Overview
We develop a foundational graph-theoretic toolkit that facilitates analysis of canonical and large-scale games, providing insights into their related topological structure in terms of their high-level strategic interactions. The prerequisite game theory background and technical details are provided in the “Methods” section, with full discussion of related works and additional details in Supplementary Note 1.
Our results are summarized as follows. We use our toolkit to characterize a number of games, first analyzing motivating examples and canonical games with well-defined structures, then extending to larger-scale empirical games datasets. For these larger games, we rely on empirical game-theoretic analysis67,68, where we characterize an underlying game using a sample set of policies. While the empirical game-theoretic results are subject to the policies used to generate them, we rely on a sampling scheme designed to capture a diverse variety of interactions within each game, and subsequently conduct sensitivity analysis to validate the robustness of the results. We demonstrate correlation between the complexity of the graphs associated with games and the complexity of solving the game itself. In Supplementary Note 2, we evaluate our proposed method against baseline approaches for taxonomization of 2 × 2 games38. We finally demonstrate how this toolkit can be used to automatically generate interesting game structures that can, for example, subsequently be used to train AI agents.
Motivating example
Let us start with a motivating example to solidify intuitions and explain the workflow of our graph-theoretic toolkit, using classes of games with simple parametric structures in the player payoffs. Specifically, consider games of three broad classes (generated as detailed in the Supplementary Methods 1): games in which strategies have a clear transitive ordering (Fig. 2a); games in which strategies have a cyclical structure wherein all but the final strategy are transitive with respect to one another (Fig. 2d); and games with random (or no clear underlying) structure (Fig. 2g). We shall see that the core characteristics of games with shared underlying structure is recovered via the proposed analysis.
a, d, and g, respectively, visualize payoffs for instances of games with transitive, cyclical, and random structure. Each exemplified game consists of two players with 10 strategies each (with payoff row and column labels, {s0, …, s9}, indicating the strategies). Despite the numerous payoff variations possible in each class of games illustrated, each shares the underlying payoff structure shown, respectively, in b, e, and h. Moreover, variations in payoffs can notably impact the difficulty of solving (i.e., finding the Nash equilibrium) of these games, as visualized in c, f, i.
Each of these figures visualizes the payoffs corresponding to 4 instances of games of the respective class, with each game involving 10 strategies per player; more concretely, entry M(si, sj) of each matrix visualized in Fig. 2a, d and g quantifies the payoff received by the first player if the players, respectively, use strategies si and sj (corresponding, respectively, to the i-th and j-th row and column of each payoff table). Despite the variance in payoffs evident in the instances of games exemplified here, each essentially shares the payoff structure exposed by re-ordering their strategies, respectively, in Fig. 2b, e and h. In other words, the visual representation of the payoffs in this latter set of figures succinctly characterizes the backbone of strategic interactions within these classes of games, despite not being immediately apparent in the individual instances visualized.
More importantly, the complexity of learning useful mixed strategies to play in each of these games is closely associated with this structural backbone. To exemplify this, consider the computational complexity of solving each of these games; for brevity, we henceforth refer to solving a game as synonymous with finding a Nash equilibrium (similar to prior works69,70,71,72, wherein the Nash equilibrium is the solution concept of interest). Specifically, we visualize this computational complexity by using the Double Oracle algorithm73, which has been well-established as a Nash solver in multiagent and game theory literature47,74,75,76. At a high level, Double Oracle starts from a sub-game consisting of a single randomly-selected strategy, iteratively expands the strategy space via best responses (computed by an oracle), until discovery of the Nash equilibrium of the full underlying game.
Figure 2c, f and i visualize the distribution of Double Oracle iterations needed to solve the corresponding games, under random initializations. Note, in particular, that although the underlying payoff structure of the transitive and cyclical games, respectively, visualized in Fig. 2a, d is similar, the introduction of a cycle in the latter class of games has a substantial impact on the complexity of solving them (as evident in Fig. 2f). In particular, whereas the former class of games are solved using a low (and deterministic) number of iterations, the latter class requires additional iterations due to the presence of cycles increasing the number of strategies in the support of the Nash equilibrium.
