The Clumsy Game-Player
You and a partner are playing an Iterated Prisoner's Dilemma. Both of you have publicly pre-committed to the tit-for-tat strategy. By iteration 5, you're going happily along, raking up the bonuses of cooperation, when your partner unexpectedly presses the "defect" button.
"Uh, sorry," says your partner. "My finger slipped."
"I still have to punish you just in case," you say. "I'm going to defect next turn, and we'll see how you like it."
"Well," said your partner, "knowing that, I guess I'll defect next turn too, and we'll both lose out. But hey, it was just a slipped finger. By not trusting me, you're costing us both the benefits of one turn of cooperation."
"True", you respond "but if I don't do it, you'll feel free to defect whenever you feel like it, using the 'finger slipped' excuse."
"How about this?" proposes your partner. "I promise to take extra care that my finger won't slip again. You promise that if my finger does slip again, you will punish me terribly, defecting for a bunch of turns. That way, we trust each other again, and we can still get the benefits of cooperation next turn."
You don't believe that your partner's finger really slipped, not for an instant. But the plan still seems like a good one. You accept the deal, and you continue cooperating until the experimenter ends the game.
After the game, you wonder what went wrong, and whether you could have played better. You decide that there was no better way to deal with your partner's "finger-slip" - after all, the plan you enacted gave you maximum possible utility under the circumstances. But you wish that you'd pre-committed, at the beginning, to saying "and I will punish finger slips equally to deliberate defections, so make sure you're careful."
The Lazy Student
You are a perfectly utilitarian school teacher, who attaches exactly the same weight to others' welfare as to your own. You have to have the reports of all fifty students in your class ready by the time midterm grades go out on January 1st. You don't want to have to work during Christmas vacation, so you set a deadline that all reports must be in by December 15th or you won't grade them and the students will fail the class. Oh, and your class is Economics 101, and as part of a class project all your students have to behave as selfish utility-maximizing agents for the year.
It costs your students 0 utility to turn in the report on time, but they gain +1 utility by turning it in late (they enjoy procrastinating). It costs you 0 utility to grade a report turned in before December 15th, but -30 utility to grade one after December 15th. And students get 0 utility from having their reports graded on time, but get -100 utility from having a report marked incomplete and failing the class.
If you say "There's no penalty for turning in your report after deadline," then the students will procrastinate and turn in their reports late, for a total of +50 utility (1 per student times fifty students). You will have to grade all fifty reports during Christmas break, for a total of - 1500 utility (-30 per report times fifty reports). Total utility is -1450.
So instead you say "If you don't turn in your report on time, I won't grade it." All students calculate the cost of being late, which is +1 utility from procrastinating and -100 from failing the class, and turn in their reports on time. You get all reports graded before Christmas, no students fail the class, and total utility loss is zero. Yay!
Or else - one student comes to you the day after deadline and says "Sorry, I was really tired yesterday, so I really didn't want to come all the way here to hand in my report. I expect you'll grade my report anyway, because I know you to be a perfect utilitarian, and you'd rather take the -30 utility hit to yourself than take the -100 utility hit to me."
You respond "Sorry, but if I let you get away with this, all the other students will turn in their reports late in the summer." She says "Tell you what - our school has procedures for changing a student's previously given grade. If I ever do this again, or if I ever tell anyone else about this, you can change my grade to a fail. Now you know that passing me this one time won't affect anything in the future. It certainly can't affect the past. So you have no reason not to do it." You believe her when she says she'll never tell, but you say "You made this argument because you believed me to be the sort of person who would accept it. In order to prevent other people from making the same argument, I have to be the sort of person who wouldn't accept it. To that end, I'm going to not accept your argument."
The Grieving Student
A second student comes to you and says "Sorry I didn't turn in my report yesterday. My mother died the other day, and I wanted to go to her funeral."
You say "Like all economics professors, I have no soul, and so am unable to sympathize with your loss. Unless you can make an argument that would apply to all rational actors in my position, I can't grant you an extension."
She says "If you did grant this extension, it wouldn't encourage other students to turn in their reports late. The other students would just say 'She got an extension because her mother died'. They know they won't get extensions unless they kill their own mothers, and even economics students aren't that evil. Further, if you don't grant the extension, it won't help you get more reports in on time. Any student would rather attend her mother's funeral than pass a course, so you won't be successfully motivating anyone else to turn in their reports early."
You think for a while, decide she's right, and grant her an extension on her report.
The Sports Fan
A third student comes to you and says "Sorry I didn't turn in my report yesterday. The Bears' big game was on, and as I've told you before, I'm a huge Bears fan. But don't worry! It's very rare that there's a game on this important, and not many students here are sports fans anyway. You'll probably never see a student with this exact excuse again. So in a way, it's not that different from the student here just before me, the one whose mother died."
