Principal Trade-off Analysis
Authors: Alexander Strang
David Robert SeWell
Alexander Kim
Kevin Alcedo
David Rosenbluth
Venue: Submitted to ICLR 2023 (2023)
Keywords: Learning theory
Representation Learning
algorithmic game theory
Functional form games
matrix decomposition

Abstract: The focus on equilibrium solutions in games underemphasizes the importance of understanding their overall structure. A different set of tools is needed for learning and representing the general structure of a game. In this paper we illustrate "Principle Trade-off Analysis" (PTA), a decomposition method that embeds games into a low dimensional feature space and argue that the embeddings are more revealing than previously demonstrated. Here, we develop an analogy to Principal Component Analysis (PCA). PTA represents an arbitrary two-player zero-sum game as the weighted sum of pairs of orthogonal 2D feature planes. We show that each of the feature planes represent unique strategic trade-offs (cyclic modes) and truncation of the sequence provides insightful model reduction. We demonstrate the validity of PTA on a pair of games (Blotto, Pokemon). In Blotto, PTA identifies game symmetries, and specifies strategic trade-offs associated with distinct win conditions. These symmetries reveal limitations of PTA unaddressed in previous work. For Pokemon, PTA recovers clusters that naturally correspond to Pokemon types, correctly identifies the designed tradeoff between those types, and discovers a rock-paper-scissor (RPS) cycle in the Pokemon generation type - all absent any specific information except game outcomes. 

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PRINCIPAL TRADE-OFF ANALYSIS
Anonymous authors Paper under double-blind review
ABSTRACT
The focus on equilibrium solutions in games underemphasizes the importance of understanding their overall structure. A different set of tools is needed for learning and representing the general structure of game. In this paper we illustrate "Prin- ciple Trade-off Analysis" (PTA), decomposition method that embeds games into a low dimensional feature space and argue that the embeddings are more revealing than previously demonstrated. Here, we develop an analogy Principal Compo- nent Analysis (PCA). PTA represents an arbitrary two-player zero-sum game as the weighted sum of pairs of orthogonal 2D feature planes. We show that each of the feature planes represent unique strategic trade-offs (cyclic modes) and trun- cation of the sequence provides insightful model reduction. We demonstrate the validity of PTA a pair of games (Blotto, Pokemon). In Blotto. PTA identifies game symmetries, and specifies strategic trade-offs associated with distinct win conditions. These symmetries reveal limitations of PTA unaddressed in previous work. For Pokemon, PTA recovers clusters that naturally correspond to Pokemon types, correctly identifies the designed tradeoff between those types, and discov- ers rock-paper-scissor (RPS) cycle in the Pokemon generation type- all absent any specific information except game outcomes.
INTRODUCTION
In recent years algorithms have achieved superhuman performance in number of complex games such as Chess, Go, Shogi, Poker and Starcraft (Silveret Hei8: & Silver 2016 Moravčík cct al.) 2017: Vinyalls et al., 2019).. Despite impressive game play, enhanced understanding of the game is typically only achieved by additional analysis of the algorithms game play post facto (Silver) 2018). Current work emphasizes the "policy problem", developing strong agents, despite growing demand for a task theory which addresses the "problem problem". i.e. what games are worth study and play (Omidshafiei et al.] 2020)] Clune, 2019). task theory requires a language that character- izes and categorizes games, namely, a toolset of measures and visualization techniques that evaluate and illustrate game structure. Summary visuals and measures are especially important for complex games where direct amalysis is intractable. In this vain tournaments are used to sample the game and to empirically evaluate agents. The empirical analysis of tournaments has a long history, in sports analytics (Lewis, 2004 [Bozóki al., (016) ecology and animal behavior (Laird & Schamp (006)] Silk 1999), and biology (Stuart-Fox et al. 2006) Sinervo & Lively 1996). While the primary in- terest these cases is typically in ranking agents/players. tournament graphs also reveal significant information about the nature of the game being played (Tuyls et al. 2018). This paper describes mathematical techniques for extracting useful information about the underlying game structure di- rectly from tournament data. While these methods can be applied to the various contexts in which tournaments are already employed in machine learning (e.g., population based training), they open up range of new research questions regarding the characterization of natural games, synthesis of artificial games (c.f. Omiishafiiii et al. (2020)), game approximation via simplified dynamies, and the strategic perturbation of games.
Heinrich
Fine structural characteristics a tournament graph can be represented by low dimensional embed- dings that map competitive relationships to embedded geometry. We review and expand on methods introduced by Balduzzi et al. (2018b), who proposed a canonical series of maps that provide a com- plete description of sample tournament in terms of sum of simple games, namely, disc games. PTA provides a simplified global understanding of a tournament compatible with a broad set of objectives beyond finding equilibrium solutions.

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Note that our objectives are as empirical as they are game theoretic. Empirical game theory, the study of games from actual game play data (e.g. sports analytics), studies games as they are played by a particular population, rather than by an idealised player. Thus, empirical game theory has its own, valid, objectives beyond finding equilibria or optimal players. Exclusive focus on optima ignores the global structure of game it is experienced by the majority of players. What decision dilemmas do they face? What game dynamics do they experience? What game space must they navigate in the process of optimization? How should they exploit chosen opponent, population, or form teams? All of these questions are more easily addressed given simplified global representation that isolates each important independent aspect of a game. PTA offers such summary.
Our contribution follows. First, we compare PCA (Pearson 1901 to disc game embedding, and show that disc game embeddings inherit key algebraic properties responsible for the success of PCA. Based on this analogy, we propose PTA as general technique for visualizing data arising from com- petitive tasks or pairwise choice tasks. Indeed, while we focus on games for their charisma, any data set representing a akew-symmetrie comparison of objects is amenable to PTA. Via series of ex- amples, we show that PTA provides a much richer framework for analyzing trade-offs in games than previously demonstrated. Our examples exhibit wide variety of strategic trade-offs that can be clearly visualized with PTA. Unlike previous work, we focus on the relation between embedding coordinates, which represent performance relations, and underlying agent attributes in order to eluci- date the principal trade-offs responsible for cyclic competition in each game. Moreover, we consider the full information content of PTA by analyzing multiple leading disc games and by studying the decay in their importance. Important strategic trade-off's can arise in later disc games, so previous empirical work's focus on the leading disc game is myopic. These examples also raise conceptual limitations not addressed in previous work, thus outline future directions for development.
2 RELATED WORK
Our work builds directly on Balduzzi et al. (2018b), which used the embedding approach to intro- duce a comprehensive agent evaluation scheme. Their scheme uses the real Schur form (PTA) in conjunction with the Hodge decomposition
overcome deficiencies in standard ranking models. Our work also compliments efforts to explore cyclic structures in competitive systems (Candogan al. 201T Sttaang et al., 2022b), economics (Linares 2009 May 1954), and tangentially as multi- class classification problems (Bilmes et al.) 2001: Huang et al. 2006). Cycles challenge traditional gradient methods and can slow training (Omidshatiei et al.J 2020; Balduzzi et al.) 2018a). More- over, cyclic structures in games are often intricate and difficult to disentangle, particularly among intermediate competitors. Games of skill frequently exhibit this "spinning top geometry" (Czarnecki et al.) 2020). By summarizing eyclic structures, PTA helps identify areas of the strategy space that cause difficulty during training, or should be targeted for diverse team design (Balduzzi et al. 2019) Garnelo et al. (022). Here, we show that PTA can identify fundamental trade-offs that summarize otherwise opaque cyclic structure. Trade-offs play an important role in decision tasks and evolu- tionary processes outside of games, so general tools that isolate and reify trade-offs are of generic utility (Omidshafici et al., 22000] Tuyls et al. 2018). In that sense, our attempt to visualize game structure is in line with generic data visualization efforts, which aim convert complicated data into elucidating graphics (c.f. Heaal (2018): Garnelo et al. (202)).
