Learning to Safely Exploit a Non-Stationary Opponent

In dynamic multi-player games , an effective way to exploit an opponent ’ s weak-1 nesses is to build a perfectly accurate opponent model . This renders the learning2 problem a single-agent optimization which can be solved by typical reinforcement3 learning . However , naive behavior cloning may not suffice to train an exploiting4 policy because opponents ’ behaviors are often non-stationary due to their adapta-5 tions in response to other agents ’ strategies . On the other hand , overfitting to an6 opponent ( i.e. , exploiting only one specific type of opponent ) makes the learning7 player easily exploitable by others . To address the above problems , we propose8 a method named Exploit Policy-Space Opponent Model ( EPSOM ) . In EPSOM,9 we model an opponent ’ s non-stationarity as a series of transitions among different10 policies , and formulate such a transition process through Bayesian non-parametric11 methods . To account for the trade-off between exploitation and exploitability , we12 train a player to learn a robust best response to the opponent ’ s predicted strategy13 by solving a modified meta-game in policy space . In this work , we consider a14 two-player zero-sum game setting and evaluate EPSOM on Kuhn poker ; results15 suggest that our method is capable of exploiting its adaptive opponent , whilst16 maintaining low exploitability ( i.e. , achieving safe opponent exploitation ) . Further-17 more , we show that our EPSOM agent has strong performance against unknown18 non-stationary opponents without further training.19 1 Introduction20 In single agent reinforcement learning ( SARL ) , an agent learns to act by iteratively interacting with21 an environment . In such a setting , an agent ’ s learning objective and its performance evaluation22 are normally clear and straightforward , e.g. , its long-term cumulative rewards gained from the23 environment . However , in multi-agent reinforcement learning ( MARL ) , one agent ’ s performance24 greatly depends on the behavior of other agents . Hence , finding a reliable learning objective and25 evaluation method become non-trivial [ 3 , 9 , 31 , 48 ] . Naive solutions of the problem using SARL26 generalize badly [ 21 ] and optimizing the joint policy of all agents does not scale . Recent approaches27 combining game theoretical analysis with deep RL have seen some success in large zero-sum28 games [ 4 , 44 ] .29 Game theory offers a mathematical framework to model strategic interactions among players [ 28 ] .30 Under perfect rationality [ 12 ] , a central solution concept is Nash equilibrium ( NE ) where no player31 benefits from deviating from their equilibrium strategy . In a two-player zero-sum game without32 any inherent advantage for either player ( e.g . as a first mover ) , a NE is a safe strategy to play ( i.e.,33 playing not to lose ) – NE guarantees a tie in the worst case in expectation . However , NE is not the34 most profitable strategy in many cases . In complex competitive games , such as poker , it is common35 that agents encounter opponents with bounded rationality , in the sense that they may at best play an36 Submitted to 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . Do not distribute . approximate Nash equilibrium strategy and often play dominated actions [ 5 , 33 ] . Therefore , playing37 a NE can potentially forego significant rewards against sub-optimal opponents . This incentivizes38 players to deviate from the NE and exploit their opponents ’ weakness ( i.e. , playing to win ) . However,39 the resulting strategy could render itself exploitable should it overfit to the current opponent . Playing40 to win can therefore lead to exploitation by other opponent strategies . In the case of deceptive41 opponents such exploitation is known as the “ get taught and exploited ” problem [ 35 ] .42 To better balance the trade-off between playing to win against the current opponents ( exploitation ) and43 not losing to unknown opponents ( exploitability ) , Johanson et al . [ 19 ] proposed a solution concept,44 named Restricted Nash Response ( RNR ) . RNR and its variants [ 18 , 19 , 33 ] assume stationary45 opponents , i.e. , the strategies they learn to exploit are unknown but fixed . However , in many real-46 world applications , opponents may adapt and change their strategies on an ongoing basis . For example,47 in Rock-Paper-Scissors when a player learns to best respond by playing Rock to an opponent ’ s