Exact Mixing - Zero-Sum Games: Minimax is Duality

When no saddle exists, the players mix. The exact mixed strategies come from equalizing the opponent's relevant options, producing a rational game value. The intuition is that randomization removes an opponent's profitable worst-case response by making the active options equally unattractive.

highlighted = computed this step

Row Equalizes

The row player equalizes the column payoffs at p equals 1/2. Why: if one column were worse for the row player, the minimizer would choose it. Randomization protects the row player by making the opponent indifferent between the columns that matter.

5p - 2 = - 3p + 2

Row Mix

The row mix is 1/2 and 1/2. Why: both columns then give the same exact value, so the column player cannot exploit a predictable row. The mix should be interpreted as a defensive guarantee, not as a claim about psychological randomness.

x^*=1/2,1/2

Column Mix

The column mix is 3/8 and 5/8. Why: it equalizes the row player's pure-row payoffs. The minimizer randomizes so the row player cannot improve by choosing one pure row over the other.

y^*=3/8,5/8

Game Value

The game value is 1/2. Why: both optimal mixtures force the same expected payoff. This value is the fair price of the zero-sum game: above it the column player can cap the payoff, and below it the row player can guarantee more.

v=1/2

Diagram note

The mixed strategies and value are exact fractions from the recomputed equalization. The absence of a saddle forced mixing in this finite zero-sum game, and the displayed fractions are the exact equilibrium strategies for this pinned payoff matrix. Pixel positions are rounded for layout; every number shown is exact.

\text{no saddle means mix in this game}