Nash Equilibrium for Terminal Maneuvers
The setup
Last year Ethan Heilman published a clever little game called Terminal Maneuvers that reads like a pocket sci‑fi thriller and teaches game theory at the same time. It has been reported that the premise simulates a missile attacking an interstellar ship equipped with a relativistic laser: the laser must aim not at the missile’s current position but where it will be, because the ship and the weapon are moving at a significant fraction of light speed. One player controls the missile and the other the laser; hit the ship and Missile wins, hit the missile and Laser wins.
The choice
The twist is fuel. Missile can perform erratic, fuel‑burning maneuvers to make its future position uncertain, but fuel is finite across five rounds. It has been reported that Heilman suggests starting the missile with seven units of fuel — a setup that allegedly gives Missile roughly a 25% chance with naïve play. If the missile runs out of fuel, Laser will predict zero and the missile is as good as dead. Desperation builds toward that final round: burn now and you might survive to fight on, burn later and you might be predictable. Tension? Oh yes.
The math
Deterministic strategies are a sucker’s bet. Heilman shows a simple deterministic plan — burn 1 fuel for four rounds, 3 in the last — and that predictable pattern lets Laser guess correctly every time, leaving Missile with only about a 4.6% chance of success. The real solution, he argues, lies in mixed strategies and backward induction: work out optimal probabilistic choices at the endgame and propagate them backward to the start. The game is constant‑sum (translatable to zero‑sum), so classic Nash equilibrium machinery applies. Mixed strategies — not bravado, but randomness used intelligently — are Missile’s lifeline.
Why it matters
This isn’t just a neat puzzle for armchair mathematicians. It’s a tidy, small finite example that forces students to confront non‑trivial mixed‑strategy equilibria without getting lost in infinite or continuous games. And it’s strangely poetic: a last‑ditch gamble against a precise, relativistic defender. Think chess, but at light speed — and with fuel gauges. If you want a compact, tangible way to teach Nash, or just a brain‑teaser that feels like a sci‑fi moral dilemma, Terminal Maneuvers is worth a look.
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