Team decision problems with classical and quantum signals.

We study team decision problems where communication is not possible, but coordination among team members can be realized via signals in a shared environment. We consider a variety of decision problems that differ in what team members know about one another's actions and knowledge. For each type of decision problem, we investigate how different assumptions on the available signals affect team performance. Specifically, we consider the cases of perfectly correlated, i.i.d., and exchangeable classical signals, as well as the case of quantum signals. We find that, whereas in perfect-recall trees (Kuhn 1950 Proc. Natl Acad. Sci. USA 36, 570-576; Kuhn 1953 In Contributions to the theory of games, vol. II (eds H Kuhn, A Tucker), pp. 193-216) no type of signal improves performance, in imperfect-recall trees quantum signals may bring an improvement. Isbell (Isbell 1957 In Contributions to the theory of games, vol. III (eds M Drescher, A Tucker, P Wolfe), pp. 79-96) proved that, in non-Kuhn trees, classical i.i.d. signals may improve performance. We show that further improvement may be possible by use of classical exchangeable or quantum signals. We include an example of the effect of quantum signals in the context of high-frequency trading.