156 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 2, MARCH 2008
A Comprehensive Survey of Multiagent
Reinforcement Learning
Lucian Bus¸oniu, Robert Babuˇska, and Bart De Schutter
Abstract—Multiagent systems are rapidly finding applications
in a variety of domains, including robotics, distributed control,
telecommunications, and economics. The complexity of many tasks
arising in these domains makes them difficult to solve with preprogrammed
agent behaviors. The agents must, instead, discover a
solution on their own, using learning. A significant part of the
research on multiagent learning concerns reinforcement learning
techniques. This paper provides a comprehensive survey of multiagent
reinforcement learning (MARL). A central issue in the field
is the formal statement of the multiagent learning goal. Different
viewpoints on this issue have led to the proposal of many different
goals, among which two focal points can be distinguished: stability
of the agents’ learning dynamics, and adaptation to the changing
behavior of the other agents. The MARL algorithms described in
the literature aim—either explicitly or implicitly—at one of these
two goals or at a combination of both, in a fully cooperative, fully
competitive, or more general setting. A representative selection of
these algorithms is discussed in detail in this paper, together with
the specific issues that arise in each category. Additionally, the benefits
and challenges of MARL are described along with some of the
problem domains where the MARL techniques have been applied.
Finally, an outlook for the field is provided.
Index Terms—Distributed control, game theory, multiagent
systems, reinforcement learning.
I. INTRODUCTION
AMULTIAGENT system [1] can be defined as a group of
autonomous, interacting entities sharing a common environment,
which they perceive with sensors and upon which
they act with actuators [2]. Multiagent systems are finding applications
in a wide variety of domains including robotic teams,
distributed control, resource management, collaborative decision
support systems, data mining, etc. [3], [4]. They may arise
as the most natural way of looking at the system, or may provide
an alternative perspective on systems that are originally
regarded as centralized. For instance, in robotic teams, the control
authority is naturally distributed among the robots [4]. In
resource management, while resources can be managed by a
central authority, identifying each resource with an agent may
provide a helpful, distributed perspective on the system [5].
Manuscript received November 10, 2006; revised March 7, 2007 and June
18, 2007. This work was supported by the Senter, Ministry of Economic Affairs
of The Netherlands, under Grant BSIK03024 within the BSIK project “Interactive
Collaborative Information Systems.” This paper was recommended by
Associate Editor J. Lazansky.
L. Busoniu and R. Babusaka are with the Delft Center for Systems
and Control, Faculty of Mechanical Engineering, Delft University of Technology,
2628 CD Delft, The Netherlands (e-mail: i.l.busoniu@tudelft.nl;
r.babuska@tudelft.nl).
B. De Schutter is with the Delft Center for Systems and Control, Faculty of
Mechanical Engineering and also with the Marine and Transport Technology
Department, Delft University of Technology, 2628 CD Delft, The Netherlands
(e-mail: b@deschutter.info).
Digital Object Identifier 10.1109/TSMCC.2007.913919
Although the agents in a multiagent system can be programmed
with behaviors designed in advance, it is often
necessary that they learn new behaviors online, such that the
performance of the agent or of the whole multiagent system
gradually improves [4], [6]. This is usually because the complexity
of the environment makes the aprioridesign of a good
agent behavior difficult, or even, impossible. Moreover, in an
environment that changes over time, a hardwired behavior may
become inappropriate.
A reinforcement learning (RL) agent learns by trial-and-error
interaction with its dynamic environment [6]–[8]. At each time
step, the agent perceives the complete state of the environment
and takes an action, which causes the environment to transit
into a new state. The agent receives a scalar reward signal that
evaluates the quality of this transition. This feedback is less informative
than in supervised learning, where the agent would be
given the correct actions to take [9] (such information is, unfortunately,
not always available). The RL feedback is, however,
more informative than in unsupervised learning, where the agent
would be left to discover the correct actions on its own, without
any explicit feedback on its performance [10]. Well-understood
algorithms with good convergence and consistency properties
are available for solving the single-agent RL task, both when
the agent knows the dynamics of the environment and the reward
function (the task model), and when it does not. Together
with the simplicity and generality of the setting, this makes RL
attractive also for multiagent learning. However, several new
challenges arise for RL in multiagent systems. Foremost among
these is the difficulty of defining a good learning goal for the
multiple RL agents. Furthermore, most of the times each learning
agent must keep track of the other learning (and therefore,
nonstationary) agents. Only then will it be able to coordinate its
behavior with theirs, such that a coherent joint behavior results.
The nonstationarity also invalidates the convergence properties
of most single-agent RL algorithms. In addition, the scalability
of algorithms to realistic problem sizes, already problematic in
single-agent RL, is an even greater cause for concern in multiagent
reinforcement learning (MARL).
The MARL field is rapidly expanding, and a wide variety of
approaches to exploit its benefits and address its challenges have
been proposed over the last few years. These approaches integrate
developments in the areas of single-agent RL, game theory,
and more general, direct policy search techniques. The goal of
this paper is to provide a comprehensive review of MARL. We
thereby select a representative set of approaches that allows us to
identify the structure of the field, to provide insight into the current
state of the art, and to determine some important directions
for future research.
1094-6977/$25.00 © 2008 IEEEBUS¸ONIU et al.: A COMPREHENSIVE SURVEY OF MULTIAGENT REINFORCEMENT LEARNING 157
A. Contribution and Related Work
This paper provides a detailed discussion of the MARL
techniques for fully cooperative, fully competitive, and mixed
(neither cooperative nor competitive) tasks. The focus is placed
on autonomous multiple agents learning how to solve dynamic
tasks online, using learning techniques with roots in dynamic
programming and temporal-difference RL. Different viewpoints
on the central issue of the learning goal in MARL are discussed.
A classification of the MARL algorithms along several taxonomy
dimensions is also included. In addition, we provide an
overview of the challenges and benefits in MARL, and of the
problem domains where the MARL techniques have been applied.
We identify a set of important open issues and suggest
promising directions to address these issues.
Besides single-agent RL, MARL has strong connections with
game theory, evolutionary computation, and optimization theory,
as will be outlined next.
Game theory—the study of multiple interacting agents trying
to maximize their rewards [11]—and, especially, the theory
of learning in games [12], make an essential contribution
to MARL. We focus here on algorithms for dynamic multiagent
tasks, whereas most game-theoretic results deal with
static (stateless) one-shot or repeated tasks. We investigate the
contribution of game theory to the MARL algorithms for dynamic
tasks, and review relevant game-theoretic algorithms for
static games. Other authors have investigated more closely the
relationship between game theory and MARL. Bowling and
Veloso [13] discuss several MARL algorithms, showing that
these algorithms combine temporal difference RL with gametheoretic
solvers for the static games arising in each state of
the dynamic environment. Shoham et al. [14] provide a critical
evaluation of the MARL research, and review a small set of
approaches that are representative for their purpose.
Evolutionary computation applies principles of biological
evolution to the search for solutions of the given task [15], [16].
Populations of candidate solutions (agent behaviors) are stored.
Candidates are evaluated using a fitness function related to the
reward, and selected for breeding or mutation on the basis of
their fitness. Since we are interested in online techniques that exploit
the special structure of the RL task by learning a value function,
we do not review here evolutionary learning techniques.
Evolutionary learning, and in general, direct optimization of the
agent behaviors, cannot readily benefit from the RL task structure.
Panait and Luke [17] offer a comprehensive survey of evolutionary
learning, as well as MARL, but only for cooperative
agent teams. For the interested reader, examples of coevolution
techniques, where the behaviors of the agents evolve in parallel,
can be found in [18]–[20]. Complementary, team learning techniques,
where the entire set of agent behaviors is discovered by
a single evolution process, can be found, e.g., in [21]–[23].
Evolutionary multiagent learning is a special case of a larger
class of techniques originating in optimization theory that explore
directly the space of agent behaviors. Other examples in
this class include gradient search [24], probabilistic hill climbing
[25], and even more general behavior modification heuristics
[26]. The contribution of direct policy search to the MARL
algorithms is discussed in this paper, but general policy search
techniques are not reviewed. This is because, as stated before,
we focus on techniques that exploit the structure of the RL
problem by learning value functions.
Evolutionary game theory sits at the intersection of evolutionary
learning and game theory [27]. We discuss only the contribution
of evolutionary game theory to the analysis of multiagent
RL dynamics. Tuyls and Nowé [28] investigate the relationship
between MARL and evolutionary game theory in more detail,
focusing on static tasks.
B. Overview
The remainder of this paper is organized as follows. Section II
introduces the necessary background in single-agent and multiagent
RL. Section III reviews the main benefits of MARL and the
most important challenges that arise in the field, among which
is the definition of an appropriate formal goal for the learning
multiagent system. Section IV discusses the formal goals put
forward in the literature, which consider stability of the agent’s
learning process and adaptation to the dynamic behavior of the
other agents. Section V provides a taxonomy of the MARL
techniques. Section VI reviews a representative selection of the
MARL algorithms, grouping them by the type of targeted learning
goal (stability, adaptation, or a combination of both) and by
the type of task (fully cooperative, fully competitive, or mixed).
Section VII then gives a brief overview of the problem domains
where MARL has been applied. Section VIII distills an outlook
for the MARL field, consisting of important open questions and
some suggestions for future research. Section IX concludes and
closes the paper.
