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The first part of my question is simple: Is every game continuous in time and strategy-space also a game of perfect information with a good equilibrium? For example, consider rock-paper-scissors. The discrete version has no nash equilibrium - a perfectly uniform random mixed strategy is the best option.

Continuous rock-paper-scissors, by contrast, allows players to move at some limited velocity (consider 2 cases, acceleration is limited and acceleration is infinite) through a "strategy space" s.t. R+P+S = 1, and (0.5, 0.5, 0) vs. (0,1,0) returns 0.5 to player one and -0.5 to player 2, while (1,0,0) returns 1 to player one, -1 to player 2. To avoid the "go directly to the middle" strategy, it's fine to remove (1/3,1/3,1/3) or some disk around it from the strategy plane.

So, is continuous RPS effectively a game of perfect information?

For a more dramatic example, consider the stock market as a game. If it were continuous, would randomness essentially be removed? Would a player also need to explicitly know the strategies of all other players as individuals, or only the end result of those strategies (i.e. value of stocks at a given point in time) in order to play perfectly?

For a more realistic example, consider a hunt between a dog and hare. Strategies for them are the direction they choose to run in the pursuit. The dog has reflexes r, the time it takes him to notice the hare's change in direction. The rabbit has acceleration a. Ignore the dog's acceleration for now. As r*a becomes extremely small (i.e. the dog's reflexes are swift relative to the hare's acceleration), does this effectively converge to a continuous, perfect-information game (specifically the game of the homicidal chaffeur), or is the difference still important? Specifically, suppose that the dog can only make decisions on pursuit directions at increments equal to r - I don't want it to be a continuous game with a lagging signal.

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It seems like you want more than continuous; that you want second-differentiable, with the control a player has being the ability to dictate what the acceleration of his position in strategy space is. –  Greg Muller Jun 4 '10 at 19:22
    
Does second-differentiability matter if time and strategy space are continuous? I know it matters if strategy space is continuous but time isn't (a la the last example), but I'm not clear on whether that case allows a very small discrete timestep to converge to the instance of a continuous game. –  DoubleJay Jun 4 '10 at 20:58
    
In any reasonable model of the stock market, a fundamental source of randomness is the random arrival of new information about future dividend streams. I don't see how you propose to eliminate that. –  Steven Landsburg Mar 30 '11 at 3:19
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2 Answers

I don't know the answer to your question, but perhaps we might gain insight from the following Interview with Jason Simmons, a professional rock/paper/scissors player, which appeared a few years ago on NPR.

Transcript: (Audio link available)

RENEE MONTAGNE, host:

And the election may be over, but several House races are still too close to call. Recounts are likely, but how about a different way to settle things - by playing rock paper scissors. It's quick, simple and has the charm of being non-partisan. Rock smashes scissors, scissors cuts paper, paper covers rock.

STEVE INSKEEP, host:

With so much at stake, the campaigns may want to bring in a consultant like Jason Simmons. He is a professional rock, paper, scissors player. And tomorrow he will attend the RPS World Championship in Toronto.

Mr. JASON SIMMONS (Professional Rock, Paper, Scissors Player): Under the name Master Roshambollah, I've competed on three separate continents, both in North America. I've competed in Asia. I've also competed in Australia. And over the last couple of years I've been retired.

INSKEEP: Is there money at stake here?

Mr. SIMMONS: Absolutely. Not only is there, you know, the 10,000 dollars coming up in Toronto, I was at a tournament in Vegas earlier this year where 50,000 dollars was on the line. I've heard of an Israeli tournament that had a quarter of a million dollars on the line.

INSKEEP: Let's play a game, shall we?

Mr. SIMMONS: You know, actually, I'm retired and I have to clear all of my matches through my manager. And I actually get an appearance fee for actual matches that I play.

(Soundbite of laughter)

INSKEEP: Okay, hold on, hold on. We have to clarify this. You're retired under the name Master Roshambollah.

Mr. SIMMONS: Correct.

INSKEEP: You're going to play under another name at this forthcoming championship.

Mr. SIMMONS: I have no idea what you're talking about.

INSKEEP: Explain to me what's happening here.

Mr. SIMMONS: When I retired in 2004, it was to let some of the younger players have their shot at some of the glory that I've achieved. There is a player who goes by the name of the Midnight Rider and...

INSKEEP: In fact, I'm looking at a photograph here from the New York Rock Paper Scissors Association of someone named Midnight Rider. He's wearing a mask but seems to have your hairstyle.

Mr. SIMMONS: That is correct. And although he's put up a tremendous stream of tournament victories over the last two years, I personally find his style of play to be very derivative.

INSKEEP: Derivative of?

Mr. SIMMONS: Of my own.

INSKEEP: By the way, there are people who call it paper, rock, scissors?

Mr. SIMMONS: There are. But you can even do a Google search. Rock, paper, scissors tends to be the most common.

INSKEEP: Scissor, paper, rocks?