strategy48 which always plays Scissors , the opponent may then learn to best respond to your best response by49 playing Paper . Furthermore , prior RNR approaches only provide one-off solutions in the sense that50 whenever we need to re-adjust the trade-off between exploitation and exploitability or the opponent51 uses a new fixed policy , we need to re-solve the updated game from scratch.52 In this work , we focus on problems with non-stationary opponents . An opponent ’ s learning process53 can be generally modeled as transitions among a mixture of unknown number of policies . This54 motivates the usage of a Dirichlet process mixture model . As we can only collect trajectories produced55 by the adaptive opponents online , we propose to learn our model in a streaming fashion . Given the56 predicted opponent policy from our model , we provide a general framework for training an agent57 to safely exploit the non-stationary opponent where safe exploitation means exploiting the current58 opponent with a low probability of being exploited by other opponents in future . We empirically59 demonstrate the ability of our approach to safely exploit a non-stationary opponent in Kuhn Poker , a60 simplified Poker game . Furthermore , once trained , our model can produce strong counter strategies61 to unseen opponents without any further training in new tournaments.62 2 Related Works63 A fundamental ability of an effective AI agent is the capacity to interact with other intelligent64 agents . Therefore , the capability of reasoning about other agents ’ goals [ 34 ] , private information [ 27 ] ,65 behavior [ 13 ] and other characteristics is crucial . The issue of non-stationarity in multi-agent systems66 resulting from coexisting agents is well-known and well documented [ 14 , 32 ] . Classical solutions to67 resolve the issue of non-stationarity include centralized training [ 24 ] , self-play [ 44 ] , meta-learning [ 1 ] 68 and opponent modeling [ 2 ] . When specifically applied to the issue of non-stationarity , most previous69 works focusing on opponent modeling which switches between different opponent models when a70 change in opponent ( s ) is detected . A switch of model may be triggered by a drop in opponent model71 prediction accuracy [ 10 ] or when performance in terms of reward received for a fixed policy drops72 [ 15 ] . Deep BPR+ [ 49 ] combines a measure of opponent model accuracy and reward tracking to decide73 when to learn a new policy . Significantly , most of these works limit the opponents ’ non-stationarity74 to periodically changing their policies within a finite pre-defined set of stationary policies.75 In this work , we consider non-stationarity during the training stage arising from the opponents ’ 76 concurrent learning dynamics , rather than drawing stationary opponents from a pre-defined set . The77 entire lifetime of an opponent can generally be modeled as a mixture of an unknown ( possibly infinite ) 78 number of policies . This motivates the usage of a Dirichlet Process ( DP ) mixture model [ 6 , 42 ] which79 can infer the number of mixture components from data and provide incremental model capacity80 on demand . Various approximate inference methods are reported for DP mixture models , such as81 Markov chain Monte Carlo [ 17 ] and variational inference [ 6 , 16 , 45 ] . However , these inference82 methods either do not adapt to an online setting or truncate the number of clusters to a finite value.83 Recently proposed streaming inference algorithms [ 23 , 41 ] enable the DP mixture model to solve84 online non-stationary problems in a truly non-parametric way . Applications have been reported in85 task-free continual learning [ 22 ] and model-based reinforcement learning [ 47 ] . In this paper , we86 adopt this approach to model and simulate a non-stationary opponent for MARL.87 It is well known that finding a NE is PPAD-hard even in two-player games [ 8 ] . An exception is two-88 player zero-sum games where the NE can be tractably solved by a linear program ( LP ) in polynomial89 time [ 43 ] . However , in games with extremely large action spaces , approximate NE solutions , such90 as fictitious play ( FP ) [ 7 ] and counterfactual regret minimisation ( CFR ) [ 50 ] , have to be used . An91 important design principle that underpins NE approximation is the iterative best-response dynamics.92 Two representative methods are Double Oracle ( DO ) [ 26 ] and Policy Space Response Oracle ( PSRO ) 93 [ 21 ] . In the dynamics