Note that algorithm names are typeset in italics throughout
the paper, e.g., Q-learning.
II. BACKGROUND: REINFORCEMENT LEARNING
In this section, the necessary background on single-agent and
multiagent RL is introduced [7], [13]. First, the single-agent
task is defined and its solution is characterized. Then, the multiagent
task is defined. Static multiagent tasks are introduced separately,
together with necessary game-theoretic concepts. The
discussion is restricted to finite state and action spaces, as the
large majority of MARL results is given for finite spaces.
A. Single-Agent Case
In single-agent RL, the environment of the agent is described
by a Markov decision process.
Definition 1: A finite Markov decision process is a tuple
〈X, U, f, ρ〉 where X is the finite set of environment states, U
is the finite set of agent actions, f : X × U × X → [0, 1] is the
state transition probability function, and ρ : X × U × X → R
is the reward function. 1
The state signal xk ∈ X describes the environment at each
discrete time-step k. The agent can alter the state at each time
1 Throughout the paper, the standard control-theoretic notation is used: x for
state, X for state space, u for control action, U for action space, f for environment
(process) dynamics. We denote reward functions by ρ, to distinguish them
from the instantaneous rewards r and the returns R. We denote agent policies
by h.158 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 2, MARCH 2008
step by taking actions uk ∈ U. As a result of the action uk ,
the environment changes its state from xk to some xk+1 ∈ X
according to the state transition probabilities given by f: the
probability of ending up in xk+1 given that uk is executed in xk is
f(xk ,uk,xk+1). The agent receives a scalar reward rk+1 ∈ R,
according to the reward function ρ: rk+1 = ρ(xk ,uk,xk+1).
This reward evaluates the immediate effect of action uk , i.e., the
transition from xk to xk+1. It says, however, nothing directly
about the long-term effects of this action.
For deterministic models, the transition probability function f
is replaced by a simpler transition function, ¯ f : X × U → X.It
follows that the reward is completely determined by the current
state and action: rk+1 =¯ρ(xk ,uk), ¯ρ : X × U → R.
The behavior of the agent is described by its policy h, which
specifies how the agent chooses its actions given the state. The
policy may be either stochastic, h : X × U → [0, 1], or deterministic,
¯ h : X → U. A policy is called stationary if it does not
change over time.
The agent’s goal is to maximize, at each time-step k, the
expected discounted return
⎧
⎨ ∞∑
Rk = E γ
⎩
j ⎫
⎬
rk+j+1
(1)
⎭
x ′ ∈X
j=0
where γ ∈ [0, 1) is the discount factor, and the expectation is
taken over the probabilistic state transitions. The quantity Rk
compactly represents the reward accumulated by the agent in
the long run. Other possibilities of defining the return exist [8].
The discount factor γ can be regarded as encoding increasing
uncertainty about rewards that will be received in the future,
or as a means to bound the sum that otherwise might grow
infinitely.
The task of the agent is, therefore, to maximize its long-term
performance, while only receiving feedback about its immediate,
one-step performance. One way it can achieve this is by
computing an optimal action-value function.
The action-value function (Q-function), Qh : X × U → R,
is the expected return of a state-action pair given the policy h:
Qh (x, u) =E{ ∑∞
j=0 γj rk+j+1 | xk = x, uk = u, h}. The optimal
Q-function is defined as Q∗ (x, u) = maxhQh (x, u). It
satisfies the Bellman optimality equation
Q ∗ (x, u) = ∑
f(x, u, x ′ ) [ ρ(x, u, x ′ )+γ max
u ′ Q∗ (x ′ ,u ′ ) ]
∀x ∈ X, u ∈ U. (2)
This equation states that the optimal value of taking u in x is
the expected immediate reward plus the expected (discounted)
optimal value attainable from the next state (the expectation is
explicitly written as a sum since X is finite).
The greedy policy is deterministic and picks for every state
the action with the highest Q-value
¯h(x) = arg max Q(x, u). (3)
u
The agent can achieve the learning goal by first computing Q∗ and then choosing actions by the greedy policy, which is optimal
(i.e., maximizes the expected return) when applied to Q∗ .
A broad spectrum of single-agent RL algorithms exists,
e.g., model-based methods based on dynamic programming
[29]–[31], model-free methods based on online estimation of
value functions [32]–[35], and model-learning methods that estimate
a model, and then learn using model-based techniques [36],
[37]. Most MARL algorithms are derived from a model-free
algorithm called Q-learning [32], e.g., [13], [38]–[42].
Q-learning [32] turns (2) into an iterative approximation procedure.
The current estimate of Q ∗ is updated using estimated
samples of the right-hand side of (2). These samples are computed
using actual experience with the task, in the form of
rewards rk+1 and pairs of subsequent states xk , xk+1
Qk+1(xk ,uk)=Qk (xk ,uk)
[
+ αk rk+1 + γ max
u ′ Qk (xk+1,u ′ ) − Qk (xk ,uk) ] . (4)
Since (4) does not require knowledge about the transition
and reward functions, Q-learning is model-free. The learning
rate αk ∈ (0, 1] specifies how far the current estimate
Qk (xk ,uk) is adjusted toward the update target (sample)
rk+1 + γ maxu ′Q(xk+1,u′). The learning rate is typically time
varying, decreasing with time. Separate learning rates may be
used for each state-action pair. The expression inside the square
brackets is the temporal difference, i.e., the difference between
the estimates of Q∗ (xk ,uk) at two successive time steps, k +1
and k.
The sequence Qk provably converges to Q∗ under the following
conditions [32], [43], [44].
1) Explicit, distinct values of the Q-function are stored and
updated for each state-action pair.
2) The time series of learning rates used for each state-action
pair sums to infinity, whereas the sum of its squares is
finite.
3) The agent keeps trying all actions in all states with nonzero
probability.
The third condition means that the agent must sometimes explore,
i.e., perform other actions than dictated by the current
greedy policy. It can do that, e.g., by choosing at each step a
random action with probability ε ∈ (0, 1), and the greedy action
with probability (1 − ε). Thisisε-greedy exploration. Another
option is to use the Boltzmann exploration strategy, which in
state x selects action u with probability
h(x, u) =
eQ(x,u)/τ
∑
ũ eQ(x,ũ)/τ
where τ>0, the temperature, controls the randomness of the
exploration. When τ → 0, this is equivalent with greedy action
selection (3). When τ →∞, action selection is purely random.
For τ ∈ (0, ∞), higher-valued actions have a greater chance of
being selected than lower-valued ones.
B. Multiagent Case
The generalization of the Markov decision process to the
multiagent case is the stochastic game.
Definition 2: A stochastic game (SG) is a tuple 〈X, U1,...,
Un,f,ρ1,...,ρn 〉 where n is the number of agents, X is the
(5)BUS¸ONIU et al.: A COMPREHENSIVE SURVEY OF MULTIAGENT REINFORCEMENT LEARNING 159
discrete set of environment states, Ui, i =1,...,nare the discrete
sets of actions available to the agents, yielding the joint
action set U = U1 ×···×Un, f: X × U × X → [0, 1] is the
state transition probability function, and ρi: X × U × X → R,
i =1,...,nare the reward functions of the agents.
In the multiagent case, the state transitions are the result of
the joint action of all the agents, uk =[uT 1,k ,...,uT n,k ]T, uk ∈
U, ui,k ∈ Ui (T denotes vector transpose). Consequently, the
rewards ri,k+1 and the returns Ri,k also depend on the joint
action. The policies hi: X × Ui → [0, 1] form together the joint
policy h. TheQ-function of each agent depends on the joint
action and is conditioned on the joint policy, Qh i : X × U → R.
If ρ1 = ···= ρn, all the agents have the same goal (to maximize
the same expected return), and the SG is fully cooperative.
If n =2and ρ1 = −ρ2, the two agents have opposite goals, and
the SG is fully competitive. 2 Mixed games are stochastic games
that are neither fully cooperative nor fully competitive.
C. Static, Repeated, and Stage Games
Many MARL algorithms are designed for static (stateless)
games, or work in a stagewise fashion, looking at the static
games that arise in each state of the stochastic game. Some
game-theoretic definitions and concepts regarding static games
are, therefore, necessary to understand these algorithms [11],
[12].
A static (stateless) game is a stochastic game with X = ∅.
Since there is no state signal, the rewards depend only on the
joint actions ρi: U → R. When there are only two agents, the
game is often called a bimatrix game, because the reward function
of each of the two agents can be represented as a |U1|×|U2|
matrix with the rows corresponding to the actions of agent 1,
and the columns to the actions of agent 2, where |·| denotes set
cardinality. Fully competitive static games are also called zerosum
games, because the sum of the agents’ reward matrices is
a zero matrix. Mixed static games are also called general-sum
games, because there is no constraint on the sum of the agents’
rewards.
When played repeatedly by the same agents, the static game
is called a repeated game. The main difference from a one-shot
game is that the agents can use some of the game iterations to
gather information about the other agents or the reward functions,
and make more informed decisions thereafter. A stage
game is the static game that arises when the state of an SG is
fixed to some value. The reward functions of the stage game are
the expected returns of the SG when starting from that particular
state. Since in general the agents visit the same state of an SG
multiple times, the stage game is a repeated game.
