Independent Learning in Stochastic Games

Introduction

Reinforcement learning (RL) in which autonomous agents make decisions in unknown dynamic environments has emerged as the backbone of many artificial intelligence (AI) problems. The frontier of many AI systems emerges in multi-agent settings, including playing games such as chess and Go (Silver et al., 2016, 2017), robotic manipulation with multiple connected arms (Gu et al., 2017), autonomous vehicle control in dynamic traffic and automated warehouses or production facilities (Shalev-Shwartz et al., 2016; Yang et al., 2020). Further advances in these problems critically depend on developing stable and agent incentive-compatible learning dynamics in multi-agent environment. Unfortunately, the mathematical framework upon which classical RL depends on is inadequate for multi-agent learning, since it assumes an agent’s environment is stationary and does not contain any adaptive agents.

The topic of multi-agent learning has a long history in game theory, almost as long as the discipline itself. One of the most studied models of learning in games is fictitious play, introduced by Brown (Brown, 1951), with first rigorous convergence analysis presented by Robinson (Robinson, 1951) for its discrete-time variant and for finite two-player zero-sum games. See also Miyasawa (1961); Shapley (1964); Fudenberg and Kreps (1993); Fudenberg and Levine (1995); Monderer and Shapley (1996a); Hofbauer and Sandholm (2002a) and others for the analysis of fictitious play. In fictitious play, each agent is myopic (i.e., she does not take into account the fact that her current action will have an impact on the future actions of other playersHereafter, we use player and agent interchangeably.), and therefore chooses a best response to the opponent’s strategy, which she estimates to be the empirical distribution of past play. Despite extensive study on learning in repeated play of static complete-information games (also referred to as strategic-form or normal-form games) and the importance of the issues, there is limited progress on multi-agent learning in dynamic environments (where the environments evolve over time). The key challenge is to estimate the decision rules of other agents that in turn adapt their behavior to changing non-stationary environments.

In this review paper, we first present stochastic games, first introduced by Shapley (1953), as a model for representing dynamic multi-agent interactions in §3.The preliminary information on strategic-form games and learning in strategic-form games with repeated play are provided in §2. Stochastic games extend strategic-form games to dynamic settings where the environment changes with players’ decisions. They also extend single-agent Markov decision problems (Markov decision processes) to competitive situations with more than one decision-maker. Developing simple and independent learning rules, e.g., the fictitious-play/best-response type dynamics, for stochastic games has been an open question for some time in the literature (see Condon (1990); Tan (1993); Claus and Boutilier (1998) for some negative non-convergent results due to non-stationarity).

In the second part of the paper in §4, we present recently proposed simple and independent learning rules from (Sayin et al., 2020, 2021), and show their convergence for zero-sum stochastic games. Crucially, these rules are based on fictitious play-type dynamics and unlike earlier works, do not require coordination between agents, leading to fully decentralized and independent multi-agent learning dynamics. We combine ideas from game theory and RL in developing these learning rules, and consider three different settings: model-based setting where players know their payoff functions, transition probabilities of the underlying stochastic games, and observe opponent’s actions; model-free setting where players do not know payoff functions and transition probabilities but can still observe the opponent’s actions; and the minimal information setting where players do not even observe opponent’s actions. In all three settings, the players do not know the opponent’s objective, i.e., they do not possess the knowledge that the underlying game is zero-sum. In the minimal-information setting, the players may not even know the existence of an opponent.

In §5, we have also reviewed several other algorithms/learning dynamics, and their convergence results for multi-agent learning in stochastic games. We cover both results from the game theory literature that typically assumes knowledge of the model of the players’ payoff functions, and the transition probabilities of the underlying stochastic games, and also from the RL literature which posit learning dynamics that perform updates without knowing the transition probabilities. Most of these update rules typically involve coordination and computationally intensive steps for the players. These algorithms can be viewed more as ones for computing the Nash equilibrium of the stochastic games, as opposed to natural learning dynamics that would be adopted by self-interested agents interested in maximizing their own payoffs given their inferences (as captured in our learning dynamics). Finally, we conclude the paper with open questions on independent learning in stochastic games in §6.

Preliminary: Strategic-Form Games

A two-player strategic-form game can be characterized by a tuple $\langle A^{1},A^{2},r^{1},r^{2}\rangle$, in which

The finite set of actions that player $i$ can take is denoted by $A^{i}$,

Each player $i$ takes an action from her action set $A^{i}$ simultaneously and receives the payoff $r^{i}(a^{1},a^{2})$.

We let players choose a mixed strategy to randomize their actions independently. For example, $\pi^{i}:A^{i}\rightarrow$ denotes the mixed strategy of player $i$ such that $\pi^{i}(a^{i})$ corresponds to the probability that player $i$ plays $a^{i}$. Note that we have $\sum_{a^{i}\in A^{i}}\pi^{i}(a^{i})=1$ by its definition.

We represent the strategy profile and action profile of the players by $\pi=(\pi^{1},\pi^{2})$ and $a=(a^{1},a^{2})$, respectively. Under the strategy profile $\pi$, the expected payoff of player $i$ is defined by

Note that the expected payoff of player $i$ is affected by the strategy of the opponent. We next introduce the Nash equilibrium where players do not have any (or large enough) incentive to change their strategies unilaterally.

A strategy profile $\pi_{*}$ is a mixed-strategy $\varepsilon$-Nash equilibrium with $\varepsilon\geq 0$ if we have

Furthermore, $\pi_{*}$ is a mixed-strategy Nash equilibrium if (2.1) holds with $\varepsilon=0$.

The following is the classical existence result for any strategic-form game (e.g., see (Başar and Olsder, 1999, Theorem 3.2)).

In strategic-form games (with finitely many players and finitely many actions), a mixed-strategy equilibrium always exists.

The key question is whether an equilibrium can be realized or not in the interaction of self-interested decision-makers. In general, finding the best strategy against another decision-maker is not a well-defined optimization problem because the best strategy that reflects the viewpoint of the individual depends on the opponent’s strategy. Therefore, players are generally not able to compute their best strategy beforehand. When there exists a unique equilibrium, we can expect the players to identify their equilibrium strategies as a result of an introspective thinking process. For example, what would the opponent choose? What would the opponent have chosen if she knew I am considering what she would pick while choosing my strategy? And so on. However, many empirical analyses suggest that an equilibrium would not typically be realized in one shot even with such reasoning (see, e.g., Fudenberg and Levine (1998)).

It is instructive to consider the following well-known example: Consider a game played among $n>1$ students. The teacher asks the students to pick a number between and $100$, and submit it within a closed envelope. The winner will be the one who chooses the number closest to the $2/3$rd of the average of all numbers picked. It can be seen that the unique equilibrium is the strategy profile where every player chooses . We would expect the students to pick as a result of an introspective thinking process, however, empirical studies show that they typically pick numbers other than zero such that their average end up around $30$ with its $2/3$rd around $20$ (Nagel, 1995). This results in players who have selected by strategizing their actions introspectively losing the game. However, if the game is played repeatedly with players observing chosen actions, each player will have a tendency to pick numbers closer to the winning number (or its $2/3$rd if they notice that others can also have such a tendency to pick the number closest to the winning one). This results in convergence to the equilibrium play along repeated play of the game, even when the players have not engaged in any forward-looking strategy.

Many games have multiple equilibria which makes coordination and selection through introspective thinking challenging. On the other hand, empirical studies suggest even in strategic situations equipped with multiple equilibria, individual agents reach an equilibrium as long as they engage with each other multiple times and receive feedback to revise their strategies (Fudenberg and Levine, 1998).

In the following, we review the canonical models of learning with multiple agents through repeated interactions.