Workflow
Overall, the characterization of the topological structure of games is an important and nuanced problem. To address this problem, we use graph theory to build an analytical toolkit automatically summarizing the high-level strategic interactions within a game, and providing useful complexity measures thereof. Specifically, consider again our motivating transitive game, re-visualized using a collection of graph-based measures in Fig. 3. Each of these measures provides a different viewpoint on the underlying game, collectively characterizing it. Specifically, given the game payoffs, Fig. 3b visualizes the so-called α-Rank response graph of the game; here, each node corresponds to a strategy (for either player, as this particular game’s payoffs are symmetric). Transition probabilities between nodes are informed by a precise evolutionary model (detailed in “Methods” section and Omidshafiei et al.50); roughly speaking, a directed edge from one strategy to another indicates the players having a higher preference for the latter strategy, in comparison to the former. The response graph, thus, visualizes all preferential interactions between strategies in the game. Moreover, the color intensity of each node indicates its so-called α-Rank, which measures the long-term preference of the players for that particular strategy, as dictated by the transition model mentioned above; specifically, darker colors here indicate more preferable strategies.
Given the game payoffs a, the so-called α-Rank response graph of the game is visualized in b. In c, reprojecting the response graph by using the top eigenvectors of the graph Laplacian yields the spectral response graph, wherein similar strategies are placed close to one another. In d, taking this one step further, one can cluster the spectral response graph, yielding the clustered response graph, which exposes three classes of strategies in this particular example. In e, contracting the clustered graph by fusing nodes within each cluster yields the high-level characterization of transitive games. In f, the lack of cycles in the particular class of transitive games becomes evident. Finally, in g and h, one can extract the principal components of various response graph statistics and establish a feedback loop to a procedural game structure generation scheme to yield new games.
This representation of a game as a graph enables a variety of useful insights into its underlying structure and complexity. For instance, consider the distribution of cycles in the graph, which play an important role in multiagent evaluation and training schemes41,50,53,77 and, as later shown, are correlated to the computational complexity of solving two-player zero-sum games (e.g., via Double Oracle). Figure 3f makes evident the lack of cycles in the particular class of transitive games; while this is clearly apparent in the underlying (fully ordered) payoff visualization of Fig. 3a, it is less so in the unordered variants visualized in Fig. 2a. Even so, the high-level relational structure between the strategies becomes more evident by conducting a spectral analysis of the underlying game response graph. Full technical details of this procedure are provided in the “Methods” section. At a high level, the so-called Laplacian spectrum (i.e., eigenvalues) of a graph, along with associated eigenvectors, captures important information regarding it (e.g., number of spanning trees, algebraic connectivity, and numerous related properties78). Reprojecting the response graph by using the top eigenvectors yields the spectral response graph visualized in Fig. 3c, wherein similar strategies are placed close to one another. Moreover, one can cluster the spectral response graph, yielding the clustered response graph, which exposes three classes of strategies in Fig. 3d: a fully dominated strategy with only outgoing edges (a singleton cluster, on the bottom left of the graph), a transient cluster of strategies with both incoming and outgoing edges (top cluster), and a dominant strategy with all incoming edges (bottom right cluster). Finally, contracting the clustered graph by fusing nodes within each cluster yields the high-level characterization of transitive games shown in Fig. 3e.
We can also conduct this analysis for instances of our other motivating games, such as the cyclical game visualized in Fig. 4a. Note here the distinct differences with the earlier transitive game example; in the cyclical game, the α-Rank distribution in the response graph (Fig. 4b) has higher entropy (indicating preference for many strategies, rather than one, due to the presence of cycles). Moreover, the spectral reprojection in Fig. 4d reveals a clear set of transitive nodes (left side of visualization) and a singleton cluster of a cycle-inducing node (right side). Contracting this response graph reveals the fundamentally cyclical nature of this game (Fig. 4f). Finally, we label each strategy (i.e., each row and column) of the original payoff table Fig. 4a based on this clustering analysis. Specifically, the color-coded labels on the far left (respectively, top) of each row (respectively, column) in Fig. 4a correspond to the clustered strategy colors in Fig. 4d. This color-coding helps clearly identify the final strategy (i.e., bottom row of the payoff table) as the outlier enforcing the cyclical relationships in the game. Note that while there is no single graphical structure that summarizes the particular class of random games visualized earlier in Fig. 2h, we include this analysis for several instances of such games in Supplementary Note 2.
a Game payoffs, b response graph, c cycles histogram, d spectral response graph, e clustered response graph, f contracted response graph.
