In tournament theory, a rat race is a situation in which the "race" towards a reward escalates. The "rat race" situation comes into effect when all participants focus only on being the best in the criteria that decides over who gets the reward. Usually, this leads to them using more "energy" (time, effort, etc.) than is efficient and neglecting other criteria that are not important for the reward.
The term "rat race" comes from the image of two rats spotting a piece of cheese at the other end of the room. Both start to run, and eventually, one rat increases its speed to be in the lead. The second rat increases its own speed even more, causing the first rat to increase its speed even more, and so on. In the end, both rats will have spend far more energy than could be recovered by eating the piece of cheese.
Real life situations
In real life, it is often easy to spot a potential rat race before it happens. For example, a company that makes it clear that the employee with the most billed hours will be promoted at the end of the year, this sets a clear criterion that the employees will keep in mind. The resulting rat race could appear in the form of an enormous increase in overtime, or it could lead to employees billing more hours than they actually worked. At the same time, other criteria (such as making decisions that are good for the company or the customer, being a good team player, making an effort to be nice to customers, etc.) are pushed into the background.
There are two simple ways to defuse a tournament and to avoid a rat race.
The first one is to keep the employees separated. This countermeasure uses the fact that participants in a rat race rely on closely observing the other participants (in order to find out when it is necessary to increase their own efforts). For example, giving more workers their own offices automatically slows down (or even stops) this information flow.
The second one is to reduce the reward. With a lower reward, the probability of participants going out of their way to reach it also drops. This is not always a possible solution (when the position of CEO is available, it's impossible to downgrade the reward), but it works in cases in which the reward is an adjustable incentive like higher wages. Of course, the reward should not be too low - the goal is to make the participants compete (thus increasing their productivity), just without escalating the situation.
A rat race can be compared to the prisoner's dilemma because it can also lead to a Nash equilibrium that is not pareto optimal: Both participants choose the high effort (and escalate the situation), even though it would be better for them if both chose the low effort.