The Traveler's Dilemma

May 30, 2007 8:56 AM

"He asks each of them to write down...any dollar integer between 2 and 100 without conferring together. If both write the same number...he will pay each of them that amount. But if they write different numbers, he will ... pay both of them the lower number along with a bonus and a penalty--the person who wrote the lower number will get $2 more...and the one who wrote the higher number will get $2 less.... For instance, if Lucy writes 46 and Pete writes 100, Lucy will get $48 and Pete will get $44."

What amount would you choose? And what does your answer tell us about the limits of Game Theory?
posted by empath (245 comments total)

50 bucks. I'm probably wrong, cause I always am about this sort of stuff, but I'd do it because I would think it would be the most likely to have been guessed by another person who was trying to guess what I was thinking.
posted by facetious at 9:08 AM on May 30, 2007

Damn it facetious, you just cost me $52!
posted by Pollomacho at 9:11 AM on May 30, 2007

90, and hope that my partner is also a greedy fuck.
posted by flibbertigibbet at 9:12 AM on May 30, 2007

I fail to see how logic leads us to that point (that the article says it should)--if the players are logical enough to realize they could earn a bonus, they should also be logical to realize how following this approach indefinitely will leave them with a greatly reduced reward in the end. Just because a response is "rational" in one context does not mean that rejecting that rational response is irrational or that there are not other rational choices that may give better results.

Disclaimer: I am slightly hungover and my coffee hasn't finished brewing yet. How this affects the rationality of the above statement remains to be seen.
posted by Benjy at 9:13 AM on May 30, 2007

100
posted by papercake at 9:14 AM on May 30, 2007 [2 favorites]

Yeah, 100. People, I'm telling you now, if any of you run into this situation with me in the future, let's both choose 100.
posted by Greg Nog at 9:16 AM on May 30, 2007 [9 favorites]

I'm disappointed that the article doesn't address the solution Douglas Hofstadter proposed (in his column in Scientific American, no less) ~20 years ago, to answer the question of why people don't choose the "rational" Nash equilibrium of 2.

Hofstadter proposed what he called (IIRC) hyperrationality - not only that one's behavior is rational, but that the behavior of all players is rational, and all players know that all players are rational, and all players know that all players know that all players are rational, ad infinitum.
posted by DevilsAdvocate at 9:17 AM on May 30, 2007 [3 favorites]

100 clearly. At the very least you would have the smug superiority of knowing that you were trying to do what's best for both of you.
posted by sourbrew at 9:18 AM on May 30, 2007

Why would anyone not choose $100?
posted by rusty at 9:18 AM on May 30, 2007

Sorry, double post, and without a completed comment, no less! I meant to continue as follows:

Since the game is symmetrical, a hyperrational player may conclude that another hyperrational player will come to the same choice that he does; knowing that both players will make the same choice, it is easy for the hyperrational player to see that 100 is the correct choice.
posted by DevilsAdvocate at 9:19 AM on May 30, 2007

Benjy Agree. Also I think the "logic" here requires that we think the participants have a greater desire to harm the other than they do to gain for themselves.

If I select 2, and you select any number greater than 2, I get $4 and you get squat. But why would my desire to screw you be greater than my desire to get money?

The difference between $100 and $101 is so small that I can't see anyone chosing $99 simply to a) get an extra buck, and b) screw the other guy out of $4. Why should I bother?
posted by sotonohito at 9:20 AM on May 30, 2007 [1 favorite]

The game assumes people would rather get more money than their partner than get as much money as possible, an assumption which is surely false.

Only a player obsessed with obtaining a greater reward than the other would wind up at the Nash equilibrium of 2. Otherwise, 100.
posted by BorgLove at 9:21 AM on May 30, 2007 [2 favorites]

100 ... the next time i'd play, i'd pick 100 ... and keep picking 100, as eventually the other person would catch on
posted by pyramid termite at 9:21 AM on May 30, 2007

I would choose $2, 'cause damn did you see the piece of shit car this guy drove up in?
posted by NationalKato at 9:21 AM on May 30, 2007

Why would anyone not choose $100?

Because if I think everyone will choose $100, then I can increase my payout by choosing $99 - I'll end up with $101. So, by that logic, everyone should choose $99, but then I should choose $98, etc, etc...
posted by Bort at 9:22 AM on May 30, 2007

100, right? You're assured the greatest gain if you choose the highest number. Is this not completely obvious, or am I just a super-genius?
posted by Pecinpah at 9:22 AM on May 30, 2007

100. It maximises the pain for the insurance guy, who needs to be punished for thinking up such an infernal scheme.
posted by edd at 9:22 AM on May 30, 2007

100, right? You're assured the greatest gain if you choose the highest number. Is this not completely obvious, or am I just a super-genius?

No, if the other person says any number under 100, you get penalized $2. But I'd still say 100 every time. A $2 penalty isn't high enough to deter me from trying to game the system.
posted by Terminal Verbosity at 9:25 AM on May 30, 2007

Why would anyone not choose $100?

Only a logician would be stupid enough not to choose $100.

Knowing, based on the article, that most people pick $100, I would also pick $100, since I would not be afraid that my counterpart would be an idiot logician who would compromise our position by arguing his way down to $2.

The logic used to get to $2 is the same kind of stupidity that helps MeFi threads about religion, politics, and a great many other things devolve into stupidity.
posted by The World Famous at 9:25 AM on May 30, 2007 [13 favorites]

I'd start by choosing $100, since it offers very nearly the highest possible payoff regardless of the other player's strategy.

Since there's no (apparent) incentive in this game for the other player to lowball her number, I'd assume that she, too, would choose a number close to or equal to $100.

If she chooses $100, we both maximize our payoffs.

Even if the other player maximizes her payoff at my expense and chooses $99, receiving $101, I'm still up $97, giving her only a 4% advantage. I'd be happy with $97.

If she lowballs her number to punish me, say she chooses $2, then I get nothing, an "infinite" advantage. But then she gets $4 — practically no payoff, either. It's probably not a good long-term strategy for either of us.

My strategy would assume mutual self-interest would dominate over vindictiveness, since we can all make a lot of money off this very generous researcher, if we more or less agree to get along.

Nonetheless, if we played this game repeatedly, I might change my strategy to occasionally choose $99, if my opponent does so. We might then cycle to ever-lower "bids", and probably settle on some equilibrium value, occasionally underbidding each other slightly to score a temporary advantage.
posted by Blazecock Pileon at 9:26 AM on May 30, 2007

Nthing those who are confused that anyone would chose something other than $100. I think there needs to be a bigger bonus for the under-bidder, or some other penalty for greed.

How about this:

You both have to chose a number - whoever choses the lower number gets the dollar value of the higher of the two chosen numbers.

If you both chose the same value, you both get nothing.

What number do you choose?
posted by Jon Mitchell at 9:26 AM on May 30, 2007 [5 favorites]

The problem with game theory in this instance is that it assumes perfect rationality in what is basically a lottery question. The travelers have all the information. They can make out like bandits because of the other person's ignorance. People with that option aren't going for the "perfectly rational" choice. They are going for the "lets screw that guy" choice.
posted by dios at 9:29 AM on May 30, 2007

I'd pick 100 without a second thought. If my counterpart picked 2, I'd be really pissed off. The fact of the matter is that picking 2 is only rational in the coldest, most hyper-logical HAL 9000 definition of the word, and you'd need to overthink countless plates of beans to reach that point.

Game theorists who wonder why everyone doesn't pick 2 are so wrapped up in their field that they forget how people actually think. A normal person would never think "If my counterpart picks X, I must pick X minus 1," and then chase it all the way to the bottom. Picking 100 or a number close to it may not be the "rational" choice, but it's the one that stands the highest likelihood of maximizing reward. People care about maximizing their reward a whole lot more than they care about selecting the optimal rational choice.

Oh, and as to why people pick 100 even though 99 dominates? It's because in order to determine that 99 dominates 100 you need to puzzle things through. Trying to squeeze that last dollar out of the problem takes more effort and analysis than most people would consider worth the reward. Avoidance of excess labor is a factor that most normal people tend to consider.

