Hansen K.A., Ibsen-Jensen R., Koucký M. (2016) The Big Match in Small Space. In: Gairing M., Savani R. (eds) Algorithmic Game Theory. SAGT 2016. Lecture Notes in Computer Science, vol 9928. Springer, Berlin, Heidelberg
We study repeated games with absorbing states, a type of two-player, zero-sum concurrent mean-payoff games with the prototypical example being the Big Match of Gillete (1957). These games may not allow optimal strategies but they always have \(\varepsilon \)-optimal strategies. In this paper we design \(\varepsilon \)-optimal strategies for Player 1 in these games that use only \(O(\log \log T)\) space. Furthermore, we construct strategies for Player 1 that use space s(T), for an arbitrary small unbounded non-decreasing function s, and which guarantee an \(\varepsilon \)-optimal value for Player 1 in the limit superior sense. The previously known strategies use space \(\varOmega (\log T)\) and it was known that no strategy can use constant space if it is \(\varepsilon \)-optimal even in the limit superior sense. We also give a complementary lower bound. Furthermore, we also show that no Markov strategy, even extended with finite memory, can ensure value greater than 0 in the Big Match, answering a question posed by Neyman .