fix: Enforce Subgame Perfection in Sequence Form LP (#1304)

[SH20RAJ]

Description

Enforces subgame perfection in the Sequence Form LP solver, addressing Issue #1304.
Previously, the solver could produce arbitrary policies (e.g., uniform random) in unreachable information states. This PR modifies the formulation to use dual variables (slacks) to extract game-theoretically optimal actions even in unreachable subgames.

Changes

• Modified open_spiel/python/algorithms/sequence_form_lp.py to use slack variables for policy extraction.
• Added check_subgame_perfection.py to verify the fix on Kuhn Poker (asserts Player 0 folds 'J' in unreachable subgame).

Verification

• New Test: check_subgame_perfection.py passes (Output verified locally: "Assertion logic passed: P0 folds correctly.")
• Regression: open_spiel/python/algorithms/sequence_form_lp_test.py passed (6/6 tests).
• Supersedes Implement subgame perfection in Sequence Form LP solver (#1304) #1417 (which was closed due to test environment issues).

Dependencies

Requires cvxopt (and numpy).

[SH20RAJ]

Hi @lanctot and the OpenSpiel team,

This PR serves as a core technical contribution for my GSoC 2025 candidacy. It addresses the subgame perfection issue in the Sequence Form LP solver (Issue #1304), which is critical for equilibrium finding in zero-sum games.

Summary for GSoC Context

• Problem: The LP solver previously produced arbitrary policies in unreachable states, violating subgame perfection.
• Solution: I implemented the use of dual (slack) variables to correctly infer optimal actions even when the realization probability is zero.
• Validation:
  • Reproduced the original failure from PR Implement subgame perfection in Sequence Form LP solver (#1304) #1417.
  • Fixed the environmental/dependency issues (numpy/cvxopt import errors).
  • Added check_subgame_perfection.py which formally asserts that Player 0 folds a Jack in Kuhn Poker even in the unreachable subgame (the classic test case).
  • Verified that all existing regression tests in sequence_form_lp_test.py pass.

I hope this demonstrates my ability to work with the core solver algorithms and ensure mathematical correctness in the codebase. I would appreciate any feedback!

[SH20RAJ]

For completeness regarding my GSoC application, I would also like to cross-reference my other contributions to OpenSpiel:

Merged / Internally Imported:

• PR refactor(algorithms): add type hints to cfr.py #1412: refactor(algorithms): add type hints to cfr.py
• PR refactor(examples): add type hints to tic_tac_toe_qlearner #1411: refactor(algorithms): add type hints to external_sampling_mccfr.py

Active / Ready for Review:

• PR feat: Implement Serialize for MCTSBot (#1257) #1466: MCTS Serialization (Essential for checkpointing/experiments).
• PR feat: GSoC 2025 Contributions - Core Updates & Optimizations #1454: Core Updates (Includes optimizations for Euchre/UniversalPoker and standardized Kuhn/PhantomTTT fixes).

This set of PRs covers core algorithms, game logic correctness, infrastructure/serialization, and code quality (typing).