"Suppose you get one point for every round you win." says Bengt.
"Sounds reasonable."
"And suppose you play forever."
"Sounds a lot less reasonable."
"Oh, you know what I mean. You play for a really long time so that everything becomes statistical and stuff."
"Okay, sure."
"So, question: What is the optimal strategy?" asks Bengt.
"That depends on what the other guy does."
"Yeah, but assuming you don't know anything about the enemy's strategy."
"Enemy?"
"Or opponent, whatever."
"There isn't one."
"Right. There isn't any strategy that makes sure you win. So we just slightly redefine 'optimal', just for the sake of this problem, to mean the strategy which makes sure you play even. In the long run."
"Actually, it's very unlikely that you'll play even, in a very long game."
"Oh, you know what I mean. You'll sort of play even, almost."
"It's not like you to be so vague."
"All right, how about this: The optimal strategy is the strategy such that regardless of the enemy's strategy the expectation value of one's score minus the enemy's score is zero."
"Sounds good. And that strategy is to pick a sign at random."
"Specifically, with the same probability for each sign. That's important."
"Okay, sure. What's your point?"
"Well, it's boring. It's no fun if you can't win. Maybe we could ban the use of that strategy?"
"I don't think that would work."
"Fine. Let's think of some other games. Here's one: It works the same, except that if you win with a rock, you get two points instead of one."
"Eh, sounds like fun... Here's one I've heard of: Rock-Paper-Scissors-Lizard-Spock. It's like the normal game but with five signs. Lizard beats paper and Spock, Spock beats rock and scissors."
"Oh, I have another one: Rock-Paper-Water-Lichen-God."
"Okay, how does that work?"
"Well, paper beats rock as usual, and it also beats lichen, because, I don't know, you can scrape it up with the paper or something. Lichen beats rock and water, because it grows on the rock and drinks water. Water beats paper, obviously, and it also beats rock, because dripping water can wear down the rock."
"Wait, so they all beat rock?"
"Yup. And God beats all of them. But rock beats God."
"How does rock beat God?"
"It's the rock from the Omnipotence Paradox."
"Oh, that rock. I see."
- In the original game, the so called optimal strategy is, as Alban said, to pick each sign with equal probability. But what happens when you make the modification that winning with rock gives two points? What distribution should you use to be "optimal"?
- How about the five-sign game, Rock-Paper-Scissors-Lizard-Spock? With equal points, the optimal strategy is again of course to use equal distribution, but what if winning with rock scores double?
- Finally, Bengt's own invention. Now we give one point per win again. What is the optimal distribution?
- Alban is of course right in saying that you are unlikely to play even, after a large number of games. After n games, what is the expectation value of the (absolute) point difference?
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