In a static or repeated game, the policy loses the state argument
and transforms into a strategy σi: Ui → [0, 1]. An agent’s
strategy for the stage game arising in some state of the SG is its
policy for that state. MARL algorithms relying on the stagewise
2 Full competition can also arise when more than two agents are involved.
In this case, the reward functions must satisfy ρ1 (x, u,x ′ )+···+
ρn (x, u,x ′ )=0∀x, x ′ ∈ X,u ∈ U. However, the literature on RL in fully
competitive games typically deals with the two-agent case only.
approach learn strategies separately for every stage game. The
agent’s overall policy is, then, the aggregate of these strategies.
Stochastic strategies (and consequently, stochastic policies)
are of a more immediate importance in MARL than in singleagent
RL, because in certain cases, like for the Nash equilibrium
described later, the solutions can only be expressed in terms of
stochastic strategies.
An important solution concept for static games, which will
be used often in the sequel, is the Nash equilibrium. First, define
the best response of agent i to a vector of opponent strategies
as the strategy σ∗ i that achieves the maximum expected reward
given these opponent strategies
E{ri | σ1,...,σi,...,σn} ≤E{ri | σ1,...,σ ∗ i ,...,σn}∀σi.
(6)
A Nash equilibrium is a joint strategy [σ∗ 1,...,σ∗ n] T such that
each individual strategy σ∗ i is a best response to the others (see
e.g., [11]). The Nash equilibrium describes a status quo, where
no agent can benefit by changing its strategy as long as all other
agents keep their strategies constant. Any static game has at least
one (possibly stochastic) Nash equilibrium; some static games
have multiple Nash equilibria. Nash equilibria are used by many
MARL algorithms reviewed in the sequel, either as a learning
goal, or both as a learning goal and directly in the update rules.
III. BENEFITS AND CHALLENGES IN MARL
In addition to benefits owing to the distributed nature of the
multiagent solution, such as the speedup made possible by parallel
computation, multiple RL agents may harness new benefits
from sharing experience, e.g., by communication, teaching,
or imitation. Conversely, besides challenges inherited from
single-agent RL, including the curse of dimensionality and the
exploration–exploitation tradeoff, several new challenges arise
in MARL: the difficulty of specifying a learning goal, the nonstationarity
of the learning problem, and the need for coordination.
A. Benefits of MARL
A speedup of MARL can be realized thanks to parallel computation
when the agents exploit the decentralized structure of
the task. This direction has been investigated in, e.g., [45]–[50].
Experience sharing can help agents with similar tasks to learn
faster and better. For instance, agents can exchange information
using communication [51], skilled agents may serve as teachers
for the learner [52], or the learner may watch and imitate the
skilled agents [53].
When one or more agents fail in a multiagent system, the
remaining agents can take over some of their tasks. This implies
that MARL is inherently robust. Furthermore, by design, most
multiagent systems also allow the easy insertion of new agents
into the system, leading to a high degree of scalability.
Several existing MARL algorithms often require some additional
preconditions to theoretically guarantee and to fully
exploit the potential of these benefits [41], [53]. Relaxing these
conditions and further improving the performance of the various
MARL algorithms in this context is an active field of study.160 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 2, MARCH 2008
B. Challenges in MARL
The curse of dimensionality encompasses the exponential
growth of the discrete state-action space in the number of state
and action variables (dimensions). Since basic RL algorithms,
like Q-learning, estimate values for each possible discrete state
or state-action pair, this growth leads directly to an exponential
increase of their computational complexity. The complexity of
MARL is exponential also in the number of agents, because
each agent adds its own variables to the joint state-action space.
Specifying a good MARL goal in the general stochastic game
is a difficult challenge, as the agents’ returns are correlated and
cannot be maximized independently. Several types of MARL
goals have been proposed in the literature, which consider stability
of the agent’s learning dynamics [54], adaptation to the
changing behavior of the other agents [55], or both [13], [38],
[56]–[58]. A detailed analysis of this open problem is given in
Section IV.
Nonstationarity of the multiagent learning problem arises because
all the agents in the system are learning simultaneously.
Each agent is, therefore, faced with a moving-target learning
problem: the best policy changes as the other agents’ policies
change.
The exploration–exploitation tradeoff requires online (singleas
well as multiagent) RL algorithms to strike a balance between
the exploitation of the agent’s current knowledge, and
exploratory, information-gathering actions taken to improve that
knowledge. The ε-greedy policy (Section II-A) is a simple example
of such a balance. The exploration strategy is crucial for
the efficiency of RL algorithms. In MARL, further complications
arise due to the presence of multiple agents. Agents explore
to obtain information not only about the environment, but also
about the other agents (e.g., for the purpose of building models
of these agents). Too much exploration, however, can destabilize
the learning dynamics of the other agents, thus making the
learning task more difficult for the exploring agent.
The need for coordination stems from the fact that the effect
of any agent’s action on the environment depends also on the
actions taken by the other agents. Hence, the agents’ choices
of actions must be mutually consistent in order to achieve their
intended effect. Coordination typically boils down to consistently
breaking ties between equally good actions or strategies.
Although coordination is typically required in cooperative settings,
it may also be desirable for self-interested agents, e.g., to
simplify each agent’s learning task by making the effects of its
actions more predictable.
IV. MARL GOAL
In fully cooperative SGs, the common return can be jointly
maximized. In other cases, however, the agents’ returns are different
and correlated, and they cannot be maximized independently.
Specifying a good MARL goal is, in general, a difficult
problem.
In this section, the learning goals put forward in the literature
are reviewed. These goals incorporate the stability of the learning
dynamics of the agent on the one hand, and the adaptation
to the dynamic behavior of the other agents on the other hand.
Stability essentially means the convergence to a stationary policy,
whereas adaptation ensures that performance is maintained
or improved as the other agents are changing their policies.
The goals typically formulate conditions for static games,
in terms of strategies and rewards. Some of the goals can be
extended to dynamic games by requiring that the conditions are
satisfied stagewise for all the states of the dynamic game. In
this case, the goals are formulated in terms of stage strategies
instead of strategies, and expected returns instead of rewards.
Convergence to equilibria is a basic stability requirement [42],
[54]. It means the agents’ strategies should eventually converge
to a coordinated equilibrium. Nash equilibria are most
frequently used. However, concerns have been voiced regarding
their usefulness. For instance, in [14], it is argued that the link
between stagewise convergence to Nash equilibria and performance
in the dynamic SG is unclear.
In [13] and [56], convergence is required for stability, and
rationality is added as an adaptation criterion. For an algorithm
to be convergent, the authors of [13] and [56] require that the
learner converges to a stationary strategy, given that the other
agents use an algorithm from a predefined, targeted class of
algorithms. Rationality is defined in [13] and [56] as the requirement
that the agent converges to a best response when the
other agents remain stationary. Though convergence to a Nash
equilibrium is not explicitly required, it arises naturally if all the
agents in the system are rational and convergent.
An alternative to rationality is the concept of no-regret, which
is defined as the requirement that the agent achieves a return that
is at least as good as the return of any stationary strategy, and
this holds for any set of strategies of the other agents [57]. This
requirement prevents the learner from “being exploited” by the
other agents.
Targeted optimality/compatibility/safety are adaptation requirements
expressed in the form of average reward bounds [55].
Targeted optimality demands an average reward, against a targeted
set of algorithms, which is at least the average reward of a
best response. Compatibility prescribes an average reward level
in self-play, i.e., when the other agents use the learner’s algorithm.
Safety demands a safety-level average reward against all
other algorithms. An algorithm satisfying these requirements
does not necessarily converge to a stationary strategy.
Significant relationships of these requirements with other
properties of learning algorithms discussed in the literature can
be identified. For instance, opponent-independent learning is
related to stability, whereas opponent-aware learning is related
to adaptation [38], [59]. An opponent-independent algorithm
converges to a strategy that is part of an equilibrium solution
regardless of what the other agents are doing. An opponentaware
algorithm learns models of the other agents and reacts to
them using some form of best response. Prediction and rationality,
as defined in [58], are related to stability and adaptation,
respectively. Prediction is the agent’s capability to learn nearly
accurate models of the other agents. An agent is called rational
in [58] if it maximizes its expected return given its models of
the other agents.
Table I summarizes these requirements and properties of the
MARL algorithms. The stability and adaptation properties areBUS¸ONIU et al.: A COMPREHENSIVE SURVEY OF MULTIAGENT REINFORCEMENT LEARNING 161
TABLE I
STABILITY AND ADAPTATION IN MARL
given in the first two columns. Pointers to some relevant literature
are provided in the last column.
Remarks: Stability of the learning process is needed, because
the behavior of stable agents is more amenable to analysis
and meaningful performance guarantees. Moreover, a stable
agent reduces the nonstationarity in the learning problem of
the other agents, making it easier to solve. Adaptation to the
other agents is needed because their dynamics are generally unpredictable.
Therefore, a good MARL goal must include both
components. Since “perfect” stability and adaptation cannot be
achieved simultaneously, an algorithm should guarantee bounds
on both stability and adaptation measures. From a practical
viewpoint, a realistic learning goal should also include bounds
on the transient performance, in addition to the usual asymptotic
requirements.
