully expanded; next, it adds a new
random child of this node to the tree; it then performs a Monte
Carlo rollout from the new child (randomly sampling actions
until the end of the game or a predetermined depth L = 10);
the state reached at the end of the rollout is evaluated with
a heuristic function, and this score is backed up through all
the nodes visited during the iteration. The process is repeated
until the budget is exhausted, and the most visited child of
the root is chosen as the action to play.
        </p>
        <p>The version implemented in the framework is closed-loop:
it stores game states in each of the nodes. Further, the rollout
step was removed after initial experiments showing an
increased performance without it; therefore, the forward model
of the game is only used when expanding a leaf node. The
resulting node is immediately evaluated using the heuristic
and its value is backed up through the tree.</p>
        <p>4</p>
      </sec>
    </sec>
    <sec id="sec-3">
      <title>Discussion</title>
      <p>This section gives an overview of game analytics which can
be extracted from all games, as well as some preliminary
results for the sample AI players described in Section 3.4.
4.1</p>
      <sec id="sec-3-1">
        <title>Game Analysis</title>
        <p>All games in the framework can be analysed to illustrate the
challenge they provide for AI players, with the following
metrics currently readily available:</p>
        <p>Action space size: the number of actions available for a
player on their turn (e.g. Figure 2). Branching factor: the
number of distinct game states reached through player actions
from a given state. State size: the number of components
in a state. Hidden information: the percentage of
components hidden from players on their turn. Game speed: the
execution speed of 4 key functions (in number of calls per
second): setup, next, available action computation and state
copy. Game length: measured as the number of decisions
taken by AI players, the total number of game loop iterations
(or ticks), the number of rounds in the game and the number
of actions per turn for a player. Reward sparsity: granularity
of the heuristic functions provided by the game, measured by
min, max and standard deviation of rewards seen by players.</p>
        <p>When looking at the games currently implemented, the
first thing to note is that all games are very fast to execute:
most games can execute over 1 million calls per second to the
(usually) most expensive functions (next and copy). The
games vary in length, with only 7:61 ticks for the simplest
game, Tic-Tac-Toe, but 540:78 for Uno. We further see
variations in the state size, with Pandemic showing most complex,
while Uno includes the most hidden information. Love Letter
shows its strategic complexity through the higher branching
factor (10:78), while Exploding Kittens boasts one of the
largest spread of rewards.
We tested the performance of the sample agents on each of
the implemented games. For Tic-Tac-Toe, we report the
winrate per agent when playing 100 times against every possible
opponent. Since Pandemic is a cooperative game, we report
results from games played with a team of 4 instances of the
same agent, e.g. 4 MCTS players. For the remaining games,
we report the average win rate per agent when playing in a
4player match against one instance of all other agents. All but
the random agent are using state evaluation functions that are
provided with each game. For both search-based algorithms,
we use a budget of 4000 calls to the FM.next() function.</p>
        <p>The average win-rate per games is shown in Table 1. Our
results indicate the MCTS agent to be the best, achieving the
highest average win rate in 5 out of 6 competitive games, with
an overall win rate of 45:5% in these games - thus clearly
dominating the other agents. While RHEA also outperforms
random in most of the games (with the exception of Uno), it
still falls behind the OSLA agent. This could be due to the
large uncertainty built up in its rigid sequences of actions (as
opposed to the flexible game trees built by MCTS) in these
games with partial observability and stochasticity, where
a greedy approach appears to be preferable. Further, some
games tested are simple enough that a greedy approach works
best and are highly advantaged by the heuristics provided by
each game (e.g. Tic-Tac-Toe).</p>
        <p>Additionally, we note that no agent is able to win in the
cooperative game Pandemic, as they are unable to perform
the multi-faceted long-term planning required to avoid all of
the loose conditions and win the game. Further analysis, such
as the distance from winning game states for each AI team,
could show interesting insights into these agents’ capabilities.
Moreover, we observe Uno and Colt Express as cases where
the performance between all players, including random, is
very close (22 26% in Uno and 19 29% in Colt Express).
