Recently (August 1998) on Skip Hughes video poker mailing list, an assertion was made that the meter movement of a progressive jackpot should be treated as a direct addition to EV (comparable to cash back). I disagreed with this, and I will explain my reasons here.
First, there is a circumstance where meter movement can be considered equivalent to added cash back. The following three conditions must hold:
Now, these three conditions can hold for VP teams. They may be able to monopolize a bank of linked progressives until the jackpot is hit, playing around the clock and with access to enough funds to see them through the bad runs. But for most of us, at least one of the three conditions is false. In that case, treating meter movement as equivalent to cash back can lead to incorrect strategy decisions.
To simplify things enough to make them amenable to analysis, while not distorting things too much, let's assume we are playing a game where each hand has only two possible outcomes: a loss or winning the jackpot. But instead of a loss being a single unit, the amount of the loss depends on the strategy. This is similar to what is often done when people calculate the EV of a game excluding the Royal. What I have done is essentially remove all of the variance due to non-jackpot hands.
Let's work with a specific example. The starting Jackpot is at $12000. We will consider two strategies. Strategy A (Aggressive), costs $1 per hand and has a cycle of 10000 (that is, the chance of hitting the jackpot in one hand is 1/10000). Strategy P (Passive) costs only $0.54 per hand, but has a cycle of 20000. The Aggressive strategy simulates one that goes strongly for the Royal, getting there much more quickly at the expense of a higher loss rate while getting there. [This example is not realistic, but will serve to illustrate my points.]
Which strategy is preferred using the max-profit/min-cost criterion? Usually, we simply multiply the cost per hand by the expected number of hands, so Strategy A has a cost of $10000 and a net expected profit of $2000, while Strategy P has a cost of $10800 and an expected profit of $1200. Since Strategy A is also "faster", it is clearly preferred.
Now suppose we suddenly get an extra $0.10 cash back per hand. The cost of Strategy A is reduced to $9000, while Strategy P goes to $8800. Strategy P has become preferred with an expected profit of $3200 compared to $3000 for A. Is this also true for a comparable amount of meter movement? [Note that it is true that it will take much longer to play Strategy P. Playing time is ignored using the max-profit criterion. We can take care of the cost and aggravation of playing hands by increasing the cost of playing a hand. Let's ignore this or assume it has already been done.]
Suppose instead of the extra cash back, the meter movement increasing the jackpot is $0.10 per hand. As stated above, if all three conditions hold, then Strategy P is again preferred. Now let's see what happens when one of the three conditions does *not* hold.
For the added cash back case, Strategy P is always preferred. This is under the assumption that cash back is immediately available (and so helps to extend playing time).
For the meter movement case, the choice of strategy depends on the size of the bankroll. With an infinite bankroll, we have already seen that Strategy P is preferred. But with a small bankroll, Strategy A is superior. For example, with a $1000 bankroll, Strategy A will earn an expected $195, while Strategy P will earn only $114. [Note that in both cases you will almost always lose your stake.] Unlike the cash back case, Strategy P does not benefit from getting extra hands to play caused by the additional meter movement that it creates compared to Strategy A. For this particular example, the cross-over point where Strategy P becomes better than A is a playing bankroll of about $34,000.
In Case 1, Strategy P had the advantage of being able to play a lot more hands than Strategy A (because of the slower drain). If we are time limited (modeled as playing a fixed number of hands), Strategy A is superior to Strategy P even for the cash back case (for a small number of hands). For 1000 hands, Strategy A has an expectation of $286 and Strategy P only $156 for the cash back assumption. Strategy P is better if at least 55000 hands will be played.
For the meter movement case, the advantage of Strategy A is even more apparent ($195 for 1000 hands versus $61 for Strategy P). Strategy P doesn't catch up until the playing time more than 81000 hands.
If you are playing on a linked jackpot machine, there is always the chance that someone else will hit the jackpot before you do. It should be clear that meter movement is of much less value than cash back in this case. For this study, I assume that the probability of an opponent hitting the jackpot between any two hands we play is 1/1000. (This is probably not unreasonable if there are a dozen or more linked machines.) How does this affect things?
Even if we have unlimited time and money, we can't be sure of collecting the jackpot (and the added meter movement). For the cash back case, Strategy A is always preferred. The expected net earnings of Strategy A are $1016, while Strategy P gives only $551. This assume we will keep playing until someone hits the jackpot.
It shouldn't be too surprising that the results are similar for the meter movement case. Strategy A expects to earn $1016 and Strategy P expects $551 (there are small differences between the meter movement and cash back cases, but the final results differ less than one dollar). The reason is simply that the dominating effect in both cases is the chance that the jackpot will be hit by someone else.
I hope this explanation makes it clear that we can't just add in the meter movement of a progressive as we would cash back in making strategy decisions (unless we can "lock up" the jackpot). As can be seen from the analysis, the details can be complex, but I believe, in most cases, it is prudent to design strategies (for max-profit) that ignore meter movement.
--jazbo