Solving Football by Exploiting Equilibrium Structure of 2p0s Differential Games with One-Sided Information
Authors: Mukesh Ghimire
Lei Zhang
Zhe Xu
Yi Ren
Venue: ICLR 2026 Poster (2026)
Keywords: Differential Game
Incomplete-Information Game
Game Theory
Primary area: reinforcement learning
TL;DR: The paper highlights the limitations of current state-of-the-art when applied to solving one-sided incomplete information differential games with continuous actions, such as Football, and proposes scalable methods to solve the problem.

Abstract: For a two-player imperfect-information extensive-form game (IIEFG) with $K$ time steps and a player action space of size $U$, the game tree complexity is $U^{2K}$, causing existing IIEFG solvers to struggle with large or infinite $(U,K)$, e.g., differential games with continuous action spaces. To partially address this scalability challenge, we focus on an important class of 2p0s games where the informed player (P1) knows the payoff while the uninformed player (P2) only has a belief over the set of $I$ possible payoffs. Such games encompass a wide range of scenarios in sports, defense, cybersecurity, and finance. 
We prove that under mild conditions, P1's (resp. P2's) equilibrium strategy at any infostate concentrates on at most $I$ (resp. $I+1$) action prototypes. When $I\ll U$, this equilibrium structure causes the game tree complexity to collapse to $I^K$ for P1 when P2 plays best responses, and $(I+1)^K$ for P2 in a dual game where P1 plays best responses. We then show that exploiting this structure in model-free multiagent reinforcement learning and model predictive control leads to significant improvements in learning accuracy and efficiency from SOTA IIEFG solvers. Our demonstration solves a 22-player football game with continuous action spaces and $K=10$ time steps, where the offense team needs to strategically conceal their play until a critical moment in order to exploit information advantage. Code is available [here](https://github.com/ghimiremukesh/cams/blob/iclr/).

Published as a conference paper at ICLR 2026
SOLVING FOOTBALL BY EXPLOITING EQUILIBRIUM STRUCTURE OF 2POS DIFFERENTIAL GAMES WITH
ONE-SIDED INFORMATION
Mukesh Ghimire
Lei Zhang Purdue University West Lafayette, IN 47907, USA zhan581 (tan58140purruee edu
Zhe Xu
Arizona State University Tempe, AZ 85281, USA
Arizona State University Tempe, AZ 85281, USA @azeel0ar @0ael0asu.. edu
mgghmire@asu.. edu
Yi Ren"
Arizona State University Tempe, AZ 85281, USA yiueneasu.edu
ABSTRACT
For two-player ceppeffct-informaatte extensive-form game (IIFFG with K time steps and a player action space size U, the game tree complexity U2K causing existing IEFG solvers to struggle with large infinite (U, K). e.g., differential games with continuous action spaces. To partially address this scalability challenge, we focus on an important class of 2p0s games where the informed player (P1) knows the payoff while the uninformed player (P2) only has a belief over the set of possible payoffs. Such games encompass wide range of scenarios in sports, defense, cybersecurity, and finance. We prove that under mild conditions, Pl's (resp. P2's) equilibrium strategy at any infostate concentrates on most (resp. I + 1) action prototypes. When I < U, this equilibrium structure causes the game tree complexity to collapse to K for P1 when P2 plays best responses, and (I+ 1)K for P2 in dual game where P1 plays best responses. We then show that exploiting this structure in model-free multiagent reinforcement learning and model predictive control leads to significant improvements in learning accuracy and efficiency from SOTA IIFG solvers. Our demonstration solves a 22-player football game with continuous action spaces and K 10 time steps, where the offense team needs to strategically conceal their play until a critical moment in order to exploit information advantage. Code is available here
INTRODUCTION
The strength of game solvers has grown rapidly in the last decade, beating elite-level human players in Chess (iiverr et al. 2017a), Go (Silver et al. 2017b), Poker (Brown & Sandholm 2019 Brown let al. 2020b), Diplomacy (FAIRT et al. 2022). Stratego (Perolat al. 2022), among others with increasing complexity. However, most of the existing solvers with proven convergence, either based on regret matching (Tammelin) 2014 Burch et al.) 2014 Moravčík et al. 2017 Brown et al. 2020bb Lanctot et al. (2099) or gradient descent- ascent (McMahan) 2011 Perolat et al. [021 Sokota et al. 2022| et al.] 202T Vieillard et al. 2020). have computational complexities increasing with the size of the finite action set, and suffer from game-tree complexities growing exponentially with both the action size U and the tree depth K. Real-worle games, however, often have continuous actions and happen in continuous time and state spaces, making them differential in nature. Applying existing solvers to differential games would require game-specific insight or extra computational overhead for automated abstraction (Kroer& & Sandholm 2015 Hawkin et al.) 2011 Brown & Sandholm 2015).
In this paper, we address this scalability challenge for an K subset of 2p0s differential games where the informed player (PI) knows the payoff type while the uninformed player (P2) only holds a public
Conresponding author.

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Published as a conference paper at ICLR 2026
(4)
HEFG Game Tree (PI P2)
2ptst PI
Apli
(b) Initial State
Type 0: RB Power Push
Type QB Throw
K
RRB
12
EI Offense
I
+
02k
(I + 1yx
QB
(c)
belief
Type 0
P
conceals till 0.5s
0.0
0.5
1.0
1.5
2.0
2.5
Type
!
time
Figure 1: (a) LIFG with U actions per player per infostate and K time steps has game- tree complexity of U2K For 2p0s1 with 7 payoff types, deterministic dynamics, and Isaacs' condition, we show that the NE is T-atomic for P1 and (T + --atomic for P2, leading to game-tree complexity of 7K for P1 the primal game where P2 plays best responses and (T + 1)K for P2 in the dual game (b) American Football with 22 players and continuous action spaces (U = 0o) with = 10 time steps. PI (red) attacks with two private game types (/ = 2): Running back (RB) power-runs through the space created by blockers, and quarterback (QB) throws the ball to the leading wide receiver (WR). See animation
where P1 plays best responses.
(c) At NE, P1 conceals type until 0.5 sec., similar to the reported 1.0 sec. Due to significant tree size reduction, the game can be solved in 30 minutes.
belief po E (1) over the finite set of / possible types. At the beginning of the game, nature draws game type according to po and informs Pl about the type. As the game progresses, the public belief about the true game type is updated from Po based on the action sequence taken by P1 and its strategy via the Bayes' rule. Pl's (resp. P2's) goal is to minimize (resp. maximize) the expected payoff over po. Due to the zero-sum nature, P1 may need to delay information release or manipulate P2's belief to take full advantage of information asymmetry. While restricted, such games represent a wide range of attack-defense scenarios including football set-pieces where the attacker has private information about which play is to be executed, and missile defense where multiple potential targets concerned The setting of one-sided information, i.e., P1 knows everything about P2, is necessary for P2 to derive defense strategies in risk-sensitive games. We call this focused set of games "2p0s1""
We claim the following contributions:
We prove two unique Nash equilibrium (NE) structures for 2p0s1: :: (1) The equilibrium behavioral strategy for P1 (resp. P2) is I-atomic (resp. (I + 1)-atomic) on their continuous action space, and (2) the equilibrium strategies for P1 and P2 can be computed via separated primal and dual reformulations of the game. Together, these structures collapse the game- tree complexity to most K for P1 and (I + 1)K for P2. In comparison, solving the same game through the lens of IIEFG would have a game-tree complexity of U2K with a discretized action size of U (Fig. Th).
We demonstrate how this structural knowledge can significantly accelerate game solving: (1) For value and policy approximation settings where the ground-truth NE is available, we achieve qualitative improvements on solution accuracy and efficiency from SOTA normal- and behavioral-fome further demonstrate the practical value of the equilibrium structure of 2p0ss by solving an American football setting where the attacking team needs to strategically conceal their true intention between "RB power push" and "QB throw". While this IIFGG has a complexity of 10440 even with coarse mesh of 10 values per action dimension, the atomic structure of its NE makes the game solvable in less than 30 minutes on a MI Pro Macbook. See Fig. D...
solvers (CFR+, MMD, Deep-CFR, JPSPG, PPO, R-NaD). (2) We
2 RELATED WORK
2p0s games with incomplete information. Games where players have missing information only about the game types are called incomplete-ing cammpleet-informatiinn. These are a special case of imperfect- information games where nature only plays a chance move the beginning Haasanyil (1967). The seminal work Aumann et al. (1995) developed equilibrium strategies for a repeated game with one- sided incomplete information through the "Cav u" theorem, which reveals that delief-manipulating
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behavioral strategies optimize value through value convexification. Building on top of this, De Meyer (1996) introduced a dual game reformulation where the behavioral strategy of P2 becomes Markov. This technique helped Cardaliaguet (2007); Ghimire et al. (2024) to establish the value existence proof for 2p0s differential games with incomplete information. Unlike repeated games where belief manipulation occurs only in the first round of the game, differential games may have multiple critical Lime-ttate-belie points where belief' manipulation is required to achieve equilibrium, depending on the specifications of system dynamics, payofl', and state constraints (Ghimire et al. (024). Our paper builds on top of Cardaliaguet (2007); Ghimire et al. (2024), providing a thorough analysis of the convexification nature of behavioral strategies and, as a consequence, their atomic structure.
IEFGSs. IIEGss are
stochastic games (POSGs) with finite horizons. Signif- icant efforts have been taken approximate equilibrium of large IIFGGs with finite U (Koller &
Megiddo| 1992 Billings et al., [003] Gilpin & Sandholm 2006 Gilpin et al. 2007; Saanhhll 2010
Brown Sandholm, 2019: Brown et al. 2020a]] | Perolat et al. 2022 Scchid et al.] 2023). Regret
matching algorithms (Zinkevich et al.] 2007: Lanctot et al. |209]] Abernethy et al. [011 Brown
ct al. 2019 Tammelin 2014 2014] Johanson et al., 2012) have computational complexity O to
N-Nash, while gradient-baser based solvers (MeMahan 2011: Perolat al., [021 Sokota et al. 2022) have
O (In(U)== In (e to E-QRE. All these complexities increase with U, and convergence guaran-
tee only exists if the NE lies in the interior of the simplex (U)) (Perolat et al. 2021). Critically,
this latter assumption does not hold for 2p0s1, as we explain in Sec. [4] For games with continuous
action spaces, existing
based approaches (Martin & Sandholm 2023 2024) lack
convergence guarantee, and perform poorly in our case studies in both normal- and extensive-form settings.
Multigrid for value approximation. Since solving 2p0s essentially requires solving Hamilton- Jacobi PDE, we briefly review multigrid methods for accelerating PDE 2000). In a typical V-cycle solver (Braess & Hackbusch] 1983), fine-mesh solve is first performed brieffy, the residual is then restricted to a coarser mesh where a correction is solved and prolongated back the fine mesh. Essentially, multigrid uses coarse solves reduce the low-frequency approx- imation errors in the PDE solution at low costs, leaving only high-frequency errors to be resolved through the fine mesh. Multigrid was successfully applied to solving Hamilton-Jacoh PDEs for optimal control and differential games (Han & Wan (013). although characterizing its computational complexity for such nonlinear PDEs is yet to be done. During value approximation, we extend multigrid to solving Momilton-Jacobb PDEs underlying 2p0s1.
et al.
PROBLEM STATEMENT
We denote by A(I) the simplex in RR, [4] := (I...., J. ali] the ith element of vector a, VpV and V.V the respective gradient and subgradient of function V with respect to p. || lla the 12-norm and ||u]|a= = V Ar. Consider a time-invariant dynamical system that defines the evolution of the joint state x E X C dd of Pl and P2 with control imputs E 2 C Rd= and E V C Rd,, respectively: (()) f(((((, v).
(1)
The initial belief of players is set to nature's 's distribution about the game type. Denote by { the joint sets of I behavioral strategies of P1, and Z, the set of behavioral strategies of P2. The subscript
r indicates the random nature of behavioral strategies: At any infostate (1, x) E [0, x X x A(T) and with type i, P1 draws an action based on 1) E HH which is a probability measure over 2 parameterized by (t, x. p). P2's strategy ( E Z, is a probability measure over V independent of i. Denote by X 6, 00 the random state arrived at 4 E (4n, T from (lo, following (TK, S) and the system dynamics Eq. [ With mild abuse of notation, let (1,, (()) denote the random controls
at time induced by (n. (). Pl accumulates rumning cost (z((), m. (t), (()) during the game and receives terminal cost gi (X Mo............ Together, a type-i P1 minimizes:
1" L(x(s), n((), (()))ss
9: (x+p:.0....))
J Jt Foi 17. () := En:.
while P2 maximizes J [to, zo, poi () = Ei-po [J]. We say the game has value V if and only if the upper value V V To, Po) = fn sup, J and the lower value V- To Po) = sup, (n.) (o.) J are equal: V= V -V-, The game is proven to have value under the following sufficient conditions (Cardaliaguet 2007):
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A1. 4 _ Rd and V _ Rd are compact and
A2. f xxxxuxx + X is C' and has bounded value and first-order derivatives.
A3. 9: : : R and 4 : X' x u x V - R are bounded and Lipschitz continuous.
A4. Isaaes' condition holds for the Hamiltonian H X x R (T) - IR: min max (2r u, v) E Ex (1(,; u)
H(r, ==
well vEV
(2)
max min (I, u, v) ç (1 u, e)]-
VVV well"
Isaacs' condition allows any equilibria, including nonrevealing games where neither player knows the actual game type.
information versions of this game to have pure Nash
A5. Both players have full knowledge about f. {g:) U4iH=1 Po- Control inputs and states are fully observable. Players have perfect recall.