Convergence and rationality have been used in dynamic
games in the stagewise fashion explained in the beginning of
Section IV, although their extension to dynamic games was not
explained in the papers that introduced them [13], [56]. Noregret
has not been used in dynamic games, but it could be
extended in a similar way. It is unclear how targeted optimality,
compatibility, and safety could be extended.
Fig. 1.
Breakdown of MARL algorithms by the type of task they address.
V. TAXONOMY OF MARL ALGORITHMS
MARL algorithms can be classified along several dimensions,
among which some, such as the task type, stem from properties
of multiagent systems in general. Others, like awareness of the
other agents, are specific to learning multiagent systems.
The type of task targeted by the learning algorithm leads to
a corresponding classification of MARL techniques into those
addressing fully cooperative, fully competitive, or mixed SGs. A
significant number of algorithms are designed for static (stateless)
tasks only. Fig. 1 summarizes the breakdown of MARL
algorithms by task type.
The degree of awareness of other learning agents exhibited
by MARL algorithms is strongly related to the targeted learning
goal. Algorithms focused on stability (convergence) only are
typically unaware and independent of the other learning agents.
Algorithms that consider adaptation to the other agents clearly
need to be aware to some extent of their behavior. If adaptation
is taken to the extreme and stability concerns are disregarded,
algorithms are only tracking the behavior of the other agents.
The degree of agent awareness exhibited by the algorithms can
be determined even if they do not explicitly target stability or
adaptation goals. All agent-tracking algorithms and many agent-
Fig. 2. MARL encompasses temporal-difference reinforcement learning,
game theory, and direct policy search techniques.
aware algorithms use some form of opponent modeling to keep
track of the other agents’ policies [40], [76], [77].
The field of origin of the algorithms is a taxonomy axis that
shows the variety of research inspiration benefiting MARL.
MARL can be regarded as a fusion of temporal-difference RL,
game theory, and more general, direct policy search techniques.
Temporal-difference RL techniques rely on Bellman’s equation
and originate in dynamic programming. An example is the
Q-learning algorithm. Fig. 2 presents the organization of the
algorithms by their field of origin.
Other taxonomy axes include the following. 3
1) Homogeneity of the agents’ learning algorithms: the algorithm
only works if all the agents use it (homogeneous
learning agents, e.g., team-Q, Nash-Q), or other agents
can use other learning algorithms (heterogeneous learning
agents, e.g., AWESOME, WoLF-PHC).
3 All the mentioned algorithms are discussed separately in Section VI, where
references are given for each of them.162 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 2, MARCH 2008
TABLE II
BREAKDOWN OF MARL ALGORITHMS BY TASK TYPE
AND DEGREE OF AGENT AWARENESS
Fig. 3. (Left) Two mobile agents approaching an obstacle need to coordinate
their action selection. (Right) The common Q-values of the agents for the state
depicted to the left.
2) Assumptions on the agent’s prior knowledge of the task: a
task model is available to the learning agent (model-based
learning, e.g., AWESOME) or not (model-free learning,
e.g., team-Q, Nash-Q, WoLF-PHC).
3) Assumptions on the agent’s inputs. Typically, the inputs
are assumed to exactly represent the state of the environment.
Differences appear in the agent’s observations of
other agents: an agent might need to observe the actions of
the other agents (e.g., team-Q, AWESOME), their actions
and rewards (e.g., Nash-Q), or neither (e.g., WoLF-PHC).
VI. MARL ALGORITHMS
This section reviews a representative selection of algorithms
that provides insight into the MARL state of the art. The algorithms
are grouped first by the type of task addressed, and
then by the degree of agent awareness, as depicted in Table II.
Therefore, algorithms for fully cooperative tasks are presented
first, in Section VI-A. Explicit coordination techniques that can
be applied to algorithms in any class are discussed separately in
Section VI-B. Algorithms for fully competitive tasks are reviewed
in Section VI-C. Finally, Section VI-D presents algorithms
for mixed tasks.
Algorithms that are designed only for static tasks are given
separate paragraphs in the text. Simple examples are provided
to illustrate several central issues that arise.
A. Fully Cooperative Tasks
In a fully cooperative SG, the agents have the same reward
function (ρ1 = ···= ρn) and the learning goal is to maximize
the common discounted return. If a centralized controller were
available, the task would reduce to a Markov decision process,
the action space of which would be the joint action space of
the SG. In this case, the goal could be achieved by learning the
optimal joint-action values with Q-learning
Qk+1(xk , uk )=Qk (xk , uk )
+ α [ rk+1 + γ max
u ′ Qk (xk+1, u ′ ) − Qk (xk , uk ) ]
and using the greedy policy. However, the agents are independent
decision makers, and a coordination problem arises even if
all the agents learn in parallel the common optimal Q-function
using (7). It might seem that the agents could use greedy policies
applied to Q∗ to maximize the common return
¯h ∗ i (x) =arg max max Q
u i
∗ (x, u). (8)
u 1 ,...,ui −1 ,ui +1,...,un
(7)
However, the greedy action selection mechanism breaks ties
randomly, which means that in the absence of additional mechanisms,
different agents may break ties in (8) in different ways,
and the resulting joint action may be suboptimal.
Example 1: The need for coordination. Consider the situation
illustrated in Fig. 3: Two mobile agents need to avoid an obstacle
while maintaining formation (i.e., maintaining their relative
positions). Each agent has three available actions: go straight
(Si), left (Li), or right (Ri).
For a given state (position of the agents), the Q-function can
be projected into the space of the joint agent actions. For the state
represented in Fig. 3 (left), a possible projection is represented
in the table on the right. This table describes a fully cooperative
static (stage) game. The rows correspond to the actions of agent
1, the columns to the actions of agent 2. If both agents go left,
or both go right, the obstacle is avoided while maintaining the
formation: Q(L1,L2) =Q(R1,R2) =10. If agent 1 goes left,
and agent 2 goes right, the formation is broken: Q(L1,R2) =
0. In all other cases, collisions occur and the Q-values are
negative.
Note the tie between the two optimal joint actions: (L1,L2)
and (R1,R2). Without a coordination mechanism, agent 1
might assume that agent 2 will take action R2, and therefore,
it takes action R1. Similarly, agent 2 might assume that
agent 1 will take L1, and consequently, takes L2. The resulting
joint action (R1,L2) is largely suboptimal, as the agents
collide.
1) Coordination-Free Methods: The Team Q-learning algorithm
[38] avoids the coordination problem by assuming that
the optimal joint actions are unique (which is rarely the case).
Then, if all the agents learn the common Q-function in parallel
with (7), they can safely use (8) to select these optimal joint
actions and maximize their return.
The Distributed Q-learning algorithm [41] solves the cooperative
task without assuming coordination and with limited
computation (its complexity is similar to that of single-agent Q-
learning). However, the algorithm only works in deterministic
problems. Each agent i maintains a local policy ¯ hi(x), and a
local Q-function Qi(x, ui), depending only on its own action.
The local Q-values are updated only when the update leads to
an increase in the Q-value
Qi,k+1(xk ,ui,k) = max { Qi,k(xk ,ui,k),
rk+1 + γ max Qi,k(xk+1,ui)
u i
} . (9)
This ensures that the local Q-value always captures
the maximum of the joint-action Q-values:
Qi,k(x, ui) = maxu 1 ,...,ui −1 ,ui +1,...,un Qk (x, u) at all k, whereBUS¸ONIU et al.: A COMPREHENSIVE SURVEY OF MULTIAGENT REINFORCEMENT LEARNING 163
u =[u1,...,un] T with ui fixed. The local policy is updated
only if the update leads to an improvement in the Q-values:
⎧
⎪⎨
ui,k if max Qi,k+1(xk ,ui)
u i
¯hi,k+1(xk )=
> max Qi,k(xk ,ui) (10)
⎪⎩
u i
¯hi,k(xk ) otherwise.
This ensures that the joint policy [ ¯ h1,k,..., ¯ hn,k] T is always
optimal with respect to the global Qk . Under the conditions
that the reward function is positive and Qi,0 =0∀i, the local
policies of the agents provably converge to an optimal joint
policy.
2) Coordination-Based Methods: Coordination graphs [45]
simplify coordination when the global Q-function can be additively
decomposed into local Q-functions that only depend
on the actions of a subset of agents. For instance, in an
SG with four agents, the decomposition might be Q(x, u) =
Q1(x, u1,u2)+Q2(x, u1,u3)+Q3(x, u3,u4). The decomposition
might be different for different states. Typically (like in
this example), the local Q-functions have smaller dimensions
than the global Q-function. Maximization of the joint Q-value
is done by solving simpler, local maximizations in terms of
the local value functions, and aggregating their solutions. Under
certain conditions, coordinated selection of an optimal joint
action is guaranteed [45], [46], [48].
In general, all the coordination techniques described in
Section VI-B next can be applied to the fully cooperative MARL
task. For instance, a framework to explicitly reason about possibly
costly communication is the communicative multiagent
team decision problem [78].
