Tuesday, July 3, 2012

Jewish Poker


“For quite a while the two of us sat at our table, wordlessly stirring our coffee. Ervinke was bored. ‘All right,’ he said. ‘Let's play poker.’

‘No,’ I answered. ‘I hate cards. I always lose.’
‘Who's talking about cards?’ thus Ervinke. ‘I was thinking of Jewish poker.’
He then briefly explained the rules of the game. Jewish poker is played without cards, in your head, as befits the People of the Book.

‘You think of a number, I also think of a number,’ Ervinke said. ‘Whoever thinks of a higher number wins. This sounds easy, but it has a hundred pitfalls. Nu!’”


Jewish Poker is one of “those” games, games that both players play as a joke, smirks on their faces as they announce numbers like “negative four” and “square root of pi” and “Ackermann function of one billion, one billion.” The rules are really no more complicated than Ervinke made them out to be: each player thinks of a (finite!) number, and whoever thought of the higher number wins. Players can reveal their numbers simultaneously, or one after the other, whichever they prefer.


“We plunked down five piasters each, and, leaning back in our chairs began to think of numbers. After a while Ervinke signaled that he had one. I said I was ready.”


The game is stupid. (That’s what I meant when I said it was one of “those” games.) It’s stupid no matter how you play it, but it becomes even more stupid when the numbers are not revealed simultaneously, because the second player can just add one to the number that the first player revealed.


“‘All right,’ thus Ervinke. ‘Let's hear your number.’
‘Eleven,’ I said.
‘Twelve,’ Ervinke said, and took the money.”


It makes sense that there should be a second-player advantage in a game like this, but somehow it seems a little strange, doesn’t it? Because, well, eleven loses to twelve, so why not pick a number higher than twelve? That’s a number that beats all the numbers eleven would have beaten, and would have had the additional advantage of not losing to twelve. Maybe it wasn’t going first that sunk him; maybe our narrator just played badly.


“I could have kicked myself, because originally I had thought of Fourteen, and only at the last moment had I climbed down to Eleven, I really don't know why. ‘Listen.’ I turned to Ervinke. ‘What would have happened had I said Fourteen?’
‘What a question! I'd have lost. Now, that is just the charm of poker: you never know how things will turn out. But if your nerves cannot stand a little gambling, perhaps we had better call it off.’”


Of course this logic doesn’t make any sense. If the narrator had said fourteen, then Ervinke, the clever bastard, would have said fifteen. No matter how colossal the number you pick, it’s always going to be smaller than your opponent’s could be.


“Without saying another word, I put down ten piasters on the table. Ervinke did likewise. I pondered my number carefully and opened with Eighteen.
‘Damn!’ Ervinke said. ‘I have only Seventeen!’
I swept the money into my pocket and quietly guffawed. Ervinke had certainly not dreamed that I would master the tricks of Jewish poker so quickly. He had probably counted on my opening with Fifteen or Sixteen, but certainly not with Eighteen. Ervinke, his brow in angry furrows, proposed to double the stakes.”


So, what if we try playing simultaneous Jewish Poker, each player writing down their number before revealing? Something’s still wrong. Depending on your level of mathematical sophistication, you’ll cram your paper with nines, or you’ll search the Internet for the fastest-growing function you can find, and call it on a trillion. (In fact, it’s probably better to cram your paper with ones; you can fit more in.)


Now this is starting to look like a weird version of Rock-Paper-Scissors. In fact, it looks a lot like Rock-Paper-Scissors, doesn’t it? In Rock-Paper-Scissors, each strategy loses to and beats one other strategy. In Jewish Poker, each strategy loses to and beats an infinite number of other strategies…


“‘As you like,’ I sneered, and could hardly keep back my jubilant laughter. In the meantime a fantastic number had occurred to me: Thirty-five!
‘Lead!’ said Ervinke.
‘Thirty-five!’
‘Forty-three!’
With that he pocketed the forty piasters. I could feel the blood rushing into my brain.”


So are all strategies equivalent here, like in Rock-Paper-Scissors? Can we pick “at random”? (Whatever that means…) That doesn’t seem right. Some strategies just dominate others. Surely “Ackermann of one billion, one billion” isn’t equivalent to “Negative one times Ackermann of one billion, one billion.” Yes, they’re both much smaller than infinity, but from experience, I can tell you: one will win you more games of Jewish Poker than the other.

Of course, maybe your opponent doesn’t know fancy concepts like the Ackermann function; maybe you’re just better at writing small. But those seem like silly constraints, independent of the game itself, forced upon us by our sad, physical existence. One could conceive of deities playing this game, deities that could conceive of any, any, number they wanted, unconstrained by the finite bounds of our universe, and somehow even they seem to have problems with this game. Every number is an infinitesimal grain of sand on the vast beach of infinity; every number is so, so much smaller than what could have been said.

And yet a winner must be declared…


“‘Listen,’ I hissed. ‘Then why didn't you say Forty-three the last time?’
‘Because I had thought of Seventeen!’ Ervinke retorted indignantly. ‘Don't you see, that is the fun in poker: you never know what will happen next.’
‘A pound,’ I remarked dryly, and, my lips curled in scorn, I threw a note on the table. Ervinke extracted a similar note from his pocket and with maddening slowness placed it next to mine. The tension was unbearable. I opened with Fifty-four.
‘Oh, damn it!’ Ervinke fumed. ‘I also thought of Fifty-four! Draw! Another game!’”


It’s possible to make versions of Rock-Paper-Scissors with more strategies, with each strategy losing to half of the other strategies and beating the other half. See Rock-Paper-Scissors-Lizard-Spock, for instance. (Like the original, except Lizard beats Paper and Spock and loses to Rock and Scissors; Spock beats Rock and Scissors but loses to Paper and Lizard.) There’s even a version with 25 elements. So is that what Jewish Poker is? The limit of these games, approaching infinity?

Well, no, because you can make games of Rock-Paper-Scissors with infinite elements and not suffer from Jewish Poker’s bizarre defects. Take, for instance, the following game:

-Each player says a number simultaneously.
-The number must be positive.
-If the difference between the two numbers is greater than one, the player who said the smaller number wins. Otherwise, the player who said the larger number wins.

After a bit of thought, it becomes clear that saying numbers greater than 2 is pointless: You might as well have said the number you just said, but minus 2. (For example, don’t say 2.7; say 0.7. That beats everything 2.7 would have beaten, but avoids losing to 0 through 0.7 and 2.7 through 3.7.) And now we have a kind of “cycle,” looping back to 0 when it reaches 2, of an uncountably infinite number of different plays, each one beating and losing to half of the other plays. That’s what we really wanted to see, when we talked about Rock-Paper-Scissors with infinite elements.


“My brain worked with lightning speed. ‘Now you think I'll again call Eleven, my boy,’ I reasoned. ‘But you'll get the surprise of your life.’ I chose the sure-fire Sixty-nine.

‘You know what, Ervinke’- I turned to Ervinke –‘you lead.’
‘As you like,’ he agreed. ‘It's all the same with me. Seventy!’

Everything went black before my eyes. I had not felt such panic since the siege of Jerusalem.”


So what’s Jewish Poker then? Well, I don’t really know, but I’d say it’s probably a game best left to the deities.


“‘Nu?’ Ervinke urged. ‘What number did you think of?’
‘What do you know?’ I whispered with downcast eyes. ‘I have forgotten.’”




The full story, by Ephraim Kishon, can be found here.



-Adam

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