Our goal is to compute a Nash equilibrium (NE) (tu)),t)) that attains V. given the game G {xt, (24, V), (9r), f.T.
4 A PRIMAL-DUAL REFORMULATION OF THE GAME
In this section, we introduce discrete-time primal and dual reformulations of G, denoted by G and G;, respectively, for which dynamic programming principles (DP) exist. We show that P1's equilibrium behavioral strategy (ml,-) in G, is I-atomic, i.e., n/.-- concentrates on at most / actions in U, and P2's strategy ç in G is (I + 1)-atomic. Then we show that (1r), ).))) (+) approaches the Nash equilibrium of the differential game G as the time interval
0++
The primal game GT. G- is a discrete-time Stackelberg version of G where P2 plays a pure best response after P1 announces its next action. Let the value of G, be V,, V,, satisfies the following DP for (t, x, E [0, T x x (()):
Vt, {m
min E Binparo
max Vi + T, x' (u, u), p))) - Th(a,, NEV
- w))
(3)
with terminal boundary V= x.)) E; (()) For small enough т, x' (u,
T f((, u, v)- ('(u) is the Bayes update of the public belief after P1 announces u: p'(u][]] =
mi (u; p)plil/n(ee ((u t, x., p). where t, x, p) Eien (u t, Ir, p)pli is the marginal over U across types. Note that P2's equilibrium behavioral strategy cannot be derived from Eq. 3
The dual game G;, For P2's strategy, we need a separate DP where P2 announces its next action and P1 best responses to it. We do so by first introducing the convex conjugate V of the value:
V"(to, To. 20. == maxp"' po V(oo xo. p) = mmxxp p" po sup inf CE2, (m.E((H.))'
Em J
(4)
max inf sup pToo C. [J.] inf sup max P CEZ. (m)E(H.)
Poli] E
J
<EZ, mjE H fE(.......
Eq. 4 describes a dual game G with complete information, where the strategy space of P1 becomes H x [I]. i.e., the game type is now chosen by P1 rather than the nature. We prove in App. A that P2's equilibrium in the dual game is also an equilibrium in the primal game if pa E V To, Po), and such po[i] represents the loss of type-i P1 should it play best responses to P2's equilibrium strategy. Therefore Pu[i] Ec[J.] measures P2's 's risk, and P2's ss equilibrium strategy minimizes the worst-case risk across game types. We now introduce a discrete-time version of the dual game G where P1 plays a pure best response after P2 announces their action at each time step. Let the value of G be Vir the dual DP is:
max V, (1 7, x' u), T((x,, u, (.)) uEll
(1, min Exsc (.)
with terminal boundary V (T Φ) = max {pli] - 9 (x)). Here p'(-) -RR is constrained
by E (x)) = á (similar to the martingale nature of p in Gr), and ((u, [6] = =).
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Published as a conference paper at ICLR 2026
Equilibrium strategies of G, and G are atomic. Our first theoretical result is Thm. 4. which states that P1's strategy that solves G, I-atomic, and P2's strategy that solves G² is (1 atomic:
Theorem 4.1. The RHS of Eq. can be reformulated as
>> < Tf uh ).)))'))) + TBuuph Pe ut, ,,,,
min max {u MA (oM)
afpli
s.t. a Eu, ot Ev, EV E 10, Σ = =1, k Σ a pli, p0] [
Vi, E
X
k=1
(P1)
i.e, Tay concentrates on actions
{ur Ik=1 for [T]. The RHS of Eq. can be reformulated as
I+1
x (V (t+ Tf uh, )0 Tl u., ,0*)))
min ( 3
max {u
(P2)
1+1
u E u, t e ev, x [0.1], Σ 1
=
> x 11 [++]
k=1
Proof sketch. (1) Using Isaacs' condition, show that P2's best response in Eq. 3 is implicitly governed by Pl's action u, and u is in turn governed by the posterior belief p'. (2) With this insight, we can rewrite Eq. as V- (t, x, p) = min, Sa(n V=(t, x. (d)'',,,, where we control a pushforward density v(dp') for the posterior belief to be p'. V is subjected Sa(n p'u(dp') p. For each p', o(p)) is found by solving vt,, min maxvev (t ,,,',p')) E.-p/ Here (() is Pl's loss should both players play pure strategies at (1, 21, p') within [t, L T), and its existence is guaranteed by Isaacs' condition. Since playing pure strategies does not change the public belief, we call V- the non-revealing value. (3) We can then show that min, J V. (dd))(dp)) is
convexification of V. in (TT). Since convexification requires at most I vertices in A(7), is at most I-atomic. Since v determines vlyy- the latter is also T-atoiic. (4) A similar argument can be made for G+, in which case the convexification is with respect to the dual variable Since is defined on R rather than constrained on ((T), ç is at most (7 + -atomii.. Proof in App.
An example. To support the proof sketch, we use Fig. 2 to il- lustrate why equilibrium behavioral strategy is atomic. Here = 2. thus the value is defined on A(2) for fixed (t, z). To solve min, SA(2) (dd)), we scan the non-revealing value V. across A(2). One notices that if V is not convex in it is al- ways possible for P1 to achieve a lower loss by convexifying V through a mixed strategy, leading to P1 In the figure. P1 identifies [X*,X]] 1 (22 and {n" ,p) such that x + >Xpp p. Solving the non-revealing pure NE (u x*) for each E {a,b), b). P1's mixed strategy that convexifies V is to play action x² with probability af = p* [ /pli] if P1 is type-i. By announcing this strategy, the public belief shifts to p* via the Bayes' rule when P2 observes action u² As result, P1 receives (pp) (p") + (""("²) on the convex hull of (p))
vt,,x,p) v(t,x,p)
77
1
p
Figure 2: Value convexifica-
tion causes NE to be atomic.
Remarks. (1) The atomic structure of the equilibrium strategies was first discovered for 2p0s repeated games with one-sided information (Aumann al., 1995, De Meyer 1996). Our new contribution is in explaining its presence in differential games and in demonstrating its significant utilities in improving the effectiveness of multiagent reinforcement learning and model predictive control schemes. (2) Pl's actions in 2p0s1 simultaneously advance system states and achieve signaling. The deterministic dynamics allows precise belief control (e.g., (p", p²) p²) in Fig. for value convexification. For games with stochastic synamies/observations. however, P1 will not be able to precisely control the belief, and the convex Bellman operators (P) and P2) become lower bounds of the value. Finding tight upper-bounding operators that preserve atomic NES is left for future work.
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((n).-). approaches NE of G. Next we present Thm. 4.2 which proves that (f7) St) computed from Pi and 22 approaches the equilibrium of G when r is sufficiently small. The theorem uses Pl's loss in G when they adapt {π,,-} 3 to G by playing a constant action within each time interval. This loss is denoted by (t I, p) == maxçez (t,, x. pi (nl,...)) Similarly, P2's loss in G" when they adapt ç to G is denoted by 2t,, == max ( J (t, P; {m. (t).
Theorem 4.2. If A1-5 hold, then for any e > 0, there exists T > 0, such that for any T E (0, and the corresponding (4y), ()), (1,,,,p) (I, p) E [0, <) for any (l, (1,x,)) E [0, x X ( and (t, x. V x) E (0, c) for any (t, P) E [0, T x X x RR.
Proof sketch. We first note that V s V1 because Pl's NE in G, is not necessarily a NE in G. i... PI can get a better value in G using {t.] than (vk.-). We also have, from Cardaliaguet (0099) , the uniform convergence of V- + V as r + 0+. Then, it remains to show that V1 is "not far" from V, To do so, first show that the per-stage value gap between V1 and V, is within ( which accumulates over time into (T)) V12 S V, + ( Then as T + 0+, combined with the uniform convergence of V,, we conclude that the gap Vi - V [0, e]. This implies that the equilibrium {n/ derived in G, is e-optimal in the original game G as long as 7 is small enough. Full proof in App. C The same techniques can be applied to G².
With Thms. 4.1 and 4.2 and given that G' is differential (T + 0+), we now proved that NEs of are atomic, and can be approximated via Eqs. P and P2 with sufficiently small
INCORPORATING THE ATOMIC STRUCTURE INTO MARL AND CONTROL
We study the efficacy of atomic NEs when applied to (1) value approximation, (2) model-free MARL,
and (3) control settings. For (I), we approximate value and strategies the entire [0, T] X X A(I) through Pi and P2 and introduce multigrid to further reduce the compute. We compare with discrete- action solvers CFR+, MMD. CFR-BR, and continuous-actio
solver JPSPG. For (2) and (3), we solve NE strategies for a fixed initial (to, To. Po). We compare with discrete-action solvers MMD, PPO, and R-NaD. We call the proposed continuous-action mixed-strategy solvers CAMS, CAMS-DRI, and CAMS-MPC for value approximation, MARL, and control, respectively.
5.1 VALUE APPROXIMATION
CAMS for G,, We discretize time as {KT LK 7 T/K. Let = (ie,)) be collocation
set. We solve P1 starting from -= (K - 1)T at all collocation points in S. The resultant nonconvex- nonconcave minimax problems have size (T(((T+dw), (TT,)) and are solved by DS-GDA (Zheng nonconvex-noncone
ct al.) 2023). which guarantees sublinear convergence generalize value and strategies across X' x A(1), a value network is trained on the minimax solutions and used to formulate the next round of minimax at t 7. G² is solved similarly.
problems. To
Computational challenge. From Thm. 4.2 large K (small 7) is required for strategies derived from Pi and P2 to be good approximations of the NE. Yet suppressing the value prediction error at / = 0
requires computational complexity exponential in K. Specifically, let Vo p) : X x A(I) + RR be the trained value networks at 0, we have the following result (see proof in App.
Theorem 5.1. Given K, a minimax approximation error c > O, a prediction error threshold > O.
there exists C 1, such that with computational complexity of O(K3 P2 CAMS achieves
max (1)))
(Vo(z, - (O, x. p)1 5
(6)
similar result applics to the dual game. Zanette et al. (2019) discussed a linear value approximator that achieves C = 1. However, their method requires solving linear program (LP) for every inference V2 p) if (x, p) does belong to the training set S. In our context, incorporating their method would require auto-differentiaaing through the LP solver for each descent and ascent steps in minimax, which turned out to be too expensive. To this end, introduce a multigrid scheme to reduce the cost for games with large K.

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Published as a conference paper at ICLR 2026
Multigrid. Since strategies time are implicitly nonlinear functions of the value at t + the Hamilton-Jacobi PDEs underlying P1 and P2 are nonlinear. Our method extends the Full Approximation Scheme (FAS) for solving nonlinear PDEs (Tooteenbrrg et al. 2000 Henson et al.) 2003). A two-grid FAS has four steps: (1) Restrict the fine-grid approximation and its residual; (2) solve the coarse-grid problem using the fine-grid residual; (3) compute the coarse-grid correction; (4) prolong the coarse-grid correction to fine-grid and add the correction to fine-grid approximation. For conciseness, we focus on the primal problem. Let V be the value network for time on grid size (time interval) l. Let R be the restriction operator to a coarser grid with size 21: R (V,) (V + V/+1)/2 is the value restriction from to 21. Similarly, we define the prolongation operators p2i as (V??) V if & E 7 or V otherwise, where 7 : {n 2/ n E No, n < T//2I). Let (t,, x, p; V) PI (t, x. where V is the value at + and outputs an approximation for v,, The dataset {(1, x00,, pl)) (1, x00), pb)); V(+1))} is used to train (.((,,). Let Let (fx, (z. p) R 0 for all 2,, E T' X'X (()) we restrict the fine grid approximations and residuals to the coarse grid and solve to determine the conrections. To do so, let el (z. p) be the correction in grid at (t) The coarse-grid problem is
(=) (t. x. pi V/41) be the residual. To achieve (z,
RT 02 1,,
1+21 + 2422) - (R + 2e (x.
- RV))
(7)
residual
grid eq. corrections
eq. w/o corrections
where (²" p) is computed backward from T - 21 using e7 =
R'V4 1421 + +e2ta)) 0²0 ef (x, -02) (1, (.
(2 (22) - RR'44
(8)
This correction ensures consistency: If ViJ v-,,,,) for all E (..)) 0 for all I T The coarse grid corrections are prolonged to the fine grid to update the fine-grid value approximation. Note that from Eq. 8 computing the coarse correction in case requires two separate minimax calls with similar loss formulations. We further accelerate the multigrid solver by warm-starting these minimax problems using the recorded minimax solution derived from the fine grid (during the residual computation).
5.2 MODEL-FREE MULTIAGENT REINFORCEMENT LEARNING
Recent studies (Rudolph et al. 2025 Sokota et al. 2022) showed that policy gradient methods for MARL, such as PPO and MMD, can effectively solve IIEFGs with properly tuned hyperparameters. We show that the atomic structure can be directly applied to this unified model-free framework with minimal code changes, while yielding significant solution improvement for 2p0s1. For CAMS-DRL, the policy network of P1 takes in the infostate (t, p) and outputs / logit vectors lg E R and T action prototypes = E l for & E [4]. Each logit vector lu is transformed via softmax to define the behavioral strategy of type-i Pl, i.e., the probability πą(u* h;; f, x),)) of choosing action w². The policy network for P2 takes in (t, x. p) and outputs a single logit & E R [+1 and action prototypes fr for k E [I + 1]. We directly solve the NE of these policy models using PPO and MMD.
