Optimal mix of currency tokens

Generally I start by looking at the MAX possible currency needed, not just the average (often having just a couple of large bills solves any "uncommon situation" problems, so if your game averages $250, but can top out at $500, you should probably have some $50's to handle that). Then you look at the number of players, and make sure there are at least nine $1's for each player (this is generally an adequate "change-making buffer" to avoid fiddliness). If you were jumping from $5's to $20's (with no $10's), you'd do something similar (seven $5's per player), but that doesn't appear to be the case here. Then just increase your denominations in a 4:2:1 ratio for $1's/$5's/$10's until you pass your "expected needs" by a couple of ticks, and fill out with larger bills until you hit your max.

Assumption #1: Stick with familiar denominations if possible. Denominations of 1, 5, 10, and 20 are familiar. Denomination of 2, less so. Denominations of 4, 8, and 16 are very efficient, but not familiar at all to most people.

Assumption #2: A player can exchange equivalent bills with the bank at any time with no charge. Otherwise, requiring everyone to have exact change on hand inflates the needed number of bills.

If you stick with only three denominations ($1, $5, $10), you should have at least four $1s and one $5 for each player. More are only useful as replacements, and with fewer, you risk being unable to have exact change for all players. The number of $10s is then dependent on how many players your game supports. If you can have six players, you'll have $54 in thirty necessary bills, so you'd need $200 in $10s, or twenty of them. That'd be twenty-four $1s, six $5s, and twenty $10s, for $254 in fifty bills total. With four players, you'll have $36 in twenty small bills, so you'd want $220 in $10s. That'd be sixteen $1s, four $5s, and twenty-two $10s, for $256 in forty-two bills total.

If you increase the denominations, you'll be definitely be able to get away fewer bills, but the total cost to produce the game might actually be higher. The money saved by a reduced quantity of tokens might not offset having additional unique tokens to produce. Adding $2s would change the four $1s per person into three bills, two $1s and a single $2, reducing only the number of bills. For six players, that only saves six bills, requiring you to have 44 bills. For four players, you save four bills, leaving you with thirty-eight.

Adding $20s instead would reduce the necessary $10s to one each, giving you $19 each, or $114 total, in 'necessary' bills. That means that you'd need another $140 in $20s, or seven of them. That'd be a total of $254 in forty-three bills (twenty-four $1s, six $5s, six $10s, and seven $20s). With four players, the necessary bills would only be $76, so you'd need nine $20s. That'd give you sixteen $1s, four $5s, four $10s, and nine $20s for a total of $256 in thirty-three bills.

Adding both $2s and $20s makes the required bills (per person) into two $1s and one each of $2, $5, and $10, for $19 in five bills per person. With six players, you'll have $254 in thirty-eight bills (twelve $1s, six $2s, six $5s, six $10s, and seven $20s). With four players you'll have $256 in twenty-nine bills (eight $1s, four $2s, four $5s, four $10s, and nine $20s).

My recommendation is (only if it's cost effective) to go with denominations of $1, $5, $10, and $20. If it would cost more to print with four denominations, stick with your original three denominations. A benefit of sticking with fewer denominations would mean each player having more bills on hand, so players would be more likely to have exact sums on hand, letting transactions go faster. I recommend $20 over $2 primarily because $20s are more familiar to most people than $2s, but also because they do allow the game to require fewer bills than $2s do.

Also, if we knew how many tokens you could get on a sheet, it might change things. Printing a sheet of eight paper bills might cost the same as a sheet of seven, and having that extra one as a replacement can be invaluable. Bills will get lost. Games that have spare components get a thumbs-up from me.

Akanucho, your math is correct in the sense that $2 tokens only slightly reduce the number of tokens needed and the number of tokens player will use during the game. However, having $2 tokens does make making change faster and more intuitive.

I was assuming the OP used tokens and not bills, in which case the numner of different tokens shouldn't matter much. If he does use bills, then I agree it will add to the production cost and you might want to not use the $2 denomination.

The $2 bill might be rare in the US, but the denomination is actually very common in Europe and it's used in a lot of games as well, so I think familiarity is not a huge problem. I agree, though, that using $4 and $16 bills would be awkward, even if they are technically more efficient :)

All in all, it's a fairly minor point. The most important thing is that you have enough tokens/bills to cover all the worst-case scenarios, and then a few extra just to be sure.

Also certain things change depending on how the game is played. If money is committed to a pool that is separate from the player's money for a time, you'll need to make sure each player has enough tokens to continue to pay into the pool without having to do too much money changing. This is especially important in a zero sum game.

Having only 4 1's per player is all well and good, but if people have to pay a dollar for a round, and then spend a dollar on 5 or 6 different places on the board to place workers (as a brief example), then the players need more 1 dollar bills.

How often do people need 5's vs. 20's will help make that determination in your game.

I think that players should get to decide, how many copies of each currency token they begin with, as long as they have exactly the required amount of starting money (for example, you could choose to take either 1x $10, 1x $5 and 5x $1 or 3x $5 and 5x $1 if the starting money happened to be $20). Players might have different strategies and some of them might have more use for the smaller tokens and more use for the bigger tokens and vice versa.

I think that you should have 1, 10 and one token between these (pretty sure 5 is the best). Any other tokens don't probably add much to the game, but they might give the players the impression that they are always supposed to get those bigger tokens during the game, and that might affect their strategy in a negative way: sometimes ignoring the big tokens would be another step new players would have to learn in order to do well in this particular game, and when it doesn't actually give the experienced player more options, I believe it's a bad thing. If the big tokens are really needed pretty much every time, then they are worth having, but because you don't have them already, I think you shouldn't add them.

Also, there isn't a mathematical way to figure out the best mix. I suppose you can do a really complicated calculation which would give the optimal numbers, but even after that, you would have to test and tweak it.

EDIT: One thing that also comes into mind is that you can make a rule that says that a player can't have more than X money. X should be big enough that it really doesn't affect the game at all, but that way you can be 100% sure that nobody is going to need more money than your game has. I put a money limit rule in my game for whole other reasons after some ridiculous "for the first 50 minutes, I do nothing but gather money, and then proceed to spend them all in one turn and win the game" strategies saw play in testing, but I also found out that it made having currency tokens WAY easier.