Security Levels & Saddles

Payoff and Security Levels

A zero-sum payoff matrix gives gains to the row player and losses to the column player. The first question is adversarial: what can a player guarantee against the opponent's worst response? This lesson computes each player's security level and shows why a gap between maxmin and minmax means no pure-strategy solution.

highlighted = computed this step

Payoff Matrix

The entries are payoffs to the row player. Why: the row player maximizes while the column player minimizes, so every cell is a conflict between a gain and an equal loss. Before mixing or linear programming appears, the matrix asks a guarantee question: what can each player force if the opponent responds in the most damaging way?

row player maximizes, column player minimizes\text{row player maximizes, column player minimizes}
payoff securityC1C2R13-1R2-22

Row Security

The row minima are -1 and -2; the row security level is -1. Why: the row player asks which row has the best worst-case payoff. That number is a promise the row player can keep against any column choice, not a prediction of what the column player will actually choose.

max⁡imin⁡jAij=−1\max_i\min_j A_{ij}=-1
Security levelsThe security values come from row minima and column maxima.security levelsfirstsecondrow min-1-2col max32levels-12

Column Security

The column maxima are 3 and 2; the column security level is 2. Why: the column player chooses the column with the smallest possible loss cap. This is the opponent's version of a guarantee: hold the row player's payoff below the best cap that a pure column can enforce.

min⁡jmax⁡iAij=2\min_j\max_i A_{ij}=2
Security levelsThe security values come from row minima and column maxima.security levelsfirstsecondrow min-1-2col max32levels-12

Security Gap

The gap is 3, so no pure strategy settles the game. Why: maxmin and minmax disagree, leaving room for each side to exploit a fixed pure choice. That gap is the first signal that randomization is not decorative; it is needed to remove predictable weaknesses.

gap=3\text{gap}=3
Security levelsThe security values come from row minima and column maxima.security levelsfirstsecondrow min-1-2col max32levels-12

Diagram note

The matrix marks row-minimum and column-maximum roles; every security value is recomputed from the payoff entries. Security levels are adversarial guarantees, not average-case summaries. The exact claims here are for finite two-player zero-sum games only. Pixel positions are rounded for layout; every number shown is exact.

security levels are exact matrix scans\text{security levels are exact matrix scans}
payoff securityC1C2R13-1R2-22