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Subject: Game theory: playing illogically reaps a larger reward rss

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Nate Straight

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Except it's all wrong. Any idiot can see that 2 is, under any definition, not the most logical choice. Having a possible range of outcomes between 2 and 4 is never more logical or rational than having a possible range of outcomes between 99 and 101. If she really thought the other guy was going to reason through the whole process they describe and then select 2 as his answer, she wouldn't stop her reasoning there... she'd go back up and pick 100, because even the 98 she'd get after the penalty would be better than the 2 she'd get otherwise. This infinite regress could continue indefinitely and is nowhere even remotely close to being rational.

The most rational choice is somewhere between 98 and 100, depending on your tolerance for risk. All of those choices are guaranteed to get you more money than any other choice. You're not going to start reasoning your way on down to choices that net you between, say, 49 and 51, because you've already passed up choices that give you twice that much. Game theory is silly business.
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Jon W
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No, the game is set up so that if X picks 100 and Y picks 2, X gets 0 and Y gets 4. If the numbers are different, the reward is set at the lower number, with the person choosing that number getting a +2 bonus, and the high chooser getting a -2 ding (from the lower choice).

But I agree the problem makes more sense when the range has a higher lower boundary (as in the real-world experiments described).

(Edited for better clarity, and a direct use of Nate's actual example.)
 
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Nate Straight

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Jon Waddington wrote:
No, the game is set up so that if X picks 98 and Y picks 2, X gets 0 and Y gets 4. If the numbers are different, the reward is set at the lower number, with the person choosing that number getting a +2 bonus, and the high chooser getting a -2 ding.

Ah... now I see.

I still think a higher number is the more rational choice. Reason isn't just about cold logic and mathematics. It includes experience and empirical observation. No one in their right mind is going to sit down and go through this whole process and select 2 as their final answer. Knowing this (just from knowing human nature and sensibilities), you can safely select a higher number, up to 100 (or, more than likely, the two will just write down their best guess as to the correct price... but that's a moral / normative issue).

You'd only write down 2 if you were ridiculously risk averse and paranoid about what other people might do to you. Otherwise, you'd write down a high number and anticipate that the other person would be sensible and not try to niggle their way into an extra buck and cause the whole system to come crashing down.

I still think it's all silly business with no tie to reality.
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MK
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This is actually not too far from the "math problem" that appears in Beautiful Mind.

The way the logic in this problem is posed, each person is competing against each other to reap a slightly higher dollar award, but the reality is, they are both competing against the clueless airline manager. If the players view the game as playing against each other, then they both make the wrong choice - the one that lets the airline manager win. But if they view it correctly as playing against the airline manager, then they both win $100 and the airline manager loses.

In "Beautiful Mind" the problem was posed as there being 4 guys and 5 girls at the bar. Only 1 of the girls was a knockout blonde bombshell, the others were all plain-jane brunettes. Nash is portrayed as realizing the following:

- If the guys compete against each other for the blonde, then only one of them can win, the other three must settle for one of the Janes, but the Janes will feel slighted and won't give the guys what they want.

- If the guys agree not to compete for the blonde and let one of them take her, the other three guys will still feel they got the short-end of the stick, which the Janes will pick up on, and again, nobody gets what they want.

- But if all four guys agree not to compete, and each takes a brunette (leaving the blonde completely out of the equation), then each guy gets a girl, no one of the Janes feels slighted, and the only one that loses is the blonde.


Perspective on who you are really competing against, and whom you should be cooperating with, is the essence of this kind of problem.
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Neil Whyman
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I have to agree with Nate. 2 is only the logical answer if you assume that the two travellers are competing against each other and nobody else. But the scenario proposed has them competing against the airline. They would even consider themselves a team and would probably both reason out what was best for the team, putting that ahead of what is best for the individual. So better to have $200 between them than $4 between them no matter what the distribution of the cash.

And then in my mind it is better to have a small chance of getting $100 than a large chance of getting $4. You can't even buy a decent game with $4!
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Jorge Arroyo
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It's funny that this "logic" they talk about is always about betraying the other person. If I know the other person, I can trust they'll choose 100 and so will I. What do I care if I don't earn 1$ more than my friend...

