The ending effect happens all the time in experiments. If people play the repeated prisoner's dilemma, they'd cooperate UNTIL the last round. If people play the public good game, they'd contribute generously UNTIL the last round.
Because if the prioner's dilemma and public good game are played one shot, the rational thing to do is to cheat right away (defect / not contribute). Being cooperative (in hope to make the world a better place) goes against the selfish motives.
So, at the last round, because there is no tomorrow, it's just like the game is played one-shot, people'd cheat. The theory of game goes too far to say that, if people are rational, they work the game backward (backward induction) and at the 9th round, they'd cheat, and at the 8th round, they'd cheat...-> they'd cheat right from the first round.
But experiments show that people are myopic or at least dont cheat right in the first round, they wait until last round. Because of experiments, theorists have to add the social image. Social image can be built in the first 9 round, but in the last round, not anymore.
Which is to say, I wonder if this effect might change if the participants were offered more money?
Or it could be that people don't realize there's no way for the experimenters to know if they're cheating, so they're on their best behavior until they become familiar with the system.
I think this is true in a lot of domains (jobs, dating, etc). In the beginning, people follow the rules. When they're comfortable with the ins and out of the system, they start gaming it.
One that occurs to me if that someone has unlimited tosses, then the number of correct guesses doesn't matter that much for their "expected income stream" - if they want more money, they can just toss more coins instead of cheating. And oppositely, if the experiment is coming to an end, the only way to get more money is by cheating and so now the incentive to cheat becomes a factor.
This isn't fundamentally that different than kazinator's approach [1] but it is a little different.
The position I take on all this is that psychological experiment always involve some particular, reproducible series of behaviors that are then given an abstract intuitive interpretation (distorting heads-tails results becomes "cheating-in-the-abstract"). But naturally any particular behavior actually can have legion abstract interpretations and explanation. I suspect only a much large collection of behaviors and abstraction interpretations could serve as a real compelling explanation. The problem is the intuitive human "toolbox" of abstract explanation - such as "cheating"(in the abstract) might not be the real toolbox we need to use to explain human behavior.
(This is in a different setting, when there is some limited time to achieve some predetermined, measurable goal.)
When the pressure sets in is when people start cutting corners, and that can come simply from being closer to a deadline while lagging in progress.