Who is losing in an arbitrage?

Nobody has to loose in an arbitrage. Economic relationships are not necessarily zero-sum (in fact often they will not be zero-sum). For example, if apples in city A are sold for ${$}5$ and apples in city B can be sold for ${$}8$, and we assume zero transaction cost there will be an arbitrage opportunity to earn ${$}3$ riskless profit per apple by buying apples in city A and selling them in city B. But nobody looses in the transaction.

The apple producers in city A clearly value ${$}5$ more (or at least indifferent) than an apple otherwise they would not trade it and just keep the apple. So giving them ${$}5$ will make them better off (or at least not worse off). Then in city B people who buy apples must value apples more than ${$}8$ they are paying for it otherwise they would just keep the money instead of buying them (or at least they must be again indifferent).

Consequently, when people take advantage of arbitrage opportunities generally speaking nobody looses anything. There must be some other issues present that would turn the problem into zero-sum game.

Regarding the microeconomic implications the main one is that people will take advantage of the arbitrage opportunities until the prices on the markets equalize - in the example above until the prices in both cities would become equal. The law of one price is based on the concept of arbitrage. There might be other implications depending on precise setting where arbitrage occurs, to explore all of them would be beyond the scope of SE answer.

I would note that there is a dispute about the approaches to answer this question. I will give an answer based on finance theory, which may or may not be better placed to be on the quantitative finance board. That said, “finance” is a tag here, so the answer appears to appropriate. Note that I am using an idealised definition of profit that is used in finance and accounting, which might not align with that of economics.

I am using the formal definition of arbitrage - an opportunity to simultaneously enter trades that generate risk-free (supernormal profits). Somewhat equivalently, the ability to enter into a position with zero initial cost, and in all (future) states of the world, the future value of the portfolio is greater than or equal to zero - with strictly positive profits in some states. (My reference is page 80 of Rebonato’s “Interest-Rate Option Models,” but should be in any text on mathematical finance.) Many people use a less formal version of “arbitrage,” but I am referring to the formal definition.

Update: Varian uses the definition I use (or the methamtical equivalent): link to Varian article “The Arbitrage Principle in Financial Economics”.

(The zero entry cost definition of arbitrage eliminates the issue of defining “excess risk free profits”: one can buy a default-risk free money market instrument and get a “risk-free return”. An arbitrage has to generate a higher return. The zero cost definition eliminates the issue; you sell short the money market instrument to get zero initial investment.)

We then assume that we have observed market quotes at all time, and that we can find a set of theoretical prices from an arbitrage-free model that best fits those prices. For simplicity, I will assume that these are mid-prices. Note that observed prices might have arbitrage opportunities, and there are presumably many fittings. All that matters is that we can find one fitting.

We can then do a mark-to-market of all instruments versus that theoretical fitting at all times. Any time an entity trades away from that mid price, they get a corresponding mark-to-market gain/loss, which is mirrored by their counterparty.

Note that this is an idealised version of how financial accounting treats mark-to-market of securities. In the real world, everything is marked to market (bid, mid, depending on convention) at the close. In this example, the pricing is continuous - to match the simultaneous transactions. It’s also pricing versus an arbitrage-free model that is calibrated/fit against benchmarks, which is also standard practice for the derivatives that represent the bulk of the arbitrage trading. Finally, all entities use the same prices for this hypothetical mark-to-market, which is needed for trading activity to be zero sum.

When an entity enters into an arbitrage trade, it is clear that the gains/losses of all trades have to add up to a profit. The counter parties will in aggregate generate a loss, but the exact distribution is obviously unknown (unless a single counterparty was willing to arbitrage itself).

This framework also explains how market-makers earn a profit: they hope to continuously be trading at prices on the “correct” side of the mid price, and thus generate continuous trading profits.