Suppose that players know the primitives of the game, i.e., $\langle A^{1},A^{2},r^{1},r^{2}\rangle$. If players knew the opponent’s strategy, computation of the best strategy is a simple optimization problem where they pick one of the maxima among linearly ordered finitely many elements. However, players do not know the opponent’s strategy. When they play the same game repeatedly and observe the opponent’s actions in these games, they have a chance to reason about what the opponent would play in the next repetition of the game. Therefore, they can estimate the opponent’s strategy based on the history of the play. However, the opponent is not necessarily playing according to a stationary strategy since she is also a strategic decision-maker who can adapt her strategy according to her best interest.

Fictitious play is a simple and stylist learning dynamic where players (erroneously) assume that the opponent plays according to a stationary strategy.It is called fictitious play because Brown (1951) introduced it as an introspective thinking process that a player can play by herself. This assumption lets players form a belief on the opponent’s strategy based on the history of the play, e.g., the empirical distribution of the actions taken. Then, the players can adapt their strategies based on the belief constructed.

Fictitious play, since its first introduction by Brown (1951), has become the most appealing best-response type learning dynamics in game theory. Formally, at iteration $k$, player $i$ maintains a belief on the opponent’s strategy, denoted by $\hat{\pi}^{-i}_{k}\in\Delta(A^{-i})$.We represent the probability simplex over a set $A$ by $\Delta(A)$. For example, the belief can correspond to the empirical average of the actions taken in the past. Note that we can view an action $a^{i}$ as a deterministic strategy in which the action is played with probability $1$, i.e., $a^{i}\in\Delta(A^{i})$ with slight abuse of notation. Then, the empirical average is given by

The belief $\hat{\pi}_{k+1}^{-i}$ can be computed iteratively using bounded memory according to

with arbitrary initialization $\hat{\pi}_{0}^{-i}\in\Delta(A^{-i})$. In other words, players do not have to remember every action taken by the opponent in the past. Moreover, player $i$ selects her action following

with an arbitrary tie-breaking rule, playing a greedy best-response to the belief she maintains on opponent’s strategy.

The two-player zero-sum strategic-form games have fictitious play property (Robinson, 1951).

The $n$-player identical-interest strategic-form games have fictitious play property (Monderer and Shapley, 1996a).

Alternative to the insightful proofs in (Robinson, 1951) and (Monderer and Shapley, 1996a), we can establish a connection between fictitious play and continuous-time best response dynamics to characterize its convergence properties. For example, Harris (1998) provided a proof for the continuous-time best-response dynamics in zero-sum strategic-form games through a Lyapunov function formulation. This convergence result also implies the convergence of fictitious play in repeated play of the same zero-sum strategic-form game. We next briefly describe Harris (1998)’s approach to convergence analysis for continuous-time best-response dynamics.

In continuous-time best response dynamics, the strategies $(\pi^{1},\pi^{2})$ evolve according to the following differential inclusion

for $i=1,2$. We highlight the resemblance between (2.3) and (2.5) because we can view (2.5) as the limiting flow of (2.3) as $1/(k+1)\rightarrow 0$. Note also that there exists an absolutely continuous solution to this differential inclusion (Harris, 1998). To characterize the convergence properties of this flow, Harris (1998) showed that the function

Generally, the convergence of the limiting flow would not lead to the convergence of the discrete-time update. However, based on tools from differential inclusion approximation theory (Benaim et al., 2005), the existence of such a Lyapunov function yields that the fictitious play dynamics converge to an equilibrium since its linear interpolation after certain transformation of the time axis can be viewed as a perturbed solution to the differential inclusion (2.5) with asymptotically negligible perturbation while the existence of Lyapunov function yields that any such perturbed solution also converges to the zero-set of the Lyapunov function, i.e., $\{\pi:V(\pi)=0\}$.

The fictitious play dynamics enjoy the following desired properties (Fudenberg and Levine, 1998): $i)$ The dynamics do not require knowledge of the underlying game’s class, e.g., the opponent’s payoff function, and is not specific to any specific class of games; $ii)$ Players attain the best-response performance against an opponent following an asymptotically stationary strategy, i.e., the learning dynamics is rational; $iii)$ If the dynamics converge, it must converge to an equilibrium of the underlying game.

Unfortunately, there exist strategic-form games that do not have fictitious play property as shown by Shapley (1964) through a counter-example. The classes of strategic-form games with fictitious play property have been studied extensively, e.g., see (Robinson, 1951; Miyasawa, 1961; Milgrom and Roberts, 1991; Monderer and Sela, 1996; Monderer and Shapley, 1996a, b; Sela, 1999; Berger, 2005, 2008). Variants of fictitious play, including smoothed fictitious play (Fudenberg and Kreps, 1993) and weakened fictitious play (Van der Genugten, 2000) have also been studied extensively. However, all these studies focus on the repeated play of the same strategic-form game at every stage. There are very limited results on dynamic games where players interact repeatedly while the game played at a stage (called stage-game) evolves with their actions. Note that players need to consider the impact of their actions in their future payoffs as in dynamic programming or optimal control when they have utilities defined over infinite horizon.

In the next section, we introduce stochastic games, a special (and important class) of dynamic games where the stage-games evolve over infinite horizon based on the current actions of players.

Stochastic Games

Stochastic games (also known as Markov games), since its first introduction by Shapley (1953), have been widely used as a canonical model for dynamic multi-agent interactions (e.g., see the surveys (Busoniu et al., 2008; Zhang et al., 2021)). At each time $k=0,1,\ldots$, players play a stage game that corresponds to a particular state of a multi-state environment. The stage games evolve stochastically according to the transition probabilities of the states controlled jointly by the actions of both players. The players receive a payoff which is some aggregate of the stage payoffs; a typical model is to assume the players receive a discounted sum of stage payoffs over an infinite horizon.

Formally, a two-player stochastic game is characterized by a tuple $\langle S,A^{1},A^{2},r^{1},r^{2},p,\gamma\rangle$, in which

The finite set of states is denoted by $S$,

The finite set of actions that player $i$ can take at any state is denoted by $A^{i}$,The formulation can be generalized to the case where the action spaces depend on state in a rather straightforward way.

For any pair of states $(s,s^{\prime})$ and action profile $a\in A$, we define $p(s^{\prime}|s,a)$ as the transition probability from $s$ to $s^{\prime}$ given action profile $a$.

The players also discount the impact of future payoff in their utility with the discount factor $\gamma\in[0,1)$.

The objective of player $i$ is to maximize the expected sum of discounted stage-payoffs collected over infinite horizon, given by

where $a_{k}\in A$ denotes the action profile played at stage $k$, $\{s_{0}\sim p_{o},s_{k+1}\sim p(\cdot|s_{k},a_{k}),k\geq 0\}$ is a stochastic process representing the state at each stage $k$ and $p_{o}\in\Delta(S)$ is the initial state distribution. The expectation is taken with respect to randomness due to stochastic state transitions and actions mixed independently by the players.

The players can play an infinite sequence of (mixed) actions. When they have perfect recall, they can mix their actions independently according to a behavioral strategy in which the probability of an action is taken depends on the history of states and action profiles, e.g., $h_{k}=\{s_{0},a_{0},s_{1},a_{1},\ldots,s_{k-1},a_{k-1},s_{k}\}$ at stage $k$. This results in an infinite-dimensional strategy space, and therefore, the universal result for the existence of an equilibrium, Theorem 2.2, does not apply here. On the other hand, stochastic games can also be viewed as a generalization of Markov decision processes (MDPs) to multi-agent cases since state transition probabilities depend only on the current state and current action profile of players. Behavioral strategies that depend only on the final state of the history (which corresponds to the current state) are known as Markov strategies. Furthermore, we call a Markov strategy by a stationary strategy if it does not depend on the stage, e.g., see (Shoham and Leyton-Brown, 2008, Section 6.2). In (discounted) MDPs, there always exists an optimal strategy that is stationary, e.g., see (Filar and Vrieze, 2012). Shapley (1953) showed that this can be generalized to two-player zero-sum stochastic games.