So let's break it down. Picking 2 requires a lot of thought and drastically reduces potential gain. Picking 99 requires some thought and maximizes potential gain. Picking 100 requires the least thought and has a potential gain almost as high as 99. It's no wonder that 100 is the most common answer.

When your game theory overrides common knowledge of human nature, it's time to get your head out of your spreadsheets and take a walk.
posted by Faint of Butt at 9:30 AM on May 30, 2007 [17 favorites]

$100. The value of possibly getting more than the other guy isn't high enough for me to race him to the bottom to do it.
posted by tula at 9:30 AM on May 30, 2007 [2 favorites]

Traveler's Dilemma (TD) achieves those goals because the game's logic dictates that 2 is the best option,

Only if your desire is to "win" (get more than the other person) rather than win (get the most money you can).

This "dilemma" seems completely stupid to me.
posted by papakwanz at 9:31 AM on May 30, 2007 [2 favorites]

I think there are really only two strategies, Low and High. If I thought there was a better than 10% chance of the other guy going high, I'd go high. If I was POSITIVE that the other guy was an asshole, I'd choose 2.
posted by empath at 9:33 AM on May 30, 2007

tit for tat is the way to go.
posted by SaintCynr at 9:34 AM on May 30, 2007 [1 favorite]

It seems that game theorists assume that each player wants to "beat" the other player, whereas a participant in the traveler's dilemma just wants as much compensation as possible for his broken item. If the other traveller gets more money than the other, who cares? It's better to be "beaten" by the other traveler than to have less money in your pocket.

The "game theory" model inappropriately casts the other traveler as the opponent. In fact, the airline manager is the opponent, and the goal is to extract as much money as possible from him. That can only occur when both participants choose "100."
posted by deanc at 9:34 AM on May 30, 2007 [3 favorites]

Btw, I think the world would be a much better place if we all agreed that if we ever get into this one, or an analogous one, we would all pick $100.

And also, that if anyone picks 2, we stone them.
posted by empath at 9:35 AM on May 30, 2007 [3 favorites]

hunnerd.
posted by BlackLeotardFront at 9:36 AM on May 30, 2007

I would love to have an explanation of how some game-theorists are so lacking in meta-cognition or insight into real games and real rationality. Is it simply a fetishisation for mathematical models or something deeper and more perverse? Perhaps a panic-defense at the inadequecy of their ability to represent actual rational cognition in simple toy models.
posted by MetaMonkey at 9:36 AM on May 30, 2007 [1 favorite]

Seems to me I'd pick 100, and then when the other person picked 2, I'd shoot them in the chest and pick 100 again. The next participant would be far more cooperative.
posted by aramaic at 9:36 AM on May 30, 2007 [3 favorites]

(uh...into this situation or an analogous one)
posted by empath at 9:36 AM on May 30, 2007

$100. Duh.

Given a choice of maximizing the reward for both of us or simply beating my "opponent" I would maximize the reward for both of us. Beating your opponent only matters if you're playing with monopoly money or if there is some secondary greater real cash reward for winning the game.

I must admit, though, I know diddly about game theory.
posted by lordrunningclam at 9:37 AM on May 30, 2007

$100
posted by autodidact at 9:38 AM on May 30, 2007

The scary part is that game theory was what our government was depending on to save us all from nuclear annihilation at the hands of the Soviets.
posted by empath at 9:39 AM on May 30, 2007

God, I hate rational choice theory.

Since, in the scenario, the players are asked to imagine that they are playing to recoup the loss of a broken antique, wouldn't they assume that they had to guess a number higher than the imagined cost of the antique in order to actually make a profit? If they chose $2 because it is the "rational choice," then they would have to also assume that would be losing money because the airline damaged a possibly valuable belonging.

I think that the outcome of the game would change if the item in question were something clearly less valuable, but the possible payout was the same, if for no other reason than the players would feel ashamed at the fact that they were obviously liars. In the real world, decisions are based on emotions, not just on logical calculations. Is this fact really that shocking?
posted by exacta_perfecta at 9:39 AM on May 30, 2007

Sure, if you choose $2, you can definitely get $2 and maybe even $4. But if you went with $100, or $50, or anything high, there's the chance that you would end up with nothing, but there's also the chance that you would end up with $100. A potential $100 is more fun than a certain $2.
posted by drezdn at 9:39 AM on May 30, 2007

You both have to chose a number - whoever choses the lower number gets the dollar value of the higher of the two chosen numbers.

If you both chose the same value, you both get nothing.

What number do you choose?

first time 100 ... second time 99 ... 3rd 100 ... 4th 99 ... and so on

it would take a brain dead opponent not to come up with the obvious counterstrategy ... one that lets both of us win
posted by pyramid termite at 9:40 AM on May 30, 2007 [1 favorite]

I must admit, though, I know diddly about game theory.
I think in this case that means you'll end up with more money than someone who does.
posted by ElmerFishpaw at 9:41 AM on May 30, 2007 [4 favorites]

Drezdn: I think you nailed it. Rational Choice theory would seem to predict that no one would ever play slots or roulette, and yet they do, in vast numbers.
posted by empath at 9:41 AM on May 30, 2007 [2 favorites]

Man, if I chose $100 and my partner chose $99, I'd be like, "Congratulations. You are willing to look like a dick for the princely sum of one dollar."
posted by Greg Nog at 9:43 AM on May 30, 2007 [7 favorites]

"I must admit, though, I know diddly about game theory."

He's playing possum, folks. This is like ""Big Deal In Dodge City," only with nerds instead of cowboys.
posted by Jofus at 9:43 AM on May 30, 2007 [2 favorites]

For chrissake, $100 is the only right answer. Nobody cares about winning $98 rather than $102.
posted by hodyoaten at 9:44 AM on May 30, 2007

Follow up question:

Would you do this differently if you knew for a fact that this was a one time 'game' and you would never see any of these people again than you would if you were going to play 5 more times with the same people?
posted by empath at 9:45 AM on May 30, 2007

Part of the problem with this article and it's use of the Traveler's Dilemma is it doesn't consider risk and that's why the results don't match the Nash Equilibrium.

Compare using the value of 2 (NE) to 100. Worst (logical) case with 2 is I get 2 dollars (this assumes nobody would pick 1 or 0). Worst case with 100 is I get no dollars (other person picks 2).

So the difference in worst-case, in terms of money, between the two choices is 2 dollars. The difference between best-case is 98 dollars. Those willing to take risk will have higher rewards. So what level of risk is a person willing to take? In this case two bucks doesn't mean much to me. 100 bucks does. So I'll always pick 100.

Another miss is on goal assumption. What's the goal of the game? To get more money than the other guy or to get the as much money for yourself as possible? I pick 100 and you pick 99? Great, you win 101 but I also get 98 bucks. Why do I care if you get 3 bucks more? I've maximized the amount of money I could walk away with.

I think this whole article is narrow-minded and lacks factoring in risk versus reward and assumed goal.
posted by ruthsarian at 9:45 AM on May 30, 2007 [2 favorites]

On reading more of the article, it seems they did other experiments where the payoff ranged from 80 to 200. In that case, since the goal of a real-live player is to end up with some compensation, it becomes more rational to go down to 80, which is what the participants did after multiple trials.

As drezdn said, a potential $100 is more fun than a certain $2. However, a certain $80 isn't that much worse than a possible $200, depending on how large the penalty is.
posted by deanc at 9:47 AM on May 30, 2007

The sociology of this scenario and the Prisoner's Dilemma are interestingly different, and on that account, this Dilemma seems far more an illogical failure. Why would any traveler desire $2 more than her companion, at the companion's expense of $2? Better to milk the corporate cash cow whose error it was, right? On the other hand, just being arrested and in a jail cell tends to cast a pall of deserved punishment on a Prisoner, justifying somewhat any vindictive moves a player in that Dilemma might make against the other.
posted by Ambrosia Voyeur at 9:47 AM on May 30, 2007

Why would anyone not choose $100?

My understanding from the article: because the penalty isn't very large. If the range was from $50 to $100 with a $50 reward/penalty, you might be tempted to underbid my 100 by saying 99. Result: $149 for you, $49 for me

Problem with Hofstater's hyperrationality: I don't trust the other guy to act that way, e.g. facetious and flibbertigibbet.