3) Indirect Coordination Methods: Indirect coordination
methods bias action selection toward actions that are likely to
result in good rewards or returns. This steers the agents toward
coordinated action selections. The likelihood of good values is
evaluated using, e.g., models of the other agents estimated by
the learner, or statistics of the values observed in the past.
a) Static tasks: Joint Action Learners (JAL) learn jointaction
values and employ empirical models of the other agents’
strategies [62]. Agent i learns models for all the other agents
j ̸= i, using
ˆσ i C
j (uj )=
i j (uj )
∑
ũ j ∈U j Ci j (ũj
(11)
)
where ˆσ i j is agent i’s model of agent j’s strategy and Ci j (uj )
counts the number of times agent i observed agent j taking action
uj . Several heuristics are proposed to increase the learner’s
Q-values for the actions with high likelihood of getting good
rewards given the models [62].
The Frequency Maximum Q-value (FMQ) heuristic is based
on the frequency with which actions yielded good rewards in the
past [63]. Agent i uses Boltzmann action selection (5), plugging
in modified Q-values ˜ Qi computed with the formula
˜Qi(ui) =Qi(ui)+ν Ci max(ui)
Ci (ui) rmax(ui) (12)
where rmax(ui) is the maximum reward observed after taking
action ui, Ci max(ui) counts how many times this reward has
been observed, C i (ui) counts how many times ui has been
taken, and ν is a weighting factor. Compared to single-agent
Q-learning, the only additional complexity comes from storing
and updating these counters. However, the algorithm only works
for deterministic tasks, where variance in the rewards resulting
from the agent’s actions can only be the result of the other
agents’ actions. In this case, increasing the Q-values of actions
that produced good rewards in the past steers the agent toward
coordination.
b) Dynamic tasks: In Optimal Adaptive Learning (OAL),
virtual games are constructed on top of each stage game of
the SG [64]. In these virtual games, optimal joint actions are
rewarded with 1, and the rest of the joint actions with 0. An algorithm
is introduced that, by biasing the agent toward recently
selected optimal actions, guarantees convergence to a coordinated
optimal joint action for the virtual game, and therefore,
to a coordinated joint action for the original stage game. Thus,
OAL provably converges to optimal joint policies in any fully
cooperative SG. It is the only currently known algorithm capable
of achieving this. This, however, comes at the cost of increased
complexity: each agent estimates empirically a model of the SG,
virtual games for each stage game, models of the other agents,
and an optimal value function for the SG.
4) Remarks and Open Issues: All the methods presented earlier
rely on exact measurements of the state. Many of them also
require exact measurements of the other agents’ actions. This is
most obvious for coordination-free methods: if at any point the
perceptions of the agents differ, this may lead different agents to
update their Q-functions differently, and the consistency of the
Q-functions and policies can no longer be guaranteed. Communication
might help relax these strict requirements, by providing
a way for the agents to exchange interesting data (e.g., state
measurements or portions of Q-tables) rather than rely on exact
measurements to ensure consistency [51].
Most algorithms also suffer from the curse of dimensionality.
Distributed Q-learning and FMQ are exceptions in the sense
that their complexity is not exponential in the number of agents
(but they only work in restricted settings).
B. Explicit Coordination Mechanisms
A general approach to solving the coordination problem is to
make sure that ties are broken by all agents in the same way. This
clearly requires that random action choices are somehow coordinated
or negotiated. Mechanisms for doing so, based on social
conventions, roles, and communication, are described next
(mainly following the description of Vlassis [2]). The mechanisms
here can be used for any type of task (cooperative, competitive,
or mixed).
Both social conventions and roles restrict the action choices
of the agents. An agent role restricts the set of actions available
to that agent prior to action selection, as in, e.g., [79]. This
means that some or all of the ties in (8) are prevented.
Social conventions encode aprioripreferences toward certain
joint actions, and help break ties during action selection. If
properly designed, roles or social conventions eliminate ties
completely. A simple social convention relies on a unique164 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 2, MARCH 2008
ordering of agents and actions [80]. These two orderings must
be known to all agents. Combining them leads to a unique ordering
of joint actions, and coordination is ensured if in (8) the
first joint action in this ordering is selected by all the agents.
Communication can be used to negotiate action choices, either
alone or in combination with the aforementioned techniques,
as in [2] and [81]. When combined with the aforementioned
techniques, communication can relax their assumptions
and simplify their application. For instance, in social conventions,
if only an ordering between agents is known, they can
select actions in turn, in that order, and broadcast their selection
to the remaining agents. This is sufficient to ensure coordination.
Learning coordination approaches have also been investigated,
where the coordination structures are learned online, instead
of being hardwired into the agents at inception. The agents
learn social conventions in [80], role assignments in [82], and
the structure of the coordination graph together with the local
Q-functions in [83].
Example 2: Coordination using social conventions in a fully
cooperative task. In the earlier Section VI-A (see Fig. 3), suppose
the agents are ordered such that agent 1 < agent 2 (a <b
means that a precedes b in the chosen ordering), and the actions
of both the agents are ordered in the following way:
Li <Ri <Si, i ∈{1, 2}. To coordinate, the first agent in the
ordering of the agents, agent 1, looks for an optimal joint action
such that its action component is the first in the ordering of
its actions: (L1,L2). It then selects its component of this joint
action, L1. As agent 2 knows the orderings, it can infer this
decision, and appropriately selects L2 in response. If agent 2
would still face a tie [e.g., if (L1,L2) and (L1,S2) were both
optimal], it could break this tie by using the ordering of its own
actions [which because L2 <S2 would also yield (L1,L2)].
If communication is available, only the ordering of the agents
has to be known. Agent 1, the first in the ordering, chooses an
action by breaking ties in some way between the optimal joint
actions. Suppose it settles on (R1,R2), and therefore, selects
R1. It then communicates this selection to agent 2, which can
select an appropriate response, namely the action R2.
C. Fully Competitive Tasks
In a fully competitive SG (for two agents, when ρ1 = −ρ2),
the minimax principle can be applied: maximize one’s benefit
under the worst-case assumption that the opponent will always
endeavor to minimize it. This principle suggests using opponentindependent
algorithms.
The minimax-Q algorithm [38], [39] employs the minimax
principle to compute strategies and values for the stage games,
and a temporal-difference rule similar to Q-learning to propagate
the values across state-action pairs. The algorithm is given
here for agent 1
h1,k(xk , ·) =arg m1(Qk ,xk ) (13)
Qk+1(xk ,u1,k,u2,k) =Qk (xk ,u1,k,u2,k)
+ α[rk+1 + γ m1(Qk ,xk+1)
− Qk (xk ,u1,k,u2,k)] (14)
Fig. 4. (Left) An agent (◦) attempting to reach a goal (×) while avoiding
capture by another agent (•). (Right) The Q-values of agent 1 for the state
depicted to the left (Q2 = −Q1 ).
where m1 is the minimax return of agent 1
m1(Q, x) = max
h 1 (x,·) min
∑
h1(x, u1)Q(x, u1,u2). (15)
u 2
u 1
The stochastic strategy of agent 1 in state x at time k is
denoted by h1,k(x, ·), with the dot standing for the action argument.
The optimization problem in (15) can be solved by linear
programming [84].
The Q-table is not subscripted by the agent index, because the
equations make the implicit assumption that Q = Q1 = −Q2;
this follows from ρ1 = −ρ2. Minimax-Q is truly opponent independent,
because even if the minimax optimization has multiple
solutions, any of them will achieve at least the minimax return
regardless of what the opponent is doing.
If the opponent is suboptimal (i.e., does not always take the
action that is most damaging the learner), and the learner has a
model of the opponent’s policy, it might actually do better than
the minimax return (15). An opponent model can be learned
using, e.g., the M ∗ algorithm described in [76], or a simple
extension of (11) to multiple states
ˆh i C
j (x, uj )=
i j (x, uj )
∑
ũ j ∈U j Ci j (x, ũj )
(16)
where C i j (x, uj ) counts the number of times agent i observed
agent j taking action uj in state x.
Such an algorithm then becomes opponent aware. Even agentaware
algorithms for mixed tasks (see Section VI-D4) can be
used to exploit a suboptimal opponent. For instance, WoLF-
PHC was used with promising results on a fully competitive
task in [13].
Example 3: The minimax principle. Consider the situation
illustrated in the left part of Fig. 4: agent 1 has to reach the
goal in the middle while still avoiding capture by its opponent,
agent 2. Agent 2, on the other hand, has to prevent agent 1 from
reaching the goal, preferably by capturing it. The agents can
only move to the left or to the right.
For this situation (state), a possible projection of agent 1’s
Q-function onto the joint action space is given in the table on
the right. This represents a zero-sum static game involving the
two agents. If agent 1 moves left and agent 2 does likewise,
agent 1 escapes capture, Q1(L1,L2) =0; furthermore, if at
the same time, agent 2 moves right, the chances of capture
decrease, Q1(L1,R2) =1. If agent 1 moves right and agent 2
moves left, agent 1 is captured, Q1(R1,L2) =−10; however,
if agent 2 happens to move right, agent 1 achieves the goal,
Q1(R1,R2) =10. As agent 2’s interests are opposite to those
of agent 1, the Q-function of agent 2 is the negative of Q1. For
instance, when both agents move right, agent 1 reaches the goal
and agent 3 is punished with a Q-value of −10.BUS¸ONIU et al.: A COMPREHENSIVE SURVEY OF MULTIAGENT REINFORCEMENT LEARNING 165
The minimax solution for agent 1 in this case is to move left,
because for L1, regardless of what agent 2 is doing, it can expect
a return of at least 0, as opposed to −10 for R1. Indeed, if agent
2 plays well, it will move left to protect the goal. However, it
might not play well and move right instead. If this is true and
agent 1 can find it out (e.g., by learning a model of agent 2),
it can take advantage of this knowledge by moving right and
achieving the goal.