This highlights the difficulty of the types of problems
proposed, as well as the importance of the heuristic chosen for a
game, as some features of a game state may prove deceiving.</p>
        <p>However, the statistical forward planning methods
described here (MCTS and RHEA) benefit from ample
literature with a large parameter space each, which could be tuned
for increased performance.</p>
        <p>5</p>
      </sec>
    </sec>
    <sec id="sec-4">
      <title>Challenges and Opportunities</title>
      <p>
        The presented framework opens up several directions of
research and proposes a variety of challenges for AI players,
be it search/planning or learning algorithms. Its main focus
is to promote research into General Game AI that is able
to play many tabletop games at, or surpassing, human level.
Relatedly, the agents should be able to handle both
competitive (most common testbeds in literature), cooperative
and even mixed games. For instance, a future planned
development is the inclusion of the game Betrayal at House
on the Hill
        <xref ref-type="bibr" rid="ref17">(Glassco and others 2004)</xref>
        , in which the
players start off playing cooperatively to later split into teams
mid-way through the game, from which point on they are
competing instead with newly given team win conditions and
rules. Most tabletop games include some degree of hidden
information (e.g. face-down decks of cards) and many more
players compared to traditional video-game AI testbeds,
introducing higher levels of uncertainty. However, such games
often make use of similar mechanics, even if in different
forms: thus knowledge transfer would be a fruitful area
to explore, so that AI players can pick up new game rules
more easily based on previous experiences, similar to how
humans approach the problem. Some tabletop games
further feature changing rules (e.g. Fluxx
        <xref ref-type="bibr" rid="ref27">(Looney and Looney
1997)</xref>
        ) which would require highly adaptive AI players, able
to handle changes in the game engine itself, not only the
game state. Many others rely on large amounts of content and
components, for which the process of creating new content
or modifying the current one for balance, improved synergies
etc. could be improved with the help of Procedural
Content Generation methods (e.g. cards for the game Magic the
Gathering
        <xref ref-type="bibr" rid="ref14">(Garfield 1993)</xref>
        were previously generated in a
mixed-initiative method by
        <xref ref-type="bibr" rid="ref36">(Summerville and Mateas 2016)</xref>
        ).
      </p>
      <p>Specific types of games can also be targeted by research,
an option highlighted by TAG’s categorisation and labelling
of games and their mechanics. Thus AI players could learn
to specialise in games using certain mechanics or in areas
not yet explored, such as Role-Playing or Campaign games
(i.e. games played over several linked and progressive
sessions). These games often feature asymmetric player roles,
with a special one highlighted (the dungeon master) whose
aim is to control the enemies in the game in order to not
necessarily win, but give the rest of the players the best
experience possible and the right level of challenge. Strategy AI
research could see important applications in this domain, as
many tabletop games include elements of strategic planning.
Role-playing games focused more on the story created by
players (e.g. Dungeons and Dragons) rather than combat
mechanics (e.g. Gloomhaven) would also be a very engaging
and difficult to approach topic for AI players, where Natural
Language Processing research could take an important role.</p>
      <p>The framework enables research into parameter
optimisation: all parameter classes for games, AI players or
heuristics can implement the ITunableParameters interface;
parameters can then be automatically randomised, or more
intelligently tuned by any optimisation algorithm. This
allows for quick and easy exploration of various instances of
a problem, a potential increase in AI player performance, or
adaptation of AI player behaviour to user preference.</p>
      <p>We have mentioned previously that the games
implemented offer reduced observations of the game state to the
AI players, based on what they can currently observe. These
hidden information states (usually) do not keep a history of
what was previously revealed to a player. Instead, the AI
players should learn to memorise relevant information and build
belief systems, as humans would in a real-world context - a
very interesting direction of research encouraged by TAG.</p>
      <p>Lastly, the framework includes the possibility for games
to define their states in terms of either vector observations
(IVectorObservation), which enables learning
algorithms to be easily integrated with the framework; or
featurebased observations (IFeatureRepresentation),
which allows for more complex algorithms which can
perform a search in the feature space of a game, rather than
the usual game state space approached.</p>
      <p>6</p>
    </sec>
    <sec id="sec-5">
      <title>Conclusion</title>
      <p>This paper introduces the new Tabletop Games (TAG)
framework, which aims to promote research into general
Artificial Intelligence with features for easy implementation and
