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It was written in the current opener of Arbitrage in Wikipedia:

In economics and finance, arbitrage is the practice of taking advantage of a difference in prices in two or more markets;

To me, dual pricing can be a private case of that, although in dual pricing the price is something created anew by the price maker (so there is no "taking advantage of a difference in (existing) prices").

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Dual/discriminatory pricing is not arbitrage. At most you could say that dual pricing sometimes creates arbitrage opportunities.

In situations where the goods/assets are tradeable without any friction, and good A is sold to somewhere/someone at price $P_A$ while good B is sold to someone at price $p_B$ people may be able to take advantage of this and buy only at the lower price.

Usually however dual pricing is applied when the seller can hinder resale. E.g., services such as loans and airplane travel are usually sold to a specific person, who cannot freely change the owner of the loan/airplane ticket. Then even though dual prices exist, there is no arbitrage opportunity for the buyer.

One might ask if the seller could benefit by creating a single price. The answer is that if the seller can identify market segments with different price elasticites and hinder resale then selling at separate prices can indeed be profitable. For a detailed discussion see the Price Discrimination chapter of an industrial organization or microeconomics textbook (e.g., Varian).

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