by Dr Catalin Barboianu PhD | Date of Publishing: 30 June 2023
Slots are known as casino games that do not offer players many strategic possibilities. The only available strategy is that of choosing: choosing a game or another, choosing the bet size, choosing the paylines to enable, and so on. It’s essentially like in a lottery, just that the slot game is more attractive as it is conceived and offers prizes more frequently.
This lack of strategic possibilities is true if talking about optimal strategies, which are mathematically-devised strategies proven to give the player an advantage against the house in certain circumstances. Unlike in blackjack, for example, where playing in a certain way in certain situations may increase your winning probability in the game or expectation over the long run, in slots, the only choices you can make besides choosing the game are to play and not to play.
And yet there is a hardly seen exception to this principle, which applies to progressive jackpot slots and is related to the so-called breakeven value and advantage play. The effectiveness of the advantage play is usually overestimated by slots players and not understood adequately. In this article, we will see the breakeven value, its strategic role, and how we should correctly interpret its involvement in advantage play.
Table of Contents
The term ‘jackpot’ is traditionally used in games of chance to express the top prize of a game. Like we have the jackpot in a lottery as the prize associated with hitting all the numbers in the drawn combination, in slots, it is the prize paid if you hit the top-winning symbol combination in the base prize schedule of the game. Usually, this top combination consists of the same top-award symbol on each reel showing on the payline and – as seen in the pay window – it has the highest payout odds associated, which are fixed. This base jackpot can be easily determined by multiplying the winner’s wager with the payout odds of that prize.
Since the 1980s, a new prize feature has been added to some slots games, namely the progressive jackpot. This additional prize works like a side bet, independently of the base prize schedule of the game. A certain rate of the players’ bets or wins adds continuously to the progressive jackpot fund until it is released to a winner and then reset. This rate is known as the rate of the meter rise and varies across systems.
The most common values are 1% and 2%. The progressive jackpot may cumulate from one machine or several linked in a network. The jackpot is triggered by algorithms based on RNG from a poll of eligible players. Eligibility usually requires the player to hit the base (or regular) jackpot. Still, there are also systems requiring any prize or even no prize (any player – losing or winning – may receive the jackpot). A common eligibility requirement is for the player to place bets of a certain size or denomination; a minimum bet usually does not qualify for the jackpot. Unlike the regular jackpot, the progressive jackpot is not fixed and increases with every bet.
When a player’s contribution to the jackpot from their bet or win (depending on the rules) meets or exceeds the jackpot value randomly selected for release, the player is declared the progressive jackpot winner.
There are two kinds of progressive slots systems concerning triggering the jackpot. In one of them, the jackpot increases unlimitedly until a winner occurs. In wide-area network progressives, such unlimited jackpots may reach values in order of millions and tens of millions, making the game highly attractive. In unlimited jackpot systems, the triggering algorithm may be based on several variables (attributes of the meter values, timeframes of the day, identifiers of the machines in the network, and so on) and is not known except by the producers or owners of the systems.
The other category of progressive systems is that the jackpot may increase to a certain pre-established value before it must be triggered. They are called must-hit-by progressives (or mystery progressives). The jackpot may be triggered at any value between an initial lower limit (called seed) and the maximal value. A seed also stands for unlimited progressives.
The algorithm of triggering is simpler than that of the unlimited progressives. After every reset, usually, the RNG randomly chooses a value between the seed and the maximal (must-hit-by) value over a uniformly distributed selection of numbers in that interval (or a smaller interval included in it). Many algorithms for triggering a must-hit-by jackpot choose the hit value in the second half of the main interval, and some even on narrower intervals at the upper end of the main interval. As in the case of the unlimited progressive, no one except the owners know that algorithm.
Slots are known as having a special status among casino games with respect to transparency – the parametric configuration of their internal design is kept secret by their producers. Some games were “deconstructed” by experts, and their parameters were retrieved and exposed through statistical methods based on tracking and recording; for others, this information was retrieved by legal intervention.