bridge-building between tabletop games and artificial players,
with some examples already included. We further analyse the
games in the framework, showing a wide variety of action
spaces, information available to the AI agents, duration etc.,
as well as tasks and challenges introduced. The AI player
performance analysis shows Monte Carlo Tree Search to
dominate all other sample agents in the framework, with
simple greedy methods being surprisingly competitive in some
of the games. Overall, however, the problems proposed are
far from being solved.</p>
      <p>The framework opens up and facilitates many directions
of research, and yet many more developments are possible
and planned. More measurements for both games and AI
players can be added, to paint a more complete picture of
the challenges the players would face, as well as the current
state of available methods for such games: skill depth, overall
state space size, stochasticity, size of search trees, player role
asymmetry - all would give a much more in-depth view of
the framework as a whole, aiding in future developments of
both tabletop games and general AI players.</p>
      <p>Further, we aim to facilitate interfacing external games
with our framework in order to gain the full benefits of the
indepth analysis and interaction with the implemented players
without the need to re-implement everything from scratch:
this would open up the framework to many more already
existing games, and also increase the number and complexity
of environments the AI players can be exposed to, improving
their quality as well.</p>
    </sec>
    <sec id="sec-6">
      <title>Acknowledgments</title>
      <p>This work was partly funded by the EPSRC CDT in
Intelligent Games and Game Intelligence (IGGI) EP/L015846/1
and EPSRC research grant EP/T008962/1.</p>
    </sec>
  </body>
  <back>
    <ref-list>
      <ref id="ref1">
        <mixed-citation>
          <string-name>
            <surname>Bard</surname>
            ,
            <given-names>N.</given-names>
          </string-name>
          , et al.
          <year>2020</year>
          .
          <article-title>The Hanabi Challenge: A New Frontier for AI Research</article-title>
          .
          <source>Artificial Intelligence</source>
          <volume>280</volume>
          :
          <fpage>103216</fpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref2">
        <mixed-citation>
          2019.
          <article-title>Rinascimento: Optimising Statistical Forward Planning Agents for Playing Splendor</article-title>
          .
          <source>In 2019 IEEE Conference on Games (CoG)</source>
          ,
          <fpage>1</fpage>
          -
          <lpage>8</lpage>
          . IEEE.
        </mixed-citation>
      </ref>
      <ref id="ref3">
        <mixed-citation>
          <string-name>
            <surname>Browne</surname>
            ,
            <given-names>C. B.</given-names>
          </string-name>
          , et al.
          <year>2012</year>
          .
          <article-title>A Survey of Monte Carlo Tree Search Methods</article-title>
          .
          <source>IEEE Transactions on Computational Intelligence and AI in games 4</source>
          (
          <issue>1</issue>
          ):
          <fpage>1</fpage>
          -
          <lpage>43</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref4">
        <mixed-citation>
          <string-name>
            <surname>Cabrero</surname>
            ,
            <given-names>D.</given-names>
          </string-name>
          , et al.
          <year>2015</year>
          . Virus! El Drago´n Azul.
        </mixed-citation>
      </ref>
      <ref id="ref5">
        <mixed-citation>
          <string-name>
            <surname>Campbell</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          , et al.
          <year>2002</year>
          .
          <string-name>
            <given-names>Deep</given-names>
            <surname>Blue</surname>
          </string-name>
          .
          <source>Artificial Intelligence</source>
          <volume>134</volume>
          (
          <issue>1-2</issue>
          ):
          <fpage>57</fpage>
          -
          <lpage>83</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref6">
        <mixed-citation>
          <string-name>
            <surname>Cazenave</surname>
            ,
            <given-names>T.</given-names>
          </string-name>
          , and
          <string-name>
            <surname>Ventos</surname>
            ,
            <given-names>V.</given-names>
          </string-name>
          <year>2019</year>
          .
          <article-title>The alpha-mu Search Algorithm for the Game of Bridge</article-title>
          . arXiv preprint arXiv:
          <year>1911</year>
          .07960.
        </mixed-citation>
      </ref>
      <ref id="ref7">
        <mixed-citation>
          <string-name>
            <surname>Chaco´n</surname>
          </string-name>
          , P. S., and
          <string-name>
            <surname>Eger</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          <year>2019</year>
          .
          <article-title>Pandemic as a Challenge for Human-AI Cooperation</article-title>
          .
