A common way of analyzing markets is through the lens of arbitrage: the ability to exploit price differences in order to make essentially risk-free profit. From before, we will write g for the forward exchange function of a constant function market maker, used by the batching design presented above.

Existence. Assuming g is differentiable at 0, we can interpret g 0 (0) as the marginal amount of asset B that one would receive for a marginal amount of A. (The function g is often not differentiable at 0, but is one-sided differentiable at 0+, which suffices.) If g 0 (0) is larger than the price of an external market, say c > 0, then anyone who can directly trade with g can make risk-free profit by trading some (potentially small) amount, t > 0 of asset A for g(t) of asset B, and then sell this amount of asset B to get g(t)/c − t > 0 of profit. (One simple way to see this is true is to use the definition of a derivative on g(t)/c and send t ↓ 0.)

Optimal arbitrage. Since an agent can make risk-free profit in these cases, it is reasonable to ask: what is the maximum amount of profit an agent can make with this strategy? This is known as the optimal arbitrage problem, written:

maximize g(t) − ct

subject to t ≥ 0,

The (aggregated) arbitrage game. In the batched exchange above, arbitrageurs cannot directly trade with the constant function market maker, but must instead go through the batching process. Assuming there are n arbitrageurs competing to maximize their profit, the next question is: what are the properties of this game? Defining

f(t) = g(t) − ct,

then this game is a concave pro-rata game with function f, since the payoff (1) for player i is

Note that this is exactly the amount received from the DEX with forward exchange function g, minus the cost of trading xi with the external market, for player i. This game inherits all of the properties derived in §1. We show some numerical simulations of iterated behavior for some utility functions of this form in appendices A and B.