Current literature in machine learning holds that unaligned, self-interested agents do not learn to use an emergent communication channel. We introduce a new sender-receiver game to study emergent communication for this spectrum of partially-competitive scenarios and put special care into evaluation. We find that communication can indeed emerge in partially-competitive scenarios, and we discover three things that are tied to improving it. First, that selfish communication is proportional to cooperation, and it naturally occurs for situations that are more cooperative than competitive. Second, that stability and performance are improved by using LOLA , especially in more competitive scenarios. And third, that discrete protocols lend themselves better to learning cooperative communication than continuous ones. Multi-agent RL is used for many fully-competitive, zero-sum games (; ;) with the objective of finding a single, best learned agent (or team) at test time. In these games, a "best" agent is one that can outplay all opponents . We cannot evaluate its play by looking at the reward that agent achieves against a given opponent because the reward received will be a function of the choice of opponent. 1 In contrast, MARL in fully cooperative games has usually been able to make the assumption that we get to pick our team, and so can use self-play to try and achieve a maximal possible reward. (b). In this way, fully-cooperative MARL compares the maximum performance between learning algorithms and architectures by optimising the joint reward of the team. We investigate the space of partially competitive games known as general-sum games, where there is some amount of common interest and some amount of conflict. In this case, care must be taken in defining the "best" agent: it is not necessarily the agent that does as well or better than all opponents because, to achieve the highest possible reward in a general-sum game, agents might have to cooperate to some extent . For example, consider an agent playing iterated prisoner's dilemma. The agent that always defects will never have a reward worse than its opponent, but, when playing against a tit-for-tat agent, it will also not achieve the total reward of an agent that is also playing a tit-for-tat strategy. Since the maximum expected reward may only be possible by cooperating, agents must learn how to coordinate with each other. This can be done by training together or learning to understand opponent intentions at test time by observing their actions. The latter allows for ad-hoc comparison of learned agents at test time; the former requires comparing learning algorithms trained together. The latter should then require a sequential/iterative game, so there is time to infer opponent intentions before acting . It may also require meta-learning or other modifications to understand and adapt to these intentions as it seems current self-play methods are insufficient to adapt to ad-hoc play even against different versions of their own architecture . Work in general-sum MARL has mostly worked on analysing learning algorithms' ability to cooperate and resolve social dilemmas (a; ;) To date, investigations of emergent communication have remained mostly in the realm of fullycooperative games; ). For continuous communication, claimed to learn in mixed cooperative-competitive scenarios. However, their setup uses parameter sharing between opponents; their "mixed" case is non-competitive (and implicitly cooperative); their competitive game is actually two stages-one fully cooperative and one fully competitive; and, their in the competitive scenario are simply to mask out all communication. Previous attempts to learn discrete emergent protocols by selfish agents in competitive games have failed unless additional, more complex learning rules are adopted . In the latter case, they compare learning algorithms trained together as opposed to learned agents at test time to avoid a significant issue of comparing emergent communication agents-i.e., different protocols. Since an emergent protocol only has meaning between the agents that learned it together, comparing two learned agents at test time would require that they infer each others' protocols without training with them. This seems impossible, and it is more reasonable to frame it as a meta-learning problem where agents can have a brief adaptation period to synchronise their protocols. Meta-learning is beyond the scope of this paper, so we follow in comparing learning algorithms trained together. We also take a more principled approach than previous work by guaranteeing no communication through the action space, using more rigorous, quantitative criteria for evaluating communication, and precisely setting the levels of competitiveness. Notably, this work does not propose new architectures or learning rules but aims to take a critical look at existing beliefs and draw important distinctions as did for natural language emergence. Our experiments use the simple emergent-communication framework known as a sender-receiver game (or "referential game" ), which finds extensive use in economics and philosophy among others. In the classic game, the sender is given a target value to be communicated to the receiver via a message. The receiver receives the sender's message and must decode it to predict the target value. Both players are rewarded according to the negative of the receiver's prediction error. In this fully-cooperative setting, players often coordinate a protocol to transfer information as effectively as possible . The messages between the sender