In the real world, mark-to-market is not done instantaneously, it is done at the end of the day. Furthermore, the pricing used for the mark-to-market procedure used by different entities need not be the same. (In fact, some entities will not do a daily mark-to-market, such as in held-to-maturity accounting.) This implies that actual trading profit and loss will no longer be exactly zero sum.

The main point (already made by 1muflon1) is that no one needs to lose. _{(The presumption that someone must lose in any transaction or exchange is an example of the zero-sum fallacy. This is a common mistake by non-economists.)}

The comments to 1muflon1's answer seem to contain some objections/confusion. To clear these up, here's an example where everyone wins and no one loses:

---

Example. Each nail clipper usually trades for $1 in city A and $1.05 in city B (10000 km away). On a particular day, Bob happens to be driving in his car from A to B to visit a dying relative. His car can load up to 10000 nail clippers. So, his plan is to buy 10000 nail clippers in A and sell them when he arrives at B.

It would however take him some time to buy and sell the 10000 nail clippers if he simply offers the usual prices. So, he decides to offer to buy the nail clippers for $1.01 each in A and sell them for $1.04 each in B. By doing so, he is able to quickly buy and sell the 10000 nail clippers in A and B.

Altogether, everyone wins and no one loses:

• Bob makes $300, which exceeds his costs (time spent buying and selling, loading/unloading his car, additional gasoline).

• Sellers in city A make an additional $100 (compared to what they'd usually have made).

• Buyers in city B save $100 (compared to what they'd usually have paid).

Now, if there are many individuals who regularly drive from A to B, then we'd expect the $0.05 price difference to be arbitraged away. But if Bob's drive from A to B is a rare occurrence, then this price difference can persist over time, because driving from A to B is costly.

---

I want to also emphasize a second point:

The costs and benefits of arbitrage may differ across individuals, so that only particular individuals may find it worthwhile to execute the arbitrage.

In the above Example, for most individuals, the costs (time and money spent driving 10000 km, time taken to buy and sell 10000 nail clippers) outweigh the benefits ($300).

But for Bob though, he has the additional benefit of visiting his dying relative, so that benefits happen to outweigh costs. It may be that this is true only of Bob on this particular day and that this is the only occasion where the nail-clipper arbitrage is ever executed.

---

Other points:

• Information -- it may be that Bob is the only one aware of the price discrepancy, so that even though many others regularly drive from A to B, only Bob knows to take advantage of it.
• Alertness -- it may be that many are aware of the price discrepancy and regularly drive from A to B, but only Bob is "alert" and proactive enough to take action. _{(This point about "entrepreneurial alertness" was emphasized by the Austrian-school economist Israel Kirzner. I think it's an important point that's been neglected as it's not easily formalized, modeled, quantified, tested, and so not very useful for pumping out papers.)}

It's important to distinguish between the effects of arbitrage on: a) the direct parties to arbitrage transactions; b) other agents in the markets in which the arbitrage takes place.

Suppose arbitrageurs buy a good in market A in which its price is $1 and sell in market B where its price is $2. Assume further that in each market those prices have freely come about through the interaction of upward-sloping supply and downward-sloping demand curves. Sellers in market A will not lose by selling to an arbitrageur at $1 or more, and buyers in market B will not lose by buying from an arbitrageur at $2 or less. So there is scope for arbitrage to occur with no loss (and indeed some gain) to the direct parties to arbitrage transactions.

However, if the volume of arbitrage is sufficient to move the prices in the two markets, so that the price in market A becomes more than $1, and that in market B becomes less than $2, then there are also consequences for agents who are not parties to the arbitrage transactions, namely, buyers in market A and sellers in market B. The higher price in market A reduces the consumer surplus to its buyers, while the lower price in market B reduces the producer surplus to its sellers. Thus there is a welfare loss to both these groups.

This is not to make a case that arbitrage is undesirable or a zero-sum game: its overall effect may well be to raise welfare. It is only to assert that, even so, it may lower welfare for some groups.