The Hanabi Challenge: A New Frontier for AI Research

02/01/2019 ∙ by Nolan Bard, et al. ∙ 54

From the early days of computing, games have been important testbeds for studying how well machines can do sophisticated decision making. In recent years, machine learning has made dramatic advances with artificial agents reaching superhuman performance in challenge domains like Go, Atari, and some variants of poker. As with their predecessors of chess, checkers, and backgammon, these game domains have driven research by providing sophisticated yet well-defined challenges for artificial intelligence practitioners. We continue this tradition by proposing the game of Hanabi as a new challenge domain with novel problems that arise from its combination of purely cooperative gameplay and imperfect information in a two to five player setting. In particular, we argue that Hanabi elevates reasoning about the beliefs and intentions of other agents to the foreground. We believe developing novel techniques capable of imbuing artificial agents with such theory of mind will not only be crucial for their success in Hanabi, but also in broader collaborative efforts, and especially those with human partners. To facilitate future research, we introduce the open-source Hanabi Learning Environment, propose an experimental framework for the research community to evaluate algorithmic advances, and assess the performance of current state-of-the-art techniques.

READ FULL TEXT VIEW PDF

Authors

page 21

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

Throughout human societies, people engage in a wide range of activities with a diversity of other people. These multi-agent interactions are integral to everything from mundane daily tasks, like commuting to work, to operating the organizations that underpin modern life, such as governments and economic markets. With such complex multi-agent interactions playing a pivotal role in human lives, it is essential for artificially intelligent agents to be capable of cooperating effectively with other agents, particularly humans.

Multi-agent environments present unique challenges relative to those with a single agent. In particular, the ideal behaviour for an agent typically depends on how the other agents act. Thus, for an agent to maximize its utility in such a setting, it must consider how the other agents will behave, and respond appropriately. Other agents typically are the most complex part of the environment: their policies are commonly stochastic, dynamically changing, or dependent on hidden information that is not observed by everyone. Furthermore, agents generally need to interact while only having a limited time to observe others.

While these issues have typically made inferring the behaviour of others a daunting challenge for AI practitioners, humans routinely make such inferences in their social interactions using theory of mind  (PremackWoodruff78, ; Rabinowitz18, ): reasoning about others as agents with their own mental states – such as perspectives, beliefs, and intentions – to explain and predict their behaviour. 141414Dennett uses the phrase intentional stance to refer to this “strategy” for explanation and prediction Dennett87 . Alternatively, one can think of theory of mind as the human ability to imagine the world from another person’s point of view. For example, a simple real-world use of theory of mind can be observed when a pedestrian crosses a busy street. Once some traffic has stopped, a driver approaching the stopped cars may not be able to directly observe the pedestrian. However, they can reason about why the other drivers have stopped, and infer that a pedestrian is crossing.

In this work, we examine the popular card game Hanabi, and argue for it as a new research frontier that, at its very core, presents the kind of multi-agent challenges that humans solve using theory of mind. Hanabi won the prestigious Spiel des Jahres award in 2013 and enjoys an active community, including a number of sites that allow for online gameplay (BoardGameArena, ; keldon, ). Hanabi is a cooperative game of imperfect information for two to five players, best described as a type of team solitaire. The game’s imperfect information arises from each player being unable to see their own cards (i.e. the ones they hold and can act on), each of which has a color and rank. To succeed, players must coordinate to efficiently reveal information to their teammates, however players can only communicate though grounded hint actions that point out all of a player’s cards of a chosen rank or colour. Importantly, performing a hint action consumes the limited resource of information tokens, making it impossible to fully resolve each player’s uncertainty about the cards they hold based on this grounded information alone. To overcome this limitation, successful play involves communicating extra information implicitly through the choice of actions themselves, which are observable by all players. In Section 2 we provide a more detailed description of the game and what human gameplay looks like.

Hanabi is different from the adversarial two-player zero-sum games where computers have reached super-human skill, e.g., chess (campbell2002deep, ), checkers (schaeffer1996chinook, ), go (silver16mastering, ), backgammon (tesauro95temporal, ) and two-player poker (Moravcik17DeepStack, ; Brown17Libratus, ). In those games, agents typically compute an equilibrium policy (or equivalently, a strategy) such that no single player can improve their utility by deviating from the equilibrium. While two-player zero-sum games can have multiple equilibria, different equilibria are interchangeable: each player can play their part of different equilibrium profiles without impacting their utility. As a result, agents can achieve a meaningful worst-case performance guarantee in these domains by finding any equilibrium policy. However, since Hanabi is neither (exclusively) two-player nor zero-sum, the value of an agent’s policy depends critically on the policies used by its teammates. Even if all players manage to play according to the same equilibrium, there can be multiple locally optimal equilibria that are relatively inferior. For algorithms that iteratively train independent agents, such as those commonly used in the multi-agent reinforcement learning literature, these inferior equilibria can be particularly difficult to escape.

The presence of imperfect information in Hanabi creates another challenging dimension of complexity for AI algorithms. As has been observed in domains like poker, imperfect information entangles how an agent should behave across multiple observed states Burch14CFRD ; Burch17PhD . In Hanabi, this manifests in the communication protocol 151515In pure signalling games where actions are purely communicative, policies are often referred to as communication protocols. Though Hanabi is not such a pure signalling game, when we want to emphasize the communication properties of an agent’s policy we will still refer to its communication protocol. We will use the word convention to refer to the parts of a communication protocol or policy that interrelate, such as the situations when a signal is used and how a teammate responds to that signal. Technically, these can be thought of as constraints on the policy to enact the convention. Examples of human conventions in Hanabi will be discussed in Section 2. between players, where the efficacy of any given protocol depends on the entire scheme rather than how players communicate in a particular observed situation. That is, how the other players will respond to a chosen signal will depend upon what other situations use the same signal. Due to this entanglement, the type of single-action exploration techniques common in reinforcement learning (e.g., -greedy, entropy regularization) can incorrectly evaluate the utility of such exploration steps as they ignore their holistic impact.

Hanabi’s cooperative nature also complicates the question of what kind of policy practitioners should seek. One challenge is to discover a policy for the entire team that has high utility. Most of the prior research on Hanabi has focused on this challenge, which we refer to as the self-play setting. However, it is impractical for human Hanabi players to expect others to strictly abide by a preordained policy. Not only would it be hard to convey and memorize nuanced policies, but humans also routinely play with ad-hoc teams that may have players of different skill levels. Even without agreeing on comprehensive conventions, humans are still able to play successfully in this ad-hoc setting. Unfortunately, unlike the aforementioned two-player zero-sum setting, good policies for self-play have no meaningful worst-case performance guarantee in the ad-hoc team setting.

Humans approach Hanabi differently than current learning algorithms. Instead of searching for conventions in advance to produce complete policies, humans seem to establish conventions to search for their next action, relying on others to reason about their intent and view of the game to infer salient signals. These basic conventions for search tend to be relatively intuitive: ensure the next player has enough information to make a useful move or has an information token available so they can provide a hint. Humans combine these basic conventions for search with pragmatic reasoning, enabling more effective use of grounded messages (e.g. implicitly signalling a card is playable). Pragmatic reasoning thus represents an implicit convention for how the vocabulary of the game’s grounded messages should be used and interpreted when referring to a specific card. Humans also rely on theory of mind for Hanabi’s more advanced “finesse” plays: these plays often appear irrational on the surface, which makes them vulnerable to misinterpretation, in order to cue players to infer additional information by reasoning about what the acting player could have observed that would make such a move rational. Relying on simple conventions for search, pragmatic reasoning, and theory of mind makes it possible for humans to join an ad-hoc team where complete policies are not agreed upon and memorized in advance. This makes theory of mind central to Hanabi’s gameplay, particularly in such an ad-hoc team setting.

The combination of cooperation, imperfect information, and limited communication make Hanabi ideal for investigating the importance of incorporating theory of mind into AI agents. Whether it is for efficiency of communication or for artifical agents to better understand and collaborate with human partners, machine learning practitioners will need new techniques for designing agents capable of reasoning about others. In Section 2 we describe the details of the game and how humans approach it. In Section 3 we elaborate on the challenge it poses for artificial intelligence and why theory of mind is expected to be critical for effective play. We facilitate agent implementation and experimentation by providing an open source code framework (Section 4.1). Furthermore, to promote consistent and detailed empirical evaluation in future research, we propose an evaluation benchmark for both the self-play (Section 4.2) and ad-hoc (Section 4.3) team settings. We evaluate the performance of current state-of-the-art reinforcement learning methods in Section 5. Our results show that although these learning techniques can achieve reasonable performance in self-play, they generally fall short of the best known hand-coded agents (Section 5.3). Moreover, we show that these techniques tend to learn extremely brittle policies that are unreliable for ad-hoc teams (Section 5.4). These results suggest that there is still substantial room for technical advancements in both the self-play and ad-hoc settings, especially as the number of players increases. Finally, we highlight connections to prior work in Section 6.

2 Hanabi: The Game

Hanabi is a game for two to five players, best described as a type of cooperative solitaire. Each player holds a hand of four cards (or five, when playing with two or three players). Each card depicts a rank (1 to 5) and a colour (red, green, blue, yellow, and white); the deck (set of all cards) is composed of a total of 50 cards, 10 of each colour: three 1s, two 2s, 3s, and 4s, and finally a single 5. The goal of the game is to play cards so as to form five consecutively ordered stacks, one for each colour, beginning with a card of rank 1 and ending with a card of rank 5. What makes Hanabi special is that, unlike most card games, players can only see their partners’ hands, and not their own.

Figure 1: Example of a four player Hanabi game from the point of view of player 0. Player 1 acts after player 0 and so on.

In Hanabi players take turns doing one of three actions: giving a hint, playing a card from their hand, or discarding a card. We call the player whose turn it is the active player.

Hints. On their turn, the active player can give a hint to any other player. A hint consists of choosing a rank or colour, and indicating to another player all of their cards which match the given rank or colour. Only ranks and colors that are present in the hand of the player can be hinted for. For example, in Figure 1, the active player may tell Player 2, “Your first and third cards are red.” or “Your fourth card is a 3.” To make the game interesting, hints are in limited supply. The game begins with the group owning eight information tokens, one of which is consumed every time a hint is given. If no information tokens remain, hints cannot be given and the player must instead play or discard.

Discard. Whenever fewer than eight information tokens remain, the active player can discard a card from their hand. The discarded card is placed face up (along with any unsuccessfully played cards), visible to all players. Discarding has two effects: the player draws a new card from the deck and an information token is recovered.

