Concave Pro-rata Games: Additional Numerics

Table of Links

Abstract and Introduction

1 The concave pro-rata game

1.1 Symmetric pure strict equilibrium

1.2 Uniqueness of equilibrium

1.3 Equilibrium payoff

2 Batched decentralized exchanges

2.1 Arbitrage

3 Conclusion and References

A Numerics

B Additional Numerics

C Relaxing strict concavity

D Rosen condition

B Additional Numerics

Here we expand on the simulations introduced in appendix A using a class of utility function that allows us to express many quantities of interest in closed form.

Game setup. We consider the following three scenarios:

Simulation results. In our simulations, we fix β = 0.5 and γ = 0.05. We average each reported value over 100 trials. In figure 4, the intial strategy of each player is drawn uniformly at random from the interval (0, w/n), where w is a value such that f(w) = 0.

The left plot of figure 4 illustrates that the number of iterations needed to reach the unique equilibrium, in the absence of budget constraints, scales superlinearly in the number of players. The right plot demonstrates that in the scenario of bounded strategy updates, for small values of δ, the number of iterations required to reach equilibrium increases significantly when compared to the unbounded strategy update scenario.

Price of Anarchy The equilibrium payoff can easibly be found to be

Similarly, it can be show that the optimal payoff conditioned on every agent receving the same payoff is given by

We obtain the price of anarchy by taking the ratio of the equilibrium payoff and the optimal payoff:

We again fix β = 0.5 and γ = 0.05. The left plot of figure 6 illustrates the optimal payoff function and the equilibrium payoff function as a function of the number of players n while the right plot of figure 6 illustrates the price of anarchy as function of n.