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NfgTransformer: Equivariant Representation Learning for Normal-form Games
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Proposes NfgTransformer, a permutation-equivariant transformer that encodes normal-form games as action embeddings and reaches state-of-the-art on Nash equilibrium solving, deviation-gain estimation, and payoff prediction in DISC games.
Problem
Learning general-purpose, equivariant representations of normal-form games that serve many game-theoretic tasks (equilibrium solving, deviation gain estimation, ranking) and handle incomplete payoff tensors and varying game sizes.
Approach
Zero-initialised action embeddings are iteratively refined by attention blocks over joint-action "plays" and other actions; attention's order-invariance over keys and equivariance over queries makes the encoder permutation-equivariant, and masked attention handles unobserved payoffs. Parameter count is independent of game size.
Key results
- On 3-player 16x16x16 NE solving, NfgTransformer (D=128,K=8,A=1) reaches NE Gap 0.0297 versus 0.0929 for EquivariantMLP and 0.1140 for NES.
- On 2-player 16x16 max-deviation-gain estimation, NfgTransformer (D=128,K=8,A=1) achieves MSE 0.0001 versus 0.0488 for NES.
- In DISC games with Z=1, NfgTransformer recovers the latent game near-perfectly at 10% observed joint-actions, with error an order of magnitude below the second-best method.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use NfgTransformer instead of the strongest baseline | Synthetic-game NE solving · Synthetic games, N=2, T=16 | NE Gap: 0.0829 -> 0.0243 | Helped |
| Use NfgTransformer instead of the strongest baseline | Synthetic-game NE solving · Synthetic games, N=2, T=64 | NE Gap: 0.1635 -> 0.0502 | Helped |
| Use NfgTransformer instead of the strongest baseline | Synthetic-game NE solving · Synthetic games, N=3, T=8 | NE Gap: 0.1431 -> 0.0412 | Helped |
| Use NfgTransformer instead of the strongest baseline | Synthetic-game NE solving · Synthetic games, N=3, T=16 | NE Gap: 0.0929 -> 0.0297 | Helped |
| Use NfgTransformer instead of the strongest baseline | Synthetic-game deviation-gain estimation · Synthetic games, N=2, T=16 | Max-deviation-gain MSE: 0.0488 -> 0.0001 | Helped |
| Use NfgTransformer instead of the strongest baseline | Synthetic-game deviation-gain estimation · Synthetic games, N=2, T=64 | Max-deviation-gain MSE: 0.4860 -> 0.0161 | Helped |
| Use NfgTransformer instead of the strongest baseline | Synthetic-game deviation-gain estimation · Synthetic games, N=3, T=8 | Max-deviation-gain MSE: 0.4047 -> 0.0487 | Helped |
| Use NfgTransformer instead of the strongest baseline | Synthetic-game deviation-gain estimation · Synthetic games, N=3, T=16 | Max-deviation-gain MSE: 0.6480 -> 0.3641 | Helped |
| Increase transformer blocks from K=2 to K=8 (ablation) | NfgTransformer (D=32, A=1) · Synthetic games, NE solving | NE Gap: 0.2239 -> 0.0344; 0.1685 -> 0.0554; 0.1344 -> 0.0484; 0.0796 -> 0.0334 | Helped |
| Increase action embedding size from D=32 to D=128 (ablation) | NfgTransformer (K=8, A=1) · Synthetic games, NE solving | NE Gap: 0.0344 -> 0.0308; 0.0554 -> 0.0502; 0.0484 -> 0.0412; 0.0334 -> 0.0297 | Mixed |
| Remove action-to-action self-attention (A=0) (ablation) | NfgTransformer (D=64, K=8) · Synthetic games, NE solving | NE Gap: 0.0308 -> 0.0332; 0.0545 -> 0.0661; 0.0478 -> 0.0636; 0.0325 -> 0.0384 | Mixed |
| Add a second action-to-action self-attention operation (A=2 instead of A=1) (ablation) | NfgTransformer (D=64, K=8) · Synthetic games, NE solving | NE Gap: 0.0308 -> 0.0243; 0.0545 -> 0.0542; 0.0478 -> 0.0437; 0.0325 -> 0.0314 | Mixed |
Surprising findings
- Two-player 2x2 coordination and anti-coordination games receive identical action embeddings, so the embedding cannot disambiguate them due to equivariance.
- Equivariant MLP is only marginally better than plain MLP and remains ineffective at NE solving in the reported tables.
Limitations
- Equivariance makes strategically equivalent games (e.g. coordination vs anti-coordination 2x2) indistinguishable from embeddings alone.
- Interpretability analysis uses simplified single-head, no action-to-action attention architecture on 10x10 games.
Experimental setup
Datasets: synthetic normal-form games (equilibrium-invariant subspace) DISC games GAMUT games (22 classes)
Benchmarks: synthetic NFG NE solving (NE Gap) synthetic max-deviation-gain estimation (MSE) DISC game payoff prediction (MSE) GAMUT games Colonel Blotto (out-of-distribution)
Metrics: NE Gap mean squared error
Baselines: MLP EquivariantMLP NES Elo mElo xElo
Builds on: Transformer self- and cross-attention NES (Neural Equilibrium Solver) Elo xElo mElo
Closest papers by meaning
- Large Scale Learning of Agent Rationality in Two-Player Zero-Sum Games (AAAI 2019 · similarity 0.86)
- Kernelized Multiplicative Weights for 0/1-Polyhedral Games: Bridging the Gap Between Learning in Extensive-Form and Normal-Form Games (ICML 2022 · similarity 0.86)
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