of DO [ 26 ] , players are initialized with restricted strategy sets ; then at each94 iteration , a NE will be computed over the current restricted sets . These sets will be expanded by95 adding the best-response strategy to the NE computed over the full strategy sets . The iterative96 process continues until the best response is in the restricted strategy pool . PSRO approximates DO by97 interleaving empirical game-theoretic analysis ( EGTA ) with deep RL . In contrast with DO , the game98 with restricted strategy sets has to be estimated through simulation . Furthermore , the exact analytical99 best-response oracle is replaced in PSRO by a deep RL oracle which calculates an approximate best100 response . PSRO is a general self-play framework for MARL and many approaches built upon it have101 been proposed to improve its performance [ 25 , 29 , 30 , 39 ] . Our approach , EPSOM , is not limited to102 the self-play setting . In addition , although we favor solutions with low exploitability ( i.e . solutions103 close to NE ) as PSRO does , our ultimate goal is to find a robust best response to a non-stationary104 opponent rather than solving the game for an ( approximate ) NE.105 3 Preliminaries106 We consider the decentralized training and decentralized execution ( DTDE ) setting in zero-sum107 games where we have access to interaction trajectories ⌧ between our agent and the opponenti .108 Whilst our approach can be extended to games with multiple opponents , in this work we focus on109 2-player zero-sum games . Before introducing our algorithm , we present some necessary preliminary110 concepts and notation in the remainder of this section.111 3.1 Meta Normal-Form Game112 We consider opponent modeling in policy space and learn to respond to the predicted distribution113 of the opponents ’ policies . We formulate this problem as solving a two-player normal-form game114 ( NFG ) between our agent and its opponents as a whole with notation adapted to our presentation . We115 denote a 2-player NFG by a tuple ( ⇧ , U , N ) where ⇧i is player i ’ s set of policies and i 2 N where116 N = { 1 , 2 } . For ease of notation , we take player 1 as the training agent and player 2 as its opponent.117 We use ⇧ = Q i2N ⇧ i to denote the set of joint policies ( strategy profiles ) . U ( ⇡ ) : ⇧ ! < n is a118 payoff table of utilities for each joint policy ⇡ played by all players . ui ( ⇡ ) denotes the utility value119 for player i and joint policy ⇡ . A player can choose a policy ⇡i from ⇧i or sample from a mixture120 ( meta-strategy ) over them i 2 ⇧i where is a probability simplex . In the terminology of game121 theory , i is a mixed strategy and each policy ⇡i is a pure strategy.122 Each player in the game is assumed to maximize their utility . The most well-known steady-state123 concept of a game is the Nash equilibrium ( NE ) . NE is a strategy profile ⇡ such that no player has an124 incentive to deviate from its current strategy given the strategies of the other players . Namely , each125 player ’ s strategy is a best response to others ’ BR ( ⇡ i ) = argmax⇡i ui ( ⇡i , ⇡ i ) 8i 2 N . We call a126 set of policies ✏-best responses to a joint opponents ’ policy ⇡ i , when there exists an ✏ > 0 , such127 that BR✏ ( ⇡ i ) = { ⇡i : ui ( ⇡i , ⇡ i ) ui ( BR ( ⇡ i ) , ⇡ i ) ✏ } . An ✏-Nash equilibrium is a strategy128 profile that satisfies : ui ( ⇡ ) max⇡i0 ui ( ⇡i0 , ⇡ i ) ✏ 8i 2 N .129 3.2 Exploitability and Exploitation130 To evaluate our learned policy ⇡1 , we use two metrics . An agent ’ s policy ’ s ⇡1 exploitation of an131 opponent ’ s policy ⇡2 is the extra gain obtained by the agent compared to its NE value v1:132 ! ( ⇡1 , ⇡2 ) = u1 ( ⇡1 , ⇡2 ) v1 . This measures how much the policy ⇡1 exploits the weakness of the opponent ’ s policy ⇡2 . However,133 in general , there is no guarantee that the learned policy ⇡1 has no weakness . Therefore , we also134 define the exploitability of a policy ⇡1 which measures the loss incurred when the agent faces the135 iA detailed definition of the trajectory in a stochastic game [ 38 ] can be found in Appendix A.1 . best opponent policy ⇡2 = BR ( ⇡1 ) compared to the agent ’ s Nash equilibrium value v1:136 ✏ ( ⇡1 ) = v1 u1 ⇡1 , BR ( ⇡1 ) = max ⇡10 min ⇡2 u1 ( ⇡10 , ⇡2 ) min ⇡2 u1 ( ⇡1 , ⇡2 ) . ( 1 ) From Equation 1 we can see that the exploitability of a policy is non-negative and represents the137 distance of policy ⇡1 to an equilibrium.138 3.3 Restricted Nash Response ( RNR ) 139 Johanson et al . [ 19 ] consider a modified zero-sum game where an opponent has a restricted strategy140 space ⇧2p , ⇡fix such that it plays a fixed policy ⇡fix with probability p and plays any possible policy141 from the original strategy space ⇧2 with probability 1 p. Given ( p , ⇡fix ) , they define a restricted142 Nash equilibrium as a strategy profile ( ⇡1⇤ , ⇡2⇤ ) such that ⇡1⇤ 2 BR ( ⇡2⇤ ) and ⇡2⇤ 2 BRp , ⇡fix ( ⇡1⇤ ) ,143 where : BRp , ⇡fix ( ⇡1⇤ ) = argmax⇡22⇧2p , ⇡fix u 2 ( ⇡1 , ⇡2 ) . It is shown that ⇡1⇤ is the best response to144 ⇡fix among strategies which have equal or lower exploitabilities than ⇡1⇤ , i.e . : ⇡1⇤ = BR✏ ( ⇡fix ) ,145 where ✏ = ✏ ( ⇡1⇤ ) . Therefore , ⇡1⇤ is called a p-restricted Nash response ( RNR ) to ⇡fix . An RNR can146 be computed by solving the modified game , we present a linear programming solver implementation147 for NFGs in Appendix A.2.148 4 Dirichlet Process Mixture Opponent Modeling149 This section presents our non-parametric Bayesian method for modeling a non-stationary opponent.150 We consider an opponent ’ s learning process as consecutive transitions from one policy to another151 such that one opponent can theoretically adopt infinitely many policies during its life-time . Therefore,152 we propose to use a Dirichlet process ( DP ) mixture to model the learning process as it has the ability153 to model an infinite number of clusters ( policies in this case ) while inferring the current number of154 policies from the data collected thus far . As our agent interacts with the opponent online , we learn a155 model with a sequential maximum-a-posteriori approach.156 We model an opponent policy as a parameterized function ⇡2 and denote the parameter space as157 . To avoid cluttered notation in this section , we use to represent the modeled opponent ’ s policy.158 DP ( ↵H ) is a stochastic process with a concentration parameter ↵ and a base distribution H over .159 A random draw G ⇠ DP ( ↵H ) is itself a distribution over , satisfying:160 ( G ( A1 ) , ... , G ( Ar ) ) ⇠ Dirichlet ↵H ( A1 ) , ... , ↵H ( Ar ) for every finite measurable partition A1 , ... , Ar of . The full graphical model for opponent modeling161 is shown in Figure 1a . It illustrates a generative process where at step m , the opponent first samples a162 policy ̂m ⇠ G and then rolls-out this policy to collect a trajectory ⌧m.163 To facilitate Bayesian inference , two representations of DP are considered . The stick-breaking164 representation in Figure 1b reveals the discrete nature of G. G ⇠ DP ( ↵H ) can be constructed165 as G = P1 k=1 k k where ⇠ GEM ( ↵ ) is an infinite-dimensional random variable sampled166 from the Griffiths-Engen-McCloskey ( GEM ) distribution and { k } 1k=1 are i.i.d . sampled policies167 from H . At step m , the opponent samples a policy index zm ⇠ Categorical ( ) and rolls-out the168 policy zm . Inference with the stick-breaking representation is required in order to handle the infinite169 dimensional . Therefore , the truncation method [ 6 ] is commonly used to limit the model capacity170 to a K mixture and infer the actual number of policies by collapsing redundant ones . This requires171 tracking all K policies simultaneously and does not adapt well to online settings.172 The Chinese restaurant process ( CRP ) representation in Figure 1c can be obtained by integrating173 out . This introduces temporal dependencies between the policies , which can be expressed by the174 conditional distribution:175 p zm+1 = k|z1 : m = 8 > > > < > > > : ↵ m+ ↵ , k = Km + 1 |k|m m+ ↵ , 1 k Km ( 2 ) where |k|m = Pm i=1 I ( zi = k ) is the total number of trajectories from the k-th policy and Km is176 the number of realized policies up until step m. Inference with the CRP representation does not177 need to handle the infinite dimensional . Furthermore , at step m , we only need to track at most m178 policies ( Km m ) while all policies beyond Km are independent from the collected trajectories179 ⌧1 : m , and thus can be discarded from the model . In addition , the temporal dependencies between180 policies introduced by CRP can be used to develop an online learning algorithm.181 Given sampled trajectories ⌧1 : m , the target182 of our opponent model is to assign each183 trajectory to a policy and update existing184 policies with assigned trajectories . This185 can be achieved by seeking maximum-a-186 posteriori ( MAP ) estimations of z1 : m and187 1 : Km . To deal with streaming trajectories,188 the MAP algorithm should operate in an189 online