We have two urns $A$ and $B$. $A$ urn contains $2$ white and $1$ black balls while urn $B$ contains $2$ black and $1$ white balls. Urns are called states of nature and can each happen with the probability of $0.5$.
An urn is selected randomly and $n$ players need to sequentially guess which urn has been selected. Before guessing, they have to pick one of the two choices that should facilitate their guess:
1) To draw a ball from an urn (get a signal), look at its color and make a guess about the true state of nature. For example, if urn $A$ was randomly selected and a player drew a white ball, an inference that urn $A$ is the true state of nature can be made with the probability of $0.66$.
2) To look at the history of guesses made by other players. For example, player number $2$ can see the guess of player number $1$. Player number $3$ can see the guesses by players number $1$ and $2$ and so on (note that only guesses which urn is being used can be seen, not the signals).
Edit: the players are rational, therefore inferences about the signals received can be made from the observed guesses. Additionally, when indifferent between the social information and the private signal, player opts for the latter one.
How can we prove that players $1$, $2$, and $3$ get at least as good chance in guessing the true state of nature by receiving a private signal rather than observing the history of choices, while the rest of the players starting from player $4$ are better off seeing the social information? Note that the player $1$ cannot see the history, therefore the choice is between guessing randomly and drawing a ball.
The Bayesian updating should be used in this case but I fail to go beyond the analysis for player $1$ who, obviously, chooses between 0.5 for a random guess and $0.(66)$ for a draw.