3 BACKGROUND
3.1 FUNCTIONAL. FORM GAMES
A two-player zero-sum functional form game, is defined by an attribute space Ω C RT and an evaluation function f that returns the advantage of one agent over another given their attributes. Agents in the game can be represented by their attribute vectors x, E ? the entries of which could represent agent traits, strategic policies, weights a neural net governing their actions, or more generally, any attributes that influence competitive behavior. The function f is of the form f Q x . + R. The value fxx y). quantifies the advantage of agent x over y with a real number. The evaluation function must be fair, that is, the advantage of one competitor not depend on the order they are listed in. Consequently. f must be skew symmetric, f(x, y) (f( (Strang et al. 2022b). If fxx v) > we say that beats y and the outcome is tie if
another should
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f(x, y) 0. The larger (f. y) the larger the advantage one competitor possesses another. We do not specify how advantage is measured, since the appropriate definition may depend on the setting. Possible examples include expected return in zero-sum game. probability of win minus a half, or log odds of victory. With a set of agents X, pairwise comparisons of all agents gives a N x N evaluation matrix F where Fij =[(;;; FTy). Any such matrix can be separated into transitive and cyclic components, F and Fe. via the Helmhoolz-Hodge decomposition (HHD) (Balduzzi et al., 2018b Strang et al. 2022b Lim 2020). The HHD writes F' F. + Fe where Fili = Ti rj, and where r are least squares ratings that evaluate the average performance of each agent. These matrices can be analyzed study the structure of the game among those competitors, i.e. the resulting tournament.
3.2 DISC GAMES
The cyclic component of tournament be visualized using a combination of simple cyclic games (Balduzzi et al. 2019: 2018b). The simplest cyclic functional form game is a disc game, which acts
as a continuous analog to pock-paper-seissors (RPS) in wwo-dimensional attribute spaces. The disc game evaluation function is the cross product between competitor's 's embedded attributes,
1 0
1
disc(E, 9) y T192 D2y1
T
y
x Ry
(I)
where R is the ninety degree rotation matrix (Balduzzi et al. 2018b). The cross product models a basic trade-off between the two attributes. Any evaluation matrix can be represented with
a sum of pointwise embeddings into sequence of disc games. The necessary construction follows.
4 PRINCIPAL TRADE-OFF ANALYSIS (PTA)
PTA decomposes an arbitrary performance matrix F into a sum of simpler performance matrices by embedding each agent into series of disc games that model important strategic trade-offs.
Any real, m m, skew-symmetric matrix A admits Schur decomposition (real Schur form), QUQT. Here Q is an orthonormal m x rank(A) matrix, U is block diagonal with rank ())/2, 2 x blocks of the form U(A)) = Wi R. and where Wt 2 0 is a nonnegative scalar. Each pair of consecutive columns of Q, [92k-1, {2k]. correspond to the real and imaginary parts of an eigenvector of A scaled by V2. The scalars w are the nonnegative imaginary part of the corresponding eigenvalues, listed in decreasing order (Youla, [961 Zumino 1962). A linear algebra exercise demonstrates that the columns of Q are also proportional to the singular vectors of A, and the sequence of scalars match the singular values of A. A truncated expansion of A using only the first r blocks is equivalent to truncating the singular value decomposition at 2r singular vectors, so equals the optimal rank 2r approximation to A under the Frobenius norm via the Eckart- Young-Mirsky theorem (Eckart & Young 1936
Mirsky 1960] Strang 2019).
When A is replaced with the performance matrix F, each block in the Schur decomposition acts as a scaled version of a disc game where each competitor is assigned embedding coordinates via Q. The performance matrix F us skew symmetric, so admits a Schur decomposition:
F F=QUQ"
(2)
As in PCA, we consider low rank approximation of F associated with expansion onto the first disc games, where & is chosen large enough to satisfy desired reconstruction accuracy. Low rank approximation allows model reduction and mixed equilibria approximation in
time
(Lipton et al. 2003). The closest rank 2r approximation to F' in Frobenius norm is given by replacing Q with Q and U with (1:22) in Equation 2 where ((1:2r) is the first 2r columns of Q. and (1:24) is the upper 2r by 2r minor of U.
The matrix (1::2r) provides a set of basis vectors. Projection onto those basis vectors define a new set of coordinates, thereby embedding the competitors. Specifically, let:
ÿ (1:22)) F ((1:2r) ((1:22)((1:22) (1:22))
(3)
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and scale each pair of embedding coordinates by the associated eigenvalue (1)
(128-1,44 s2 = wi 1/1 1928-1.5 U22..l = wk 1/2221 92H,a). Then (i) maps from competi- tor indices, i, to points in IR2,, and the set Y - {Th,) is a collection of planar embeddings, where y/ is given by projection onto feature plane spanned by 92k-1 and 92k-
user interested in the transitive and cyclie components of F separately, could begin by breaking F into F3 and Fe (Strang al.) (022b). The transitive component can be represented on line via the ratings, so does not require additional visualization (Balduzzi et al. 2018b). The cyclic component Fe is skew symmetric, so be represented via PTA. Then, performance is represented by com- bination of two components. The first compares the overall quality of the agents, as quantified by set of ratings. The second represents any cyclic relations as combination of principal trade-offs. We apply PTA to F. not F. in all of our experiments.
Since PTA depends only on F, the cost of performing PTA is independent of the complexity of the underlying game or agents. Once F is formed PTA proceeds at the same cost (O ) for n sampled agents Gollb & Van Loan )2013)) as standard low-rank matrix decompositions, like PCA, which seek the leading singular vectors of a symmetric matrix. While iterative algorithms may be more efficient when only few leading columns of Q are required, the computational cost of performing PTA in an empirical setting will almost always be swamped by the cost of gathering the data for forming F. which requires evaluating O(n²) pairs of agents. The cost of performing PTA is inconsequential since any computational cost comes from all simulations needed to compute F. The simulation cost could be reduced if low-rank completion methods were applied to fill in missing data. We leave sampling considerations and matrix completion methods (Meka et al. 2009 Gleich & Lim 2011
Chen & Joachims 2016) to future work.
Note that the Schur decomposition is only unique up to rotation within each feature plane, since complex conjugate pairs of eigenvectors of F are only uniquely defined up to their complex phase. Thus, two embeddings are equivalent if they agree up rotation within each planar embedding.
The evaluation sj product (1) x (i) (see Appendix B). That is:
between agent i and agent j equals a sum over each embedding Te. of the cross
F(2r)
(1e X Tk(1) disc
Th (c))
Thus, restricted to each planar embedding ((r)) is a disc game and the optimal rank 2r approxima- tion of F(2r) a linear combination of disc games applied to the sequence of planar embeddings (ū
This decomposition is useful for two reasons. First, it depends on a spectral decomposition of F, so inherits the key properties that account for the success of PCA. An equivalent construction is intro- duced in [Chen& & Joachims (0166) where it called the "blade-chest-inner" model. The construction in Chen & Joachims (2016) is not based spectral decomposition, so lacks orthogonality low rank optimality.
