The Finite Setting
Although our primary interest in this monograph is with the infinite, we begin with a discussion of hat problems in which the set A of agents is finite and visibility is given by a directed graph on A (the visibility graph). Most of what is known in the finite case (where agents cannot pass) can be found in a single paper entitled Hat Guessing Games, by Steve Butler, Mohammad Hajiaghayi, Robert Kleinberg, and Tom Leighton. This chapter considers minimal predictors (which guarantee at least one correct guess), optimal predictors (which achieve the most correct guesses possible in the given context), a relationship between the Tutte-Berge formula and how successful predictors can be in symmetric visibility graphs, hat problems with a variable number of colors, and other variations on the standard hat problem.
- [Fei04]Feige, U.: You can leave your hat on (if you guess the color). Technical report MCS04-03, Computer Science and Applied Mathematics, The Weizmann Institute of Science, p. 10 (2004)Google Scholar
- [Vel11]Velleman, D.J.: The even-odd hat problem (2011, preprint)Google Scholar
- [Win01]Winkler, P.: Games people don’t play. In: Wolfe, D., Rodgers, T. (eds.) Puzzlers’ Tribute, pp. 301–313. A K Peters, Natick (2001)Google Scholar
- [Yip94]Yiparaki, O.: On some tree partitions. PhD thesis, University of Michigan (1994)Google Scholar