John von Neumann reportedly said that pure and applied mathematics have a symbiotic relationship: not only does applied math draw heavily on the tools developed on the pure side, but, correspondingly, pure math cannot exist in the rarefied atmosphere of abstract thought alone; if it is not somehow rooted in the real world, it will wither and die.

The work before us – which certainly qualifies as beautiful, subtle, pure mathematics – is a case in point. It originated half a century ago, at the height of the Cold War between the United States and the Soviet Union, indeed as a direct result of that conflict. The US and SU were trying to keep the Cold War from getting hot; to minimize the damage if it did; and to cut down the enormous expenses that the nuclear arms race entailed. To that end, they met repeatedly in Geneva to negotiate mutual reductions in their nuclear arsenals. Regarding these arsenals, both sides were in the dark. Neither knew how many weapons the other had; and clearly, it was the number retained, rather than destroyed, that mattered. In Princeton, Oskar Morgenstern and Harold Kuhn had just founded the mathematics consulting firm “Mathematica.” The United States Arms Control and Disarmament Agency (ACDA) was responsible for conducting the Geneva negotiations for the US; it turned to Mathematica to see whether the Theory of Games – created two decades earlier by John von Neumann and Oskar Morgenstern (the same as the Mathematica principal) – could help in addressing the strategic issues raised by these negotiations. Mathematica responded by assembling a team of theorists that included Gerard Debreu, John Harsanyi, Harold Kuhn, Mike Maschler, Jim Mayberry, Herb Scarf, Reinhard Selten, Martin Shubik, Dick Stearns, and the writer of these lines. Mike and I took charge of the informational aspect (Dick joined us later): whether one side could glean any information about the size of the other's nuclear arsenal from its tactics in previous negotiation rounds.