In PTA, the embeddings are projections onto orthogonal planes, so each embedding encodes in- dependent information about cyclic competition. Here independence means that the embedding coordinates of a randomly sampled agent in two distinct planes are uncorrelated. Note that this definition depends on the density of sampled agents. The planes act like feature vectors, and are typically associated with some strategic trade-off (see Section 5). Therefore, PCA identifies prin- cipal components, PTA identifies principal trade-offs: orthogonal planes associated with a sequence of fundamental cyclic modes. The two decompositions differ since PCA uses the singular value de- composition, while PTA uses the Schur real form. Nevertheless, the sequence of embeddings form optimal low rank approximations to F, where the importance of each embedding is quantified by the magnitude of the associated eigenvalue. Thus, the sequence of eigenvalues determines the number of disc game embeddings, т. required to achieve a sufficiently accurate approximation of F. The number of disc games is half the effective rank of F, and is a natural measure of the complexity of cyclic competition. The complexity is distinct from the overall intensity of cyclic competition or the game balance, which depend on [|] instead of its rank (Strang et al. 2022b). Instead, it counts the number of distinct cyclic modes in the evaluation matrix. It 18 possible to have many distinct, yet weak cyelic trade-offs, or one, very strong, cyclic trade-off
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Rating
Zone 1 Allocation
Zone 2 Allocation
Zone a Allocation
Angle
Radius
Game
TUOP
Low
Techose
Low
High
Indienett
Dise
Figure 1: Disc games 1 to 3 of Blotto [1,1,1] game with N 75. Rows: disc game number. Columns 4: disc game embeddings colored according to agent rating, then agent allocation to zones to 3. Column 5: the angle (measured counterclockwise from the horizontal axis), of the embedding of each strategy. Advantage flows clockwise in angle. so blue beats purple beats yellow beats green beats blue. Column 6: the radius of the embedding of each strategy. High radius corresponds to strong involvement in a trade-off (yellow), while small radius corresponds to low involvement (blue). Each triangle in the fifth and sixth columns represents the space of available allocations. High allocations to zone cluster near the bottom left corner, high allocations to zone 2 cluster the bottom right corner, and high allocations zone three cluster the top corner. Labels: Representative allocations defining the underlying trade-offs, indicated with bold arrows (see Table 2).
Second, the disc game encodes performance relations via geometry. Given coordinates 7(() and Tl), the advantage of competitor over competitor given by 77() x T(1 equals twice the signed area of the triangle with vertices the origin, 7(), and 7(j). polar coordinates, each agent is assigned radius and angle in each disc game, (rx(i), 0& (i)). Then, disc()((i) 7()) (i())x(() (j)
(i)). So, the larger (t), the more the 4th cyclic mode influences the performance of agent i. For fixed radius, one competitor gains the most advantage when it is embedded 90" clockwise from its opponent, and possesses an advantage as long as is embedded clockwise of its opponent. Thus, advantage flows clockwise about the origin. We visualize this flow with a circulating vector field, ((()) 1/2. -v1]- These geometric properties allow dise games to encode a variety of eyclic structures in interpretable visuals.
EXPERIMENTS
Here, we illustrate the graphical power of PTA via Blotto and Pokemon. Both exhibit interesting cyclic structure. We emphasize the interpretation of each principal trade-off in terms of game strat- egy to show that PTA reveals diverse, fine-grained game structure based only on empirical game data. simpler example is provided in Appendix D for reference.
5.1 COLONEL BLOTTO
Colonel Blotto a zero-sum, simultancous action, two player resource allocation game (Kovenock & Roberson 2021). Each player possesses N troops to distribute across K zones. Each zone has an associated payout Zr. zone is conquered by a player if they allocate more troops to that zone than their opponent. The conquering player receives the payout. Ties result in both players receiving
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Principal trade-offs associated with the first three disc games. The columns list the allocation types involved in each disc game (D.G.), their location in the simplex of possible allocations, provide an example set of allocations, and the competitive relations between the types. The letters H, M. and Under review as a conference paper at ICLR 2023
Table 1: Principal Trade-Off's
D.G.
Allocation Types
Location in Simplex
Example
Advantage Relation
(1) = allocate to I zone
corners
[70, 0, 5]
(2) allocate to 2 zones
midpoints of edges center
38. 37, 0]
(2)<())((2><())
(3) = allocate to 3 zones
[25, 25, 25]
R [H, M, L
corners
[50, 25, 0]
2
= [M. L, H
shifted
25, 50
R<P<S<R
P L, H, M)
counter clockwise
[0,50, 25]
R' = [L. M, H.
corners
25, 50
3
S' H, M.M
shifted
50, 0, 25
R'<P'<<'<
P M, L
clockwise
25, 50
L,, are used to denote high, medium, and low allocation. Note that the allocations involved in the trade-off defined by disc game correspond locations in the radius panels in Figure shaded green or yellow. Advantage in disc game flows clockwise, so can be inferred from the angle panel.
0 payout. The player with the highest total payout wins the match. All allocations revealed simultaneously.
At simplest, the payouts uniform across zones, the player who conquers the most zones wins the game. Unweighted Blotto is a highly cyclic game since there is no dominating strategy. Every strategy admits a counter. Unless K N or K <= 2, all allotments lose some other allotment. To defeat an allotment, adopt the maxim, "lose big, win small". Mimic the allotment, then redistribute all the units from the zone with the most units uniformly as possible across the remaining zones. Then, unless all zones were allotted one unit, the exploiting strategy sacrifices loss one zone win more than one other zone. In general, the more an allotment commits to single zone, the more easily it is defeated. Unweighted Blotto is also complex, since the zones are indistinguishable. Thus unweighted Blotto admits K! fold symmetries with respect to zone labels.
We consider each unique strategy as a separate "agent", parameterized by the corresponding allot- ment. We generate agents by randomly sampling over the strategy space using Dirichlet distri- bution with the support equal to the number of zones. After sampling, we compare each pair of strategies in the population. Each match-up is deterministic and results in win, loss tie, which we assign scores (0.5,0,-0.5). We construct the associated evaluation matrix by setting Fig to the score of strategy against strategy
PTA allows elegant visualization of relevant game structure by reducing a game to a small set of key trade-offs. We start by looking at the K 3, N 75 blotto game with uniform payouts. Table summarizes the principal trade-offss associated with each disc game. These trade-offs are the most important sources of cycles in the tournament, accounting for 80% of its structure.
In general, the number of distinct allotments in a K battlefield, N troop blotto game grows at (N but the complexity, which reflects the underlying number of cyclic modes, converges to constant value associated with continuous Blotto game, where commanders can allocate an arbi- trary fraction their force to each zone. Unweighted K 3, N = 75 blotto admits 2926 allotments, but has 3! fold exchange symmetry under permutations of the battlefield labels, leaving roughly 488 distinct allotments. Three disc games reconstruct the evaluation matrix to 80% accuracy, to Rs 90% accuracy, and 12 to > 95% accuracy, the game has complexity 12 95% standard. Trade-offs 4- 12 represent refinements of the trade-offs present in the first three disc games, so PTA really allows reduction in complexity from 2926 allocations (absent prior knowledge regarding symmetries), to 3 fundamental cyclic modes. Thus, PTA can effectively separate the underlying complexity of game from the size of its strategy space. See Appendix E for further discussion.

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Oix Game
Diee Game
Disc Game
Dbcc Game4
Dit Game
Disc Game
Disc Game?
Disc Game
Disc Game
1d3c
R=S+PHI
I<3<bd
143<2
RodaPaR
Low Frequency, Large Scale Trade-offs
Mid Frequescy, Mil Scale Trade oft
High Frequency, Small Scale Trade-offt
665
Percent
Figure 2: The first nine disc games of the N = 75, [1,1,1] blotto game. Each column is separate disc game. The first and second rows show the angle and radius assigned to each allocation. The disc games are labelled by trade-off type. Consecutive sets of three share the same eigenvalue and are grouped by spatial scale with eigenvalue and percent recovery of provided beneath.
The exchange symmetry of the zones is apparent in the sequence of eigenvalues, Wt, representing disc game importance. Exchanges introduce permutations under which the evaluation matrix is invariant. Consequently, 4x come in sets of three, where each w/ represents pair of eigenvectors. Eigenvectors associated with identical eigenvalues are not uniquely defined. Instead, they are drawn from a subspace of dimension equal the multiplicity of the repeated eigenvalue. Consequently, all of the cigenvectors Q are chosen arbitrarily from six dimensional spaces.