          <source>In Proceedings of the AIIDE workshop on Experimental AI in Games.</source>
        </mixed-citation>
      </ref>
      <ref id="ref8">
        <mixed-citation>
          <string-name>
            <surname>de Araujo</surname>
            ,
            <given-names>L. J. P.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Charikova</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Sales</surname>
            ,
            <given-names>J. E.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Smirnov</surname>
            ,
            <given-names>V.</given-names>
          </string-name>
          ; and
          <string-name>
            <surname>Thapaliya</surname>
            ,
            <given-names>A.</given-names>
          </string-name>
          <year>2019</year>
          .
          <article-title>Towards a Game-Independent Model and Data-Structures in Digital Board Games: an Overview of the State-of-the-Art</article-title>
          .
          <source>In Proceedings of the 14th International Conference on the Foundations of Digital Games</source>
          ,
          <fpage>1</fpage>
          -
          <lpage>8</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref9">
        <mixed-citation>
          <string-name>
            <surname>de Mesentier Silva</surname>
            ,
            <given-names>F.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Lee</surname>
            ,
            <given-names>S.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Togelius</surname>
          </string-name>
          , J.; and
          <string-name>
            <surname>Nealen</surname>
            ,
            <given-names>A.</given-names>
          </string-name>
          <year>2017</year>
          .
          <article-title>AI-Based Playtesting of Contemporary Board Games</article-title>
          .
          <source>In Proceedings of the International Conference on the Foundations of Digital Games - FDG'17</source>
          . ACM Press.
        </mixed-citation>
      </ref>
      <ref id="ref10">
        <mixed-citation>
          <string-name>
            <surname>Eger</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          , and
          <string-name>
            <surname>Martens</surname>
            ,
            <given-names>C.</given-names>
          </string-name>
          <year>2019</year>
          .
          <article-title>A Study of AI Agent Commitment in One Night Ultimate Werewolf with Human Players</article-title>
          .
          <source>In Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment</source>
          , volume
          <volume>15</volume>
          ,
          <fpage>139</fpage>
          -
          <lpage>145</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref11">
        <mixed-citation>
          <string-name>
            <surname>Engelstein</surname>
            ,
            <given-names>G.</given-names>
          </string-name>
          , and
          <string-name>
            <surname>Shalev</surname>
            ,
            <given-names>I.</given-names>
          </string-name>
          <year>2019</year>
          .
          <article-title>Building Blocks of Tabletop Game Design: An Encyclopedia of Mechanisms.</article-title>
        </mixed-citation>
      </ref>
      <ref id="ref12">
        <mixed-citation>
          <string-name>
            <given-names>Fantasy</given-names>
            <surname>Flight</surname>
          </string-name>
          <string-name>
            <surname>Publishing</surname>
          </string-name>
          , Inc.
          <year>2012</year>
          .
          <article-title>Descent: Journeys in the Dark 2nd Edition</article-title>
          . Diamond Comic Distributors.
        </mixed-citation>
      </ref>
      <ref id="ref13">
        <mixed-citation>
          <string-name>
            <surname>Gaina</surname>
            , R. D.; Balla,
            <given-names>M.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Dockhorn</surname>
            ,
            <given-names>A.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Montoliu</surname>
          </string-name>
          , R.; and
          <string-name>
            <surname>Perez-Liebana</surname>
            ,
            <given-names>D.</given-names>
          </string-name>
          <year>2020</year>
          .
          <article-title>Design and Implementation of TAG: a Tabletop Games Framework</article-title>
          . arXiv preprint arXiv:
          <year>2009</year>
          .12065.
        </mixed-citation>
      </ref>
      <ref id="ref14">
        <mixed-citation>
          <string-name>
            <surname>Garfield</surname>
            ,
            <given-names>R.</given-names>
          </string-name>
          <year>1993</year>
          .
          <article-title>Magic: The Gathering</article-title>
          .
          <article-title>Wizards of the Coast</article-title>
          .
        </mixed-citation>
      </ref>
      <ref id="ref15">
        <mixed-citation>
          <string-name>
            <surname>Genesereth</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Love</surname>
          </string-name>
          , N.; and
          <string-name>
            <surname>Pell</surname>
            ,
            <given-names>B.</given-names>
          </string-name>
          <year>2005</year>
          .
