# Simulated Transfer Mechanism


📗 Suppose the players have true types:
➭ \(X_{0}\) = , \(X_{2}\) = .
📗 They are restricted to report \(Y_{1} \in X_{1} = t X_{0}\) and \(Y_{2} = X_{2}\) for,
➭ \(t\) between and .
📗 Suppose \(Y_{1}\) = , the following plot has the incentive compatible allocations.

➭ Black curve: mechanism; Blue curve: types; Green curve: optimal.
📗 Projection: , Number of types:
📗 Show all types; Show circles for reports.


# Random Dictator Mechanism


Plot of approximation ratio of RD with \(n = 2, d = 2, k = 1\). Move \(X_{2}\) = and \(X_{1}\) = will be updated to the position that maximizes (or minimizes) the approximation ratio.
  {{1000,,1000}}
📗 Approximation ratio \(\dfrac{\mathbb{E}\left[\left\|X - \hat{X}_{\text{RD}}\right\|_{2}\right]}{\left\|X - \hat{X}_{\text{SVD}}\right\|_{2}}\): max = , min =
➭ at \(X_{1}\) = , \(X_{2}\) = .
📗 Discretization:

# Low-Rank BIG



(left: best response of orange player ; right: best response of green player )
(squares: targets; circles: actions; domain is \([-1, 1] \times \left[-1, 1\right]\), discretization = )

📗 Targets: \(t_{1}\) = , \(t_{2}\) =
📗 Fixed actions: \(x_{1}\) = , \(x_{2}\) =

📗 (Approximate) best response: \(br_{1}\left(x_{2}\right)\) = , \(br_{2}\left(x_{1}\right)\) =

📗 Debug Information
➭ Input \(X\) = = '
➭ Prediction \(\hat{X}\) = = '
➭ Utilities \(u_{1}\) = , \(u_{2}\) = .

➭ The receiver gets data \(X\) = \(\begin{bmatrix} x_{1} & x_{2} \end{bmatrix}\) = \(\begin{bmatrix} x_{11} & x_{21} \\ x_{12} & x_{22} \end{bmatrix}\) = \(\begin{bmatrix} u_{11} & u_{12} \\ u_{21} & u_{22} \end{bmatrix} \begin{bmatrix} \sigma_{1} & 0 \\ 0 & \sigma_{2} \end{bmatrix} \begin{bmatrix} v_{11} & v_{12} \\ v_{21} & v_{22} \end{bmatrix}\) and computes \(\hat{X}\) = \(\begin{bmatrix} \hat{x}_{1} & \hat{x}_{2} \end{bmatrix}\) = \(\begin{bmatrix} u_{11} & u_{12} \\ u_{21} & u_{22} \end{bmatrix} \begin{bmatrix} \sigma_{1} & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} v_{11} & v_{12} \\ v_{21} & v_{22} \end{bmatrix}\).
➭ The influencers (players) choose \(x_{i}\) to minimize \(\left\|t_{i} - \hat{x}_{i}\right\|^{2}\). 





Last Updated: December 03, 2025 at 11:46 PM