With the exception of the Panavia Tornado, fighters do not have thrust reversers. Why is this? Why does the Panavia Tornado have them?
Thrust reversal is a complex system (equals money) and it uses significant amount of fuel (also equals money). It is not a very efficient means to reduce the speed of a landing aircraft either, and other methods (e.g. brakes or drogue parachutes) are much better suited.
The biggest advantage of thrust reversal is that it cancels out the idle forward thrust working against the aircraft. Canceling out idle thrust isn't an issue on most fighters, as they have an engine nozzle that is open at ground idle to prevent any thrust from being produced. This nozzle probably also makes it a bit difficult to integrate Thrust reversals, but the Panavia Tornado proves it can be done.
Fighter aircraft are specialized tools optimized for a specific purpose: to destroy enemy forces (in the air and on the ground). Everything that is not vital to that mission is quite literally dead weight that reduces the performance of the aircraft as it relates to its specific purpose. Jet fighters are not required to "taxi back" on the tarmac, and simpler "speed brakes" are more effective at slowing down in flight.
I don't think is has anything to do with cost or wasting fuel: a military jet is notoriously designed to do the best it can, to hell with how much it costs. Fuel efficiency is only a concern in terms of being economical as possible in order to have the best range to attack the enemy, or to loiter on station on patrol. Wasting a little fuel on landing is inconsequential; why else would jet fighters have fuel dumps? It's common practice to jettison any few thousand extra pounds of fuel to get landing weight down, and it's also common to take extra fuel on from the tanker just in case. So you take three times as much fuel as you need to get back to base, then when you get there safely, you just dump it so you're light enough to land. If fuel expense was a concern, this would not be the way they operate. If expense of construction was a concern, they'd think it was ridiculous to build from exotic components just to shave a few pounds. The reason they don't include reversers is because it's extra weight, and it takes up space that could be better used for something else, like more fuel or avionics. No, military jets, especially fighters basically burn money like fuel. Ridiculously expensive to operate, not just buy. Commercial aircraft, on the other hand, need to do things as cheaply as they can, and they need as little maintenance and the shortest turn-around times possible. Passenger aircraft and airlifters are far larger and heavier than fighters; that's why they need reverse thrust. A parachute big enough to slow down a 747 is a real pain to pack back up again, and replacing worn out brakes is expensive. A thrust reverser is extra initial expense, but it pays for itself. In the military they aren't concerned with cutting expenses. They use thrust reversers on large airlifters because they need to be able to unload and return for more very quickly, and they need to be able to slow down and land on very short runways; that's rarely a concern for a fighter. They may also need to turn themselves and ready for takeoff again; while a fighter could be pushed by hand in a pinch, a C-5 Galaxy, not so easy. It's not surprising, in the few cases where a fighter aircraft has a reverser (the Tornado and the Saab Viggen), being able to safely land on short runways was a primary motivation, coupled with a desire for rapid turn-around time, and the fact that regular brakes don't work on an icy runway, and parachutes are only effective at high speeds).
It is because they already have several ways of braking, so they don't need reverse exhaust.link This should explain to you the braking systems used by the f-16 instead of reverse exhaust. This jet you are describing may not have as many braking systems as a normal jet does so it uses reverse exhaust.