I still choose $100 though.

With a $2 reward/penalty, I think: "What have I got to lose? If I choose the $2, John Nash can't screw me out of that, but the most I'll get is enough to buy a burger and fries. But if I choose $100, Doug Hofstadter and I could both walk away with $100, and all I stand to lose is my cheeseburger money"

But if the reward/penalty was $50, and I don't know if it's John or Doug in the other room, it's a much tougher choice. I might guarantee myself at least the minimum by choosing the Nash strategy, rather than gambling that on the maximum by choosing the Hofstadter strategy
posted by rossmik at 9:48 AM on May 30, 2007 [2 favorites]

Adam Curtis' documentary The Trap (discussed previously) explores some of the issues around this; how these kind of simple and narrowly-applicable mathematical-economic theories of rationality have been abstracted far beyond their usefulness, to become not only accepted but pervasive, economically (and politically) accepted dogmatic truth in spite of their self-evident contradiction to observed reality. Also it's now on google video.
posted by MetaMonkey at 9:50 AM on May 30, 2007 [9 favorites]

Would you do this differently if you knew for a fact that this was a one time 'game' and you would never see any of these people again than you would if you were going to play 5 more times with the same people?

I wouldn't have the opportunity to learn the other's strategy. So I'd base my decision, again, on maximum payoff, since the minimum payoff is nearly worthless.

There's practically no return on choosing $2, but I'd be happy bidding $100 and winning either $100 or $97, on the assumption that the other player shares my dollar valuation system.
posted by Blazecock Pileon at 9:50 AM on May 30, 2007

On the other hand, just being arrested and in a jail cell tends to cast a pall of deserved punishment on a Prisoner, justifying somewhat any vindictive moves a player in that Dilemma might make against the other.

Nash didn't consider the shiv strategy.
posted by Blazecock Pileon at 9:52 AM on May 30, 2007 [2 favorites]

I think this whole article is narrow-minded and lacks factoring in risk versus reward and assumed goal.

I think the point of the article was to point out that game theorists are narrow minded. The author himself doesn't seem that surprised that people choose 100 or 99-- in fact, he seems to have set up the experiment knowing that this would be the outcome. What the article points out is that, as obvious at this might be to us, if you plug the scenario into a game theory spreadsheet, it predicts that everyone should choose $2. Game theory itself then (if the author is to be believed) apparently lacks factoring in risk versus reward and assumed goal.
posted by deanc at 9:53 AM on May 30, 2007 [3 favorites]

empath has it. Followup games are where game theory comes into it. If you can play an unlimited number of times, $2 is the rational choice. If you can play once, $100 is the obvious choice. Figuring out the optimum choice for a given number of games is left as an exercise to the reader.
posted by Skorgu at 9:57 AM on May 30, 2007

Methinks their choice of back story is getting in the way of their theory. Yes? No? Reframe the question in terms of points, where whoever has the most points at the end "wins" and it might make more sense, at least to me, but then they wouldn't have a Scientific American article touting their research.
posted by Skwirl at 9:59 AM on May 30, 2007 [1 favorite]

$100. Btw, wasn't game theory used in developing the mutual assured destruction doctrine of nuclear war strategy? Those uys were sure smart.
posted by RussHy at 10:01 AM on May 30, 2007

Here's the thing: the theory doesn't serve this contrived example well, and the realm of ways in which game theoretic models end up at odds with real human behavior is a fascinating one, but the example can be put into a context where it does make sense.

Assume one trial a day. Assume you need medicine that costs $2 a day. Assume you will die if you do not get your medicine, that you are broke, and that you cannot get medicine by any alternate route.

What number do you choose?

The breakdown of the model in real life situations is interesting, and there's a great big yawning gap between classical game theory and observable human behavior, but the model isn't junk. It just doesn't incorporate all these meta-model things we're talking about as presented: there's not enough at stake in the TD to drive truly mercenary, rational behavior on the part of the participants.
posted by cortex at 10:01 AM on May 30, 2007 [5 favorites]

I would choose $69, followed by a chest-bump and a HOO HAH.
posted by The Straightener at 10:02 AM on May 30, 2007 [1 favorite]

The only thing worse than the generalizations made by game theory about people is the generalizations that people make about game theorists.
posted by vacapinta at 10:02 AM on May 30, 2007 [4 favorites]

guys, those guys were sure smart. I, on the other hand, am not so smart.
posted by RussHy at 10:02 AM on May 30, 2007

If you can play an unlimited number of times, $2 is the rational choice.

Only if 1) you want to have more money than the other guy, and 2) you want to amass your fortune really slowly and 3) you can't trust the other guy.

If you want to get alot of money quickly, and you can trust the other dude, the rational choice is $100.

Isn't it?
posted by 23skidoo at 10:04 AM on May 30, 2007

This somehow reminds me of the judge who tells the convict: "You are to be executed next week at 6 in the morning, on a day that will be a surprise to you."

The convict reasons "I can not be executed Saturday (the last day of the week, because I would know on Friday that I must be executed Saturday, and so it would not be a surprise. Therefore, knowing I can not be executed Saturday, I can not be executed Friday, because it would not be a surprise."

And reasoning thus, he is convinced he can not be executed.

So he is quite surprised when, on Wednesday morning, he is taken out and executed.
posted by hexatron at 10:05 AM on May 30, 2007 [4 favorites]

I'd have to side with the hyperrationals on this one. $100, and that's my final offer!

My first reaction to this was that it was a "dating game" type of problem: I assumed that Pete and Lucy were a couple, and that the goal was for them to maximize their collective income. The insurance guy's evil scheme, clearly, was to penalize them for disloyalty to each other!

On a related note: don't these researchers know that altruism is hard-wired?
posted by otherthings_ at 10:05 AM on May 30, 2007

I was about to post an argument that the game has a false premise. $100 seemed right to me. But then I actually read the whole article and found that the point is that game theory itself has flaws. $2 is the "rational" response according to formal rules. But only a psychopath would give that answer. :)
posted by stiggywigget at 10:06 AM on May 30, 2007

I'm more interested in the version that pays between $2 and $1,000,000.
posted by Clamwacker at 10:07 AM on May 30, 2007

Man, if I chose $100 and my partner chose $99, I'd be like, "Congratulations. You are willing to look like a dick for the princely sum of one dollar."

Yup.
posted by Bookhouse at 10:07 AM on May 30, 2007

Reframe the question in terms of points, where whoever has the most points at the end "wins" and it might make more sense

and

Assume one trial a day. Assume you need medicine that costs $2 a day. Assume you will die if you do not get your medicine, that you are broke, and that you cannot get medicine by any alternate route.

These are both situations in which "winning" is a necessary outcome. For many scenarios, winning isn't necessary. What's desired by players in the TD is simply an "acceptable" outcome, rather than a winning outcome. Game theory assumes that each player wants to win, whereas in many real-life situations, each player merely wants to "get by," which apparently creates better outcomes in some cases.

Now that I'm done with graduate school, I can confess that I was supposed to read a book on game theory for my qualifying exams, but I never made it past the first couple of chapters. Thus, I'm a bit out of my depth here.
posted by deanc at 10:09 AM on May 30, 2007

I agree with deanc that the blinkered me-vs-other bidder game theoretic approach is the downfall here. If you cast it as me-vs-pilot, with the other player a completely irrational lunatic who's likely to spit out any number whatever, I make the expected payoff for a $2 bid to be $3.98 and the expected payoff for a $100 bid to be $49.52.
posted by ormondsacker at 10:12 AM on May 30, 2007

I think the "rationality" here assumes that we're all motivated by money over anything else. If I'm more interested in being a socialized human being than an extra buck or two, then choosing $100 is perfectly rational.

If I choose $100 and you choose $100, then we both get $100 yey.

If I choose $100 and you choose $99 just to get an extra buck, I roll my eyes at how much of a dick you are, but at least I get $97.

If I choose $100 and you choose $2, I marvel at how you're not only a dick, you're a self-defeating dick.