fashion . Therefore , given the CRP190 representation , we decompose the posterior191 into the product of the posterior from the192 last step , the current priors and the likeli-193 hood , which leads to a recursive form:194 p ( z1 : m , 1 : Km |⌧1 : m ) / ✓Km 1+1Y k=1 p ( k ) ◆ p ( z1 ) p ( ⌧1| z1 ) ✓ mY i=2 p ( zi|z1 : i 1 ) p ( ⌧i| zi ) ◆ , / p ( z1 : m 1 , 1 : Km 1 |⌧1 : m 1 ) p ( Km ) p ( zm|z1 : m 1 ) p ( ⌧m| zm ) , ( 3 ) where p ( k ) = H is the base distribution of the DP.195 The opponent model either assigns the current trajectory ⌧m to a previous policy k or creates a196 new policy Km 1+1 to model ⌧m . The choice is made according to the MAP trajectory assignment197 z⇤m = argmaxzm p ( zm , z ⇤ 1 : m 1|⌧1 : m ) :198 p ( zm = k , z ⇤ 1 : m 1|⌧1 : m ) / ( R k ↵p ( k ) p ( ⌧m| k ) d k , k = Km 1 + 1 p ( zm = k|z⇤1 : m 1 ) p ( ⌧m| m 1 k ) = |k| ⇤ m 1p ( ⌧m| m 1 k ) , otherwise ( 4 ) where |k|⇤m 1 = Pm 1 i=1 I ( z ⇤ i = k ) . Here , the hard assignment z⇤m for ⌧m is based on previous199 assignments z⇤1 : m 1 and policies m 1 k , which is equivalent to applying assumed density filtering200 ( ADF ) [ 41 ] to approximate the true posterior in Eq . ( 3 ) with a Delta distribution ( z⇤1 : m ) . The hard201 assignment prevents creating a new policy at each step if ⌧m is assigned to an existing policy , which202 significantly reduces the memory usage . Furthermore , the MAP estimations for all existing policies,203 except z⇤m , remain unchanged , which dramatically accelerates the algorithm . We then optimize the204 policy z⇤m by maximizing the likelihood of all trajectories assigned to it:205 mz⇤m = argmax z⇤m log p ( z⇤m ) Y z⇤i =z ⇤ m p ( ⌧i| z⇤m ) . ( 5 ) Where finding the global optimum is not tractable in non-conjugate cases , we take gradient steps to206 update nz⇤m as207 nz⇤m = m 1 z⇤m + r z⇤m log p ( z ⇤ m ) Y z⇤i =z ⇤ m p ( ⌧i| z⇤m ) . The entire algorithm fits into the general expectation-maximization ( EM ) framework . See Appendix208 A.4 for a detailed derivation.209 The original CRP in Eq . ( 2 ) encapsulates a prior that the distribution of the next policy mimics the210 empirical policy distribution from the history . This prior is not consistent with our knowledge of the211 policy evolution process since a new opponent policy is commonly updated from the previous one.212 Therefore , we adopt a sticky variant in Eq . ( 6 ) to incorporate the belief that the opponent tends to213 persist in the latest policy [ 11 , 47 ] .214 p zm = k z1 : m 1 = 8 > > > < > > > : ↵ m 1 + ↵+ , k = Km 1 + 1 |k|m 1 + ̂ ( Km 1 , k ) m 1 + ↵+ , 1 k Km 1 ( 6 ) where 0 is a ‘ stickiness ’ parameter and ̂ is the Kronecker delta function.215 Following Eq . ( 4 ) , the probability of creating a new policy for ⌧n is given by:216 p ( z⇤m = k ) / Z k ↵p ( k ) p ( ⌧m| k ) d k , ( 7 ) where k = Km 1 + 1 . We use a Monte Carlo method to estimate Eq . ( 7 ) by sampling new policies217 from p ( k ) . However , sampling new policies from a data-independent prior p ( k ) is likely to yield a218 low trajectory likelihood p ( ⌧m| k ) , which prevents the new policy creation . Therefore , we update the219 sampled policies to increase the likelihood p ( ⌧m| k ) by taking a few gradient steps before estimating220 the integration in Eq . ( 7 ) .221 According to Eq . ( 6 ) , the CRP prior encourages the opponent model to create redundant policies at the222 early stage when the number of trajectories n is small and ↵ dominates . Redundant policies could hurt223 the algorithm ’ s performance as it incurs extra cost in terms of computation and memory . Trajectories224 from the same ground truth policy could be assigned to different ks and these assignments never225 revisited . Therefore , an error correction mechanism has to be introduced . Here , we adopt a symmetric226 distance metric between two policies and develop a policy merge procedure based on the metric.227 Given a set of states S , we define d ( k , j ) = Es⇠Uniform ( S ) h JS ⇣ k ( ·|s ) j ( ·|s ) ⌘i , where228 JS ( ·||· ) is the Jensen–Shannon divergence and k ( ·|s ) is the action distribution given state s under229 the policy k. When the distance between two policies is below a pre-defined threshold ⌘ , the merge230 procedure simply re-assigns all trajectories of k to j .231 With the opponent model developed in this section , at step m , we can construct an opponent policy set232 ⇧̃2 = { mk } Km k=1 and a distribution ̃ 2 over ⇧̃2 . The distribution ̃2 ( = k ) / |k|m + ̂ ( Km , k ) 233 is essentially the