Folklore research into this suggests that there were several early variations that did not mention the product, Coke. To explain to those who had not heard of this (which includes me), when two children say the same word or phrase at the same time, the first one to declare "Jinx" ( followed by a set phrase), was the victor, the loser then becoming jinxed.

Early forms of this, which date at least to the 1950s, include "Jinx! you owe me a soda" and what seems to be a later, "pinch, poke, you owe me a Coke" (exploiting a rhyme with poke and coke). In this case, the winner could punch the loser. A diffusion from this rhyme to replace "soda" may be the origin of the word "Coke" in the phrase beginning with "Jinx".

Uncovering the origins of folk sayings - in this case something that circulates internationally among children - is notoriously difficult. There is, simply, no way of saying where it came from, but I see no evidence to suggest that the Coca-Cola company was at the heart of this. These things are rarely introduced from the "top down." They typically well up from subterranean reservoirs of unknown location and character.

Research on this, as with so much of children games and sayings, comes from Iona Opie and Peter Opie, folklorists who specialized in these things. The folklorist Steve Roud also published on this.

edit: There is a problem inherent with discussing traditions - particularly modern/recent traditions on this platform, a subreddit devoted to documentary evidence that serves the discipline of history so well. It is difficult for a folklorist to evaluate an "informant" under the best of circumstance and doing so on a subreddit that is vigorously monitored (as it should be) causes many of the responses/contributions to be deleted. We cannot be sure, of course, if the contributions are fraudulent or sincere. One of the ways so "smooth out" the possibilities is to rely on an abundance of testimony, but this is not the platform to conduct this sort of research; rather it is here to report on what can be known based on documentary evidence.

Unfortunately, when it comes to folklore - and children's folklore especially - there is often a lag time between when the tradition exists and when it is documented. This makes the historical process problematic, and it will inevitably suggest later dates than are accurate for specific traditions. Neither my wife nor I - born in the American West in the mid to early 1950s - were familiar with this tradition, hinting at a late arrival of the tradition in that region. Documentation by the work of the Opies suggests a children's tradition that may have served as a foundation of this practice dates to at least the 1950s in Britain. Without additional information, it is not possible to determine the direction of diffusion, but the implication is clear that it may have originated in Britain by at least the 1950s and then diffused to the US at a later date.

Sadly, many of the contributions by our informants among our fellow redditors have been deleted, so we cannot witness their contributions. This is appropriate for reasons indicated, but also because a lack of time and place are not generally indicated by the deleted responses. Among those now deleted, I was intrigued by the suggestion that there was a college tradition that involved beer rather than Coke (suggesting that the tradition was adapted by the aging process). It is also meaningful that a redditor attested to the use of "soda" even though their dialect refers to this drink as "pop." Here we can see that the phrase was irresistible as it diffused, even though the word was not normal for that place: this also hints at a direction of diffusion.

This is how the folkloristic process might unfold in a different venue. Here, we must rely on what primary literature that exists and what limited research on the topic has revealed - although to underscore again, documents are likely to miss an earlier existence of the tradition.

According to the OED, the first documented use of jinxing as a children's game is in 1973. An article called "The Jinx Game: A Ritualized Expression of Separation-Individuation" by Jerome D. Oremland was published that year in The Psychoanalytic Study of the Child. Oremland describes the jinx game as "a sophisticated, stereotyped ritual". The version of the game he analyzes uses only the cry of "Jinx!" without any reference to soda or another reward. That the game may be much older is suggested by its equivalents in other linguistic environments: "Though the Game is played in a remarkably identical manner in various geographical areas, the word used to induce the spell varies widely, e.g., Israeli children shout, Ain, the Arabic word for ghost or evil eye". There is also a French variant called Chips, although it's unclear whether this is derivative of the English "jinx".

Older versions of the game involved linking fingers together, which has its roots in earlier word ritual games using words other than "jinx". In his book The Lore of the Playground: One Hundred Years of Children's Games, Rhymes and Traditions, Steven Roud includes the jinx game under the category of the 'same word' ritual. Drawing on the work of Iona and Peter Opie in their 1959 book The Lore and Language of Schoolchildren, he links the game to earlier versions that involved holding pinkie fingers together and repeating the name of a poet, such as this example from County Antrim, Ireland, in 1905:

Little fingers were a very important part of a special charm. If you and your friend happened to say the same word at the same time, you immediately linked your wee fingers together without uttering a single word until you both had silently wished. When each of you had said the name of a poet, you might safely disengage your fingers. Shakespeare and Wordsworth were very much favoured poets, perhaps because they often appeared in the Readers of the day.

