Variable Bound Tightening for Nash Equilibrium Computation in Multiplayer Imperfect-Information Games

Abstract: There has been significant recent progress in algorithms for approximation of Nash equilibrium in large two-player zero-sum imperfect-information games and exact computation of Nash equilibrium in multiplayer strategic-form games. While counterfactual regret minimization and fictitious play are scalable to large games and have convergence guarantees in two-player zero-sum games, they do not guarantee convergence to Nash equilibrium in multiplayer games. Recently, an approach has been presented for exact computation of Nash equilibrium in multiplayer imperfect-information games that solves a quadratically constrained program based on a nonlinear complementarity problem formulation derived from the sequence-form game representation. This formulation was solved using Gurobi's nonconvex quadratic solver, which employs spatial branch-and-bound to iteratively refine variable bounds by solving convex relaxations of bilinear terms via McCormick envelopes. During presolve, Gurobi introduces auxiliary variables and, in some cases, binary variables, leading to an internal MIQCP reformulation. This approach was demonstrated to outperform prior algorithms from the Gambit software suite and quickly solve three-player Kuhn poker after removal of dominated actions; however, the algorithm was not able to solve the full version of the game within 24 hours. In this paper, we derive finite bounds on slack and multiplier variables in the nonlinear complementarity formulation. These bounds strengthen the convex relaxations used within spatial branch-and-bound and lead to substantial computational improvements. We demonstrate the impact of the proposed bounds on exact Nash equilibrium computation in three-player Kuhn poker.

• The paper introduces an exact Nash equilibrium computation method using a sequence-form NLCP representation with rigorously derived variable bounds.
• It derives finite, problem-specific upper bounds for slack and multiplier variables that strengthen convex relaxations and improve spatial branch-and-bound efficiency.
• Experimental results on three-player Kuhn poker show dramatic reductions in computation time compared to prior methods, validating the approach.

Variable Bound Tightening for Efficient Nash Equilibrium Computation in Multiplayer Imperfect-Information Games

Sequence-Form NLCP Formulation for Multiplayer Games

The paper presents an exact Nash equilibrium computation method for multiplayer imperfect-information games via a nonlinear complementarity problem (NLCP) based on the sequence-form representation. Unlike strategic-form, sequence-form leverages action sequences and flow conservation constraints, drastically reducing representation size for extensive-form games with perfect recall. The NLCP formulation extends previous results for two-player games to arbitrary $n$-player settings, encapsulating player-specific realization plans and Lagrange multipliers to enforce sequence-form constraints. For $n$ players, bilinear and multi-linear terms are introduced and modeled via auxiliary variables, enabling tractable quadratically-constrained feasibility programs. This representation allows direct input into Gurobi's nonconvex quadratic solver, which internally performs spatial branch-and-bound with convex relaxations constructed from variable bounds, predominantly using McCormick envelopes.

Derivation and Impact of Variable Bound Tightening

A central contribution is the rigorous derivation of finite, problem-specific upper bounds for NLCP slack and multiplier variables. Previously, slack variables were set as $[0,\infty)$ and multipliers as $(-\infty,\infty)$, offering negligible pruning power in spatial branch-and-bound. The paper establishes much tighter bounds for slack variables: each slack component $r_i^1$ is bounded above by $V_1^{\max}-U_i$, where $V_1^{\max}$ is the unconstrained best-response value and $U_i$ is the best-response value with sequence $i$ forced. The universal bound $u_1^{\max}-u_1^{\min}$ is simple to compute and suffices for practical efficiency improvements. For multiplier variables, bounds dependent on subtree sizes, payoff extrema, and absolute payoffs are derived via backward induction.

These theoretical bounds strengthen convex relaxations for bilinear constraints, allowing solvers to more aggressively eliminate infeasible regions, perform deeper pruning in spatial branch-and-bound, and reduce the search space for global optimality. The slack bounds, in particular, have the largest impact due to their direct appearance in quadratic constraints arising from complementarity slackness, whereas multiplier bounds are less critical unless extremely tight.

Experimental Evaluation on Three-Player Kuhn Poker

The approach is evaluated on three-player Kuhn poker, a well-known non-trivial imperfect-information game with bluffing and trapping motivations. The NLCP solver is compared against the Gambit suite (notably the logit quantal response method) using Gurobi's solver across both the reduced (dominated actions removed) and full game versions. Previously, the reduced game was solved in 2.47 seconds while the full game remained intractable ($n$024 hours).

Results demonstrate that imposing only slack bounds (from Corollary~co:slack) enabled the full game to be solved in 1.160 seconds. Introducing additional multiplier bounds slowed down computation: using Corollary~co:lambda bounds produced a 3.299 second solve time, and tighter $n$1 bounds performed even worse. Imposing only multiplier bounds (no slack bounds) yielded a 9.257 second solve time. The best observed performance is with slack bounds alone and unconstrained multipliers, a finding supported by the solvers' internal handling of quadratic constraints.

No player is incentivized to deviate by more than $n$2 in the computed strategies, validating exact equilibrium computation within numeric tolerances. Only Gambit's logit approach solves the full game, but at a far higher computational cost (2.5 minutes vs. seconds for the improved NLCP implementation). The bounds derived allow solving previously intractable imperfect-information multiplayer games without relying exclusively on dominated action removal.

Implications and Future Directions

The results highlight the practical importance of variable bound tightening in quadratically-constrained NLCP-based methods for Nash equilibrium computation. By integrating theoretically principled duality-derived bounds, the spatial branch-and-bound procedure in solvers is amplified, facilitating tractable solution of larger and more complex multiplayer games. The methodology is generalizable to other imperfect-information domains and could impact areas such as security games, auction design, and multi-agent learning. The bounds' derivations create potential for further exploration of sequence-form-based optimization, auxiliary variable reductions, and tailored relaxations for games with richer structure (e.g., games with limited information sets or high payoff variance).

The work suggests that future developments may incorporate adaptive bound refinement, dynamic auxiliary variable allocation, and tighter integration with solver internals to handle multi-linear terms for even larger $n$3-player games. Theoretical advances in equilibrium computation can thus directly translate into practical computational gains as solver architectures mature.

Conclusion

The paper establishes a robust framework for exact Nash equilibrium computation in multiplayer imperfect-information games via NLCP sequence-form representations, complemented by rigorously-derived variable bounds for slack and multiplier variables. Imposing bounds on slack variables delivers a significant reduction in computational overhead, enabling solution of games formerly regarded as intractable. The bounds facilitate stronger convex relaxations, enhance spatial branch-and-bound pruning, and outperform all prior approaches for Kuhn poker. The theoretical and computational advances present substantial implications for exact equilibrium computation and open avenues for solver-driven expansion in large-scale multiplayer game analysis (2606.25997).