5.3 MODEL PREDICTIVE CONTROL
When dynamics f is known and differentiable, and an initial state (to, zo. Po) given, we can formulate the primal (resp. dual) minimax objective as the sum over I (resp. (I+ + +1 1)*) game tree paths (Fig. [T). CAMS-MPC builds the computational graph for the entire tree where the policy networks follow CAMS-DRL, and applies DS-GDA to this differentiable loss. This method is feasible thanks to the atomic structure of NEs and for small I. Since P1 in 2p0sl games reveals their payoff mid-game, the game tree further collapses after information revelation. E.g., Hexner's game (see below) has a proven game-tree tree complexity of I, where P1 plays a fixed nonrevealing strategy before splitting to type-dependent revealing strategies. Due to this collapse, modeling strategies using neural networks turns out to be more effective for the convergence to NEs than using infostate-wint wise parameterization.
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Published as a conference paper at ICLR 2026
6 EMPIRICAL VALIDATION
Hexner's game. We introduce Hexner's game (Hexner 1979. Ghimire et al. 2024 to compare
CAMS variants with baselines on solution quality and computational cost. This game has an analytical NE. The dynamics is 2; = Az + Byu; for [2). where 2j E X,, Uy E E,, and A; and B3 are known matrices. The target state of Pi is 28 where is drawn with distribution po from e, | = I. and 2 E Rd is fixed and public. The expected payoff to P1 is:
((m)).5) = Binupo [ [m(()) (I((t) dt + [Tx 20: 12,, - [lx2(T) 20, 11:,
where R1, R2, K1, K2 > are control- and state-penalty matrices. The goal of P1 is to get closer to the target 20 than P2. To take information advantage, P1 needs to decide when to home-in and reveal the target. Analytical NE: There exists a critical time 4, == 6, (T, {Ay}, {Br}, [Rz}. {K₃}), H) If I, E (0, T). P1 moves towards [E(O as if it does not know the actual target until 4 when it fully reveals the target, i.e., value convexification happens at ty. If t, I 0, P1 homes towards the actual
target at = O. P2's NE mirrors Pl. See proof in App. ET
Time Complexity
Algorichm Iterations
Action Error
Action Error
109
10F
100
-Beepp-FFF: I7 15]
SDDD -CAMS (asen)
-CCAS (coz4)
Ieegratiogee
time
10+
30
10a
30+
100
10a
CFR- RR: Prinal
10a
-AMMS (oers)
CAMS (curv)
16
96
64
144
10
10
36
64
144
16 96
64
144
0.25
0.6
0.75
(a)
(b)
(c)
(d)
Figure 3: (a-c) Comparisons b/w CAMS, JPSPG, CFR+, MMD, JFR-BR-Primat on I-stage Hexner's
game. (d) Comparison b/w CAMS, JPSPG, and DeepCFR on 4-stage Hexner's w/ similar compute.
6.1 EFFECT OF ATOMIC NE ON VALUE APPROXIMATION
Action-size-invariane convergence of CAMS. We use a normal-form Hexner's game with 7 T and a fixed initial state Tp E X' to demonstrate that baseline algorithms suffer from increasing discrete action sizes while CAMS does not. The baselines are SOTA normal-form solvers including CFR+ (Tammelin] 2014), MMD (Sokota et al. 2022). JPSPG (Martin & Sandholm 2024), and a modified CFR-BR (Johanson et al. 2012) (dubbed CFR-BR-Primal), where we focus on con- verging P1's strategy and only compute P2's ' best responses. Among these, only JPSPG naturally handles continuous action spaces. All baselines except JPSPG are standard implementations in OpenSpiel (Lanctot et al. 2019). JPSPG implementation details are in App. G.4 The normal-form primal game has a trivial ground-trutt strategy where P1 goes directly to its target. For visualization, use d₂ = 4 (position and velocity in 2D). For baselines except JPSPG, use discrete action sets defined by 4 grid sizes so that U = |2,11 E {16, 36, 64, 144). All algorithms terminate when a max;; Vi(=(() Vi(T.) is met. For conciseness, we only consider solving Pl's s strategy and thus use P1's NashConv. We set the threshold to 10-3 3 for baselines and use more stringent threshold of 10 for CAMS. We then use DeepCFR and JPSPG as baselines for a 4-stage game where T = and r = 0.25. DeepCFRs were run for 1000 CFR iterations (resp.
threshold of NashConn((..)
CAMS explnitabdlity
CAMS epplnitability
JPSCG eepleitahiltiy
10a
30
16
10a
C 16
101.
JPSNG
CAMS
10
64
364
10-1
64
10
10-4
1500
2500
560
1500
2000
Berations
Recations
Sterations
Figure 4: Exploitability comparison the normal-form Hexner's 's game.
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Published as a conference paper at ICLR 2026
(a) Schomatics
(b) Primal, Goal-1
(c) Primal, Goal-2
(d) Prinal-Dual, Goal-1
(e) Primal-Dual, Goal-2
10
0
02
0.5
0.5 0.5
0.5
0.5 X 0.5
Figure 5: (a) Hexner's game schematics: one goal is selected out of two possible goals: Goal-1 and Goal-2, and communicated to P1. (b-e) Sample trajectories for the primal game (b-c) where P1 plays Nash and P2 plays best response, and primal-dual game (d-e) where both players play Nash. Dotted lines are ground-truth Nash. Color shades indicate evolution of public belief (Pr[Goal is IJ). Filled Magenta circle represents the true goal. Initial position pairs are marked with the same markers.
100) with 10 (resp. 5) traversals for U = 9 (resp. 16). The wall-time costs for game solving 17 hours using CAMS (baseline), 24 hours for JPSPG, 29 hours (U = 9) and 34 hours (U = 16) using DeepCFR, all on A A100 GPU. More details experiment settings can be found in App. G.3 Furthermore, JPSPG was run for 2- 10 iterations, where each iteration consists of solving a game with random initial state and type, and performing strategy update.
Fig. 3 summarizes the comparisons with baselines. For the normal-form game, we compare both computational cost and the expected action error from the ground-truth action of Pl: ((x))
|4| k= 0ual| uz (zn))l2) where ut (Fo) is the ground truth for type i at zg. In the 4-stage
E Bippp
game, compare the expected action errors each time-step: é, == Bizee (((z)) where is the strategy learned by the respective learning method. For each strategy, we estimate (&))__ by generating 100 trajectories with initial states uniformly sampled from X. In terms of computational cost, all baselines (except JPSPG) have complexity and wall-time costs increasing with U, while CAMS is invariant to U. We also compare exploitability of CFR+, MMD, and JPSPG against CAMS for the normal-form Hexner's game in Fig. For fair comparison, best responses are computed in the continuous action spaces.
a
With similar or less compute, CAMS achieves significantly better strategies than DeepCFR and JPSPG in the 4-stage game. Sample trajectories for the 4-stage game are visualized in App. G Fig. compares the ground-truth vs. approximated NEs for 10-stage games with different initial states. While approximation errors exist, CAMS successfully learns the target-concealing behavior of PI. Averaging over 50 trajectories derived from CAMS, Pl conceals the target until Ey- = 0.60s + 0.06s (compared to the ground-truth. t, 0.5s).
CAMS scalability with multigrid. We solve Hexner's games using CAMS with and without multi- grid to demonstrate the scalability of our approach and the effect of multigrid. We report the runtime on one H100 GPU for 4-, 10-, and 16-stage games
Table 1:
Runtime comparison of CAMS with and without multigrid
time steps
no multigrid
multigrid +
4
9.3 hrs
2.3 hrs
in Tab. D We run the 2-level multigrid on the 4- and 10-stage games, and 4-level multigrid on the 16-stage game (see Alg. for pseudo code). We report result- ing trajectories in App.
10
27.6 his
10.9 hrs
16
46.2 hrs
17.8 hrs
6.2 EFFECT OF ATOMIC NE ON MODEL- FREE MARL
For model-free MARL, we compare CAMS-DRL with MMD, PPO and R-NaD with discrete actions
of size 100 formed by pairing each of the ten lincarly spaced z-direction acceleration between (-1, 1) and y-direction acceleration between (-4, 4). We test the policy convergence in the normal-form
game by comparing Pl's I learned policy every 8192 steps (corresponds to 1 iteration in Fig. ((a)) against the ground truth policy. MMD. PPO, and R-NaD are implemented as in Rudolph et al. (0025).
Results in Fig. 6 (a-b), which uses the same comparison metric as outlined in Sec. 6. show that HAMS-DRLE approximates NEs accurately. while PPO and MMD fail. R-NaD, on the other hand, shows better performance than MMD and PPO with little to hyperparameter tuning (see App. H)). This contradicts with observations in Rudolph et al. (202)). We conjecture that PG methods such as PPO and MMD have difficulty converging to non-interior solutions, which is the case in 2p0s I due to

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Published as a conference paper at ICLR 2026
(a) Strategy Convergence
(b) Action Error
(c) Football Exploitability Estimate
3.5
MMD (130reete)
35
MMM (Duerete)
(Ntereta)
B.Nab (Tarrete))
30
- R.a
2.5
PP0 (CAMS
DRE. nit.
25
-MDrD (CAMI.)RR))
exploitability
20
4
No
10
0.5
400
800
1,200
0.25
0.5
0.75
5,000
10,000
iters
ilers
Figure 6: Comparisons b/w CAMS-DRL and standard PG methods on (a) 1-stage and (b) 4-stage games. (c) Exploitability estimate of the strategies in the Football game different iterations.
the atomic structure of NEs. We performed the same comparison for 4tage game, with a discrete action size of 100 composed by pairing linearly spaced x- and y-direction accelerations, all between (-12, 12). Results in Fig. 6 show significant improvement in solution accuracy when the atomic structure is exploited.
6.3 EFFECT OF ATOMIC NE ON MPC
Hexner's game. With known differentiable dynamics and 2, 10-stage Hexner's (primal) game has at most 20 rollouts, making exact policy gradient computation feasible. Using DS-GDA, the convergence to the ground-truth NE takes 5 minutes on a MI Pro Macbook. See results in App. [
Football game. The independence of the game-tree complexity from the action space allows us to solve games beyond what IIEFG solvers can afford. Here we model an 11-vs-11 American Football play as a short, two-team game in a 2D plane. The horizontal axis represents "downfield" progress and the vertical axis is sideline-to-iideline. Each player follows double-integrator kinematics and chooses acceleration at discrete time steps. Physical contact is captured by smooth "merge" weight that grows opponents approach, softly blending their velocities and attenuating their ability to apply new acceleration, so tackles emerge continuously rather than through hard impulses. The offense has two types (T = 2): an "RB power push" that prefers the RB to advance straight upfield, and "QB throw" that rewards whichever offensive player gets furthest downfield. With loose bounds on velocity and acceleration, the soft tackle dynamics control-affine, allowing Isaacs' condition to hold. Together with differentiable rollouts, the game settings satisfy A1-5, leading to atomic NE. With K 10 and known differentiable dynamics, it is feasible to solve Pl's 's (and then P2's s) strategy by applying DS-GDA directly to the minimax objective constituting all I ((I + 1)*) rollouts. See App. [ for detailed game settings. The convergence takes 30 minutes on a M1 Pro Macbook. The resulting plays are summarized in Fig. Tb.c and animated in the github repo) Qualitatively, the results resemble real football tactics: the offense either tries to push through the defense (aka inside zone play), or goes out wide by faking a move (aka waggle play), while the defense, in response, either close-in on the player with the ball possession or stay back to guard. Importantly, our results show that the offense conceal their play selection for 0.5 seconds, which is comparable coaching analyses that estimate roughly one-second window before the play becomes clear (Grabowski 2020). Quantitatively, we estimate the exploitability of the strategies and plot its convergence in Fig. 66. We also provide a comparison of resulting trajectories in App. when each player switches to their respective BR policy to show that the final policies are minimally exploitable.
CONCLUSION
Unlike general IEFGs where NE strategies are distributions over the entire action space, we showed that 2p0s1 games enjoy an atomic NE structure when P1 can precisely control the public belief, leading to an exponentially reduced game-tree complexity. We demonstrated the utilities of this NE structure in solving games with continuous action spaces in model-free and model-based modes, in terms of computational cost and solution quality. Our methods enable fast approximation of deceptive and counter- -eceptto team strategies, with potential applications to sports, missile/drone defense, and risk- sensitensitvve robotics applications, tailored for specific team dynamics, action spaces, and task specifications.
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ETHICS STATEMENT
LLM Usage. Large language models (LLMs) were sparingly used for (1) polishing the writing, (2)
code generation for visualization, and (3) some proof steps for Thm. 4.1 and Thm. 4.2 All codes and proofs are verified by authors.
Impact. This work examines strategic behavior in sequential decision-making and how deceptive tactics can arise autonomous agents. The findings have dual-use implications; all experiments are simulation-only and involve no human subjects or personal data.
REPRODUCIBILITY STATEMENT
The paper provides all the relevant details needed setup the experiments and reproduce the results. These artifacts include: 1) value approximation algorithm and the necessary assumptions to achieve the theoretical guarantees, 2) detailed description of the game setups in the appendix, and, 3) fully executable code with adequate documentation shared via github repository.
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APPENDIX
A CONNECTION BETWEEN PRIMAL AND DUAL GAMES
Here we use the infinitely-repeated game setting explain the connection between the primal and dual games and the interpretation of the dual variable p. Please see Theorem 2.2 in De Meyer (996) and the extension to differential games in Cardaliauuet (2007).