Play. Finally, the active player may pick a card (known or unknown) from their hand and attempt to play it. Playing a card is successful if the card is the next in the sequence of its colour to be played. For example, in Figure 1 Player 2’s action would be successful if they play their yellow 3 or their blue 1; in the latter case forming the beginning of the blue stack. If the play is successful, the card is placed on top of the corresponding stack. When a stack is completed (the 5 is played) the players also receive a new information token (if they have fewer than eight). The player can play a card even if they know nothing about it; but if the play is unsuccessful, the card is discarded (without yielding an information token) and the group loses one life, possibly ending the game. In either circumstances, a new card is drawn from the deck.

Game Over. The game ends in one of three ways: either because the group has successfully played cards to complete all five stacks, when three lives have been lost, or after a player draws the last card from the deck and every player has taken one final turn. If the game ends before three lives are lost, the group scores one point for each card in each stack, for a maximum of 25; otherwise, the score is 0.

2.1 Basic Strategy

There are too few information tokens to provide complete information (i.e., the rank and colour) for each of the 25 cards that can be played through only the grounded information revealed by hints161616Since the deck contains 50 cards, 25 of which players aim to play, and when the deck runs out the players will always hold at least 10 cards that cannot recover usable information tokens, this means at most 15 cards can be discarded to recover information tokens. Combined with the eight initial information tokens, and the four recovered through completing colour stacks, this means players can hint at most 27 times during the game (and fewer times as the number of players increases).. While the quantity of information provided by a hint can be improved by revealing information about multiple cards at once, the value of information in Hanabi is very context dependent. To maximize the team’s score at the end of the game, hints need to be selected based on more than just the quantity of information conveyed. For example in Figure 1, telling Player 3 that they hold four blue cards reveals more information than telling Player 2 that they hold a single rank-1 card, but lower-ranked cards are more important early on, as they can be played immediately. A typical game therefore begins by hinting to players which cards are 1s, after which those players play those cards; this both “unlocks” the ability to play the same-colour 2s and makes the remaining 1s of that colour useful for recovering information tokens as players can discard the redundant cards.

Players are incentivized to avoid unsuccessful plays in two ways: first, losing all three lives results in the game immediately ending with zero points; second, the card itself is discarded. Generally speaking, discarding all cards of a given rank and colour is a bad outcome, as it reduces the maximum achievable score. For example, in Figure 1 both green 2s have been discarded, an effective loss of four points as no higher rank green cards will ever be playable. As a result, hinting to players that are at risk of discarding the only remaining card of a given rank and colour is often prioritized. This is particularly common for rank-5 cards since there is only one of each colour and they often need to be held for a long time before the card can successfully be played.

2.2 Implicit Communication

While explicit communication in Hanabi is limited to the hint actions, every action taken in Hanabi is observed by all players and can also implicitly communicate information. This implicit information is not conveyed through the impact that an action has on the environment (i.e., what happens) but through the very fact that another player decided to take this action (i.e., why it happened). This requires that players can reason over the actions that another player would have taken in a number of different situations, essentially reasoning over the intent of the agent. Human players often exploit such reasoning to convey more information through their actions. Consider the situation in Figure 1 and assume the active player (Player 0) knows nothing about their own cards, and so they choose to hint to another player. One option would be tell Player 1 about the 1s in their hand. However, that information is not particularly actionable, as the yellow 1 is not currently playable. Instead, they could tell Player 1 about the red card, which is a 1. Although Player 1 would not explicitly know the card is a 1, and therefore playable, they could infer that it is playable as there would be little reason to tell them about it otherwise, especially when Player 2 has a blue 1 that would be beneficial to hint. They may also infer that because Player 0 chose to hint with the colour rather than the rank, that one of their other cards is a non-playable 1. This is an example of the type of pragmatic reasoning that humans commonly use.

An even more effective, though also more sophisticated, tactic commonly employed by humans is the so-called “finesse” move. To perform the finesse in this situation, Player 0 would tell Player 2 that they have a 2. By the same pragmatic reasoning as above, Player 2 could falsely infer that their red 2 is the playable white 2 (since both green 2s were already discarded). Player 1 can see Player 2’s red 2 and realize that Player 2 will make this incorrect inference and mistakenly play the card, leading Player 1 to question why Player 0 would have chosen this seemingly irrational hint. The only rational explanation for the choice is that Player 1 themselves must hold the red 1 (in a predictable position, such as the most recently drawn card) and is expected to rescue the play. Using this tactic, Player 0 can reveal enough information to get two cards played using only a single information token. There are many other moves that rely on this kind of reasoning about intent to convey information (e.g., bluff, reverse finesse) ZamiellHanabiConventions . We will use finesse to broadly refer to this style of move.

3 Hanabi: A Challenge for Communication and Theory of Mind

Due to Hanabi’s limited communication, players cannot rely solely on the grounded hints to perform well: players, including artificial agents, will need to convey extra information through their actions. We argue this makes Hanabi a novel challenge for artificial intelligence practitioners that requires algorithmic advancements capable of learning to communicate and exploiting the theory of mind reasoning commonly employed by humans. This challenge connects research from several communities, including reinforcement learning, game theory, and emergent communication. Section 6 has additional discussion of related work from these communities.

One avenue to approach this problem is to establish conventions prior to play. Abiding by such conventions enables agents to communicate information about their observations (i.e., their perspective) beyond the grounded information contained in hints. To illustrate, one convention (possibly a poor one) could be that players never hint about a card unless they observe that it is playable. It then follows that if an agent receives a hint about one of their cards, they know that card is playable. Note that conventions can communicate information either with or without a direct signal: if players expect some behaviour in certain situations, then the absence of that behaviour indicates they are not in one of these situations. For example, if players expect to be warned when they are at risk of discarding a valuable card, like a 5, a lack of warning would indicate they are not at risk. Conventions can also be used to relay more detailed information through hints by encoding it into a player’s choice of colour or rank hint and the target of their hint. In fact, this idea forms the basis of Cox and colleagues’ Hanabi agents (Cox15, ), which we discuss in Section 5.2.

Conventions are obviously useful in the self-play setting, where all players know others will act exactly as they would. Although experienced human players develop and exploit conventions ZamiellHanabiConventions , humans are clearly able to play effectively outside of the self-play setting: human teams consist of distinct individuals, each with their own capabilities, and are often formed on the fly at online websites BoardGameArena or social events. Notably, humans collaborate successfully without relying on prior coordination or knowing the policy of their teammates, commonly referred to as the ad-hoc team setting (05aaai-pickup, ; Stone10AdHocTeamwork, ). Consequently, for artificial agents to achieve truly superhuman performance in Hanabi’s collaborative setting, they will need to master both settings.

3.1 A Challenge for Self-Play Learning

One way to learn conventions is to optimise a joint policy to maximize reward in self-play. This “search over conventions” will naturally develop conventions to overcome Hanabi’s information scarcity. However, such an optimisation is difficult in its own right 171717

Hanabi is an instance of a decentralized Markov decision process (DEC-MDP) since the players jointly observe the full state of the game. Bernstein and colleagues 

(BernsteinGIZ02:DEC-MDP, ) showed that solving DEC-MDPs is in the nondeterministic exponential time (NEXP) complexity class, i.e., requiring exponential time even if P=NP. Furthermore, Baffier and colleagues show that even a single player version of Hanabi where the player has perfect information of all cards (including the order of cards in the deck) is NP-complete, despite removing any need to communicate (Baffier17, )., as Hanabi’s policy space is rife with locally optimal equilibria. One such equilibrium is the babbling equilibrium where players do not intentionally communicate information to other players, and ignore what other players tell them. In that case, there is no incentive for a player to start communicating because they will be ignored, and there is no incentive to pay attention to other players because they are not communicating.

For learning algorithms to improve over such inferior equilibria, players will need to jointly deviate, potentially changing established (suboptimal) conventions. However, to change conventions a learning algorithm needs to change behaviour across many decisions points simultaneously, since what can be inferred from a signal depends on all of the situations in which that signal is given. Additionally, the receiver of the signal has to simultaneously change how they respond for the overall deviation to result in a performance improvement. Previous multi-agent research has explored “cheap talk” channels as a mechanism to communicate directly to other agents (Foerster16RIAL, ; Lewis17, ; Cao18, ), however Hanabi’s restricted communication structure prevents such cheap talk. Every signal affects performance not just through what is communicated but also by its affect on the game state itself, which creates a further challenge for learning algorithms. Section 5.3 will present experimental results on how current state of the art algorithms fair on this challenge.

3.2 The Human Approach

Human players appear to approach Hanabi differently: using “conventions over search” that guide how they choose their immediate action and how they expect other players will interpret their action. Part of these “conventions over search” come from how humans use theory of mind (ToM) to ascribe mental states to agents, such as perspectives, beliefs, and intentions. Humans rely on theory of mind not only to reason about the potential mental states that could cause an agent’s behaviour, but also in forming expectations for how other agents will interpret their behaviour using the same reasoning.

Importantly, this type of theory of mind reasoning is central to not just how humans approach Hanabi, but more generally to how humans handle communication and multi-agent interactions. For example, when communicating with natural language, humans exploit theory of mind through their use of pragmatics: conveying meaning based on not only what was literally said, but also what is implicated (i.e., suggested) by the speaker based on the context grice1975:logic . By relying on a listener to disambiguate a speaker’s meaning through mechanisms like pragmatic reasoning, communication can be more efficient PIANTADOSI2012280 . Notably, computational models of human pragmatic reasoning depend on the assumption that speakers are being cooperative and have the intention of communicating useful information Frank998 .

Analogously, when humans play Hanabi, players rely on the same mutual assumption of communicative intent and the acting player (the “speaker”) expects other players (the “listeners”) to infer the information implicated by their specific choice of action and the context in which it was taken. Relying on such inference enables players to efficiently signal about the hidden state of players’ cards (as in the example of humans hinting about playable cards) despite typically not having explicit conventions about what an agent’s actions should communicate, let alone having access to their policy.

Theory of mind and the assumption that other agents have communicative intent provide useful mechanisms for making communication more efficient and form the basis for the “conventions over search” employed by humans. While this assumption imposes conventions in a manner analogous to those learned by the “search over conventions” approach described for self-play, they are naturally limited to conventions that can be inferred with communicative intent. As a result, humans can interpret sophisticated plays without explicit agreement because theory of mind makes them “self-explaining” instead of relying on nuanced information theoretic encoding schemes. For instance, the finesse described in Section 2 is possible without established agreement because players assume the hint revealing the initial two was intended to communicate useful information. These self-explaining conventions suggest a more efficient and possibly easier-to-learn approach to the self-play learning challenge.

