Alice and Bob play the following game with two six-sided dice (numbered $1$ to $6$):
- Alice rolls both dice; she can see the rolled values but Bob cannot
- Alice chooses one of the dice and reveals it to Bob
- Bob chooses one of the dice: either the one he can see, or the one he cannot
- 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:
- Alice rolls three dice; she can see the rolled values but Bob cannot
- Alice chooses two of the dice and reveals both to Bob
- Bob chooses one of the three dice: either one of the two visible dice, or the one hidden dice
- 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.
solutions/s982.md.