But anyway, everyone knows the right thing to do is tell the true price!
 
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Nate Straight

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maka wrote:
But anyway, everyone knows the right thing to do is tell the true price!

Game theorists will tell you "who cares about the right thing to do, this isn't ethics," but I think that's actually a rather sad thing that we can't include morality in our consideration of reason / logic. The obvious implication is that moral behavior is somehow irrational and that the two people who chose to give their best estimate of the price of the antique were somehow acting "irrationally." It's sheer insanity.
 
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Barak Engel
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nwhyman wrote:
I have to agree with Nate. 2 is only the logical answer if you assume that the two travellers are competing against each other and nobody else. But the scenario proposed has them competing against the airline.

And then in my mind it is better to have a small chance of getting $100 than a large chance of getting $4. You can't even buy a decent game with $4!

Actually, I'll go further and say that BECAUSE it is obvious they are playing against the airline, it is rather highly likely that they will both pick $100, and not even think about further implications. The $2 won't come into it.

The whole example is flawed.
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Jon W
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NateStraight wrote:
No one in their right mind is going to sit down and go through this whole process and select 2 as their final answer.
From the article:

Of the 51 players, 45 chose a single number to use in every game (the other six specified more than one number). Among those 45, only three chose the Nash equilibrium (2), 10 chose the dominated strategy (100) and 23 chose numbers ranging from 95 to 99. Presumably game theorists know how to reason deductively, but even they by and large did not follow the rational choice dictated by formal theory.

So three of 45 weren't in their right mind, I guess.

But I take your point. That's why the problem becomes much more interesting when the goalposts are moved, say from 80-150. Or better yet, assuming Bill Gates is funding the experiment, from $2,000,000 - $100,000,000. Note that the article indicates that the average choice declines as the reward increases, even when it's still a ludicrously low number ("With a reward of 5 cents, the players' average choice was 180, falling to 120 when the reward rose to 80 cents.").
 
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Nate Straight

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Jon Waddington wrote:
So three of 45 weren't in their right mind, I guess.

I would wholeheartedly agree with that assessment, so I'm not sure why the wink.
 
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Evan Stegman
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Geez, the whole thing smacks of ivory tower thinking:

"Obviously ensuring you get $2 is more logical than trying to get $100!"

Um, ok. Do actually know any humans?
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Nate Straight

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EvanMinn wrote:
Um, ok. Do actually know any humans?

Game theory is supposedly based on basic tenets of human behavior (we always try to get more rather than less, etc), but it carries them out to ridiculously absurd conclusions.
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Eric Jome
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A fun problem!

Does the airline manager tell them the rules? If he does, things go like this;

"If I write a high number and the other player writes a high number, we'll both get paid a lot. If I write a high number and they write a low number, I'll get paid just a bit less than they do. And, if I write a low number, I'll make a bit more than they do, but it won't be very much."

What does this line of reasoning say? I'm better off writing a high number. If I write a low number, I've already automatically lost. I've made little money. Instead, I'll write high and hope the other person does too. This is the safest bet - I might get paid a lot. If I write low, I can be very sure I won't get paid much at all.

The regression posed in the actual article isn't reasonable - I proclaim that the truly rational choice is what I've described. I should prefer $101 to $100? Well, I suppose. But, I can't be sure that my opposite will write a high number. Instead, I should just write the highest number I can. Either I'll get $100 or $98 or some low value that is not in the interest of my fellow player. End of chain. My payoff can be expected and is always as high as it can be.

Funny eh?

The article is very entertaining read, even if I don't really agree with the traditional game theorist or the author. A fun problem nonetheless.
 
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Jon W
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NateStraight wrote:
Jon Waddington wrote:
So three of 45 weren't in their right mind, I guess.

I would wholeheartedly agree with that assessment, so I'm not sure why the wink.
First, it's 3 of 51 (oops, I need to read more closely blush ). But to your point, all 51 were "professional game theorists." I think they were in their right minds, unless perhaps the real experiment was by the university's psychology department, and the poor fools who headed up the game theory experiment were hoodwinked.
 