empirical distribution of z⇤1 : m altered by the stickiness factor .234 5 Exploit Policy-Space Opponent Model235 In this section , we present how to learn a safe best response to this meta strategy , given a predicted236 distribution ̃2 ii over opponent ’ s policies . The advantages of our approach of focusing on policy space237 are two-fold : first , we do not need to assume the access to the opponent ’ s learning characteristics such238 as its training algorithm , its neural network ’ s architecture or its update frequency ; we only require past239 trajectories . Additionally , the distribution of the types of opponent policy ̃2 gives us an approximate240 stable overview of the current opponent ’ s playing behavior compared to the opponent ’ s current241 policy whose updates greatly depend on the opponent ’ s learning characteristics and randomness from242 playing ( e.g . exploration behavior ) and training ( e.g . stochastic gradient descent ) . Therefore , learning243 a response to this meta-strategy ̃2 will rely on less prior knowledge about the opponent ’ s learning244 characteristics and is more robust to noise.245 However , there is no guarantee that our learned meta strategy 1 has no ( or at least low ) exploitability.246 It has been shown that overfitting to an opponent strategy ̃2 often renders the resulting learned247 strategy brittle [ 19 , 21 , 46 ] . Such a brittle strategy performs badly when playing against different248 opponent strategies ̃20 . Therefore , a more desirable goal is to learn a safe best response to an249 opponent meta-strategy ̃2 . RNR solutions consider cases where the game to solve is fixed and250 known and the opponent ’ s policy is stationary . However , when we consider non-stationary opponent251 exploitation on policy space , the size of the meta-game to solve increases with the number of252 interactions between the training agent and the adaptive opponent . Furthermore , each player is free to253 learn and update its policy at any time point during the process.254 iiTo simplify our notation , we will ignore the subscript henceforth . Algorithm 1 : Exploit Policy-Space Opponent Model ( EPSOM ) input : Hyper-parameter p , H , E , an adaptive opponent i output : Policy ⇡i1 , ... , E and meta-policy i Initialize learning agent i ’ s policy ⇡i0 Initialize a memory buffer B Initialize opponent meta-policy i ( · ) = 1 for epoch e in { 1 , 2 , . . . , E } do for episode h 2 { 1 , 2 , . . . , H } do Play an episode against the opponent with strategy 1RNR Collect the trajectory ⌧e , h and save them into B end ̃2 , ⇧̃2 = opponent_modeling ( B ) p̄ = 1|⇧̃2| P j p j ̃2 ( j ) Compute missing entries in U ⇧̃ from ⇧̃ = ⇧1 ⇥ ⇧̃2 by simulations _ , 2RNR = RNR_solver ( U ⇧̃ , p̄ , ̃2 ) for episode h 2 { 1 , 2 , . . . , H } do Sample ⇡̃2 from 2RNR Train oracle ⇡1 over ⇢ ⇠ ⇡1 , ⇡̃2 end ⇧1 = ⇧1 [ ⇡1 Compute missing entries in U ⇧̃ from ⇧̃ = ⇧1 ⇥ ⇧̃2 by simulations 1RNR , _ = RNR_solver ( U ⇧̃ , p̄ , ̃2 ) end To address the above issues , we combine DO with RNR to solve a meta-game built from EGTA where255 the opponent ’ s policies are predicted by the opponent model from Section 4 . Pseudo-code explaining256 our approach is presented in Algorithm 1 . We maintain a utility table U ⇧̃ wherein rows represent257 learned policies for the training agent and columns represent a modeled policy of the opponent258 respectively . An epoch is defined as a fixed amount of episodes of games where we play against the259 opponent holding our strategy 1 fixed . At each epoch , we run our opponent model to predict the260 current distribution ̃2 of the opponent ’ s policies . If a new policy is detected , we will add it into ⇧̃2.261 Given ̃2 , we run a p-RNR solver to obtain the opponent ’ s RNR meta-strategies 2RNR which is a262 restricted Nash solution to the current meta-game assuming that the opponent is playing according to263 ̃2 with probability at least p. Then we train an ( approximate ) best-response policy to 2RNR and add264 the new policy into ⇧1 . We re-run a p-RNR solver to obtain our RNR meta-strategies 1RNR which265 we use to mix the policies in population ⇧1 for the next epoch ’ s playing policy.266 When a new type j of modeled policy ⇡2 , j is added by our opponent model , we initialize a p-value267 pj = pinit to this type . Its p-value is incremented proportionally to the probability that the opponent268 plays this policy in the following epochs , ̃2 ( j ) , and clipped at 1 . At each epoch , we calculate the269 average