The Mathematics of Near-Misses: Why Your Brain Thinks You "Almost Won"

The Psychology Behind Near-Misses

The psychology of gambling is a fascinating area of research for scientists and mathematicians: behaviours exhibited by gamblers defy logic and appear paradoxical to an outside observer. After all, everyone knows that “the house always wins”, so why bother gambling at all?

Among these irrational behaviours is the reaction to a near-miss. In the world of slot machines, a near-miss is defined as an outcome that falls just short of a win, like landing two jackpot symbols on a payline and just missing out on the third, or matching symbols landing adjacent to the payline.

Multiple studies have been carried out to determine whether or not a near-miss impacts players’ subsequent decisions, and whether they could potentially lead to an increase in problem gambling behaviour. Some researchers have found links between gamblers falling just short of a winning combination and physiological and psychological effects that impact risk assessment and decision making.

Why Near-Misses Feel Like Progress

Near-miss events aren’t exclusive to gambling, and can be found in other areas, including natural disasters and industrial accidents. There is experimental evidence from a 2008 study by Dillon and Tinsley that suggests that sometimes near-misses are actually viewed as successes, and result in people making riskier decisions. A follow-up study in 2011 found that if a house survived a hurricane with no flood damage (a near-miss) then subjects were less likely to buy flood insurance, or evacuate in the event of another hurricane. There were no material changes to the quality or structure of the house or its environment, and therefore no reduction in the objective probability of damage. However, the subjects now felt differently about the odds, and believed they were more likely to land in their favour in the future.

This behaviour has also been observed in gamblers. Despite representing a monetary loss, near-misses are generally viewed less negatively than total losses. Some researchers theorised that actually a near-miss can be perceived as positive feedback. The player is learning the ropes of the game, and if they just keep playing they might learn enough to crack it. This implies a belief that wins are a result of improved ability or acquired skill, even though slot machine results are random.

This belief is more than just psychological. Another experiment showed evidence of a physiological response in gamblers who experienced near misses. Participants showed increased heart rate acceleration and skin conductance responses compared to wins or losses. Near-miss outcomes have even been associated with increased activity in the brain’s reward centres.

The reason behind these chemical responses is, believe it or not, evolutionary. Near-misses help us stay motivated when we are engaged in activities that require skill. It’s how our brains register progress: being a few marks off a perfect score on a test, or getting closer to the bullseye with every throw of a dart. If we keep studying, or practising, we will eventually achieve success.

The fact that we can’t turn off these evolved responses for luck-based activities is obviously very appealing for casinos. Players still feel positive about their outcomes while the house keeps winning – and taking in more money. Many software providers have started taking this into consideration when designing their games, using specific tricks to trigger these near-misses in an attempt to keep people playing for longer.

We talk about slots as games of chance when that’s not entirely accurate, they are governed by statistics and probability. Not all outcomes have an equal chance of occurring, but they must still be random. Most regulators require casinos to use a random number generator (RNG) for their slots, a mathematical formula which controls the game’s outcomes. Theoretically, these RNGs can be manipulated to help the house maintain its advantage. 

One such case of tampering was brought to light by the Nevada Gaming Control Board in 1988. The regulator brought disciplinary action against a licensed manufacturer after discovering that their devices contained a “near-miss feature”. After the machine had decided that the outcome was a loss, the positions of the other reels were then selected from a group of losing combinations where the jackpot symbol appeared more frequently than statistically average. This gave players the illusion of having come close to a win. The regulator argued that because of this feature, the games no longer met the standard for randomness that they imposed.

While the manufacturer insisted that these features weren’t explicitly prohibited by any laws, and that they received fewer complaints from players using those machines than others – likely due to the dopamine release of a near-miss – they were eventually forced to remove this feature from the devices. 

Research Reveals the Mathematics of Manipulation

Gambling has always been a rapidly-evolving industry, and even more so now with the rise of online casinos. Regulators are struggling to keep up with the tricks casinos and developers come up with to skew the odds and their effects on players. In particular, studies around the relationship between near-misses and problem gambling are relatively recent. 

In 2009, two Canadian researchers, Harrigan and Dixon, decided to take a closer look at variable characteristics of slot games. Some of these characteristics included speed of play, stop buttons and, crucially, near-misses. The study discussed how certain techniques, like clustering or reducing the amount of winning symbols on certain reels, increased the amount of near-miss outcomes. These near-misses were found to contribute to the illusion of control in players, a form of irrational thinking strongly associated with problem gambling.

Harrigan and Dixon used Probability and Accounting Report (PAR) sheets (accessed via FOI requests) in their research to analyse the fairness of four different slot machines. A PAR sheet is a tool that defines the mathematical model of a slot machine game. These have existed since the very first modern slot machine, the Liberty Bell, was built in the late 1800s, but they have evolved alongside the games they represent. 

PAR sheets act as a blueprint for slots, listing all of the symbols on each reel along with how frequently they appear, and the payouts for each winning combination. They use mathematical formulas based on variables determined by the manufacturer to provide information about the probability of certain results and, perhaps most importantly for players, the RTP of a slot. 

Here’s an example of a PAR sheet for a simple three-reel, one-payline slot. Starting with a detailed breakdown of the symbols on each reel strip:

This table summarises the symbol distribution on each reel to inform the algorithm:

The paytable is the most important part of a PAR sheet. Here the game designer can toggle with the data to influence the RTP:

Using the PAR sheets, Harrigan and Dixon were able to see examples of the clustering technique being used to increase the probability of a near-miss outcome compared to a total loss. On the PAR sheet for Double Diamond Deluxe, you can see how game designers put more blanks next to the high-paying symbols, creating a higher frequency of near-misses.