In all cases, I've reached my goal of being the person I want to be, so I'm perfectly rational. Money on this scale is irrelevant to that. You might as well offer me fifty bucks to punch a puppy and then declare me "irrational" because I didn't take it.
posted by L. Fitzgerald Sjoberg at 10:14 AM on May 30, 2007 [3 favorites]

You guys! You obviously can't pick $100. It says any integer between 2 and 100. So you can pick 3 up to and including 99.

Sheesh.
posted by ODiV at 10:15 AM on May 30, 2007 [2 favorites]

I'm more interested in the version that pays between $2 and $1,000,000.

I know you're trying to be funny, but the dynamics aren't any different than the version that pays between $2 and $100. What if there were a version that pays between $100,000 and $1,000,000? What if you were penalized $90,900 for guessing too high, and you could only play once? Then what would you do? The guarantee of $100,000 is worth a lot more to me than the possibility of getting $1,000,000 given a risk of getting only $100. People with the net worth of high-level investment bankers may make different decisions, though.
posted by deanc at 10:18 AM on May 30, 2007

cortex: Here's the thing: the theory doesn't serve this contrived example well, and the realm of ways in which game theoretic models end up at odds with real human behavior is a fascinating one, but the example can be put into a context where it does make sense.

But the fascinating thing that this article highlights is that some professional game theorists (3 from a sample of 50) playing this game against other professional game theorists, would actually bid $2.

I think most people would accept that the narrowly-rational strategy could make sense in some specific circumstances (playing poker, for example), but what is astonishing is that apparently some people who do this for a living don't realise that it doesn't work outside those circumstances.
posted by MetaMonkey at 10:21 AM on May 30, 2007

Who really needs that extra few dollars by playing this game asshole style? 100 works because people operate on a social level: "Will this guy beat the crap out of me if I pick 2?" On an emotional level "100 bucks and we all win. yah!"

The only person I can think of who would really think an extra 20 is worth it is probably too dumb to figure out the math to play properly.

The problem with these games is that there's nothing to lose. You didnt invest to play this and the monetary amounts are too small to make most people get all hot and bothered about the chance to get 20-30% more. The success of showsl ike Deal or no Deal is the fact that its such a big fantasy you might as well keep going to the top. I would stop at a comfortable 10 or 20k because, hey, thats free money and after that point comes a whole lot of risk.
posted by damn dirty ape at 10:21 AM on May 30, 2007

or perhaps you will, after 10 rounds enter an auction with other players for something of value that you can buy with your monopoly money?
posted by Megafly at 10:26 AM on May 30, 2007

But the fascinating thing that this article highlights is that some professional game theorists (3 from a sample of 50) playing this game against other professional game theorists, would actually bid $2.

It's fascinating to me that only 6% of a group composed of people so thoroughly grounded in the literature would make such a move: the stakes are (presumably) low, since professional game theorists probably aren't living on the street; they know the context well enough to anticipate a wide variety of play from the other folks; and their choice of $2 could be spawned from stubborn self-aware loyalty to the model/principles, to mischieviousness, to value-neutral point-making, etc.
posted by cortex at 10:28 AM on May 30, 2007

Yeah, I think this game's flawed, not the theory.
It doesn't state the goals. It appears to be a one-shot choice.
And it looks like it was designed specifically to make game theory look bad.

I like Jon Mitchell's game much better.
posted by MtDewd at 10:28 AM on May 30, 2007

I'm with most of the above. The puzzle assumes that it's better to come out ahead of the other player than to get a bunch of money. I'd rather get a bunch of money, even if it's only $97, or whatever, than lowball and end up with $7 or something lame like that.

Not terribly difficult, imho.
posted by HighTechUnderpants at 10:29 AM on May 30, 2007

The success of showsl ike Deal or no Deal is the fact that its such a big fantasy you might as well keep going to the top. I would stop at a comfortable 10 or 20k because, hey, thats free money and after that point comes a whole lot of risk.

I wonder about that. Jeopardy (tried to) screen out the superbrains along with the mediocre; does Deal or No Deal screen out the modest-return pragmatists?
posted by cortex at 10:30 AM on May 30, 2007

100 would be the obvious choce, and hope the other person has the damn common sense to do the same.. it's the only way to get the largest amount of money for both of you! I see the screwy 'logic' behind choosing 2... but that only gives you a maximum possible reward of $4. Not very logical, IMO.
posted by triolus at 10:31 AM on May 30, 2007

I prefer Phillipe's game.
posted by L. Fitzgerald Sjoberg at 10:33 AM on May 30, 2007

Man, if I chose $100 and my partner chose $99, I'd be like, "Congratulations. You are willing to look like a dick for the princely sum of one dollar."

I've done plenty of things for free which made me look like a dick. The extra $1 is just icing on the cake.
posted by DevilsAdvocate at 10:35 AM on May 30, 2007 [2 favorites]

I'm more interested in the version that pays between $2 and $1,000,000.

I think that version is better known as "Deal or No Deal."

does Deal or No Deal screen out the modest-return pragmatists?

From what I've heard about the contestant selection "test" - a series of situations with "would you take this offer?" - it seems the answer is yes.
posted by djlynch at 10:37 AM on May 30, 2007 [1 favorite]

Sorry, haven't gotten through all the comments yet hope I'm not being completely redundant...

The article states that they apply backwards induction. Which means you start with the last action of the game and work backwards. So, you start with $100 then work back and find the best action in response for the other person is to choose $99. But if the reaction to that is to adjust the original $100 to $98, which means your maximum winning is still only $100. So you have NOT improved at all and only at best hurt your opponent. Ergo, induction fails and other logic must be applied (say expectation values).

Thus with this scenario, game theory would NOT lead to a choice of $2. Interestingly, if they upped the difference in values to $3 then backwards induction would actually work as stated sending the value down...I don't know enough about game theory to know if you're ONLY supposed to apply backwards induction though and I somehow doubt it.
posted by kigpig at 10:39 AM on May 30, 2007

If the dilemma is expanded to an iterated traveler's dilemma, Tit for Tat is very nearly an optimum strategy, in which you'd choose $100 until the other guy chose $2 in which case you'd fall back to $2 as long as you were playing him. For the single case though, it all depends on the individuals playing and their own personal risk level. As with any theory a situation can be developed that makes the theory look dumb. Game theory is applicable (and effectively so) in a wide range of areas, but it does not purport to be able to accurately predict all human responses in all possible game-like areas.
posted by Skorgu at 10:39 AM on May 30, 2007

$100

The criticisms of game theory are off the mark. Game theory doesn't factor in psychology, that's not a flaw. The intent is to give the optimal strategy for winning according to the parameters of the contest.

Game theory does have some relevance for psychology in showing how human behavior departs from the rational solution. Once those instances have been identified then some guess can be made about human motivations and cognitive biases. But game theory certainly isn't wrong and humans don't give a better answer.

In this case it appears that humans value a personal monetary reward over the possibility of victory over an opponent. Another interpretation is that humans recognize their opponents are less than rational.

Whether the game is played once or multiple times makes no difference. The strategy is the same. An optimal solution does not change because of an increase in the number of trials. If you have a human opponent and know the guy is liable to change for an irrational reason then you can make a counter strategy reflecting that. But you could also build that 'read' of your opponent into the question and game theory would give you the optimal solution. But if you don't know that your opponent will change, you keep the same strategy. It's like flipping a coin for all your money if you lose vs. 10 times the amount if you win. If you chose to play at all you might say you would only do it one time but wouldn't commit to a series of 5. That's fine but recognize that when you do so there are assumptions, perhaps unconscious, about how much that windfall is worth to you at your current level of wealth and how you feel about the 50% possibility of going broke vs. the 3% possibility of going broke. If those assumptions were built into the equation, game theory would come up with the same solution. But without that specification the assumption going in is that money doesn't change value based on how much you have. For humans that generally doesn't hold true. People are more likely to bet everything when they are only worth $3,000 vs. $3,000,000 even if the proportional reward is the same.

Game theory does not lack in comparing risk, reward and assumed goal. That is its forte.

Finally, I think one of the proposed game theory solutions to the cold war was to preemptively bomb the Soviet Union before they could catch up to us enough in nuclear weapons to really hurt us. Whether that is the optimal solution depends, like all other game theory questions, on whether you agree with the assigned relative weights of victory, defeat and stalemate.
posted by BigSky at 10:41 AM on May 30, 2007 [1 favorite]

The puzzle assumes that it's better to come out ahead of the other player than to get a bunch of money.

