Instead you should be thinking in probability distributions. When someone asks for your P90 or P50 of project completion, you know they are a serious estimator, worth your time to give a good thoughtful answer. What is the date at which you would bet 90:10 that the project is finished? What about 99:1? And 1:99? Just that frameshift alone solves a lot of problems. The numbers actually have agreed-upon meaning, there is a straightforward way to see how bad an estimate really was, etc.
At the start of a project have people give estimates for a few different percentiles, and record them. I usually do it in bits, since there is some research that humans can't handle more than about 3 bits +/- for probabilistic reasoning. That would be 1:1, 2:1, 4:1, 8:1, and their reciprocals. Revisit the recorded estimates during the project retrospective.
You can make this as much of a game as you want. If you have play-money at your company or discretionary bonuses, it can turn into a market. But most of the benefit comes from playing against yourself, and getting out of the cognitive trap imposed by single number/date estimates.