The code in this repository is meant to find Nash equilibria for the following model:

We assume that n players produce safety, s, and performance, p, as

$$s_i = A_i X_{s,i}^{\alpha_i} p_i^{-\theta_i}, \quad p_i = B_i X_{p,i}^{\beta_i}$$

for i = 1, ..., n. The X are inputs chosen by the players, and all other variables are fixed parameters.

In a Nash equilibrium, each player i chooses X_s,i and X_p,i to maximize the payoff

$$u_i := \sum_{j=1}^n \sigma_j(s) q_j(p) \rho_{ij}(p) - \left( 1 - \sum_{j=1}^n \sigma_j(s) q_j(p) \right) d_i - c_i(X_s, X_p)$$

subject to the other players' choices of X_s and X_p. The components of this expression will be explained more below, but the basic parts are as follows:

• q_i(p) is the probability that player i wins a contest between all players.
• σ_i(s) is the probability of a safe outcome given that player i wins the contest.
• ρ_ij(p) is player i's payoff if player j wins the contest, and the outcome is safe.
• d_i is the cost incurred by player i in the event of an unsafe (disaster) outcome.
• c_i(X_s, X_p) is the price that player i pays for the inputs.

Note that the weighted sum Σ_i σ_i q_i is the unconditional probability of a safe outcome (no disaster).

If you don't have Julia, download it from https://julialang.org/downloads/ and install. (You'll need at least version 1.6.7.)

Next, you'll need to install the project code. The easiest way to do this is to open a new Julia REPL session, then run

] add https://github.com/quevivasbien/AIIncentives.jl

which will automatically download and install the project as a package on your computer. (You'll only need to do this last step once, but after that you can run ] update AIIncentives every now and then to make sure you're up to date with the latest available version.)

You should now be able import the package:

using AIIncentives

You can then create and solve a scenario like

scenario = Scenario(
    n = 2,
    A = 10.,
    α = 0.5,
    B = 10.,
    β = 0.5,
    θ = 0.25, 
    d = 1.,
    r = range(0.01, 0.1, length = 20),
    varying = :r
)

solution = solve(scenario)

[

A couple of tangential tips here:

• To use math-related characters in most Julia code, you can typically just type the Latex code then press Tab. For example, \alpha + [Tab] becomes α. I've also allowed the spelled-out versions of Greek letters to work in most places in this code; for example, you could use alpha, beta, and theta instead of α, β, and θ in the example above.

• You can speed up a lot of solve calls by enabling multithreading. To do that, follow the instructions here.

]

This will find Nash equilibria over the given parameterization and range of values for r. To generate a plot of the result, you can execute

plot(solution)

The first time you run something in the Julia session, it will take a while, since Julia compiles your code the first time you run it in a new setting. It should run a lot faster after that.

For more detailed documention, see this document.