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Three verdicts from the field. Professional footballers mix their penalty kicks almost exactly as minimax says. Wimbledon servers hit the right frequencies but switch sides a shade too often to be truly random. And in the centipede game, where the theory says grab the money at once, almost everyone passes. Theory wins some tests, half-wins others, and loses one outright.

Field tests and boundaries

Behavioral game theory is not “people are irrational.” It is a map of where the sharp theory holds and where it breaks. Zero-sum minimax survives its hardest test among elite athletes under pressure, missing only slightly on true randomness. Backward induction in the centipede fails outright. Knowing which is which is the point.

From earlier lessons

Two quick ones before we start.

In a zero-sum game, optimal play means mixing to your...

Backward induction solves a game tree starting from the...

Predict, then play

Here is a centipede game. At each step a player can take the pot (ending it) or pass, growing it for the other. Walk it, then solve it backward.

Why did that happen?

Take a guess, then reveal the reason

The evidence does not line up neatly. Penalty kicks are a field setting with expert players, and there minimax performs well. Tennis serves pass the frequency test but show slight excess switching. Centipede evidence comes from laboratory experiments, where players often pass. Backward induction follows from the specified payoffs and common knowledge of rationality; departures may reflect social preferences, bounded reasoning, or uncertainty about how others reason, not an assumption of distrust.

The math

Definition, what a field test of minimax checks

Equilibrium implies two separate restrictions, and a good field test checks both. Equal success rates: a player who mixes must be indifferent, so every option in the mix has to pay the same. Serial independence: each choice must carry no information about the next. Passing the first and failing the second still leaves money on the table, so both matter.

Three verdicts, not two. Palacios-Huerta (2003) analysed 1,417 professional penalty kicks: his estimated scoring rates put the kicker’s equilibrium mix just under 39 percent to one side and the keeper’s at about 42 percent, and observed play came in at 40 and 42 percent, with no serial correlation he could detect. Walker and Wooders (2001) ran the same two tests on serve direction at Wimbledon: winning rates by direction came out equal, exactly as the theory demands, but servers alternated sides slightly more often than chance, a small negative correlation and the familiar human failure to produce a random sequence. And McKelvey and Palfrey (1992) ran the centipede in the lab: subgame-perfect play says take at the first node, yet almost no one does, and most pairs pass several times.

So the boundary is not a clean line between “holds” and “fails”. Minimax passes on frequencies across both sports and very nearly passes on randomness; backward induction fails outright.

Fold the centipede backward: at the last node the mover takes ( $5 > 4$ for Them), so the previous mover takes ( $4 > 3$ for You), and so on to the first node, where You take for $(2,0)$ . That is the subgame-perfect prediction. But mutual passing reaches $(6,4)$ , which both prefer to $(2,0)$ . Real players capture much of that surplus by passing, contradicting the fold. The penalty-kick result, by contrast, confirms the theory exactly where it should be hardest: strategic, high-pressure, expert play.

Work it out

Fold the centipede, then compare to the field

Solve backward. Last node (Them): Take gives Them $5$ , Pass gives $4$ , so Take, outcome $(3,5)$ . Prior node (You): Take gives You $4$ , passing leads to $(3,5)$ where You get $3$ , so Take, outcome $(4,2)$ . Next (Them): Take gives $3$ , passing leads to $(4,2)$ giving Them $2$ , so Take. First node (You): Take gives $2$ , passing leads to Them taking for You’s $1$ , so Take. Backward induction: take immediately, $(2,0)$ .

Now compare the evidence. In the laboratory centipede, pairs often pass repeatedly and capture more of the growing surplus than the subgame-perfect prediction. In field data, professional penalty takers mix close to the minimax frequencies with little readable pattern. These are different environments, so the comparison maps where the models perform rather than delivering one verdict on all game theory.

Key idea

The sharp theory has boundaries, and they are not tidy. Zero-sum minimax survives among elite athletes, missing only slightly on true randomness; backward induction fails outright in the centipede, where real players pass to reach outcomes both prefer.

Check yourself

Check yourselfapply

What does backward induction predict in this centipede, and what do real players do?

Check yourselfanalyse

Sport data largely confirm minimax, while centipede data reject backward induction. What is the lesson?

Check yourselfbrings back: minimax

Why is the penalty-kick result considered a strong confirmation of minimax?

Sources

Palacios-Huerta, I. (2003). Professionals play minimax. Review of Economic Studies, 70(2), 395-415.
Walker, M., and Wooders, J. (2001). Minimax play at Wimbledon. American Economic Review, 91(5), 1521-1538.
McKelvey, R. D., and Palfrey, T. R. (1992). An experimental study of the centipede game. Econometrica, 60(4), 803-836.