http://www.epjournal.net/blog/2013/05/are-all-dictator-game-results-artifacts/

You walk into a laboratory, and you read a set of instructions that tell you that your task is to decide how much of a $10 pie you want to give to an anonymous other person who signed up for the experimental session.

This describes, more or less, the Dictator Game, a staple of behavioral economics with a history dating back more than a quarter of a century. The Dictator Game (DG) might not be the drosophila melanogaster of behavioral economics – the Prisoner’s Dilemma can lay plausible claim to that prized analogy – but it could reasonably aspire to an only slightly more modest title, perhaps the e. coli of the discipline. Since the original work, more than 20,000 observations in the DG have been reported.

[...]

How much would participants in a Dictator Game give to the other person if they did not know they were in a Dictator Game study? Simply following me around during the day and recording how much cash I dispense won’t answer this question because in the DG, the money is provided by the experimenter. So, to build a parallel design, the method used must move money to subjects as a windfall so that we can observe how much of this “house money” they choose to give away.

And that is what Winking and Mizer did in a paper now in press and available online (paywall) in Evolution and Human Behavior, using participants, fittingly enough, in Las Vegas. Here’s what they did. Two confederates were needed. The first, destined to become the “recipient,” was occupied on a phone call near a bus stop in Vegas. The second confederate approached lone individuals at the bus stop, indicated that they were late for a ride to the airport, and asked the subject if they wanted the $20 in casino chips still in the confederate’s possession, scamming people into, rather than out of money, in sharp contradiction of the deep traditions of Las Vegas. The question was how many chips the fortunate subject transferred to the nearby confederate.

[...]

In a second condition, the confederate with the chips added a comment to the effect that the subject could “split it with that guy however you want,” indicating the first confederate. This condition brings the study a bit closer, but not much closer, to lab conditions, In a third condition, subjects were asked if they wanted to participate in a study, and then did so along the lines of the usual DG, making the treatment considerably closer to traditional lab-based conditions.

The difference between the first two treatments and the third treatments is interesting, but, as I said at the beginning, the DG should be thought of as a measuring tool. Figure 1 shows how many chips people give away in the DG in the three treatments. In conditions 1 and 2, the number of people (out of 60) who gave at least one chip to the second confederate was… zero. To the extent you think that this method answers the question, how much Dictator Game giving is due to people knowing they’re in an experiment, the answer is, “all of it.”

Link to paper (paywalled).

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Suppose we consider the players' actions in the Dictator and Ultimatum games not as attempts to gain or distribute money as an end in itself, but as (sometimes expensively-signaled) communications with the experimenter, the counterpart, or some other audience — possibly including the self.

In other words, the goal of my immediate act is not to gain for myself nor to give to others; it is to make a statement about what sort of person I am (generous! fair! realistic! not a sucker!), or what sort of society or economy I expect (or desire?) to live in.

Condition 3 was a more traditional DG utilizing the same
population and currency and offering anonymity in a manner very
similar to that employed by Hoffman and colleagues (Hoffman et al.,
1994). Individuals waiting alone at a bus stop within a block of a
casino in Las Vegas, Nevada were approached and asked if they
wished to participate in a study that would involve them receiving
some money. If they agreed, they were then consented only to the DG
portion of the study and not the questionnaire (this was because the
participants in Conditions 1 and 2 were unaware of the $10 payment
when they made their donation decisions). In this experiment,
participants were handed two envelopes, a yellow envelope with
the word “KEEP” on it, and a white envelope with the word “GIVE” on
it. They were then given $20 worth of chips as well as six fake chips.
The chips were handed to them in a small chip-holder box with a
divider in the middle and “FAKE” written on one side and “REAL”
written on the other. While the researcher's back was turned, they
divided the real chips between the two envelopes as they desired, and
then distributed the fake chips between the two envelopes so as to
make them feel roughly the same. In this way, the researcher
remained totally unaware as to the composition of the chips in the
donation envelope. It was explained to them that the chips in the
“GIVE” envelope were to be recorded by a separate research assistant
at a later date and handed out to a random individual waiting at a bus
stop. They were to take those in the “KEEP” envelope home with them.

I think this renders your story a bit less plausible, since apparently the subject isn't having face-to-face contact with anyone who learns their choice.

(Disclaimer: I was just skimming over the paper and happened to see this bit--in the same way I made you seem wrong by finding this, someone else may make me seem wrong by reading the entire paper or the DG literature in depth. Unknown unknowns and whatnot.)

Simply saying 'you could share it with this guy' is not like the other case, because the two people haven't been called out as special. You don't share windfall with everyone you meet, why should this guy be different? In the DG, the other player was brought into the game and spent time waiting around just like you. It calls them out.

I'm trying to find a way of constructing a closely analogous situation without self-consciousness, and am having a tough time.

Perhaps something like this: participant & confederate are waiting at the bus stop. Guy with chips approaches them in a hurry and says "I'm late for my ride to the airport, does someone want my chips?" Participant & confederate both say that they do and move closer to the guy with chips (confederate attempts to act similarly to the participant). Guy with chips hands them all to the participant (who is perhaps slightly closer), says "here, take them - you can share them with the other guy if you want", and rushes off.

That way the confederate is someone who has been through essentially the same experience as the participant up until the moment the participant was handed the chips. He is not just some guy who happened to be standing nearby.

problem: this introduces social pressure and possible consequences( this person might recognize you in future and penalize you in the same way), part of the dictator game is that you don't have to look the person who you're splitting the cash with in the eye.

You don't share windfall with everyone you meet

I usually do.

Well, for some value of “share”, “windfall”, “everyone”, and “meet”, at least.

My impression is that many people are reluctant to start conversations with strangers. It seems like the trivial inconvenience of/subconscious aversion to starting a conversation with a stranger could have dissuaded people from sharing their chips.