D. Mixed Tasks
In mixed SGs, no constraints are imposed on the reward
functions of the agents. This model is, of course, appropriate
for self-interested agents, but even cooperating agents may encounter
situations where their immediate interests are in conflict,
e.g., when they need to compete for a resource. The influence
of game-theoretic elements, like equilibrium concepts, is the
strongest in the algorithms for mixed SGs. When multiple equilibria
exist in a particular state of an SG, the equilibrium selection
problem arises: the agents need to consistently pick their
part of the same equilibrium.
A significant number of algorithms in this category are designed
only for static tasks (i.e., repeated, general-sum games).
In repeated games, one of the essential properties of RL, delayed
reward, is lost. However, the learning problem is still nonstationary
due to the dynamic behavior of the agents that play the
repeated game. This is why most methods in this category focus
on adaptation to other agents.
Besides agent-independent, agent-tracking, and agent-aware
techniques, the application of single-agent RL methods to the
MARL task is also presented here. That is because single-agent
RL methods do not make any assumption on the type of task,
and are therefore, applicable to mixed SGs, although without
any guarantees for success.
1) Single-Agent RL: Single-agent RL algorithms like Q-
learning can be directly applied to the multiagent case [69].
However, the nonstationarity of the MARL problem invalidates
most of the single-agent RL theoretical guarantees. Despite its
limitations, this approach has found a significant number of
applications, mainly because of its simplicity [70], [71], [85],
[86].
One important step forward in understanding how singleagent
RL works in multiagent tasks was made recently in [87].
The authors applied results in evolutionary game theory to analyze
the dynamic behavior of Q-learning with Boltzmann policies
(5) in repeated games. It appeared that for certain parameter
settings, Q-learning is able to converge to a coordinated equilibrium
in particular games. In other cases, unfortunately, it seems
that Q-learners may exhibit cyclic behavior.
2) Agent-Independent Methods: Algorithms that are independent
of the other agents share a common structure based on
Q-learning, where policies and state values are computed with
game-theoretic solvers for the stage games arising in the states
of the SG [42], [61]. This is similar to (13) and (14); the only
difference is that for mixed games, solvers can be different from
minimax.
Denoting by {Q.,k (x, ·)} the stage game arising in state x
and given by all the agents’ Q-functions at time k, learning takes
place according to
hi,k(x, ·) =solvei{Q.,k (xk , ·)} (17)
Qi,k+1(xk , uk )=Qi,k(xk , uk )
+ α [ ri,k+1 + γ · evali{Q.,k (xk+1, ·)}
− Qi,k(xk , uk ) ]
(18)
where solvei returns agent i’s part of some type of equilibrium
(a strategy), and evali gives the agent’s expected return given
this equilibrium. The goal is the convergence to an equilibrium
in every state.
The updates use the Q-tables of all the agents. So, each agent
needs to replicate the Q-tables of the other agents. It can do that
by applying (18). This requires two assumptions: that all agents
use the same algorithm, and that all actions and rewards are
exactly measurable. Even under these assumptions, the updates
(18) are only guaranteed to maintain identical results for all
the agents if solve returns consistent equilibrium strategies for
all agents. This means the equilibrium selection problem arises
when the solution of solve is not unique.
A particular instance of solve and eval for, e.g., Nash Q-
learning [40], [54] is
{
evali{Q.,k (x, ·)} = Vi(x, NE{Q.,k (x, ·)})
(19)
solvei{Q.,k (x, ·)} = NEi{Q.,k (x, ·)}
where NE computes a Nash equilibrium (a set of strategies),
NEi is agent i’s strategy component of this equilibrium, and
Vi(x, NE{Q.,k (x, ·)}) is the expected return for agent i from
x under this equilibrium. The algorithm provably converges to
Nash equilibria for all states if either: 1) every stage game encountered
by the agents during learning has a Nash equilibrium
under which the expected return of all the agents is maximal
or 2) every stage game has a Nash equilibrium that is a saddle
point, i.e., not only does the learner not benefit from deviating
from this equilibrium, but the other agents do benefit from
this [40], [88]. This requirement is satisfied only in a small class
of problems. In all other cases, some external mechanism for
equilibrium selection is needed for convergence.
Instantiations of correlated equilibrium Q-learning (CE-Q)
[42] or asymmetric Q-learning [72] can be performed in a similar
fashion, by using correlated or Stackelberg (leader–follower)
equilibria, respectively. For asymmetric-Q, the follower does not
need to model the leader’s Q-table; however, the leader must
know how the follower chooses its actions.
Example 4: The equilibrium selection problem. Consider the
situation illustrated in Fig. 5, left: Two cleaning robots (the
agents) have arrived at a junction in a building, and each needs
to decide which of the two wings of the building it will clean. It
is inefficient if both agents clean the same wing, and both agents
prefer to clean the left wing because it is smaller, and therefore,
requires less time and energy.
For this situation (state), possible projections of the agents’ Q-
functions onto the joint action space are given in the tables on the
right. These tables represent a general-sum static game involving166 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 2, MARCH 2008
Fig. 5. (Left) Two cleaning robots negotiating their assignment to different
wings of a building. Both robots prefer to clean the smaller left wing. (Right)
The Q-values of the two robots for the state depicted to the left.
the two agents. If both agents choose the same wing, they will
not clean the building efficiently, Q1(L1,L2) =Q1(R1,R2) =
Q2(L1,L2) =Q2(R1,R2) =0. If agent 1 takes the (preferred)
left wing and agent 2 the right wing, Q1(L1,R2) =3,
and Q2(L1,R2) =2. If they choose the other way around,
Q1(R1,L2) =2, and Q2(R1,L2) =3.
For these returns, there are two deterministic Nash equilibria
4 : (L1,R2) and (R1,L2). This is easy to see: if either agent
unilaterally deviates from these joint actions, it can expect a
(bad) return of 0. If the agents break the tie between these two
equilibria independently, they might do so inconsistently and
arrive at a suboptimal joint action. This is the equilibrium selection
problem, corresponding to the coordination problem in fully
cooperative tasks. Its solution requires additional coordination
mechanisms, e.g., social conventions.
3) Agent-Tracking Methods: Agent-tracking algorithms estimate
models of the other agents’ strategies or policies (depending
on whether static or dynamic games are considered)
and act using some form of best-response to these models. Convergence
to stationary strategies is not a requirement. Each agent
is assumed capable to observe the other agents’ actions.
a) Static tasks: In the fictitious play algorithm, agent i acts
at each iteration according to a best response (6) to the models
ˆσ i 1,...,ˆσ i i−1 , ˆσi i+1 ,...,ˆσi n [65]. The models are computed
empirically using (11). Fictitious play converges to a Nash equilibrium
in certain restricted classes of games, among which are
fully cooperative, repeated games [62].
The MetaStrategy algorithm, introduced in [55], combines
modified versions of fictitious play, minimax, and a gametheoretic
strategy called Bully [89] to achieve the targeted optimality,
compatibility, and safety goals (see Section IV).
To compute best responses, the fictitious play and MetaStrategy
algorithms require a model of the static task, in the form of
reward functions.
The Hyper-Q algorithm uses the other agents’ models as
a state vector and learns a Q-function Qi(ˆσ1,...,ˆσi−1, ˆσi+1,
...,ˆσn,ui) with an update rule similar to Q-learning [68]. By
learning values of strategies instead of only actions, Hyper-
Q should be able to adapt better to nonstationary agents. One
inherent difficulty is that the action selection probabilities in
4 There is also a stochastic (mixed) Nash equilibrium, where each agent
goes left with a probability 3/5. This is because the strategies σ1 (L1 )=
3/5,σ1 (R1 )=2/5 and σ2 (L2 )=3/5,σ2 (R2 )=2/5 are best responses
to one another. The expected return of this equilibrium for both agents is 6/5,
worse than for any of the two deterministic equilibria.
the models are continuous variables. This means the classical,
discrete-state Q-learning algorithm cannot be used. Less understood,
approximate versions of it are required instead.
b) Dynamic tasks: The Nonstationary Converging Policies
(NSCP) algorithm [73] computes a best response to the models
and uses it to estimate state values. This algorithm is very similar
to (13) and (14) and (17) and (18); this time, the stage game
solver gives a best response
hi,k(xk , ·) =arg bri(Qi,k,xk ) (20)
Qi,k+1(xk , uk )=Qk (xk , uk )+α[ri,k+1 + γbri(Qi,k,xk+1)
− Qk (xk , uk )] (21)
where the best-response value operator br is implemented as
∑
bri(Qi,x) = max hi(x, ui)·
Qi(x, u1,...,un)
h i (x,·)
u 1 ,...,un
n∏
j=1,j̸=i
ˆh i j (x, uj ). (22)
The empirical models ˆ hi j are learned using (16). In the computation
of br, the value of each joint action is weighted by
the estimated probability of that action being selected, given the
models of the other agents [the product term in (22)].
