## Rock, Paper, Scissors Robot Redux: Brash Is Right, You're Wrong

Posted on by Benjamin Chabot-Hanowell (Brash Equilibrium)URL for sharing: http://thisorth.at/1v7j

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Earlier this month, I posted about my epic Rock, Paper, Scissors tournament with the New York Times R.P.S. robot. The point of the New York times article is to demonstrate how humans are really bad at games like R.P.S. because our behavior is highly predictable. To the New York Times robot, who has near perfect recall of your and (if you played the veteran robot) 200,000 other players' tendencies in the game, your strategy very quickly becomes predictable. Because the robot gets very good at predicting your highly predictable "strategy," and you as a human aren't as good (unless you are Alan Turing or something), you lose hardcore. But by using a little trick with a random number generator, I did much better against that evil robot. I tied!

I got a lot of comments from people saying that I suck at Rock, Paper, Scissors and that beating the computer (instead of tying) is easy. These comments imply that I am dumb and wrong. Those of you who follow me on This or That know that my ego is bigger than this celestial object:

Check out the schematic above for a spell. No one strategy is dominant. If you're paper, scissors beats you. If you're rock, paper beats you. If you're scissors, rock beats you. It's a cycle. When you work the math out and assume two equally good players with perfect memories who never make mistakes and know the rules of the game and want to win it, the best strategy for both players is this:

The outcome of this game between two perfect players is, as you might have guessed:

My "win" of three points is, statistically speaking, a tie. Specifically, if we assume that I'm right (as we should always assume!) and my strategy should result in me winning only a third of the time, then the probability of me getting 79 wins or greater out of 233 rounds is a whopping 45% (from the binomial probability distribution, where a "success" is defined as me winning). So, yeah, I tied the computer. Nobody's arguing about that.

Now let's look at the claims some others have made regarding how "easily" they "beat" the New York Times robot. I'll start with the most recent claim from rdsoze, who writes, "all I can say is that it ain't difficult to beat the veteran comp...My stats were...My Wins-100, Ties -94, Comp wins- 91." He even posted a screen shot to prove it. (Don't be fooled by the screen shot, like I was, that rdsoze only got 10 wins. The New York Times article can only handle two digits.)

Wow! Victory is rdsoze's!

Not so fast, sucker! What's the cumulative probability that you'd get 100 or more wins out of 285 rounds, assuming you should win only a third of the time? Hold on a second. I can do cumulative probability calculations like this in my head if I sit down and Euro-style it like a boss, like I always do. And like I always $%#&ing will.

That's better. Hold on, I gotta get my thinking cap on, too.

Oh, snap! That's classic Brash Equilibrium, baby!

Drum roll, mother $%#&er!

The cumulative probability that I'm right is 28%! I don't know about you, but I'd bet my money that rdsoze tied the computer (and I'd be right more than a fourth of the time). Thing is, rdsoze still did damn well for a human, who should have gotten stomped by the computer (because we're so predictable). So the question is whether rdsoze used some kind of randomization device or if his brain actually is a randomization device. Or, if he didn't use a randomization device and played with a naked brain, the question is: is he even human? Open question, man. Open question.

Sorry, rdsoze.

What about danielgrad, who went against my post's advice (Lesson #2: Don't think you're cleverer than the machines) and started lying to the robot? Basically, danielgrad proposes that if you learn the machine's beliefs about your tendencies and then do the opposite, you can game the system. His screen shot evidence comes from a game of only 9 rounds, of which he won 8. If he kept that up, he's doing well. Even with the small number of rounds, he's winning even in a statistical sense (the chances of winning 8 rounds or more out of 9, assuming I'm right, are about 1 in 1,000). He claims that he won 60% of the rounds in a tournament consisting of almost 30 rounds, but provides no evidence. If he's not lying, he might have a point: the chances that I'm right are only about 1 in 500.

My suspicion, however, is that even if danielgrad is telling the truth, he'll only be able to game the system for so long (unless he is cheating and monitoring what the robot is "thinking," which you can do on the New York Times site). The reason is that, as danielgrad keeps switching strategies, the way in which his strategies shift will become predictable. Or, at best, he will start shifting his strategies so often that, in the limit of a very large number of rounds, it will be as if he is playing a random strategy. Which means that, in this limit, the tournament will become a tie.