When F' has repeated eigenvalues, the associated disc game embeddings are not unique. Any uni- tary transform of the set of eigenvectors sharing eigenvalue defines a valid embedding. Thus, symmetry presents an unusual challenge: degeneracy.
our case, the disc games come sets of three, each representing an arbitrary rotation of a six dimensional object. Consequently, we consider multiple disc games simultaneously. This issue was not addressed previous work, which largely only considered the leading disc game. Generic games should not exhibit strong symmetries, so such degeneracy will be rare and confined to toy examples. That said, generic games also require more than disc game, it is essential consider more than the leading disc game.
We analyze the three leading disc games to identify the most important allocation trade-offs. Figure I shows the first three disc games colored by rating, allocation the three zones, and the mapping to angle and radius each disc game as a function of allocation. Each share the same eigenvalue, so are equally important and could be mixed. Nevertheless, these three disc games represent distinct trade-offs in allocations that can be easily explained.
The specific trade-off's can be identified directly from the disc games when colored by allocation. Consider the points labelled 1, 2, and 3 in the first disc game. Each maximize the radius of the scatter cloud while moving along its boundary, low rated points at the bottom of the scatter allocate primarily to one zone (yellow in panels 2 4).
represent the allocations most involved in the cycle. The
Moving clockwise, the next extrema occurs at the top of the scatter. It is high rated, and has nearly equal allocation across all three zones (colored green in panels 2 4). Uniform allocations are rated highly since they perform well against most randomly sampled allocations, particularly those lying along line connecting corner of the simplex to its center. This induces transitive trend among the bulk of the allocations moving from allocations that prioritize one zone, to allocations that treat the zones equally. This transitive trend is represented by the general shift of the disc game leftward off the origin. This subset of allocations compete transitively, producing the regular gradient from purple to yellow in rating when moving clockwise from the bottom to the top in the scatter.
Not all allocations satisfy this transitive trend. Allocations that prioritize two zones counter the uni- form strategy, and countered by allocations that prioritize single zone. For example, allocation [70,0,5] defeats [38.3700]. Thus, allocations lying on the midpoints an edge of the simplex lose to allocations either neighboring endpoint. These counters close the cycle, and represented

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Disc Game
Game 2
Disc Game
Rating
1ype
Figure 3: Disc games 1,2 and 4 for pokemon. Disc game one is colored by rating. Disc game two is colored by type, then generation. Disc game four is colored by rating, then generation.
by the rightmost pair of corners labelled 2 in disc game 1. Panels 2 4 show that each such comner receives an intermediate allocation in two zones (green). but little none in the third (dark blue).
Similar visual analysis identifies the RPS cycles among cyclic permutations of allocations [H,M,L] and [L.M.H] shown in disc games 2 and 3. For example, the leftmost corner of the scatter cloud shown in disc game 2 receives high allocation in zone 1 (teal), an intermediate allocation in zone (blue-green). and a low allocation in zone (dark blue). Walking from R to to S, the allocation patterns shifts cyclically. The same analysis applies to disc game 3, starting from [L,M.H].
Figure shows the angle and radius assigned to each allocation in the simplex. Strikingly, subse- quent disc games imitate the disc game 1-3 trade-offss, only at higher frequency on smaller spatial scale in allocation. This suggests that the disc games may act like Fourier modes, where early disc games capture low frequency, global trade-offs, and later disc games capture high frequency, local trade-offs. It also suggests that orthogonality may not be the appropriate notion of independence for trade-offs. A sharper notion of equivalency is needed. Methods like nonnegative matrix factoriza- tion, which address similar issues among PCA features (Lee & Seung] 1999), suggest an avenue for further development. An example that produces explicit sine series is discussed in the Appendix.
5.2 POKEMON
We conclude by analyzing Pokemon. Pokemon originated from the Nintendo Game Boy console, but has since been played on a variety of mediums including playing cards. Pokemon is of considerable interest from game design perspective since the creators must design certain trade-offs to keep the game balanced and engaging. The game is made up of creatures, called Pokemon, that come in many varieties. Players are rewarded for collecting diverse teams. Thus, each Pokemon has different type, and each type has its own set of strengths and weaknesses. These different types satisfy interlocking cyclic relationships.
The data used in this analysis comes from an open-source Kaggle data set (Bouchet 2017). The original data has 800 Pokemon, but we removed the 65 "legendary" Pokemon simplify the anal- ysis. The data consists of battle outcomes and pokemon attributes. Battle outcomes were converted into an evaluation matrix by logistic regression (see Appendix FF)
Figure 3 shows three of the first four disc games, chosen for their significance. The first disc game the most important, and clearly transitive since all points fall on curve that does not enclose the origin. Position along the curve is closely correlated with speed, so speed determines rating.
We query by attribute to interpret the remaining disc games. To start, consider the "type" attribute. The second disc game is clearly clustered by type (see Figure A variety of RPS relationships are apparent among the type clusters. Any loop of clusters containing the origin corresponds to a cycle of type advantage. The intensity of the corresponding cycle (curl) is proportional to its area. Focus on the large clusters most involved in the trade-off, i.e. furthest from the origin. Figure4Isummarize the RPS relations between these clusters. First, notice the highlighted triangle formed by the Water- Fire-Gree Grass clusters. The disc game shows the expected advantage cycle since the triangle contains the origin. Thus, PTA identifies known game structures without domain specific knowledge.
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Disc RPS relatonships
Paxemmn Performane Matti
Skow Altsck Matric
Grass Water Ground Rock
Waler Ground Rack
-0.18
-1.0
0.18
0
-0.11
0.299
0.085
0.34
1.5
0.0
-1.0
-1.5
-0.5
1.0
0.11
-0.39
0.34
0.0
0.13
0.29
0.39
0
-0.34
-0.3
1.0
1.5
1.5
0.0
-1.5
-1.5
Fire Bug
-0.24
-0.065
0.042
0.34
-0.33
Bug
-1.0
0.5
0.0
1.5
0.0
-1.5
-0.34
-0.34
-0.34
0:3
0.33
-1.5
-1.0
-1.5
1.5
Rock
Ground Water
Grass
Fire
Rock
Ground Water
Grass
Bug
Fire
Figure 4: Left: RPS sub-game discovery. Each cluster type is represented by a matching pokemon, Middle: Empirical performance matrix, Right: Performance matrix derived from (tppe chart
Additional clusters on the outer ring satisfy more intricate relations. The other three types are "bug", "rock" and "ground". To summarize these relations construct coarse grained evaluation matrix, F. Specifically Fij is the average performance of Pokemon from type vs the Pokemon from type in the second disc game. The associated matrix heat-map is shown in the middle panel of Figure The types are ordered by angle moving clockwise about the origin.
We compared these relations with available game design matrices known as "attack matrices". At- tack matrix entries list the advantage of one Pokemon type over the other. We use the attack matrix from (type chart). An attack matrix is written in terms of multiples, so Pokemon that are evenly matched have 1 x advantage. We bucket the range of i. attack multipliers azj into 5 bins ranging from 0x 2x, symmetmttr via (A A²). The result the rightmost panel in Figure 4
The coarse grained summary F strikingly similar to the provided attack matrix. The apparent structural parity in these two matrices highlights the virtues of PTA. Without any domain knowledge, access to attributes, or any explicit instruction to identify clusters, PTA clustered Pokemon by their most relevant attributes (type) then encoded game mechanism (type specific attack multipliers) directly from the cluster locations. Conversely, the second disc game shows how cyclic relations introduced at the mechanism level are realized in actual performance.