Many other variants of this game appeared in the first half of the 20th century. In Nottingham in the 1950s, for example, the magical word was "crosses" which was explicitly said to ward off the bad luck that came with saying the same word at the same time. This is where the link to the idea of a jinx comes in. The word "jinx" is originally an American spelling variant of a much older word, "jynx", going back to at least the 17th century. A jynx was a charm or a spell. The name originally came from a type of bird associated with witchcraft.

The "jinx" spelling is first attested in the early 20th century. According to the OED, the word first appears in this sense in 1911 in the Chicago Daily News with the sentence, "Dave Shean and 'Peaches' Graham [...] have not escaped the jinx that has been following the champions". It also appears in the 1910 short story "The Jinx" by Allen Sangree. This is its first attested association with baseball, a game in which players were notoriously superstitious and worried that an opponent was "jinxing" the game:

But the ball players instantly knew the truth. "A jinx, a jinx," they whispered along the bench. "Cross-eyed girl sittin' over there back o' third. See her ? She's got Th' Dasher. Holy smoke, look at them eyes!" Like the discreet and experienced manager he was, McNabb did not chasten his men in this hour of peril. He treated the matter just as seriously as they, condoling with The Dasher, bracing up the Yeggman, execrating the jinx and summoning all his occult strategy to outwit it.

In this excerpt, you can see that a person is called a jinx for bringing bad luck or a curse. So the jinx can refer to the bad luck itself or the person causing it. It may be the popularity of baseball that led to the American usage of the word as we know it today. Why the word "jinx" became associated with same-word rituals in the 1970s, however, remains unexplained. Roud merely says,

All this has changed dramatically in recent decades [...] In modern playgrounds, there is no talk of wishes or poets, but instead it is a question of 'jinx'. [...] The change seems to have occurred in the 1970s, and the new style appears now [in 2010] to be known all over Britain.

The jinx game has always had variants in the exact wording and rules. The number of times the jinxed person's name had to be said varies from place to place, as do the rules about who is allowed to release them by saying their name.

I suspect it's this inherent variability that brings us to the variant you're familiar with, "Jinx! You owe me a coke". In many parts of the United States, it's "Jinx! You owe me a soda". The insertion of the word "coke" may be because coke is the generic name for a soft drink in the American South. Most famous soda brands were invented in the South, with Coca-Cola being among the oldest from 1886. More importantly, Coca-Cola was the first mass produced soda or soft drink. Due to a bungled contract negotiation, a bottle of Coca-Cola cost only 5 cents until 1959; this was part of what it enabled it to become so ubiquitous in American culture. "Coke" was being used as a generic term for soft drinks by the early 20th century, and although the Coca-Cola company initially resisted this, by the 1940s the company accepted it and even started using "coke" as a generic term for soda themselves. It's essentially the "Kleenex" effect, where the most popular brand becomes accepted as a generic. Outside the South, "pop" and "soda" predominate, but "coke" remains in a solid third place.

The use of a soft drink as a prize in the jinx game is part of the competition of dominance inherent in the game. In his article "Trickster's Economics: Conservation and Innovation in the Game of Jinx", Jeffrey G. Howard speculates that the competitive element entered the game after the 1940s because of the influence of WWII. Whether or not this is true, dominance exerted through the demands of compensation or physical attacks such as pinching or arm-punching are a key component of the game, in contrast to the earlier pinky-linking versions which focused on mutual cooperation to overcome the bad luck. A soda is not the only possible compensation in the game, however. Howard records his niece in 2013 saying, "Jinx! The jinx machine is out of order. Please insert another quarter", thereby demanding money rather than a beverage. The rhyming element of this is another common component of the game (e.g. "pinch poke, you owe me a coke", recorded in 1973). This is probably based on a variant where the demand for a soda is countered with the rhyme "Soda machine is out of order, please insert another quarter".

(1/2)