But, listen, I'm not arguing it isn't possible to game the computer program. I'm just arguing it is nearly impossible to do so if you are a human, rather than an even better strategy prediction algorithm. It remains to be seen if there actually is a computer algorithm that can do much better than the New York Times robot. But eventually, the best strategy prediction algorithm will be found (or something close to it). Hell, I bet some computer scientist has already figured it out by now.

But when two computers using that best algorithm play against each other, you know what the outcome will be?

I add another caveat. I'm not arguing that the best thing to do in any case when playing against a smart computer is to draw your strategy randomly from a hat. It's easy to think of a situation in which that would be a terrible thing to do.

Here is an example. Say that you're playing another simple game against a smart computer with a perfect memory. In this game, you have two choices. Your first choice is that you can engage the robot in a coin flipping contest. If the robot also chose (simultaneously) to do the coin flipping contest, you flip a coin. If it lands on heads, you get 1 point and the robot loses four points. If it lands on tails you lose 4 points and computer gains 1 point. If the robot decides not to participate in the coin flipping contest, you get 1 point with certainty.

Your second choice is to allow the robot, if it decides to participate in the coin flipping contest, to take 1 point while you take nothing. If instead the robot also decides (simultaneously) to forgo a coin flipping contest, you split 1 point with the robot (taking a half point).

In this case, it turns out that you should be willing to play the coin flipping contest a fourth of the time, and forgo coin flipping the rest of the time. In other words, you should draw your strategy from a hat in which there are three "forgo coin flipping" cards for every "coin flip contest" card. For you geeks out there, the game I just described is equivalent to the famous hawk-dove game.

So, I challenge you, danielgrad, and others, to showing me that you are cleverer than the New York Times robot in an epic game of at least 200 rounds. Post your results as a comment to this feature article. If I check the numbers and it is legit, I will create an epic ToT in your honor.

*The tournament was about as epic as this Uruki, bull-stomping hero whose name*

*starts with a "G."*

I also love a challenge, especially when it is intellectual, and especially when it comes to game theory, which I have studied at a graduate level for four years. I'm not saying this means I am 100% sure I'm right. I'm just saying it means that, when it comes to people saying I suck at a game for which I have myself computed (not originally, I admit) the Nash equilibrium, well. Let's put it this way.

I'm sorry, that was Brash of me. Let me class it up a bit:

I'm sorry, that was Brash of me. Let me class it up a bit:

So, let's review the problem. For those who've forgotten, here are the rules to Rock, Paper, Scissors:

Check out the schematic above for a spell. No one strategy is dominant. If you're paper, scissors beats you. If you're rock, paper beats you. If you're scissors, rock beats you. It's a cycle. When you work the math out and assume two equally good players with perfect memories who never make mistakes and know the rules of the game and want to win it, the best strategy for both players is this:

- play rock a third of the time
- play paper a third of the time
- play scissors a third of the time

The outcome of this game between two perfect players is, as you might have guessed:

- Player 1 wins a third of the time.
- There is a tie a third of the time.
- Player 2 wins a third of the time.

My "win" of three points is, statistically speaking, a tie. Specifically, if we assume that I'm right (as we should always assume!) and my strategy should result in me winning only a third of the time, then the probability of me getting 79 wins or greater out of 233 rounds is a whopping 45% (from the binomial probability distribution, where a "success" is defined as me winning). So, yeah, I tied the computer. Nobody's arguing about that.

Now let's look at the claims some others have made regarding how "easily" they "beat" the New York Times robot. I'll start with the most recent claim from rdsoze, who writes, "all I can say is that it ain't difficult to beat the veteran comp...My stats were...My Wins-100, Ties -94, Comp wins- 91." He even posted a screen shot to prove it. (Don't be fooled by the screen shot, like I was, that rdsoze only got 10 wins. The New York Times article can only handle two digits.)

Wow! Victory is rdsoze's!

Not so fast, sucker! What's the cumulative probability that you'd get 100 or more wins out of 285 rounds, assuming you should win only a third of the time? Hold on a second. I can do cumulative probability calculations like this in my head if I sit down and Euro-style it like a boss, like I always do. And like I always $%#&ing will.

That's better. Hold on, I gotta get my thinking cap on, too.

Oh, snap! That's classic Brash Equilibrium, baby!

Drum roll, mother $%#&er!