Coloring the disc games by "generation", i.c. pokemon release date, reveals design choices. The game is frequently updated by the addition of new Pokemon. Updates present design challenge. Game designers must introduce desirable new Pokemon without upsetting the game balance. The fourth disc game, shown in the far right plot of Figure 3 is balanced in that rating does not predict angle, and instead correlates with radius. Strong and weak Pokemon are closer to the origin, while Pokemon of intermediate rating are more involved in the trade-off. This reveals a spinning top structure characteristic of many games (Czarnecki et al. 2020). Rather, generation predicts angle. Each generation possesses an advantage over its predecessor, as illustrated by the fade from purple to yellow. Balance is retained since generational advantage is periodic. The same clockwise generation shift reappears the second dise game. Within type, new beats old, For example, the bottom-most cluster (grass) clearly trends old to young. Cross type relations are largely unchanged.
6 CONCLUSION
Following Balduzzi Balduzzi et al. (2018)), we have demonstrated that all evaluation matrices admit an expansion onto a sum of disc game embeddings. We suggest the name PTA based on the close analogy with PCA. Through examples, we have demonstrated that embeddings produced by PTA can reveal a surprising variety competitive structures from outcome data alone. Future work could provide more general methods for finding embeddings, such functional theory connecting per- formance with attribute space, or could seek a sparser representation via extensions of sparse PCA (Zou 2006). Future work should also investigate automated methods that summarize the trade- offs identified by PTA without the need for visual inspection, and that leverage the representation for game classification, construction, and exploration.

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7 REPRODUCIBILITY
In the supplementary material we have included the necessary code to be able to reproduce the experiments from the main section of the paper. Included is a README file with instructions how to run the code.
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A APPENDIX (SUPPLEMENTARY MATERIAL)
B PRINCIPAL TRADE-OFF ANALYSIS
B. SCHUR DECOMPOSITION IS SUM OF DISC GAMES
Here prove that the Schur decomposition (real Schur form), is equivalent to sum of disc games applicd to the embedding maps yk-
Recall the embedding construction. Given skew symmetric matrix F, write F QUQ where Q real, orthonormal, is block diagonal with diagonal blocks wy R, and the two by two ninety degree rotation matrix. Let Th (i)
1/2 wi [gi,2k-1, V.22k.. Then, the rank 2r approximation to F is:
F (2)) e/ Σ wk 12 (2x) Rly22- 926) e
wi 14 Rl0,,28--11 98,28]
(5)
w. (9.28--19922 9i.2k 9j.2k
91,2k x www.ga.2k-1.9.22
95.2k]
Recalling the embedding construction, write:
((22)
Ja(1) x x()) disc(T ()).).
(6)
Thus, restricted to each planar embedding F. (2r) is a disc game and the optimal rank 2r approxima- tion of ((2)) a linear combination of disc games applied to the sequence planar embeddings (Tt 1k-1-
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Des
Law Trequency, Large Scale Trace-offa
High Trequency, Small Scala Trade
135
Recovery: $0.1%
95.0%
37%%% 8..
$8.AN
60.8% 50.0%
Figure 5: Top row: The first 9 disc games for [1, 2, 4| blotto, colored by rating. Bottom row: Phase as function of allocation each disc game. The top corner of the triangle corresponds to exclusive allocation to zone 3. Note that phase allocation increases in frequency with increasing disc game. Colors denote phase, with advantage flowing clockwise. The cigenvalues associated with each disc game provided, along with the percent of F recovered by the running sum of the disc games. Note that the first disc game is responsible for 90% of the structure of F, and the first two disc games account for 95% of its structure.
B.2 PTA AND FOURIER SERIES
Both the [1, 1] and the [2, 3, 4] blotto examples exhibit strikingly modal disc games that repeat at increasing frequency, and on smaller spatial scales, with increasing disc game number. These patterns suggest an analogy Fourier series. To make the analogy more concrete we present one last example.
Consider [1, 4] blotto. Since the net value of the first two zones is less than the value of the fourth zone, player wins the overall game if they they win the third zone, or, tie in the third zone and win the second. Otherwise they tie in all zones or lose. Thus, the performance function f[[11, (2, 23), [5/1, 1/2 wal) sign(æ3 ya) + X= y) sign((22 - y2) where x(=) is the in- dicator function for the event in the subscript. Performance can be reduced to a comparison of bign(w(x)) w(y)). Thus, performance is step function applied to the difference w(x) w(y). The difference is dominated by the difference in allocations to the third zone.
single agglomerated trait, w(x) 23 + NI x2. Then f(z
Figure shows the first nine disc games. Notice that all allocations are embedded onto circles, or, in the first case, half circles. Moreover, phase (position along each circle), is entirely function of the agglomerated trait w(x) = 23 -- This form is apparent in the phase panels, where phase close to constant for fixed allocation to zone 3, but tilted slightly account for allocation to zone 2. The first disc game is transitive and completes one half circle moving clockwise from zero allocation to zone 3 to exclusive allocation to zone Discs games 2 and 3 complete 1.5 and 2.5 rotations each when moving from zero allocation to zone 3 to exclusive allocation to zone 4. The pattern continues for the first disc games. Moreover, the circle radii decay geometrically (as shown by the sequence of gold scatter points in the last panel of Figure g).
These features are hallmarks of sine series embedding. We show below that any translationally invariant performance function of single attribute can be represented by sum of disc games, where the embedding into each disc game maps to a circle, the attribute maps to a phase coordinate around the circle, and the radii of the circles in each disc game are controlled by the coefficients of sine series expansion. The performance function fx. y) only depends on the difference allocations, y, is translationally invariant, and is function of single trait, w(x). Thus, it admits sine series expansion. Note, the subsequent analysis does not guarantee low rank optimality, so only shows that disc game embedding via sine series is possible, not that PTA will necessarily produce such an embedding.
Consider performance function of the form:
[(x, v) A sin (2n(2 v))
(7)
for x. y E Ω IR, for arbitrary amplitude A, and frequency w.
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Performance functions of this form are easy to embed, since the disc game product. The cross product between two points on a plane, expressed in polar coordinates, the product of their radii times the sine of the difference in their phases. Therefore, if:
((x) = VIA (2 (A) A)wz),
(A A)wx)]
(8)
then:
disc (ff(z). v sin (2rsiig((A))( -y) -iig( (A) (assiin((((((( y)) = = Asin((nT(( - v)) = f(r, v).
(9)
Lemma 1: [Trigonometric Performance Functions of One Trait] If fx. y) A (2in(2 y)) for 2, y both in a one- mie-dimensooal trait space, then f is disc game embeddable using the embedding:
(x) v [cos (A)war) sin(mnsign((A)ur)
Notice that this construction maps the intervals of length 1 1/W in Ω to the circle radius VIA centered the origin. It follows that, if performance function is embeddable onto circle centered at the origin then there exists a mapping from trait space to the real line where performance is of the form
This result extends easily to linear combinations of sinusoidal functions with varying frequencies. Consider performance function of the form:
f(z, y) E A& sin(2rwuxr v))
(10)
Then, f can be recovered using sum of n disc game embeddings, where the 1 embedding has the form:
The A& [cos((riinA (Ax)sr))))) sin (( Ax)wa=)]
(11)
Note that, all the performance functions of this kind are translation invariant since they are functions of the difference x y. which does not change if after shifting x and by some amount 8.