No, it doesn't, and the classical game theory argument which leads to choosing 2 does not rely on any assumption of a desire to "beat" the other player, and it does assume that each player wishes to maximize his own profit, regardless of what the other player gets.

There's plenty of reasons to trash classical game theory here, but please at least understand the classical argument so you don't go off and attack a different argument entirely, which no one, not even classical game theorists, is making.
posted by DevilsAdvocate at 10:42 AM on May 30, 2007

cortex: It's fascinating to me that only 6% of a group composed of people so thoroughly grounded in the literature would make such a move

Really? That 6% of people who are paid to figure out how to win games would make a bid that could not possibly win does not surprise you? It makes about as much sense to me as if you held a competition of 100 chefs to see who can make the best dish, and 6 of them shat on the plate and went home (even if shitting on a plate may be optimal in a very specific context). Wouldn't professional pride motivate you not to end up at the bottom of the table (as those 3 $2 dudes will have done).
posted by MetaMonkey at 10:42 AM on May 30, 2007 [2 favorites]

chiming in with the 100 people. this is an opportunity to cooperate with the other player in your position to take the fool asking the question for as much money as you can. once you realize this, you don't need no game theory.
posted by bruce at 10:46 AM on May 30, 2007

Traveler's Dilemma (TD) achieves those goals because the game's logic dictates that 2 is the best option, yet most people pick 100 or a number close to 100

Two is the best option? WTF? How does that even remotely make sense? If you pick two, the most you can hope for is $4, but if you pick $100, you might get $100. In fact, you probably will. The only thing "illogical" is that the author of the article is apparently a moron.
posted by delmoi at 10:49 AM on May 30, 2007

So, you start with $100 then work back and find the best action in response for the other person is to choose $99. But if the reaction to that is to adjust the original $100 to $98, which means your maximum winning is still only $100. So you have NOT improved at all and only at best hurt your opponent. Ergo, induction fails and other logic must be applied (say expectation values).

But this is incorrect. If you choose 98 rather than 100, and you know for a fact that your opponent will always choose a higher number than you if possible, sure—but you can't possibly know that in the problem as presented.

The rational goal, in the jargon sense of the word, is not to improve your maximum payout, it is to maximize your predicted payout vs. a similarly rational opponent. Your predicted payout has to take into account the payout for either a win or a loss, and the probability that each case will occur.

If you choose anything other than $2, so the reasoning goes, your oppenent can underbid you. If that happens, you incur a loss in your predicted payout because of your choice of bids. That is why induction leads you both down to the bottom of the limit. At $2, you are sure to take $2 against a like-minded rational opponent, and $4 against an opponent who is not rational. At anything other than $2, you are risking your bid vs. either $0 if your opponent is rational, or ??? if your opponent is not.

Modelling the rationality itself of your opponent is not within the scope of the model.

The model clearly doesn't capture actual human behavior here, but it's a failure of application, not a failure of the model logic.
posted by cortex at 10:49 AM on May 30, 2007 [2 favorites]

That 6% of people who are paid to figure out how to win games would make a bid that could not possibly win does not surprise you?

You have made the unwarranted assumption here that receiving $2 (and very likely $4) is "not winning." The game allows a wide variety of possible outcomes, from $0 to $101, and there is no clear dividing line between "winning" and "not winning." It is your interpretation, not part of the game setup, that getting only $2 would be "not winning."

I'm not saying I would pick 2 - after reading the thread so far, I'd probably pick 99 - but if I were given the opportunity to participate in a game/experiment which took only a few seconds of my time, and came away $2 richer than I was before, I wouldn't consider that losing.
posted by DevilsAdvocate at 10:52 AM on May 30, 2007

The logic of the game looks much more sensible if, instead of dollars, you are playing for nuclear missiles (which the guys who developed this math were).
posted by localroger at 10:53 AM on May 30, 2007 [2 favorites]

I choose ten. I just like the number 10, it's kind of lucky for me. So take your $8 and get off my case, ok?

(Or should I have chosen 8, so I'd win 10 when you pick 100? Math is hard)
posted by Crash at 10:53 AM on May 30, 2007

Really? That 6% of people who are paid to figure out how to win games would make a bid that could not possibly win does not surprise you? It makes about as much sense to me as if you held a competition of 100 chefs to see who can make the best dish, and 6 of them shat on the plate and went home (even if shitting on a plate may be optimal in a very specific context). Wouldn't professional pride motivate you not to end up at the bottom of the table (as those 3 $2 dudes will have done).

You can reduce 'professional pride' to the monomaniacal goal to Get The Most Money in a deeply omphaloskeptic exercise? That's like saying professional pride would motivate any working artist to paint as strictly realistic a portrait as they could if someone gave them the lark instructions to Paint The Queen.

It does surprise me, yes.
posted by cortex at 10:55 AM on May 30, 2007

As an aside, readers of this thread may like the Martingale.

In a classroom, ask your students how far they would go in betting on the Martingale. Most would probably go as high as $8 or $16.

But if you look at the math, the longer you play the Martingale (assuming you're really rich and can keep playing), the exponential payoff grows to be astronomical.

What's interesting is that human psychology does not deal with mathematical situations in a rational manner — we're particularly bad at rational assessments of probabilistic models in gambling, game theory, and other non-zero strategic behaviors.

Just as game theorists would bet $2 in empath's game, while regular folks would bet $100, those same regular folks might stop betting against the Martingale after about $8-$16 in losses, but knowledgable (and rich) mathematicians would probably keep betting higher amounts — knowing that the long-term payoff is worth it.

The interesting question is why we're wired to ignore the odds, or why we're wired to be so bad at it. Is it smarter for the population to be dumb about risk assessment, if the payoff is large for a few individuals?
posted by Blazecock Pileon at 10:56 AM on May 30, 2007

DevilsAdvocate, I direct you to TFA:

51 members of the Game Theory Society, virtually all of whom are professional game theorists, played the original 2-to-100 version of TD. They played against each of their 50 opponents by selecting a strategy and sending it to the researchers. The strategy could be a single number to use in every game or a selection of numbers and how often to use each of them. The game had a real-money reward system: the experimenters would select one player at random to win $20 multiplied by that player's average payoff in the game. As it turned out, the winner, who had an average payoff of $85, earned $1,700.

posted by MetaMonkey at 10:56 AM on May 30, 2007

I took an entire semester on game theory and I'm relying on common sense here, not choice matrices. The increments are so small here that you'd be an idiot not to choose the highest number possible. The reason Deal or No Deal works is because you are dealing with sums ranging from a penny to a million dollars.

But that said, the terms in the article are INCREDIBLY significant changing your logic versus the way it's presented in the FPP: for the latter you're setting it up as a game, while the article sets up a scenario where an actual item of tangible value is what is being evaluated. Only one of these sets up the dilemma as a test of morals as well as logic skill.

In fact, with asking for a number in the context of an actual item's value in play, the dilemma seems even simpler to solve. Unless you believe your partner will say the artifact was worth LESS than it actually was, then your maximum threat of loss is two dollars.
posted by XQUZYPHYR at 10:56 AM on May 30, 2007 [1 favorite]

Does your answer change if the other guy is Hitler?
posted by Caviar at 10:57 AM on May 30, 2007 [4 favorites]

"the classical game theory argument which leads to choosing 2 does not rely on any assumption of a desire to "beat" the other player, and it does assume that each player wishes to maximize his own profit, regardless of what the other player gets."

Yup. That's it. Change my third paragraph to:

One interpretation here is that humans recognize a like minded opponent, one who is more hopeful about taking home a bunch, rather than devoted to making sure he does as well as possible against someone who is also rationally pursuing their own self interest.
posted by BigSky at 10:59 AM on May 30, 2007

Blazecock Pileon, I once watched an actual human being playing with real money run out a Martingale starting at 25 cents and finally running out the table limit of $1000. He was betting Red. I think he lost 13 bets in a row. On the very next spin after he ran out of money, his color came in.