Game setting, In an infinitely-repeatee normal-form game with one-sided information, there is set of I possible payoff tables {A,);- A table is initially drawn from Po and shown to Pl, while P2 only knows Po. At stage & E [ the players draw actions from their strategies 1/2 and ç, respectively, under the current belief Pt. The actions are public, which trigger belief updates, yet the actual payoffs are hidden (or otherwise P2 may infer the true payoff easily). At the terminal T. average payoff is received by P1 from P2. This game is proven to have a value (Aumann et al. 1995) for finite and infinite T. Here only consider the latter case where the value is defined
T
(Po) = max min lim {nd S T T
B Bimpo T A.C).
(9)
Notice that this is special case the differential games we are interested in, where the incomplete information is on the running loss, the action spaces are discrete, and the dynamics is identity.
Primal-dual properties. Let the primal game be G(p) for p A(T), the dual game be G for á E RR, and let (m)) be the set of strategies for P1 and the strategy for P2. 1/: E (d.)) and Ç E (dd)). We note that Pl's strategy {n-H/z can also be together represented in terms of V.a. such that Σx- Tay pii] and nb1 Tt//pli], i.e., nature's 's distribution is the marginal of ग and Pl's strategy the conditional æ. Let G be the payoff to PI of type i for strategy profile (1), 5). From De Meyer 1996) we have the following results connecting G(p) and ()):
1. If is Nash for P1 in G(p) and á (p then (n is also Nash for P1 in GD)
2. If x is Nash for P1 in G* (p) and p is induced by æ, then E Ov* (p) and æ is Nash for P1 in G(p).
3. If Ç is Nash for P2 in G* (p) and p E (p), then ( is also Nash for P2 in G(p).
4. If Ç is Nash for G(p), and let p¹ == max,e G and p = lp .p......''T p'IT. then p (f) and Ç is also Nash for P2 in (r).
From the last two properties we have: If ( is Nash for G(p) and G' (0), then P max G ie., plb] is the payoff of type if Pl plays best response for that type to P2's Nash.
B PROOF OF THEOREM 4.
Theorem 4.1
(A splitting reformulation of the primal and dual DPs). The RHS of Eq. 3 can be
reformulated as
Σ ( ,D') BB Di(a,
min max (at)) (ok)
sa. n² Eu,,
=²= 7f uf )n eeve
(Pt)
on E 10,1), Σ am = 1, x Σ ampli,
ampli x
p
Vi,k E [4].
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Published as a conference paper at ICLR 2026
And the RHS of Eq. can be reformulated
I+1
min ("*),(2"))1p"
max Σ x* (V' :((x,t,,0))
(u*) k=1
s.t. "º E u, EV,, =
(P2)
2 e 10, 11,
Σ xx = p.
k
Proof. Recall that the primal DP is:
V- vo. p) min Ezpp, won' max (V-(to + T, 7,,'' (1, (l)) rl, {9:}
NEV
I ((2)) max (Va 7, ())) + =E TE (1) (1) u, du (10)
min
NE =min [ i(u)a(u,,' (')))du, (x)) Ju
{wr} Ju
where the last equality uses Isaaes' condition reduce e as an implicit function of u:
NE ((u,p'(u) max V+ + v,,'(u, (u, v), (u)) + TBe-p' (0sspp((()lll [e(2, (1)
v)].
(11)
Now we introduce pushforward measure on A(T) for any E C A(T): v(E)
= (WEp) ATu) du. Let up be the conditional measure on U for each p'. Then we have
min [ ('u)) = min f min (n.
f Uy(W) dh'
alu,p') mo (du) v(dp')
I min a(u, ('),p')((pp)) well
min
Sa(n)
[ a(z')u(dz').
min Jan
This leads to the following reformulation of Vg:
V=(lo.J0,P) = min JA(I) a(p')u(dp)
(12)
s.t. Exlp'==
Now we show that the RHS of Eq. 12 computes the convexificiation of ā(p') at p' = pp
To do so, we first show V, is convex on A(T:: Let two probability measures 1/ and u2 be the
solutions for (to, zo. pp) and V,(to, Zo. zo.p²). p²). respectively. For any 0 E [0,1] and nd = Op'
(1 - 0)pp, the mixture " == 0v + (1 f)v² satisfies SAin p'v(dp') SaUy p'v''' +
8) Jain (dp') pd and is feasible solution to Eq. 12 for p.. Therefore
V(T") S / JAI
alF)) 0V (1
Then we show V(p) S ä(p) for any p A(T): The Dirae measure 6p concentrated at p sat- isfies Sain (dp')) = p. Thus 5, is feasible solution to Eq. 12 and by definition V(p) S
SACD) 0(p')6,
ā(p).
Lastly, we show V is the largest convex minorant of ã: Let h be any convex function on A()) such that h((p) S a(p) for all p. Given p E (()), for any probability measure v that satisfies SaCTy p'v(d p. have
h(p) ht (AI)
p'v(dd')) S { h(p')v(dp') & S a(p')v(dp').
Jain
A())
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Since this inequality holds for arbitrary including the optimal ones that define (p)) through Eq. it follows that hp)) I (p)). With these, (--) in Eq. 12 is the convexification of a(-).
12
Since convexification in ((T) requires most vertices, 1/ that solves Eq. 12 is I-atomic. We will denote by {or kr the set of "splitting" points that have non-zero probability masses according to v², and let X := v" (p"). Using Isaacs' condition, arg min a(u, p) is non-empty for any E A(),, and therefore each p is associated with (at least) one action in arg min p"). As a result, {nr} also concentrated on a common set of I actions in U. Specifically, denote this set by [u")ke[]; we should have Or: == mn(")) A [p]]].. Thus we reach P1 The same proof technique can be applied the dual DP derive
C PROOF OF THEOREM 4.2
Theorem 4.2
For any (1, x. p) E [0, X x A()), if-5 hold, then for any e > O, there exist
C> 0, and
(0, 7"), such that
max J(,, (nl,)))) v(t,,,)) CE2
(13)
Similarly, for any (1, a. p) E [0, x X' x R!
0
max (m
J' - Se
(14)
C.1 SETTINGS
We recall all necessary settings for the proof.
From Thm. the Bellman backup for G, is which can be written an operator Tç:
Σ Ext.,tt,,p')
(t +T, V xt,
min (A#),(**)
(15)
where (t, p²) min m V₁ (t + T, + u,) + TEinpe (4 u, v)] is the
non-revealing value at (t,x,p" and El= A =1 Ek= X²p² p, p² E A()), Vk E [4].
From Cardaliaguet (007), the value V of the original game G is the unique viscosity solution the following Hamilion-Jacee Jacobi equation
Vew + H(I, x, Vzw) = 0,
(16)
with terminal boundary wTT, x. p) El plilg (z). and is convex in p.
C.2 PRELIMINARY LEMMAS
The following lemmas will be used in the main proof.
Lemma C.1 (Uniform Convergence (Thm 3.1 of Cardaliaguet (2009)). Assuming (AI-A5) hold, the
map V, converges uniformly to the value function on compact subsects of [o, T) x X A(I), in the following sense:
lim
Vr .D) = (tt,2,)) V(t,2,p) [0. x X x A(T).
T-00 by +t, xf +2, D¹ pp
Proof. Proved in Cardaliaguet 2009 for both V- and V
Lemma C.2. Let (lo) (v: Noo T] V} denote set of open-loop controls for P2. Then, given f. a constant control on [k, 1++1), a time- varying control u E (L), there exists a w such that
w E (xf(x, u, V)
(17)
where Co denotes the convex hull.
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Proof. Let
1 (k11 S. f(z, u, v(s) ds
(18)
We show that as defined above lies in the convex hull of the set f(x u, V).
Let S = f(x u, V) = [/(x,, u, 8) U VEVV V). For any time 8, since v(s) E V, we have:
flz,u, s for all E [t., +x+1]
(19)
Case 1: when v(-) is piecewise-constant. Assume first that v(-) is piecewise constant on the partition [t,, tx+1]:
U Nj,
[N = Alj,
Σ
Alj =
j=1
and r(s) Dj on Nj. Then
fu+1
((s)ds = Σ (x,
Divide by to get
m
m At (t Σ (r,
Atj
where Oj
t j=1
Øj 2 0 and 21=1 95 = 1, so w is finite weighted average of points in S. Hence,
Co(S) Caf
(20)
Case 2: general measurable case. For a general measurable v(-), we can approximate it by piccewise-constant (-) such that
fl f(z, u,,, (z in L' ([t++
This is true because f(x, .) is bounded and measurable through v(-).
Define
Wn := 5 1//0001 f(x, u, (s) ds.
fett
From Case
each Wn E Co(S).
Moreover,
| ///h" (f(z,u,u(s))) - f(z, ds
1 (011
[w, w/
< 1/m
(f(z,u, f(x u v(s)) ds +
So Wn, -> w
Finally, S is compact because V is compact and f is continuous. Hence, Co(S) is compact and therefore closed. Thus, the limit of w,, E Co( also belongs to Co(S):
Co(S) Cof(x, (,,,u,11
(21)
Lemma C.3 (Local Quadratic Error). For any {0..... L - 1}, any x, a fixed control u E U4 any control v E Vtg)), there exists some E Cof(x, V). such that
Ix (x Tw)) s 7
(22)
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Proof. We have, from the ODE,
X (tx+1) = X 1 = =I L. f(X(s), u(s)) ds.
Mkll
(23)
From the definition 18 of wt
x =+ x l(
(/-
[(,, u())d,
(24)
Subtract 24
from 23
px+1
X(1e+1) (xr Tw)- = (I(((s), - (x,,, ds
= (I((()) s(x, u, o(5) ds
X(4x-1) -(x Tw)
[(X((s) - f(x, u, v(s) ds From Lipschitz continuity of (x u())) f(a, u, v(8)) s ()(x(s) x/.
X(I++1) (xc + & { 1x+1 Ls(X(s) - x ds.
From boundedness of f: Let |1/1 F. Then for & [x,,
X(s) ( ((X(r), dr S f (5(-) dr dr Jmn
1" Fdr F(s 1x),
which leads to
((I++1) (x TW) < f Lg. F. (s (x) ds Ls F [ dr Lg.
J
which yields equation 22 with C = L...F.
Lemma C.4 (Grid Error). There exists some constant C > independent of such that, for any k E (0...., L}, RRd and p (T)), have
max (tu, x.p; (. {) (ts, C(T tt) CEZ
(25)
Proof. T/ ( be the random strategy associated with the feedback strategy (u² and) in the game G,, and Ç E Z. Then, for any {O...., L.J, x E Rdd, and p E A(T), claim that:
J(4,, x. ( V+(4,, C(L k)7² CEZ
(26)
We use backward induction to prove the equation 26 At = L, the inequality holds trivially as there is no time left in the game and the error term goes to 0. Then, assume that (26) holds for + 1:
max J(Le+1, J., (.l.)).)) V, (1e+11,2, C(T (k: 1))
(27)
(E2(()+1)
Now, to prove for any k let us analyze what happens on [Kx, 41-1). From the feedback given by P for fixed (xr,)) p) we have:
Aua A Ent
x x (
x
with probability (1)r(1)/l
where
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(np..- } is the continuation of the random strategy from time 4 associated with the same
feedback but starting at (fx+1, x'.'),
ç² E 2(t++1) is the continuation of the non annaciicipative Ç after w¹ chosen.
Then, using Lemma
C.3 there is some w E Cof(z, u², V) such that:
|x' - (x + rwy) s C1T²
(28)
From the definition of J,
J(k,, F, (.
plJ Ent R 19 xx [ (x + (" l (x(()), (1h. (o).)s)) ds)
(r. 96 (x ( ((t
dat1 (((())) ,(w()) ds
S 4 (1)))) as
1+1
Σr'
(ne [or
(xteeerrt frhe ( ((w))) ds
x
max
C'E
1)
J=1
f (x(6)) as
Combining the above result with (7 we get:
J(l,, v, (.p) S
X V= C C(L 1))7?
Finally, applying 28 and assuming Ci-Lipschitz continuity of V,, arrive at:
21,,, (nl.
x V +
Colod (x Tw))l+
(k--(k-) 1))r
X + +CC +k+ 1))r
1+1
Σ
X
V C + C(L (k +))>²]
max
(eCo/
1=1
$ VV(,,, + (C, 2)7 + C(I. (k + 1))
The term 27 arises from two approximation steps: first 72 from selecting an approximately optimal convex decomposition in the definition of (tx, x. p), and the second 72 from choosing u² to be only optimal for min,, max,( (-) so that they are Borel measurable with respect to (x.p).
Then, choosing C' (C₂ C2 2) satisfics 26
Since we have, L = T/T, and t = kt. We get
max J(t,, x. p; In 9) S V=(t,,,,)) + C(T 12)r CE2
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Note that compared with the proof provided in Cardaliauuet (2009) Theorem 4.1, proof incorpo-
rates instantaneous costs and clarified the stage-wise ( error in
C.3 PROOF OF THEOREM
Proof. Our proof focuses on the primal game. The same technique can be applied to the dual game. First, it is easy to see (t,,,, I maxçez Jt, P; (m,,)),)) (,5) because (nk,7 is not necessarily NE in G.
Next, we need to show
0 S max [t,, 2. 6E2
((t, E
(29)
ie., applying [nl,, (which solves G2) to G will yield a value not far from the true value of G.