Theorem 1: [Translation Invariant One Trait Performance Functions] Suppose that Z is dimensional trait space, and (z, g) translation invariant. Then there exists a function such that f(x = h(x y). Suppose in addition that l(xx) is periodic with period P, Ω is contained inside an interval with length P/2. Then f is disc game embeddable using countably infinite sequence disc games, which correspond to the sine series expansion same conditions as the sine series. Moreover, each disc game represents a term in the sine series, and maps to a subset of circle centered at the origin with radius fixed by the corresponding coefficient in the sine series.
and converge under the
h(x- y) for some function h. Since fzz y) = =f((y.2) h must be an odd function. If Ω2 is contained inside an interval of length P/2, then i can be extended to an odd, continuous, 2P periodic function, or an odd periodic function. If not, then, by assumption, h is periodic with period P.
Proof: If f(x y) is translation invariant then f(z. v)
All integrable periodic functions on the real line can be approximated with a Fourier series. If the function is real valued and odd, then the Fourier series is a sine series of the form:
k
h(x)
A sin (2mwwx)) Wk
(12)
P
Each term in the sine series can be reproduced by a dise game embedding using the method for emhedding sinusoidal functions introduced before. Specifically, let:
(x) = VI [coss(2tsiign (Aa Jw (inssnig ( Aw)w)))]
(13)
where A is the th amplitude in the sine series of h:
4 11100
P/2 h(x)) sin(2
A =
(14)
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Then, partial expansion in terms of r disc games equals the term sine series expansion of h(x-y):
disc (w)) = A sin (2Twu (x y)).
(15)
Thus, convergence of the sequence disc game embeddings follows convergence of the sine series expansion.
remains to show that, after sampling a finite set of agents, the result of PTA recovers the sine series representation. Sine series are low rank optimal in this case since, if the agents are ordered by increasing w(x)), the evaluation matrix is of the form:
1
-1 0
F=
-1 -1 0
(16)
: -1
This matrix is real, skew-symmetric, Toeplitz, and is diagonalized by the discrete Fourier transform, so PTA produces a sine series. The sequence of disc games act as sine series expansion of step function, with each higher order disc game corresponding to higher order correction of an approximation to the step function. The first disc game is transitive and captures 90% of the structure of the evaluation matrix. Subsequent disc games correct the first disc game in order to produce step function. While it only takes two disc games to recover 95% of the structure of F, so the 95% complexity of [1, 2, blotto is 2, the subsequent eigenvalues decay slowly, so stricter accuracy requirements lead to large complexities. The slow decay of the eigenvalues is a natural consequence of the slow convergence of sine series a step function. Here it is clear that the complexity predicted by PTA overstates the complexity of the underlying game, since subsequent disc games are best interpreted as corrections that gradually finesse the first disc game, not distinct trade-offs.
More general proof and exploration is saved for future work.
C
DISC GAMES
C.1 GEOMETRY
Principal trade-off analysis is useful visualization technique since disc games encode performance relations via embedding geometry. Reading disc game requires familiarity with this geometry- Namely, familiarity with the various interpretations of a cross product. Here we review some relevant relations.
Cross products are closely related to area in the embedding. Given a pair of competitors with embed- ding coordinates (i) and yk (j), the performance of against j in disc game k. disc (6), Te (i)), equals the signed area of the triangle with vertices (i). Z% (j. and the origin. Further, the degree of cyclicity on loop of competitors can be computed by evaluating a path sum of the advantages around the loop, i.c. curl(C) (Strang et al. 2022b)) The curl on loop C equals the signed area of the loop traced out in each embedding (Strang al. 2022a), summed over the embeddings. It follows that curl inherits the invariances of areas. In particular, curl is translation invariant.
The cyclic component F. on given edge i.j equals the average curl over all possible triangles formed by drawing a random k. Since curl is translation invariant, the cyclic component of compe- tition is translation invariant. In contrast, the transitive component of competition is not translation invariant, and translation in disc game induces transitive component of competition (Balduzzi ct al., 2018b). By subtracting F. from F' to recover F, we center all the rows and columns, so the scatter cloud of embedded competitors will be centered the origin. In contrast, if embed F directly, then transitivity arises from translation of each seatter cloud away from the origin. the origin is not included in the convex hull of the embedded agents, then competition is transitive.
In contrast, scaling the embedding coordinates does change the predicted performance relations. Area is proportional to length squared, scaling the embedding coordinates by /s scales the asso-
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Paper
Schsom
Liarrd
Spock
Interred Groph
Figure 6: The two disc games that represent RPS+2 are shown. Starting from the first column the points are colored by the label for that column. The graph on the left shows a reduced representation of the game by using representative points for each strategy (red for rock, blue for paper, green for scissors, yellow for lizard, orange for spock).
ciated cyclic component of competition by 8. The scaling from Y to Y, was adopted to ensure that unit area in embedding generates unit curl. It follows that the area encompassed by set of points in disc game embedding directly represents the amount of cyclic competition among those agents, and hence the importance of that embedding.
D ROCK PAPER SCISSORS + 2
Here we provide an extended example of the popular rock-paper-scissen -cipsssssiisors (RPS) game. We consider poock-pppeessissoollaa---p-o (RPS+2). The utility matrix for (RPS+2) is shown below:
0
-1
-1
17
0
-1
URPS+2
-1
1
0
-1
1
(17)
0
1 -1 1
We generate a population of agents using fictitious self play (FSP). Each agent in the population is defined by a vector length representing a distribution over the strategy space. We start with an initial random agent and generate best response agents using FSP. Each best response agent becomes a new agent in the population. We then create F by computing the expected value each match-up using the utility matrix equation [7
What should we expect to see? All mixed strategies are an interpolation of the 5 pure strategics so each mixed strategy should be contained inside the convex hull of a polygon formed by embedding the pure strategies. The utility matrix is invariant under cyclic permutations, so the polygon must be regular and centered at the origin. Thus, the pure strategies must be the vertices of regular pentagon. The full game includes two, equally important, interlocking cyclic relations among the pure strategies. These are, rock > paper > scissors > lizard spock rock, and rock lizard > paper > spock scissors > rock, where > denotes loses to. Both cycles contribute equally to the utility matrix, and the utility matrix is unchanged under a permutation that rearranges the strategy labels so that rock is follow by lizard, then paper, then spock, then scissors.
These two cycles are represented by pair of disc games, shown in Figure In both, the pure strategies are vertices of regular pentagon centered at the origin. The eigenvalues for these games are similar, but not identical, since the sampled population is not entirely regular. The Inferred Graph in the rightmost column plots a representative point for each of the pure strategies. Advantage relations that are not accounted for in the first disc game are represented in the second. In the first disc game the sequence of pure strategies, Rock. Paper, Scissor, Lizard, Spock, spaced by 144 degrees about the pentagon. In the second, the sequential pure strategies are spaced 72 apart, so occupy adjacent corners of the pentagon.
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Full Graph
Disc Game 1 Graph
Disc Game 2 Graph
Full Performance
Disc Game 1 Performance
Disc Game 2 Performance
+
Figure 7: The first row shows the abstracted performance graphs for the full game as well as each disc game. Pure strategies are represented by color (red for rock, blue for paper, green for seissors, yellow for lizard, orange for spock). Arrows represent dominance relations. For example, the arrow from green to red the middle column denotes the strong advantage rock possesses over scissors due the first dise game. The heat maps in the second represent performance matrices for the full game and for each disc game. The full performance relation is sum of the two advantage cycles produced by each disc game.
Intransitivity Comparison
Complexity Comparison
Eigenvalues us Disc Games
# Disc gamess
payout
1,
2,
[2, 3, 4]
875
20
15
N=
Troops
N Troops
Logidisc gamee m)
Figure 8: For a variety different blotto games we have Left:
[[]]]] vs allocation (troops) N Middle:
|[
complexity vs N where complexity is the number of disc games & takes to reach 0.05% Frobenius norm error rate, Right: For each game we list the eigenvalues in decreasing order using log scale
To clarify the ad advantage relations encoded by each disc game, we use the inferred graph to construct an abstract representation of performance graph for the pure strategies. The result is shown in Fig- u Colors denote strategy, and arrows denote advantage. The full game graph can be decomposed into the two separate game graphs embedded in each of the disc games.
