WHY A HAND WITH A HIGHER ER MAY BE RANKED UNDER A HAND WITH A LOWER ER by Jim Wolf In "Frugal Video Poker" (FVP), a hand with a lower ER may be ranked above a hand with a higher ER. There are two reasons for this. For most situations it has to do with groupings. Other times it is simply more accurate. First let’s take a look at groupings… Groupings Groupings condense a chart to create the most efficient chart possible. By this I mean the least amount of rankings. The less rankings you have, the shorter the chart, and the easier it is to reference. Let's create a Double Bonus 10/7 machine that pays only 90 for 4 of a kind, 5 through King. In other words, 4 Aces pay 800, 4 - 2,3,4 pays 400. 4 of everything else pays 90. It is possible that by ranking solely by ERs that the chart could come out like this... 4 Aces ER 800 4 - 2,3,4 ER 400 Straight Flush ER 250 4 Card Royal Flush ER 94.71 4 - 5 through King ER 90 The chart is much smaller if I say... 4 of a Kind ER 216.15 Straight Flush 250.00 4 Card Royal Flush ER 94.71 In this example, {4 – 5 through King} does not compete with either the {4 Card Royal Flush} or the {Straight Flush}, so I can push it up in ranking and make the chart smaller. I do this with all fields - straights, flushes, etc. If a hand doesn't compete with a ranking above it, and I can push it up and not sacrifice chart EV, I will do it. The chart is cleaner, and again, easier to reference. You can always examine the full chart in the "Tweak Strategy Charts" window. If you click the "Detail View" key, it will display all the individual strategy combinations that make up the chart, and their ERs, or whatever statistics you desire. ER Abnormalities The simple fact is that the ER does not always make that strategy combination the best ranking play. First, where did I get the ERs, and how do I rank the strategy combinations? I analyze the machine in perfect play mode, which is always the correct play on a given hand. The strategy combinations are given perfect play ERs. This is the ER of that combination when it is the best play on a hand, when playing all possible hands. Except for some minor floating point decimal precision issues, VPTutor, BDWP, other experts, and I all agree on calculating these ERs, and the resultant EVs of the machines. In a perfect world, this is the ER that should be the sole basis for arranging the rankings in the charts. This does work to some degree, but problems occur. Strategy chart rankings are concrete. If I say {Hold the King} over {Hold 2 Card RF, 10, King}, this means in every situation, not just sometimes, if you are playing strictly according to the charts, and not supplementing your play with perfect play (though FVP will not signal an error if it was the correct play in perfect play). When the ERs for the combinations were determined in perfect play, there was flexibility. In perfect play, sometimes {Hold the King} is the best play. Sometimes {Hold 2 card RF 10, King} is the best play. This flexibility is not available when designing basic strategy charts. I have to pick one or the other. For example, in 1 to 4 credits 9/6 Jacks or Better, the combination {Hold 1 Card, King}, had a slightly higher ER than {Hold 2 Card RF, 10, King}. I have to choose one ranking over the other. If I go by ER, I should choose {Hold 1 Card, King}, but I have found that the overall chart EV will be higher if I choose {Hold 2 Card RF, 10, King}. What went wrong with our ERs? Well, let's look at how they were determined. The {Hold the King} strategy combination, of course, will appear in a lot more hands (when playing all possible hands), than {Hold 2 Card RF, 10, King}. Therefore, it is possible that it is going to win a lot more hands - and since its ER is an averaged statistic over all won hands, it can be inflated by sheer volume. What you also have to look at is how the rankings compete against each other - what I like to call hand to hand combat (pun is intended). Let's forget all the hands played that determined the ER. Let's add a filter that just looks at when the 2 hands went up against each other, and then decide whom is the best-damned hand in the land. Actually, I have a couple of other filters, and I put all the rankings through them. So now what? We play all possible hands again, but this time through the rigid standards of our chart. Now what is our EV? How does it compare to the perfect play EV? This is the strategy EV of the chart. This is why I have both strategy and perfect play designations both to the machine statistics and to the individual strategy combinations. For further reading, in the FVP help file there are two topics that explain this in a different way, "Why Are The Returns Different?" and "Perfect and Strategy Designations in Strategy Charts".
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