We denote the stationary mixed strategy of player $i$ by $\pi^{i}:S\rightarrow\Delta(A^{i})$, implying that she takes actions according to the mixed strategy specific to state $s$, i.e., $\pi^{i}(s)\in\Delta(A^{i})$. We represent the strategy profile of players by $\pi:=\{\pi^{1},\pi^{2}\}$. Correspondingly, the expected discounted sum of stage payoffs of player $i$ under the strategy profile $\pi$ is defined by

where $a_{k}\sim\pi(s_{k})$, and the expectation is taken with respect to the all randomness. We next introduce the Nash equilibrium (more specifically Markov perfect equilibrium (Maskin and Tirole, 1988a, b)) where players do not gain any utility improvement by unilateral changes in their stationary strategies regardless of the initial state, e.g., see (Shoham and Leyton-Brown, 2008, Section 6.2).

We say that a stationary strategy profile $\pi$ is a stationary mixed-strategy $\varepsilon$-Nash equilibrium with $\varepsilon\geq 0$ if we have

We say that $\pi$ is a stationary mixed-strategy Nash equilibrium if (3.3) holds with $\varepsilon=0$.

We next state an important existence result for discounted stochastic games.

In stochastic games (with finitely many players, states and actions, and discount factor $\gamma\in[0,1)$), a stationary mixed-strategy equilibrium always exists.

The proof for two-player zero-sum stochastic games is shown by (Shapley, 1953) while its generalization to $n$-player general-sum stochastic games is proven by (Fink, 1964) and (Takahashi, 1964) concurrently. Shapley (1953) had also presented an iterative algorithm to compute the unique equilibrium value of a two-player zero-sum stochastic game. To describe the algorithm, let’s first note that in a zero-sum strategic-form game, there always exists a unique equilibrium value for the players (though there may exist multiple equilibria). For example, given a zero-sum strategic-form game $\langle A^{1},A^{2},u^{1},u^{2}\rangle$, we denote the equilibrium values of player $1$ and player $2$, respectively, by

Shapley (1953) showed that if we follow this backward induction, we can always compute the equilibrium values associated with a stationary equilibrium. To this end, he introduced the operator $\mathcal{T}^{i}$ defined by

starting from arbitrary $v_{(0)}^{i}$ converges to the unique fixed point of the operator. Further inspection of the fixed point reveals that it is indeed the equilibrium values of states associated with some stationary equilibrium of the underlying two-player zero-sum stochastic game. There does not exist a counter-part of this iteration for the computation of equilibrium values in general-sum stochastic games, since the value of a game is not uniquely defined for general-sum stochastic games, and involves a fixed point operation, which is hard to compute at each stage of an algorithm. However, Shapley’s iteration is still a powerful method to compute equilibrium values in a two-player zero-sum stochastic game.

In the following section, we examine whether a stationary equilibrium would be realized as a consequence of non-equilibrium adaptation of learning agents as in Section 2.1 but now for stochastic games instead of repeated play of the same strategic-form game.

Learning in Stochastic Games

At each stage $k$, player $i$ has a belief on player $-i$’s strategy, which we denote by $\hat{\pi}_{k}^{-i}$. Player $i$ also forms a belief on the payoff function for the auxiliary game, or the $Q$-function, denoted by $\hat{Q}_{k}^{i}$. Let $s$ be the current state of the stochastic game. Then, player $i$ selects her action $a_{k}^{i}$ according to

Observing the opponent’s action $a_{k}^{-i}$, player $i$ forms her belief on player $-i$’s strategy for the current state $s$ as a weighted empirical average, which can be constructed iteratively as

Here $\alpha_{c}\in$ is a step size and it vanishes with $c_{k}(s)$ indicating the number of visits to state $s$ rather than time. Note that if there was a single state, $c_{k}(s)$ would correspond to the time, i.e., $c_{k}(s)=k$, as in the classical fictitious play. The update (4.4) can also be viewed as taking a convex combination of the current belief $\hat{\pi}_{k}^{2}(s)$ and the observed action $a_{k}^{2}$ while the step size $\alpha_{c_{k}(s)}$ is the (vanishing) weight of the action observed. Vanishing step size as a function of the number of visits implies that, the players give less weight to their current belief than the observed action by using a large step size if that state has not been visited many times. This means that the players will still give less weight to their current belief even at later stages if the specific state has not been visited many times, and indicating, they have not been able to strengthen their belief enough to rely more on it.

Simultaneously, player $i$ updates her belief on her own $Q$-function for the current state $s$ according to

and $\beta_{c}\in$ is another step size that also vanishes with $c_{k}(s)$. Similar to (4.4), the update of the belief on the $Q$-function (4.5) can be viewed as a convex combination of the current belief $\hat{Q}_{k}^{i}(s,a)$ and the new observation $r^{i}(s,a)+\gamma\sum_{s^{\prime}\in S}p(s^{\prime}|s,a)\hat{v}_{k}^{i}(s^{\prime})$. Such vanishing step size again implies that the players are relying on their beliefs more if they have had many chances to strengthen them.

The key feature of this learning dynamic is that the players update their beliefs on their $Q$-functions at a slower timescale than the update of their beliefs on the opponent strategy. This is consistent with the literature on evolutionary game theory (Ely and Yilankaya, 2001; Sandholm, 2001) (which postulates players’ choices to be more dynamic than changes in their preferences) since we can view $Q$-functions in auxiliary games as slowly evolving player preferences. Particularly, the two-timescale learning framework implies that the players take smaller and smaller steps at (4.5) than the steps at (4.4) such that the ratio of the step sizes, $\beta_{c}/\alpha_{c}$, goes to zero with the number of visits to the associated state. Note that this implies that $\beta_{c}$ goes to zero faster than $\alpha_{c}$ does, implying slower update of the $Q$-function estimate compared to the opponent’s strategy estimate. This weakens the dependence between evolving beliefs on opponent strategy and $Q$-function.

We say that this two-timescale fictitious play dynamics converge to an equilibrium if beliefs on opponent strategies converge to a Nash equilibrium which associates with the auxiliary games while the beliefs on $Q$-functions converge to the $Q$-functions for a stationary equilibrium of the underlying stochastic game. Particularly, given an equilibrium $\pi_{*}$, the associated $Q$-function of player $i$ satisfies

Recall that players are playing a dynamically evolving auxiliary game at each state repeatedly, but update their beliefs on the $Q$-functions and opponent strategies only when that state is visited. Therefore, the players are updating their beliefs on the opponent strategy and $Q$-function specific to that state only during these visits. Hence, we make the following assumption ensuring that players have sufficient time to revise and improve their beliefs specific to a state.

Stochastic games reduce to the repeated play of the same strategic-form game if there exists only one state and the discount factor is zero. Correspondingly, Assumption 4.1 always holds in such a case. However, when there are multiple states, Assumption 4.1 does not necessarily hold, e.g., since some states can be absorbing by preventing transitions to others. In the following, we exemplify four Markov chain configurations with different generality:

Case $i)$ The probability of transition between any pair of states is positive for any action profile. This condition is also known as irreducible stochastic games (Leslie et al., 2020).

Case $ii)$ The probability of transition between any pair of states is positive for at least one action profile. Case $ii)$ includes Case $i)$ as a special case.Another possibility in between Case $i)$ and Case $ii)$ is that the probability of transition between any pair of states is positive for at least one action of one player and any action of the opponent. In other words, the opponent cannot prevent the game to transit from any state to any state.

