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Matt_Stevenson comments on Eight Short Studies On Excuses - Less Wrong
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What Thomas Schelling would do. Partly tongue-in-cheek.
The Clumsy Game-Player: agree to the deal, then perform an identical "finger slip" several turns later.
The Lazy Student, The Grieving Student, The Sports Fan: make the deadline for reports a curve instead of a cliff. Each day of delay costs some percentage of the grade.
The Murderous Husband: if you really don't want these things to happen, make the wife partially responsible for the murder in such cases, by law. (Or the lover, if the husband chooses to murder the wife.)
The Bellicose Dictator: publicly threaten sanctions unless the invading army withdraws immediately. Do this before any negotiations.
The Peyote-Popping Native, The Well-Disguised Atheist: when the native first comes to you, offer to balance out the permission to smoke peyote with some sanction against the Native American church. Then the atheists won't bother asking for a free lunch.
I've always liked the "drop the n lowest scores" strategy. For example, 10 assignments given with the lowest 2 scores ignored.
You are pre-committing to a set of rules, where any excuse would have a much lower probability of being true. Any excuse would need to include 3 excuses. Combining the probabilities of each of the excuses will likely bring the total under your acceptable threshold. Basically, it's lowering the likelihood that you will want to violate the rules.
You can also look at like this. Your model of people predicts that they are scoundrels, and will try to violate the rules, maximizing their utility at your expense. So build a system where procrastinators can maximize their utility at no expense to you.
"Any excuse would need to include 3 excuses." - not as such; there is then the possibility that someone will wish to have an excuse to turn in an assignment they expect to do well on late to replace the grade of one of the other assignments which they did poorly on (or had no excuse).