Let us fix some t',, x', and p', and some such that E [tt-1, tx). Then,
x't',''p'' (1..).))))) = max J(t x' x',''''''' (vl.r.))) (EZ(1)
max (E2,(r)
max J(tk, x. p': ( 1. 7|10]bb
max
1z-z' FF (E2(te)
where (n/_1} is the strategy at (tg, x, x,)') is a bound on f. and ||0|/ term due to bounded
instantaneous cost. Hence, from Lemma C.4
max J(L x.. .p', (n.,)) ..C) () (E(P)2
max V1 v, 2') O(T) 1r-r'SSF
Here, O(T) absorbs C(T t4)T =|11]100. Together with 7 T/L 0 and Lemma C.I max V- (Ix, v. 1/) converges to V(L', a ,/) and ((r) + 0
Therefore, for any > 0 there exists T² > 0 such that for all T E inequality 29 holds.
D PREDICTION ERROR OF VALUE APPROXIMATION
Here we show that CAMS shares the same exponential error propagation as in standard approximate value iteration (AVI). The only difference from AVI is that the measurement error in CAMS comes from numerical approximation of the minimax problems rather than randomness in state transition and rewards. To start, let the true value be V(I, pt. Following Zanette al. (2019), the prediction error c = max [V((..)) V(t, p)] is affected by (1) the prediction error 2++7 propagated back from + т, (2) the minimax error e minmax caused by limited iterations in solving the minimax problem each collocation point: emmimaaz max |V (t, x, p) - (tt,, x, x)), and (3) the approximation error due to the fact that (t,,,.) may not lie in the model hypothesis space Ve of Ve: app max,,p min [V(x,,)) xI,,,
Approximation error. For simplicity, will abuse the notation by using in place of (z, p) and omit time dependence of variables when possible. In practice we consider V, neural networks that share the architecture and the hypothesis space. Note that V+(()) = V(T,-) analytically defined by the boundary condition and thus PPP -hiiss = 0. To enable the analysis on neural networks, we adopt the assumption that V is infinitely wide and that the resultant neural tangent kernel (NTK) is positive definite. Therefore from NTK analysis (Jacot al. 2018). V' can be considered kernel machine equipped with kernel function (() x(1) =
defined by feature map X' + Rdd Given training data S = {(x()) V(0)) let r(x) == ()) (x),, Raj == r(a(1) ,.)), Vs [V(1)... ...., V(N)JT Ts (d(x(11), (x)()))), and ws = $5($505) -Vs be model parameters learned from S. then
((x) = rxx) R- R--V = (0(x), ws)
(30)
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is linear model in the feature space. Let (0(2) : r(x) ( let S := arg ming [((x),, ws} - (z)) and wi := Wgt i.e., wi represents the best hypothetical model given sample size N. Since N is finite, the data-dependent hypothesis space induces an approximation error tt" == max, ( - [(+)l- From standard RKHS analysis, we have G x N-
R- and C == max, ||00()) ||2. Further,
Error propagation. Recall that we approximately solve P1 each collocation point. Let 2 {A, u, of be the collection of variables and ă be the approximated saddle point resulting from DS-GDA. Let (tt, 2) be the approximate value at (t, and let (tt, 2²) be the value at the true saddle point z". Lemma D.I bounds the error of V (1,, xr, Z): Lemma D.I. max, (1,, v(1,,,="") S bios + eqnsmmas
Proof. Note that E A Then
max max xt, V x, 2*)l S max IV x,, Vt, xt,,,,2)| max ((t,,,,") vt,,,,2") 2")l
minmar + maxx
Σ ('( V( ,x')
minmar
bias -t-T
(31)
Now we can combine this measurement error with the inherent approximation error " to reach the following bound on the prediction error hiias
Lemma D.2. max, Vx(x) (tt,, xt) I (e (eminmaa + GC7 + CPP) c,
Proof.
max Vi(z) - [tt, & max [V(x)) (d(z), w0)1 + max (0(z)) w²) V(t,z))
2
I max ((d()) V(t,x) V(t, (t,, + max (()(()) V(t,z) x(x) T wnt) + c:P
(32)
S C(e minmar + El+
bios
+ opp) + E
Lem. D33 characterizes the propagation of error:
Lemma D.3. Let avp < cPp,, eminmox & minmaa and C. S for all E (T). If of = 0, then
hios < <TC (c + C(Eminmar + c))
Proof. Using Lem. D.2 and by induction, have
blas < PP + Ci + (PP) 1 - C S STC (E + Ol ePP)).
(33)
1-C
We characterize the computational complexity of the baseline algorithm through Thm. by taking into account the number of DS-GDA iterations and the per-iteration complexity:
D.4
Theorem D.4. For fixed T and some error threshold 8 > O, with computational complexity of C 12--8-- 18-2), CAMS achieves
[o(a,p) V00, p) 8
(34)
(XA))
D.3 and using the fact that aP 1122, achieving prediction error of d at = 0 requires N O(C2TT28-2) CAMS solves TN minimax problems, each requires a worst-case O(e iterations, and each iteration requires computing gradients of dimension O(1²). considering the dimensionalities of action spaces as constants. This leads to a total complexity of C2T 12-2)
Proof. From Lem.
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E MULTIGRID ALGORITHMS AND RESULTS
Fig. shows typical 2-1evel multigrid (Full-Aprrxximation Scheme FAS) approach. As discussed, FAS has four steps, namely: (1) restriction of the fine-grid approximation and its residual into the coarse grid (red arrows in Fig. 7) (2) computation of the coarse-grid solution by incorporating re- stricted fine-grid residuals; (3) computation of the coarse-grid correction; and finally, (4) prolongation of the coarse-gric grid correction the fine-grid (shown by the blue arrows in Fig. 7). This can be further extended to n-level multigrid by recursively reducing the coarse-grid gridd size until the desired coarsest grid reached. Alg. presents the n-level multigrid algorithm.
0.25
0.5
0.75
Fine Grid
Restriction
Prolongation
Coarse Grid
0.5
Figure 7: Illustration 2-level multigrid method.
Algorithm 1: n-Level Multigrid for Value Approximation
Input: kmaxs kmin, O (minimax solver), I" (time horizon), N (number of data points), R (restriction operator), (prolongation operator)
Initialize: T 20, ... , T 4], VI (2- Kmaa ..... ,2 2- t 2
Initialize: Value networks V7, VIE TT, vI (2 kmax ...., 2-kmm }, policy set +- Ø
while resources not exhausted or until convergence do R + Ø, EL 0, S S <+0; Ø;
Initialize coarsest-grid correction networks e,, VLETL,
Sit] + sample N (t,z,p), Vt Time
// down-cycle
for Is t kmax down to kmin + do Compute target via 0t (init. with n, if [[kk] # Ø), and store updated policies n, in
[[ke), V1 E T Compute residuals r VtE Th; if k * kmax then
r t pk Rr R+ Vte Th;
Store r in R[k]]
for 1 + T L to do
// coarsesollve backwards in time
et - + (-(()+)) R Rr kmi+1, Store of in EL: el to et;
11 eh 0, of Ø
1/ up-cycle for kmin + 1 to kmax do + VLE
Update V.7 + 4 + +
// post smoothing (for all t's and 1's)
target, m + ('(())))) (initialized with m.)
Fit V target and replace h, 11[0]:
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In Fig. 8 compare learned trajectories via the multigrid approach against the ground truth. The learned trajectories closely resemble the ground truth as successfully concealing its payoff type until critical time. In Fig. g we visualize the NE trajectories of P2 by solving the dual game.
(a) 4-stage
(c) 10-stage
(c) 16-stage
P2
... GT
0
0
-1
0
0.5
0.5
0.5
0.5
-0.5
0.5
X
Figure 8: Comparison of trajectories generated using value learned via multigrid method vs the ground truth.
(a) 4-stage
(b) 10-stage
P1 P2
GT
0
-1
0.5
0 X
0.5
0.5
0.5
X
Figure 9: Trajectories when P1 and P2 play their respective primal and dual game. P1's actions result of the primal value function whereas P2's actions are a result of the dual value function. Both primal and dual values learned using multigrid approach.
F ANALYTICAL EXAMPLES
In this section, we walk through the derivation of analytical NEs for two problems: Hexner's game and zero-sum variant of the classic beer-quiche game. The former is differential where players take actions simultaneously; the latter is dynamic and turn-based based. Both have one-sided payoff information and finite time horizons. These examples are reproduced from Ghimire et al. (2024) with permission.
F. HEXNER'S S GAME: ANALYTICAL SOLUTION
Here we discuss the solution to Hexner's game using primal and dual formulations (i.e., P and P2 differential game proposed in Hexner (197)). Consider players with linear dynamics
3; = A + Bas
for i 2, where 2t) &d are system states, (t(t) E 2 control inputs belonging to the admissible set U, A,, B, E xd,, Let 0 E {-1,1} be Player I's type unknown to Player 2. Let Pa be Nature's probability distribution of 0. Consider that the game is to be played infinite many times,
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the payoff' is an expectation over A:
J(u, w2) u2) = Ea I/ (luille - -Iluallka )et +
(35)
| (T) 2 (iEITTT || =0 (c
where, 2 E IRd, Rí and R2 are continuous, positive-definite matrix-valued functions, and (T) and (T) are positive semi-definite matrices. All parameters are publicly known except for 0, which remains private. Player I's objective is to get closer to the target a0 than Player 2. However, since Player 2 can deduce 0 indirectly through Player I's control actions, Player I may initially employ non-revealing strategy. This involves acting as though it only knows about the prior distribution pe (rather than the true 0) hide the information, before eventually revealing A.
First, it can be shown that players' control has ID representation, denoted by 0, E. IR through:
U, R, B K, R B. K, K,
for i 1, where é A,d; with boundary condition (T")) 1, and K; - A K; K;A; K B B K₁.
Then define a quantity d, as:
d = -= " K,B.R, B K. 2,2.
(36)
With these, the game can be reformulated with the following payoff function:
/
T
(t,.0,,02 Eg
(01(T) ()) - - - d))
(37)
where d1, d2, po common knowledge; A is only known to Player 1; the scalar 0g (resp. 8₂) is Player I's (resp. Player 2's) strategy. We consider two player types E {-1,1} and therefore Po (2).
Then by defining critical time:
4, arg min [ (d)(s) d2(s))ds,
we have the following equilibrium:
(s) (2(s) =0 Vs E [0, tr] (s) = 02(s) = E E
(38)
(39)
To solve the game via primal-dual formulation, we introduce few quantities. First, introduce time stamps [T>]?", as roots of the time-dependent function d1 d2, with To = 0, T24+1 fr, and Ter41 T. Without loss of generality, assume:
di - d2 0
Vt E (T2k, T24+1) Vk: 0, O,...TT
(40)
d1 - d2 2 0
Vt E [T2x- 1, T26l Vk = I,...,r.
(41)
Also introduce D2 = 17 Te+t (d1 - d2 ds) and
D = { D + DR.
if Dx + De < 0 otherwise
(42)
0,
with D2r+1 0.
The following properties are necessary (see Ghimire et al. 2024 for details):
1. [ 24+1 1 (di d2)ds Σd 24 Dj < 0. Vk 0,... 2g:
2. f (dh - dz)dss E2gt1 Dr > 0, Vk = 2g + 2, .... 2r 1;
3. D29t2 + D2g+1 > 0;
4. Dg 0, Vk < 2q +
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Primal game.
We start with V(T, p) 0 where p po[1] Pr(0 -1). The Hamiltonian is as
follows:
(v) min max Eg [00 - -0)e -02--0)2
0
0
4p(1 - (d1 - d2).
The optimal actions for the Hamiltonian are 0, 02 = 1 -2p. From Bellman backup, we can get
V(Tx, = 4p(1 D)
Therefore, at T I, we have
V(T2++1,P) Vexp (V( + 4p(1 -p)D2q+1
= =Verp (pp(1 () + D24+1))
Notice that D2042 + D20+1 > (property 3) and Dy < for < 2q + (property 4), T2g+1 is the first time such that the right-hand side term inside the convexification operator, i.e. 4p(1 p) + D2y+1), becomes concave. Therefore, splitting belief happens at T24+1 with p² = and p² = 1. Player I plays 0, = -1 (resp. 0 = 1) with probability if = -1 (resp. 8 1), i.e., Player 1 reveals its type. This result is consistent with Hexner's.
Dual game.
To find Player 2's strategy, we need to derive the conjugate value which follows
max;e (1,2) p[a]
Vt 2 T2g+1
212] D. (1 -11--111
V2 < T24+1, 4D, S p[1] [[]]]-p2]- S -4D;
(*(1 =
AD,
D[1]
VI < 1, P[1] -|2] 2 4D;
2(2)
Vt < T2 D[1] [(2) < 4D;
Here á E Pr, V(.. Po) and V(0, po) 4p[1]p/2] For any particular p. E A(2), from the [2] = 4p. 11p D and p[1] 6[2] = 4(p.
definition of subgradient, we have
(.]))oo Solving these to get P [2p 4p. [1] Do) Therefore $[1] p[2] = 4Do
[1] E [4Dp, -DD.. and
V'(O,)) -11] -Do (1
1 p 1(2] 2
4D.
Notice that (t, is convex to P since Do < 0 (property 4) for all t E (0 Therefore, there is no splitting of during the dual game, i.e., 02 2p. This result is also consistent with results in Hexner (1979).
