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I can't say for all pilots but in WWII, there certainly was some mutual respect among fighter pilots. In his book The Big Show , Pierre Clostermann (France's greatest "ace" in WW2) talks about this relationship between allied and axis pilots. In this book which regroups his diary entries during the war, he never mentions any games, however there are several entries about battles during which they knew they would be flying against certain Luftwaffe aces. Allied intel would know about certain top fighters being moved around on the battlefield or certain enemy squadrons taking off, and generally they would become familiar with certain enemy squadron compositions, so if they saw certain types of planes come from a specific direction, they they would know which squad it was.
Now concerning mutual respect between allied and axis pilots, his view on this is that all pilots are brothers who happen to be seperated by frontiers, but that when it comes down to it, they share the same passion for aeronautics and flight. He also describes what you talk about in your post, which is that most pilots have a very competitive and sportsman attitude in their job. This impression comes up several times, hell he even named his book The big show because that's what his impression of dogfights were. Also in his foreword dedicates his book to the fallen heroes of the Luftwaffe, Rudel, Hartmann, Galland, Krupinsky, etc...And later on he describes a night where himself and several RAF pilots learn about the death of Walter Nowotny, one of the best luftwaffe aces. All the pilots saw it as a tragedy and expressed their appreciation for one of the greats, lamenting the fact he wasn't on their side.
I highly suggest his book if you're interested about this subject because it constitutes a very thorough account of the life of an RAF pilot (he was french but trained and flew with the RAF for most of the war).
Edit: I've went back looking for other anecdotes in his book so here are some other passages which I think are interesting:
Clostermann highlights the impressive industrial capabilities of Germany even during the end of the war. Despite heavy bombing, alot of factories were able to be rebuilt and quickly put out several dozens of planes per week, or if they couldn't be rebuilt, they would turn to alternatives like in January of 1944 where Hans Kammler identified all the underground quarries which could be used for plane production, and by April of the same year they were able to output 300 finished planes per factory in their first three underground factories. This hidden but very productive manufacturing of planes kind of killed the moral of allied fighters that had the impression that the more planes they shot down, the more appeared.
According to him, the luftwaffe was divided into 2 categories by the end of the war. 15 to 20% of it were highly trained pilots that survived the Spanish civil war and the early campaigns on the western but also eastern front. End of 1944, and in 1945, the west had alot of trouble in dog fights because they were almost sure to encounter enemy pilots of the highest caliber accompanied by younger less trained but very motivated pilots. This is in contrast to the early years of the war where alot of western pilots would be sent to the eastern front, leaving mediocre ones behind, and it's only in the later years they sent those in numbers to the east and brought back highly trained ones to the west.
British, French and Polish pilots recognized the technical superiority of german fighters, only the americans refused to admit and instead boasted as being the best. To RAF pilots the number of air victories claimed by american fighter pilots was laughable and everyone knew they were more or less propaganda. He then goes into detail about how after the war RAF and Luftwaffe records show the dominance and importance german pilots had even till the end. If anyone is interested I can translate the whole passage with numbers (he goes over overall loss records from western, german and soviet parties and then illustrates these with key air battles)
He describes an incident right after Germany surrenders where he ends up taking care of a german prisoner pilot. The german pilot speaks impeccable french due to living close to the french border (this proximity is something he brings up in the foreword too as he discovered long after the war that a cousin from the german branch of his family was also a pilot but in the luftwaffe, adding to this feeling of pilot brotherhood). The pilot salutes him in the traditional way (he insists here but also in his foreword that the Luftwaffe was like the military aristocracy and only did the military salute, never the nazi one). It's all very surreal as they compare logbooks, the german pilot has over 100 victories and asks Clostermann if he could go see a Tempest up close. I think he talks about this to show once again that there's a real passion for aviation. They spend the end of night all drinking beer, german and allied pilots alike, looking over a map of Germany and trying to figure out if they ever met in the air. This just highlights how in the end they're the same breed of men.
There are a lot of WW2 answers here, so I'll expand on WW1.
There was a great deal of mutual respect and 'sportsmanlike conduct' between pilots from opposing nations in the First World War on the western front, right from the beginning of hostilities - before weapons were mounted on aircraft it was common for pilots to wave to their opposites as they flew past one another. Such an attitude was the beginning of aircrew considering themselves of a different calibre to the infantry; It was commonplace when a plane was downed near an airbase for squadron staff to meet the now captive pilot and take him back to their mess hall for a drink and a smoke before sending him off to a POW camp. Oswald Boelcke in particular is famous for such behaviour, regularly landing next to an aircraft he had just defeated to shake the pilot's hand and wish them well. Boelcke was even known to drop letters written by his defeated foes at their airfields, to let them know that they had survived and were in good health. Aviation is seen as the technological replacement of cavalry at the start of the war, and some of that cavalry swagger and bravado is very much evident in Boelcke.
The exchanging of information in this way was common practice, though not in quite as personal a way as Boelcke's gentlemanly behaviour. Flying to opposing airbases to drop notes was, in an era where aircraft communicated by flare gun, the only way a squadron could confirm aircrew had been killed or captured, and subsequently played an important role in pilots 'scoring' their victories. The British-German battles away from the western front even go so far as to have squadrons inviting their counterparts for tea and exchanging necessities.
In terms of personal challenges/duels/recognising individuals, this tended to be somewhat one-sided. German aircrew were given much more freedom in changing the appearance of their planes and subsequently individual pilots were more easily recognised by their Entente counterparts, whose aircraft were more uniform. German doctrine was to keep its best pilots in Jasta squadrons which were moved around the front constantly in response to events, and because of this it would be difficult for the truly skilful German fighter pilots to arrange any kind of 'duel'.
It should be made clear that not all aircrew lived up to this ideal of sportsmanlike conduct and camaraderie. Many of the very high-scoring pilots whom you would suspect of being most likely to seek duels and challenges, Manfred Von Richtofen and Mick Mannock being an example from each side of the conflict, took a very cold-hearted approach to combat, no doubt in an attempt to create some emotional detachment from the ever-increasing numbers of men they killed. No matter the intentions, this ideal of the 'knights of the air' was to fade away with the reality and brutalities of war as the years dragged onward, the evermore desperate situation for the central powers accelerating the demise of the era.
Sources:
Edit:formatting
Squadrons operating the same airfields or flight paths often encountered each other repeatedly, several times a week. Bomber crews that flew routes through air defenses often came across the same fighters a second time during the return trip home. These units took detailed notes and poured over missions identifying enemy planes and elements, assigning nicknames, etc. Most Aces did not have unique markings or garish paint jobs, keeping their planes deceptively stock, but most were wing commanders or often recognizable by specific maneuvers or tactics.
An Ace's notoriety was well publicized at home (and learned by the enemy of course, typically a nickname) http://query.nytimes.com/gst/abstract.html?res=9902E7DB1331E23ABC4C51DFB066838E659EDE
In WW1 specifically, air combat was more intimate, honorable and 'chivilrous' due to the lower speeds, reduced range and exclusivity/prestige of the post. The movie Flyboys (based off https://en.wikipedia.org/wiki/Lafayette_Escadrille) shows American pilots amid these challenges, including recurring engagements with specific enemy counterparts and (fictitious) enemy Ace, showing combat as quite personal.
There was however no "game" about it, and only a few occasional examples like Douglas Bader where mercy was so deliberate. In general pilots showed eachother respect under some circumstances you'd not see in other services. Many personal memoirs from pilots attest to a depth of tactics, rumors and repeated recognition of enemy elements, planning around these types of engagements. There were no recorded 'famous' duels that were orchestrated or the result of a challenge or game though.