Montaa combines classical solution concepts with modern simulation and self-play methods — the same families of techniques used in economics, auction design, and superhuman game-playing AI.
Nash equilibrium
uᵢ(sᵢ*, s₋ᵢ*) ≥ uᵢ(sᵢ, s₋ᵢ*)
Find stable outcomes where no party gains by unilaterally changing course.
Bayesian Nash equilibrium
σᵢ(θᵢ) ∈ argmax E[uᵢ | θᵢ]
Reason about what others privately know, believe, and are hiding.
Subgame-perfect equilibrium
backward induction on the move tree
Discard empty threats; keep only commitments that stay credible at every stage.
Nash bargaining solution
max Πᵢ (uᵢ − dᵢ)
The principled split of surplus relative to each side's walk-away point.
Kalai–Smorodinsky solution
uᵢ − dᵢ ∝ maxᵢ − dᵢ
A fairness benchmark proportional to what each side could ideally achieve.
Shapley value
φᵢ = Σ |S|!(n−|S|−1)!/n! · [v(S∪i) − v(S)]
Attribute value to each party by their average marginal contribution to every coalition.
The Core
Σᵢ∈S xᵢ ≥ v(S) ∀ S ⊆ N
Test whether any sub-group could profitably walk away and break the deal apart.
Monte Carlo simulation
E[U] ≈ (1/N) Σₖ U(ωₖ)
Sample thousands of futures to estimate expected value, variance and tail risk.
Monte Carlo tree search
UCT = X̄ᵢ + c√(ln N / nᵢ)
Efficiently explore deep move sequences, focusing compute where it matters.
Counterfactual regret minimization
σ ← regret-matching(Rᵀ)
Self-play to converge on robust strategies in games with hidden information.
Quantal response equilibrium
P(a) ∝ exp(λ·u(a))
Model real counterparts as noisy and boundedly rational, not perfect calculators.
Minimax regret
min_a max_ω [ u*(ω) − u(a, ω) ]
Choose the action you'll least regret across the scenarios you can't rule out.
Grounded in the literature
Nash (1950), The Bargaining Problem · Harsanyi (1967–68), Games with Incomplete Information Played by “Bayesian” Players · Shapley (1953), A Value for n-Person Games · Kalai & Smorodinsky (1975), Other Solutions to Nash's Bargaining Problem · McKelvey & Palfrey (1995), Quantal Response Equilibria for Normal Form Games · Kocsis & Szepesvári (2006), Bandit Based Monte-Carlo Planning · Zinkevich et al. (2007), Regret Minimization in Games with Incomplete Information