BLOTTO: INTRANSITIVITY AND COMPLEXITY ANALYSIS
Here we look at how PTA can be useful for comparing different blotto games. New blotto games are
easily created by varying the game parameters. The number of zones, troops, and payouts can be changed to generate games with varying structure. Other Blotto variants, such as Boolean Blotto, or Colonel Lotto, are not considered here. We focus small number of zones K = 3 or K
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to illustrate how the intransitivity number of different payout structures.
( )
and complexity change for varying N and K across
We propose three hypotheses. First, while the number of distinct strategies grows combinatorially in N and K, the allotment problem converges, in the limit of large N, continuous problem in which each commander can allot an arbitrary fraction of their total force to any zone. Therefore, our structural measures should converge finite values representing the structure of a Blotto game allowing any fractional unit allotment on the interval [o. 1. Both measures should increase with increasing N towards the limiting case, as the space of available allotment strategies grows with N. Second, complexity should increase with increasing K, since the number of distinct strategic trade- offs should increase as the number of distinct, exploitable, win conditions increases. Third, small changes in game structure can lead to large differences in both intransitivity and complexity. Small changes can break underlying symmetries, and, the majority functions that determine performance discontinuous in nearby allocations small changes in battlefield weights can produce sudden changes in the set of win conditions.
To test these hypotheses, we vary N between 5 and 45 while holding K fixed and 4. We consider three distinct payout structures for K=3 and 2 for K=4. For each game we construct F [Full and compute the intransitivity T[F:l using the HHD. The complexity of each game is computed by finding the number of disc games required reach an error tolerance of 0.05 (measured in relative Frobenius error). All of the games considered are highly cyclic, with FFlll > F& in all but the [1, 2, case. The [1, 2, 4] case is transitive case, since the only win condition is victory in. Note that [ is not zero for the [1, 2, 4] case since the chosen performance measure cannot be expressed as a linear function of difference
ratings, so not perfectly transitive. Nevertheless, if the step function used to assign battlefield outcomes is replaced with any sigmoid s(z), then the [1, 2, 4] case perfectly transitive with respect to s where rating equals allotment to zone 4.
the results, and confirms our three hypotheses. First, the intransitivity and complexity for all games plateaus for large enough N. This indicates that, beyond a certain N, the game reaches its "strategic capacity"; all meaningful types of trade-offs have been expressed. In both cases, the limiting complexity is strikingly small relative to the size of the strategy space, which grows rapidly in N. For a game with K zones and N units, there are (N + K - choose K - 1) O(N*) distinct strategies to consider. In the most extreme case tested, N = 45 and K = 4, so the strategy space contains O(10)) distinet allotments, which can be reduced to (110) distinct trade-offs. Thus PTA can effectively separate the underlying complexity of a game from the size of its strategy space.
Second, in the right side of Figure the complexity of the K-4 games much higher than the K=3 games.
Third, the difference in complexity from game to the next in stark. At N 45 for example there are about 15 additional disc games needed to reach 95 % accuracy when going from the [1,2,4] transitive game to the uniform [1,1,1] game. Furthermore having similar [[A]]| complexity. The [1,1,1] and [2.3.4] games have similar intransitivity but have a difference of 20 in complexity.
|| does not imply similar
F POKEMON
Here we describe in further detail the construction of the Pokemon data. In the full game player (or trainer) captures Pokemon to compete against the Pokemon of other players, usually with teams of 6 Pokemon chosen at the player's discretion. Players also choose the order in which their Pokemon compete since the actual combat is done pairwise. This pairwise interaction is what allowed us to ignore the team aspect and still learn important aspects of the game. Each of the pokemon had set of attributes. The attributes are shown below.
1. Type 1: Main Type- Fire, Water, Grass, ect...
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Ratings
Speed
Evaluation
High
Disc Game
speed
speed
Disc Game
Low
High
Disc Game 3
Sppedd
speed
Rating
High
Slow
Speed
Fast
Slow
Fast
Dise game 3 colored by speed. Right: Eval- uation matrices generated by disc games (top) and 3 (bottom), with agents ordered by increasing speed. Both evaluation matrices are close Toeplitz, produce evaluation that depend primarily on the difference in speed between agents. The function that returns performance given speed difference approximated by sampling the evalutation matrix along the dios--aa marked in white. The subpanels containing scatter plots show the sampled evaluations.
Figure 9: Left:
Disc game colored by rating. Middle:
2. Type 2: Secondary Type Not all pokemon have two types but did not find this to contribute to any performance tradeoffs in a significant way
3. HP: Hit points Indicated how much damage a pokemon can endure before losing the match.
4. Attack: Base modifier for normal attacks
5. Defense: The base damage resistance against normal attacks
6. Special attack
7. Special Defense
8. Speed: This stat largely determines which pokemon get to attack first. As combat is turn based, this constitues a large advantage which we saw disc game
There were 50,000 pairwise interactions among the 735 pokemon that were used. The data for each interaction consisted of the name of the first and second pokemon as well the winner of the match. In an individual matchup, each Pokemon has certain level of HP or health. The two Pokemon take turns attacking one another until one of the them loses all of their HP and is declared the loser. The first to attack is determined by some set of attributes that is not explicitly given by the data set, but speed known to be a large contributing factor. Since we did not filled in any missing data using logistic regression, producing win probability matrix. We obtained the evaluation matrix F via the logistic link function commonly used in Elo rating. The evaluation for competitor competitor j is then given by ft, log( Pis 1 where Piy is the probability that Pokemon i beats Pokemon j. We applied the Schur decomposition directly F show that disc game embedding can successfully isolate a dominant transitive component (speed).
the full interaction graph We
In Figure we show the third dise game left with a full inner circle and half outer circle. Like disc game 1, disc game 3 is, approximately, a curve parameterized by speed. As in the Fourier examples discussed before, the double loop represents a higher order correction to disc game 1. Disc game confers transitive, monotonically increasing advantage to faster agents. The faster an agent relative to their opponent, the larger their advantage. Disc game 3 adds nuance to this relation by discounting the advantage conferred by small differences in speed, increasing the advantage conferred by intermediate differences in speed, discounting the advantage conferred by large speed differences, and strongly rewarding maximal
the main analysis. shows a double loop structure
19

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Under review as a conference paper at ICLR 2023
speed differences (see the evaluation matrix and associated subpanel in the rightmost column of Figure 9
Note, these conrections to disc game 1 are very small. The eigenvalue for disc game is roughly 15 times larger than the eigenvalue for disc game 3, hence the relationship between speed and
performance is largely determined by disc game 1. We did not discuss disc game 3 in the main text for this reason.
