The Game as an LP
The Row Player's LP
The row player's mixed strategy can be written as a maximin linear program. A visible value shift keeps the LP positive while preserving the optimal strategies. The motivation is to turn an adversarial guarantee into a bounded geometric object whose optimal vertex is still the game's strategy.
Value Shift
Add 3 to every payoff. Why: the strategies do not change, and the shifted value is positive. This is bookkeeping, not a new game: every pure or mixed payoff is lifted by the same amount, so the ranking of strategies remains the same.
Row Envelope
The row player chooses p and maximizes v below both shifted payoff lines. Why: v is the guaranteed payoff against every column, so it must sit under the payoff the row mix receives no matter which column is chosen. The best row strategy is the one that lifts this worst-case floor as high as possible.
Optimal Vertex
The optimal vertex is (1/2, 7/2). Why: the best guarantee is where the two active payoff bounds meet. If one bound were lower, the minimizer would use that column; at the meeting point the row player has balanced the two threats.
Subtract the Shift
The shifted value is 7/2; subtracting 3 gives the original value 1/2. Why: adding a constant to every payoff shifts the value by that constant and nothing else. The displayed subtraction is the honesty check that returns from the LP-friendly shifted matrix to the original game.
Diagram note
The row LP diagram shows the bounded envelope feasible region and the recomputed optimal vertex. The vertical coordinate is the shifted guarantee, and the final value is recovered only after subtracting the shift. This is still a finite two-player zero-sum game calculation, not a general-sum equilibrium claim. Pixel positions are rounded for layout; every number shown is exact.