Eccoci.

Alice and Bob play the following game with two six-sided dice (numbered $1$ to $6$):

1. Alice rolls both dice; she can see the rolled values but Bob cannot
2. Alice chooses one of the dice and reveals it to Bob
3. Bob chooses one of the dice: either the one he can see, or the one he cannot
4. Alice pays Bob the value shown on Bob's chosen dice

Each player devises a (possibly non-deterministic) strategy. An example strategy for each player could be:

• Alice chooses to reveal the dice with value closest to $3.5$, or if both are equidistant she chooses randomly with equal probability
• Bob chooses the revealed dice if its value is at least $4$; otherwise he chooses the hidden dice

In fact, these two strategies together form a Nash equilibrium. That is, given that Bob is using his strategy, Alice's strategy minimises the expected payment; and given that Alice is using her strategy, Bob's strategy maximises the expected payment.

With these strategies the expected payment from Alice to Bob is $\frac{145}{36}\approx 4.027778$.

To make the game more interesting, they introduce a third (six-sided) dice:

1. Alice rolls three dice; she can see the rolled values but Bob cannot
2. Alice chooses two of the dice and reveals both to Bob
3. Bob chooses one of the three dice: either one of the two visible dice, or the one hidden dice
4. Alice pays Bob the value shown on Bob's chosen dice

Supposing they settle on a pair of strategies that form a Nash equilibrium, find the expected payment from Alice to Bob, and give your answer rounded to six digits after the decimal point.