3.3 A Challenge for Ad-hoc Teams

The human approach suggests a second challenge presented by Hanabi — ad-hoc teams — where agents must collaborate with an arbitrary team, possibly an established team with existing conventions. This ad-hoc team setting is important, especially for exploring collaboration with humans, since it is unreasonable for an agent to impose its policy on others or for human players to memorise entire policies before playing. In the most extreme case for ad-hoc play, an agent would have no knowledge of the other players’ conventions or policy to rely on and they must attempt to infer them on the fly. The less extreme case allows an agent to observe a small number of examples of the team playing together, before playing with the team, giving the agent an opportunity to first infer intent from examples of collaboration.

Self-play training allows agents to expect their teammates share the exact same policy and conventions. Indeed, as we demonstrate in Section 5.4, the resulting policies of state of the art algorithms rely heavily on this fact. As a result they often perform poorly in the ad-hoc team setting. Human Hanabi players provide inspiration for the ad-hoc team setting as well. In particular, the use of theory of mind and reasoning about other players as agents with communicative intent may lead to artificial agents that are capable of playing with other intentional agents as well as collaborating with humans by communicating with them on their terms.

Both the self-play and ad-hoc settings present difficult challenges for current artificial intelligence techniques that will require long term algorithmic innovation. In the next section, we introduce an open-source learning environment to facilitate research and propose an experimental framework for the research community to evaluate algorithmic advances in both settings.

4 Hanabi: The Benchmark

We propose using Hanabi as a challenging benchmark problem for AI, with a set of unique properties, that distinguish it from other benchmarks. It is a multi-agent learning problem, unlike, for example, the Arcade Learning Environment (bellemare13arcade, ). It is also an imperfect information game, where players have asymmetric knowledge about the environment state, which makes the game more like poker than chess, backgammon, or Go. The cooperative goal of Hanabi sets it further apart from all of these other challenge problems, which have players competing against each other. As mentioned above, this combination of partial observability and cooperative rewards creates unique challenges around the learning of communication protocols and policies. Moreover, unlike signalling games (lewis2008convention, ) the communication in Hanabi does not use a separate channel, but rather mixes communication and environment actions. Finally, the resulting coordination and communication problem in Hanabi was designed to be challenging to human players.

The previous section suggests two different challenges presented by Hanabi. The first, easier, problem is to learn a fixed policy for all players in self-play. In this case, the learning process is in control of all players, and the objective is to maximise the expected utility of the resulting joint policy. The second problem, ad-hoc team play, is learning to play with a set of unknown partners, within a human timescale of a few games rather than millions of offline interactions.

In the spirit of the proposed evaluation protocols (Mochado18, ) for the Arcade Learning Environment, which discuss recommendations for learning in Atari games, we will make some recommendations on how research should be carried out under each challenge.

4.1 Open Source Environment

To help promote consistent, comparable, and reproducible research results, we have released an open source Hanabi RL environment called the Hanabi Learning Environment.181818The code release is being finalized. It will appear on the DeepMind github. Written in Python and C++, the code provides an interface similar to OpenAI Gym (OpenAIGym, ). It includes an environment state class which can generate observations and rewards for an agent, and can be advanced by one step given agent actions. An agent only needs to be able to generate an integer action, given an observation bitstring.

The default agent observation in our Hanabi environment includes card knowledge from previous hint actions, which includes both positive and negative information (e.g., “the first card is red” also says all other cards are not red). This removes the memory task from the challenge, but humans tend to find remembering cards to be an uninteresting distraction, and the experimental results in Section 5 show that the game remains challenging without requiring agents to learn how to remember card knowledge. For researchers interested in memory, we provide the option to request a minimal observation, which does not include this card knowledge.

For debugging purposes, the code includes an environment for a small game with only two colors, two cards per player, three information tokens, and a single life token. There is also a very small game with a single color.

4.2 Challenge One: Self-Play Learning

The first challenge is focused on finding a joint policy that achieves a high expected score entirely through self-play learning. A practical advantage of the Hanabi benchmark is that the environment is extremely lightweight, both in terms of memory and compute requirements, and fast (around 0.1ms per turn on a CPU). It can therefore be used as a testbed for RL methods that require a large number of samples without causing unreasonable compute requirements. However, developing sample efficient algorithms is also an important goal for RL algorithms in its own right. With this in mind we propose two different regimes for the Hanabi benchmark:

Sample limited regime (SL). In the sample limited regime, we are interested in pushing the performance of sample-efficient algorithms for learning to play Hanabi. To that extent, we propose to limit the number of environment steps that the agent can experience to be at most 100 million. Here environment steps count the total number of turns taken during training. If the current episode does not end at 100 million steps, then we let the agent finish the episode before terminating training. This regime is similar to the evaluation scheme for Atari 2600 games proposed by (Mochado18, ). The 100 million step limit was chosen based on the learning curves of the Rainbow agent presented in Section 5 to make sure that the current state-of-the-art agents can achieve a decent score in the given amount of time.

Unlimited regime (UL). In the unlimited regime there are no restrictions on the amount of time or compute. The unlimited regime describes research where the focus is on asymptotic performance, such as achieving high performance using large-scale computation. However, we encourage all work on Hanabi to include numbers about the compute requirements and run-time of their methods alongside the final results.

For every -player game (where ), we recommend the following details be reported. Here the best agent is the training run with the highest average score under test conditions at the end of training, e.g., when disabling exploration and picking the greedy action.

  • The training curves for all random number generator seeds, highlighting the best agent.

  • The histogram of game scores for the best agent and the percentage of perfect games.

  • The mean and standard deviation of the performance of the best agent, computed as an average performance across at least 1000 trials (

    i.e., full games). In the future, as performance increments become smaller this number should be increased to allow for significant results.

As we show in Section 5, Hanabi is difficult for current learning algorithms. Even when using a large amount of data and computation time (UL regime), learning agents have trouble approaching the performance of hand-crafted rules in four player games, and fall far short of hand-crafted rules for three and five players.

4.3 Challenge Two: Ad-hoc Teams

The second challenge Hanabi poses is that of ad-hoc team play. The ultimate goal is agents that are capable of playing with other agents or even human players. For this, a policy which achieves a high score in self-play is of little use if it must be followed exactly by teammates. Good strategies are not unique, and a robust player must learn to recognize intent in other agents’ actions and adapt to a wide range of possible strategies.

We propose to evaluate ad-hoc team performance by measuring an agent’s ability to play with a wide range of teammates it has never encountered before. This is measured via the score achieved by the agent when it is paired with teammates chosen from a held-out pool of agents. 191919For two-player games forming a team is straightforward: we pair the evaluated agent with a random player from the pool and randomly permute their order in the team. For three to five-player games, the presence of more than one player from the pool is a potential confounding factor: it could be that the team fails because of the interaction between these players and through no fault of the evaluated agent. We therefore recommend to limit teams to two unique players — the evaluated agent, in one seat, and one agent from the pool — which is replicated in the remaining seats. The composition of that pool should be such that the players exhibit diverse strategies, which can be hard-coded or learned by self-play.

We recommend the evaluated agents be given ten random self-play games of its ad-hoc teammates prior to play. While other alternatives may be more challenging (e.g., ten “warm-up” games, or average performance in the first ten games with no prior information), this focuses the challenge on an agent’s ability to recognize intent in other agents’ behaviour, as they can observe examples of the intended properly coordinated behaviour.

We recommend that the mean and standard deviation of the performance be reported across at least 1000 trials for each hold-out team. Specifically, each trial for a particular hold-out team should be evaluated by giving the agent a set of ten self-play games for the team, followed by the agent playing a single game in place of a player from the hold-out team in a random position, and finally resetting the agent so it does not retain memory across trials. The 1000 trials should also involve at least 100 different random sets of self-play games provided to the agent. These results should be reported along with mean and standard deviations of the performance of the hold-out teams in self-play (as a baseline comparison). We further recommend crosstables of the hold-out teams’ performance when paired with each other as a method of assessing the diversity of the pool (e.g., see Figure 5).

In the future we expect to see canonical agent pools of pre-trained or hard-coded self-play agents be made available for training and hold-out sets to allow for consistent comparisons.

5 Hanabi: State of the Art

Hanabi presents interesting multi-agent learning challenges for both discovering conventions and adapting to an ad-hoc team of players. In this section, we provide empirical evidence that both tasks are challenging for state-of-the-art learning algorithms, even with an abundance of computational resources. We examine the performance of three multi-agent reinforcement learning algorithms using deep learning for function approximation, and contrast them with a few of the best known hand-coded Hanabi “bots”.

5.1 Learning Agents

Actor-Critic-Hanabi-Agent. The family of asynchronous advantage actor-critic algorithms (Mnih16asynchronous, ) demonstrates stability, scalability and good performance on a range of single-agent tasks, including the suite of games from the Arcade Learning Environment (bellemare13arcade, ), the TORCS driving simulator (Wymann2015TORCS, ), and 3D first-person environments (Beattie16DMLab, )

. In the original implementation, the policy is represented by a deep neural network, which also learns a value function to act as a baseline for variance reduction. Experience is accrued in parallel by several copies of the agent running in different instantiations of the environment. Learning gradients are passed back to a centralized server which holds the parameters for a deep neural network.

Since the environment instantiations and the server interact asynchronously, there is a potential for the learning gradients to become stale, which impacts negatively on performance. ACHA uses the Importance Weighted Actor-Learner variant to address the stale gradient problem (Espeholt18IMPALA, ) by adjusting the stale off-policy updates using the V-trace algorithm. The variant has been successfully applied to the multi-agent task of Capture-the-Flag, achieving human-level performance (Jaderberg18CTF, ). ACHA also incorporates population-based training (Jaderberg17PBT, )

, providing automatic hyperparameter optimization.

For our experiments, ACHA was ran with a population size of to per run,

actors generating experience in parallel, and hyperparameter evolution over the learning rate and entropy regularisation weight. ACHA also used parameter-sharing across the different players in combination with an agent-specific ID that is part of the input. Parameter sharing is a standard method which increases learning speed, while the agent-specific ID allows for some level of specialisation between agents. Our neural architecture consisted of the following. All observations were first processed by an MLP with a single 256-unit hidden layer and ReLU activations, then fed into a 2-layer LSTM with 256 units in each layer. The policy

was a softmax readout of the LSTM output, and the baseline was a learned linear readout of the LSTM output. We refer to this method as the Actor-Critic-Hanabi-Agent (ACHA).