and the receiver can be categorised as "cheap talk": messages are costless, non-binding, non-verifiable and may affect the receiver's beliefs . Our work benefits greatly from -a seminal work in classical game theory. They study possible fixed communication equilibria under competition by giving the sender and receiver different targets and creating a conflict of interest. They perform a static analysis and prove the existence of a Nash equilibrium where the amount of information communicated is proportional to the alignment between the players' interests; however no informative equilibrium exists when interest diverge too greatly. In contrast, we do not look for existence of an equilibrium but do a dynamic analysis and show the feasibility of communication using standard learning rules in RL. We do not explicitly aim for equilibria but look at the information transfer of communication protocols in flux (and therefore out of equilibrium). This is more in line with previous work in emergent communication as well as evolutionary signalling . To investigate a range of competitive scenarios, we introduce a modified sender-receiver game with a continuous-bias variable, b, that represents the agents' conflict of interest, ranging from fully cooperative to fully competitive. The two players-the Sender (S) and the Receiver (R)-have corresponding targets (T s and T r), which are represented by angles on a circle that are b degrees apart: T r = (T s + b) mod 360 •. The game starts with the sender's target being sampled uniformly from the circle T s ∼ Uniform. The sender is given its target as input and outputs a message, m = S(T s), consisting of a single, discrete token from a vocabulary m ∈ V. The receiver is given the message and outputs a scalar action a = R(m). The goal of each agent is to make the receiver's action as close as possible to its own target value. After the receiver acts, both players get a loss between the action and their respective targets,. By using an L 1 loss between the angle of the target and action ) and a game with the maximum bias b = 180 • is fully competitive or constant-sum (a generalisation of zero-sum, see Appendix A.1 for proof). All values in-between, b ∈ (0 •, 180 •), represent the spectrum of partially cooperative/competitive general-sum games. Figure 1a gives an instance of this game; the game's algorithm is given in Algorithm 1 in Appendix B. This can be seen as the game from modified to cover the range of cooperative/competitive games. Both agents are implemented as MLPs with two hidden layers and ReLU nonlinearities between all layers. The targets are sampled from the circle, the sender takes its target, T s, as input and outputs a categorical distribution over a vocabulary from which we sample a message-its output. The receiver takes the message as input and deterministically outputs its action, a. Errors are calculated using the L 1 loss on the circle. The sender estimates its loss using the score function estimator -also known as REINFORCE -and has an added entropy regularisation term. Since the loss is differentiable with respect to the receiver, it is trained directly with gradient descent, so we are training in the style of a stochastic computation graph . We train for 30 epochs of 250 batches, with batch size 64, and set the circumference of our circle to 36 (so that a loss of 90 • is an error of 9). Both agents are trained using Adam . To evaluate, we use a fixed test set of 100 equidistant points and take the arg max of output distributions instead of sampling. We do all hyperparameter searches with Oríon , using random search with a fixed budget of searches. We perform a hyperparameter search to over both agents' learning rates, hidden layer sizes, the vocabulary size, and entropy regularisation (when used). We always report for given hyperparameters averaged over 5 random seeds, and we average our metric for hyperparameter search over the last 10 epochs to capture some level of stability as well as performance. All hyperparameter search spaces are available in the config files of the code repository. To evaluate the communication emerged with a cheap-talk channel, we can simply look to the sum of agents' L 1 losses. Under non-communication (or uninformative communication), we know that the receiver will just guess a point at random, and the average loss for both players is the expected value of the loss-given that it is drawn uniformly • Therefore, any error for either agent below 90 • is evidence of information transfer . Furthermore, since there is no other action space for agents to communicate in, the information transfer must be happening in the emergent communication space . Therefore, the lower L r 1 + L s 1 is, the more informative the learned protocol is; and, the most informative protocol will have the lowest loss min To show this comparison, we always plot the loss under uninformative communication (90 •) and the loss for each agent if they were to both fairly split the bias (b/2). While we have found evidence of information transfer, does that necessarily mean our agents have learned to communicate? For example, our hyperparameter search could find a minimal learning rate for the sender, such that it is essentially static, and a normal configuration for the receiver. The game would then become not one of learning a protocol between two agents, but rather just a receiver learning the sender's initial random mapping of targets to messages. The receiver could then dominate the sender by always choosing a = T r, which would yield L r 1 + L s 1 = b; namely, the optimal