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Game theory alone cannot explain it, psychology is as important. Because when you play another human, you play the little "I guess that you guess that I guess..." game, but you might still have hope in the goodness of human nature, and bet they agree with you on a high payout.

The results from the Hohenheim study (mentioned in the article, but the result was not stated):

In the original 2 to 100 example with a +2/-2 payoff, the winner had taken 99 in each round.
 
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Nate Straight

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cosine wrote:
If I write low, I can be very sure I won't get paid much at all.

Well, as has been pointed out, the whole problem changes considerably (or, at least, our approach to the problem, as human beings, would change considerably) when the lower-end of the range of answers is greater and when the reward/penalty is greater.

Let's say the lower end is $1M and the higher end is $1M and 1 cent, but the penalty was $1M for having the higher answer. It's essentially and mathematically the same problem (just with different absolute values), but it's now pretty clear that no one would choose the high value.

The whole problem leaves out risk (which is, if I'm remembering correctly, one of the other basic tenets of game theory: that people are risk averse), which is why it always prescribes the lowest response as the "best" and "most rational" response. It works for this $1M example, of course.

People don't always respond the same way to risk, however. Everyone has a different tolerance for risk. The "most rational" response to the problem is based on your personal risk aversion, basically. Weigh the possible outcomes, weigh the risks involved with each choice, and then make a choice. That's textbook reasoning technique.
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Kris Verbeeck
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kimapesan wrote:
This is actually not too far from the "math problem" that appears in Beautiful Mind.

The way the logic in this problem is posed, each person is competing against each other to reap a slightly higher dollar award, but the reality is, they are both competing against the clueless airline manager. If the players view the game as playing against each other, then they both make the wrong choice - the one that lets the airline manager win. But if they view it correctly as playing against the airline manager, then they both win $100 and the airline manager loses.

In "Beautiful Mind" the problem was posed as there being 4 guys and 5 girls at the bar. Only 1 of the girls was a knockout blonde bombshell, the others were all plain-jane brunettes. Nash is portrayed as realizing the following:

- If the guys compete against each other for the blonde, then only one of them can win, the other three must settle for one of the Janes, but the Janes will feel slighted and won't give the guys what they want.

- If the guys agree not to compete for the blonde and let one of them take her, the other three guys will still feel they got the short-end of the stick, which the Janes will pick up on, and again, nobody gets what they want.

- But if all four guys agree not to compete, and each takes a brunette (leaving the blonde completely out of the equation), then each guy gets a girl, no one of the Janes feels slighted, and the only one that loses is the blonde.


Perspective on who you are really competing against, and whom you should be cooperating with, is the essence of this kind of problem.


I don't agree with that kind of reasoning.
If three guys go with a brunette and leave the place. I'm sure i'm the fourth guy who is left with the brunette and the blonde...

You see, i suffer from AP

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Nate Straight

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Jon Waddington wrote:
But to your point, all 51 were "professional game theorists." I think they were in their right minds, unless perhaps the real experiment was by the university's psychology department, and the poor fools who headed up the game theory experiment were hoodwinked.

Well, they may have been "in their right minds" but it was a flawed experiment because it had nothing to do with typical human behavior. They KNEW they were playing a game and they KNEW they had to try to come up with the "best" / "most rational" / "most mathematical" equilibrium, and furthermore they KNEW how to arrive at that according to their various theories.

People given a similar situation in the real world won't react in the same way 51 professional game theorists would. That much should be patently obvious. The results of this little experiment are about as useless as the results of a survey of the approval rating of the president would be if taken from his extended family, for example. It's ridiculously biased to the point of being unusable as a sample.
 
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Jon Waddington wrote:
That's why the problem becomes much more interesting when the goalposts are moved, say from 80-150.

It's only an illusion that the stakes matter. You will, presumably, always choose the best payoff, be it a nickel or a billion.

The best payoff here is obvious, the problem is that regression or recursive thinking isn't normal... or even rational in this case.