p-value p̄ = 1|⇧̃2| P j p j ̃2 ( j ) for solving current RNR strategies . In the extreme case where270 p̄ = 0 , iRNR is the same as the Nash strategy in the current meta game . At another extreme where271 p̄ = 1 , 2RNR = ̃ 2 and 1RNR = BR ( ̃ 2 ) . Therefore , when we have low confidence in ̃2 ( p̄ is272 low ) , we learn an approximate best response to opponent ’ s current Nash mixture which will enlarge273 our current empirical gamescape [ 3 ] and thus help to find strategies with lower exploitability . At the274 same time , the training agent becomes risk-adverse and the next epoch strategy 1RNR becomes a275 strategy closer to the Nash strategy of the current meta game.276 When p̄ is high , it means that our opponent model has high confidence that the opponent is playing ̃2277 and an approximate best counter strategy to ̃2 will be added into our policy population . The training278 agent becomes profit-driven and 1RNR becomes a strategy closer to BR ( ̃ 2 ) in the next epoch.279 Therefore , mixing the playing policy by 1RNR flexibly switches the agent between risk-adverse and280 profit-driven depending on the confidence of the opponent model . In contrast with previous RNR281 solution , EPSOM can always recover a strategy with approximately the lowest exploitability it has282 seen so far as we maintain a population of policies.283 6 Experiments284 In this section , we empirically investigate whether the proposed method can ( 1 ) exploit an unknown285 non-stationary opponent while still maintaining a strategy with low exploitability , ( 2 ) improve its286 performance by continued training against different opponents and ( 3 ) exploit previously unseen287 opponents without further training . We verify EPSOM ’ s performance in Kuhn Poker [ 20 ] , a simplified288 version of poker which importantly retains strategic elements useful for game-theoretic analysis . We289 use an agent learning to play using the PPO [ 37 ] algorithm as our opponent , which suffices to provide290 a non-stationary setting . More details about the game and our experiment implementation can be291 found in Appendix A.3.292 Figure 3 : Opponent ’ s learning process modeled by trained CEPSOM : ( left ) CEPSOM adjusts playing strategy online ; ( right ) CEPSOM plays an approximate Nash strategy . Each point is 2-D embedding of a modeled policy to which a trajectory produced by the opponent is assigned by our opponent model . Color bar indicates time sequence . We select 3 representative algorithms as baselines . PSRO is a popular algorithm which guarantees the294 convergence to an approximate NE . As PSRO is a self-play algorithm , its is trained before playing295 against any adaptive opponents . Behavior cloning ( BC ) models the opponent ’ s policy by taking296 maximum likelihood estimation of history trajectories stored in a sliding-window buffer and learns297 an ( approximate ) best-response policy to it . PPO represents a canonical choice among many SARL298 algorithms . In our work , as agents and their opponents update asynchronously , we always evaluate299 each algorithm ’ s performance right after the opponent ’ s update for a more robust evaluation . The300 following results reported with mean and standard deviation ( std ) are all obtained by repeating the301 corresponding experiment over 5 random seeds.302 As shown in Figure 2 , EPSOM can achieve a safe strategy with relatively low exploitability while still303 being able to exploit its opponent . Though PSRO plays a strategy with the lowest exploitability ( ⇡ 0 ) 304 it also has very low exploitation against its opponent . In contrast , BC can exploit its opponent to a305 similar extent as EPSOM but it comes with the cost of high exploitability . The PPO algorithm has306 large variance and performs badly on average in this non-stationary setting . We also test a continual307 learning version of EPSOM which we name CEPSOM . It is implemented by training an EPSOM agent308 against 5 different opponents without re-initialization thereby building up a richer set of modeled309 opponent policies and a more robust best-response policy population . Its average performance over310 these opponents is also reported in Figure 2 . In our experiments , we use an analytical method to311 calculate a best response to a given policy . We obtain similar results for approximate best response312 learned by RL which are reported in Appendix A.5.313 Next , we test these agent ’ s performance against two adaptive opponents