For a long time I've been idly thinking about a competitive game with strong collection elements with such a rule that players will win or loose resources at the expense of each other during each match.
Sorry if this seems unclear; let me provide a concrete example: Chess with an ecosystem.
Each player starts with a standard set of a king, a queen, two bishops, two knights, two rooks, eight pawns. Each time, during a match, a figure or pawn is captured, it permanently goes to the player who captured it (promoted pawns count as pawns for this purpose, not as whatever they were promoted to). This is the only way players can earn pieces¹.
The game is divided into several leagues: each league restricts the total value of pieces both players may start with. As long as each player only uses pieces they own and their value doesn't exceed their league's maximum, they can start with any pieces they desire. Thus each player who's just founded their account will start at a mediocore league that enforces the total value of pieces of up to 43 (8 pawns * 1 + 2 bishops * 3 + 2 knights * 3 + 2 rooks * 5 + 1 queen * 9 + one king * 4 = 8 + 6 + 6 + 10 + 9 + 4 = 43); if they loose pieces they will loose access to the starting league and be demoted to lower leagues, down to the lowest league that only accepts the total value of pieces of 5 (meaning that, aside from the mandatory king, it only allows one pawn); if they earn pieces they are promoted to higher leagues that allow them to start with more total material. (Players may also play lower leagues than their league if they wish, but not higer). If they loose so much pieces they cannot even play at the lowest league, they loose permanently and have to start with a fresh account.
Now from this follows that since there are no sinks and the total material of the game will only be increasing with people founding more and more accounts. From this we can expect that as more and more players loose permanently, we will have more and more players having hundreds and thousands of pieces, so inflation happens.
But if, regardless of their league, players are always matchmade according to their elo and if there are so many players that the matchmaking works and is unpredictable (granted, big 'if's...), their winrates should sit at 50%! Thus their earn/loose balance should be around zero. Pieces have a fixed cost: Getting them requires winning against an opponent of equal skill.
Ugh... Looking at the problem from these two different perspectives yields conflicting results. What am I missing? Would inflation happen in such a game?
Note this chess example is really only an example: I only provided it because it is the only concrete example I could muster. However, we can easily think about a collectible card game with a system similar to this, and I suppose such a TCG example would be even better here.
¹ Well, almost. If a player looses a king and have no more kings they must buy another king. To this end they must sell other pieces: pawns for one gold piece each, knights and bishops: three pieces each, rooks for 5 gold pieces each, queens for 9 gold pieces each; this in-game currency goes to the 'king fund' that will be used to buy kings from the in-game shop, 4 gold pieces for each king. No other pieces can be purchased for gold gained in such a way.
The average winrate between a player who loses all their pieces and one who gains all those pieces is 50%, but the player who loses all their pieces then goes on to make a new account, so now amongst those two people there are three complete sets of pieces, and you have inflation.
As for the middle and high-rank players accumulating more pieces over time despite not getting any better in rank: Imagine a ladder of people who are each just a tiny bit better than the one before them, close enough that they get matched against the adjacent players: on average, each player will collect pieces from the slightly worse player, and give pieces to the slightly better player, so over (a probably pretty long) time pieces will go up the chain, from the bottom where new pieces are generated, to the top where the greatest players are.