If I write $100, I might get $100 if my counterpart writes $100. If I write $100, I might get $98 if my counterpart writes $99. If I write $100 and my counterpart writes any lower number, I'll get $2 less than he does.

So, I write $100. I am in control of all the factors I can be in control of. I am acting rationally and hoping for the best. I don't want to recurse... it's self-defeating.

Now, change the problem in one small way. Let's say the person writing the higher number gets nothing. What should I write? Now, I should probably write $1. Anything else runs a high risk of getting nothing. So, in this situation, I do recurse... but how rational is this all or nothing game?

Recursive thinking is rarely rational or believable.
 
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Eric Jome
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NateStraight wrote:
Let's say the lower end is $1M and the higher end is $1M and 1 cent, but the penalty was $1M for having the higher answer. It's essentially and mathematically the same problem (just with different absolute values), but it's now pretty clear that no one would choose the high value.

Sure they would.

The problem here is that I have nothing to lose choosing a high number. That is playing rationally. You have to severely penalize me for choosing something else.

Take the original problem, but instead of +2/-2 go with all/nothing. If I write high, I get nothing. If I write low, I get something. Rationally, I write low now.

Thus, the game becomes one of "where is the optimal range to get people to consider thinking recursively/rationally?"... the boundaries aren't the limiting factor, the penalties are. What would you do at +10/-10? +25/-25? +50/-50?
 
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Neil Whyman
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When the answers are 2 and 100 the player writing 100 gets nothing anyway.

But I see how writing 99 can be effective. It just seems to depend on whether you are scoring wins or accumulating points from game to game. In the former case then 2 is the best option.

Looked at another way, the players risk loosing a certain $2 in order to have a chance at $100 (or whatever amount they bid). Personally $2 isn't going to change my life enough to worry about not getting it. But move the range up to $2M to $100M and I'd always select $2M.
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Nate Straight

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cosine wrote:
NateStraight wrote:
Let's say the lower end is $1M and the higher end is $1M and 1 cent, but the penalty was $1M for having the higher answer. It's essentially and mathematically the same problem (just with different absolute values), but it's now pretty clear that no one would choose the high value.

Sure they would.

The problem here is that I have nothing to lose choosing a high number. That is playing rationally. You have to severely penalize me for choosing something else.

Take the original problem, but instead of +2/-2 go with all/nothing. If I write high, I get nothing. If I write low, I get something. Rationally, I write low now.

Thus, the game becomes one of "where is the optimal range to get people to consider thinking recursively/rationally?"... the boundaries aren't the limiting factor, the penalties are. What would you do at +10/-10? +25/-25? +50/-50?

Reread more carefully... my penalty was $1M.
 
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Nate Straight

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nwhyman wrote:
Looked at another way, the players risk loosing a certain $2 in order to have a chance at $100 (or whatever amount they bid). Personally $2 isn't going to change my life enough to worry about not getting it. But move the range up to $2M to $100M and I'd always select $2M.

Exactly... your behavior is all about the relative values of the risks and rewards and has absolutely nothing to do with the mathematical setup of the problem. You won't always pick the lowest number, because that's not always the most rational choice.
 
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Eric Jome
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My mistake... quite right. A high penalty (all/nothing) will induce recursive thinking.
 
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NateStraight wrote:
Exactly... your behavior is all about the relative values of the risks and rewards and has absolutely nothing to do with the mathematical setup of the problem. You won't always pick the lowest number, because that's not always the most rational choice.

I stand by the idea that the actual range doesn't matter. It's the penalties that matter.

If the payoff is from $1 million to $1 billion and the penalty is $100,000, I'm going to write $1 billion. $100,000 is not a significant penalty even though I agree it's a lot of money. If the penalty is $1 million, I will write $1 million because, even though in the scope of the range it isn't that much, I run the risk of getting nothing with any other value.

Is this rational? The article talks about people acting from altruism, socialization, or faulty reasoning. I contend I am being rational by judging the penalties in the context of the rewards. Real world factors (ie $100,000 is a lot of money) don't factor into it.
 
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