Rainbow-Agent Rainbow  (Hessel17Rainbow, ) is a state of the art agent architecture for deep RL on the Arcade Learning Environment. It combines some of the key innovations that have been made to Deep Q-Networks (DQN) (mnih15human, ) over the last few years, resulting in a learning algorithm that is both sample efficient and achieves high rewards at convergence. In our benchmark we use a multi-agent version of Rainbow, based on the Dopamine framework (castro18dopamine, ). In our code the agents controlling the different players share parameters. Our Rainbow agent is feedforward and does not use any observation stacking outside of the last action, which is included in the current observation.

Our Rainbow agent uses a 2-layer MLP of hidden units each to predict value distributions using distributional reinforcement learning (bellemare17distributional, ). Our batch size, i.e., the number of experience tuples sampled from the replay buffer per update, is , and we anneal the of our -greedy policy to 0 over the course of the first training steps. We use a discount factor, , of and apply prioritized sampling (schaul16prioritized, ) for sampling from the replay buffer. Finally, our value distributions are approximated as a discrete distribution over uniformly spaced atoms.

BAD-Agent (Foerster18BAD, ). For the two player self-play setting we also include the results of the Bayesian Action Decoder since it constitutes state-of-the-art for the two-player unlimited regime. Rather than relying on implicit belief representations such as RNNs, the Bayesian Action Decoder (BAD) uses a Bayesian belief update that directly conditions on the current policy of the acting agent. In BAD all agents track a public belief, which includes everything that is common knowledge about the cards, including the posterior that is induced from observing the actions different agents take. BAD also explores in the space of deterministic policies, which ensures informative posteriors while also allowing for randomness required to explore. Further details for the BAD agent are provided in (Foerster18BAD, ).

5.2 Rule-Based Approaches

For benchmarking we provide results of a number of independently implemented rule-based strategies. These provide examples of the quality of play that can be achieved in Hanabi.

SmartBot (SmartBot, ). SmartBot is a rule-based agent that tracks the publicly known information about each player’s cards. Tracking public knowledge allows SmartBot to reason about what other players may do, and what additional knowledge it gains from its specific view of the game. Among other things, this enables SmartBot to play/discard cards that its partners don’t know it knows are safe to play/discard, thereby preventing partners from wasting a hint to signal as much. However, this tracking assumes all other players are using SmartBot’s convention. When this assumption does not hold, as in the ad-hoc team setting, SmartBot can fall into false or impossible beliefs. For example, SmartBot can believe one of its cards has no valid value as all possible cards are inconsistent with the observed play according to SmartBot’s convention. Finally, note that SmartBot has a parameter specifying if it should attempt uncertain plays that may cost a life. Risking lives increases the frequency of perfect games while reducing average score, except in two player games where it is better on both criteria. Our SmartBot results only risks lives in the two player setting.

HatBot (Cox15, ) and WTFWThat (wtfwt, ). HatBot uses a technique often seen in coding theory and “hat puzzles”. When giving hints, HatBot uses a predefined protocol to determine a recommended action for all other players (i.e., play or discard for one of a player’s cards). This joint recommendation is then encoded by summing the indices for the individual recommendations and using modular arithmetic. The encoded joint recommendation is mapped to different hints that HatBot could make, specifically, whether it reveals the color or rank of a card for each other player. Since each player can view everyone’s cards but their own, they can reconstruct the action recommended to them by figuring out what would have been recommended to the other players based on HatBot’s convention, which HatBot assumes they know and use. Although this convention is not very intuitive for human players, it would still be possible for humans to learn and follow. Cox and colleagues also introduce an “information strategy” using a similar encoding mechanism to directly convey information about each player’s cards (as opposed to a recommended action), however it requires additional bookkeeping that makes it impractical for humans to use. As originally proposed, both the recommendation and information strategies were tailored for playing 5-player Hanabi. However, a variant of the information strategy, called WTFWThat (wtfwt, ), can play two through five players.

FireFlower (FireFlower, ).

FireFlower implements a set of human-style conventions. The bot keeps track of both private and common knowledge, including properties of cards that are implied by the common knowledge of what the conventions entail. Using this, FireFlower performs a 2-ply search over all possible actions with a modeled probability distribution over what its partner will do in response, and chooses the action that maximizes the expected value of an evaluation function. The evaluation function takes into account the physical state of the game as well as the belief state. For example, if there is a card in the partner’s hand that is commonly known (due to inference from earlier actions) to be likely to be playable, then the evaluation function’s output will be much higher if it is indeed playable than if it is not. FireFlower implements a few additional conventions for three and four players, but avoids the hat-like strategies in favor of conventions that potentially allow it to partner more naturally with humans assuming the human players exactly follow its strategy. Also, according to the creator, FireFlower is designed with a focus on maximising the win probability, rather than average score. Further details are in

A.1.

5.3 Experimental Results: Self-Play

Table 1 shows the experimental results of the baseline agents and state-of-the-art learning algorithms with each number of players.

Regime Agent 2P 3P 4P 5P
SmartBot 22.99 (0.00) 23.12 (0.00) 22.19 (0.00) 20.25 (0.00)
29.6% 13.8% 2.076% 0.0043%

FireFlower 22.56 (0.00) 21.05 (0.01) 21.78 (0.01) -
52.6% 40.2% 26.5%
HatBot 22.56 (0.06)
14.7 %
WTFWThat 19.45 (0.03) 24.20 (0.01) 24.83 (0.01) 24.89 (0.00)
0.28% 49.1% 87.2% 91.5%
SL Rainbow 20.64 (0.22) 18.71 (0.20) 18.00 (0.17) 15.26 (0.18)
2.5 % 0.2% 0 % 0 %
UL ACHA 22.73 (0.12) 20.24 (0.15) 21.57 (0.12) 16.80 (0.13)
15.1% 1.1% 2.4% 0%
UL BAD 23.92 (0.01)
58.6%
Table 1: Shown are the results for the three learning agents, Rainbow, ACHA and BAD, compared to the rule-based agents, SmartBot, FireFlower and HatBot, for different numbers of players in self-play

. For each algorithm and number of players we show mean performance of the best agent followed by (standard error of the mean) and percentage of perfect (

i.e., 25 point) games. Error of the mean differs based on different number of evaluation games.

First, note that except for BAD in the two-player setting, the best learning agent does not reach the performance of the best hand-coded agents. The information-theoretic strategy based on Cox et al. achieves scores over 24 points when playing with three to five players (including a remarkable 24.89 points and 91.5% perfect games in the five-player game), showing a large performance gap in what is possible and what state-of-the-art learning algorithms achieve. Even rule-based strategies aimed at attempting to codify more human-like conventions achieve scores higher than the learning algorithms, particularly in the three and five player setting.

The ACHA agent in the unlimited regime (using over 20 billion steps of experience) achieved higher scores than Rainbow (using 100 million steps of experience) across all number of players. This quite naturally may be due to less training experience, but may also be due to Rainbow’s feed-forward network architecture with no history of past actions possibly making it harder to learn multi-step conventions. Both agents saw a decline in performance as the number of agents increased with Rainbow’s more gradual, while ACHA saw a precipitous drop in performance with five players.

Figure 2: ACHA results for two to five players, from top to bottom respectively. Performance curves (left) are training-time results from the current soft-max policy. Average scores and distributions (right) are test-time results from 1000 episodes generated using the greedy policy from the agent with the best training score.

Figure 2 shows training curves for one run of ACHA showing the performance of the multiple “agents” within the population. Note that these curves are linked together through parameter evolution and are not independent. In all but the four player setting, it is clear ACHA has found a local minimum in policy space it is unable to escape even with further training. However, parameter evolution is hiding the true extent of the local minima problem. Figure 3 gives ACHA training curves for two and four player games when evolution is disabled, so each curve is an independent self-play trial, using the same fixed hyper-parameters found as the best in the earlier experiment. Here we can see that independent learning trials find a wide array of different local minima, most of which are difficult to escape. For example, in the two player setting, roughly a third of the independent agents are below 15 points and appear to no longer be improving.

Figure 3: ACHA results without evolution, using 50 independent runs for two players (left), and 30 independent runs for four players (right).

We further find that even ACHA runs with similar final performance can learn very different conventions. For example, one agent uses color hints to indicate that the 4th card of the teammate can likely be discarded, while another agent uses the different color hints to indicate which card of the teammate can be played. Different agents also use the rank-hints to indicate playability of cards in different slots. Details examining specific examples of learned policies are in A.2.

In contrast to ACHA, the Rainbow agents exhibit low variance across different independent training runs, as shown by the learning curves in Figure 4. In this case each line represents an independent training trial. In addition, Rainbow agents tend to converge to similar strategies, seemingly identifying the same local minima. In particular, Rainbow agents are 3-4 times less likely to hint for color than ACHA, and when they do there is no evidence of specific conventions associated with the color hinted. Instead all Rainbow agents we looked at primarily hint for rank, typically indicating that the most recent card is playable. See A.2 for details. We speculate that two factors contribute to this consistency across different runs. First, Rainbow has a one-step memory for the past action and no memory of past observations. This drastically reduces the space of possible strategies. Second, Rainbow is a value-based method and starts out with a high exploration rate. During the initial exploration, since agents fail to successfully finish any games without running out of lives, Q-values will tend towards zero. This might cause agents to learn from effectively the same starting Q-values, even across different independent runs.

Figure 4: Rainbow results for two to five players, from top to bottom respectively. Performance curves (left) are training-time results from the current policy. Average scores and distributions (right) are test-time results from 1000 episodes generated using the agent with the best training score. in -greedy for all agents is set to zero.

5.4 Experimental Results: Ad-Hoc Team Play

The policies learned by the aforementioned ACHA, BAD and Rainbow agents are moderately effective in self-play. We now investigate these agents in the ad-hoc team setting, where teammates are using different conventions. In particular, we examine the performance of different ad-hoc teams of ACHA and Rainbow agents. Since we established in Section 5.3 that Rainbow agents learn nearly identical strategies across different runs, for the rest of the section we consider a population of agents made up of the best performing Rainbow agent and a pool of independently trained ACHA agents taken from the top ten agents according to final training-time performance from Figure 3.

