Next, adapt an innovate BOARD GAME MECHANIC (I just love when I actually write a post that goes with the theme of my blog) from
Settlers of Catan to suit your needs.
Ok, so the adaptation of the Settlers of Catan mechanic takes a little bit more explanation. Allow me to summarize.
See the following picture to jog your memory about the circular "chits" that are used in Settlers to denote probability:

When you roll two dice, the probability of rolling a "7" is the highest and of rolling a "12" or a "2" the lowest. In Settlers, numbers 2-12 are placed randomly on the board to demonstrate the probability of any given resource being "produced." So, a player would want to locate his towns near resources that have numbers close to 7 to produce the most resources.
Remember how that works?
Good!
So, to apply the same principle, count the number (n) of people you want to arrive to your staff meeting on time (in my case 8) and assign a probability to n outcomes along a continuum, such that a roll of two dice will have one outcome.
Next, make a board! Like this...

And assign each staff member's pawn to a location on the board according to when they arrived at the meeting. Those arriving first are placed at the low levels of probability and those arriving last are placed in the high levels of probability. In this instance, green arrived first!
Last, explain that you will ask for volunteers during the meeting to help with projects A, B and C. If nobody volunteers say that you will roll the dice and the person who's number(s) are rolled have to volunteer. (Alternative: Allow staff members who volunteer before the die is rolled move up in position on the board)
This works! But why????
One more model should explain:

The arrival window has been expanded! Instead of having incentive to arrive 1 SECOND before the meeting time, incentive has been changed to arrive SOONER.
As a result, spillage is depleted to ZERO:

And so my staff are SPRINTERS!
...and they are also PAWNS.
But they make it to staff meeting on time, and yours will too!
(Of course, you will occasionally have to make good on your threat to roll the dice, but it is
ever so fun to be able to let fate (and their own decision to be later than others) decide who gets punished).