          <article-title>General Game Playing: Overview of the AAAI Competition</article-title>
          .
          <source>AI</source>
          magazine
          <volume>26</volume>
          (
          <issue>2</issue>
          ):
          <fpage>62</fpage>
          -
          <lpage>62</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref16">
        <mixed-citation>
          <string-name>
            <surname>Gibson</surname>
            ,
            <given-names>R.</given-names>
          </string-name>
          ; Desai, N.; and
          <string-name>
            <surname>Zhao</surname>
            ,
            <given-names>R.</given-names>
          </string-name>
          <year>2010</year>
          .
          <article-title>An automated technique for drafting territories in the board game Risk</article-title>
          .
          <source>In Sixth Artificial Intelligence and Interactive Digital Entertainment Conference.</source>
        </mixed-citation>
      </ref>
      <ref id="ref17">
        <mixed-citation>
          <string-name>
            <surname>Glassco</surname>
            ,
            <given-names>B.</given-names>
          </string-name>
          , et al.
          <year>2004</year>
          .
          <article-title>Betrayal at House on the Hill</article-title>
          .
        </mixed-citation>
      </ref>
      <ref id="ref18">
        <mixed-citation>
          <string-name>
            <surname>Guzdial</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Acharya</surname>
            ,
            <given-names>D.</given-names>
          </string-name>
          ; Kreminski,
          <string-name>
            <given-names>M.</given-names>
            ;
            <surname>Cook</surname>
          </string-name>
          ,
          <string-name>
            <given-names>M.</given-names>
            ;
            <surname>Eladhari</surname>
          </string-name>
          ,
          <string-name>
            <given-names>M.</given-names>
            ;
            <surname>Liapis</surname>
          </string-name>
          ,
          <string-name>
            <given-names>A.</given-names>
            ; and
            <surname>Sullivan</surname>
          </string-name>
          ,
          <string-name>
            <surname>A.</surname>
          </string-name>
          <year>2020</year>
          .
          <article-title>Tabletop Roleplaying Games as Procedural Content Generators</article-title>
          .
        </mixed-citation>
      </ref>
      <ref id="ref19">
        <mixed-citation>
          <string-name>
            <surname>Henry</surname>
            ,
            <given-names>J.</given-names>
          </string-name>
          <year>2015</year>
          .
          <string-name>
            <given-names>Tabletop</given-names>
            <surname>Simulator</surname>
          </string-name>
          . Berserk Games.
        </mixed-citation>
      </ref>
      <ref id="ref20">
        <mixed-citation>
          <string-name>
            <surname>Inman</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          , et al.
          <year>2015</year>
          .
          <string-name>
            <given-names>Exploding</given-names>
            <surname>Kittens</surname>
          </string-name>
          . Ad Magic, Inc.
        </mixed-citation>
      </ref>
      <ref id="ref21">
        <mixed-citation>
          <string-name>
            <surname>Justesen</surname>
            ,
            <given-names>N.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Uth</surname>
            ,
            <given-names>L. M.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Jakobsen</surname>
            ,
            <given-names>C.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Moore</surname>
            ,
            <given-names>P. D.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Togelius</surname>
          </string-name>
          , J.; and
          <string-name>
            <surname>Risi</surname>
            ,
            <given-names>S.</given-names>
          </string-name>
          <year>2019</year>
          .
          <article-title>Blood Bowl: A New Board Game Challenge And Competition For AI</article-title>
          .
          <source>In 2019 IEEE Conference on Games (CoG)</source>
          ,
          <fpage>1</fpage>
          -
          <lpage>8</lpage>
          . IEEE.
        </mixed-citation>
      </ref>
      <ref id="ref22">
        <mixed-citation>
          <string-name>
            <surname>Kanai</surname>
            ,
            <given-names>S.</given-names>
          </string-name>
          <year>2012</year>
          .
          <string-name>
            <given-names>Love</given-names>
            <surname>Letter</surname>
          </string-name>
          . Alderac Entertainment Group.
