Optimal pricing for Crypto-Currency Exchanges I have a general understanding of game theory, and want to try to apply it to crypto-currency exchanges, which are completely decentralised systems. I come from a finance background so I will define some key terms.

BID Price: Price at which the exchange is willing to buy an asset from you.

ASK Price: Price at which the exchange is willing to sell an asset from you.

ASK>BID, so as a trader, you will always pay the higher ASK price when buying and will get the lower BID price when selling.

The BID-ASK spread is ASK price minus BID price.

Consider the following autonomous Bitcoin exchanges with the following prices on a certain date:



As you can see each exchange quotes its own price and spread.

In a frictionless world, a trader can buy Bitcoins at the cheapest ASK price (in this case exchange B at 1477) and sell the coins on the exchange with the highest BID price (exchange A at a BID price of 1603). This arbitrage opportunity is also 
found on the other exchanges as long as the ASK price paid by the trader is less than the BID price on the other exchange.

Now let's say I want to create my own exchange. If I want to eliminate all arbitrage opportunities, I would quote a BID price equal to the lowest BID (exchange B: 1475) and an ASK price equal to the highest ASK (exchange A: 1604). While this eliminates arbitrage on my exchange, it also makes me very uncompetitive given that my spread will equal 129. As such, no one will trade on my exchange. 

Given that I want to keep my spread competitive, with a goal of minimising arbitrage opportunities. What is the best course of action to take for quoting Bitcoin prices?

This is a real case, and the figures in the table are actual Bitcoin\USD prices taken on a particular date for these anonymised exchanges.
You are missing the key element, transfer fees.  If the transfer fees are large enough, then the marginal cost to move between exchanges just needs to stay inside the gross bid-ask spread.  It can probably move outside that spread because there is an additional risk for one who moves between exchanges.  Not all exchanges move back and forth between real world bank accounts equally well, nor do all have equal trading depth.  Additionally, the time value of money between the exchanges is probably sufficient to eat up any arbitrage.  It may be worth giving up arbitrage profits if they are traded for the time value of money profits.  If the arbitrage is small and the cost is great, you can carry a wider bid-ask spread and get away with it as a market maker.

It sounds like you need decision theory and not game theory unless you are considering the behaviors of the other actors as part of the game.  You should start with de Finetti's Coherence Principle.  It is a weaker principle than the "no arbitrage" principle.  In its weakest form, it only states that odds are coherent if a bookie cannot take a sure loss in all states of nature through the clever combination of bets either by a single or by multiple actors.

What makes it weaker is that it never says that no arbitrage opportunity exists, but it would imply that if it did exist, then the market makers would capture all or nearly all the profits.  I would be surprised to find that the exchanges did not profit greatly from the arbitrage opportunities on each other's exchanges.  They are perfectly placed and have deep inventory that they could borrow from their customers.  They probably allow some to go by to get speculators into their markets.  They are profit maximizers.