You respond "It may be true that very few people will be able to say both that they're huge Bears fans, and that there's a big Bears game on the day before the report comes due. But by accepting your excuse, I establish a precedent of accepting excuses that are approximately this good. And there are many other excuses approximately as good as yours. Maybe someone's a big soap opera fan, and the season finale is on the night before the deadline. Maybe someone loves rock music, and there's a big rock concert on. Maybe someone's brother is in town that week. Practically anyone can come up with an excuse as good as yours, so if I accept your late report, I have to accept everyone's.
"The student who was here before you, that's different. We, as a society, already have an ordering in which a family member's funeral is one of the most important things around. By accepting her excuse, I'm establishing a precedent of accepting any excuse approximately that good, but almost no one will ever have an excuse that good. Maybe a few people who are really sick, someone struggling with a divorce or a breakup, that kind of thing. Not the hordes of people who will be coming to me if I give you your exemption.
The Murderous Husband
You are the husband of a wonderful and beautiful lady whom you love very much - and whom you just found in bed with another man. In a rage, you take your hardcover copy of Introduction To Game Theory and knock him over the head with it, killing him instantly (it's a pretty big book).
At the murder trial, you plead to the judge to let you go free. "Society needs to lock up murderers, as a general rule. After all, they are dangerous people who cannot be allowed to walk free. However, I only killed that man because he was having an affair with my wife. In my place, anyone would have done the same. So the crime has no bearing on how likely I am to murder someone else. I'm not a risk to anyone who isn't having an affair with my wife, and after this incident I plan to divorce and live the rest of my days a bachelor. Therefore, you have no need to deter me from future murders, and can safely let me go free."
The judge responds: "You make a convincing argument, and I believe that you will never kill anyone else in the future. However, other people will one day be in the position you were in, where they walk in on their wives having an affair. Society needs to have a credible pre-commitment to punishing them if they succumb to their rage, in order to deter them from murder."
"No," you say, "I understand your reasoning, but it won't work. If you've never walked in on your wife having an affair, you can't possibly understand the rage. No matter how bad the deterrent was, you'd still kill the guy."
"Hm," says the judge. "I'm afraid I just can't believe anyone could ever be quite that irrational. But I see where you're coming from. I'll give you a lighter sentence."
The Bellicose Dictator
You are the dictator of East Examplestan, a banana republic subsisting off its main import, high quality hypothetical scenarios. You've always had it in for your ancestral enemy, West Examplestan, but the UN has made it clear that any country in your region that aggressively invades a neighbor will be severely punished with sanctions and possible enforced "regime change." So you decide to leave the West alone for the time being.
One day, a few West Examplestanis unintentionally wander over your unmarked border while prospecting for new scenario mines. You immediately declare it a "hostile incursion" by "West Examplestani spies", declare war, and take the Western capital in a sneak attack.
The next day, Ban Ki-moon is on the phone, and he sounds angry. "I thought we at the UN had made it perfectly clear that countries can't just invade each other anymore!"
"But didn't you read our propaganda mouthpi...ahem, official newspaper? We didn't just invade. We were responding to Western aggression!"
"Balderdash!" says the Secretary-General. "Those were a couple of lost prospectors, and you know it!"
"Well," you say. "Let's consider your options. The UN needs to make a credible pre-commitment to punish aggressive countries, or everyone will invade their weaker neighbors. And you've got to follow through on your threats, or else the pre-commitment won't be credible anymore. But you don't actually like following through on your threats. Invading rogue states will kill a lot of people on both sides and be politically unpopular, and sanctions will hurt your economy and lead to heart-rending images of children starving. What you'd really like to do is let us off, but in a way that doesn't make other countries think they'll get off too.
"Luckily, we can make a credible story that we were following international law. Sure, it may have been stupid of us to mistake a few prospectors for an invasion, but there's no international law against being stupid. If you dismiss us as simply misled, you don't have to go through the trouble of punishing us, and other countries won't think they can get away with anything.
"Nor do you need to live in fear of us doing something like this again. We've already demonstrated that we won't go to war without a casus belli. If other countries can refrain from giving us one, they have nothing to fear."
Ban Ki-moon doesn't believe your story, but the countries that would bear the economic brunt of the sanctions and regime change decide they believe it just enough to stay uninvolved.
The Peyote-Popping Native
You are the governor of a state with a large Native American population. You have banned all mind-altering drugs, with the honorable exceptions of alcohol, tobacco, caffeine, and several others, because you are a red-blooded American who believes that they would drive teenagers to commit crimes.
A representative of the state Native population comes to you and says: "Our people have used peyote religiously for hundreds of years. During this time, we haven't become addicted or committed any crimes. Please grant us a religious exemption under the First Amendment to continue practicing our ancient rituals." You agree.
