A simple mathematical definition of arbitrage

External question — Economics Stack Exchange Author: Julius Hamilton Original post: https://economics.stackexchange.com/questions/59078 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I am trying to formulate an understanding of arbitrage to ensure I comprehend it. There are many reference materials online, of course. I envision a set of commodities C and a set of owners O . Each commodity is associated with its quantity type Q_t . (Water can be quantified by positive real numbers, apples by natural numbers, Bitcoin by positive rational numbers with a minimum increment of 1 Satoshi). For example, owner o_1 is associated to a set {(c_1, q_1), (c_2, q_2), ...} , where c_1, c_2 are commodities they own, and q_1, q_2 are quantities of those commodities (which are values in the set of quantities which that commodity can take). An exchange E is a vendor with a set of commodity buy-sell pairs: a set of tuples (b, q, r) where b is a base currency, q is a quote currency, and r is an exchange rate. If b is Bitcoin, q is USD, and r is 5000, that means that the vendor will buy or sell 1 Bitcoin for 5000 dollars. (Perhaps exchanges have asymmetrical exchange values?) A trade occurs when a quantity of one commodity and a quantity of another commodity switch owners. Let there be a trade function which accepts two owner-commodity-quantity triples from the set of all owners, and returns an updated set of all owners: trade: (O x C x Q) x (O x C x Q) -> O_all , such that trade((o_1, c_1, q_1), (o_2, c_2, q_2)) equals the set of all owners O_all , where for owner o_1 , the commodity c_1 now has value q_c_1 - q_1 (the original quantity minus the traded quantity), and the commodity c_2 now has value q_c_2 + q_2 , and analogously for o_2 . Consider that the exchange rates r1, r2 for the same currency-pair (b, q) on exchanges E1, E2 are not equal; r1 < r2 for (b, q, r1) in E1 , (b, q, r2) in E2 . Maybe one way we can define how "buying low and selling high" is profitable is by defining the value of each currency in reference to some "stable third currency". Maybe we can define the "average market value exchange rate" for all currency pairs. Selling a commodity on an exchange where the rate is above market rate, and buying a commodity on an exchange where the rate is below market rate, is therefore profitable. I haven't come close to making this formal definition complete, but I was wondering if someone could comment on it, critique it, or enhance it.