Guess WHO game. Dominated by a little girl!

[pzhon]

I'm not sure what the rules are from your description. I assumed that you win when you ask a question that narrows the field to one candidate (no additional question is necessary) and your hidden card gives you no information about your opponent's hidden card. Slight variations are possible.

The optimal strategy lets the first player win 91/144=63.2% of the time from (24,24) candidates left. The first player is equally successful on average after aiming to narrow the field to 8, 9, 10, 11, or 12 candidates, but trying to narrow the field to 7 has an equity of only 173/288=60.1%.

If you have 24 candidates left, and your opponent has narrowed the field to n, here are the optimal responses:

n response

2 1
3 2
4 2
5 4
6 4
7 4
8 4
9 8
10 8
11 8
12 8
13 8
14 8
15 8
16 8
17 8-12
18 8-12
19 8-12
20 8-12
21 8-12
22 8-12
23 8-12
24 8-12

I feel a bit bad for your opponent...

Here is the Mathematica code I used:

<ul type="square">
Clear[guessing];
guessing[mine_, opps_] := guessing[mine, opps] =
If[opps &lt;= 1, 0, Max[Table[i/mine(1 - guessing[opps, i]) +
(mine - i)/mine(1 - guessing[opps, mine - i]),
{i, 1, Ceiling[mine/2]}]]]

guesseqs[mine_, opps_] :=
Sort[Table[{i/mine(1 - guessing[opps, i]) + (mine - i)/mine(1 - guessing[opps, mine - i]), i},
{i, 1, Ceiling[mine/2]}]][/list]
I think there is probably a simple heuristic that explains the optimal strategy and why the equities come out to such nice numbers with small denominators. I think you need to distinguish whether you need to get very lucky to win, or just need your opponent not to get very lucky. The right target amount for a "Hail Mary" question is not instant victory, but whatever would force your opponent to start making such aggressive questions.

By the way, with 32 candidates, the unique best first choice would be to try to narrow the field to 16. With 40 candidates, it is equally good to aim for 16-20. With 48, 16-24. With 56, 24-28. With 60, 28-30. With 64, only 32. The value of going first is not monotone, and may have a supremum of 2/3, which might be exceeded at 3, 5, 7, and powers of 2.