The Martingale is basically a self-made negative lottery; you'll probably win a little, but if you lose you will lose a lot.
posted by localroger at 11:00 AM on May 30, 2007 [2 favorites]

Ok, here is his explanation:

When studying a payoff matrix, game theorists rely most often on the Nash equilibrium, named after John F. Nash, Jr., of Princeton University. (Russell Crowe portrayed Nash in the movie A Beautiful Mind.) A Nash equilibrium is an outcome from which no player can do better by deviating unilaterally. Consider the outcome (100, 100) in TD (the first number is Lucy's choice, and the second is Pete's). If Lucy alters her selection to 99, the outcome will be (99, 100), and she will earn $101. Because Lucy is better off by this change, the outcome (100, 100) is not a Nash equilibrium.

But the problem is, once Lucy gets to 98, she has no reason to continue lowering her estimate. She if she continued to reduce her estimate, she would get less money then if she picked $100 and her opponent picked $99. So (100,99) is a better outcome for her then (97,98). Only the first three values 100, 99, or 98 make any sense to choose. If this guy thinks the math works out to $2, he's doing the math wrong.
posted by delmoi at 11:03 AM on May 30, 2007 [1 favorite]

I write $100. You write $99. Those are close enough to being the same price. Wouldn't the insurance guy just assume I was rounding up? And who says we bought the identical antiques in the same store?
posted by yeti at 11:03 AM on May 30, 2007

DevilsAdvocate, I direct you to TFA

The part you quoted has absolutely no relevance to the point I was making. If the person who had been randomly selected had only had an average payout of $10, would the fact that he gained $200 make him a "loser?" Perhaps in your eyes it would, but that's your interpretation, not part of the game, and there's no particular reason why game theorists should share your subjective dividing line between "winning" and "losing" in this game where you are virtually guaranteed some profit.
posted by DevilsAdvocate at 11:06 AM on May 30, 2007

I once watched an actual human being playing with real money run out a Martingale starting at 25 cents and finally running out the table limit of $1000.

Of course, Martingale is part of the reason you have table limits. Without that, someone with an unrestricted bankroll could beat the house every time.
posted by cortex at 11:06 AM on May 30, 2007

"The interesting question is why we're wired to ignore the odds, or why we're wired to be so bad at it. Is it smarter for the population to be dumb about risk assessment, if the payoff is large for a few individuals?"

Because the money does not maintain the same level of value.

The money or points in these games is just an abstraction. For us it has utility. We aren't necessarily dumb about risk assessment, although we might be. But it's that we aren't measuring the value according to how it is perceived by the holder. We don't feel the same way about spending 5% of our annual income when it is 14,000 vs 280,000.

Assuming you could play the Martingale until you reached a win, the value of the win does not exponentially grow, it stays at the value of the original wager. It is the amount wagered on each step that grows exponentially.

The Martingale is not a negative lottery, it has positive expectation.
posted by BigSky at 11:08 AM on May 30, 2007

I do agree with both DevilsAdvocate and cortex that since the game does not specify the conditions of winning, their bids are irrelevant. I had assumed maximising money to be the clear implicit goal. But in playing a game without an onbjective, what is the point of considering rationality whatsoever? They may as well pick their birthday.

You can reduce 'professional pride' to the monomaniacal goal to Get The Most Money in a deeply omphaloskeptic exercise? That's like saying professional pride would motivate any working artist to paint as strictly realistic a portrait as they could if someone gave them the lark instructions to Paint The Queen.

I disagree, I believe most artists would assume the goal is to paint a picture that is in line with the preferences of themselves and the people they care about.
posted by MetaMonkey at 11:09 AM on May 30, 2007

Only the first three values 100, 99, or 98 make any sense to choose. If this guy thinks the math works out to $2, he's doing the math wrong.

What if your opponent picks 97? How do any of 100, 99, or 98 then make sense? What if he picks 50? What if he picks 2?

You don't get to pick one static opponent choice: you have to evaluate his reaction to your choice, and iterate until it stabilizes. Whether that's how you and your opponent would truly analyze the situation is up for debate, but the math is fine.
posted by cortex at 11:09 AM on May 30, 2007

I disagree, I believe most artists would assume the goal is to paint a picture that is in line with the preferences of themselves and the people they care about.

And likewise game theorists. Much as this article paints a dumb-as-rocks portrait of the field for rhetorical effect, game theorists are not dead-eyed robots.
posted by cortex at 11:12 AM on May 30, 2007

Blazecock Pileon, I once watched an actual human being playing with real money run out a Martingale starting at 25 cents and finally running out the table limit of $1000. He was betting Red.

Of course it could happen. And the stupid thing about that "Algorithm" is that the most you can win is the first bet (in this case 25¢) and the most you can loose is, erm, infinite.

Of course, Martingale is part of the reason you have table limits. Without that, someone with an unrestricted bankroll could beat the house every time.

That's not true either. If someone won, say $10,000,000 they would have already paid the casino $999,999.75. And, because there is one spot that's not red or black, the probability is not actually even 50/50. Even with a billion dollar bankroll you still would only get 32 spins.
posted by delmoi at 11:12 AM on May 30, 2007

But the problem is, once Lucy gets to 98, she has no reason to continue lowering her estimate. She if she continued to reduce her estimate, she would get less money then if she picked $100 and her opponent picked $99. So (100,99) is a better outcome for her then (97,98).

The Nash equilibrium is a point where neither player can improve by choosing something else. It's true that Lucy can't improve from (99,100) by changing her number, but Pete can, by changing his 100 to 98. (99,100) only gives Pete $97, but (99,98) gives Pete $100. Likewise, (99,98) isn't a Nash equilibrium, because Lucy can do better against Pete's 98, by choosing 97 instead of 99. etc., etc., until you get to (2,2), and neither player can improve their position by choosing something else.

(Not defending the "2" choice, just explaining how game theory arrives at that conclusion.)
posted by DevilsAdvocate at 11:12 AM on May 30, 2007 [1 favorite]

THIS IS GAME THEORY!

Quote: "At the end of the war, if there are two Americans and one Russian, we win!"


General Thomas Power, commander in chief of the Strategic Air Command from 1957 to 1964 and Director of the Joint Strategic Target Planning Staff from 1960 to 1964, ranked near the top of the U.S. Armed Forces waging the Cold War.
posted by yoyo_nyc at 11:13 AM on May 30, 2007 [1 favorite]

No clearly defined goal = no basis for determining rationality.
posted by yesster at 11:17 AM on May 30, 2007

I had assumed maximising money to be the clear implicit goal.

"maximising money" is not well-defined when you have a game with many possible outcomes. Do you mean 1) maximizing the highest possible payout you could potentially receive? 2) maximizing the lowest possible payout you could potentially receive? 3) maximizing the average payout? 4) maximizing yet some other function of the various possible payouts and their probabilities?

The person who seeks to maximize the highest possible payout plays the lottery; the person who seeks to maximize the average, or the lowest payout, does not. Which one of these is "maximising money?"

In fact, it's worth noting that if your goal is to maximize the lowest possible payout--that is, to maximize the amount you are absolutely guaranteed to get, regardless of what the other player does - 2 is the correct choice!
posted by DevilsAdvocate at 11:19 AM on May 30, 2007

What if your opponent picks 97? How do any of 100, 99, or 98 then make sense? What if he picks 50? What if he picks 2?

If he picks $97, I'd get $95. As you know, $95 is more then $2. So picking $100, 98, or 99 still make far more sense then picking $2. Same as if he picks $50. You get $48, which is still more then $2.

Think about this way. Let's say you were going to play the game against 10 different opponents, who will go on to pick $100, $50, $2, $99, $98, $100, $100, $2, $8, and $20.

If you always pick $100, you'll make $564. If you pick $2 every time, you'll make $34. So, picking a high number is better then picking a low number, because it gets you more money.