A leader of your state's atheist community breaks into your office via the ventilation systems (because seriously, how else is an atheist leader going to get access to a state governor?) and says: "As an atheist, I am offended that you grant exemptions to your anti-peyote law for religious reasons, but not for, say, recreational reasons. This is unfair discrimination in favor of religion. The same is true of laws that say Sikhs can wear turbans in school to show support for God, but my son can't wear a baseball cap in school to show support for the Yankees. Or laws that say Muslims can get time off state jobs to pray five times a day, but I can't get time off my state job for a cigarette break. Or laws that say state functions will include special kosher meals for Jews, but not special pasta meals for people who really like pasta."
You respond "Although my policies may seem to be saying religion is more important than other potential reasons for breaking a rule, one can make a non-religious case justifying them. One important feature of major world religions is that their rituals have been fixed for hundreds of years. Allowing people to break laws for religious reasons makes religious people very happy, but does not weaken the laws. After all, we all know the few areas in which the laws of the major US religions as they are currently practiced conflict with secular law, and none of them are big deals. So the general principle 'I will allow people to break laws if it is necessary to established and well-known religious rituals" is relatively low-risk and makes people happy without threatening the concept of law in general. But the general principle 'I will allow people to break laws for recreational reasons' is very high risk, because it's sufficient justification for almost anyone breaking any law."
"I would love to be able to serve everyone the exact meal they most wanted at state dinners. But if I took your request for pasta because you liked pasta, I would have to follow the general principle of giving everyone the meal they most like, which would be prohibitively expensive. By giving Jews kosher meals, I can satisfy a certain particularly strong preference without being forced to satisfy anyone else's."
The Well-Disguised Atheist
The next day, the atheist leader comes in again. This time, he is wearing a false mustache and sombrero. "I represent the Church of Driving 50 In A 30 Mile Per Hour Zone," he says. "For our members, going at least twenty miles per hour over the speed limit is considered a sacrament. Please grant us a religious exemption to traffic laws."
You decide to play along. "How long has your religion existed, and how many people do you have?" you ask.
"Not very long, and not very many people," he responds.
"I see," you say. "In that case, you're a cult, and not a religion at all. Sorry, we don't deal with cults."
"What, exactly, is the difference between a cult and a religion?"
"The difference is that cults have been formed recently enough, and are small enough, that we are suspicious of them existing for the purpose of taking advantage of the special place we give religion. Granting an exemption for your cult would challenge the credibility of our pre-commitment to punish people who break the law, because it would mean anyone who wants to break a law could just found a cult dedicated to it."
"How can my cult become a real religion that deserves legal benefits?"
"You'd have to become old enough and respectable enough that it becomes implausible that it was created for the purpose of taking advantage of the law."
"That sounds like a lot of work."
"Alternatively, you could try writing awful science fiction novels and hiring a ton of lawyers. I hear that also works these days."
Conclusion
In all these stories, the first party wants to credibly pre-commit to a rule, but also has incentives to forgive other people's deviations from the rule. The second party breaks the rules, but comes up with an excuse for why its infraction should be forgiven.
The first party's response is based not only on whether the person's excuse is believable, not even on whether the person's excuse is morally valid, but on whether the excuse can be accepted without straining the credibility of their previous pre-commitment.
The general principle is that by accepting an excuse, a rule-maker is also committing themselves to accepting all equally good excuses in the future. There are some exceptions - accepting an excuse in private but making sure no one else ever knows, accepting an excuse once with the express condition that you will never accept any other excuses - but to some degree these are devil's bargains, as anyone who can predict you will do this can take advantage of you.
These stories give an idea of excuses different from the one our society likes to think it uses, namely that it accepts only excuses that are true and that reflect well upon the character of the person giving the excuse. I'm not saying that the common idea of excuses doesn't have value - but I think the game theory view also has some truth to it. I also think the game theoretic view can be useful in cases where the common view fails. It can inform cases in law, international diplomacy, and politics where a tool somewhat stronger than the easily-muddled common view is helpful.
If you encounter this situation in the real world, I suggest asking for clarification, such as, "Does that mean we can skip it, or that we'll fail?" Once you encounter the phrase many times, and always with the latter definition, updating would suggest that any future uses will have the same (not quite literal) meaning, allowing you to be more confident in your understanding. If there is still enough doubt remaining, then ask every time.
This method works even if you can't think of alternatives but realize that people don't always mean what they say. "Does that mean we can skip it?" is an adequate way to express, "This is my understanding of what action I am allowed to take without penalty; please check it and let me know if I am correct."
Encountering many situations where you require this question/answer method to determine what someone really means would show that it's a necessary component of understanding neurotypical communication. As someone who has problems with literal-mindedness, constantly asking questions and updating based on the responses changed my relationships significantly for the better.