Case $iii)$ There is positive probability that any state can be reached from any state within a finite number of stages for any sequence of action profiles taken during these stages. Case $iii)$ includes Case $i)$ as a special case but not necessarily Case $ii)$.

Case $iv)$ There is positive probability that any state can be reached from any state within a finite number of stages for at least one sequence of action profiles taken during these stages. Case $iv)$ includes Cases $ii)$ and $iii)$ as special cases.Another possibility in between Case $iii)$ and Case $iv)$ is that there is positive probability that any state can be reached from any state within a finite number of stages for at least one sequence of actions of one player and for any sequence of actions taken by the opponent during these stages. In other words, the opponent cannot prevent the player to reach any state from any state within a finite number of stages.

Note that Assumption 4.1 holds under Case $iii)$ but not necessarily under Cases $ii)$ or $iv)$.

Recall that in the classical fictitious play, the beliefs on opponent strategy are formed by the empirical average of the actions taken by the opponent. The players can also form their beliefs as a weighted average of the actions while the weights may give more (or less) importance to recent ones depending on the player’s preferences, e.g., as in (4.4). In other words, we let $\alpha_{c}$ take values other than $1/(c+1)$ for $c=0,1,\ldots$. Furthermore, the two-timescale learning scheme imposes that $\beta_{c}/\alpha_{c}$ goes to zero as $c$ goes to infinity. In the following, we specify conditions on step sizes that are sufficient to ensure convergence of the two-timescale fictitious play in two-player zero-sum stochastic games under Assumption 4.1.

The step sizes $\{\alpha_{c}\}$ and $\{\beta_{c}\}$ satisfy the following conditions:

They vanish at a slow enough rate such that

while $\alpha_{c}\rightarrow 0$ and $\beta_{c}\rightarrow 0$ as $c\rightarrow\infty$.

They vanish at two separate timescales such that

The following theorem shows that the two-timescale fictitious play converges in two-player zero-sum stochastic games under these assumptions.

Given a two-player zero-sum stochastic game, suppose that players follow the two-timescale fictitious play dynamics (4.4) and (4.5). Under Assumptions 4.1 and 4.2, we have

as $k\rightarrow\infty$ for some stationary equilibrium $\pi_{*}=(\pi_{*}^{1},\pi_{*}^{2})$ of the underlying stochastic game and $(Q_{*}^{1},Q_{*}^{2})$ denote the associated $Q$-functions.

Before delving into the technical details of the proof, it is instructive to compare the two-timescale fictitious play with both the classical fictitious play and the Shapley’s iteration. For example, the update of $\hat{\pi}^{-i}_{k}$, described in (4.4), differs from the classical fictitious play dynamics (2.3) since the auxiliary game depends on the belief $\hat{Q}^{i}_{k}$ while the belief (and therefore the payoffs of the auxiliary games) evolves in time with new observations, quite contrary to the classical scheme (2.4). In general, this constitutes a challenge in directly adopting the convergence analysis for the classical scheme to stochastic games. However, the two-timescale learning scheme weakens this coupling, enabling us to characterize the asymptotic behavior specific to a state separately from the dynamics in other states as if $(\hat{Q}_{k}^{1}(s,\cdot),\hat{Q}_{k}^{2}(s,\cdot))$ is stationary.

Moreover, even with the two-timescale learning scheme, we still face a challenge in directly adopting the convergence analysis of fictitious play specific to zero-sum games, e.g., Robinson (1951); Harris (1998). Particularly, players form beliefs on their $Q$-functions independently based on the backward induction that they will always look for maximizing their utility against the opponent strategy. Due to this independent update, the auxiliary games can deviate from the zero-sum structure even though the underlying game is zero-sum. Hence we do not necessarily have $\hat{Q}_{k}^{1}(s,a)+\hat{Q}_{k}^{2}(s,a)=0$ for all $a\in A$ and for each $s\in S$. This poses an important challenge in the analysis since an arbitrary general-sum game does not necessarily have fictitious play property in general.

Next, we compare the two-timescale fictitious play with Shapley’s value iteration. We can list the differences between the update of $\hat{Q}_{k}^{i}$, described in (4.5), and the Shapley’s iteration (3.8) as follows:

The Shapley’s iteration is over the value functions, however, it can be turned into an iteration over the $Q$-functions with the operator

as derived in (Szepesvári and Littman, 1999). The transformed iteration is given by $Q_{(n+1)}^{i}=\mathcal{F}^{i}Q_{(n)}^{i}$ starting from arbitrary $Q_{(0)}^{i}$. Furthermore, the Shapley’s iteration does not involve a step size, however, a step size can be included if we view $Q_{(n+1)}^{i}=\mathcal{F}^{i}Q_{(n)}^{i}$ as the one

with the step size $\beta_{(n)}=1$ for all $n$.

The Shapley’s iteration updates the value function at every state at each stage while (4.5) takes place only when the state is visited. Therefore, we face the asynchronous update challenge in the convergence analysis of (4.5) together with (4.4), which can take place only when the associated state is visited. To address this, we can resort to the asynchronous stochastic approximation methods, e.g., see Tsitsiklis (1994) (also upcoming Theorem 4.5).

for all $(s,a)\in S\times A$ and $i=1,2$. The function (2.6) presented in Harris (1998) for continuous-time best response dynamics in zero-sum games is no longer a valid Lyapunov function since $\sum_{i=1,2}Q^{i}(s,a)$ is not necessarily zero for all $s$ and $a$. Therefore, we modify this function to characterize the asymptotic behavior of this flow in terms of the deviation from the zero-sum structure, e.g., $\max_{a\in A}|\sum_{i=1,2}Q^{i}(s,a)|$. The new function is defined by

where $\lambda$ is a fixed scalar satisfying $\lambda\in(1,1/\gamma)$. The lower bound on $\lambda$ plays a role in its validity as a Lyapunov function when $\max_{a\in A}\left|\sum_{i=1,2}Q^{i}(s,a)\right|\neq 0$ while the upper bound will play a role later when we focus on the evolution of $\sum_{i=1,2}Q^{i}(s,a)$ to show that the sum converges to zero, i.e., the auxiliary stage games become zero-sum, almost surely.

Note that $V_{*}(\cdot)$ reduces to $V(\cdot)$, described in (2.6), if $\sum_{i=1,2}Q^{i}(s,a)=0$ for all $a\in A$. Furthermore, it is a valid Lyapunov function for any $Q^{1}(s,\cdot)$ and $Q^{2}(s,\cdot)$ since we have

where $a_{*}=(a_{*}^{1},a_{*}^{2})$ are the maximizing actions in (4.10), and we always have

if it is not zero-sum, since $\lambda>1$. In other words, the term inside $(\cdot)_{*}$ in the new Lyapunov function always decreases along the flow when it is non-negative and cannot be positive once it becomes non-positive.

If we let $\bar{v}_{k}:=\hat{v}_{k}^{1}+\hat{v}_{k}^{2}$ and $\bar{Q}_{k}:=\hat{Q}_{k}^{1}+\hat{Q}_{k}^{2}$, the new Lyapunov function yields that

as $k\rightarrow\infty$ for each $s\in S$. On the other hand, we always have $\bar{v}_{k}(s)\geq-\lambda\max_{a\in A}|\bar{Q}_{k}(s,a)|$ by the definition of $\hat{v}_{k}^{i}$. These bounds imply that $\bar{Q}_{k}(s,a)\rightarrow 0$, and therefore $\bar{v}_{k}(s)\rightarrow 0$ for all $(s,a)\in S\times A$, because the evolution of $\bar{Q}_{k}$ for the current state $s$ is given by

by (4.5) while the upper bound on $\lambda$ ensures that $\lambda\gamma\in(0,1)$, and therefore, $\bar{Q}_{k}(s,a)$ contracts at each stage until it converges to zero for all $s\in S$ and $a\in A$. The asynchronous update and the asymptotic upper bound on $\bar{v}_{k}$, as described in (4.13), constitute a technical challenge to draw this conclusion, however, they can be addressed via asynchronous stochastic approximation methods, e.g., see Tsitsiklis (1994).