As neither of these algorithms make use of sample play of their ad-hoc teammates, the ad-hoc team’s performance will simply depend on the compatibility of the different protocols. The purpose of this section, then, is primarily to illustrate the difficulty in ad-hoc team play, and suggest a source for creating a diverse pool of agents for future evaluation.

Two Players. Figure 4(a) shows a crosstable of the agents’ test performance in the ad-hoc setting. The entry for the -th row and -th column shows the mean performance of agent playing in an ad-hoc team consisting of agent , evaluated for 1000 games with a random player starting each game. The scores of 20 or greater along the diagonal entries show that the agents indeed perform well in self-play. However, when paired with other agents, performance drops off sharply, with some agents scoring essentially zero in any of these ad-hoc teams.

(a) Ad-hoc results for two players.
(b) Ad-hoc results for four players.
Figure 5: Ad-hoc team results. Teams were constructed using the 10 best independently trained ACHA agents (1–10; see Figure 3) and the best Rainbow agent (R; see Figure 4). Mean scores over 1000 trials are shown. Standard error of the mean was less than and for two and four players, respectively.

Four Players. We observe analogous results when evaluating ad-hoc teams in the four player setting. We used the top ten ACHA agents from Figure 3. Similar to the two player ad-hoc results, the entry for the -th row and -th column of Figure 4(b) shows the mean performance of agent when playing with an ad-hoc team consisting of three other agent players, evaluated for 1000 games with a random player starting each game. As in the two player results, the agents fare relatively well in self-play but performance dramatically decreases once we introduce a second unique agent to the team.

Developing agents that can learn from, adapt, and play well with unknown teammates represents a formidable challenge.

6 Hanabi: Related Work

In this section, we discuss prior work on the game of Hanabi. We also briefly discuss relevant work in multi-agent reinforcement learning.

6.1 Prior Work on Hanabi AI

To the best of our knowledge, the earliest published work on Hanabi was in 2015. Osawa described some of the unique elements of Hanabi, and showed that simulated strategies that try to recognize the intention of the other players performed better than a fixed set of static strategies in two-player Hanabi (Osawa15, ). Later in the same year, Cox et al. developed the hat strategy described in Section 5.2 (Cox15, ).

Several studies focused on techniques to achieve strong Hanabi play. First, van den Bergh et al. described fixed rules whose trigger thresholds were optimized by manual tuning via simulation (vandenBergh16, ); the action selection is also improved during play using Monte Carlo planning. Several rule-based agents and Monte Carlo tree search players were evaluated in (WaltonRivers17, )

: again, here, a predictor version that modeled the other players was found to perform better than bots without this ability. Finally, the best-performing heuristic players in their study combined search with the hat strategy to improve the frequency of the highest scores 

(Bouzy17, ). In the five-player game when ignoring the rule in that hints cannot refer to an empty set of cards, they reported achieving 24.92 points and a perfect score 92% of the time on average. While some of these ideas are used in the agents that we describe in Section 5.2, the agents we benchmark have superior performance under the complete Hanabi rules to the numbers reported in these papers.

There are two related works that are not about (directly) increasing performance. The first was on the complexity of the generalized game, which found optimal gameplay in Hanabi to be NP-hard even with a centralised, cheating player playing all seats and knowing every hand (Baffier17, ). The second describes how to model knowledge and how it is revealed through actions using dynamic epistemic logic (Eger17, ). These epistemic formalizations within game theory attempt to quantify what players know and assume others know, how players can reason about this knowledge, and what rationality means in this context (Pacuit17, ). These ideas are also reflected in the BAD agent Foerster18BAD , where they are combined with scalable deep multi-agent RL, and have resulted in the strongest two player Hanabi agent.

Recently, there was a Hanabi competition that took place at the IEEE Computational Intelligence in Games Conference in Maastricht (CIG2018Competition, ). There were a total of five agents, three submissions and two samples . There were two tracks, called “Mirror” and “Mixed” which broadly match the two categories we propose in Section 4

. Similarly to van den Bergh, the second-place player used a genetic algorithm to evolve a sequence of rules from a fixed rule set. This agent achieved an average score of 17.71 points in the Mirror competition, while the first place agent, “Monte Carlo NN”, achieved a score of 20.56. According to the website these scores are averaged across the results for two to five players. No further detail on this agent was available on the website.

6.2 Reinforcement Learning (RL) and Multi-Agent RL

While there has been much work in rule-based and search-based players using various heuristics based on domain knowledge, we are unaware of any prior approaches on learning to play Hanabi directly from experience given only the rules of the game. The framework for this approach is reinforcement learning, where an agent chooses actions in its environment, receives observations and rewards (SuttonBarto18, ). The goal is to learn a policy that achieves a high expected sum of rewards, i.e. score.

The setting with multiple reinforcement learning agents was first investigated in competitive games (Littman94markovgames, ). This was the start of the foundational work on multiagent reinforcement learning (MARL), which focused on algorithms and convergence properties in Markov/stochastic (simultaneous move) games. Independent learners in the cooperative case, even in these simpler game models, already face several coordination problems (Matignon12, ). Several years of work on MARL gave rise to many algorithms, extensions, and analyses (bowling2003simultaneous, ; Panait05, ; Busoniu08Comprehensive, ; Nowe12Game, ; BloembergenTHK15, ). For a more recent survey on multi-agent deep reinforcement learning, see (HernandezLeal18Survey, ).

Model-free methods, which learn a direct mapping from observations to actions, are appealing in traditional RL because the agent need not understand the dynamics of the environment in order to act. In games, however, the perfect model (i.e., the rules) is given. This leads to methods than can combine planning with RL, as first demonstrated by TD-Leaf (Baxter98, ) and TreeStrap (VenessSilver09, ). Recently, Monte Carlo tree search (kocsis06bandit, ) was combined with deep neural networks in AlphaGo (Silver16Go, ), a computer Go-playing agent that was stronger than any preceding program and defeated the best human Go players. Shortly thereafter AlphaGo was surpassed by AlphaGo Zero, which used less knowledge (Silver17AGZ, ), and then AlphaZero which also learned to play chess and shogi at super-human levels (Silver17AZ, ).

Many successes above were in applications of RL to perfect information games (e.g., Go, chess, shogi). Partially-observable environments offer new challenges since the world cannot be perfectly simulated: agents must reason about information that is not known to them. In the competitive case, such as Poker, strong play required new algorithmic foundations (CFR, ; Bowling15Poker, ; Brown17Libratus, ; Moravcik17DeepStack, ).

Finally, there has been recent interest in the problem of emergent communication. These problems include an arbitrary or structured communication channel, and agents must learn to communicate to solve a cooperative problem. Algorithms have been developed to learn to solve riddles and referential games (Foerster16RIAL, ; Lazaridou16, ), gridworld games requiring coordination Sukhbaatar16 , object identification via question-and-answer dialog (deVries17GuessWhat, ), and negotiation (Lewis17, ; Cao18, ). The main difference in Hanabi is that there is no “cheap-talk” communication channel: any signalling must be done through the actions played. It is therefore more similar in spirit to learning how to relay information to partners in the bidding round of bridge (Yeh16, ).

6.3 Beliefs, Bayesian Reasoning, and Theory of Mind

There has been much work that proposes to model beliefs about the intentions or plans of other agents. Several formalisms have been proposed for this, such as I-POMDPs Gmytrasiewicz05 and Bayesian games (Shoham09, ). However, algorithms to compute exact solutions quickly become intractable for complex problems.

Classical models to predict human behavior in games include ways to deal with imperfect rationality (Yoshida08, ; Wright17, ). In recent years, several mechanisms have been proposed to learn these models using deep learning, from human data in one-shot games (Hartford16, ), learning players with expert features (DRON, ), and end-to-end prediction of fixed policies in gridworld games (Rabinowitz18, ). When predictions are used to exploit or coordinate, this is generally called opponent/teammate modeling (Albrecht18Modeling, ). Several MARL algorithms have been recently proposed that learn from models of others, by using a training regime based on cognitive hierarchies (Lanctot17PSRO, ), by defining a learning rule that shapes the anticipated learning of other agents (Foerster17LOLA, ), or by training architectures that incorporate agent identification as part of the objective and conditioning the policy on these predictions (Grover18, ). Lastly, an approach in poker (DeepStack) introduced function approximators that represent belief-conditional values and explicitly representing and reasoning about manipulating agents’ beliefs  (Moravcik17DeepStack, ), similar to the approach taken in BAD Foerster18BAD .

7 Conclusion

The combination of cooperative gameplay and imperfect information make Hanabi a compelling research challenge for machine learning techniques in multi-agent settings. We evaluate state-of-the-art reinforcement learning algorithms using deep neural networks and demonstrate that they are largely insufficient to surpass current hand-coded bots when evaluated in a self-play setting. Furthermore, we show that in the ad-hoc team setting, where agents must play with unknown teammates, such techniques fail to collaborate at all. Theory of mind appears to play an important role in Hanabi, not only for the efficiency and interpretability of agent communication, but also for agents to be capable of collaborating with human partners that rely on such reasoning. This suggests that there is considerable room to improve in both settings and that novel techniques will be required to close this gap, hopefully advancing our understanding of how our agents can develop something like a theory of mind. To promote effective and consistent comparison between techniques, we provide a new open source code framework for Hanabi and propose evaluation methodology for practitioners.

Acknowledgements

We would like to thank many people: Matthieu d’Epenoux of Cocktail Games and Antoine Bauza, who designed Hanabi, for their support on this project; Alden Christianson for help with coordinating across three different time zones, and discussions with Cocktail Games; Angeliki Lazaridou for discussion and feedback on pragmatics and theory of mind; Ed Lockhart for many discussions on small Hanabi-like games and belief-based reasoning; Daniel Toyama for assistance with writing clear, readable code for the Hanabi research environment used in our experiments; Kevin Waugh for helpful comments and feedback; David Wu for providing details on the FireFlower bot; and Danny Tarlow for early feedback on the project.