        </mixed-citation>
      </ref>
      <ref id="ref23">
        <mixed-citation>
          <string-name>
            <surname>Kowalski</surname>
            ,
            <given-names>J.</given-names>
          </string-name>
          ; Mika,
          <string-name>
            <given-names>M.</given-names>
            ;
            <surname>Sutowicz</surname>
          </string-name>
          , J.; and Szykuła,
          <string-name>
            <surname>M.</surname>
          </string-name>
          <year>2019</year>
          .
        </mixed-citation>
      </ref>
      <ref id="ref24">
        <mixed-citation>
          <string-name>
            <given-names>Regular</given-names>
            <surname>Boardgames</surname>
          </string-name>
          .
          <source>In Proceedings of the AAAI Conference on Artificial Intelligence</source>
          , volume
          <volume>33</volume>
          ,
          <fpage>1699</fpage>
          -
          <lpage>1706</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref25">
        <mixed-citation>
          <string-name>
            <surname>Lanctot</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          , et al.
          <year>2019</year>
          .
          <article-title>OpenSpiel: A Framework for Reinforcement Learning in Games</article-title>
          . CoRR abs/
          <year>1908</year>
          .09453.
        </mixed-citation>
      </ref>
      <ref id="ref26">
        <mixed-citation>
          <string-name>
            <surname>Leacock</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          <year>2008</year>
          .
          <string-name>
            <surname>Pandemic. Z-Man</surname>
            <given-names>Games</given-names>
          </string-name>
          , Inc.
        </mixed-citation>
      </ref>
      <ref id="ref27">
        <mixed-citation>
          <string-name>
            <surname>Looney</surname>
            ,
            <given-names>A.</given-names>
          </string-name>
          , and
          <string-name>
            <surname>Looney</surname>
            ,
            <given-names>K.</given-names>
          </string-name>
          <year>1997</year>
          . Fluxx. Looney Labs.
        </mixed-citation>
      </ref>
      <ref id="ref28">
        <mixed-citation>
          <article-title>Moravcˇ´ık</article-title>
          ,
          <string-name>
            <surname>M.</surname>
          </string-name>
          , et al.
          <year>2017</year>
          .
          <article-title>Deepstack: Expert-Level Artificial Intelligence in Heads-Up No-Limit Poker</article-title>
          .
          <source>Science</source>
          <volume>356</volume>
          (
          <issue>6337</issue>
          ):
          <fpage>508</fpage>
          -
          <lpage>513</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref29">
        <mixed-citation>
          <string-name>
            <surname>Perez-Liebana</surname>
            ,
            <given-names>D.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Samothrakis</surname>
            ,
            <given-names>S.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Lucas</surname>
            ,
            <given-names>S. M.</given-names>
          </string-name>
          ; and Rolfshagen,
          <string-name>
            <surname>P.</surname>
          </string-name>
          <year>2013</year>
          .
          <article-title>Rolling Horizon Evolution versus Tree Search for Navigation in Single-Player Real-Time Games</article-title>
          .
          <source>In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO)</source>
          ,
          <fpage>351</fpage>
          -
          <lpage>358</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref30">
        <mixed-citation>
          <string-name>
            <surname>Piette</surname>
            ,
            <given-names>E.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Soemers</surname>
            ,
            <given-names>D. J.</given-names>
          </string-name>
          ; Stephenson,
          <string-name>
            <given-names>M.</given-names>
            ;
            <surname>Sironi</surname>
          </string-name>
          ,
          <string-name>
            <given-names>C. F.</given-names>
            ;
            <surname>Winands</surname>
          </string-name>
          ,
          <string-name>
            <given-names>M. H.</given-names>
            ; and
            <surname>Browne</surname>
          </string-name>
          ,
          <string-name>
            <surname>C.</surname>
          </string-name>
          <year>2019</year>
          .
          <article-title>Ludii-The Ludemic General Game System</article-title>
          . arXiv preprint arXiv:
          <year>1905</year>
          .05013.
        </mixed-citation>
      </ref>
      <ref id="ref31">
        <mixed-citation>
          <string-name>
            <surname>Raimbault</surname>
            ,
            <given-names>C.</given-names>
          </string-name>
          <year>2014</year>
          .
          <string-name>
            <given-names>Colt</given-names>
            <surname>Express</surname>
          </string-name>
          . Ludonaute.