4) Agent-Aware Methods: Agent-aware algorithms target
convergence, as well as adaptation to the other agents. Some
algorithms provably converge for particular types of tasks
(mostly static), others use heuristics for which convergence is
not guaranteed.
a) Static tasks: The algorithms presented here assume the
availability of a model of the static task, in the form of reward
functions. The AWESOME algorithm [60] uses fictitious play,
but monitors the other agents and, when it concludes that they
are nonstationary, switches from the best response in fictitious
play to a centrally precomputed Nash equilibrium (hence the
name: Adapt When Everyone is Stationary, Otherwise Move to
Equilibrium). In repeated games, AWESOME is provably rational
and convergent [60] according to the definitions from [56]
and [13] given in Section IV.
Some methods in the area of direct policy search use gradient
update rules that guarantee convergence in specific classes of
static games: Infinitesimal Gradient Ascent (IGA) [66], Winor-Learn-Fast
IGA (WoLF-IGA) [13], Generalized IGA (GIGA)
[67], and GIGA-WoLF [57]. For instance, IGA and WoLF-IGA
work in two-agent, two-action games, and use similar gradient
update rules
⎧
⎪⎨
∂E{r1 | α, β}
αk+1 = αk + δ1,k
∂α
⎪⎩ βk+1 = βk + δ2,k
∂E{r2 | α, β}
.
∂β
(23)
The strategies of the agents are sufficiently represented by
the probability of selecting the first out of the two actions, α
for agent 1 and β for agent 2. IGA uses constant gradient steps
δ1,k = δ2,k = δ, and the average reward of the policies converges
to Nash rewards for an infinitesimal step size (i.e., whenBUS¸ONIU et al.: A COMPREHENSIVE SURVEY OF MULTIAGENT REINFORCEMENT LEARNING 167
δ → 0). In WoLF-IGA, δi,k switches between a smaller value
when agent i is winning, and a larger value when it is losing
(hence the name, Win-or-Learn-Fast). WoLF-IGA is rational by
the definition in Section IV, and convergent for infinitesimal
step sizes [13] (δi,k → 0 when k →∞).
b) Dynamic tasks: Win-or-Learn-Fast Policy Hill-Climbing
(WoLF-PHC) [13] is a heuristic algorithm that updates Q-
functions with the Q-learning rule (4), and policies with a WoLF
rule inspired from (23)
hi,k+1(xk ,ui) =hi,k(xk ,ui)
δi,k =
⎧
⎪⎨
δi,k if ui = arg max Qi,k+1(xk , ũi)
ũ i
+
⎪⎩ −
δi,k
otherwise
|Ui|−1
{
δwin if winning
δlose
if losing.
(24)
(25)
The gradient step δi,k is larger when agent i is losing than
when it is winning: δlose >δwin. For instance, in [13], δlose is
two to four times larger than δwin. The rationale is that the agent
should escape fast from losing situations, while adapting cautiously
when it is winning, in order to encourage convergence.
The win/lose criterion in (25) is based either on a comparison
of an average policy with the current one, in the original version
of WoLF-PHC, or on the second-order difference of policy
elements, in PD-WoLF [74].
The Extended Optimal Response (EXORL) heuristic [75] applies
a complementary idea in two-agent tasks: the policy update
is biased in a way that minimizes the other agent’s incentive to
deviate from its current policy. Thus, convergence to a coordinated
Nash equilibrium is expected.
5) Remarks and Open Issues: Static, repeated games represent
a limited set of applications. Algorithms for static games
provide valuable theoretical results; these results should however
be extended to dynamic SGs in order to become interesting
for more general classes of applications (e.g., WoLF-PHC [13]
is such an extension). Most static game algorithms also assume
the availability of an exact task model, which is rarely the case
in practice. Versions of these algorithms that can work with imperfect
and/or learned models would be interesting (e.g., GIGA-
WoLF [57]). Many algorithms for mixed SGs suffer from the
curse of dimensionality, and are sensitive to imperfect observations;
the latter holds especially for agent-independent methods.
Game theory induces a bias toward static (stagewise) solutions
in the dynamic case, as seen, e.g., in the agent-independent
Q-learning template (17)–(18) and in the stagewise win/lose criteria
in WoLF algorithms. However, the suitability of such stagewise
solutions in the context of the dynamic task is currently
unclear [14], [17].
One important research step is understanding the conditions
under which single-agent RL works in mixed SGs, especially in
light of the preference toward single-agent techniques in practice.
This was pioneered by the analysis in [87].
VII. APPLICATION DOMAINS
MARL has been applied to a variety of problem domains,
mostly in simulation but also to some real-life tasks. Simulated
domains dominate for two reasons. The first reason is that results
in simpler domains are easier to understand and to use for
gaining insight. The second reason is that in real life, scalability
and robustness to imperfect observations are necessary, and few
MARL algorithms exhibit these properties. In real-life applications,
more direct derivations of single-agent RL (see Section
VI-D1) are preferred [70], [85], [86], [90].
In this section, several representative application domains are
reviewed: distributed control, multirobot teams, trading agents,
and resource management.
A. Distributed Control
In distributed control, a set of autonomous, interacting controllers
act in parallel on the same process. Distributed control
is a meta-application for cooperative multiagent systems: any
cooperative multiagent system is a distributed control system
where the agents are the controllers, and their environment is the
controlled process. For instance, in cooperative robotic teams,
the control algorithms of the robots identify with the controllers,
and the robots’ environment together with their sensors and actuators
identify with the process.
Particular distributed control domains where MARL is applied
are process control [90], control of traffic signals [91],
[92], and control of electrical power networks [93].
B. Robotic Teams
Robotic teams (also called multirobot systems) are the most
popular application domain of MARL, encountered under the
broadest range of variations. This is mainly because robotic
teams are a very natural application of multiagent systems, but
also because many MARL researchers are active in the robotics
field. The robots’ environment is a real or simulated spatial domain,
most often having two dimensions. Robots use MARL to
acquire a wide spectrum of skills, ranging from basic behaviors
like navigation to complex behaviors like playing soccer.
In navigation, each robot has to find its way from a starting
position to a fixed or changing goal position, while avoiding
obstacles and harmful interference with other robots [13], [54].
Area sweeping involves navigation through the environment
for one of several purposes: retrieval of objects, coverage of as
much of the environment surface as possible, and exploration,
where the robots have to bring into sensor range as much of the
environment surface as possible [70], [85], [86].
Multitarget observation is an extension of the exploration
task, where the robots have to maintain a group of moving
targets within sensor range [94], [95].
Pursuit involves the capture of moving targets by the robotic
team. In a popular variant, several “predator” robots have to
capture a “prey” robot by converging on it [83], [96].
Object transportation requires the relocation of a set of objects
into given final positions and configurations. The mass or size of
some of the objects may exceed the transportation capabilities168 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 2, MARCH 2008
of one robot, thus requiring several robots to coordinate in order
to bring about the objective [86].
Robot soccer is a popular, complex test-bed for MARL, that
requires most of the skills enumerated earlier [4], [97]–[100].
For instance, intercepting the ball and leading it into the goal
involve object retrieval and transportation skills, while the strategic
placement of the players in the field is an advanced version
of the coverage task.
C. Automated Trading
Software trading agents exchange goods on electronic markets
on behalf of a company or a person, using mechanisms such
as negotiations and auctions. For instance, the Trading Agent
Competition is a simulated contest where the agents need to arrange
travel packages by bidding for goods such as plane tickets
and hotel bookings [101].
MARL approaches to this problem typically involve
temporal-difference [34] or Q-learning agents, using approximate
representations of the Q-functions to handle the large
state space [102]–[105]. In some cases, cooperative agents represent
the interest of a single company or individual, and merely
fulfil different functions in the trading process, such as buying
and selling [103], [104]. In other cases, self-interested agents
interact in parallel with the market [102], [105], [106].
D. Resource Management
In resource management, the agents form a cooperative team,
and they can be one of the following.
1) Managers of resources, as in [5]. Each agent manages one
resource, and the agents learn how to best service requests
in order to optimize a given performance measure.
2) Clients of resources, as in [107]. The agents learn how
to best select resources such that a given performance
measure is optimized.
A popular resource management domain is network routing
[108]–[110]. Other examples include elevator scheduling [5]
and load balancing [107]. Performance measures include average
job processing times, minimum waiting time for resources,
resource usage, and fairness in servicing clients.
E. Remarks
Though not an application domain per se, game-theoretic,
stateless tasks are often used to test MARL approaches. Not
only algorithms specifically designed for static games are tested
on such tasks (e.g., AWESOME [60], MetaStrategy [55], GIGA-
WoLF [57]), but also others that can, in principle, handle dynamic
SGs (e.g., EXORL [75]).
As an avenue for future work, note that distributed control
is poorly represented as an MARL application domain. This
includes not only complex systems such as traffic, power, or
sensor networks, but also simpler dynamic processes that have
been successfully used to study single-agent RL (e.g., various
types of pendulum systems).
VIII. OUTLOOK
In the previous sections of this survey, the benefits and challenges
of MARL have been reviewed, together with the approaches
to address these challenges and exploit the benefits.
