Stickin’ It!

Over the past few years, we have become acquainted with slot games featuring variations on the stick’n’respin theme. In one of the most popular of those, a 3×5 machine, the wild symbol is present on Reels 2-4. When it lands in the window, it expands vertically to the entire visible part of the reel. The paylines are then evaluated (both left-to-right and right-to-left!) and the wild reel sticks during a free respin. If during a respin an additional wild symbol appears, it too will expand to the whole reel before line evaluation, and stick for yet another free respin. The process continues until no additional wild symbols appear. Thus, a player can get up to 4 spins with an increasing number of reels covered by wild symbols, all by staking and pressing the PLAY button only once.

As a continuation of the previous post, it is briefly explained below how this can be solved by means of Markov chains. Each spin in a “stick’n’respin cycle” is started with each of Reel 2, Reel 3 and Reel 4 being either wild or not. We will therefore work with 9 states numbered from 0 to 8, where 0-7 represent spinning with one of the possible wild distributions (think binary!) and 8 represents “no more spins”. Knowing the probabilities that a wild symbol appears on a specific reel, we readily get the transition probabilities between the states. Always performing the initial spin in State 0, it turns out that we will surely end up in State 8 no later than prior to the 5th consecutive spin (which is thus not executed).

If we consider a 3-bit binary number, where the least significant bit represents Reel 2, then we get a natural enumeration of the states 0-7. If the “wild probabilities” are, say, 0.02, 0.05 and 0.03, for reels 2, 3 and 4, respectively, and we repeatedly multiply the resulting transition matrix from the left by an initial vector with probability 1 for State 0 and probabilities 0 for all other states, we end up at the following table.

Expanding wild stick'n'respin probabilities

Expanding wild stick’n’respin probabilities

It is then a matter of calculating the conditional averages given that some of Reels 2-4 are wild which, to a large extent, is a matter of copy and paste (once we have the average for State 0). The nth column in the above table contains the probabilities that the nth spin is executed in State k, where k=8 means that the spin is not executed at all.

Another popular concept involving sticky wilds is to let all wild symbols popping up during a freespin session stick in their positions for the remainder of the freespins, thereby overriding any symbol that appears underneath. Assuming a possibly random but fixed number of freespins, we can solve this in a similar manner. Note that we may consider one payline at a time, and expand the average return linearly to all lines. If the wild symbol is present on all five reels, we have to deal with 32 states and need the conditional average for each of them. The corresponding probability table dictates how often the nth freespin is played with wild configuration k. These are then multiplied with the corresponding conditional averages and the sum of the products is the average total session win. Watch out if retriggers are possible!

A third variation on the same theme is to let the triggering scatter symbols turn into wilds and remain in their positions during an entire freespin session. It is standard that three or more scatter symbols are required (anywhere in the symbol window) in order to start the freespins. There are a number of differences compared to the previous concept, the most important ones being that we do not start at State 0, and different paylines are in general affected differently by the wild symbols. For simplicity, we assume that no additional wild symbols can appear during the freespins. Suppose that the game has 20 paylines. Then, given a triggering scatter/sticky wild combination, some of the paylines may not be affected at all while others have one or more wild symbols given.

Paylines are affected differently

Paylines are affected differently

Thus, in general, for each possible triggering combination we get one “line count” for each of the 32 possible wild configurations, adding up to 20. Again, we need the conditional averages for all possible wild configurations. Once those are in place, we weigh them with the line counts mentioned above to get the average total win per spin.

In order to complicate things, the latter two can be combined. It gets a little messier if we allow retriggers, since then we don’t know in advance how many freespins will be played. Also, we need to define whether or not a scatter symbol shall count should it appear underneath a sticky wild. All in all, however, it is usually possible to solve the game analytically with a bit of caution.