implemented by PPO and314 TRPO [ 36 ] without further training and results are presented in Table 1 . Relying on an opponent model315 to predict the current opponent ’ s policy type and flexibly adjusting its playing strategy accordingly,316 CEPSOM achieves the highest average exploitation against adaptive opponents . EPSOM also obtains317 positive average exploitation but with a much lower value , since EPSOM has only ever been trained318 with one opponent . PSRO plays a safe strategy and performs slightly better than EPSOM in terms of319 opponent exploitation . BC and PPO perform badly as they overfit to one opponent , and thus they are320 exploited by other adaptive opponents . Note that , in this zero-shot learning tournament , although we321 do not train EPSOM and CEPSOM , they still need to predict the opponent ’ s policy and solve the322 meta-game for a RNR solution given the prediction.323 Facing an adaptive opponent , a trained CEPSOM ’ s opponent model can assign trajectories collected324 during learning into modeled policies it has built . Therefore , we can visualize an opponent ’ s learning325 process by presenting a sequence of those modeled policies on a 2-dimensional plot . In Figure 3,326 we present two opponent ’ s visualized learning process when it faces ( a ) a trained CEPSOM agent327 which adjusts its playing strategy online based on its opponent model prediction , and ( b ) a trained328 CEPSOM agent which always plays an approximate Nash equilibrium strategy . We can see that in a329 non-stationary environment ( left ) , the opponent ’ s learning exhibits a cyclic pattern . In contrast , the330 opponent ’ s learning is much more transitive ( i.e. , monotonically moving in one direction in the 2D331 space ) in a stationary environment ( right ) .332 7 Conclusion333 In this work , we propose a framework for training an agent to safely exploit its opponent . Compared334 to RNR and its variants , our work focuses on non-stationary opponents . We consider the opponent ’ s335 learning as a series of policy transitions and model such a process by a Dirichlet Process . Safe336 exploitation means that an agent can exploit an opponent ’ s weakness to maximize our utility while337 simultaneously maintaining a strategy which has low exploitability . This property is desirable as338 naively overfitting to one type of opponent could easily lead to exploitation by other opponents . We339 empirically verify our algorithm ’ s performance on Kuhn Poker , a simplified version of Poker.340 Opponent modeling based MARL algorithms typically require extra computation for learning a341 good opponent model . This cost often scales dramatically with the number of opponents , action342 space dimensionality and the complexity of the problem . It can be a heavy burden on an agent if343 it learns an opponent model from scratch online . Therefore , a more realistic way for utilizing the344 power of an opponent model is offline training and online prediction . We build CEPSOM based on345 this idea where we train one EPSOM agent across different opponents and aggregate knowledge by346 maintaining a never-reinitialized opponent model and policy population . Our experiment results show347 that CEPSOM can achieve high exploitation against a new adaptive opponent without further training,348 outperforming other representative baselines from SARL and MARL . In complex competitive games,349 a strong player can often encounter sub-optimal opponents and playing a Nash strategy can potentially350 forego significant profit . EPSOM , alongside many prior works , shows the potential of an opponent351 modeling based approach for solving this problem , and our preliminary results from CEPSOM352 demonstrate the possibility of a trained agent beating an as yet unseen adaptive opponent.353 EPSOM is limited by its computation and memory complexity . Naively applying EPSOM to more354 complex problems requires a great amount of resources . 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Summary: The paper proposes to use the Dirichlet process for opponent modelling in games. It argues that the approach is particularly suitable for non-stationary opponents. It then proposes an algorithm to compute a counter-strategy to the model based on the restricted Nash response. Empirical evaluation on Kuhn poker shows that this approach can exploit a non-stationary learning opponent similarly to one of pre-existing methods, while maintaining relatively slow exploitability.