Traveler's Dilemma
So, Scientific American just ran a short piece by Kaushik Basu about the Traveler's Dilemma. Basu describes the game and explains why its interpretation causes some problems. He also mentions a few possible ways out. (Coincidentally, Scott Aaronson recently mentioned a similar issue in a totally different context, noting that people only tend to apply modus ponens a couple of times before they figure the outcome's good enough.)
I feel like there's a couple of pretty obvious features that a good interpretation will have, features that Basu didn't really mention. First, it's pretty important to be clear about what the real goal of the game is. The goal, as stated, is simply to make as much money as possible, but the interpretations seem fairly quickly to introduce the idea of making more than the other player. This is a really important distinction, and if the goal of the game is to make as much as possible (regardless of the other player's return), we don't really need to talk about altruism at all. We'd be better off framing the "choose from the high range" strategies as a form of sacrifice rather than a form of altruism, the thought process being, "Perhaps (s)he'll make $101 to my $97, but $97 is still a hell of a lot more than $2."
We can understand the "honesty bonus" essentially the same way, which explains why the tendency to choose lower goes up as the honesty bonus is increased. Effectively the honesty bonus serves as a meta-game, a slider changing the emphasis of the base game from "making as much as possible, period" to "beating your opponent at all costs." It's interesting that for large honesty bonuses, failure to undercut your opponent will certainly drive your payoff to zero.
Second, it's pretty important to understand how many times we'll be playing this game. If you tell me I'll be iterating the game infinitely, I'll just go ahead and start choosing $2 every time. (And in fact, choices do converge to the low end as the game is repeated. The author vaguely attributes this to people slowly starting to "get it," but that's not the only possible reason.) But if I know I'm only going to play once, I'll certainly make a higher-risk choice. This also brings me to three: our interpretation ought to somehow account for the fact that there is a large qualitative difference between making $2 and making $100. As an example, I happened to go to the horse races a few weeks ago, and was given $4 to bet. Not being much of a betting man, I figured I'd bet my $4 and call it a day. So I was faced with a similar choice: make a safe bet and possibly walk out with $10, or make long-shot bets. I went with the long-shot bets and predictably walked out with nothing. Why? I looked at it like this: $10 isn't much money, I've already got more than that in my wallet, and honestly it's just not that great of a story to tell ("Hey guys, I played it safe and won $10 at the track today!"). On the other hand, making a long-shot and turning $4 of someone else's money into $1000 would be a big thrill and make a great story.
So I think a good interpretation ought to account for the fact that if the payoff is very small, the qualitative difference is great enough that the game's not even worth playing, assuming it's played only once. It's essential to my choice at the track that I haven't been to the horse races in fifteen years, and it'll probably be another fifteen before I place another bet.
I feel like there's a couple of pretty obvious features that a good interpretation will have, features that Basu didn't really mention. First, it's pretty important to be clear about what the real goal of the game is. The goal, as stated, is simply to make as much money as possible, but the interpretations seem fairly quickly to introduce the idea of making more than the other player. This is a really important distinction, and if the goal of the game is to make as much as possible (regardless of the other player's return), we don't really need to talk about altruism at all. We'd be better off framing the "choose from the high range" strategies as a form of sacrifice rather than a form of altruism, the thought process being, "Perhaps (s)he'll make $101 to my $97, but $97 is still a hell of a lot more than $2."
We can understand the "honesty bonus" essentially the same way, which explains why the tendency to choose lower goes up as the honesty bonus is increased. Effectively the honesty bonus serves as a meta-game, a slider changing the emphasis of the base game from "making as much as possible, period" to "beating your opponent at all costs." It's interesting that for large honesty bonuses, failure to undercut your opponent will certainly drive your payoff to zero.
Second, it's pretty important to understand how many times we'll be playing this game. If you tell me I'll be iterating the game infinitely, I'll just go ahead and start choosing $2 every time. (And in fact, choices do converge to the low end as the game is repeated. The author vaguely attributes this to people slowly starting to "get it," but that's not the only possible reason.) But if I know I'm only going to play once, I'll certainly make a higher-risk choice. This also brings me to three: our interpretation ought to somehow account for the fact that there is a large qualitative difference between making $2 and making $100. As an example, I happened to go to the horse races a few weeks ago, and was given $4 to bet. Not being much of a betting man, I figured I'd bet my $4 and call it a day. So I was faced with a similar choice: make a safe bet and possibly walk out with $10, or make long-shot bets. I went with the long-shot bets and predictably walked out with nothing. Why? I looked at it like this: $10 isn't much money, I've already got more than that in my wallet, and honestly it's just not that great of a story to tell ("Hey guys, I played it safe and won $10 at the track today!"). On the other hand, making a long-shot and turning $4 of someone else's money into $1000 would be a big thrill and make a great story.
So I think a good interpretation ought to account for the fact that if the payoff is very small, the qualitative difference is great enough that the game's not even worth playing, assuming it's played only once. It's essential to my choice at the track that I haven't been to the horse races in fifteen years, and it'll probably be another fifteen before I place another bet.