This question seems fairly straightforward and I thought I would just easily find you some answers. However, very little is written about it (especially in secondary sources) and very little seems straightforward and clear. Let us explore some information:

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Piers Plowman

The first mention that I could find is in the medieval poem called Piers Plowman. It was written somewhere between 1370 and 1390, by the author William Langland. The game is mentioned in lines 5.320 - 5336 and they read as follows:

Clement the Cobelere caste of his cloke,

And at the newe feire nempned it to selle.

Hikke the Hakeneyman hitte his hood after,

And bad Bette the Bocher ben on his syde.

Ther were chapmen ychose this chaffare to preise:

Whoso hadde the hood sholde han amcndes of the cloke.

Tho risen up in rape and rouned togideres,

And preised the penyworthes apart by hemselve.

There were othes an heep, for oon sholde have the werse];

Thei kouthe noght by hir conscience acorden in truthe,

Til Robyn the Ropere arise the[i by]sou[ght]e,

And nempned hym for a nounpere, that no debat nere.

Hikke the Hostiler hadde the cloke

In covenaunt that Clement sholde the cuppe fille

And have Hikkes hood the Hostiler, and holden hym yserved;

And whoso repented rathest shoulde aryse after

And greten Sire Gloton with a galon ale.

Maybe your Middle English is a little rusty, so I included a translation which you will find in note (1). In short, a hood is traded for a cloak. A neutral umpire decides that the cloak-giver should also offer a cup of drink to make the trade a fair one.

You will see that "new fair" is mentioned, written as "newe feire". To me this part is problematic. In the translation and also in some other writings we see that the 'new fair' is interpreted as the 'new fair game', new fair being the name of the game. However, in the original text, there is no mention of game. In fact, reading the text itself, you could see it as a location or occasion. Medieval fairs were big and important affairs in a town, where there was opportunity to trade, feast and make acquaintances. I'm not a translator of Middle English and don't pretend to know better than a translator, however it struck me and it deserves mention.

Sir Gawain and the Green Knight

The next mention of the trading system with a neutral third party is in Sir Gawain and the Green Knight, a chivalric romance from the late 14th century. Here, the game has not received a name. We read that the Green Knight from the title hangs his hood from a speak to start a mock auction, as part of festivities around Christmas. An academic journal (Harwood, see sources) mentions this passage and says that this game is elsewhere called the New Fair. The 'elsewhere' is probably in Piers Plowman and indeed there is a similarity in the trading.

There is no mention of this trading system being called hand in cap, or any mentions of hands or caps. We only know that there was a trading system in place where a third party acted as an arbiter. From these sources it seems it was more of a game or festive activity more so than an daily trade use. But then, we only have two examples, which is not a lot.

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Sir Samuel Pepys

Now, we need to skip a few centuries, to 1660. Here we find the first mention of a game called Handicap or handycapp. Samuel Pepys in his valuable diary makes mention of playing it with friends. The full text of the diary entry you will in note (2).

Sadly, he does not explain what the game is about or how to play it. For this we have another literary source, be it from much later. in 1875, Charles Hindley writes Tavern Anecdotes and Sayings: Including the Origin of Signs, and Reminiscences Connected with Taverns, Coffee-houses, Clubs, Etc., Etc, which in itself is a joy to read and a link to the full text you will find in the sources below. How to play Handicap according to Hindley:

Handicap.—A game at cards not unlike Loo, but with this difference—the winner of one trick has to put in a double stake, the winner of two tricks a triple stake, and so on. Thus: If six persons are playing, and the general stake is 1s., and A gains three tricks, he gains 6s., and has to “hand i’ the cap” or pool 3s. for the next deal. Suppose A gains two tricks and B one, then A gains 4s. and B 2s., and A has to stake 3s. and B 2s. for the next deal.

You will see that there is a very small similarity between this game and the trading system in Piers Plowman, in the sense that the "loser" is compensated in a way: in Piers Plowman the person offering an item of lesser value also has to offer a drink, in the card game the person winning has to bet more, making the chances more even. However there is no longer a neutral third party playing the arbiter, so while there are similarities they are not extremely clear.

The Sporting Magazine; or Monthly Calendar of the Transactions of the Turf, the Chace, and every other Diversion interesting to the Man of Pleasure and Enterprize (sic) of February 1793 gives us yet another variation of the game:

A handy cap match is for A B and C to put an equal sum into a hat; C, which is the handy capper, makes a match for A and B, which, when perused by them they put their hands into their pockets, and draw them out closed; then they open them together, and if both have money in their hands the match is confirmed, if neither have money it is no match. In both cases the handy capper draws all the money out of the hat; but if one has money in his hand, and the other none, then it is no match; and he that has money in his hand is entitled to the deposit in the hat.

That's nice because this game is nothing whatsoever like either Piers Plowmans or Pepys's. However this is a version you see on the internet a lot. The third party arbiter is back in this version. The date is also problematic on this one: in 1660 Pepys calls Handicap a card game, in 1875 it is called a card game. In 1793 it is not called a card game. So it's not a matter of slowly changing over time, this version truly is an outlier.

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Dictionaries and etymologies

What's adding even more to this chaos is different dictionaries and etymologies giving different explanations to the word handicap. In a paper by Lehman and Cook, different dictionaries and etymologies from the later 19th century (around 1870) are compared, the word handicap is used as an example. They focus on the dictionaries of Skeat, Onions and Klein.

Skeat writes that the word was formerly the name of a game, he then mentions Pepys' diary entry. According to Skeat the card game Handicap was originally the same as New Fair in Piers Plowman, which shows it was customary to barter articles and to set the value by a neutral arbiter, determine which was the more valuable and what value should be given to the holder of the inferior item. The etymology is from hand i' cap, from the mode of drawing lots.

Klein's dictionary: handicap - from hand i' cap, originally a kind of lottery game in which the forfeit money was held in the hand in the cap.

Onion's dictionary: lottery in which one person challenged an item belonging to another, for which he offered something in return, and an umpire determined the respective values. the word comes from the money being in the hand or cap.

The problem with these dictionaries is that they show no bibliography or sources, so it's hard to determine where they found this information. It might be that some source texts are lost, or that sources were not as factual as they should be.

Another academic article (Baynton) gives yet another explanation:

A popular game of chance in the 17th century which involved drawing money or slips of paper from a cap. The rules ensured that each time a player lost, their odds of winning the next time improved; over a succession of games, wins and losses would more or less even out.

This seems a variation on other games we have discussed before, and most closely resembles the one from Sporting Magazine.

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