I think, as someone else said, the "game" here isn't to get more money then your "opponent" it's to maximize your own return, you're playing against the airline, not the other passenger.
posted by delmoi at 11:20 AM on May 30, 2007

Point taken cortex, if you are suggesting the $2 bidders were doing it for in-joke japery. I had assumed the game to be a reward-maximisation competition, since the job of game-theorists is reward-maximisation, assuming the reward is well defined (which in this game, it is). If those 3 game-theorists are playing the meta-game of supra-game-utility maximisation, and having a laugh is of greatest utility to them, then fine. But as yesster says, if there is no goal, there is no rationality, and in fact no game, rendering all this pointless.
posted by MetaMonkey at 11:22 AM on May 30, 2007

"maximising money" is not well-defined when you have a game with many possible outcomes. Do you mean 1) maximizing the highest possible payout you could potentially receive? 2) maximizing the lowest possible payout you could potentially receive? 3) maximizing the average payout? 4) maximizing yet some other function of the various possible payouts and their probabilities?

The goal is to maximize the amount of money you actually make after the game is over. Anything less then $98, and you're not doing that.
posted by delmoi at 11:23 AM on May 30, 2007

Think about this way. Let's say you were going to play the game against 10 different opponents, who will go on to pick $100, $50, $2, $99, $98, $100, $100, $2, $8, and $20.

But that's thinking about a different problem than the one presented. If you could know in advance what your opponents would pick—or even just the average value of their individually unspecified picks—you would have more information than you have in the problem we're dealing with.
posted by cortex at 11:23 AM on May 30, 2007 [2 favorites]

VIZZINI: But it's so simple. All I have to do is divine it from what I know of you. Are you the sort of man who would put the poison into his own goblet or his enemies? Now, a clever man would put the poison into his own goblet because he would know that only a great fool would reach for what he was given. I am not a great fool so I can clearly not choose the wine in front of you ... But you must have known I was not a great fool; you would have counted on it, so I can clearly not choose the wine in front of me.

THE MAN IN BLACK: You've made your decision then?

VIZZINI: [Happily] Not remotely! Because Iocaine comes from Australia. As everyone knows, Australia is entirely peopled with criminals. And criminals are used to having people not trust them, as you are not trusted by me. So, I can clearly not choose the wine in front of you.

THE MAN IN BLACK: Truly, you have a dizzying intellect.

VIZZINI: Wait 'til I get going!! ... [via]

posted by mosk at 11:29 AM on May 30, 2007 [2 favorites]

DevilsAdvocate, I understand and agree with your logic, but in the game in question there is a specific reward; a multiple of your average. If this isn't the goal, and there is no well defined goal, there isn't much of a game.
posted by MetaMonkey at 11:29 AM on May 30, 2007

And the stupid thing about that "Algorithm" is that the most you can win is the first bet (in this case 25¢) and the most you can loose is, erm, infinite.

You can choose to win anything you want. It would be foolish to only bet 25 cents over your losses. Betting more does hit the limit faster of course.
posted by smackfu at 11:30 AM on May 30, 2007

I do agree with both DevilsAdvocate and cortex that since the game does not specify the conditions of winning

Also, you completely missed my point. You seem to have the notion that there must be some value X, and if you gain $X or more, that's "winning," and if you gain less than $X, that's "losing."

You are misunderstanding the concept of a "game" in the sense that game theorists use the term. You may be familiar with games such as Risk, or Monopoly, or baseball, where there's one or more "winners" and one or more "losers."

The possible outcomes of games that game theorists deal with are not necessarily "winning" or "losing." This game has 102 possible outcomes for each player: get nothing; get $1; get $2; get $3; ... get $100; get $101.

The point I was making was not "oh, you can define winning however you like and then you can do anything you like and any attempt to analyze 'rationality' goes out the window," as you seem to think. I was objecting to your attempt to reduce a game with 102 possible outcomes for each player - with a clear, well-defined preference ordering - to a simple binary "win-or-lose" scenario.
posted by DevilsAdvocate at 11:31 AM on May 30, 2007

But that's thinking about a different problem than the one presented. If you could know in advance what your opponents would pick—or even just the average value of their individually unspecified picks—you would have more information than you have in the problem we're dealing with.

What do you mean? We can't even guess? If I was going to guess, I would guess that they would pick the same number that I would in order to maximize their payoff just as I would pick a number to maximize mine. So, I would guess that they would pick 98, 99, or 100. I would naturally assume they would think the same way I would. I don't see any reason to pick anything else.

In fact, assuming the null hypothesis, which is that all choices from $0 to $100, then actually $100 would still be the best choice. The only way $2 makes sense if you do have some prior information, namely information that they'll always pick two.

With a "normal human" model, $100 is the best answer. With a null hypothesis, you still pick $100. Only if you use the bizarre, broken model that the author came up with is $2 the best choice.
posted by delmoi at 11:34 AM on May 30, 2007

You seem to have the notion that there must be some value X, and if you gain $X or more, that's "winning," and if you gain less than $X, that's "losing."

No, my notion is that winning is reward-maximisation.
posted by MetaMonkey at 11:42 AM on May 30, 2007

Fascinating. I thought about why I instinctively go for $100 and what I came up with was this: My motivation was to maximize the total payout by the airline guy, not to beat the other person. If we both choose $100, the total payout is $200. If one of us chooses $99, the total payout is now only $198. And so on. The total payout of this game is always (low bid * 2), so both choosing the highest allowed bid is naturally the only way to maximize total winnings.

I was surprised the article didn't mention this as a theory for why people tend to choose high in this game. It's the rational choice if you are able to imagine you and the other player as one unit, rather than pure competing individuals. Given the social nature of humanity, this seems like it would be a natural strategy for most people. If I come up with a name for this principle, can I have it? Like "Rusty's Law of Collaborative Airline Screwage"?
posted by rusty at 11:43 AM on May 30, 2007 [2 favorites]

Only if you use the bizarre, broken model that the author came up with is $2 the best choice.

But the model is not bizarre or broken, it's just badly applied. I agree—pretty much everyone in this thread agrees—that actualy humans put to this trial in a realistic setting would pick something other than $2, but that says nothing about the soundness of the model when applied to situations better fit to it.

Someone mentioned replacing dollars with nuclear weapons. How does that change the answer? It's less inflexible than my medicine example: the other guy having 1+ nukes is not instant death for you, just a stand-off. Still and all, how does 2 and 2 compare to 0 and n? Is there a sum total of nukes you could potentially have that would make it worth it for you to also potentially end up with none against an armed opponenent?
posted by cortex at 11:45 AM on May 30, 2007

100. 100, 100, 100!

So... anyone want to sign up for a game-theory trial with me?
posted by bicyclefish at 11:47 AM on May 30, 2007

I thought this was a pretty interesting piece. Of course the point of the mathematics is not

"you should pick $2 or you're stupid,"

but

"standard models of rational behavior don't adequately reflect some aspects of the way real people make decisions -- how do we deal with this?"

I don't think the answer to that question is "junk the idea of thinking mathematically about human behavior."

And for people who think it's obvious that $100 is the right bid and $2 an insane bid -- do ask yourself why, if maximizing the money you take home is the goal, you would bid $100 rather than $99. The latter results in either more money or the same amount for you no matter what the other player does. If your answer is "the extra $2 I might make doesn't mean anything to me," then keep in mind that there are lots of real-life decisions made by economic actors in which a 2% difference in revenue really does mean something.

(Not defending the $2 bid, which I think people are perfectly right not to make -- just trying to emphasize that there really is some serious content here.)
posted by escabeche at 11:48 AM on May 30, 2007

At anything other than $2, you are risking your bid vs. either $0 if your opponent is rational, or ??? if your opponent is not.

I don't quite agree. The assumption required to decide on $2 is not that your opponent is rational, it's that your opponent is following this particular theory, which in this case is clearly wrong. Seems to me that in order to prove this theory rational, you have to assume that it's rational.
posted by sfenders at 11:50 AM on May 30, 2007

If you can play an unlimited number of times, $2 is the rational choice.

You keep doing that, I'll come along, say $1 and get $3, and you'll lose $1.
posted by oaf at 11:56 AM on May 30, 2007

'Rational', in this case, is jargon rather than lay terminology, which is partly what the article is getting at and partly what the article kind of misses in it's presentation.

No one is saying it's rational in the sense that it characterizes correctly what a cross-section of reasonably intelligent people would do, and that's why the model is such a bad fit for this proposition. But if you want to create a model that actual produces a decision, you have to rigorously define good vs. bad choices somehow, and to do so you create an artificial system of values and call the correct maximization of results in that system 'rational'.