Specific discussions have been provided for each particular
subject. In this section, more general open issues are given,
concerning the suitability of MARL algorithms in practice, the
choice of the multiagent learning goal, and the study of the joint
environment and learning dynamics.
A. Practical MARL
Most MARL algorithms are applied to small problems only,
like static games and small grid worlds. As a consequence,
these algorithms are unlikely to scale up to real-life multiagent
problems, where the state and action spaces are large or even
continuous. Few of them are able to deal with incomplete, uncertain
observations. This situation can be explained by noting that
scalability and uncertainty are also open problems in singleagent
RL. Nevertheless, improving the suitability of MARL
to problems of practical interest is an essential research step.
Next, we describe several directions in which this research can
proceed, and point to some pioneering work done along these
directions. Such work mostly combines single-agent algorithms
with heuristics to account for multiple agents.
Scalability is the central concern for MARL as it stands
today. Most algorithms require explicit tabular storage of the
agents’ Q-functions and possibly of their policies. This limits
the applicability of the algorithms to problems with a relatively
small number of discrete states and actions. When the state
and action spaces contain a large number of elements, tabular
storage of the Q-function becomes impractical. Of particular
interest is the case when states and possibly actions are continuous
variables, making exact Q-functions impossible to store.
In these cases, approximate solutions must be sought, e.g., by
extending to multiple agents the work on approximate singleagent
RL [111]–[122]. A fair number of approximate MARL
algorithms have been proposed: for discrete, large state-action
spaces, e.g., [123], for continuous states and discrete actions,
e.g., [96], [98], and [124], and finally for continuous states and
actions, e.g., [95], and [125]. Unfortunately, most of these algorithms
only work in a narrow set of problems and are heuristic in
nature. Significant advances in approximate MARL can be made
if the wealth of theoretical results on single-agent approximate
RL is put to use [112], [113], [115]–[119].
A complementary avenue for improving scalability is the discovery
and exploitation of the decentralized, modular structure
of the multiagent task [45], [48]–[50].
Providing domain knowledge to the agents can greatly help
them in learning solutions to realistic tasks. In contrast, the large
size of the state-action space and the delays in receiving informative
rewards mean that MARL without any prior knowledge is
very slow. Domain knowledge can be supplied in several forms.
If approximate solutions are used, a good way to incorporate domain
knowledge is to structure the approximator in a way that
ensures high accuracy in important regions of the state-action
space, e.g., close to the goal. Informative reward functions, alsoBUS¸ONIU et al.: A COMPREHENSIVE SURVEY OF MULTIAGENT REINFORCEMENT LEARNING 169
rewarding promising behaviors rather than only the achievement
of the goal, could be provided to the agents [70], [86].
Humans or skilled agents could teach unskilled agents how to
solve the task [126]. Shaping is a technique whereby the learning
process starts by presenting the agents with simpler tasks,
and progressively moves toward complex ones [127]. Preprogrammed
reflex behaviors could be built into the agents [70],
[86]. Knowledge about the task structure could be used to decompose
it into subtasks, and learn a modular solution with,
e.g., hierarchical RL [128]. Last, but not the least, if a (possibly
incomplete) task model is available, this model could be
used with model-based RL algorithms to initialize Q-functions
to reasonable, rather than arbitrary, values.
Incomplete, uncertain state measurements could be handled
with techniques related to partially observable Markov decision
processes [129], as in [130] and [131].
B. Learning Goal
The issue of a suitable MARL goal for dynamic tasks with
dynamic, learning agents, is a difficult open problem. MARL
goals are typically formulated in terms of static games. Their
extension to dynamic tasks, as discussed in Section IV, is not
always clear or even possible. If an extension via stage games
is possible, the relationship between the extended goals and
performance in the dynamic task is not clear, and is to the
authors’ best knowledge not made explicit in the literature. This
holds for stability requirements, like convergence to equilibria
[42], [54], as well as for adaptation requirements, like rationality
[13], [56].
Stability of the learning process is needed, because the behavior
of stable agents is more amenable to analysis and meaningful
performance guarantees. Adaptation to the other agents
is needed because their dynamics are generally unpredictable.
Therefore, a good multiagent learning goal must include both
components. This means that MARL algorithms should neither
be totally independent of the other agents, nor just track their
behavior without concerns for convergence.
Moreover, from a practical viewpoint, a realistic learning goal
should include bounds on the transient performance, in addition
to the usual asymptotic requirements. Examples of such bounds
include maximum time constraints for reaching a desired performance
level, or a lower bound on instantaneous performance
levels. Some steps in this direction have been taken in [55]
and [57].
C. Joint Environment and Learning Dynamics
The stagewise application of game-theoretic techniques to
solve dynamic multiagent tasks is a popular approach. It may,
however, not be the most suitable, given that both the environment
and the behavior of learning agents are generally dynamic
processes. So far, game-theory-based analysis has only been applied
to the learning dynamics of the agents [28], [87], [132],
while the dynamics of the environment have not been explicitly
considered. We expect that tools developed in the area of
robust control will play an important role in the analysis of the
learning process as a whole (i.e., interacting environment and
learning dynamics). In addition, this framework can incorporate
prior knowledge on bounds for imperfect observations, such as
noise-corrupted variables.
IX. CONCLUSION
MARL is a young, but active and rapidly expanding field of
research. It integrates results from single-agent reinforcement
learning, game theory, and direct search in the space of behaviors.
The promise of MARL is to provide a methodology and an
array of algorithms enabling the design of agents that learn the
solution to a nonlinear, stochastic task about which they possess
limited or no prior knowledge.
In this survey, we have discussed in detail a representative set
of MARL techniques for fully cooperative, fully competitive,
and mixed tasks. Algorithms for dynamic tasks were analyzed
more closely, but techniques for static tasks were investigated as
well. A classification of MARL algorithms was given, and the
different viewpoints on the central issue of the MARL learning
goal were presented. We have provided an outlook synthesizing
several of the main open issues in MARL, together with
promising ways of addressing these issues. Additionally, we
have reviewed the main challenges and benefits of MARL, as
well as several representative problem domains where MARL
techniques have been applied.
Many avenues for MARL are open at this point, and many
research opportunities present themselves. In particular, control
theory can contribute in addressing issues such as stability of
learning dynamics and robustness against uncertainty in observations
or the other agents’ dynamics. In our view, significant
progress in the field of multiagent learning can be achieved by a
more intensive cross fertilization between the fields of machine
learning, game theory, and control theory.
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Lucian Bus¸oniu received the M.Sc. degree and the
Postgraduate Diploma from the Technical University
of Cluj Napoca, Cluj Napoca, Romania, in 2003
and 2004, respectively, both in control engineering.
Currently, he is working toward the Ph.D. degree at
the Delft Center for Systems and Control, Faculty of
Mechanical Engineering, Delft University of Technology,
Delft, The Netherlands.
His current research interests include reinforcement
learning in multiagent systems, approximate
reinforcement learning, and adaptive and learning
control.
Robert Babuˇska received the M.Sc. degree in control
engineering from the Czech Technical University
in Prague, Prague, Czech Republic, in 1990, and the
Ph.D. degree from the Delft University of Technology,
Delft, The Netherlands, in 1997.
He was with the Department of Technical Cybernetics,
Czech Technical University in Prague and
with the Faculty of Electrical Engineering, Delft University
of Technology. Currently, he is a Professor
at the Delft Center for Systems and Control, Faculty
of Mechanical Engineering, Delft University of
Technology. He is involved in several projects in mechatronics, robotics, and
aerospace. He is the author or coauthor of more than 190 publications, including
one research monograph (Kluwer Academic), two edited books, 24 invited
chapters in books, 48 journal papers, and more than 120 conference papers. His
current research interests include neural and fuzzy systems for modeling and
identification, fault-tolerant control, learning, and adaptive control, and dynamic
multiagent systems.
Prof. Babuˇska is the Chairman of the IFAC Technical Committee on Cognition
and Control. He has served as an Associate Editor of the IEEE TRANSAC-
TIONS ON FUZZY SYSTEMS, Engineering Applications of Artificial Intelligence,
and as an Area Editor of Fuzzy Sets and Systems.
Bart De Schutter received the M.Sc. degree
in electrotechnical-mechanical engineering and the
Ph.D. degree in applied sciences (summa cum
laude with congratulations of the examination jury)
from Katholieke Universiteit Leuven (K.U. Leuven),
Leuven, Belgium, in 1991 and 1996, respectively.
He was a Postdoctoral Researcher at the ESAT-
SISTA Research Group of K.U. Leuven. In 1998,
he was with the Control Systems Engineering
Group, Faculty of Information Technology and Systems,
Delft University of Technology, Delft, The
Netherlands. Currently, he is a Full Professor at the Delft Center for Systems
and Control, Faculty of Mechanical Engineering, Delft University of Technology,
where he is also associated with the Department of Marine and Transport
Technology. His current research interests include multiagent systems, intelligent
transportation systems, control of transportation networks, hybrid systems
control, discrete-event systems, and optimization.
Prof. De Schutter was the recipient of the 1998 SIAM Richard C. DiPrima
Prize and the 1999 K.U. Leuven Robert Stock Prize for his Ph.D. thesis.