sum of losses and, therefore, optimal information transfer. This situation is clearly not what we are looking for, but it would be permissible, or potentially even encouraged, under an information-transfer objective (as measured by the sum of agents' L 1 losses). It is, therefore, necessary to delineate the differences in communication; here, we can look to extant in signalling . One perspective on information transfer is that of manipulation of receivers by senders or vice-versa ; this manifests as the domination of one agent over the other. We note that these situations are modelled as cue-reading or sensory manipulation, respectively, and are distinct from signalling-i.e., communication . Accordingly, communication requires both agents to receive a net benefit , which implies some degree of cooperation . For the fully-cooperative case, previous metrics of joint reward , or even influence of communication , are sufficient to drive the hyperparameter search. But for competitive scenarios, neither of these can distinguish between manipulation and cooperation . Since our focus is on the emergence of cooperative communication, we are looking for settings where both agents perform better than either their fully-exploited losses (L 2) as our hyperparameter-search metric. We can view our partially competitive scenario as having a common-interest loss (180 • − b), in which both agents are fully cooperative, and a conflict-of-interest loss (b), in which both agents are fully competitive. The sum of L 1 losses optimises only for the common interest, whereas L 2 prefers a more fair division of the conflict-of-interest loss in addition to optimising common interest (see proof in Appendix A.2). We use the L 2 metric only on hyperparameter search and keep L 1 as our game's loss to maintain a constant-sum game for the fully competitive case. • because the game is constant-sum and therefore trivially L •, but for completeness you can see the of a hyperparameter search with b = 180 • in Appendix C Figure 16. We report our in Figure 2 and find that agents do learn to cooperatively communicate without any special learning rules contrary to current literature. We can see that the performance decreases proportionately to the bias, meaning the sender is less informative with messages, forcing the receiver to be less accurate in its own guesses. This matches the theoretical of; information transfer with communication is inversely proportional to the conflict of interest. Plots for each b are in shown in Appendix C Figure 6. For the curve, we still plot the L 1 losses to maintain consistency and to make clearer the comparison to the no-communication baseline and the optimal information transfer (common interest maximisation). We find that our are basically unchanged between the different hyperparameter metrics; a relatively fair and useful protocol is learned by the agents, but this deteriorates in more competitive scenarios. This is clear when comparing the stability and relative efficacy of protocols in b = 30 •, 60 •, shown in Figures 2b, 2c, and that of b = 90 • shown in Figure 2d. We can understand this through the lens of honest communication, which can be taken advantage of in highly competitive scenarios. If, for example, the sender communicates, with complete honesty, its own coordinates, then the receiver can take advantage of this and choose its location exactly so that L r = 0 and L s = b. Comparing this situation to non-communication (L r = L s = 90 •), it is clear that even fully-exploited communication is a strictly dominant strategy for b < 90 • (i.e, when the game is more cooperative than competitive). For more competitive cases, fully-exploitable communication is no longer dominant, and active communication now requires both agents to cautiously cooperate. To achieve this cooperation, we • ] we show the training curve of the best hyperparameters found in 2b,2c,2d. We plot the test loss over training epochs and showing the mean and standard deviation over 5 seeds, finding that for b < 90 • we find stable and relatively fair communication is naturally learned propose using LOLA (a)-a learning rule, resembling theory-of-mind, that allows us to backpropogate through n steps of the opponent's learning. LOLA was able to emerge cooperative behaviour in an iterated prisoner's dilemma, so it is a prime candidate for resolving our game in a similar situation. We experiment with LOLA in three configurations-LOLA on the sender, LOLA on the receiver, LOLA on both-and do a similar hyperparameter search, with the added search space of the LOLA learning rate. Per the improvements made by , we replace the receiver's score function estimate with the DiCE estimator, and we backpropogate through exact copies of opponents as opposed to using opponent modelling. We show our in Figure 3a with extended plots in Figures 8, 9, 10 in Appendix C. We find that LOLA on the sender is ineffective, but LOLA on the receiver and on both agents does indeed lead to better performance. This implies that emerging communication in competitive scenarios necessitates cooperation and that this cooperation can be found through explicit opponent modelling. Furthermore, comparing the curves of basic agents (Figure 2d) with those of LOLA agents (Figure 3c) shows that gains in performance are not from one agent dominating the other, but from both agents improving and increasing stability. We also look at the performance of n-step LOLA, which backpropogates through n > 1 steps of opponent learning. Figure 3b demonstrates that 2-step LOLA slightly outperforms 1-step LOLA, but 3-step LOLA does not provide any increase over 2-step. We see from Figure 3d that the increase comes mostly from stability of learning and slight improvement on the part of the sender. • where our original setup does very poorly. 