• 2 players. 2 stages.
• Each player observes a state $x^i \in {1, 2}$, determining the cost of contribution:
  • $v(1)$ is the cost for state 1, $v(2)$ is the cost for state 2. $0 < v(1) < 1 < v(2)$.
• With probability $q$, player $i$ observes $x^i=2$.
  • Therefore, the initial common belief can be represented by $\pi_1=[q, q]$.
• At each time $t$, if player $i$ contributes, she receives $1-v(x^i)$ dollars. If player $i$ does not contributes, she receives $1$ dollar if the other player contributes, and receives $0$ dollar if the other player does not contribute.
  • $a^i_t \in {1, 2}$, where 1 represents "not contribute", and "2" represents "contribute".

The reader may refer to the sPCE.py directly. The class PublicInvestProblem includes the detailed comments for each module. test.ipynb presents some sample codes.

• script_env.m: a script for initialization. It creates all the required global parameters.

  • x_vals: $[v(1), v(2)]$
  • pi2_vals: all the possible values of $\pi_2$. Note that $\pi_1$'s values are also included.

• V2i_gen.m: a function that generates $V^1_2, V^2_2$, given the global vars x_vals, pi2_vals, and a given $\hat{\phi}^C_2$.

  • The output consists of two matrices. Each matrix is indexed by the state $x^i$ and the index of $\pi_2$ in pi2_vals.
  • NOTE THAT both value functions treat the player as the player 1, so for V2, if you want to find the value under certain $[\pi^1_2, \pi^2_2]$, the inputted belief index should be the one of $[\pi^2_2, \pi^1_2]$ instead of $[\pi^1_2, \pi^2_2]$.

• next_belief.m: a function that finds the index of $\pi_2$

  • Inputs: the index of $\pi_1$, index of $\gamma^1_1, \gamma^2_1, a^1_1, a^2_1$, and the global var pi2_vals.
  • $\gamma^i_t$ is indexed by 1, 2, 3, 4, corrresponding to $\gamma_{00}, \gamma_{01}, \gamma_{10}, \gamma_{11}$.
  • BTW, $\gamma_t$ as a prescription profile, is indexed by $1, \ldots, 16$, corresponding to $(\gamma_{00}, \gamma_{00}), (\gamma_{00}, \gamma_{01}), \ldots, (\gamma_{01}, \gamma_{00}), \ldots, (\gamma_{11}, \gamma_{10}), (\gamma_{11}, \gamma_{11})$.

• A_gen.m: a function that generates the coefficient matrix $A$ for the rationality constraints.

  • Inputs: time $t$, the index of the current belief $\pi_t$, and global vars x_vals, pi2_vals. If $t=1$, it requires the pre-calculated value functions V1, V2.
  • Output: the coefficient matrix for the rationality of $\psi_t = \hat{\phi}^C_t[\pi_t]$.

• f_gen.m: a function that generates the coefficient matrix $f$ for the calculation of social welfare.

  • Inputs: time $t$, the index of the current belief $\pi_t$, and global vars x_vals, pi2_vals. If $t=1$, it requires the pre-calculated value functions V1, V2.
  • Output: a vector $f$. $f^T \cdot \hat{\phi}^C_t[\pi_t]$ is the expected social welfare.

• expected_sw.m: a function that generates that evaluates the expected social welfare (to go) of a given device.

  • Inputs: device $\hat{\phi}^C_t$, global vars x_vals, pi2_vals. The precalculated value functions from last stage V1,V2.
  • Output: a social welfare vector. Each row corresponds to an initial belief.

• main_linprog.m: a main file for the experiment. Use linear programming solver to find a correlation device.

• main_social.m: a main file for the experiment. Use linear programming solver to find a correlation device, whose objective function is set to be the expected social welfare to go.

• main_quadprog.m: a main file for the experiment. Use quadratic programming solver to find a correlation device.

• main_team.m: a main file for the experiment. Use linear programming to solve the team problem.

• main_test.m: for test purpose. May be deleted in later versions.

• $v(1) = 0.2$, $v(2) = 1.2$.
• Consider $q = 0.1 : 0.01 : 0.95$.

• Set global parameters in script_env.m
• Run main_linprog.m, main_social.m or main_quadprog.m:
  • Use script_env.m for initialization.
  • Choose a random seed for reproducibility of the experiment results.
  • Find a correlation device for Stage 2. For all the possible $\pi_2$ values, find a $\psi_2 = \hat{\phi}^C_2[\pi_2]$:
    • Generate the coefficient matrix $A$ by A_gen.m
    • Generate an objective function for linear programming or quadratic programming.
    • Find a solution of the linear/quadratic programming, subject to the constraints given by $A$, and the fact that $\psi_t$ is a probability distribution.
    • Save the result in the matrix phi2.
  • Evaluate the value functions using V2i_gen.m
  • Find a correlation device for Stage 1.
    • All the steps are similar to those in Stage 2.
    • Note that the $\pi_1$ values are the first length(q_vals) rows in pi2_vals.
    • It is possible that for some $\pi_1$ the solution does not exist. Store those indices of failure in an array.

• Belief update: suppose we know $\pi^i_1$,
  • If $\gamma^i_1 = \gamma_{00}, \gamma_{11}$, then $\pi^i_2 = \pi^i_1$.
  • If $\gamma^i_1 = \gamma_{01}$ or $\gamma_{10}$, from $a^i_1$ one may infer the exact $x^i$, so $\pi^i_2 = 0$ or $1$.
• Adjust the order of gamma profile vector when players 1 and 2 switch:
  • For each $\gamma$, the $index-1$ is exactly the decimal form of the subscript of $\gamma^1, \gamma^2$. For example, $\gamma$-index=5 implies the subscripts of the profile is $4 = (0100)2$, so it corresponds to $(\gamma{01}, \gamma_{00})$.
  • To switch the order:
    • The binary form of $\gamma^1$ is fix(($\gamma$-index - 1) / 4), so the MATLAB index is gamma1_ind = fix((gamma_ind - 1) ./ 4) + 1;
    • The binary form of $\gamma^2$ is mod(($\gamma$-index - 1), 4), so the MATLAB index is gamma2_ind = mod(gamma_ind, 4) + 4 * (mod(gamma_ind, 4) == 0);
    • The switched $\gamma$-index is (gamma2_ind - 1) * 4 + gamma1_ind;
• Given the index of $\gamma^i \in {1,2,3,4}$ and $x^i \in {1,2}$, how do we calculate $a^i \in {1, 2}$?
  • The index of $\gamma^i$ minus 1 corresponds to its binary form 00, 01, 10, 11.
  • The first bit implies the action at $x^i=1$, and the second bit implies the action at $x^i=2$.
  • Notice that $3-x^i$'s binary form is 10 and 01.
  • Therefore, if we do bitand(gammai - 1, 3 - xi), then we will get 0 if "not contribute" is taken, get 2 or 1 if "contribute" is taken.