The cumulative probability that I'm right is 28%! I don't know about you, but I'd bet my money that rdsoze tied the computer (and I'd be right more than a fourth of the time). Thing is, rdsoze still did damn well for a human, who should have gotten stomped by the computer (because we're so predictable). So the question is whether rdsoze used some kind of randomization device or if his brain actually is a randomization device. Or, if he didn't use a randomization device and played with a naked brain, the question is: is he even human? Open question, man. Open question.

Ooh, baby. Now that is one naked brain.

Sorry, rdsoze.

What about danielgrad, who went against my post's advice (Lesson #2: Don't think you're cleverer than the machines) and started lying to the robot? Basically, danielgrad proposes that if you learn the machine's beliefs about your tendencies and then do the opposite, you can game the system. His screen shot evidence comes from a game of only 9 rounds, of which he won 8. If he kept that up, he's doing well. Even with the small number of rounds, he's winning even in a statistical sense (the chances of winning 8 rounds or more out of 9, assuming I'm right, are about 1 in 1,000). He claims that he won 60% of the rounds in a tournament consisting of almost 30 rounds, but provides no evidence. If he's not lying, he might have a point: the chances that I'm right are only about 1 in 500.

My suspicion, however, is that even if danielgrad is telling the truth, he'll only be able to game the system for so long (unless he is cheating and monitoring what the robot is "thinking," which you can do on the New York Times site). The reason is that, as danielgrad keeps switching strategies, the way in which his strategies shift will become predictable. Or, at best, he will start shifting his strategies so often that, in the limit of a very large number of rounds, it will be as if he is playing a random strategy. Which means that, in this limit, the tournament will become a tie.

But, listen, I'm not arguing it isn't possible to game the computer program. I'm just arguing it is nearly impossible to do so if you are a human, rather than an even better strategy prediction algorithm. It remains to be seen if there actually is a computer algorithm that can do much better than the New York Times robot. But eventually, the best strategy prediction algorithm will be found (or something close to it). Hell, I bet some computer scientist has already figured it out by now.

But when two computers using that best algorithm play against each other, you know what the outcome will be?

# A $%^*ing tie!

I add another caveat. I'm not arguing that the best thing to do in any case when playing against a smart computer is to draw your strategy randomly from a hat. It's easy to think of a situation in which that would be a terrible thing to do.

Here is an example. Say that you're playing another simple game against a smart computer with a perfect memory. In this game, you have two choices. Your first choice is that you can engage the robot in a coin flipping contest. If the robot also chose (simultaneously) to do the coin flipping contest, you flip a coin. If it lands on heads, you get 1 point and the robot loses four points. If it lands on tails you lose 4 points and computer gains 1 point. If the robot decides not to participate in the coin flipping contest, you get 1 point with certainty.

Your second choice is to allow the robot, if it decides to participate in the coin flipping contest, to take 1 point while you take nothing. If instead the robot also decides (simultaneously) to forgo a coin flipping contest, you split 1 point with the robot (taking a half point).

In this case, it turns out that you should be willing to play the coin flipping contest a fourth of the time, and forgo coin flipping the rest of the time. In other words, you should draw your strategy from a hat in which there are three "forgo coin flipping" cards for every "coin flip contest" card. For you geeks out there, the game I just described is equivalent to the famous hawk-dove game.

This is a relevant graph that Brash made in Mathematica (back when he had a user license), which serves to make him look smart. If you know what it is, tell Brash by asking him a question on his ToT profile. If you're right, he'll make a ToT in your honor.

So, I challenge you, danielgrad, and others, to showing me that you are cleverer than the New York Times robot in an epic game of at least 200 rounds. Post your results as a comment to this feature article. If I check the numbers and it is legit, I will create an epic ToT in your honor.

# Is Brash right...or -always- right?

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## Debate It! 11

Posted By lockheed40, (3 years)

Posted By Karla, (3 years)

I love you, Honeysuckle. Here are some virtual Gerbera daisies. And you have something nice to look forward to for our upcoming anniversary!

Reason:de-biggened the image

Posted By Brash Equilibrium, (3 years)

Again.

Posted By Brash Equilibrium, (3 years)

Reason:stupid look of disapproval won't display; had to move to Plan B

Posted By Rebecca, (3 years)

Posted By Brash Equilibrium, (3 years)

Posted By Rebecca, (3 years)

Posted By Brash Equilibrium, (3 years)

Posted By smidge3, (3 years)

Posted By Brash Equilibrium, (3 years)

Posted By elnwood, (2 years and 1 months)

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