3b shows that higher step LOLA improves slightly further but not past 2-step. Best feasible communication protocols found for b = 90 • using 1-step (3c) and 2-step LOLA (3d) on both agents demonstrates that the gains in performance over the basic setup shown in Figure 2d are not just from one agent doing better (though the sender is doing better) but both agents improving in performance and stability. Shaded area is standard deviation over 5 seeds Another axis to consider is whether discrete or continuous communication lends itself better to learning with selfish agents. To compare, we make the sender's message a real-valued scalar and appropriately change its output distribution to be a Gaussian, for which it learns the mean and variance (concretely described in Algorithms 2 in Appendix B). We, again, run hyperparameter searches, and we consider training our baseline training with a REINFORCE Sender and deterministic receiver as well as training both agents with 1-step LOLA. Our in Figure 4a suggest that the learned protocols for continuous communication are all highly informative and near optimal. However, in all cases, the receiver is learning to manipulate the sender, and there is little evidence of cooperative communication. Indeed, we found no cases of both agents having a net benefit (L r, L s < 90 •) in any of the hyperparameter runs for continuous REINFORCE-deterministic agents past b = 90 •, and we only two cases of net benefit for LOLA-1 agents. Comparing this to discrete communication with the same LOLA-1 agents in Figure 4f, we can clearly see that they have a preference for more cooperative behaviour. Thus, we find that discrete messages are an important component in emerging cooperative self-interested communication. First and foremost, we show evidence against the current notion that selfish agents do not learn to communicate, and we hope our findings encourage more research into communication under The comparison between discrete and continuous communication for both the REINFORCE-deterministic setup as well as 1-step LOLA agents is shown in Figure 4a. We see that though overall continuous communication can achieve highest information transfer, the gains in performance seem to mostly from manipulation of the sender by the receiver. Two examples are shown for REINFORCE agents in Figures 4b,4c. To find a trend, we plot all 100 hyperparameter runs for b ∈ between continuous and discrete communication using 1-step LOLA agents in Figures 4d,4e,4f,4g. We find that manipulation is the common in continuous communication though individual cooperative points can sometimes be found. In general, continuous communication does not lend itself to cooperative communication competition. We have shown three important properties of communication. First, a game being more cooperative than competitive is sufficient to naturally emerge communication. Second, we've clarified the distinction between information transfer, communication, and manipulation, providing motivation for a better quantitative metric to measure emergent communication in competitive environments. Next, we've found that LOLA improves effective selfish communication and, using our metric, we find it does so by improving both agents' performance and stability. Finally, we've shown that using a discrete communication channel encourages the learning of cooperative commu-nication in contrast to the continuous communication channel setting, where we find little evidence of cooperation. In fully-cooperative emergent communication, both agents fully trust each other, so cooperatively learning a protocol is mutually beneficial. In competitive MARL, the task is using an existing protocol (or action space) to compete with each other. However, selfish emergent communication combines these two since the inherent competitiveness of using the protocol to win is tempered by the inherent cooperativeness of learning it; without somewhat agreeing to meanings, agents cannot use those meanings to compete . Thus, the agents must both learn a protocol and use that protocol simultaneously. In this way, even while competing, selfish agents emerging a communication protocol must learn to cooperate. A.1 PROOF OF FULLY COOPERATIVE/FULLY COMPETITIVE GAME For b = 0, T s = T r so trivially L s = L r and the game is fully cooperative. For b = 180 •, T r = T s + 180 mod 360 • we provide a visual demonstration in Figure 5 that the sum is always L s + L r = 180 • and therefore the game is constant-sum and fully competitive. We can also think of this as moving the actions distance d towards one agent's target means moving it distance d away from the other agent's target. •. Assume without loss of generality T s < T r so T r = T s + 180 Figure 5: The game with maximal bias 180 • showing the sum of L 1 losses L r + L s = 180 • We can extend the proof by symmetry (on the circle) for |T s − a| ≥ 360 • − |T s − a|, so the sum of losses L r + L s always equals 180 • so the game is constant-sum and therefore fully competitive. A.2 PROOF OF L 2 FAIRNESS Assume without loss of generality T s < T r, we are minimizing the sum of L 2 losses Sum of L 2 losses is minimized when the action is T s + b/2 or halfway between both agents' targets. Algorithm 1 Circular Biased Sender-Receiver Game
We manage to emerge communication with selfish agents, contrary to the current view in ML