F.2 EXAMPLE OF A TURN-BASED GAME
We present zero-sum variant of the classic beer-quiche game, which is turn-based incomplete- information game with a perfect Bayesian equilibrium. Unlike in Hexner's game, Player 1 in beer-quiche game wants to maximize its payoff, and Player 2 wants minimize it; hence, Vex becomes Cav. We solve the game through backward induction (from t = 2, 1, 0) of its primal and dual values (denoted by V and /* respectively). Players and 2 make their respective decisions at t= O and = 1, and the game ends at 2. The state at time / encodes the history of actions taken until t.
Primal game: First, we compute the equilibrium strategy of Player using the primal value. At the terminal time step (t = 2), based on Fig, 10 the value for Player is the following:
4pr 2 if (B, b)
V(2, x.
PT
if r. = (B, d)
(43)
2pr
1
if (Q,b)
2 - 2pT
if (Q,d)
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(1, 11,-1)
(0,0)
(-1, 1)
(2, 2)
IQ
4
Tough
Weak
B
TB
b
y
(2, -2)
(1,
(-2, 2)
(0, 0)
Figure 10: Zero-Sum Beer-Quiche Game
At the intermediate time step (t = 1), it is Player 2's turn to take an action. Therefore, the value is a function of Player I's action at = 0 and Player 2's current action. And for the same reason, the value is not concavification (Cav) over the RHS term.
V(1, x. p) = min V(2, u).
(44)
eE{b,d}
We can find the best responses of Player for both actions of Player 1. This leads
PT
if B, 3pr 2 2 0
(v* d)
4pr 2
if = B, 3pr
2 <<6
(r* =
(1,,,p)
(45)
2 2pr
if Q, 4pT 3
(r* d)
2pr 1
if = Q, 4pr 3<0
(o* = b)
Finally, at the beginning of the game = 0), we have
V(0, 0, p) Cav
V(1, u,
(46)
(E(B..)
Cav is achieved by taking the concave hull with respect to pr:
5pr/2-1 -1 if pr < 2/3
V(0,0,p) = pr if 2 2/3
(47)
When E 2/3),
V00,0.p p) = Amax (L,, up + (1- - A) max V(1, u, p²),
where p² [0, 11 /2/3, 1/3]" and Ap (1 A)p² p.
Therefore, when pr = 1/3, X 1/2, Player I's strategy is:
Pr(u Q[7) = Ap [1]
Pr(u = Q(W) = Ap
r[ O,
3/4,
p|2]
(48)
Pr(u B(T)= (1 - A)P"[]]
(1 - 1-)ppp22 (2
= 1, Pr(u = BW)
1/4.
p[]]
p[2]
Dual game: To solve the equilibrium of Player 2, we first derive the dual variable ý E
for = [1/3,2 2/3)T. By definition, fT defines the concave hull of V(0, 0, p). and therefore we have
[1/3, 2/33p V( 0. p) -1/6
(49)
0.. 1] = (0. (0. 1)) = 11
This leads (3/2, -17T.
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At the terminal time, we have
V x2, (in gr - gw (x)}
min - 2. + 2] if = (B.D)
min(pl -1p
if = == (Q,b)
(B.d)
(50)
1amppp11.ppp +1}
min (f(1).022 -] if dQ
At f 1, we have
(1 u. Cavp max (2, (u, .)
(51)
When = B, the conjugate value is concave hull of piece-wise linear function:
1 - 0
2 > [[1 -
(v*
p2
E [p[1 2, p[1 - 1)
(v* = b)
(*1 B) 3
p1
-
p2
E [p[ 4, 11
-
(v*
2
+
if
< < -4
(0" b)
if
2 11
(v" d)
2/3p [1] + 1/3p12] 2/3
if
E [0
[[1
(mixed strategy)
2 +
<
(u* = b)
(52)
For ý p= 3/2, -1"" which satisfies p|2] E [p[1]
4, p|1] 1). Player 2 follows a mixed strategy
determined based {A1, X2 da} E (3) and p E R for 1, such that:
(i) At least one of ه³ for j 1, 2, should satisfy p|2] = [[[]] 1 and another p p[1] 4. These conditions are necessary for (1, B, p)
to be a concave hull:
3 V*(1, Σ X max (B).
(53)
(ii Σラ Apps p.
These conditions lead to A 1 1/2 and X2 + A = 1/2. Therefore, when Player 1 picks beer, Player 2 chooses to defer and bully with equal probability.
When Q. we similarly have
P[1]
if
11
(u* = d) (mixed strategy)
V((1
2
2, p[1 22)
(54)
2 +1
1 2
(u* =-=bb b)
The derivation of the concave hull when [12] E [+[]] 2, 2. + 2) is omitted, because for P = [3/2,- -1]"", V*(1 (. 6|2] + 11 = 0 while v- = b. if Player picks quiche, Player 2 chooses to bully with a probability of
G HEXNER'S GAME SETTINGS, BASELINES, AND GROUND TRUTH
G.1 GAME SETTINGS
The players move in an arena bounded between [-1, 1] in all directions. All games in the paper follow 2D/3D point dynamics as follows: ž = An, + Buj. where zy a vector of position and velocity and uy is the action for player j. Note that we
and in the optimization problems P1 and Pg to represent player and player 2's actions respectively. The type independent effort loss for each player j is defined as (u,) u. R2u,, where R1 d di.000 0.025) and R2 di d 0.1). For the higher dimensional case, R1 di ag (0.05, 0.05, 0.025) and R₂ di (0.005 0.05, 0.1). Note that, in the incomplete information case, P1 is able to get better payoff by hiding the target because P2 incurs higher effort cost, and hence cannot accelerate as fast as P1.
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G.2 GROUND TRUTH FOR HEXNER's GAME
For the 4-stage and 10-stage Hexner's game, there exists analytical solution to the equilibrium policies via solving the HJB for respective players.
U - -
based the reformulation outlined below in which players' action B, IR become 1D and decoupled from the state: where 1 is ttte-transition matrix that solves b, A, with (T) being an identity matrix, and K is a solution to continuous- -iiinuuus--tmeeeeeee differential Ricatti equation:
K2 -1 Kj K5 K3 ByR B. K₁,
(55)
Finally, by defining
dj = = Ry BJ KJ0,2 x,z
and the critical time
4 = arg min ( (dl - d2(
and
(t()) = f0, t t0,
0 t (tr,
As explained in Sec 6 P1 chooses 0 = O until the critical time f, and P2 follows.
Note that in order to compute the ground truth when time is discretized with some t, we need the discrete counterpart of equation 55 namely the discrete-time Ricatti difference equation and compute the matrices recursively.
G.3 OPEnSPIEL IMPLEMENTATIONS AND HYPERPARAMETERS
We use OpenSpiel (Lanctot et al. 2019), a collection of various environments and algorithms for solving single and multi-agent games. We select OpenSpiel due to its ease of access and availability of wide range of algorithms. The first step write the game environment with simultaneous moves for the saage-game and the multi-stage games (with 4 decision nodes). Note that to learn the policy, the algorithms in OpenSpiel require conversion from simultancous sequential game, which can be done with a built-in method.
In the single-stage game, P1 has two information states representing its type, and P2 has only one information state (i.e., the starting position of the game which is fixed). In the case of the 4- stage game, the information state (or infostate) a vector consisting of the Pl's type (2-D: [0, 1] for type- 1, [1, 0] for type-2), states of the players (8-D) and actions of the players at each time step (4 x 2 x U). The 2-D "type" vector for P2 is populated with 0 as has no access to Pl's type. For example, the infostate at the final decision node for type-1 1 P1 could be [0, 1 (8) 1 (U) (U)
(U) d2 0()) 0(0)), and
[0, O, (8), (U) 1 12)))
1 (()) o(U) for P2, where ug, d, represent the index of the actions ht decision node, kt 2,
The hyperparameters for DeepCFR is listed in table
G.4 JOINT-PERTURBAN
SIMULTANEOUS PSEUDO-GRADIENT (JPSPG)
The core idea in the JPSPG algorithm is the use of gradient of the utility to update players' strategies. By perturbing the parameters of a utility function (which consists of the strategy), an unbiased estimator of the gradient of a smoothed version of the original utility function is obtained. Computing pseudo-gradient can often be cheaper than computing exact gradient, and at the same time suitable in scenarios where the utility (or objective) functions are box"" or unknown. Building top of pseudo-gre gradient, Martin & Sandholm (2024) proposed method that estimates the implication of this is that instead of multiple calls estimate the pseudo-gradient, we estimate the
instead of computing the actual
with respect to all players' strategies simultaneously. The
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Table Hyperparameters for DeepCFR Training
Policy Network Layers
(256, 256)
Advantage Network Layers
(256, 256)
Number Iterations
1000 (100, for U 16)
Number of Traversals
5 (10, for U = 16)
Learning Rate
1e-3
Advantage Network Batch Size
1024
Policy Network Batch Size
10000 (5000 for = 16)
Memory Capacity
1e7 (le5 for U = 16)
Advantage Network Train Steps
1000
Policy Network Train Steps
5000
Re-initialize Advantage Networks
True
pseudo-gradient in a single evaluation. More formally, let f: TRd - IR be vector-valued function. Then, its smoothed version is defined as:
(x) Ex f(x oz),
(56)
where f is a d-dimensional standard normal distribution, a * E IR is a scalar. Then, extending the pseudo-gradient of scalar-valued function to vector-valued function, have the following pseudo-Jacobian:
Vf, (x) = EEzn f(x + oz) &
(57)
where & is the tensor product.
Typically, in game, the utility function returns utility for each player given their strategy. Let : Rxd + R" be the utility function in game with players, where each player has a d- dimensional strategy. Then, the simultaneous gradient of u would be a function v RRad -> Raxd That is, row i of vuu) is the gradient of the utility of the player with respect to its strategy. v; V,u, As result, we can rewrite v concisely as: v = diag( Vu), where the Jacobian. With these we have the following:
vx = diggg( (x),
= diag
Ezzz
u, xx (x oz) 0z)
(58)
Ezz -iag
u Ux oz) 8z
Evaaa up oz) Oz.
where is element-wise product and result of the fact that di ag & b) b. Hence, by evaluating Eq. 588 once, we get the pseudo-gradien associated with all players, making the evaluation constant as opposed linear in number of players.
Once the pseudo-gradients are evaluated, the players update their strategy in the direction of the geeuo-gradient, assuming each player interested in maximing their respective utility.
JPSPG Implementation. In games with discrete-action spaces, where strategy is the probability distribution over the actions, JPSPG can be directly applied to get mixed strategy. However, for continuous-action games, a standard implementation would result in pure strategy solution than mixed. In order to compute a mixed strategy, we can turn into neural network as strategy with an added randomness that can be learned described in Martin & Sandholm (2023) 2024). We similarly define two strategy networks for each player, the outputs of which are scaled based on the respective action bounds with the help of hyperbolic tangent (tanh) activation the final layer. The
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input to the strategy networks (a single hidden layered neural network with 64 neurons and output neuron of action-space dimension) are the state the player and a random variable whose mean and variance trainable parameters. We follow the architecture outlined by Martin & Sandholm Goofspiel. We would like to thank the authors
(2024) in their implementation of continuous-actio for providing example implementation of JPSPG normal-form game.
In the normal-form Hexner's game, PI's state x1 = { U1, type), and P2's state X2 = {m2, U2)- Ty and yi denote the x-y coordinates of the player i. In 4-stage case, we also include x-y ve- locities in the state and append the history of actions chosen by both PI and P2 into the in- put to the strategy network. As an example, PI's input the very last decision step vector [21, M1, Uei Uy, 22: 1/2, U22 Py2: type, U1 "1,, d1, di,, U2 U2,,, d2 d2,, U3, 43y da.- ds,, E i2i, where Uj and dj represent actions of Pl and P2, respectively, jth decision point. P2's input, on the other hand, is the same without the yppe information making it vector in R
G.5 SAMPLE TRAJECTORIES
Here we present sample trajectories for three different initial states for each P1 type. The policies learned by CAMS results in trajectories that significantly close to the ground truth than the other two algorithms.
CAMS
Deep CFR (IAI = 9)
1
1
PlrGool
(Goal-2)
-0.5
0
0.5
1
-0.
0
0.5
X
X
Deep CFR (IA 16)
JPSPG
Y 0
Y
0.5
0.5
0.5
0 X
0.5
X
Trajectories generated using CAMS (primal game), DeepCFR, and JPSPG. The initial position pairs are marked with same marker and the final with star. The trajectories from CAMS are close the ground-truth while those from DeepCFR and JPSPG not.
Figure 11:
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G.6 VALUE NETWORK TRAINING DETAILS
Data Sampling: At each time-step, we first collect training data by solving the optimization
problem (P or P2 Positions are sampled uniformly from [-1, 1] and velocities from [- Rr Dt.
computed as
1. Umaaz where Umaar is the maximum acceleration. For the unconstrained
game, Umar 12 for both P1 and P2. For the constrained case, Uzmos = 6, Uymas = 12 for P1 and Ugmaz 6, yme= = 4 for P2. During training, the velocities are normalized between [-1, 1]. The belief p is then sampled uniformly from [0, 1]. For the dual value, we first determine the upper and lower bounds of by computing the sub-grate aub-gradient O₂ (tooro and then sample uniformly from [ (5)..
Training: We briefly discuss the training procedure of the value networks. As mentioned in the main paper, both the primal and the dual value functions are convex with respect to p and p respectively. As a result, we use Input Convex Neural Networks (ICNN) (Amos et al. 2017) as the neural network architecture. Starting from T r, solutions of the optimization problem P1 for sampled (X, p) is saved and the convex value network is fit to the saved training data. The model parameters are saved and are then used in the optimization step at 2T. This repeated until the value function at f 0 is fit. The inputs to the primal value network are the joint states containing position and velocities the players X and the belief p.