Crucially, a key benefit of this analysis is that the game structure exposed is identical for all instances of the transitive and cyclical games visualized earlier in Fig. 2a, d, making it significantly easier to characterize games with related structure, in contrast to analysis of raw payoffs. Our later case studies further exemplify this, exposing related underlying structures for several classes of more complex games.
Analysis of canonical and real-world games
The insights afforded by our graph-theoretic approach apply to both small canonical games and larger empirical games (where strategies are synonymous with trained AI agents).
Consider the canonical Rock–Paper–Scissors game, involving a cycle among the three strategies (wherein Rock loses to Paper, which loses to Scissors, which loses to Rock). Figure 5a visualizes a variant of this game involving a copy of the first strategy, Rock, which introduces a redundant cycle and thus affects the distribution of cycles in the game. Despite this, the spectral response graph (Fig. 5d) reveals that the redundant game topologically remains the same as the original Rock–Paper–Scissors game, thus reducing to the original game under spectral clustering.
In Redundant RPS, the redundant copy of the first strategy (Rock) is clustered in the spectral response graph. In 11–20, seven clusters of strategies are revealed, exposing the cyclical nature of this game.
This graph-based analysis also extends to general-sum games. As an example, consider the slightly more complex game of 11–20, wherein two players each request an integer amount of money between 11 and 20 units (inclusive). Each player receives the amount requested, though a bonus of 20 units is allotted to one player if they request exactly 1 unit less than the other player. The payoffs and response graph of this game are visualized, respectively, in Fig. 5g, h, where strategies, from top-to-bottom and left-to-right in the payoff table, correspond to increasing units of money being requested. This game, first introduced by Arad and Rubinstein79, is structurally designed to analyze so-called k-level reasoning, wherein a level-0 player is naive (i.e., here simply requests 20 units), and any level-k player responds to an assumed level-(k − 1) opponent; e.g., here a level-1 player best responds to an assumed level-0 opponent, thus requesting 19 units to ensure receiving the bonus units.
The spectral response graph here (Fig. 5j) indicates a more complex mix of transitive and intransitive relations between strategies. Notably, the contracted response graph (Fig. 5l) reveals 7 clusters of strategies. Referring back to the rows of payoffs in Fig. 5g, relabeled to match cluster colors, demonstrates that our technique effectively pinpoints the sets of strategies that define the rules of the game: weak strategies (11 or 12 units, first two rows of the payoff table, and evident in the far-right of the clustered response graph), followed by a set of intermediate strategies with higher payoffs (clustered pairwise, near the lower-center of the clustered response graph), and finally the two key strategies that establish the cyclical relationship within the game through k-level reasoning (19 and 20 units, corresponding to level-0 and level-1 players, in the far-left of the clustered response graph).
This analysis extends to more complex instances of empirical games, which involve trained AI agents, as next exemplified. Consider first the game of Go, as played by 7 AlphaGo variants: AG(r), AG(p), AG(v), AG(rv), AG(rp), AG(vp), and AG(rvp), where each variant uses the specified combination of rollouts r, value networks v, and/or policy networks p. We analyze the empirical game where each strategy corresponds to one of these agents, and payoffs (Fig. 6a) correspond to the win rates of these agents when paired against each other (as detailed by Silver et al.7 Table 9). The α-Rank distribution indicated by the node (i.e., strategy) color intensities in Fig. 6b reveals AG(rvp) as a dominant strategy, and the cycle distribution graph Fig. 6c reveals a lack of cycles here. The spectral response graph, however, goes further, revealing a fully transitive structure (Fig. 6d, e), as in the motivating transitive games discussed earlier. The spectral analysis on this particular empirical game, therefore, reveals its simple underlying transitive structure (Fig. 6f).