20


Reviews:
[{"review_id": "pI_3jxxd2_", "reviewer": "Reviewer_pi9j", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors study the overall structure of games. They propose the Principle Trade-off Analysis (PTA) a decomposition of two-player zero-sum game. Similar to Principal Component Analysis PTA represents a game as the weighted sum of pairs of orthogonal plane. Where each plane is associated with a strategic trade-off from a simple cyclic game. Furthermore truncation to the first planes provides a meaningful reduction of the game. The authors illustrate the usefulness of the method on two games Blotto and Pokemon. In particular in  Blotto allows to recover specific symmetries whereas in Pokemon PTA recovers clusters that naturally correspond to Pokemon\ntypes and correctly identifies a strategic trade-off in the Pokemon generation type.", "summary_of_the_review": "#Review\nUnderstanding the overall structure of a game rather that focusing only on computing equilibrium is a interesting research direction. And in particular the provided decomposition method is a valuable contribution. But the paper lacks of clarity in order to fully present the PTA method. In particular I find the presentation a bit too verbose and it could be improved with supporting precise quantitative statement and definition of the introduced quantities. It would be also useful to illustrate the different visual properties on a very simple example before jumping to the Blotto and Pokemon games. Finally the literature section could be improve by explaining what are the other methods of reductions, compare them to PTA in the experimental section.\n\n\n#Specific comments:\n\n- P2, related work: It is not very clear at this point what are the other methods and what are the differences between them and PTA.\n\n- P3, Section 3.1: Can you detail the Helmholtz-Hodge decomposition and precise what is 'least squares ratings that evaluate the average performance of each agent'.\n\n- P3, Section 3.1: Often we do not have access to the exact advantage F but rather samples, how does you  model can be adapted to this case?\n\n- P3 Section 4: 'Each pair of consecutive columns' of which matrix?\n\n- P3, Section 4: define SVD.\n\n- P3, Section 4: It'll be easier to follow if you define precisely what are F_t and F_c.\n\n- P3, (2): Ok but you already used the notations Q and U for the decomposition of F when you said you replaced A by F.\n\n- P3 below (2):  Can you define what is the optimal rank 2r approximation for F_c.\n\n- P4, Section 4: Actually you never defined properly what is PTA.\n\n- P4, above (4): Add a comma for the lower index of F_{c,i,j}.\n\n- P4, last paragraphs: Can you  details these properties quantitatively?\n\n- P5, last paragraph: How do you get discrete allocation from the sample of a Dirichlet?\n\n- P6, Blotto Example 1: At this point it is hard to understand Table 1, especially since some notations used in it are still not introduced.\n\n- P6: What is exactly a trade-off? Is it the same thing as a disc-game?\n\n- P6, exchange symmetry: I'm not sure I follow this part. Why such subtlety is not  treated in the previous section presenting PTA? And since generic games should not exhibit such strong symmetries why choosing this game as first example?\n\n- P7 Second paragraph: At this point it is not completely clear for me what you plot in Figure 1. Can you detail precisely which quantity in function of what is plotted.\n\n- P8, Section 5.2: What are the agents in this case?", "strength_and_weaknesses": "Strengths:\n\n- Understanding the game structure rather than focusing on the equilibrium computation.\n\n- PTA reduction of a game in simple cyclic games.\n\n\nWeaknesses:\n\n- Lack of clarity.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "ok1CBoXZLok", "reviewer": "Reviewer_enae", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper suggests using a representation similar to PCA to highlight important tradeoffs associated with different winning conditions in arbitrary games in a way that lends itself to 2D visualization.", "summary_of_the_review": "Overall, I appreciated the high-quality writing and persuasiveness of the paper. I don't have any particular concern with the paper, or reason to believe the proposed method has obvious shortcomings that would prevent it from being useful beyond the games considered in the paper. The typesetting quality needs to be improved for the final version.", "strength_and_weaknesses": "I really enjoyed the paper. To use the same wording as the authors, my research mostly focuses on the \"policy problem\", not on the \"problem problem\", so my point of view is that of a non-expert in the specific research niche of the paper. I found the writing style engaging, never boring, very approachable, and exceptionally persuasive.\n\nFor those reasons, assuming that none of the reviewers that are expert on the topic have issues with the positioning (e.g., all relevant literature has been correctly and fairly represented) and claims of the paper, I recommend acceptance of the paper.\n\nThat being said, I think that the authors should revise the paper to improve certain aspects:\n- throughout the paper, a majority of the citations should have been contained in parentheses and instead are just part of the text; this is rather distracting. Please make sure you're using the \\citep and \\citet commands adequately.\n- many plots have axes whose tick labels are straight up unreadable.\n- many (all?) plots are raster images, rather than vectorial graphic. Please consider re-exporting all your plots to be vectorial.\n- the bibliography section seems to be using the wrong style file. This is the only paper I'm reviewing that has underlined venues in the bibliography. Please align the bibliographic style to the rules of the conference.\n- some of the bibliographic entries have wrong capitalizations: for example, I believe Go and Blotto should be spelt with capitals, to name two examples ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "njX4FLbhPd", "reviewer": "Reviewer_1u6d", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors introduce principal trade-off analysis, which is in some sense an analogue of PCA for zero-sum games. They propose an algorithm for computing PTA, and run it on several games of interest, producing informative visualizations.", "summary_of_the_review": "A solid paper, introducing a new and seemingly very powerful method of analyzing two-player zero-sum games. I vote to accept.", "strength_and_weaknesses": "The concept of applying a PCA-like analysis to zero-sum games is very intriguing and, to my knowledge, novel. The method presented is simple yet powerful, and creates very interesting analyses of games. I think this is a good contribution to the literature, and as such I vote to accept.\n\nOne potential avenue of improvement is to include explicitly-worked decompositions of simple zero-sum games. For example, what is the decomposition of generalized RPS (i.e., 2k+1 actions of each player; action i beats action j if (i-j) mod (2k+1) <= k) or generalized matching pennies (normal-form game, n actions per player, payoff = identity matrix)? (of course, properly symmetrized.) That may give some intuition for what the method is doing--for example, we know exactly what the equilibria should be in both cases (uniform random).\n\nI have only a few minor questions:\n\n1. Why decompose $F_c$ instead of directly decomposing $F$ itself? That is, why ever explicitly separate out the transitive component? The decomposition could also be used on $F$ itself, and the transitive component would naturally appear as a component of the decomposition if it is prominent enough. This is what happened in the Pokemon example, and it produced a nice-looking plot of the transitive component. It would be interesting, in my opinion, to see that component explicitly in the Blotto example.\n\n1. Does this method have any implications for *solving* games? For example, if a straightforward method exists to solve games with low rank, one could apply that method to a low-rank approximation of a game that is otherwise too complex to be solved, and hopefully recover a good strategy in that game as well. \n\n1. How dependent are the embeddings $\\vec y_k(i)$ on the particular choice of (low-dimensional) embedding of the strategy space? For example, in the example Blotto game, one could hypothetically simply represent each pure strategy $i$ as a basis vector $\\vec e_i \\in \\mathbb{R}^{2926}$. Would that result in different visualizations? How different?\n\nPost-response: Thank you for the detailed response. I thought before that this was a strong paper, and I still do. I maintain my score.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "EgbA2xrNN1G", "reviewer": "Reviewer_d7g3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper introduces a method of decomposing games in low dimensional feature spaces called PTA. This method allows some characterization of game structure and allows for a general technique for visualizing data arising from competitive tasks or pairwise choice tasks. The relationships between embedding coordinates represent performance relations and elucidate the principal trade-offs responsible for cyclic competition in each game.\n", "summary_of_the_review": "The visualization aspect of disc games is insightful. However, the results are mostly empirical. Formulating the PCA analogy for game structures allows the reconstruction of the matrix Fc with minimal disc game information. However, as the authors state, the difficulty in many games is creating the cyclic component Fc altogether. It would be interesting to answer how someone can optimally use the first k linear combinations of disc games without necessarily requiring visual feedback to understand.\n", "strength_and_weaknesses": "Good visualization and coherent writing. I particularly liked the game balancing concept introduced and how we can visualize it through disk games. Similar insights are valuable for game understanding, beyond the utility function. It would be even more interesting to depart from the consideration of the reconstruction of the matrix Fc with a given accuracy and focus more on how to maximally utilize the information given by the first k disc games and capture elements of the game without the need of visualization. Higher dimensions wouldn’t allow visualization of the disc games.\n\nQuestions: \nGiven a reconstruction accuracy delta, how many disc games are needed to extract the useful information presented by the experiments? (such as the attack matrix). How does one argue about the redundancy of disc games in higher dimensions when visualization and understanding by humans is not possible?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject