wiggin is correct.

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/thread

If Vermont is the opponent, I divide my two-dice stacks into two groups of one, and make two attacks each 1 v 2.

Rock paper scissors is a game of incomplete information. Like kdice.

Yes, Wiggin is correct and I gotta give him bonuspoints for noting that optimal play in a 8vs8 will result in the opponent doing a 8+3 vs 8+4 attack.

But yeah, attacking 2vs2 is optimal. The proof for it is quite simple as well:
Assume 2vs2 is optimal, then your opponent will also 2vs2 if you lose, so your winchance becomes the infinite series montecarlo mentioned, with a ~64% winchance for the first player.
Now the only thing we need to proof is that there doesn't exist a better strategy than winning ~64%. Attacking 3vs3 through to 8vs8 will all result in lower winchances (though the chance of winning the first strike are higher than 2vs2, if you lose your chances are drastically lower).
So we come to the 8vs8 stage and without looking at it in depth we can already place an upper-bound on the first player's winchance. If we assume the first player can get an advantage (>53%) then the optimal opponent will always attack first, reducing that advantage to just under 53%. And this is lower than the ~64% you could get by attacking 2vs2, QED!


But looking at the 8vs8 stage in itself is pretty interesting as well. For the first player it is not in his best interest to attack first or with a little reserve. But does that mean it's best to build up to 8+6? NO! Your opponent could attack 8+5 vs 8+6 and get a ~55% chance of winning.
Well, it turns out that optimal play will result in your opponent attacking 8+3 vs 8+4, resulting in a ~52% winchance for the first player.

Anyway, it's been fun seeing some people breaking their heads over it... ;)

skrum, I'm not sure you understand what 'complete information' means as it pertains to game theory.

Per gametheory.net:
"A game is one of complete information if all factors of the game are common knowledge. Specifically, each player is aware of all other players, the timing of the game, and the set of strategies and payoffs for each player."

Per wikipedia:
"Complete information is a term used in economics and game theory to describe an economic situation or game in which knowledge about other market participants or players is available to all participants. Every player knows the payoffs and strategies available to other players."

Nothing in kdice is hidden. Everyone knows (or at least has the ability to know) the odds of any particular action, hence it is a complete information game. A random die roll as part of the game does not prevent this.

Also, comparing kdice to roshambo is difficult because kdice is what is known as a 'sequential' game while roshambo is 'simultaneous.' Simultaneous games often employ a different set of strategies (such as taking a low percentage strategy occasionally, which is what you suggested above.) That same tactic does not apply to kdice, a sequential complete information game.

(Not too sound too dorky (too late?) but game theory is very cool stuff.)

Vermont:

Thanks for your point on complete information. If kdice is a game of perfect information, does that mean that it is possible to design a single best kdice-playing robot?

I have thought that there might be more than one successful "species" of player in kdice. From time to time I have brought this up, mentioning four kinds: sphinx (or lion), dromedary, hyena, or horse. MadHatSam is a horse, since he is good at cutting his losses in a bad start. He tends to have a high percentage of 7th place finishes.

So with perfect information, would it be possible to figure out in advance if there is more than one successful strategy, and how many different succesful strategies there could be, and how many players would adopt a particular strategy?

roll off?

Vermont,

Another query. If a random dice roll does not make a game have imperfect information, would a player strategy that has randomness also not make a game have imperfect information? Suppose I were to design a robot that would attack in any one-on-one confrontation with the probability in the odds table (for example, in a 2 v 2 the robot would attack 44.4% of the time). Would the presence of this robot in the game make the game one with imperfect information?

Skrum, you mentioned perfect information. Keep in mind that in game theory that perfect information is different than complete information. I'm not sure if you were making that distinction on purpose or not.

A perfect games is one where all players know all moves that have taken place. Kdice would count as a perfect game as none of us make our moves in secret.

ok, now moving onto your questions.

Q1: If kdice is a game of perfect information, does that mean that it is possible to design a single best kdice-playing robot?

Sure, it is possible. Extremely difficult given the percentage of random play involving starts, attacks, and restacks, but theoretically possible. You also have to take into account that bot can't play the social part of the game, but that's not an issue if we're in the context of Ryan's iphone app. We'd also need to define best, as even kdice players have different definitions - highest 1st percentage, highest overall percentages, maximum score, etc.. That would probably affect the species you're talking about as well. {I could go on about this question in far more detail, but this post will be long enough as it is.)

Q2: So with perfect information, would it be possible to figure out in advance if there is more than one successful strategy, and how many different succesful strategies there could be, and how many players would adopt a particular strategy?

The most successful bot would most likely take different tacks at different times, but given the exact same gameplay to a certain point in a game, the "ideal" bot would make the best move each time, whatever it is, and that "best" move would not change in an identical game. Keep in mind that even in a game as analyzed as chess there are still debates as to the best lines in certain openings. However, checkers has been solved.

Q3: If a random dice roll does not make a game have imperfect information, would a player strategy that has randomness also not make a game have imperfect information?

Again, I'm not sure if you mean complete or perfect information, but it would not affect the status of either. Assuming that roll is a good move, a bot that took it 100% of the time should outperform a bot that only took it 44% of the time. You can do a simple matrix contrasting attacking/sitting vs results to see what I'm talking about. (Now, if you could guarantee that you'd only attack the 44% where you win, that's a different story!)

Hopefully, I was clear. Rereading this post in the little box is tough. I can clarify anything needing it later on.