Furthermore, the saddle point equilibrium yields that

and the right-hand side is bounded from below by

where the tracking error $\epsilon_{(n)}^{i}$ is asymptotically negligible almost surely and the operator $\mathcal{F}$, as described in (4.8), is a contraction similar to the Shapley’s operator, described in (3.7). This completes the sketch of the proof for Theorem 4.3.

We next consider scenarios where players do not know the transition probabilities and their own stage payoff function, however, they can still observe their stage payoffs (associated with the current action profile), the opponent’s action, and the current state visited. Therefore, the players can still form beliefs on opponent strategy and their $Q$-functions.

The update of the belief on opponent strategy does not depend on the model knowledge. Therefore, the players can update their beliefs $\hat{\pi}_{k}^{-i}$ as in (4.4) also in the model-free case. However, the update of $\hat{Q}_{k}^{i}$ necessitates the model knowledge by depending on the stage payoff function and transition probabilities. The same challenge arises also in model-free solution of Markov decision processes (MDPs) - a single player version of stochastic games.

For example, $Q$-learning algorithm, introduced by Watkins and Dayan (1992), can be viewed as a model-free version of the value iteration in MDPs and the update rule is given by

where the triple $(s,a,\widetilde{s})$ denote respectively the current state $s$, current action $a$ and the next state $\widetilde{s}$, the payoff $r_{k}$ corresponds to the payoff received, i.e., $r_{k}=r(s,a)$, and $\beta_{k}(s,a)\in$ is a step size specific to the state-action pair $(s,a)$. The entries corresponding to the pairs $(s^{\prime},a^{\prime})\neq(s,a)$ do not get updated, i.e., $\hat{q}_{k+1}(s^{\prime},a^{\prime})=\hat{q}_{k}(s^{\prime},a^{\prime})$.

Watkins and Dayan (1992) provided an ingenious (direct) proof for the almost sure convergence of $Q$-learning algorithm. Alternatively, it is also instructive to establish a connection between $Q$-learning algorithm and the classical value iteration to characterize its convergence properties. For example, the differences between them can be listed as follows:

In $Q$-learning, agents use the value function estimate for the next state $\widetilde{s}$, i.e., $\hat{v}_{k}^{i}(\widetilde{s})$, in place of the expected continuation payoff $\sum_{s^{\prime}\in S}p(s^{\prime}|s,a)\hat{v}_{k}^{i}(s^{\prime})$. This way, they can sample from the state transition probabilities associated with the current state-action pair by observing the state transitions. Correspondingly, this update takes place only after the environment transitions to the next state.

The update can take place only for the current state-action pair because the agent can sample only from the transition probabilities associated with the current state-action pair by letting the environment do the experimentation.

Therefore, the $Q$-learning algorithm can be viewed as an asynchronous $Q$-function version of the value iteration

where the $Q$-function version of the Bellman operator is given by

and the stochastic approximation error $\omega_{k+1}$ is defined by

with $\widetilde{s}$ denoting the next state at stage $k$. Note that (4.21) turns into an asynchronous update if $\beta_{k}(s,a)$ is just zero when $\hat{q}_{k}(s,a)$ is not updated. Though these error terms $\{\omega_{k}\}_{k>0}$ do not form an independent sequence, they form a finite-variance Martingale difference sequence conditioned on the history of parameters. The following well-known result shows that the weighted sum of such Martingale difference sequences vanishes asymptotically almost surely.

vanishes to zero asymptotically almost surely, i.e., $\lim_{k\rightarrow\infty}W_{k}=0$ with probability $1$, provided that $\alpha_{k}\in$ is a vanishing step size that is $\mathcal{F}_{k}$-measurable, square-summable $\sum_{k=0}^{\infty}\alpha_{k}^{2}<\infty$ while $\sum_{k=0}^{\infty}\alpha_{k}=\infty$ with probability $1$.

This is a powerful result to characterize the convergence properties of stochastic approximation algorithms having the structure

Given an MDP, let an agent follow the $Q$-learning algorithm, described in (4.20), with vanishing step sizes $\beta_{k}(s,a)\in$ satisfying $\sum_{k\geq 0}\beta_{k}(s,a)=\infty$ and $\sum_{k\geq 0}\beta_{k}(s,a)^{2}<\infty$ for each $(s,a)\in S\times A$. Suppose that the entries corresponding to each $(s,a)$ gets updated infinitely often. Then, we have

for each $(s,a)\in S\times A$, as $k\rightarrow\infty$, where $q_{*}$ is the unique $Q$-function solving the MDP.

(Tsitsiklis, 1994) considered a more general case where agents receive random payoffs. In general, such randomness can result in unbounded parameters. However, this is not the case for $Q$-learning algorithm, i.e., the iterates in the $Q$-learning algorithm remains bounded. Furthermore, the boundedness of the iterates plays a crucial role in the proof of Theorem 4.5. Particularly, consider the deviation between the iterate $\hat{q}_{k}$ and the unique solution $q_{*}$, i.e., $\widetilde{q}_{k}=\hat{q}_{k}-q_{*}$, which evolves according to

by (4.21) and since $\mathcal{F}_{o}q_{*}=q_{*}$. Boundedness of the iterates $\hat{q}_{k}$ yields that $\widetilde{q}_{k}$ is also bounded. For example, let $|\widetilde{q}_{k}(s,a)|\leq D$ for all $(s,a)$ and $k$. Furthermore, by the contraction property of $\mathcal{F}_{o}$ with respect to the maximum norm, we have

Therefore, we can show that the absolute value of new iterates are bounded from above by

where $\{Y_{k}(s,a)\}_{k\geq 0}$ and $\{W_{k+1}(s,a)\}_{k\geq 0}$ are two sequences evolving, respectively, according to

starting from $W_{1}(s,a)=0$ for all $(s,a)$. For each $(s,a)$, the sequence $\{Y_{k}(s,a)\}_{k\geq 0}$ converges to $\gamma D$ while $\{W_{k+1}(s,a)\}_{k\geq 0}$ converges to zero with probability $1$ by Lemma 4.4 due to the assumptions on the step size and the infinitely often update of every entry. Letting $k\rightarrow\infty$ for both sides of (4.27), we obtain that the shifted iterates are asymptotically bounded from above by $\gamma D$. This yields that there exists a stage where the iterates remain bounded from above by $(\gamma+\epsilon)D$ where $\epsilon>0$ is sufficiently small such that $\gamma+\epsilon<1$. By following the same lines, we can find a smaller asymptotic bound on the iterates. Therefore, we can induce that the shifted iterates converge to zero and the iterates converge to the solution of the MDP even with the asynchronous update.

Similar to the generalization of the value iteration to $Q$-learning for model-free solutions, Littman (1994) generalized the Shapley’s iteration to Minimax-$Q$ learning to compute equilibrium values in two-player zero-sum stochastic games in a model-free way. The update rule is given by

for the current state $s$, current action profile $a$, and next state $\widetilde{s}$ with a step size $\beta_{k}(s,a)\in$ vanishing sufficiently slow such that $\sum_{k\geq 0}\beta_{k}(s,a)=\infty$ and $\sum_{k\geq 0}\beta_{k}(s,a)^{2}<\infty$ with probability $1$. The payoff $r_{k}^{i}$ corresponds to the payoff received for the current state and action profile, i.e., $r_{k}^{i}=r^{i}(s,a)$. The Minimax-Q algorithm converges to the equilibrium $Q$-functions of the underlying two-player zero-sum stochastic game almost surely if every state and action profile occur infinitely often.

In model-free methods, the assumption that every state-action pair occur infinitely often can be restrictive for practical applications. A remedy to this challenge is that agents explore at random instances by taking any action with uniform probability. Such random exploration results in that every state-action pair gets realized infinitely often if every state is visited infinitely often. Indeed, random exploration will also yield that each state gets visited infinitely often if there is always positive probability that any state is reachable from any state within a finite number of stages for at least one sequence of actions taken during these stages. This corresponds to Case $iv)$ described in Section 4.

In the model-free two-timescale fictitious play, players play the best response in the auxiliary game with probability $(1-\epsilon)$ while experimenting with probability $\epsilon$ by playing any action with uniform probability. They still update the belief on the opponent strategy as in (4.4). Furthermore, they update their beliefs on the $Q$-function for the current state $s$, current action profile $a$ and next state $s^{\prime}$ triple $(s,a,s^{\prime})$ according to

where $\beta_{c_{k}(s,a)}\in$ is a step size vanishing with the number of times $(s,a)$ is realized and the payoff $r_{k}^{1}$ corresponds to the payoff associated with the current state $s$ and action profile $a$, i.e., $r_{k}^{1}=r^{1}(s,a)$.

Recall that the two-timescale learning scheme plays an important role in the convergence of the dynamics. Particularly, the step size $\alpha_{c}$ used in the update of the belief $\hat{\pi}_{k}^{-i}(s)$ goes to zero slower than the step size $\beta_{c}$ used in the update of the belief $\hat{Q}_{k}^{i}(s,\cdot)$. Since both step size depend on the number of visits to the associated state, the assumption that $\beta_{c}/\alpha_{c}\rightarrow 0$ as $c\rightarrow\infty$ is sufficient to ensure this timescale separation. However, in the model-free case, the asynchronous update of $\hat{Q}_{k}^{i}(s,a)$ for different action profiles can undermine this timescale separation because the step size $\beta_{c}$ specific to the update of $\hat{Q}_{k}^{i}(s,a)$ depends the number of times the state and action profile $(s,a)$, i.e., $c_{k}(s,a)$, is realized. Therefore, we make the following assumption ensuring that the step size in the update of $\hat{Q}_{k}^{i}(s,a)$ vanishes still faster than the step size in the update of $\hat{\pi}_{k}^{-i}(s)$ as long as $c_{k}(s,a)$ is comparable with $c_{k}(s)$, i.e., $\liminf_{k\rightarrow\infty}c_{k}(s,a)/c_{k}(s)>0$ with probability $1$.

The step sizes $\{\alpha_{c}\}$ and $\{\beta_{c}\}$ satisfy the following conditions:

They vanish at a slow enough rate such that

while $\alpha_{c}\rightarrow 0$ and $\beta_{c}\rightarrow 0$ as $c\rightarrow\infty$.We have the additional assumption that the step size $\beta_{c}$ is square summable to ensure that the stochastic approximation error terms have finite variance conditioned on the history of the parameters.

The sequence $\{\beta_{c}\}_{c\geq 0}$ is monotonically decreasing. For any $m\in(0,1]$, we havePerkins and Leslie (2012) made a similar assumption that $\sup_{c}\frac{\beta_{\lfloor mc\rfloor}}{\beta_{c}}<M$ for all $m\in(0,1)$ and $\frac{\beta_{c}}{\alpha_{c}}\rightarrow 0$ for two-timescale asynchronous stochastic approximation.

When we have $\liminf_{k\rightarrow\infty}c_{k}(s,a)/c_{k}(s)>0$ with probability $1$ for all $(s,a)$, the second part of Assumption 4.6 ensures that $\lim_{k\rightarrow\infty}\frac{\beta_{c_{k}(s,a)}}{\alpha_{c_{k}(s)}}=0$ with probability $1$ for all $(s,a)$. Indeed, Assumptions 4.2 and 4.6 are satisfied for the usual (vanishing) step sizes such as

where $1/2<\rho_{\alpha}<\rho_{\beta}\leq 1$.

When players do random experimentation in the model-free case, they do not take the best response with certain probability. Therefore, we do not have convergence to an exact equilibrium as in the model-based case. However, the players still converge to a near equilibrium of the game with linear dependence on the experimentation probability and the following theorem provides an upper bound on this approximation error.

Given a two-player zero-sum stochastic game, suppose that players follow the model-free two-timescale fictitious play dynamics with experimentation probability $\epsilon>0$. Under Assumptions 4.1 and 4.6, we have

with probability $1$, where $D=\frac{1}{1-\gamma}\sum_{i}\max_{(s,a)}|r^{i}(s,a)|$, where $v^{i}_{*}$ and $Q_{*}^{i}$ denote, respectively, the value function and $Q$-function of player $i$ for some stationary equilibrium of the stochastic game.

Even though the random experimentation can prevent convergence to an exact equilibrium, it provides an advantage for the applicability of this near-convergence result because every state gets visited infinitely often, and therefore, Assumption 4.1 holds, if the underlying Markov chain satisfies Case $iv)$, i.e., there is positive probability that any state can be reached from any state within a finite number of stages for at least one sequence of action profiles taken during these stages.

The dynamics can converge to an exact equilibrium also in the model-free case if players let the experimentation probability vanish at certain rate. However, there are technical details that can limit the applicability of the result for Case $iv)$.

2 Radically Uncoupled Learning in Stochastic Games

Finally, we consider minimal-information scenarios where players do not even observe the opponent’s actions in the model-free case. Each player can still observe its own stage payoff received and the current state visited. The players also do not know the opponent’s action set. Indeed, they may even be oblivious to the presence of an opponent. The learning dynamics under such minimal information case is known as radically uncoupled learning in the learning in games literature, e.g., see (Foster and Young, 2006).

given the opponent’s strategy $\pi^{-i}$. Then, the computation of the best response is a simple optimization problem for player $i$, given by

Player $i$ would be able to compute her best response $a_{*}^{i}$ even when she does not know the opponent strategy $\pi^{-i}$ and her payoff function $r^{i}$ if she knew the function $q^{i}(\cdot)$. Hence, the question is whether the computation of $q^{i}(\cdot)$ can be achieved without observing the opponent’s action.

Suppose that players are playing the same strategic-form game repeatedly and player $i$ makes the forward induction that the opponent will play as how he has played in the past similar to the fictitious play dynamics. If that were the case, i.e., the opponent were playing according to a stationary strategy $\pi^{-i}$, then at each stage the payoff received by player $i$ would be the realized payoff $r^{i}(a^{1},a^{2})$, where $a^{-i}\sim\pi^{-i}$ and $a^{i}$ is the current action she has taken. Correspondingly, player $i$ can form a belief about $q^{i}(a^{i})$ for all $a^{i}\in A^{i}$ and update $q^{i}(\cdot)$ associated with the current action based on the payoff she received. For example, let $\hat{q}^{i}_{k}$, $a_{k}^{i}$ and $r_{k}^{i}$ denote, respectively, the belief of player $i$ on $q^{i}$, her current action and the current payoff she received. Similar to the update of the belief on opponent’s strategy, the update of $\hat{q}_{k}^{i}$ is given by

where $\alpha_{k}(a^{i})\in$ is a vanishing step size specific to the action $a^{i}$. However, this results in an asynchronous update of $\hat{q}_{k}$ for different actions quite contrary to the synchronous belief update (2.3) in the fictitious play. There is no guarantee that it would converge to an equilibrium even in the zero-sum case. On the other hand, such an asynchrony issue is not present and the update turns out to be synchronous in expectation if players take smoothed best response while normalizing the step size by the probability of the current action taken (Leslie and Collins, 2005).

We say that a strategy profile $\pi_{*}$ is a Nash distribution if we have

which is positive for all $a^{i}\in A^{i}$.

where $\alpha_{k}\in(0,1)$ is a step size vanishing with $k$ and not specific to any action. This asynchronous update rule, also known as individual $Q$-learning, turns out to be synchronous in the expectation. Particularly, the new update rule is given by

and $\omega_{k}^{i}(a^{i})$ is the stochastic approximation error defined by

Furthermore, the stochastic approximation error term forms a Martingale difference sequence conditioned on the history of iterates while the boundedness of the iterates ensure that it has finite variance. Therefore, we can invoke Lemma 4.4 to characterize the convergence properties of (4.38) - a rewritten version of (4.37) with the stochastic approximation term $\omega_{k}^{i}$.

In two-player zero-sum (or identical-payoff) strategic-form games played repeatedly, if both player follows the individual $Q$-learning algorithm, described in (4.37), then their estimate $\hat{q}_{k}^{i}$ converges to $q_{*}^{i}$ for all $a^{i}\in A^{i}$ satisfying

for some Nash distribution $\pi_{*}=(\pi_{*}^{1},\pi_{*}^{2})$ under the assumption that the iterates remain bounded. Correspondingly, their smoothed best response also converges to $\pi^{*}$.

Recall that in stochastic games, players are playing an auxiliary stage-game specific to the current state $\mathcal{G}_{s}=\langle A^{1},A^{2},Q^{1}(s,\cdot),Q^{2}(s,\cdot)\rangle$, where $Q^{i}$ satisfies (4.1). Therefore, in the minimal information case, each player $i$ can form a belief about the associated

which is now specific to state $s$ contrary to (4.34), and update it based on the stage payoffs received as in the individual $Q$-learning dynamics. We can view $q^{i}$ as the local $Q$-function since it is defined over individual actions rather than action profiles. We denote player $i$’s belief on $q^{i}$ by $\hat{q}_{k}^{i}$. Let $s$ be the current state of the stochastic game. Then, player $i$ selects her action $a_{k}^{i}$ according to smoothed best response

where $\alpha_{c}\in(0,1)$ is a vanishing step size and recall that $c_{k}(s)$ denotes the number of visits to state $s$ until and including stage $k$. The update (4.39) differs from (4.37) due to the additional term $\gamma\hat{v}_{k}^{i}(s^{\prime})$ corresponding to an unbiased estimate of the continuation payoff in the model-free case. Due to this additional term, the individual $Q$-learning dynamics in auxiliary stage-games specific to each state are coupled with each other. A two-timescale learning framework can weaken this coupling if players estimate $\hat{v}_{k}^{i}$ at a slower timescale according to

This decentralized $Q$-learning dynamics, described in (4.39) and (4.40), have convergence properties similar to the two-timescale fictitious play even in this minimal information case. Furthermore, random exploration is inherent in the smoothed best response. Therefore, Assumption 4.1 holds if the underlying Markov chain satisfies Case $iv)$. However, due to the smoothed best response, the dynamics does not necessarily converge to an exact Nash equilibrium.

Given a two-player zero-sum stochastic game, suppose that players follow the decentralized $Q$-learning dynamics. In addition to Assumptions 4.1 and 4.6, we assume that $\sum_{c\geq 0}\alpha_{c}^{2}<\infty$ and the iterates are bounded. Let $Q_{*}^{i}$ and $v_{*}^{i}$ denote the unique equilibrium $Q$-function and value function of player $i$. Then, we have

for all $(i,s)\in\{1,2\}\times S$, with probability $1$, where $g(\lambda):=\frac{2+\lambda-\lambda\gamma}{(1-\lambda\gamma)(1-\gamma)}$ with some $\lambda\in(1,1/\gamma)$.

Furthermore, let $\hat{\pi}_{k}^{i}(s)\in\Delta(A^{i})$ be the weighted time-average of the smoothed best response updated as

for all $(i,s)\in\{1,2\}\times S$, w.p. $1$, where $h(\gamma):=g(\gamma)(1+\gamma)-1$. In other words, these weighted-average strategies converge to near Nash equilibrium strategies of the stochastic game.

Other Learning Algorithms

Previous sections have focused on a detailed description of best-response/fictitious-play type learning dynamics, together with $Q$-learning dynamics, for stochastic games. In this section, we summarize several other algorithms in the learning in games literature, with a focus on independent/decentralized learning for stochastic games (also belonging to the area of multi-agent reinforcement learning in the machine learning literature).

For stochastic games, other than $Q$-learning-type algorithms presented in §4.1, Borkar (2002) also established the asymptotic convergence of an actor-critic algorithm to a weaker notion of generalized Nash equilibrium. Another early work Brafman and Tennenholtz (2002) proposed R-MAX, an optimism-based RL algorithm for average-reward two-player zero-sum stochastic games, with polynomial time convergence guarantees. However, convergence to the actual Nash equilibrium is not guaranteed from the regret definition in the paper.

For strategic-form games, besides fictitious play, several other decentralized learning dynamics have also been thoroughly studied. A particular example is the no-regret learning algorithmsSee Cesa-Bianchi and Lugosi (2006) for formal definitions and results of no-regret learning. from the online learning literature. It is a folklore theorem that: If both players of a game use some no-regret learning dynamics to adapt their strategies to their opponent’s strategies, then the time-average strategies of the players constitute a Nash equilibrium of the zero-sum strategic-form game (Cesa-Bianchi and Lugosi, 2006; Roughgarden, 2010). Popular no-regret dynamics include multiplicative weights update (Littlestone and Warmuth, 1994; Freund and Schapire, 1999), online gradient descent (Zinkevich, 2003), and their generalizations (Shalev-Shwartz et al., 2011; McMahan, 2011). These no-regret learning dynamics are uncoupled in that a player’s dynamics does not explicitly rely on the payoffs of other players (Hart and Mas-Colell, 2003). They are also posited to be a rational model of players’ rational behavior (Roughgarden, 2009; Syrgkanis and Tardos, 2013). In addition, Leslie and Collins (2005) proposed individual $Q$-learning, a fully decentralized learning dynamics where each player’s update rule requires no observation of the opponent’s actions, with convergence to the Nash equilibrium distribution of certain two-player games. Notably, these decentralized learning dynamics are only known to be effective for strategic-form games.

2 Multi-Agent Reinforcement Learning

There has been a flurry of recent works on multi-agent RL in stochastic games with focuses on non-asymptotic performance guarantees. Pérolat et al. (2015, 2017) proposed batch RL algorithms to find an approximate Nash equilibrium using approximate dynamic programming analysis. Wei et al. (2017) studied online RL, where only one of the player is controlled, and develops the UCSG algorithm with sublinear regret guarantees that improves the results in Brafman and Tennenholtz (2002), though still without guarantees of finding the Nash equilibrium. Subsequently, Sidford et al. (2020) provided near-optimal sample complexity for solving turn-based two-player zero-sum finite stochastic games, when a generative model that enables sampling from any state-action pair is available. Under the same setting, the near-optimal sample complexity for general two-player zero-sum finite stochastic games was then established in Zhang et al. (2020). Without a generative model, Bai and Jin (2020); Xie et al. (2020) presented optimistic value iteration-based RL algorithms for two-player zero-sum stochastic games, with efficient exploration of the environment, and finite-time regret guarantees. The two players need some coordination to perform the algorithms, and the focus in these two works is the finite-horizon episodic setting. Later, Bai et al. (2020) and Liu et al. (2021) provided tighter regret bounds for the same setting, with model-free and model-based RL methods, respectively. Liu et al. (2021) has also studied the general-sum setting, with finite-sample guarantees for finding the Nash equilibrium, assuming some computation oracle for finding the equilibrium of general-sum strategic-form games at each iteration. Contemporaneously, Jin et al. (2021b); Huang et al. (2021) studied multi-agent RL with function approximation in finite-horizon episodic zero-sum stochastic games, with also the optimism principle and regret guarantees.

