[Icehouse] 2 player Zendo

[Kory Heath]

Dale Newfield wrote:
>> Student picks a number, X.
>>
>> The master picks a rule that they think will take the student X turns 
>> to figure out.
>
> The Student gets points for guessing in less than or equal to X turns,
> and is given one anyway if the master made the rule too hard:
> (1 point if not solved at X, 2 if solved at X, 3 at X-1, etc.)
>
> The Master gets two points if it's solved in exactly X turns.

On turn X, as the Student, I will purposely not guess the rule 
correctly, because if I guess it correctly, I gain no points over the 
Master, while if I fail to guess correctly, the game ends and I gain a 
point. You could just give the Master a single point instead of two 
points if it's solved in exactly X turns. But then neither player cares 
what happens on turn X, because the Student gains a single point either 
way. Since that's the best result the Master can hope for, the best 
strategy is just to purposely come up with a rule that's too hard. You 
could tweak the scoring further to insure that the Student always gains 
two or more points over the Master for guessing the rule correctly, and 
gains exactly one if the rule is too hard, but then once again the best 
strategy for the Master is to purposely come up with a rule that's too 
hard. I'm not seeing any obvious way around all of this.

I'm intrigued by Guy's idea of keeping X hidden from the Student. Here's 
my variation on that theme. Interestingly, this suggestion actually 
works for any number of players, and provides a kind of cap to keep 
games from going on too long.

---------------------

The Master secretly comes up with a rule, and secretly comes up with a 
target range of rounds during which he or she hopes a Student will guess 
the rule. (A round means a full round of play, with each Student taking 
one turn.)

If a Student correctly guesses the rule in fewer rounds than the 
beginning of the Master's target range, the game ends and that Student 
gets a point per round played.

If a Student guesses the rule correctly on a round within the target 
range, the game ends and the Master gains a number of points equal to 
the start of the target range minus the size of the target range.

If no Student has guessed correctly after the last round of the target 
range, the Master announces this fact, and no one gets any points for 
this game. (However, the Students may continue playing for fun.)

---------------------

So if, as the Master, I choose the range 10-15, the size of my range is 
6. I will win exactly 4 points (10 minus 6) if any Student guesses 
correctly on a round within my range, while Students can win between 1-9 
points by guessing correctly before that. If I choose the range 20-20, I 
will gain 19 points if any Student guesses correctly on exactly round 
20, and Students can win between 1-19 points by correctly guessing 
before that.

I can increase my chances of winning by increasing the size of my range, 
but that reduces the number of points I might win. Notice that there's 
no point in making the size of my range equal to or greater than the 
bottom end of my range. Choosing the range 10-30 is silly (and should 
probably just be illegal), because I will then win negative points if 
any Student guesses correctly within my range.

As a Student, my potential payoff for a correct guess becomes greater 
and greater as the game progresses, but of course, I'll wonder whether 
or not we've moved within the Master's range. If I become convinced that 
we have, I'll just stop making good guesses (and should probably just 
explicitly announce this fact and start passing my turns). This will 
keep the Master from suddenly becoming "helpful" in too obvious a way 
(or simply blurting out the answer!) when the game enters the target 
range. However, I would consider it perfectly legal and within the 
spirit of this game for the Master to start building more helpful 
counter-examples once the game moves into the target range. It's up to 
the Students to think about this and decide whether they want to risk 
guessing.

One potential way to salvage Mondo for the single-Student game would be 
to institute what I've called "Risky Mondo". You start the game with no 
stones, and you need to spend stones to guess. You win a stone by 
calling Mondo and answering correctly, but if you answer incorrectly, 
you lose all your stones. This wouldn't change the single-Student game a 
lot, but it would have some subtle effects, and perhaps it's more 
pleasing than just getting an infinite number of stones. A slightly more 
complex version of Risky Mondo can also be used in multi-Student games. 
See 
http://lists.looneylabs.com/pipermail/icehouse/2007-February/001324.html .

I have no idea if this would be fun. I'd like to try it sometime.

-- Kory