Note that as these are empirical games, strategies here correspond to trained AI agents. In AlphaGo, the strong transitive relationship between agents is revealed via our analysis. In MuJoCo soccer, more complex relations between similarly-performing agents are revealed in the clusters produced.
Consider a more interesting empirical game, wherein agents are trained to play soccer in the continuous control domain of MuJoCo, exemplified in Fig. 6 (second row). Each agent in this empirical game is generated using a distinct set of training parameters (e.g., feedforward vs. recurrent policies, reward shaping enabled and disabled, etc.), with full agent specifications and payoffs detailed by Liu et al.80. The spectral response graph (Fig. 6k) reveals two outlier agents: a strictly dominated agent (node in the top-right), and a strong (yet not strictly dominant) agent (node in the top-left). Several agents here are clustered pairwise, revealing their closely-related interactions with respect to the other agents; such information could, for example, be used to discard or fuse such redundant agents during training to save computational costs.
Consider next a significantly larger-scale empirical game, consisting of 888 StarCraft II agents from the AlphaStar Final league of Vinyals et al.9. StarCraft II is a notable example, involving a choice of 3 races per player and realtime gameplay, making a wide array of behaviors possible in the game itself. The empirical game considered is visualized in Fig. 7a, and is representative of a large number of agents with varying skill levels. Despite its size, spectral analysis of this empirical game reveals that several key subsets of closely-performing agents exist here, illustrated in Fig. 7d. Closer inspection of the agents used to construct this empirical payoff table reveals the following insights, with agent types corresponding to those detailed in Vinyals et al.9: (i) the blue, orange, and green clusters are composed of agents in the initial phases of training, which are generally weakest (as observed in Fig. 7d, and also visible as the narrow band of low payoffs in the top of Fig. 7a); (ii) the red cluster consists primarily of various, specialized exploiter agents; iii) the purple and brown clusters are primarily composed of the league exploiters and main agents, with the latter being generally higher strength than the former. To further ascertain the relationships between only the strongest agents, we remove the three clusters corresponding to the weakest agents, repeating the analysis in Fig. 7 (bottom row). Here, we observe the presence of a series of progressively stronger agents (bottom nodes in Fig. 7h), as well as a single outlier agent which quite clearly bests several of these clusters (top node of Fig. 7h).
Spectral analysis of the empirical AlphaStar League game reveals that several key subsets of closely-performing agents, illustrated in d. Closer inspection of the agents used to construct this empirical payoff table reveals the following insights, with agent types corresponding to those detailed in Vinyals et al.9: (i) the blue, orange, and green clusters are composed of agents in the initial phases of training, which are generally weakest (as observed in d, and also visible as the narrow band of low payoffs in the top region of the payoff table a); (ii) the red cluster consists primarily of various, specialized exploiter agents; (iii) the purple and brown clusters are primarily composed of the league exploiters and main agents, with the latter being generally higher strength than the former.
An important caveat, as this stage, is that the agents in AlphaGo, MuJoCo soccer, and AlphaStar above were trained to maximize performance, rather than to explicitly reveal insights into their respective underlying games of Go, soccer, and StarCraft II. Thus, this analysis focused on characterizing relationships between the agents from the Policy Problem perspective, rather than the underlying games themselves, which provide insights into the interestingness of the game (Problem Problem). This latter investigation would require a significantly larger population of agents, which cover the policy space of the underlying game effectively, as exemplified next.
Naturally, characterization of the underlying game can be achieved in games small enough where all possible policies can be explicitly compared against one another. For instance, consider Blotto(τ, ρ), a zero-sum two-player game wherein each player has τ tokens that they can distribute amongst ρ regions81. In each region, each player with the most tokens wins (see Tuyls et al.53 for additional details). In the variant we analyze here, each player receives a payoff of +1, 0, and −1 per region, respectively, won, drawn, and lost. The permutations of each player’s allocated tokens, in turn, induce strong cyclical relations between the possible policies in the game. While the strategy space for this game is of size \(\left(\begin{array}{l}\tau +\rho -1\\ \rho -1\end{array}\right)\), payoffs matrices can be fully specified for small instances, as shown for Blotto(5,3) and Blotto(10,3) in Fig. 8 (first and second row, respectively). Despite the differences in strategy space sizes in these particular instances of Blotto, the contracted response graphs in Fig. 8d, h capture the cyclical relations underlying both instances, revealing a remarkably similar structure.
