The travellers' dilemma

[SeriousSam]

Setting aside the fact that the question has no bearing to reality, i.e. no human being is entirely rational. The answer is £100... primarily because (a) the penalty is not significant enough to out-way the benefits of outbidding the other traveller and (b) the bonus isn't significant either. Assuming that both will take the rational position of trying to recoup their investment. "Beating the other guy" is an emotional rather than rational response.

[Gerrard]

Amusing distractions aside, I'm not sure I explained it properly - you're all (presumably) a bright bunch.

In any case. Game theory suggests the answer is $2 - assuming you start with $100, as the other traveller is entirely rational, he will be looking to maximise his own return and select $99. Knowing that will be his strategy, you underbid him with $98, and he changes with $97, all the way until $2.

This is a fairly useful analogy to a bunch of conflicts, including, most obviously, nuclear deterrence.

The 'real' answer is clearly some mix of $99 and $100. This leaves you with a range of payouts between $99 and $101. The key here is that bidding $2 does not maximize returns, but, rather, minimizes them, through the logic simply of trying to beat the other guy. What the ratios would be is a question for smarter men than me, but it could be as simple as half and half. But game theorists actually insist that the only, correct answer, is $2. Which is pretty stupid imo.

Yeah, at no point did you say "to get more than the other guy"; it was only implied to maximise the amount you get.

But the minimum amount does make sense to "win", as if the other person says the same, you draw. If he opts for anything higher, you get $4 ($2 + $2 honesty) and he gets $1 ($2 - $1 liar).

[Peter Parker]

In any case. Game theory suggests the answer is $2 - assuming you start with $100, as the other traveller is entirely rational, he will be looking to maximise his own return and select $99. Knowing that will be his strategy, you underbid him with $98, and he changes with $97, all the way until $2.

I've rewritten this comment twice because it's so confusing and counter intuitive!

OK, I read the wiki page for it hoping it might help, because right now it's reminding me that you should "never go in against a Sicilian when death is on the line"

I started writing about why an explanation based on John Nash's Nobel-winning theory is stupid, but ... as I write it starts to make sense. (edit 2 - except now it doesn't)

First I started thinking, surely if Alice bids $100, then she can get either $100, or if Bob bids $99 then she gets at worst $98. Why on earth would she bid $2 ever?! Economic theory is great in theory and dumb in practice! Look how clever I am, I'm smarter than Russell Crowe etc etc ...

But wait! It's not that simple. Bob isn't guaranteed to bid just either $100 or $99. He could easily bid $50 to be more "honest" which spoils everything. Alice has no idea if Bob has ever heard of the Prisoners Dilemma. Perhaps the most he ever reads is a tatty copy of The Metro on the way to work, and perhaps Nash is right and $2 is the best bid?

edit 2 :-

But wait! Would any real human ever do that? I did some searching and there's actually a paper where the researchers played this game with some random people on the Internet. There's a chart of the chosen bids at the top of page 6:
https://pdfs.semanticscholar.org/6c8...c441cf62b6.pdf

Some very odd people bid $31, $49 and $70. A few bid $2 or $3 - found the 1st Year Econ students! Most bid over about ~$90.

Here's a Sci Am paper by the original author of the TD problem which discusses more experiments with real people. Fairly readable, and he seems genuinely amazed that everybody doesn't behave like at least a Masters-level economist!
http://www.cs.virginia.edu/~robins/T...rs_Dilemma.pdf

And so finally ... this is where economic theory really is great in theory and dumb in practice. The theory assumes that the rules of the game as stated are perfect and pure - the two players know nothing about each other, and $2 is the mathematically provable rational choice. We are all robot economists.

But in reality both players at least know the other is a human, and believe that no rational human would really bid $2. Except those 1st Year Econ students who are idiots and know nothing about human behaviour. Those kids will be future governors of the Bank of England and future Chancellors of the Exchequer

[wazzickle]

Yeah, at no point did you say "to get more than the other guy"; it was only implied to maximise the amount you get.

But the minimum amount does make sense to "win", as if the other person says the same, you draw. If he opts for anything higher, you get $4 ($2 + $2 honesty) and he gets $1 ($2 - $1 liar).

If you follow the game theoretical logic, you maximize your payout by going with $2, even without going for the win.. I'm not saying it's right - it's clearly not - this highlights an obvious limitation, but you should hear how adamant the mathematicians are that this is the only and correct solution.

