Where does complementarity collide?

Niels Bohr introduced complementarity in his 1928 paper The Quantum Postulate and the Recent Development of Atomic Theory (first delivered at Como in 1927): the principle that a quantum object has pairs of properties, such as its wave behaviour and its particle behaviour, that are each valid but cannot both be displayed in one experiment, so a full account needs mutually exclusive setups that are never realised together. A measurement that fixes one member of such a pair, position for example, forecloses a sharp value of its partner, momentum, and the two descriptions complete rather than contradict each other. The referent is a relationship between mutually exclusive descriptions of a single physical system, not a zero-product condition on vectors or an incentive that rewards matching what other players do.[1]

Unit

A pair of mutually exclusive descriptions of one quantum system, such as wave and particle

Recovery means

Not applicable

Measured by

Demonstrated by which-path and interference setups that cannot both be run on the same event

In optimization, complementarity names a condition on two matched nonnegative vectors: for every index at most one of the two paired quantities is positive, so their componentwise product is zero throughout. Lemke's 1965 Bimatrix Equilibrium Points and Mathematical Programming gave a constructive pivoting method for systems carrying this condition, and Cottle and Dantzig's 1968 Complementary Pivot Theory of Mathematical Programming established it as the linear complementarity problem, the task of finding vectors z and w with w equal to Mz plus q, both nonnegative, and the product of z transpose and w equal to zero. The referent is an algebraic requirement that matched slack and activity cannot both be positive, and it is unrelated to a physicist's mutually exclusive measurements or an economic incentive to match others.[2][3]

Unit

A pair of matched nonnegative vectors whose componentwise product is zero (z transpose times w equals zero)

Recovery means

Not applicable

Measured by

Solved by complementary pivoting algorithms, such as Lemke's method, that return vectors satisfying the condition

Bulow, Geanakoplos, and Klemperer's 1985 Multimarket Oligopoly: Strategic Substitutes and Complements coined strategic complementarity for interactions in which one player choosing more of an action raises the payoff to others who also choose more of theirs, so best responses move in the same direction and the choices reinforce one another. Actions are strategic complements when a rival's increase makes your own increase more attractive, as often happens with price setting among firms, and strategic substitutes when a rival's increase makes yours less attractive. The referent is a property of how players' incentives co-move in a game, gauged by the slope of a best-response function, and it has nothing to do with a physicist's mutually exclusive measurements or a zero-product condition on vectors.[4]

Unit

A pair of players' actions whose best responses move in the same direction

Recovery means

Not applicable

Measured by

Identified by the sign of the cross-partial of a payoff function, or by an upward-sloping best-response curve