Despite the significant difference in sizes, both instances of Blotto yield a remarkably similar contracted response graph. Moreover, the contracted response graph for Go is notably different from AlphaGo results, due to the latter being an empirical game constructed from trained AI agents rather than a representative set of sampled policies.
For larger games, the cardinality of the pure policy space typically makes it infeasible to fully enumerate policies and construct a complete empirical payoff table, despite the pure policy space being finite in size. For example, even in games such as Tic–Tac–Toe, while the number of unique board configurations is reasonably small (9! = 362880), the number of pure, behaviorally unique policies is enormous (≈10567, see Czarnecki et al.37 Section J for details). Thus, coming up with a principled definition of a scheme for sampling a relevant set of policies summarizing the strategic interactions possible within large games remains an important open problem. In these instances, we rely on sampling policies in a manner that captures a set of representative policies, i.e., a set of policies of varying skill levels, which approximately capture variations of strategic interactions possible in the underlying game. The policy sampling approach we use is motivated with the above discussions and open question in mind, in that it samples a set of policies with varying skill levels, leading to a diverse set of potential transitive and intransitive interactions between them.
Specifically, we use the policy sampling procedure proposed by Czarnecki et al.37, which also seeks a set of representative policies for a given game. The details of this procedure are provided in Supplementary Methods 1, and at a high level involve three phases: (i) using a combination of tree search algorithms, Alpha–Beta82 and Monte Carlo Tree Search83, with varying tree depth limits for the former and varying number of simulations allotted to the latter, thus yielding policies of varying transitive strengths; (ii) using a range of random seeds in each instantiation of the above algorithms, thus producing a range of policies for each level of transitive strength; (iii) repeating the same procedure with negated game payoffs, thus also covering the space of policies that actively seek to lose the original game. While this sampling procedure is a heuristic, it produces a representative set of policies with varying degrees of transitive and intransitive relations, and thus provides an approximation of the underlying game that can be feasibly analyzed.
Let us revisit the example of Go, constructing our empirical game using the above policy sampling scheme, rather than the AlphaGo agents used earlier. We analyze a variant of the game with board size 3 × 3, as shown in Fig. 8 (third row). Notably, the contracted response graph (Fig. 8l) reveals the presence of a strongly-cyclical structure in the underlying game, in contrast to the AlphaGo empirical game (Fig. 6). Moreover, the presence of a reasonably strong agent (visible in the top of the contracted response graph) becomes evident here, though this agent also shares cyclical relations with several sets of other agents. Overall, this analysis exemplifies the distinction between analyzing an underlying game (e.g., Go) vs. analyzing the agent training process (e.g., AlphaGo). Investigation of links between these two lines of analysis, we believe, makes for an interesting avenue for future work.
Linking response graph and computational complexity
A question that naturally arises is whether certain measures over response graphs are correlated with the computational complexity of solving their associated games. We investigate these potential correlations here, while noting that these results are not intended to propose that a specific definition of computational complexity (e.g., with respect to Nash) is explicitly useful for defining a topology/classification over games. In Fig. 9, we compare several response graph complexity measures against the number of iterations needed to solve a large collection of games using the Double Oracle algorithm73. The results here consider specifically the α-Rank entropy, number of 3-cycles, and mean in-degree (with details in “Methods” section, and results for additional measures included in Supplementary Note 2). As in earlier experiments, solution of small-scale games is computed using payoffs over full enumeration of pure policies, whereas that of larger games is done using the empirical games over sampled policies. Each graph complexity measure reported is normalized with respect to the maximum measure possible in a graph of the same size, and the number of iterations to solve is normalized with respect to the number of strategies in the respective game. Thus, for each game, the normalized number of iterations to solve provides a measure of its relative computational complexity compared to games with the same strategy space size (for completeness, we include experiments testing the effects of normalization on these results in Supplementary Note 2).