References

  • (1) D. Premack, G. Woodruff, Does the chimpanzee have a theory of mind?, Behavioral and Brain Sciences 1 (1978) 515–526.
  • (2) N. C. Rabinowitz, F. Perbet, H. F. Song, C. Zhang, S. M. A. Eslami, M. Botvinick, Machine theory of mind, CoRR abs/1802.07740. arXiv:1802.07740.
    URL http://arxiv.org/abs/1802.07740
  • (3) D. C. Dennett, The Intentional Stance, The MIT Press, Cambridge, MA, 1987.
  • (4) Board Game Arena, Board game arena: Play board games online!, https://en.boardgamearena.com/#!gamepanel?game=hanabi, [Online; accessed 5-December-2018] (2018).
  • (5) Keldon, http://keldon.net/hanabi/ (2018).
  • (6) M. Campbell, A. J. Hoane Jr, F.-h. Hsu, Deep blue, Artificial intelligence 134 (1-2) (2002) 57–83.
  • (7) J. Schaeffer, R. Lake, P. Lu, M. Bryant, Chinook the world man-machine checkers champion, AI Magazine 17 (1) (1996) 21.
  • (8) D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. van den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, D. Hassabis, Mastering the game of Go with deep neural networks and tree search, Nature 529 (7587) (2016) 484–489.
  • (9) G. Tesauro, Temporal difference learning and TD-Gammon, Communications of the ACM 38 (3).
  • (10) M. Moravčík, M. Schmid, N. Burch, V. Lisý, D. Morrill, N. Bard, T. Davis, K. Waugh, M. Johanson, M. Bowling, Deepstack: Expert-level artificial intelligence in heads-up no-limit poker, Science 358 (6362).
  • (11) N. Brown, T. Sandholm, Superhuman AI for heads-up no-limit poker: Libratus beats top professionals, Science 360 (6385).
  • (12) N. Burch, M. Johanson, M. Bowling, Solving imperfect information games using decomposition, in: Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, 2014, pp. 602–608.
    URL http://www.aaai.org/ocs/index.php/AAAI/AAAI14/paper/view/8407
  • (13) N. Burch, Time and space: Why imperfect information games are hard, Ph.D. thesis, University of Alberta, Alberta, Canada (2017).
  • (14) J. Zamiell, Github - zamiell/hanabi-conventions: Hanabi conventions for the hyphen-ated group, https://github.com/Zamiell/hanabi-conventions, [Online; accessed 5-December-2018] (2018).
  • (15) C. Cox, J. D. Silva, P. Deorsey, F. H. J. Kenter, T. Retter, J. Tobin, How to make the perfect fireworks display: Two strategies for Hanabi, Mathematics Magazine 88 (5) (2015) 323–336.
  • (16) M. Bowling, P. McCracken, Coordination and adaptation in impromptu teams, in: Proceedings of the Twentieth National Conference on Artificial Intelligence (AAAI), 2005, pp. 53–58.
  • (17) P. Stone, G. A. Kaminka, S. Kraus, J. S. Rosenschein, Ad-hoc autonomous agent teams: Collaboration without pre-coordination, in: Proceedings of the Twenty-Fourth Conference on Artificial Intelligence, 2010.
  • (18) D. S. Bernstein, R. Givan, N. Immerman, S. Zilberstein, The complexity of decentralized control of markov decision processes, Math. Oper. Res. 27 (4) (2002) 819–840. doi:10.1287/moor.27.4.819.297.
    URL https://doi.org/10.1287/moor.27.4.819.297
  • (19) J.-F. Baffier, M.-K. Chiu, Y. Diez, M. Korman, V. Mitsou, A. Renssen, M. Roeloffzen, Y. Uno, Hanabi is NP-hard, even for cheaters who look at their cards, Theoretical Computer Science 675 (2017) 43–55.
  • (20) J. N. Foerster, Y. M. Assael, N. de Freitas, S. Whiteson, Learning to communicate with deep multi-agent reinforcement learning, CoRR abs/1605.06676. arXiv:1605.06676.
    URL http://arxiv.org/abs/1605.06676
  • (21)

    M. Lewis, D. Yarats, Y. Dauphin, D. Parikh, D. Batra, Deal or no deal? end-to-end learning of negotiation dialogues, in: In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, 2017, pp. 2433–2443, association for Computational Linguistics.

  • (22) K. Cao, A. Lazaridou, M. Lanctot, J. Z. Leibo, K. Tuyls, S. Clark, Emergent communication through negotiation, CoRR abs/1804.03980. arXiv:1804.03980.
    URL http://arxiv.org/abs/1804.03980
  • (23) H. P. Grice, Logic and conversation, in: P. Cole, J. L. Morgan (Eds.), Syntax and semantics, Vol. 3, New York: Academic Press, 1975.
  • (24) S. T. Piantadosi, H. Tily, E. Gibson, The communicative function of ambiguity in language, Cognition 122 (3) (2012) 280 – 291. doi:https://doi.org/10.1016/j.cognition.2011.10.004.
    URL http://www.sciencedirect.com/science/article/pii/S0010027711002496
  • (25) M. C. Frank, N. D. Goodman, Predicting pragmatic reasoning in language games, Science 336 (6084) (2012) 998–998. arXiv:http://science.sciencemag.org/content/336/6084/998.full.pdf, doi:10.1126/science.1218633.
    URL http://science.sciencemag.org/content/336/6084/998
  • (26) M. G. Bellemare, Y. Naddaf, J. Veness, M. Bowling, The Arcade Learning Environment: An evaluation platform for general agents, Journal of Artificial Intelligence Research 47 (2013) 253–279.
  • (27) D. Lewis, Convention: A philosophical study, John Wiley & Sons, 2008.
  • (28) M. C. Machado, M. G. Bellemare, E. Talvitie, J. Veness, M. Hausknecht, M. Bowling, Revisiting the Arcade Learning Environment: Evaluation protocols and open problems for general agents, Journal of Artificial Intelligence Research 61 (2018) 523–562.
  • (29) G. Brockman, V. Cheung, L. Pettersson, J. Schneider, J. Schulman, J. Tang, W. Zaremba, Openai gym, CoRR abs/1606.01540. arXiv:1606.01540.
    URL http://arxiv.org/abs/1606.01540
  • (30) V. Mnih, A. P. Badia, M. Mirza, A. Graves, T. P. Lillicrap, T. Harley, D. Silver, K. Kavukcuoglu, Asynchronous methods for deep reinforcement learning, in: Proceedings of the 33rd International Conference on Machine Learning (ICML), 2016, pp. 1928–1937.
  • (31) B. Wymann, C. Dimitrakakis, A. Sumner, E. Espié, C. Guionneau, Torcs : The open racing car simulator (2015).
  • (32) C. Beattie, J. Z. Leibo, D. Teplyashin, T. Ward, M. Wainwright, H. Küttler, A. Lefrancq, S. Green, V. Valdés, A. Sadik, J. Schrittwieser, K. Anderson, S. York, M. Cant, A. Cain, A. Bolton, S. Gaffney, H. King, D. Hassabis, S. Legg, S. Petersen, Deepmind lab, CoRR abs/1612.03801. arXiv:1612.03801.
    URL http://arxiv.org/abs/1612.03801
  • (33) L. Espeholt, H. Soyer, R. Munos, K. Simonyan, V. Mnih, T. Ward, Y. Doron, V. Firoiu, T. Harley, I. Dunning, S. Legg, K. Kavukcuoglu, IMPALA: scalable distributed deep-rl with importance weighted actor-learner architectures, CoRR abs/1802.01561.
  • (34) M. Jaderberg, W. M. Czarnecki, I. Dunning, L. Marris, G. Lever, A. G. Castaneda, C. Beattie, N. C. Rabinowitz, A. S. Morcos, A. Ruderman, N. Sonnerat, T. Green, L. Deason, J. Z. Leibo, D. Silver, D. Hassabis, K. Kavukcuoglu, T. Graepel, Human-level performance in first-person multiplayer games with population-based deep reinforcement learning, CoRR abs/1807.01281.
  • (35) M. Jaderberg, V. Dalibard, S. Osindero, W. M. Czarnecki, J. Donahue, A. Razavi, O. Vinyals, T. Green, I. Dunning, K. Simonyan, C. Fernando, K. Kavukcuoglu, Population based training of neural networks, CoRR abs/1711.09846.
  • (36) M. Hessel, J. Modayil, H. van Hasselt, T. Schaul, G. Ostrovski, W. Dabney, D. Horgan, B. Piot, M. G. Azar, D. Silver, Rainbow: Combining improvements in deep reinforcement learning, CoRR abs/1710.02298.
  • (37) V. Mnih, K. Kavukcuoglu, D. Silver, A. A. Rusu, J. Veness, M. G. Bellemare, A. Graves, M. Riedmiller, A. K. Fidjeland, G. Ostrovski, et al., Human-level control through deep reinforcement learning, Nature 518 (7540) (2015) 529–533.
  • (38) P. S. Castro, S. Moitra, C. Gelada, S. Kumar, M. G. Bellemare, Dopamine: A research framework for deep reinforcement learning, arXiv.
  • (39) M. G. Bellemare, W. Dabney, R. Munos, A distributional perspective on reinforcement learning, in: Proceedings of the International Conference on Machine Learning, 2017.
  • (40) T. Schaul, J. Quan, I. Antonoglou, D. Silver, Prioritized experience replay, in: International Conference on Learning Representations, 2016.
  • (41) J. N. Foerster, F. Song, E. Hughes, N. Burch, I. Dunning, S. Whiteson, M. Botvinick, M. Bowling, Bayesian action decoder for deep multi-agent reinforcement learning, CoRR abs/1811.01458.
  • (42) A. O’Dwyer, Github - quuxplusone/hanabi: Framework for writing bots that play hanabi, https://github.com/Quuxplusone/Hanabi, [Online; accessed 19-September-2018] (2018).
  • (43) J. Wu, Github - wuthefwasthat/hanabi.rs: Hanabi simulation in rust, https://github.com/WuTheFWasThat/hanabi.rs, [Online; accessed 29-November-2018] (2018).
  • (44) D. Wu, Github - lightvector/fireflower: A rewrite of hanabi-bot in scala, https://github.com/lightvector/fireflower, [Online; accessed 3-December-2018] (2018).
  • (45)

    H. Osawa, Solving Hanabi: Estimating hands by opponent’s actions in cooperative game with incomplete information, in: Proceedings on the 2015 AAAI Workshop on Computer Poker and Imperfect Information, 2015, pp. 37–43.

  • (46) M. J. H. van den Bergh, A. Hommelberg, W. A. Kosters, F. M. Spieksma, Aspects of the cooperative card game Hanabi, in: BNAIC 2016: Artificial Intelligence. 28th Benelux Conference on Artificial Intelligence, Vol. 765 of CCIS, Springer, Cham, 2016, pp. 93–105.
  • (47)

    J. Walton-Rivers, P. R. Williams, R. Bartle, D. Perez-Liebana, S. M. Lucas, Evaluating and modelling Hanabi-playing agents, in: Proceedings of the IEEE Conference on Evolutionary Computation (2017), 2017.

