Your question isn't completely hypothetical. In the US civil system, the standard is a "preponderance of the evidence" (50.0...01%) level of certainty that a tort has occurred. However, the jury also gets to determine the amount of damages they award, and can set that number independently of what any law or contract might say about the amount that is supposed to be due. To the extent that a contract (for example) sets a minimum and maximum but the jury can ignore either or both of those, each party has an incentive to go to trial based on the possibility that the jury could ignore the limit in their favor and a disincentive based on the possibility that the jury could go the other way.
In most cases (civil and criminal), there's also a disincentive to go to trial based on the costs to the person who wants to go to trial. There is a public defender system in the US, but not everybody qualifies and for those who do, the attorneys generally don't have the time or resources to go beyond the plea bargain, and defendants in those jurisdictions face high costs of trying to get an attorney who can prepare for and go to trial. 783
In 43 states, defendants can also have to pay the costs of their public defenders (and even the other side's attorney's fees!), especially if convicted. See <PERSON> coverage of the topic here.
The shortest answer to your question is probably that the penalty agreed to in the plea bargain is likely to be proportional to the weight of the evidence the prosecution has and what they think they can prove.
In many ways, that's not much different than the current system, especially for trials where a person might be charged with multiple crimes at different levels in the same case, where some charges might have slightly more elements than others (e.g. malice, intent) and the jury can choose the level of the charge where they think all the elements have been proven (e.g. someone might be convicted of manslaughter instead of murder). Prosecutors already factor this in when deciding what to offer in a plea bargain. 821