Each figure plots a respective measure of graph complexity against the normalized number of iterations needed to solve the associated game via the Double Oracle algorithm (with normalization done with respect to the total number of strategies in each underlying game). The Spearman correlation coefficient, ρs, is shown in each figure (with the reported two-sided p-value rounded to two decimals).
Several trends can be observed in these results. First, the entropy of the α-Rank distribution associated with each game correlates well with its computational complexity (see Spearman’s correlation coefficient ρs in the top-right of Fig. 9a). This matches intuition, as higher entropy α-Rank distributions indicate a larger support over the strategy space (i.e., strong strategies, with non-zero α-Rank mass), thus requiring additional iterations to solve. Moreover, the number of 3-cycles in the response graph also correlates well with computational complexity, again matching intuition as the intransitivities introduced by cycles typically make it more difficult to traverse the strategy space42. Finally, the mean in-degree over all response graph nodes correlates less so with computational complexity (though degree-based measures still serve a useful role in characterizing and distinguishing graphs of differing sizes84). Overall, these results indicate that response graph complexity provides a useful means of quantifying the computational complexity of games.
The landscape of games
The results, thus far, have demonstrated that graph-theoretic analysis can simplify games (via spectral clustering), uncover their topological structure (e.g., transitive structure of the AlphaGo empirical game), and yield measures correlated to the computational complexity of solving these games. Overall, it is evident that the perspective offered by graph theory yields a useful characterization of games across multiple fronts. Given this insight, we next consider whether this characterization can be used to compare a widely-diverse set of games.
To achieve this, we construct empirical payoff tables for a suite of games, using the policy sampling scheme described earlier for the larger instances (also see Supplementary Methods 1 for full details, including description of the games considered and analysis of the sensitivity of these results to the choice of empirical policies and mixtures thereof). For each game, we compute the response graphs and several associated local and global complexity measures (e.g., α-Rank distribution entropy, number of 3-cycles, node-wise in-degrees and out-degree statistics, and several other measures detailed in the “Methods” section), which constitute a feature vector capturing properties of interest. Finally, a principal component analysis of these features yields the low-dimensional visualization of the landscape of games considered, shown in Fig. 1.
We make several key insights given this empirical landscape of games. Notably, variations of games with related rules are well-clustered together, indicating strong similarity despite the widely-varying sizes of their policy spaces and empirical games used to construct them; specifically, all considered instances of Blotto cluster together, with empirical game sizes ranging from 20 × 20 for Blotto(5,3) to 1000 × 1000 for Blotto(10,5). Moreover, games with strong transitive components (e.g., variations of Elo games, AlphaStar League, Random Game of Skill, and Normal Bernoulli Game) are also notably separated from strongly cyclical games (Rock–Paper–Scissors, Disc game, and Blotto variations). Closely-related real-world games (i.e., games often played by humans in the real world, such as Hex, Tic-Tac-Toe, Connect Four, and each of their respective Misère counterparts wherein players seek to lose) are also well-clustered. Crucially, the strong alignment of this analysis with intuitions of the similarity of certain classes of games serves as an important validation of the graph-based analysis technique proposed in this work. In addition, the analysis and corresponding landscape of games make clear that several games of interest for AI seem well-clustered together, which also holds for less interesting games (e.g., Transitive and Elo games).
We note that the overall idea of generating such a landscape over games ties closely with prior works on taxonomization of multiplayer games38,85. Moreover, 2D visualization of the expressitivity (i.e., style and diversity) and the overall space of of procedurally-generated games features have been also investigated in closely related work86,87. A recent line of related inquiry also investigates the automatic identification, and subsequent visualization of core mechanics in single-player games88. Overall, we believe this type of investigation can be considered a method to taxonomize which future multiplayer games may be interesting, and which ones less so to train AI agents on.
