import numpy as np
import sympy as sp
import matplotlib.pyplot as pltEx | Strategic Interactions
This exercise will revisit the tragedy dilemma, agreements, and threshold public goods but use slightly different modeling assumptions. Doing so will give us a more robust understanding of the strategic interactions in multi-agent action situations.
Step 1 | Tragedy Dilemma
Next to the model presented in the lecture, another common parametrization of the tragedy dilemma is the following: Actors can either cooperate or defect. Each cooperator contributes \(c > 0\) to the public good at an individual cost of \(c\). The sum of all contributions is multiplied by a synergy factor \(r\) and then equally distributed among all actors. The payoff functions are given by:
\[\begin{align} R_c &= \frac{r c (N_c +1)}{N} - c, \\ R_d &= \frac{r c N_c}{N}, \end{align}\]
with \(N_c\) being the number of other actors cooperating and \(N\) being the total number of actors.
Step 1.1 | Visualization
Plot the payoff functions for cooperators and defectors as a function of \(N_c\) for different values of \(r\). Compare the results to the tragedy dilemma model presented in the lecture.
# ...Step 1.2 | Conditions
Give the conditions the parameters must hold for this model to be a tragedy dilemma.
# ...Step 2 | Agreements
Apply the reasoning from the lecture to compute how the critical participation levels depend on the parameters of the model \(r\), \(c\), and \(N\).
# ...Step 3 | Threshold Public Goods
Let us now consider a variant of threshold public good, where the catastrophic impact \(m\) occurs probabilistically, and each polluting actor increases the probability of collapse. We assume that if all actors pollute, there is a probability \(q_c\) of collapse; if all actors cooperate, there is zero probability of collapse. The collapse probability increases linearly with the number of polluting actors \(N_p\), i.e., \[ p_c = q_c \frac{N_p}{N}.\]
Furthermore, we assume that if the collapse occurs, the actors won’t receive the payoffs from the public. The payoff functions are given by:
\[\begin{align} R_a(N_a; r, c, q_c, m, N) &= (1 - q_c N_\mathsf{p}/N) \cdot (rc(N_\mathsf{a}+1)/N - c) + q_c N_\mathsf{p}/N \cdot m, \\ R_p(N_a; r, c, q_c, m, N) &= (1 - q_c (N_\mathsf{p}+1)/N) \cdot (rcN_\mathsf{a})/N) + q_c (N_\mathsf{p}+1)/N \cdot m, \end{align}\]
where \(N_a\) is the number of other actors abating and \(N_p\) is the number of other actors polluting. Thus, it must hold that the total number of actors \(N = N_a + N_p + 1\). Furthermore, \(r\) is the synergy factor, \(c\) is the cost of cooperation, \(q_c\) is the probability of collapse if all actors pollute, and \(m\) is the catastrophic impact.
Step 3.1 | Visualization
Plot the payoff functions for abating and polluting actors as a function of \(N_a\) for \(f=4\), \(c=5\), \(m=-5\) \(qc=0.4\), and \(N=5\). Compare the results to the threshold public goods model presented in the lecture.
# ...Step 3.1 | Conditions
Calculate the three critical conditions for this game of
- Dilemma, i.e., the actors are indifferent between all abating and all polluting,
- Greed, i.e., the actors are indifferent between abating and polluting, give all others abate, and
- Fear, i.e., the actors are indifferent between abating and polluting, given all others pollute.
Solve the conditions for the collapse impact \(m\).
You may do this by hand or by using the sympy library. I recommend the latter.
# ...Visualize the critical conditions for the collapse impact \(m\) as a function of \(q_c\) for \(r=1.2\), \(c=5\), and \(N=2\). Interpret the results briefly.
# ...Visualize the critical conditions for the collapse impact \(m\) as a function of \(N \in [2, 3, \dots, 15]\) for \(r=3\), \(c=5\), and \(qc=0.5\). Interpret the results briefly.
# ...