Von Neumann’s minimax theorem: optimal mixed strategies

Solves two zero-sum games as linear programs: rock-paper-scissors, whose optimal strategy is uniform with value zero, and a lopsided variant in which the row player’s rock beats scissors for double stakes, which tilts the game in the row player’s favour. In every game, the row player’s guaranteed payoff equals the column player’s guaranteed loss.

import matplotlib.pyplot as plt
import numpy as np

from mathematicskit.optimization import solve_zero_sum_game

Rock-paper-scissors and a lopsided variant

moves = ["rock", "paper", "scissors"]
games = {
    "standard": np.array([[0, -1, 1], [1, 0, -1], [-1, 1, 0]], dtype=float),
    "row's rock wins double": np.array([[0, -1, 2], [1, 0, -1], [-1, 1, 0]], dtype=float),
}

fig, ax = plt.subplots()
width = 0.35
for i, (name, payoff) in enumerate(games.items()):
    result = solve_zero_sum_game(payoff)
    guaranteed = np.min(result.row_strategy @ payoff)
    conceded = np.max(payoff @ result.col_strategy)
    print(f"{name}: p = {result.row_strategy.round(4)}, value = {round(result.value, 10) + 0.0:.4f}")
    print(f"  row player guarantees {round(guaranteed, 10) + 0.0:.4f}; column player concedes at most {round(conceded, 10) + 0.0:.4f}")
    ax.bar(np.arange(3) + (i - 0.5) * width, result.row_strategy, width, label=name)
ax.set_xticks(range(3), moves)
ax.set_ylabel("probability in the optimal mixed strategy")
ax.legend()
ax.set_title("Optimal mixed strategies (von Neumann, 1928)")

standard: p = [0.3333 0.3333 0.3333], value = 0.0000
  row player guarantees 0.0000; column player concedes at most 0.0000
row's rock wins double: p = [0.25   0.4167 0.3333], value = 0.0833
  row player guarantees 0.0833; column player concedes at most 0.0833

Text(0.5, 1.0, 'Optimal mixed strategies (von Neumann, 1928)')