Zero-Sum Game Theory: Security Strategies, Minimax and Nash

A zero‑sum game features exact conflict of interest: the sum of the players’ payoffs is always 0. In the strategic‑form representation a single payoff function (u) describes Player 1’s utility, while Player 2’s utility is (-u). The convention is that Player 1 tries to maximize (u) and Player 2 tries to minimize it. For monetary exchanges to be strictly zero‑sum the players must be risk‑neutral, which implies a linear von Neumann‑Morgenstern utility function. As Ian Ball puts it, “A zero‑sum game is a game that features exact conflict of interest.”

Worst‑Case Reasoning

Security strategies are defined by worst‑case reasoning. A player selects a strategy that maximizes the minimum payoff they can receive, regardless of the opponent’s choice. For Player 1 the worst‑case gain is

[ WG(\sigma_1)=\min_{\sigma_2} u(\sigma_1,\sigma_2). ]

Mixed strategies are essential because limiting oneself to pure strategies often yields a lower guaranteed payoff. Once one player’s mixed strategy is fixed, the opponent’s optimal response is always a pure strategy, simplifying the calculation of the worst‑case gain.

The Minimax Theorem

John von Neumann proved the Minimax Theorem in 1928. The theorem states that the lower value (v) (the best guarantee for the player who moves first) equals the upper value (v) (the best guarantee for the player who moves second). Consequently the “value of the game” is independent of who moves first, eliminating any systematic first‑move disadvantage. The theorem also implies that complex games such as chess possess a value—though determining it computationally remains out of reach. As the lecture notes, “Moving first and having your move observed is a weak disadvantage,” but the minimax result shows it does not affect the attainable value.

Nash Equilibrium and Security

In zero‑sum games a Nash equilibrium coincides with a pair of security strategies. If each player adopts a security strategy, neither can improve their outcome by unilaterally deviating, satisfying the definition of a Nash equilibrium. This equivalence lets us decompose the computation of a Nash equilibrium into two independent optimization problems—one for each player’s security strategy—providing a clear computational advantage.

Application: Von Neumann Poker

The von Neumann poker model abstracts a poker hand as a continuous value and forces players to decide whether to bet or fold. The minimax solution requires randomization over these actions, meaning that bluffing—betting with a weak hand—can be optimal. As the instructor emphasizes, “Bluffing is this thing that’s beyond mathematical reasoning. And it turns out just the minimax theorem immediately explains why bluffing is sometimes a good thing to do.” The model demonstrates that optimal zero‑sum play may involve deliberately misleading the opponent to raise their worst‑case loss.