Of course, the lack of information regarding the parametric configuration (the number of stops on each reel, the weighting of the symbols on each reel, and so on) prevents any probability computation regarding winning the prizes and, consequently, other statistical indicators cannot be obtained as well.
The only statistical information available for some games is the RTP (return to player or payback percentage), which is displayed on their help/about screen. This happened because, in many jurisdictions this information was required by gambling regulations.
The RTP is the share of all wagers placed at a machine that is paid back to the players as prizes over the long run. It is another way to express the house edge (HE), and the relationship between the two indicators is RTP = 1 – HE (or RTP = 1 + EV, if employing the negative expected value).
The RTP is not the only information available for the most known games and brands. Volatility (a statistical indicator reflecting how often a machine pays on average and how the prize amounts are distributed statistically) is usually available information from expert sources, as it has been retrieved through statistical methods.
But RTP and volatility are insufficient if we want to have the whole mathematical picture of the slots game. They cannot tell us what the probability of hitting a given prize is. The payout schedule tells nothing about it either. As long as we don’t know such probabilities, we cannot estimate the probability of hitting a progressive jackpot either since hitting a base prize is usually required for eligibility. But even if we knew it, the latter probability still cannot be computed as long as we know nothing about the algorithm for triggering the jackpot.
Then, what can we know about the progressive systems that could help us evaluate our chances of winning and build objective strategies for getting an edge over the house? We will find out in the next section.
One thing to note in the first place is that the RTP of a machine is a fixed indicator, depending on the payout schedule of that machine and its associated probabilities. However, for progressives, the progressive jackpot counts statistically as payback besides the regular prizes since any machine in the network is eligible for hitting it.
This means that the RTP of such a machine is not constant and changes with every move of the jackpot meter. As such, at every moment, we have an increasing RTP associated with both the regular prizes and the progressive jackpot (let’s denote it RTPJ), which is higher than the fixed regular RTP. But remember that the fixed RTP is lower than 100% (or 1) (otherwise, the casino would have no profit). It follows that at a certain moment, until the jackpot will be released and reset, the RTPJ will equal 100% and will further exceed it. The jackpot value at that specific moment is called the breakeven value of that system.
The statistical interpretation of the breakeven value is that any bet has zero expectation at that point and positive expectation beyond it. These are moments when, theoretically, players get an edge over the house. If placing the bets in the same circumstances repeatedly over the long run, the cumulated win will exceed the cumulated loss. Thus, playing only after that point has passed (the advantage play) seems like an optimal strategy, regardless of its effectiveness. We will explain this in more detail in the next section.
Again, the breakeven value is not something we can know in slots; we know for sure that it will occur, but will never know precisely when. In unlimited progressives, we know nothing about the algorithm of triggering the jackpot, so any estimation for the breakeven value is mathematically impossible. Things change a bit for the must-hit-by progressives: We know the jackpot’s maximal value and that the RNG selection base consists of numerical values within the min-max interval of the meter (although exceptions to that rule certainly exist). This information along with other data, gives us the possibility for a rough estimation of the breakeven value, as we will see further.
First, we must be aware that a precise formula for determining the breakeven value for must-hit-by progressives cannot be obtained. All that we can have are approximations based on various idealisations or suppositions. The main idealisation is that the jackpot is equally likely to trigger anywhere between the seed and its maximum value. This is grounded on the assumption that the point at which the jackpot will hit is randomly chosen on a uniform distribution of values between the starting and maximum value.
Of course, there are known exceptions to this rule (for example, the AGS must-hit-by progressives were found to always hit near the must-hit-by-point).
