Is it ethical/acceptable to give a lighter penalty to students who admit to cheating?
TL; DR: We adjust the penalty.
I sit on my department's Academic Misconduct Committee so unfortunately I see a lot of these cases. The University policy has created two similarly sounding, but different terms: academic misconduct and poor academic practice, with academic misconduct being much more severe than poor academic practice. Every student suspected of having engaged in poor academic practice is called into a meeting with the Academic Misconduct Committee. This committee looks at the evidence supporting academic misconduct/poor practice and hears the student's case which generally includes what training they have had about good academic practice, how the incident happened, and any evidence of extenuating circumstances (e.g., death in the family).
Based on this process we can come to one of 3 decisions:
- not poor academic practice,
- poor academic practice,
- academic misconduct.
If we decide that they did not engage in poor academic practice, the incident is essentially purged from the system. If the student is called in again in the future, we do not nominally know that they had been called in before. If we decide that it was poor academic practice, the students are warned and it is documented. We are not allowed to apply formal penalties in the case of poor academic practice. If they are called in again we know from the documentation and are unlikely to give them the benefit of the doubt a second time.
If we decide that academic misconduct has occurred we can apply one of a number of different penalties to the work: no reduction of the mark, remark the work with the offending material removed or reduce the mark commensurate with the misconduct (anywhere from a 5% penalty to a 0 for the piece of work). A second incident of academic misconduct results in the penalty being decided on by a university panel and starts with a zero for the class and can go as high as a zero for the year. In this way there is a big difference between poor academic practice and academic misconduct with no reduction in the mark since poor academic practice does not count as a first offense.
Within this framework the committee is faced with how to handle the students who admit the issue and the ones who deny the issue. Those who argue they didn't cheat and have no explanation as to how their assignment matches another's work are almost always found guilty of academic misconduct since they are not willing to help us create a case for poor academic practice. More often than not they receive a 0 since we have no evidence that mitigates the penalty. For students who admit what they did, we often consider poor academic practice as an outcome since they describe what they did and realize it was wrong. Sometimes the offense is too blatant to let off without a reduction, but generally saying what you did wrong and how you will not let it happen again reduces the penalty.
Depending on the response, you could punish subsequent lies. If someone cheats, gets a punishment. When confronted about that, if they recognize it, you don't take further action. But, if they give you a blatantly false explanation, they may get punished for it.
A (sadly) real case example: one student submitted exactly the same report than a previous one. When confronted about it, he claimed that he had done it, but when generating the PDF, it was somehow transformed in the other student's report, with only the name changed. This is being caught in a lie and try to avoid it treating the instructor as stupid.
Adding some research: Wikipedia links a paper where some subjects were falsely accused of academic cheating and offer a lenient punishment if they recognised the fact. 56% of them plead guilty to avoid the risk of bigger punishment.
Here is a criminal justice perspective (I have not had to deal with such egregious situations in the classroom.)
The motivation for punishment generally falls under the philosophies of retribution, deterrence, rehabilitation & restoration. The US's penal law system is based on retribution; that is you've committed a wrong against society, and it is societies duty to exact a punishment somehow equivalent to that wrong. My experience is people tend to frame the motivation for punishment de facto in retributive terms, even if it is not really appropriate for the situation.
All institutions I have been associated with have official committees that evaluate student misconduct - and cheating is their main calling. Assuming such a committee exists in your school, you should report the students behavior and it should be clear in your syllabus that is the consequence of cheating.
It is likely the said committee will cover any retributive actions necessary to fulfill any harms to society, so the question then becomes what do you do to the students in response to the behavior? This depends on your goals of the punishment to begin with. I strongly feel your role should not be to exact further retribution beyond what the schools official academic policy calls for - so that leaves deterrence, rehabilitation & restoration. (If you even have discretion at this point beyond your schools official policy for cheating.)
Studies in criminal justice tend to find that the severity of punishment is only weakly linked to deterrence - the probability of being caught is a much stronger influence on whether the student will commit the behavior. Pretty much any sort of note to the individual student that you caught them cheating will likely prevent future cheating. Even absent of punishment, this is a strong signal that their probability of being caught is high. My limited experience just letting students know "Your homework looks an awfully lot like the student who sits next to you." - effectively ended that behavior (although your situation is clearly beyond that point.) So you need not be worried about "letting them off the hook" (especially if they have been referred to the student misconduct panel).
Assuming letting the student know they have been caught is common-place, that then just leaves rehabilitation and/or restoration. Punishments oriented towards these perspectives often go hand in hand. One immediate example that comes to mind is to have the offending students lead a classroom discussion on the material they cheated on (this would be better for only one or two students though). Others may be extra-curricular activities, especially those that give back to the rest of the class (e.g. make them host a study session). Public shaming is an incredibly strong deterrent, e.g. just making them stand up in front of the class and admit their wrong-doing, but I am not sure if this violates other privacy mandates about grades and such things. (Restorative justice on its face may seem awkward since it is a victimless crime, but that doesn't make the potential punishments I suggest here any less reasonable.)
Anyway it is preferable that in the future to be very specific in your syllabus about what will happen. Otherwise it appears ad-hoc and can be construed as prejudiced toward any particular student.
In terms of relation to "admitting a wrong" - this philosophically should not have any bearing on the punishment that the offender receives. It is unfortunate a few conflations are being presented here in terms of plea bargaining - which is really a negative externality of the criminal justice system and the need to triage. When you plea bargain you concede to receiving a punishment for a lesser crime - you don't even have to admit you did anything wrong (e.g. you can plead "no contest").
The significance of admitting a wrong though is often placed on the other end of the system. It is often a requirement of parole that the offender admit their wrongdoing in front of the board before they are granted parole. In terms of restorative justice a key event is often just placing the offender in front of the victim and having the offender admit their wrong-doing. (Cooperation tends to be higher for offenders than you might expect - typically victims have a lower rate of cooperation.)