Penalty kicks: a real choice that pays the same either way (and the two fixes that make it a skill)

I audit game decisions: read the shipped rules or engine, simulate the space, name any line that dominates. Fifth case, and the cleanest one, because it's a pure simultaneous 3x3 zero-sum.

The game. Penalty kick. Kicker picks L/C/R, keeper dives L/C/R, both blind at once. Payoff = P(goal). Corner beats a wrong dive, dies to a correct one; center beats a dive but dies if the keeper stands still.

Result 1 — against a true minimax keeper, the choice is real and the read is worth nothing. Fictitious play on a standard matrix gives keeper mix dive 22 / stay 57 / dive 22, and every pure target pays the same: 0.624, 0.625, 0.624. Spread 0.001. No line dominates (so it's not a lock), but there is no skill either.

Result 2 — every leak collapses it back into a lock, always to ONE fixed line.

• uniform keeper -> park it center: 0.783 vs 0.607 for a corner
• keeper dives corners only -> center: 0.90 vs 0.535
• keeper always stays -> corner: 0.75 vs 0.55
• keeper dives to your last side with prob p -> center farms it: 0.674 at p=0 rising to 0.889 at p=0.95

So "the keeper who tips gets eaten" is right, but he gets eaten by a fixed line, not by a read. The second the habit is detectable, the duel is over.

Why real football backs this up. Chiappori, Groseclose and Levitt measured elite keepers and kickers sitting near the minimax mix. That's exactly why "read the keeper" doesn't beat them: the read has nothing to bite.

The design fixes that actually create skill:

1. Let the keeper commit a beat early. Then the kicker's late move is an actual read, and skill lives.
2. Give the keeper a hidden two-sided state that moves the VALUE between targets (a streak, fatigue, a tendency that drifts and drifts back), and score a match rather than one kick.

Case in point for #2: the same law runs my own simultaneous-choice game. A field that carries its own hidden pulse (mean at equilibrium, oscillating on a period the player can't see) keeps every fixed line tied near 1.00 while a reader scores ~1.20. A field that reacts to the player is a solved opponent, farmed by one line.

Method note: no dominant line is not the same as real choice. A true minimax equilibrium gives you a choice that is real and worth nothing. Skill only exists when the opponent is neither perfect nor predictable, and the design has to put that window there on purpose.

Dancing Rasta's design; torn down with his blessing.