While the primary focus of this paper is to establish a means of topologically studying games (and their similarities), a natural artifact of such a methodology is that it can enable investigation of interesting and non-interesting classes of games. There exist numerous perspectives on what may make a game interesting, which varies as a function of the field of study or problem being solved. These include (overlapping) paradigms that consider interestingness from: a human-centric perspective (e.g., level of social interactivity, cognitive learning and problem solving, enjoyment, adrenaline, inherent challenge, esthetics, story-telling in the game, etc.)89,90,91,92,93,94; a curriculum learning perspective (e.g., games or tasks that provide enough learning signal to the human or artificial learner)95; a procedural content generation or optimization perspective (in some instances with a focus on General Game Playing)21 that use a variety of fitness measures to generate new game instances (e.g., either direct measures related to the game structure, or indirect measures such as player win-rates)20,31,34,35,96; and game-theoretic, multiplayer, or player-vs.-task notions (e.g., game balance, level of competition, social equality or welfare of the optimal game solution, etc.)97,98,99. Overall, it is complex (and, arguably, not very useful) to introduce a unifying definition of interestingness that covers all of the above perspectives.
Thus, we focus here on a specialized notion of interestingess from the perspective of AI training, linked to the work of Czarnecki et al.37. As mentioned earlier, Czarnecki et al.37 introduce the notion of Games of Skill, which are of interest, in the sense of AI training, due to two axes of interactions between agents: a transitive axis enabling progression in terms of relative strength or skill, and a radial axis representing diverse intransitive/cyclical interactions between strategies of similar strength levels. The overall outlook provided by the above paper is that in these games of interest there exist many average-strength strategies with intransitive relations among them, whereas the level of intransitivity decreases as transitive strength moves towards an extremum (either very low, or very high strength); the topology of strategies in a Game of Skill is, thus, noted to resemble a spinning top.
Through the spinning top paradigm, Czarnecki et al.37 identifies several real-world games (e.g., Hex, Tic–Tac–Toe, Connect–Four, etc.) as Games of Skill. Notably, the lower left cluster of games in Fig. 1 highlights precisely these games. Interestingly, while the Random Game of Skill, AlphaStar League, and Elo game (noise = 0.5) are also noted as Games of Skill in Czarnecki et al.37, they are found in a distinct cluster in our landscape. On closer inspection of these payoffs for this trio of games, there exists a strong correlation between the number of intransitive relations and the transitive strength of strategies, in contrast to other Games of Skill such as Hex. Our landscape also seems to highlight non-interesting games. Specifically, variations of Blotto, Rock–Paper–Scissors, and the Disc Game are noted to not be Games of Skill in Czarnecki et al.37, and are also found to be in distinct clusters in Fig. 1.
Overall, these results highlight how topological analysis of multiplayer games can be used to not only study individual games, but also identify clusters of related (potentially interesting) games. For completeness, we also conduct additional studies in Supplementary Note 2, which compare our taxonomization of 2 × 2 normal-form games (and clustering into potential classes of interest) to that of Bruns38.
The problem problem and procedural game structure generation
Having now established various graph-theoretic tools for characterizing games of interest, we revisit the so-called Problem Problem, which targets automatic generation of interesting environments. Here we focus on the question of how we can leverage the topology discovered by our method to procedurally generate collections of new games (which can be subsequently analyzed, used for training, characterized as interesting or not per previous discussions, and so on).
Full details of our game generation procedure are provided in the “Methods” Section. At a high level, we establish the feedback loop visualized in Fig. 3, enabling automatic generation of games as driven by our graph-based analytical workflow. We generate the payoff structure of a game (i.e., as opposed to the raw underlying game rules, e.g., as done by Browne and Maire20). Thus, the generated payoffs can be considered either direct representations of normal form games, or empirical games indirectly representing underlying games with complex rules. At a high level, given a parameterization of a generated game that specifies an associated payoff tensor, we synthesize its response graph and associated measures of interest (as done when generating the earlier landscape of games). We use the multidimensional Elo parameterization for generating payoffs, due to its inherent ability to specify complex transitive and intransitive games41. We then specify an objective function of interest to optimize over these graph-based measures. As only the evaluations of such graph-based measures (rather than their gradients) are typically available, we use a gradient-free approach to iteratively generate games optimizing these measures (CMA-ES100 is used in our experiments). The overall generation procedure used can be classified as a Search-Based Procedural Content Generation technique (SBPCG)34. Specifically, in accordance to the taxonomy defined by Togelius et al.34, our work uses a direct encoding representation of the game (as the generated payoffs are represented as real-valued vectors of mElo parameters), with a theory-driven direct evaluation used for quantifying the game fitness/quality (as we rely on graph theory to derive the game features of interest, then directly minimize distances of principal components of generated and target games).