The process for training the dual value is similar to that of the primal value training. The inputs to the dual value network are the joint states containing position and velocities of the players and the dual variable p.
H HYPERPARAMETER SWEEP FOR PG BASELINES
Here we report a sweep of hyperparameters across different learning rates and entropy coefficient for the PG MMD and PPO algorithms. Specifically, we run the algorithms with learning rates of
{2.5e- - 5, 2.5c 4, 2.5e - 2, 2.5c 1}. and entropy coefficient of (0.01, 0.05, 0.1,0.2) We also run RNaD with all four learning rates. The sweep is reported in Fig. 12
1 SOLVING HEXNER'S 'S GAME VIA MPC
Here we solve a 2D Hexner's primal game with / = 2, K 10, and other settings following App. G With these settings and using the equilibrium in G.2 the true type revelation time is f, = 0.5 second. We directly solve the minimax problem by autodiffing the gradient of the sum of payoffs from the 20 paths of the game tree. At each infostate along each path, Pl's strategy is modeled by neural network that takes in (4, p) and outputs / action prototypes and an /-by-/ logit matrix that encodes the type- repeddepndeent probabilities of taking each of the action prototypes, and P2's best response is modeled by separate neural network that takes in (t. x, p) and outputs a single action. With a DS-GDA solver, the search successfully converges to the GT equilibrium. Fig, [3] illustrates the NE trajectories for one particular initial state and the corresponding belief dynamics.
J A DIFFERENTIABLE I 11-s--11 AMERICAN FOOTBALL GAME
We model single running/pass play 2p0s1 game between the offense (P1) and defense (P2) teams. Each player is a point mass with intteriit dynamics on a 2D plane. Time is discretised with maero step A/- r and K = T/T steps, and each macro step is resolved by naab semi-implicit Fuler substeps for stability.
State, controls, and bounds. Let N = 11 be players per team. offense positions and velocities are X(1, V(1) E I Nx2, defense x(2) 122) E Nx2 We pack them into a state vector & = [x(, (2())(2) v E R&N At each step, the teams apply accelerations U1, U2 E Nx2 (stacked later as [uy; Ug] E RAN). Kinematic saturations enforce a playable box of half-width BOX POS and box- lox-limited speeds and accelerations BOX, VEL, BOX ACC by componentwise clamping after each substep.
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Hyperparameter Sweep
HMD, RNDD
0.01
Coeff:
Coeff: 0.1
0.2 0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1,0
0.2
0.4
0.6
0.8 1.0
Traininn steps x
1e5
Training stops x
165
Training stops
1e5
Traiiiing steps x 105
Figure 12: Hyperparameter sweep for the baseline PG algorithms. RNaD, being non-standard PG algorithm, doesn't 't use entropy coefficient the way MMD and PPO do; hence, we copy the same plot across different entropy coefficient value for reference.
Differentiable tackle dynamics (smooth contact and merge). During substep with duration öt = s/Dsab. we first compute soft, pairwise "stickiness" weight between an attacker and defender j:
my = o Heaatto (r - ||x - x(3)11)).
where (2)) = 1/(1+ (1ll+e=)) e)*), Atuckle sets steepness and rír = MERGE_ MERGE_RADIUS' These weights form [o, N We then compute velucity "sharing" and contact accelerations via convex averaging across opponents:
RADIUS
(1) Σ, Wij (2
+ E Wig v
1:000
1+ E3 Wij
1+1 Σ
Wij
10 E, Wij
Σ Wi A
1
1+ E, Wij
1 1+ Wij
with A initialised at zero so the first pass merely defines a contact baseline. This produces smooth, differentiable coupling without hard impulses.
To blend control and contact accelerations we form state-dependent merge probabilities
Pm. -xxp
Σ wiy)
pmn
exp Σ wis
and set
1 --Pmn) U1 + pmm
(1)
A (1)
Aci1
( (1
Pmm.)
(2)
U2
(2) pmm
(2)
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2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.0
-1.5
-1.0
-0.
0.0
0.5
1.0
1.5
2.0
1.00
Belief p(
0.75
0.50
0.25
0.0
0.2
0.4
0.6
0.8
1.0
Time (second)
Figure 13: 2D Hexner's game solved by MPC.
We then perform semi-implicit Euler update with the shared velocities V:
v() + clip (1) + (1) St, =BOX_VBB VEL
4(2) + clip
(212 + 12) öt, BOX_VEL,
x + clip
alip(x11) (1() + ((1) 81, + BOX_POO)
x(2) + clip (2 + v22) St, + BOX LOOX_PPP (BO__POS)
Control-affine analysis Fix macro time h and a substep. and treat the current state (X(1,, y(), x2),, v()) given. The weights W, the merge probabilities ()) the shared velocities DO, and the contact terms A!) are functions of the state only at that substep. Consequently.
111) (1- (1-P!)) (2) OU + 103
A (1 p(2) @ U2 + p(2) O A
state-only
state-ooly
The semi-implicit update affine in (A(1, A(2)), hence affine in (U1, U2):
Fb+1
f(xx) + Bi(xx) u1 + B2(xx) u2
where the "input matrices" B1, Bg diagonal masks with entries (1 ph)) öt in the velocity rows and (1- - ( 8t2 in the corresponding position rows, all depending only on Fy- Thus the map is control-affine for any fixed state, and globally piecewise control-affine due the velocity/position clamping at the box limits; the latter introduces non non-smooth but everwheee differentiable saturations.
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Tackle probability and running cost. We summarise the likelihood of tackle against the ball- carrier (RB) via a differentiable probabilistic OR across all defenders. Let "rb" index the RB on offense, then with the same W.
Ptackce
1
II (1 Wyb.).
The running loss at macro step is
0.1 2 7 (voc(Ui)) R1 vec(U1) vec(U2) R2 vec(U2))
fuum
+ Auckle Puckle:
with R1 = R2 laN in our defaults, small control weight to encourage purposeful motion, and Anckle the penalty weight for RB being tackled.
Terminal payofl's: power-push (RB) vs. QB throw The hidden type i* E {0, 1} selects the objective. For the power-push run (i* = 0), (Teb, Ymb) denote the RB coordinates and Oin -0.8 The terminal loss is
Jm Crin (1mhl))
temm
which rewards downfield progress while softly encouraging inside lane. For the QB throw (i" 1), we reward the deepest downfield offensive player, regardless role:
L
Feem
max x(1)
i DE(l (I...N)
The implemented terminal function is
run throw term
0,
Lierr
1z 1.
The overall zero-sum loss the sum of running losses over = O,..., K plus Team-
Initial lineup. For N 11 we instantiate realistic I-formation offense against 4-3 base defense in a normalized field window. Coordinates use downfield (increasing towards the defense) and as lateral. offense aligns its line on the line of scrimmage at LINEUP -_OFF_F with O-line y coordinates {-0.80,- 00.40, 0.00, 0.40, 0.80} labelled LT, LG, C, RG. RT, tight end at y 1.10 (right), wide receivers at y SIHEUPOOFFE X 0.20, fullback at
#1.45 at the same x, quarterback at LINEUP OFF _ 0.30, y 0.20, and the running back at x = LINEUP OFF_X - 00.40, y = 0.00. defense places a four- LINEUP DEF X with
man line at at I = LINEUP LINEUP_DEF_ X - 0.15, y E {-0.80, 0.00, 0.80]. comerbacks slightly pressed at LINEUP BB___EE__XX_ 0.05, y 1..45, and two safeties deep at LINEUP_DER - 0.45, E { [ -0.90, 0.90).
{-0.60, -0.20, 0.20, 0.60), three linebackers
J.1 MAKING SENSE OF EXPLOITABILITY
While exploitability provides a good indication of whether or not the learned policy is converging to the equilibrium, without well known baseline outcome, it is often difficult make sense of how drastic the true nash equilibrium is compared to the resulting policy with non-zero exploitability. Hence, here we compare the resulting trajectories when each player switch to their best-response policy to get a sense how "exploitable" are the strategies, qualitatively. We plot the comparison in Fig. 14 Note that unlike in Fig. [b). in Fig. 14 plays dual game, which results in slight change in PT's policy.
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Final Policy
Pi BR Policy
P2 BR Policy
Type
Type
Figure 14: Comparison of resulting behavior from the final P1 and P2's policies vs their respective best-response (BR) policies. Both players' behavior do not shift qualitatively when switching their BR policies highlighting that the resulting policy with non-zero exploitability is indeed close to the equilibrium.
36


Reviews:
[{"review_id": "rWFBuPMJfL", "reviewer": "Reviewer_52aL", "summary": "This paper studies a class of two-player zero-sum differential games with one-sided information (2p0s1) and proves that equilibrium strategies are atomic in the number of payoff types $I$. P1 needs at most $I$ action prototypes and P2 at most $I+1$. This collapses game-tree complexity by reducing the effective branching factor from $U$ (all actions) to $I$ or $I+1$. Building on this, the authors implement three solvers to exploit this problem structure: a model-based pipeline (CAMS), a model-free RL version (CAMS-DRL using PPO/MMD), and a control variant (CAMS-MPC). The authors show learning speed and performance improvements vs methods that use large discrete action spaces. They also demonstrate behavior on an 11v11 American football domain.", "questions": "Your algorithms parameterize each player’s policy as a mixture over $I$ or $I+1$ action prototypes, relying on the paper’s existence result for atomic equilibria. In practice, how robust are the learned policies to an unconstrained best response that is not parameterized atomically (and instead operates over the raw action space)? How large is the risk that restricting your learning process to atomic mixtures can allow non-atomic opponents to induce out-of-distribution sequences and inputs in agents trained with the proposed method?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is well written with concepts presented clearly.\n- The proofs of $I$ and $(I+1)$-atomicity are useful and relevant to solving large 2p0s games with asymmetric information.\n- The authors show empirical gains by leveraging this problem structure across a range of algorithmic approaches.\n- OpenSpiel-compatible implementations are provided.", "weaknesses": "1) PPO/MMD hyperparameters and any relevant sweeps are not listed in the paper. It's possible that not enough hyperparameter tuning was done (for both the proposed method and baselines).\n  \n\nRudolph et al. (2025) describe that (well-regularized) PPO and MMD are quite sensitive to the entropy coefficient. While they generally recommend quite high coefficients like 0.1 or 0.05, if PPO and MMD have issues converging to non-interior solutions, perhaps there are lower entropy coefficients (or different learning rates) that are sufficient for this domain, which is quite different from dark hex/PTTT?\n\nIt would be great to see a minor sweep on these baselines to make sure that PPO/MMD isn't flatlining just because, e.g. the entropy regularization is too high.\n\nRudolph et al. Reevaluating policy gradient methods for imperfect-information games. 2025.\n\n--\n\n2) Adding exploitability or approximate exploitability comparisons could improve this paper to help measure convergence to NE. This is less important as the authors provide other indicators of convergence.", "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": "", "extra_scores": {}}, {"review_id": "dofQxQT7mv", "reviewer": "Reviewer_9LSw", "summary": "This paper studies two-player zero-sum differential games with one-sided incomplete information. They identify an atomic equilibrium structure, which dramastically reduces the game tree complexity. Base on this, they propose a primal-dual DP reformulation and establishes convergence guarantees. They also leverage this atomic property to develop several scalable solvers. Experiments  shows large reductions in computational cost and improved solution accuracy compared to SOTA solvers.", "questions": "- Provide a short, explicit toy example, e.g. one or two time steps in the main text to illustrate how the convexification produces the I atoms.\n- The paper mentions that P1’s inability to precisely control belief in stochastic settings and that the convexified Bellman operator becomes a lower bound. Is it possible to extend CAMS to stochastic settings?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "- The paper is generally well organized.\n- The paper identifies an atomic Nash structure that theoretically reduces the exponential complexity of imperfect-information differential games. The atomic-NE theorem is clean, leverages convexification geometry, and directly leads to algorithmic simplifications exploited throughout the paper. \n- It provides a unifying primal–dual reformulation, which bridges classical differential-game theory and RL/Control.\n- The paper validates using realistic case study. The football experiment is an impressive  demo that this theoretical structure can yield large practical gains when assumptions hold.", "weaknesses": "- Assumption A5 is kind of strong. It requires full knowledge of dynamics and perfect recall, but many interesting real-world problems might violate these. The atomic results assumes the ability of P1 to precisely control the public belief, and the paper notes the stochastic case only yields lower bounds. These limitations should be emphasized when claiming broad applicability.  \n- The CAMS value-approximation procedure requires solving many small minimax problems and training value networks. \n- Thm 5.1 gives a high-level complexity bound but constants and practical solver behaviour  are sensitive and not fully characterized.", "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": "", "extra_scores": {}}, {"review_id": "uJq8O9f4zb", "reviewer": "Reviewer_f6Dd", "summary": "This paper studies the specific case of zero-sum differential games where the only form of imperfect information is player 1's type (with a common prior). \n\nThe key contribution is the proof that even though the action space (at every state) can be large or continuous, the support in the NE (behavioral strategy) is at most I (or I+1), where I is the number of types. This characterization of the equilibria can be utilized to solve problems in MARL and MPC.", "questions": "- I am not convinced about the example given in Figure 1 and line 81. I agree that the branching factor is I instead of the (discretized) U. However, I do not think it is fair to compare the size of a IIEFG and the \"primal and dual games\" introduced by the author; specifically the way the depth of the search tree is K instead of 2K. \n- Is \"action error\" an appropriate measure of performance (Figure 5)? I would have thought the regular notion of exploitability is more appropriate in a 2p0s setting.