In addition, policy-based RL algorithms have also been developed for solving stochastic games. Zhang et al. (2019); Bu et al. (2019) developed double-loop policy gradient methods for solving zero-sum linear quadratic dynamic games, a special case of zero-sum stochastic games with linear transition dynamics and quadratic cost functions, with convergence guarantees to the Nash equilibrium. Later, Zhao et al. (2021) also studied double-loop policy gradient methods for zero-sum stochastic games with general function approximation. Note that these double-loop algorithms are not symmetric in that they require one of the players to wait the opponent to update her policy parameter multiple steps while updating her own policy for one step, which necessarily requires some coordination between players. Finally, Shah et al. (2020) developed an Explore-Improve-Supervise approach, which combines ideas from Monte-Carlo Tree Search and Nearest Neighbors methods, to find the approximate Nash equilibrium value of continuous-space turn-based zero-sum stochastic games. The two players are coordinated to learn the minimax value jointly.

Notably, as minimax $Q$-learning, these multi-agent RL algorithms are mostly focused on the computational aspect of learning in stochastic games: compute the Nash equilibrium without knowing the model, using possibly as few samples as possible. Certain level of coordination among the players is either explicitly or implicitly assumed when implementing these algorithms, even for the zero-sum setting where the players compete against each other. For human-like self-interested players, these update rules may not be sufficiently rational and natural to execute. Indeed, as per Bowling and Veloso (2001), a preferable multi-agent RL algorithm should be both rational and convergent: a rational algorithm ensures that the iterates converge to the opponent’s best-response if the opponent converges to a stationary policy; while a convergent algorithm ensures convergence to some equilibrium if all the agents apply the learning dynamics. In general, a rational algorithm, in which each player adapts to the (possibly non-stationary) behavior of other players and uses only local information she observes without the aid of any central coordinator, does not lead to the equilibrium of the game. In fact, investigating whether a game-theoretical equilibrium can be realized as a result of non-equilibrium adaptation dynamics is the core topic in the literature of learning in games (Fudenberg and Levine, 1998). These multi-agent RL works have thus motivated our study of independent learning dynamics presented in §4.

3 Decentralized Learning in Stochastic Games

Decentralized learning in stochastic games has attracted increasing research interest lately. In Arslan and Yüksel (2017), decentralized $Q$-learning has been proposed for weakly acyclic stochastic games, which include stochastic teams (identical-interest stochastic games) as a special case. The update rule for each player does not need to observe the opponent players’ actions, and is even oblivious to the presence of other players. However, the players are implicitly coordinated to explore every multiple iterations (in the exploration phase) without changing their policies, in order to create a stationary environment for each player. The key feature of the update rule is to restrict player strategies to stationary pure strategies. Since there are only finitely many stationary pure strategy, players can create a huge-game matrix for each stationary pure strategy and a pure-strategy equilibrium always exists when this huge-game is weakly acyclic with respect to best response. However, in the model-free case, players do not know the payoffs of this huge-game and the two-phase update rule addresses this challenge. Pérolat et al. (2018) developed actor-critic type learning dynamics that are decentralized and of fictitious-play type, where the value functions are estimated at a faster timescale (in the critic step), and the policy is improved at a slower one (in the actor step). Nonetheless, the learning dynamics only applies to a special class of stochastic games with a “multistage” structure, in which each state can only be visited once. In Daskalakis et al. (2020), an independent policy gradient method was investigated for zero-sum stochastic games with convergence rate analysis, where two players use asymmetric stepsizes in their updates with one updates faster than the other. This implicitly requires some coordination between players to determine who shall update faster. Contemporaneously, Tian et al. (2021) studied online RL in unknown stochastic games, where only one player is controlled and the update rule is fully decentralized. The work focused on the efficient exploration aspect of multi-agent RL, by establishing the regretThe regret defined in Tian et al. (2021) is weaker than the normal one with the best-in-hindsight comparator. See (Tian et al., 2021, Sec. 2) for a detailed comparison. guarantees of the proposed update rule. The work considered only the finite-horizon episodic setting, and it is also unclear if the learning dynamics converge to any equilibrium when all players apply it.The same update rule with different stepsize and bonus choices and a certified policy technique, however, can return a non-Markovian approximate Nash equilibrium policy pair in the zero-sum setting; see Bai et al. (2020), and the very recent and more complete treatment Jin et al. (2021a), for more details.

With symmetric and decentralized learning dynamics, Leslie et al. (2020); Wei et al. (2021); Cen et al. (2021) are to the best our knowledge the latest efforts on learning in stochastic games. Leslie et al. (2020) studied continuous-time best-response dynamics for zero-sum stochastic games, with a two-timescale update rule: at the slower timescale, a single continuation payoff (common among the players) is updated, representing time average of auxiliary game payoffs up to time $k$; at the faster timescale, each player updates its strategy in the direction of its best response to opponent’s current strategy in the auxiliary game. The common continuation payoff update ensures that the auxiliary game is always zero-sum, allowing the use of the techniques for the strategic-form game setting (Harris, 1998). The dynamics update the mixed strategies at every state at every time. Alternatively, the work also considered a continuous-time embedding of the actual play of the stochastic game where game transitions according to a controlled continuous-time Markov chain. Both Wei et al. (2021) and Cen et al. (2021) studied the genuine infinite-horizon discounted zero-sum stochastic games, and provided last-iterate convergence rate guarantees to approximate Nash equilibrium. To this end, Wei et al. (2021) developed an optimistic variant of gradient descent-ascent update rule; while Cen et al. (2021) focused on the entropy-regularized stochastic games, and advocated the use of policy extragradient methods. Though theoretically strong and appealing, these update rules assume either exact access or sufficiently accurate estimates of the continuation payoffs under instantaneous joint strategies and/or the instantaneous strategy of the opponent. In particular, to obtain finite-time bounds, the players are coordinated to interact multiple steps to estimate the continuation payoffs in the learning setting (Wei et al., 2021).

By and large, ever since the introduction of fictitious play (Brown, 1951) and stochastic games (Shapley, 1953), it remains a long-standing problem whether an equilibrium in a stochastic game can be realized as an outcome of some natural and decentralized non-equilibrium adaptation, e.g., fictitious play (except the contemporaneous work Leslie et al. (2020) with some continuous-time embeddings). Hence, our solutions in §4 serve as an initial attempt towards settling the argument positively.

Conclusions and Open Problems

In this review paper, we introduced multi-agent dynamic learning in stochastic games, an increasingly active research area where artificial intelligence, specifically reinforcement learning, meets game theory. We have presented the fundamentals and background of the problem, followed by our recent advances in this direction, with a focus on studying independent learning dynamics. We believe our work has opened up fruitful directions for future research, on developing more natural and rational multi-agent learning dynamics for stochastic games. In particular, several future/ongoing research directions include: 1) establishing convergence guarantees of our independent learning dynamics for other stochastic games, e.g., identical-interest ones; 2) establishing non-asymptotic convergence guarantees of our learning dynamics, or other independent learning dynamics, for stochastic games; 3) developing natural learning dynamics that also account for the large state-action spaces in practical stochastic games, e.g., via function approximation techniques.

References