        </mixed-citation>
      </ref>
      <ref id="ref32">
        <mixed-citation>
          <string-name>
            <surname>Robbins</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          <year>1971</year>
          . Uno. AMIGO.
        </mixed-citation>
      </ref>
      <ref id="ref33">
        <mixed-citation>
          <string-name>
            <surname>Serrino</surname>
            ,
            <given-names>J.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Kleiman-Weiner</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Parkes</surname>
            ,
            <given-names>D. C.</given-names>
          </string-name>
          ; and
          <string-name>
            <surname>Tenenbaum</surname>
            ,
            <given-names>J.</given-names>
          </string-name>
          <year>2019</year>
          .
          <article-title>Finding Friend and Foe in Multi-Agent Games</article-title>
          .
          <source>In Advances in Neural Information Processing Systems</source>
          ,
          <volume>1251</volume>
          -
          <fpage>1261</fpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref34">
        <mixed-citation>
          <string-name>
            <surname>Sfikas</surname>
            ,
            <given-names>K.</given-names>
          </string-name>
          , and
          <string-name>
            <surname>Liapis</surname>
            ,
            <given-names>A.</given-names>
          </string-name>
          <year>2020</year>
          .
          <article-title>Collaborative Agent Gameplay in the Pandemic Board Game</article-title>
          .
          <source>In International Conference on the Foundations of Digital Games (FDG).</source>
        </mixed-citation>
      </ref>
      <ref id="ref35">
        <mixed-citation>
          <string-name>
            <surname>Silver</surname>
            ,
            <given-names>D.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Schrittwieser</surname>
            ,
            <given-names>J.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Simonyan</surname>
            ,
            <given-names>K.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Antonoglou</surname>
            ,
            <given-names>I.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Huang</surname>
            ,
            <given-names>A.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Guez</surname>
            ,
            <given-names>A.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Hubert</surname>
            ,
            <given-names>T.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Baker</surname>
            ,
            <given-names>L.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Lai</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          ;
          <string-name>
            <surname>Bolton</surname>
            ,
            <given-names>A.</given-names>
          </string-name>
          ; et al.
          <year>2017</year>
          .
          <article-title>Mastering the Game of Go without Human Knowledge</article-title>
          . nature
          <volume>550</volume>
          (
          <issue>7676</issue>
          ):
          <fpage>354</fpage>
          -
          <lpage>359</lpage>
          .
        </mixed-citation>
      </ref>
      <ref id="ref36">
        <mixed-citation>
          <string-name>
            <surname>Summerville</surname>
            ,
            <given-names>A. J.</given-names>
          </string-name>
          , and
          <string-name>
            <surname>Mateas</surname>
            ,
            <given-names>M.</given-names>
          </string-name>
          <year>2016</year>
          .
          <article-title>Mystical Tutor: A Magic: The Gathering Design Assistant Via Denoising Sequence-To-Sequence Learning</article-title>
          .
          <source>In Twelfth artificial intelligence and interactive digital entertainment conference.</source>
        </mixed-citation>
      </ref>
      <ref id="ref37">
        <mixed-citation>
          <string-name>
            <surname>Szita</surname>
            ,
            <given-names>I.</given-names>
          </string-name>
          ; Chaslot, G.; and Spronck,
          <string-name>
            <surname>P.</surname>
          </string-name>
          <year>2009</year>
          .
          <article-title>Monte-Carlo Tree Search in Settlers of Catan</article-title>
          .
          <source>In Advances in Computer Games</source>
          ,
          <fpage>21</fpage>
          -
          <lpage>32</lpage>
          . Springer.
        </mixed-citation>
      </ref>
      <ref id="ref38">
        <mixed-citation>
          <string-name>
            <surname>Teuber</surname>
            ,
            <given-names>K.</given-names>
          </string-name>
          <year>1995</year>
          .
          <article-title>The Settlers of Catan</article-title>
          . Mayfair Games.
        </mixed-citation>
      </ref>
      <ref id="ref39">
        <mixed-citation>
          <string-name>
            <surname>Wrede</surname>
            ,
            <given-names>K.-J.</given-names>
          </string-name>
          <year>2000</year>
          . Carcassonne. Hans im Glu¨ck.
        </mixed-citation>
      </ref>
    </ref-list>
  </back>
</article>