  • (48) B. Bouzy, Playing hanabi near-optimally, in: Advances in Computer Games: ACG 2017, Vol. 10664 of LNCS, Springer, Cham, 2017, pp. 51–62.
  • (49) M. Eger, C. Martens, Practical specification of belief manipulation in games, in: Proceedings, The Thirteenth AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment (AIIDE-17), 2017.
  • (50) E. Pacuit, O. Roy, Epistemic foundations of game theory, in: E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy, summer 2017 Edition, Metaphysics Research Lab, Stanford University, 2017.
  • (51) J. Walton-Rivers, P. Williams, CIG 2018 Hanabi fireworks competition, http://hanabi.aiclash.com/, https://project.dke.maastrichtuniversity.nl/cig2018/competitions/, retrieved Sept. 27th, 2018 (2018).
  • (52) R. Sutton, A. Barto, Reinforcement Learning: An Introduction, 2nd Edition, MIT Press, 2018.
  • (53) M. L. Littman, Markov games as a framework for multi-agent reinforcement learning, in: In Proceedings of the Eleventh International Conference on Machine Learning, Morgan Kaufmann, 1994, pp. 157–163.
  • (54)

    L. Matignon, G. J. Laurent, N. L. Fort-Piat, Independent reinforcement learners in cooperative Markov games: a survey regarding coordination problems, The Knowledge Engineering Review 27 (01) (2012) 1–31.

  • (55) M. Bowling, M. Veloso, Simultaneous adversarial multi-robot learning, in: IJCAI, Vol. 3, 2003, pp. 699–704.
  • (56) L. Panait, S. Luke, Cooperative multi-agent learning: The state of the art, Autonomous Agents and Multi-Agent Systems 11 (3) (2005) 387–434. doi:10.1007/s10458-005-2631-2.
    URL http://dx.doi.org/10.1007/s10458-005-2631-2
  • (57) L. Busoniu, R. Babuska, B. D. Schutter, A comprehensive survey of multiagent reinforcement learning, IEEE Transaction on Systems, Man, and Cybernetics, Part C: Applications and Reviews 38 (2) (2008) 156–172.
  • (58) A. Nowé, P. Vrancx, Y.-M. D. Hauwere, Game theory and multi-agent reinforcement learning, in: Reinforcement Learning: State-of-the-Art, Springer, 2012, Ch. 14, pp. 441–470.
  • (59) D. Bloembergen, K. Tuyls, D. Hennes, M. Kaisers, Evolutionary dynamics of multi-agent learning: A survey, J. Artif. Intell. Res. (JAIR) 53 (2015) 659–697.
  • (60) P. Hernandez-Leal, B. Kartal, M. E. Taylor, Is multiagent deep reinforcement learning the answer or the question? a brief survey, CoRR abs/1810.05587. arXiv:1810.05587.
    URL http://arxiv.org/abs/1810.05587
  • (61) J. Baxter, A. Tridgell, L. Weaver, Knightcap: a chess program that learns by combining TD(lambda) with game-tree search, in: Proceedings of the 15th International Conference on Machine Learning, Morgan Kaufmann, 1998, pp. 28–36.
  • (62) J. Veness, D. Silver, A. Blair, W. Uther, Bootstrapping from game tree search, in: Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, A. Culotta (Eds.), Advances in Neural Information Processing Systems 22, Curran Associates, Inc., 2009, pp. 1937–1945.
  • (63) L. Kocsis, C. Szepesvári, Bandit based Monte-Carlo planning, 2006, pp. 282–293.
  • (64) D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. van den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, D. Hassabis, Mastering the game of Go with deep neural networks and tree search., Nature 529 (2016) 484––489.
  • (65) D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, T. H. Arthur Guez, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. van den Driessche, T. Graepel, D. Hassabis, Mastering the game of Go without human knowledge, Nature 550 (2017) 354–359.
  • (66) D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez, M. Lanctot, L. Sifre, D. Kumaran, T. Graepel, T. P. Lillicrap, K. Simonyan, D. Hassabis, Mastering chess and shogi by self-play with a general reinforcement learning algorithm, CoRR abs/1712.01815. arXiv:1712.01815.
    URL http://arxiv.org/abs/1712.01815
  • (67) M. Zinkevich, M. Johanson, M. Bowling, C. Piccione, Regret minimization in games with incomplete information, in: Advances in Neural Information Processing Systems 20 (NIPS 2007), 2008.
  • (68) M. Bowling, N. Burch, M. Johanson, O. Tammelin, Heads-up Limit Hold’em Poker is solved, Science 347 (6218) (2015) 145–149.
  • (69) A. Lazaridou, A. Peysakhovich, M. Baroni, Multi-agent cooperation and the emergence of (natural) language, CoRR abs/1612.07182. arXiv:1612.07182.
    URL http://arxiv.org/abs/1612.07182
  • (70) S. Sukhbaatar, A. Szlam, R. Fergus, Learning multiagent communication with backpropagation, CoRR abs/1605.07736. arXiv:1605.07736.
    URL http://arxiv.org/abs/1605.07736
  • (71)

    H. de Vries, F. Strub, S. Chandar, O. Pietquin, H. Larochelle, A. Courville, Guesswhat?! visual object discovery through multi-modal dialogue, in: IEEE Conference on Computer Vision and Pattern Recognition, 2017.

  • (72) C. Yeh, H. Lin, Automatic bridge bidding using deep reinforcement learning, CoRR abs/1607.03290. arXiv:1607.03290.
    URL http://arxiv.org/abs/1607.03290
  • (73) P. J. Gmytrasiewicz, P. Doshi, A framework for sequential planning in multi-agent settings, JAIR 24.
  • (74) Y. Shoham, K. Leyton-Brown, Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations, Cambridge University Press, 2009.
  • (75) W. Yoshida, R. J. Dolan, K. J. Friston, Game theory of mind, PLOS 4 (12).
  • (76) J. R. Wright, K. Leyton-Brown, Predicting human behavior in unrepeated, simultaneous-move games, Games and Economic Behavior 106 (2017) 16–37.
  • (77) J. Hartford, J. R. Wright, K. Leyton-Brown, Deep learning for predicting human strategic behavior, in: Thirtieth Annual Conference on Neural Information Processing Systems (NIPS 2016), 2016.
  • (78) H. He, J. Boyd-Graber, K. Kwok, H. D. III, Opponent modeling in deep reinforcement learning, in: Proceedings of the International Conference on Machine Learning (ICML), 2016.
  • (79) S. V. Albrecht, P. Stone, Autonomous agents modelling other agents: A comprehensive survey and open problems, Artificial Intelligence 258 (2018) 66–95.
  • (80) M. Lanctot, V. Zambaldi, A. Gruslys, A. Lazaridou, K. Tuyls, J. Perolat, D. Silver, T. Graepel, A unified game-theoretic approach to multiagent reinforcement learning, in: Advances in Neural Information Processing Systems, 2017.
  • (81) J. N. Foerster, R. Y. Chen, M. Al-Shedivat, S. Whiteson, P. Abbeel, I. Mordatch, Learning with opponent-learning awareness, CoRR abs/1709.04326. arXiv:1709.04326.
    URL http://arxiv.org/abs/1709.04326
  • (82) A. Grover, M. Al-Shedivat, J. K. Gupta, Y. Burda, H. Edwards, Learning policy representations in multiagent systems, in: Proceedings of the International Conference on Machine Learning (ICML), 2018.

Appendix A

a.1 FireFlower Details

Implemented conventions include the following:

  • Hints generally indicate play cards, and generally newer cards first.

  • Hints “chain” on to other hints, e.g., if hints to a playable red 2 as red, then might infer that it is indeed the red 2, and then hint back to a red 3 in ’s hand as red, expecting to believe it is a red 3 to play it after plays its red 2.

  • When discarding, discard provably useless cards, otherwise the oldest “unprotected” card.

  • Hints about the oldest unprotected card “protect” that card, with many exceptions where it instead means play.

  • Hints about garbage cards indicate “protect” cards older than it.

  • Deliberately discarding known playable cards signals to the partner that they have a copy of that card (with enough convention-based specificity on location that they may play it with absolutely no actual hints).

  • In many cases, hints about cards that have already been hinted change the belief about those cards in various ways such that the partner will likely change what they do (in accordance with the broader heuristic that one usually should give a hint if and only if they would like to change the partner’s future behavior).

a.2 Conditional Probability Tables

The tables below show conditional probability summaries of the learned policies from different runs of ACHA and Rainbow in the two player game. See Section 5 for a discussion of these policy summaries.

