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I'm look to find a good source for comparing mechanism designs for simple currency exchanges.

Is there a name for a category of game (e.g. Stackelberg game) that encapsulates the operation of an exchange?

What search terms should I use (the above terms don't seem to work) to find mechanism designs for simple currency exchanges?

Is there a good source text book that would cover this?

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    $\begingroup$ Other than futures, foreign exchange trading is a dealer market, with banks as dealers, who then trade with each other and their clients. I’m no expert on this, but there are presumably guides out there. You need to think quite carefully as to what a “currency” is in this context – people trade bank deposits. Derivative markets are layered upon this. $\endgroup$ – Brian Romanchuk Aug 6 at 21:24
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The area of research you are looking for is market microstructure, the study of how markets work and the process of price formation. Naturally, this means studying liquidity, information diffusion, uncertainty, and dynamic games. There is not a lot of mechanism design in market microstructure -- though work along those lines would certainly be welcomed.

For an introduction to the area, a good first read might be Foucault, Pagano, and Röell's book. (Sadly, the review articles which come to mind are old.) The Journal of Financial Markets is chock full of microstructure research and you would do well to start following it.

FX markets have been changing recently with more retail flow and trading on ECNs while the inter-dealer network has become less segregated and more affected by trading on those ECNs. Thus the old Olsen & Associates work like the book by Gençay, Dacorogna, Muller, Pictet, and Olsen (2001) might help explain the interbank market but won't be up-to-date.

More relevant (and shorter) would be to check into Evans and Rime (2019).

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