[wazzickle]

Setting aside the fact that the question has no bearing to reality, i.e. no human being is entirely rational. The answer is £100... primarily because (a) the penalty is not significant enough to out-way the benefits of outbidding the other traveller and (b) the bonus isn't significant either. Assuming that both will take the rational position of trying to recoup their investment. "Beating the other guy" is an emotional rather than rational response.

Please read through your post again and ask yourself whether there are any inconsistencies.

Change the levels of the penalty or bonus if you want, it doesn't matter. This is a theoretical question whereby improving your results by a small amount is the goal, if you can't improve your results by a bigger amount. "Not worth it" is not part of this equation.

[Saracen]

Let's see if I get this.

I say $100, and the other traveller (let's say, you) predict that and says $99. Result = I get $100-1=$99, and you get $99+2=$101.

So, in order to avoid losing $1, I'm supposed to claim the value as $2, and be called "rational" for doing so?

Instead, I'll just claim whatever I think the item is worth, and if you succeed in getting a whole buck or two more, well, good for you. Your good fortune is no material loss to me. Just don't spend it all at once.

Your definition of rational behaviour, however, leaves a lot to be desired.

[DanceswithUnix]

The $2 thing seems to be about maximising definite reward, I think most rational people would go for a risk/reward balance and on the rules here you punt for $100. I mean, if I paid $150 for something and I claim back $100 towards its replacement then that is unfortunate, but claiming back $2 to me means I lost the big picture. I think that is the problem here, I suspect most people on this forum are going to be big picture co-operative types and not the small detail competitors that will drive you down to $2. I wouldn't bother with the paperwork for $2.

Or the other way to put it: If I bid $100 then there is a chance that I will lose up to $101 of my best case $101, whereas if I bid $2 then there is a certainty that I lose $99. For most people a loss of $99 would be considered a defeat, so assuming the other guy is rational you both need to bid high. The choice really is $99 vs $100, because co-operation is a corollary of the rules.

Prisoner's dilemma is better

[wazzickle]

Ok guys, let's change the numbers if this will make it easier to envisage.

You can choose between $2,000 and $100,000, the 'honest' traveller receives the amount they claimed plus $2,000, the dishonest will receive that same amount minus $1,000.

[TheAnimus]

It's still the same thing, this is just co-operate betray prisoners dilemma.

[Ttaskmaster]

Ok guys, let's change the numbers if this will make it easier to envisage.

You can choose between $2,000 and $100,000, the 'honest' traveller receives the amount they claimed plus $2,000, the dishonest will receive that same amount minus $1,000.

You'd have to come up with a significant bonus/penalty, I think.
At the levels we're talking, the pain/gain is a pittance.

[scaryjim]

The real problem with this phrasing of the problem is that it's too unrealistic. There's no way to read it and think "oh, yes, I could imagine being in that situation".

A more sensible rephrasing of this problem (if I have understood he original problem correctly) would be:

Two people are asked to play a game. They each secretly request a prize between £2 and £100. If they both request the same prize they each receive it. If they request different prizes, the person who made the lower request receives the lower request plus £2, and the person who made the higher request receives the lower amount minus £1.

Assuming that both players are rational and know the rules, how much should they ask for to maximise their prize?

Personally, I'd also add the corollary question

How much would you ask for if paying this game?

which is an interesting experiment - would people willingly admit they're not rational by giving different answers?

IMO the "correct" answer under these conditions is to request £100. The potential gains for any other selection simply don't add up. And as a game it has some merit (although not much, as the loss/reward ratio simply isn't good enough to make it interesting). But when you start suggesting it as a method of calculating insurance settlements, it's just ludicrous.

[virtuo]

I think Churchill operate in a similar fashion

[wazzickle]

This is a thought experiment. This isn't a 'how much does money mean to you' question, it shouldn't matter whether the numbers are pennies or billions. The maths is what makes this question interesting, not the psychology.

[wazzickle]

It's still the same thing, this is just co-operate betray prisoners dilemma.

Nope

[Ttaskmaster]

This is a thought experiment. This isn't a 'how much does money mean to you' question, it shouldn't matter whether the numbers are pennies or billions. The maths is what makes this question interesting, not the psychology.

And you're getting our thoughts on it!!
The maths are simple, any rational human will vote for the maximum amount and any rational pair will reason that, if they both seek to maximise the odds of a higher payout, they will both vote the maximum, evening the odds and sacrificing the smaller bonus in order to secure the majority.

But you can't really ask for that of humans, because psychology will always govern their maths - This is why stuff is still priced at £14.99, because that is less than £15.00.

[peterb]

I did wonder what the effect would be if the reward or penalty was different, but it actually does make no difference - its the psychology of potential reward that is supposed to govern the choice.