\n- How does the policy network of P2 look like (e.g., section 5.2). I think it is nice (and consistent with the argument that CAMS-DRL works better than alternatives when NE do not lie in the strict interior) that the architecture allows for this by having action prototypes. Could the authors comment on how the I+1 action prototypes were learnt? I would think there may be a brittleness problem here. For sxample, if due to say, local minima, not all action protypes were learnt (e.g., if two action prototypes turned out to be near identical), then the restriction of outputting only a logit of size I+1 can lead to very a distribution that is very suboptimal since one prototype was \"wasted\".\n-Typos: Line 45 (uniformed vs uninformed)", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- I generally enjoyed reading the paper. Most papers regarding game solving suffer from an overwhelming complexity in notation, and this paper is no exception; however, the authors partially alleviated this by having text that explains most of the detail well. \n- Experiments are generally well done, with a reasonably objective explanation of the results involved (e.g., Sec 6.2).\n- The literature on solving continuous time, continuous action games with imperfect information is relatively limited; even though this is a very limited case, the empirical ability to scale looks like it will be a good addition to the literature.\n\nNote: This is slightly outside my field of expertise, so I cannot confidently comment on novelty nor quality.", "weaknesses": "- There are several terms which I felt were not well defined, and a simple search did not did yield satisfactory definitions, e.g., I-\"atomic\".\n- There are some technical questions I have/comparisons that I felt were unfair (see below)\n- I found the introduction on the multigrid speedup difficult to understand and perhaps difficult to follow for the average reader. Given that it is an important component of getting good performance, perhaps a figure or summary would be more effective, at least in the main paper.", "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": "", "extra_scores": {}}, {"review_id": "NBFA2OOj3Z", "reviewer": "Reviewer_4YnV", "summary": "This paper formulates the football control problem as a two-player zero-sum differential game under asymmetric information, where the attacking agent has full state observability while the defending agent has only partial observations. The authors derive the equilibrium structure of such 2P0S games and exploit it to design robust, feedback-based strategies for both agents. By leveraging this game-theoretic formulation, the proposed method computes strategies that are provably robust against worst-case responses. Empirical results in the Google Research Football Environment demonstrate the effectiveness of this approach across multiple offensive-defensive scenarios, showcasing improved coordination, defensive anticipation, and sample efficiency compared to traditional reinforcement learning baselines.", "questions": "- How does the atomic structure and the primal–dual DP extend when dynamics or observations are stochastic so P1 cannot precisely steer beliefs?\n- Can the authors elaborate on how feedback Nash equilibria are computed in practice for the 2P0S differential game setting? Is the computation exact, or does it involve discretization, function approximation, or numerical solvers? If approximations are used, how is the error controlled or bounded?\n- The paper seems to focus on relatively isolated one-vs-one or two-agent scenarios. Can the framework be extended to team-based settings with multiple attackers and defenders? How would the equilibrium computation scale or need to change?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Theory–structure: Clear, well-motivated primal–dual formulation with rigorous atomic-support results (P1 $I$-atomic, P2 $(I{+}1)$-atomic) and a consistency theorem, explaining why equilibria lie on the simplex boundary for 2p0s1.\n- Practical impact: Simple but powerful architectural change (action prototypes + logits) that plugs into value approximation, MARL, and MPC and delivers sizable accuracy/efficiency gains; compelling football demonstration otherwise out of reach for standard IIEFG solvers.", "weaknesses": "- Empirics emphasize action error and qualitative behavior. However, exploitability or best-response value gaps to certify closeness to NE are not systematically reported.\n- The theoretical framework hinges on the 2P0S game structure, where only one player lacks full observability. However, in realistic football environments, both agents typically face partial observability.", "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": "", "extra_scores": {}}, {"review_id": "PkeLW975ZM", "reviewer": "Reviewer_wDjH", "summary": "This paper tackles the problem of solving two-player zero-sum differential games with continuous state/action spaces and one-sided incomplete information (where P1 knows the payoff type, and P2 only has a prior belief). The core theoretical contribution is proving that the Nash Equilibrium (NE) strategies for both players have an \"atomic\" structure: P1's behavioral strategy concentrates on at most I actions, and P2's on at most I+1 actions, where I is the number of possible payoff types.", "questions": "1. In the problem setting of this paper, does P2 know that P1 knows its belief distribution? If P2 knows that P1 knows its belief distribution, why does P2 let P1 know its belief distribution and do not modify the distribution? I think the problem setting is not very clear.\n\n2. What are the specific steps in the LLM proof? How do you ensure the correctness of the proof? Since I am not good at theoretical analysis, I cannot judge whether the proof is valid.\n\n3. Are the value estimates used only for the CAMS algorithm, or for all comparison methods (CFR, etc.)? If only the CAMS algorithm is used, it will be an unfair comparison.\n\n4. The atomic structure is proven under the Isaacs' condition, which guarantees the existence of pure-strategy NEs in the \"non-revealing\" games. Could you discuss scenarios or game classes where this condition might fail and what the implications would be for the atomic structure of the overall NE?\n\n5. Algorithms like CFR can be parallelized. Can CAMS be parallelized? Is your experiment parallelized?\n\n6. What does Action Error mean in an experiment?\n\n7. How is I generated from U? What is the size of I?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "The proposed CAMS framework effectively utilizes the theoretical structural guarantee to bypass the curse of dimensionality associated with continuous action spaces. \n\nThe idea of solving a smaller, reformulated nonconvex-nonconcave problem independent of the original action space size is powerful and clever.\n\nThe work bridges differential game theory and computational game theory, offering a viable path to solve realistic problems.", "weaknesses": "1. The goal of the paper is to solve an IIEFG with a very large action space. I noticed that the authors claim that there is currently no insightful action-state-time abstraction method, but in my opinion there are lots of works on action abstraction and reducing the size of the game tree. The authors do not fully discuss the differences between this work and existing works [1-20] in the paper. The author should discuss why the previous method of dealing with continuous action space in IIEFGs fails in the 2p0s1 incomplete information game.\n\n2. The introduction is not very well written. The author abruptly raises the 2p0s1 problem, but I don't think this problem is very important because the assumptions are too idealistic. The author's description in the contribution is also not convincing enough. There seems to be no theoretical breakthrough, and I don't see enough innovation in the algorithm.\n\n3. In line 113, the author claims that “convergence guarantee only exists if the NE lies in the interior of the simplex ∆(U)”, which requires citing relevant literature. There are already some works [4, 21] that give bounds for infinite U.\n\n4. The authors use a gridded value approximation algorithm, but do not discuss or use common value approximation methods in IIGs [22-27]. This is because the value of the information set in IIGs changes with strategy, and simple approximations may not fully reflect the true value. While the multigrid method is used to accelerate the value approximation, the paper lacks a compelling explanation or theoretical guarantee for why it is particularly effective or necessary for solving the underlying Hamilton-Jacobi-Isaacs PDEs in this context. The benefits remain empirically observed but not deeply justified.\n\n5. The writing is often dense and confusing, with terminology and notation that are not well-aligned with the broader computational game theory literature. Key concepts, proofs, and the overall narrative are not self-contained enough for a general audience, making it difficult to assess and appreciate the contributions fully.\n\n6. The paper claims to optimize the complexity from U^{2K} to I^K. In my opinion, optimizing the metric from 2K to K is similar to the approach used in CFR. The optimized CFR [28, 29] can enumerate information sets during iterations, rather than states. The complexity reported in this paper is not fundamentally different from CFR, and the claim that the CFR algorithm is U^{2K} requires further discussion.\n\n7. There is already work that studies the Nash equilibrium strategies of both players when one player reveals her strategy [30, 31]. This work needs to further elaborate on its novelty.\n\n8. The experimental benchmarks used in the paper, including the selection of games, metrics, and the fairness of method comparisons, need to be verified. Achieving outstanding performance on widely recognized games is urgently needed (e.g. StarCraft).\n\n[1] David Schnizlein, Michael H. Bowling, and Duane Szafron. Probabilistic state translation in extensive games with large action sets. IJCAI 2009\n\n[2] John Alexander Hawkin, Robert Holte, and Duane Szafron. Automated action abstraction of imperfect information extensive-form games. AAAI 2011\n\n[3] John Alexander Hawkin, Robert Holte, and Duane Szafron. Using sliding windows to generate action abstractions in extensive-form games. AAAI 2012\n\n[4] Sam Ganzfried and Tuomas Sandholm. Action translation in extensive-form games with large action spaces: Axioms, paradoxes, and the pseudo-harmonic mapping. IJCAI 2013\n\n[5] Noam Brown and Tuomas Sandholm. Regret transfer and parameter optimization. AAAI 2014\n\n[6] Noam Brown and Tuomas Sandholm. Simultaneous abstraction and equilibrium finding in games. IJCAI 2015\n\n[7] Shuxin Li, Youzhi Zhang, Xinrun Wang, Wanqi Xue, and Bo An. CFR-MIX: solving imperfect information extensive-form games with combinatorial action space. IJCAI 2021\n\n[8] Carlos Martin and Tuomas Sandholm. Finding mixed-strategy equilibria of continuous-action games without gradients using randomized policy networks. IJCAI 2023\n\n[9] Boning Li, Zhixuan Fang, and Longbo Huang. RL-CFR: Improving action abstraction for imperfect information extensive-form games with reinforcement learning. ICML 2024\n\n[10] Linjie Xu, Diego Perez Liebana, andAlexander Dockhorn. Strategy Game-Playing with Size-Constrained State Abstraction. CoG 2024\n\n[11] Carlos Martin and Tuomas Sandholm. Joint-Perturbation Simultaneous Pseudo-Gradient. IJCAI 2025\n\n[12] Boning Li and Longbo Huang. Efficient online pruning and abstraction for imperfect information extensive-form games. ICLR 2025\n\n[13] Carlos Martin and Tuomas Sandholm. Solving Infinite-Player Games with Player-to-Strategy Networks. arxiv 2025\n\n[14] Weijun Zeng, Yinghao Li, Xiaosi Chen, Zijie Chang, Fei Ge. GNN-ReBeL: Enhancing Neural Belief Representation for Imperfect-Information Games. IEEE SMC 2025\n\n[15] Weijun Zeng, Yinghao Li, Xiaosi Chen, Zijie Chang, Fei Ge. An Investigation of Subgame Depth in ReBeL: Impact on Convergence and Performance in Imperfect-Information Games. IEEE SMC 2025\n\n[16] David Abel, Nate Umbanhowar, Khimya Khetarpal, Dilip Arumugam, Doina Precup and Michael L. Littman. Value Preserving State-Action Abstractions. AISTATS 2020\n\n[17] Noam Brown, Tuomas Sandholm and Brandon Amos. Depth-Limited Solving for Imperfect-Information Games. NeurIPS 2018\n\n[18] Trevor Davis, Kevin Waugh and Michael Bowling. Solving Large Extensive-Form Games with Strategy Constraints. AAAI 2019\n\n[19] Samuel Sokota, Gabriele Farina, David J. Wu, Hengyuan Hu, Kevin A. Wang, J. Zico Kolter and Noam Brown. The Update-Equivalence Framework for Decision-Time Planning. ICLR 2024\n\n[20] Sam Ganzfried. Algorithm for Computing Approximate Nash Equilibrium in Continuous Games with Application to Continuous Blotto. Games 2021\n\n[21] Christian Kroer and Tuomas Sandholm. Discretization of Continuous Action Spaces in Extensive-Form Games. AAMAS 2015\n\n[22] Rosemary Emery-Montemerlo, Geoffrey J. Gordon, Jeff G. Schneider and Sebastian Thrun. Approximate Solutions for Partially Observable Stochastic Games with Common Payoffs. AAMAS 2004\n\n[23] Anne Souquière. Approximation and representation of the value for some differential games with asymmetric information. Int. J. Game Theory 2010\n\n[24] Auke J. Wiggers and Frans A. Oliehoek and Diederik M. Roijers. Structure in the Value Function of Two-Player Zero-Sum Games of Incomplete Information. ECAI 2016\n\n[25] Gabriele Farina and Tuomas Sandholm. Fast Payoff Matrix Sparsification Techniques for Structured Extensive-Form Games. AAAI 2022\n\n[26] Vojtech Kovarík, Dominik Seitz and Viliam Lisý, Jan Rudolf, Shuo Sun and Karel Ha. Value functions for depth-limited solving in zero-sum imperfect-information games. Artif. Intell. 2023\n\n[27] Ratip Emin Berker, Emanuel Tewolde, Ioannis Anagnostides, Tuomas Sandholm and Vincent Conitzer. The Value of Recall in Extensive-Form Games. AAAI 2025\n\n[28] Michael Johanson, Kevin Waugh, Michael H. Bowling and Martin Zinkevich. Accelerating Best Response Calculation in Large Extensive Games. IJCAI 2011\n\n[29] Michael Johanson, Nolan Bard, Marc Lanctot, Richard Gibson, and Michael Bowling. Efficient nash equilibrium approximation through monte carlo counterfactual regret minimization. AAMAS 2012\n\n[30] Samuel Sokota, Ryan D'Orazio, Chun Kai Ling, David J. Wu, J. Zico Kolter and Noam Brown. Abstracting Imperfect Information Away from Two-Player Zero-Sum Games. ICML 2023\n\n[31] Weiming Liu, Haobo Fu, Qiang Fu and Wei Yang. Opponent-Limited Online Search for Imperfect Information Games. ICML 2023", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)