However, in certain cases the players may adapt their strategy depending on a piece of advice. The latter is delivered to all players by an advisor. This opens the possibility for the players to adopt correlated strategies, which can outperform independent strategies. There are various forms that advice can take. For example in the case of correlated classical advice, the advice is represented by a classical variable, l, with prior r(l). Each player can then choose a strategy depending on his type and on l.
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For Bayesian games, the possibility of having access to nonlocal correlations, for instance using entanglement, has important implications. First let us imagine that the players can share quantum advice, that is, the advisor is able to produce entangled particles and to send them to the players, who then perform local measurements on their particles. Since the statistics of such measurements can in general not be reproduced by any classical local model, the players now have access to strategies which would be impossible in the case of a classical advisor. Thus, players sharing quantum advice can outperform any classical players.
So, if some players have information that other players don't, they can outperform the others? Astonishing :-|