Discard Play Hint Colour Hint Rank
1 2 3 4 5 1 2 3 4 5 R Y G W B 1 2 3 4 5
Discard 1 4 4 3 1 10 2 1 1 0 1 1 6 2 2 1 12 14 11 12 11
2 2 6 5 2 12 1 0 0 0 0 1 4 2 2 1 14 18 13 11 5
3 2 6 11 2 13 1 1 1 1 2 1 3 2 2 1 18 17 11 4 1
4 1 2 5 4 19 0 1 0 0 0 1 6 1 6 2 18 14 10 6 2
5 1 1 1 0 30 1 0 0 0 0 0 3 2 2 2 10 12 16 13 6
Play 1 4 5 3 1 18 4 2 1 1 2 0 2 1 1 1 13 10 10 11 9
2 4 6 5 1 22 3 2 1 0 1 0 2 1 1 0 11 12 11 11 6
3 4 5 2 1 26 4 2 2 1 2 0 3 1 1 1 7 6 8 14 10
4 3 3 4 0 0 3 5 1 1 1 0 3 0 0 0 49 10 1 12 3
5 2 4 5 1 26 2 1 0 0 0 0 3 1 1 0 12 14 12 10 5
Hint R 6 31 25 9 3 0 0 0 0 1 9 4 2 5 2 1 2 0 1 1
Colour Y 19 31 12 5 8 1 0 1 0 0 1 7 2 3 1 2 3 2 2 2
G 20 30 18 1 9 2 0 1 0 0 0 4 3 1 1 0 2 2 2 2
W 18 18 16 10 15 0 0 0 0 0 1 4 1 8 0 2 2 1 2 1
B 12 8 11 6 37 2 0 0 0 0 0 2 2 4 3 1 1 3 3 4
Hint 1 3 3 6 8 1 8 9 0 8 50 0 0 0 0 0 3 0 1 0 0
Rank 2 3 6 1 0 0 8 23 0 0 52 0 0 0 0 0 3 1 0 0 0
3 1 5 2 0 1 11 20 10 0 47 0 0 0 0 0 0 1 1 0 0
4 1 4 3 1 1 13 22 6 1 37 0 0 0 0 0 6 2 1 1 0
5 2 4 0 0 2 29 19 7 2 28 0 1 0 0 0 0 1 1 1 2
3 5 4 2 14 5 7 2 1 17 0 2 1 1 1 9 8 7 7 4
Table 2: Conditional action probabilities for a first two player Rainbow agent.
Discard Play Hint Colour Hint Rank
1 2 3 4 5 1 2 3 4 5 R Y G W B 1 2 3 4 5
Discard 1 4 1 1 1 22 1 1 1 0 1 1 2 3 2 0 10 21 9 10 7
2 2 3 1 0 19 1 1 1 1 1 2 2 3 3 0 9 14 19 11 9
3 4 1 6 1 19 2 0 0 0 2 3 3 4 3 3 21 11 9 2 5
4 3 1 1 2 29 1 0 1 0 0 2 3 3 6 2 12 15 11 5 4
5 1 1 0 0 32 1 0 0 0 0 0 3 2 4 0 10 13 14 11 5
Play 1 4 2 1 1 24 3 2 2 0 3 1 1 1 1 0 11 12 10 11 10
2 4 2 1 1 28 3 2 1 0 2 0 1 1 1 0 12 14 12 8 7
3 4 2 1 0 23 4 2 1 0 4 1 2 1 2 0 11 13 7 12 9
4 3 2 0 0 8 4 2 1 0 6 0 0 1 1 0 35 23 5 5 4
5 3 1 1 1 33 2 2 1 0 1 0 2 2 1 0 11 15 12 8 5
Hint R 15 23 8 9 5 1 0 0 0 0 11 0 7 4 4 3 1 1 1 5
Colour Y 28 15 11 9 13 1 1 0 0 1 1 4 3 1 0 2 2 3 2 3
G 22 19 5 4 17 1 1 1 0 1 3 2 8 2 0 2 1 2 2 6
W 9 7 2 4 60 1 1 1 0 1 2 2 3 4 0 1 1 1 1 2
B 17 7 21 17 4 1 0 0 1 5 7 0 0 1 10 1 4 1 2 0
Hint 1 2 3 2 6 1 9 11 6 5 54 0 0 0 0 0 0 0 0 0 0
Rank 2 8 2 1 0 1 7 21 0 0 48 0 0 0 0 0 8 3 0 0 0
3 2 2 1 0 1 12 18 11 0 48 0 0 0 0 0 2 1 1 0 0
4 1 1 0 0 1 24 14 9 1 43 0 0 0 0 0 0 0 1 1 1
5 8 4 4 2 2 15 19 7 2 23 0 1 2 0 0 2 1 1 3 3
4 2 1 1 19 5 7 3 1 17 0 1 2 2 0 8 9 8 6 4
Table 3: Conditional action probabilities for second two player Rainbow agent.
Discard Play Hint Colour Hint Rank
1 2 3 4 5 1 2 3 4 5 R Y G W B 1 2 3 4 5
Discard 1 9 1 2 1 15 1 1 0 0 0 3 2 2 0 0 19 19 11 9 5
2 8 4 2 1 16 1 0 0 0 0 6 2 2 1 1 16 20 9 8 2
3 6 1 2 0 24 1 1 0 0 1 3 3 2 0 1 12 16 14 7 4
4 7 1 2 2 27 1 1 0 1 1 3 3 2 1 1 14 13 10 7 3
5 2 0 1 0 32 1 0 0 0 0 3 3 3 0 0 11 12 14 13 4
Play 1 9 1 2 1 21 4 3 1 1 2 1 1 0 0 1 10 12 11 12 8
2 4 1 2 1 25 5 5 1 1 3 1 3 1 0 1 6 7 9 13 12
3 8 2 2 1 22 2 2 1 1 3 1 1 0 0 0 17 13 10 10 4
4 5 1 1 2 2 6 3 2 1 8 0 1 0 0 0 33 13 1 15 3
5 7 1 1 1 29 2 1 1 0 1 2 2 1 0 0 12 15 11 10 4
Hint R 49 15 20 3 4 0 0 0 0 0 0 1 0 0 1 0 3 1 0 1
Colour Y 35 5 8 3 12 3 4 1 1 0 1 4 2 0 3 2 2 2 5 7
G 12 4 7 3 55 1 1 0 0 0 1 3 3 1 2 2 1 1 2 1
W 14 13 8 9 27 0 0 0 1 0 2 0 2 12 1 0 2 4 2 1
B 31 2 13 3 12 0 2 1 1 1 2 3 3 1 14 2 1 2 2 3
Hint 1 4 2 5 8 2 10 0 9 6 47 0 0 0 0 0 4 0 1 0 0
Rank 2 9 0 1 0 0 17 0 15 1 48 0 0 0 0 0 4 2 0 0 0
3 2 1 1 0 0 14 9 19 0 50 0 0 0 0 0 1 1 1 0 0
4 3 3 2 1 1 19 14 8 1 38 0 0 0 0 0 5 3 1 2 1
5 2 1 1 0 2 31 14 8 1 30 1 1 0 0 1 1 0 1 2 3
7 1 2 1 16 7 3 5 1 17 1 2 1 0 0 9 9 7 7 3
Table 4: Conditional action probabilities for third two player Rainbow agent.
Discard Play Hint Colour Hint Rank
1 2 3 4 5 1 2 3 4 5 R Y G W B 1 2 3 4 5
Discard 1 3 4 7 3 6 4 3 1 0 1 6 6 5 6 6 11 7 7 5 9
2 4 5 7 2 6 4 2 2 0 0 6 6 5 6 5 12 6 8 5 9
3 4 4 9 2 7 5 3 1 0 0 5 6 5 6 5 10 6 9 5 8
4 4 4 7 3 12 4 3 1 0 0 4 5 3 4 6 11 8 6 5 9
5 1 2 4 1 9 4 3 1 42 0 2 2 2 2 2 3 6 7 5 2
Play 1 3 3 5 3 6 9 5 2 0 2 4 3 4 5 4 8 8 7 6 11
2 4 4 6 4 7 8 5 1 0 1 4 4 4 5 3 9 7 8 5 11
3 3 3 4 2 7 5 3 1 0 1 5 4 4 4 4 12 15 8 4 9
4 4 4 5 4 9 7 4 1 0 1 4 5 4 5 5 9 9 9 5 5
5 5 5 8 5 10 13 7 1 0 0 3 3 3 3 3 6 7 6 4 5
Hint R 4 5 3 38 1 16 14 1 0 0 2 1 1 1 2 3 3 2 2 3
Colour Y 4 3 3 45 1 15 12 1 0 0 1 2 1 1 1 3 3 2 1 2
G 5 4 2 37 1 19 14 1 0 0 1 1 2 1 2 3 3 2 1 2
W 3 3 3 44 1 15 13 1 0 0 2 1 1 1 1 3 2 2 1 2
B 3 5 1 44 1 15 12 1 0 0 1 1 1 1 2 3 2 2 2 2
Hint 1 0 0 0 0 0 9 6 9 12 64 0 0 0 0 0 0 0 0 0 0
Rank 2 3 4 2 1 0 2 4 4 22 42 0 0 0 1 0 8 2 1 0 1
3 2 3 1 16 1 5 6 3 26 24 1 1 1 0 1 6 3 1 0 1
4 1 1 0 3 0 2 2 2 55 31 0 0 0 0 0 0 0 0 0 1
5 2 2 2 18 0 6 3 1 55 4 1 0 1 1 1 0 0 1 1 1
3 3 4 10 5 8 5 2 10 9 3 3 2 3 3 7 6 5 3 5
Table 5: Conditional action probabilities for first two player ACHA agent.
Discard Play Hint Colour Hint Rank
1 2 3 4 5 1 2 3 4 5 R Y G W B 1 2 3 4 5
Discard 1 2 2 0 1 31 6 0 0 0 0 9 3 2 3 4 4 2 6 8 16
2 2 2 1 1 33 2 1 1 1 1 8 2 3 4 4 2 2 3 8 17
3 0 2 5 2 0 53 3 4 1 2 5 1 5 2 1 2 2 1 1 10
4 0 0 1 1 0 0 71 0 0 0 4 2 1 1 2 2 2 1 2 11
5 2 1 0 1 31 1 0 0 0 0 11 3 2 6 4 5 3 6 8 16
Play 1 3 3 1 1 27 6 2 1 0 2 7 3 3 3 3 7 2 5 8 12
2 4 3 1 1 27 5 2 1 0 2 6 4 2 4 3 4 3 5 9 13
3 3 2 1 1 25 4 1 1 0 1 8 4 5 4 3 8 3 5 9 12
4 4 2 2 2 31 3 1 0 0 1 7 3 3 7 4 5 3 4 8 11
5 2 1 1 1 33 3 2 0 0 1 11 4 3 5 4 5 3 5 8 9
Hint R 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 1
Colour Y 0 0 0 1 0 0 94 0 0 0 1 0 0 0 0 1 0 0 0 1
G 0 0 0 0 0 0 0 0 99 0 0 0 0 0 0 0 0 0 0 0
W 0 0 0 0 0 0 0 0 98 0 0 0 0 0 0 0 0 0 0 0
B 0 0 0 1 0 0 95 0 0 0 0 0 0 0 0 1 0 0 0 1
Hint 1 0 0 0 1 0 23 0 73 0 0 1 0 0 0 0 0 0 0 0 1
Rank 2 0 0 0 0 0 47 0 0 0 44 1 1 0 1 1 2 0 1 0 1
3 28 4 1 1 5 43 0 1 0 1 1 1 1 1 1 2 1 2 3 1
4 11 32 1 2 9 17 0 3 0 0 1 2 2 3 2 5 1 3 4 2
5 0 0 0 0 0 0 0 0 0 97 0 0 0 0 0 0 0 0 0 1
3 3 1 1 19 6 6 3 5 15 6 2 2 3 3 4 2 4 6 8
Table 6: Conditional action probabilities for second two player ACHA agent.