In a game of survival, two players with limited resources play a zero-sum game repeatedly until one of them is ruined. The solution of the survival game gives one a measure of the value of resources in terms of survival probabilities. In this paper the zero-sum game is expressed as a finite matrix, but with (possibly) incommensurable entries; hence the number of different distributions of resources that can occur during a single play may be infinite. The existence of a value and optimal strategies is proved, using the theory of semi-martingales. A simple approximation to the solution is described, and several examples are discussed.
Milnor, John Willard and Lloyd S. Shapley, On Games of Survival. Santa Monica, CA: RAND Corporation, 1956. https://www.rand.org/pubs/papers/P622.html. Also available in print form.
Milnor, John Willard and Lloyd S. Shapley, On Games of Survival, Santa Monica, Calif.: RAND Corporation, P-622, 1956. As of September 09, 2021: https://www.rand.org/pubs/papers/P622.html