Naturally, we can maximize any individual game complexity measures, or a combination thereof, directly (e.g., entropy of the α-Rank distribution, number of 3-cycles, etc.). More interestingly, however, we can leverage our low-dimensional landscape of games to directly drive the generation of new games towards existing ones with properties of interest. Consider the instance of game generation shown in Fig. 10a, which shows an overview of the above pipeline generating a 5 × 5 game minimizing Euclidean distance (within the low-dimensional complexity landscape) to the standard 3 × 3 Rock–Paper–Scissors game. Each point on this plot corresponds to a generated game instance. The payoffs visualized, from left to right, respectively correspond to the initial procedural game parameters (which specify a game with constant payoffs), intermediate parameters, and final optimized parameters; projections of the corresponding games within the games landscape are also indicated, with the targeted game of interest (Rock–Paper–Scissors here) highlighted in green. Notably, the final optimized game exactly captures the underlying rules that specify a general-size Rock–Paper–Scissors game, in that each strategy beats as many other strategies as it loses to. In Fig. 10b, we consider a larger 13 × 13 generated game, which seeks to minimize distance to a 1000 × 1000 Elo game (which is transitive in structure, as in our earlier motivating example in Fig. 2a). Once again, the generated game captures the transitive structure associated with Elo games.
Each figure visualizes the generation of a game of specified size, which targets a pre-defined game (or mixture of games) of a different size. The three payoffs in each respective figure, from left to right, correspond to the initial procedural game parameters, intermediate parameters, and final optimized parameters. a 5 × 5 generated game with the target game set to Rock–Paper–Scissors (3 × 3). b 13 × 13 generated game with the target game set to Elo (1000 × 1000). c 13 × 13 generated game with the target game set to the mixture of Rock–Paper–Scissors (3 × 3) and Elo game (1000 × 1000). d 19 × 19 generated game with the target game set to the mixture of AlphaStar League and Go (board size = 3). Strategies are sorted by mean payoffs in b and c to more easily identify transitive structures expected from an Elo game.
Next, we consider generation of games that exhibit properties of mixtures of several target games. For example, consider what happens if 3 × 3 Rock–Paper–Scissors were to be combined with the 1000 × 1000 Elo game above; one might expect a mixture of transitive and cyclical properties in the payoffs, though the means of generating such mixed payoffs directly is not obvious due to the inherent differences in sizes of the targeted games. Using our workflow, which conducts this optimization in the low-dimensional graph-based landscape, we demonstrate a sequence of generated games targeting exactly this mixture in Fig. 10c. Here, the game generation objective is to minimize Euclidean distance to the mixed principal components of the two target games (weighted equally). The payoffs of the final generated game exhibit exactly the properties intuited above, with predominantly positive (blue) upper-triangle of payoff entries establishing a transitive structure, and the more sporadic positive entries in the lower-triangle establishing cycles.
Naturally, this approach opens the door to an important avenue for further investigation, targeting generation of yet more interesting combinations of games of different sizes and rule-sets (e.g., as in Fig. 10d, which generates games targeting a mixture of Go (board size = 3) and the AlphaStar League), and subsequent training of AI agents using such a curriculum of generated games. Moreover, we can observe interesting trends when analyzing the Nash equilibria associated with the class of normal-form games considered here (as detailed in Supplementary Note 2). Overall, these examples illustrate a key benefit of the proposed graph-theoretic measures in that it captures the underlying structure of various classes of games. The characterization of games achieved by our approach directly enables the navigation of the associated games landscape to generate never-before-seen instances of games with fundamentally related structure.
from Hacker News https://ift.tt/3erbJDH
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