Mr. SIMMONS: Never hear it.

INSKEEP: Scissor never gets to start.

Mr. SIMMONS: It never gets to start, and that's primarily because scissors is very rarely used as an opening throw. During my professional career, I noted that female players tended to open with scissors a lot more often than their male counterparts. In a lot of the media interviews that I did before my retirement, a lot of reporters invariably opened with paper.

INSKEEP: Aren't you curious what I'd open with?

Mr. SIMMONS: If I had to guess just based on, like, if I was facing you in a match of rock, paper, scissors right now, I would guess scissors.

INSKEEP: Shall we play one game?

Mr. SIMMONS: Let's play one quick one.

INSKEEP: Okay.

Mr. SIMMONS: All right. Do you have any requests for my opening throw?

INSKEEP: A request for your opening throw?

Mr. SIMMONS: Yeah, what would you like for me to open with?

INSKEEP: What would I like - and I don't know that you're actually going to do this? I'm just going to say...

Mr. SIMMONS: I'm just asking.

INSKEEP: ...and you're going to get information from me asking you what to throw, what you think I might throw?

Mr. SIMMONS: I've already heard enough. Let's go.

(Soundbite of laughter)

Mr. SIMMONS: Heard everything I need to hear.

INSKEEP: OK.

Mr. SIMMONS: (Unintelligible)

INSKEEP: All right. One...

Mr. SIMMONS: One, two, three...

INSKEEP: ...shoot.

INSKEEP: Oh! He beat me. I did a rock, he did paper.

Mr. SIMMONS: Paper over rock.

INSKEEP: OK.

Mr. SIMMONS: (Unintelligible).

INSKEEP: Two out of three, two out of three.

Mr. SIMMONS: All right, let's see what else you got.

INSKEEP: OK.

Mr. SIMMONS: One, two, three, shoot.

INSKEEP: Oh!

Mr. SIMMONS: When I say what else you got...

INSKEEP: Yeah.

Mr. SIMMONS: ...that subtly encourages you to go against me by throwing the same throw twice...

INSKEEP: But instead I lost again. I did rock again and you did paper again.

Mr. SIMMONS: Exactly.

INSKEEP: All right.

Mr. SIMMONS: The game started long before we actually threw the first throw (unintelligible).

INSKEEP: I'm not sure I was manipulated into throwing the rock. But you feel that you did.

Mr. SIMMONS: Good, yeah, that's fine. You always want to leave your opponent feeling good after a loss. Now I will say that doubling your throws the way you did is typically the sign of an intermediate or better player.

INSKEEP: Are you just praising me so I'll feel good about being defeated (unintelligible).

Mr. SIMMONS: It was a great match. It was a great match. I was honored. I broke a sweat. As possible as it is for, you know, getting taken out in two consecutive throws, it was a tough match for me.

(Soundbite of song)

Unidentified Singer: (Singing) Rock, paper, scissors; scissors, paper, rock. Tryin' to keep the hunger, but the hunger never stops.

INSKEEP: Jason Simmons is a professional rock, paper, scissors player who bears a strong resemblance to a competitor in the World Championship this weekend in Toronto. And, Renee, we need to end this part of the program by reading the credits. Who's goes first?

(Soundbite of laughter)

MONTAGNE: Who goes first? Scissors.

(Soundbite of laughter)

INSKEEP: I had rock again. I got you this time.

(Soundbite of banging)

MONTAGNE: OK, start.

(Soundbite of laughter)

INSKEEP: OK. From NPR News, this is MORNING EDITION.

For more practical advice on strategy, see this NPR story on playing techniques for the 2003 championship, including cloaking, priming the chump and paper clipping.

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Apologies to all who find this answer inappropriate. –  Joel David Hamkins Jun 4 '10 at 16:58
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If I interpret your question correctly, continuous RPS isn't a game of perfect information. I'm assuming what I know at any instant is A) my strategy at any past instant, and B) the outcome of our two strategies matched up against each other at any past instant.

But I can't deduce your strategy from knowing mine and the outcome; consider my strategy of (1,0,0) and an outcome of 0 for both players. I can't tell if you played (1,0,0) or (0,.5,.5) at that moment. In the continuous game, you could be moving anywhere you please on the line connecting those two strategies while I stayed at (1,0,0), and I would be none-the-wiser.

It is, however, true that in any continuous game where your opponent's strategy can be completely deduced from the outcome, you have perfect information. This follows by reverse-engineering his strategy at all previous moments, and taking the limit. Unfortunately, if the outcome is represented by a single real number, then the strategy space for such a game has to be essentially 1-dimensional, which means they are rather boring.

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Oh, I hadn't thought of it that way - I had kind of assumed you could reverse-engineer (or that you were just plain given the opponent's strategy). But if I'm understanding you right, the "imperfection of information" that comes from simultaneous discrete decisions IS eliminated by a continuous game. That answers my first question, thanks! –  DoubleJay Jun 4 '10 at 20:55
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