The formula gets complicated as we introduce more variables into the equation. A rule of thumb easy to be retained provides as with the breakeven value (E) of a must-hit-by progressive system when three parameters are known or estimated (or guessed): the must-hit-by value M (known and displayed on the machine), the RTP, and the rate of the meter rise r:
E = M x (1 – RTP + r) / (1 – RTP + 2r)
The next table notes the estimated value of E for various must-hit-by limits and payback percentages (RTP), for r = 2%:
| RTP M | 0.90 | 0.91 | 0.92 | 0.93 | 0.94 | 0.95 | 0.96 | 0.97 | 0.98 | 0.99 |
| 1000 | 857 | 846 | 833 | 818 | 800 | 777 | 750 | 714 | 666 | 600 |
| 2000 | 1714 | 1692 | 1666 | 1636 | 1600 | 1555 | 1500 | 1428 | 1333 | 1200 |
| 3000 | 2571 | 2538 | 2500 | 2454 | 2400 | 2333 | 2250 | 2142 | 2000 | 1800 |
| 4000 | 3428 | 3384 | 3333 | 3272 | 3200 | 3111 | 3000 | 2857 | 2666 | 2400 |
| 5000 | 4285 | 4230 | 4166 | 4090 | 4000 | 3888 | 3750 | 3571 | 3333 | 3000 |
| 7000 | 6000 | 5923 | 5833 | 5727 | 5600 | 5444 | 5250 | 5000 | 4666 | 4200 |
| 10000 | 8571 | 8461 | 8333 | 8181 | 8000 | 7777 | 7500 | 7142 | 6666 | 6000 |
| 20000 | 17142 | 16923 | 16666 | 16363 | 16000 | 15555 | 15000 | 14285 | 13333 | 12000 |
| 30000 | 25714 | 25384 | 25000 | 24545 | 24000 | 23333 | 22500 | 21428 | 20000 | 18000 |
| 40000 | 34285 | 33846 | 33333 | 32727 | 32000 | 31111 | 30000 | 28571 | 26666 | 24000 |
| 50000 | 42857 | 42307 | 41666 | 40909 | 40000 | 38888 | 37500 | 35714 | 33333 | 30000 |
Example: A system with the rate of the meter rise 2%, a must-hit-by value of 7,000, and a RTP of 0.98 has 4,666 as its breakeven value.
Advantage play only applies to must-hit-by progressives. Even a breakeven value also exists in unlimited progressives. It cannot be determined to play only after that point. Although jackpots are usually far higher in unlimited progressives, obviously, the likelihood of hitting a jackpot is higher in must-hit-by progressives due to the upper limitation.
All these conditions shape a straight recommendation for choosing this latter system for having greater chances to win. However the advantage play should be adequately interpreted for not forming misconceptions that might be harmful.
This is true for all games of chance. For example, in blackjack, playing by an optimal strategy telling you to split a pair of 3s when the dealer’s first card is 7 does not entail that you will win that hand, nor the next hands, but just that you have to play that way whenever that configuration occurs for getting the maximal possible gain from this situation over enough number of plays.
In the case of slots advantage play, you have to play by that strategy of passing the breakeven point – either at the same game or another – repeatedly, hoping to hit the jackpot in the near future. When this happens, you should not be too much worried that your cumulated loss might exceed the jackpot.
When playing progressive slots, the only optimal strategy is the advantage play based on passing the breakeven point. However, getting informed about the math behind the system and the strategy itself is not always sufficient. You must have an adequate interpretation of the mathematical facts and the real effectiveness of this strategy. If mathematics says that a certain strategy is optimal, you should play by it. However, many other real-life aspects of your play and its final outcome might shake your confidence in gambling math.
The highest progressive jackpots won in slots (including those historical hits at Megabucks) relied on investments from 25 cents to a few hundred dollars, so it is fair to assume that the winners did not play by any sophisticated strategy. Conversely, playing by a strategy might not eventually make you a winner. This is the nature of chance and gambling.
Chasing a progressive jackpot is risky, as it impacts your bankroll more than other strategies or games. Moreover, slots are recognised by problem-gambling researchers as the most addictive casino games. This is why you have to be careful with any strategic advice, even from experts.
Author: Dr Catalin Barboianu PhD
Catalin Barboianu PhD is a games mathematician and problem-gambling researcher. He authored ten books on gambling mathematics and several academic articles in the field of problem gambling and philosophy of science. Catalin is a science writer and consultant for the mathematical aspects of gambling for the gaming industry and problem-gambling institutions. Read More
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