# Progressing

Last week, I alluded to the seemingly complex math associated with games that offer progressive payouts (i.e. "progressives").  Progressives are games where the top pays are not fixed dollar amounts or odds payouts, but rather have variable payouts that increase as more wagers are made since the last time the prize was won.

Progressives have become very popular for table games sidebets.  They have long been used for some video poker machines for payouts on Royal Flushes.  Most commonly they are found on slot machines, which love to use a progressives ability to create a very large payout for a very rare occurrence.  As is always the case with a random event, the cycle between hits can frequently become far larger than 'average' and thus create an even larger than normal jackpot.

As I described last week, Progressives essentially have two different paybacks.  The first is the long-term payback which is what concerns the casino.  The second is the payback of the wager at any point in time which is what should concern the Player.  Let's take a closer look at how these are calculated and why there are two different paybacks.

Normally, to calculate payback, we take the frequency of a winning hand, multiply it by the payout of this hand which gives us the contribution rate for the hand.  We then sum up these contribution rates to arrive at the overall payback.  For most wagers, the frequency of a particular winning hand is fixed as it is unaffected by strategy.  So, if we are playing Caribbean Stud Poker, we don't have to worry about the strategy of Folding and Playing for the sidebet because you would never Fold a hand that is strong enough to earn a bonus.  Video Poker presents an additional challenge in that you can alter you strategy depending on the payouts and thus alter the frequency of winning hands.

So, to calculate the payback of a Progressive at a particular point in time, we follow the calculation I just described.  For example, let's assume the following paytable at a particular point in time for a \$1 wager:

 Hand Pays (For 1) Royal Flush \$113,473 Straight Flush 250 Four of a Kind 50 Full House 10 Flush 7 Straight 5 Three of a Kind 3 Two Pair 2

If we perform the calculation described, we get the following:

 Hand Frequency Pays (For 1) Contribution Rate Royal Flush 0.00015% \$65,473 10.07680% Straight Flush 0.00139% 2500 3.46292% Four of a Kind 0.02401% 250 6.00240% Full House 0.14406% 50 7.20288% Flush 0.19654% 20 3.93080% Straight 0.39246% 15 5.88697% Three of a Kind 2.11285% 10 21.12845% Two Pair 4.75390% 5 23.76951% Total 7.62536% 81.46074%

So, if you were to walk up to a table and see these payouts, the payback of the game at that very point in time is 81.46%

But, the payback to the casino could be vastly different.  Let's assume that the Royal Flush is seeded at \$50,000.  This means that every time someone wins the jackpot, the prize for the Royal Flush will be reset to \$50,000.  Further, let's assume that for every \$1 wager that is made, the Progressive increases by 10 cents (i.e. 10% of the wager).

There are two changes that we must now make to calculate the payback for the casino.  The first is that we always use the seed amount as the payout for that hand.  Thus, we repeat the calculation shown above but we use \$50,000 as the payout for the Royal Flush.  This is the amount that the casino itself directly paying out each time the jackpot is won.  When we do this, we find that the payback of this wager is 79.08%.

However, we must now ADD to this payback the amount of each wager this added to the Progressive meter - in this case 10%.  Eventually this 10% will go back to a Player.  It might happen while the jackpot is at \$50,000.10 or it might happen when it is at \$120,000 or more.  From the casino's standpoint, it doesn't matter.  That 10% belongs to the players.  Essentially all the Players that don't win the jackpot are handing those dimes to the person who finally does.  So, when we add that 10% to the 79.08% we find that to the casino this wager really has an 89.09% theoretical payback.  Over time, the casino will keep 10.91% of every dollar wagered.

So, if you were to play this wager while the Jackpot is \$65,473, you would actually be playing it on the 'low side' of the average jackpot.  How big is the average jackpot?  To calculate that, we take the average number of hands between jackpots (in this case 649,740) and multiply it by that 10%.  On average the jackpot will grow by \$64,974 before it is hit.  We add this to the seed amount and find that the average jackpot will be \$114, 974.  At that point, the payback of the wager is the same as theoretical payback of the wager.

If the jackpot grows to be above \$185,930 (which is very likely at times), then the payback of the wager at that point will actually be OVER 100%.  The only problem with this is that it will only be over 100% for the ONE person who actually wins the jackpot.  Everyone else will just be feeding dimes to the one person who wins.