It's a problem to suggest that because your game theoretical model says something is 'rational' that it's what reasonable, intelligent people will do. But it's just as much a problem to say that because the practical and technical definitions of 'rational' clash, the model is useless. In either direction it's a bad conflation of systems.
posted by cortex at 11:56 AM on May 30, 2007

oaf, $2 is the minimum in the stated problem.
posted by cortex at 11:58 AM on May 30, 2007

You don't get to pick one static opponent choice: you have to evaluate his reaction to your choice, and iterate until it stabilizes.

Nope, you play once. That's the key to this game, and what sets it apart from most game theory exercises. In the prisoner's dilemma, its crucial that the game be iterative, and that the players not know how many rounds will be played, because if they do know, the game breaks, and the only rational action ever is to defect. Think about it- say that you know that the game will have 5 turns. The first four turns have passed (the outcomes don't matter), and you're now on the last round. You now have no hope of influencing your opponent's behavior anymore, because the game is about to end- the only thing that will affect your score is the outcome of this turn. And in PD, the best choice in any given turn is to defect, regardless of what your opponent does. So you'll defect (and your opponent will too). Actually imagine this scenario- if you knew you were in the last turn of PD game, what would you do? Now, roll back one turn. Both of you, being rational, know that on the next turn, you're both going to defect, so that turn is now fixed, and the only real choice you have is what you'll do on this turn, turn four. And the same thing plays out. The only way to make the game work is to make the number of turns unknown. And this holds for most iterative game theory exercises- if the number of turns is known by both players, the incentive to cooperate disappears.

In fact, I'm not sure I'm convinced that this game differs in substance from a one turn PD game. PD is defined by outcomes of the following ranking: Defect when partner cooperates > Both cooperate > Both defect > Cooperate when partner defects. If you define cooperating as choosing 100, and defecting as choosing 99, then this pattern holds (101 > 100 > 99 > 97). The backwards reasoning described in the article actually feels so much like iterative PD that I wonder if by letting the players choose any integer less than 100 (literally), the normal behavior would reassert itself, and both choosing 100 would become a rational strategy.
posted by gsteff at 11:58 AM on May 30, 2007

escabeche, how about this:
"Thinking mathematically about human behavior can be useful, and sufficiently complex models of rational behavior can reflect some aspects of the way real people make decisions, but this problem as presented is just fricking stupid."

The reason most people pick something other than $2 isn't because most people are less-than-ideally rational. It's because this problem was poorly constructed. It completely fails to state what "winning" means. Different posters above have come up with completely plausible yet incompatible interpretations of winning.
posted by yesster at 11:59 AM on May 30, 2007

With a "normal human" model, $100 is the best answer.

There have been days where $2 made the difference between having a meal and not having a meal. Had I been playing the game on one of those days, I might well have chosen 2 with its guarantee of $2, rather than anything higher, which might have given me a very good chance at $100 or so, but also a small but non-zero risk of getting nothing. Was I not a "normal human" on those days?

No, my notion is that winning is reward-maximisation.

If the other player chooses 2 - which is unlikely, but not, as the article shows, unheard of - then choosing 2 yourself is reward-maximaztion, as anything else will get you nothing.

Now, if you want to say 99 (not 100!) maximizes the average reward against what humans typically play, I won't argue with that. What I am saying is that "the One And Only Goal is to maximize one's expected (in the mathematical sense of "expected") return" and "all definitions of goals are equally legitimate and any choice can be defended as rational" are not the only two possible ways to analyze this game.

One might, for example, consider the notion of utility - imagining that we could quantify how "useful" a given person would consider a given amount of money. We might agree, for example, that utility is monotonically increasing with money - everyone agrees that $64 is better than $63 - but not linear - people have different valuations of how much more useful $64 is than $63.

Given that different people have different utility functions, it's entirely rational that some people might choose 2, and some people might choose 99. At the same time, it's not saying "anything goes" - given the model of utility I described above, it's never rational to choose 100, as 99 always gets you at least as much money than 100, and sometimes more.

(If you want to take "not being a dick" into account, then 100 might be a rational choice, but then the utility model I described no longer applies.)
posted by DevilsAdvocate at 12:00 PM on May 30, 2007

Nope, you play once.

Yes, but you can evaluate the play to hell and back before you make your one play. You iterate your strategy in your head before you move, and that is where your analysis of your opponent's move keeps changing, unless (a) you're psychic, (b) you have extra information, or (c) you're braindead.
posted by cortex at 12:00 PM on May 30, 2007

It completely fails to state what "winning" means.

Again, games (in the game theoretical sense) don't have to have "winning" or "losing." If it helps, don't think of it as a "game," just think of it as a "situation." One with 102 possible outcomes, from gaining $0 to gaining $101.

Game theory may not be able to adequately describe what people either should or will do in this situation, but it's not because the conditions don't define "winners" and "losers."
posted by DevilsAdvocate at 12:04 PM on May 30, 2007

69

*air guitar solo*
posted by mrgrimm at 12:06 PM on May 30, 2007

100. 100, 100, 100!

So... anyone want to sign up for a game-theory trial with me?

Sure!

99. 99, 99, 99!

(And it's worth noting that I don't care whether I "beat" bicyclefish or not, I'm just trying to maximize my own payout.)
posted by DevilsAdvocate at 12:07 PM on May 30, 2007

(A very large number of people would also do the non-imaginative thing and just claim exactly what the antique actually cost them.)
posted by onlyconnect at 12:08 PM on May 30, 2007 [3 favorites]

But that's an egocentric view of the "situation." There are more than 102 possible outcomes for the situation, even if there are only 102 possible outcomes for me.
posted by yesster at 12:09 PM on May 30, 2007

the article sets up a scenario where an actual item of tangible value is what is being evaluated

That's kinda what threw me off this stupid "game," anyway. If I'm a god-fearing Christian man who doesn't lie, isn't my choice easy?

I tell the traveler exactly what I paid for it.
posted by mrgrimm at 12:09 PM on May 30, 2007 [3 favorites]

Gaming theory is concerned with winning games, not maximizing profits or scores. Think of it this way, if your choice was to pick a # between 2-100, and you're only goal is to have a higher value than a person picking against you, the correct answer is 2 (because you can't lose).

In the scenario presented, most people don't see it as a competition, but a chance at free money. In that case, you're trying to maximize your profit/score and we can throw game theory out.
posted by Crash at 12:11 PM on May 30, 2007

But that's an egocentric view of the "situation." There are more than 102 possible outcomes for the situation, even if there are only 102 possible outcomes for me.

Fair enough; my point still stands that any failure of game theory here is not due to the fact that the game does not define "winners" or "losers."

Gaming theory is concerned with winning games, not maximizing profits or scores.

Either a)"gaming theory" is something completely different than "game theory" (and we're talking about game theory here), or else b) you don't have the slightest clue what you're talking about.
posted by DevilsAdvocate at 12:16 PM on May 30, 2007

As a rule of thumb it seems to require extraordinary circumstances before it is reasonable to expect people will make deductions requiring of two or more steps (ie, A -> B -> C).

This is both why real people don't often act by backreasoning from Nash equilibrium and why the logic behind why a $2 choice could be considered optimal seems so obscure.
posted by little miss manners at 12:21 PM on May 30, 2007 [2 favorites]

I thought the "goal" was to get compensated for your broken antique that you had purchased on your trip. I would write down a number a couple bucks more than what I paid for it because of the stupid rules that the untrusting airline manager with no sense of customer service has created. If the other person chooses a higher number we both get fairly compensated. If the other person chooses $2, I get a little irate and say "Two dollars? Where the fuck did you buy yours??? First I have to deal with this airline dick and his stupid fucking game and now this? Two Fucking Dollars!? Are you serious!?" and so on. Then I walk away with nothing but a story to tell about two fucking assholes I met on the way back from my Pacific holiday. Or am I taking it too literally?
posted by DanielDManiel at 12:23 PM on May 30, 2007 [5 favorites]

jinx, mrgrimm (plus you don't need to be a Christian to tell the truth).
posted by onlyconnect at