Clarification on my network public good model, current simulation approach, and whether I should use a full best-response iteration
Emine Özge Yurdakurban
Dear colleagues,
I hope this message finds you well. I am writing to seek your advice on my network public good game model and the method I have used so far to compute what I am calling the “Nash equilibrium,” followed by the simulation results. I then learned about a second approach (Best-Response Iteration or Fixed-Point) and would like to know whether I should switch to that method or if my current approach is acceptable.
1. My Model and How I Currently Obtain a “Nash” Contribution
Setup:
- I have a set of players (nodes) in a network. Each player has a (possibly) different altruism parameter , income , and a local neighborhood of size .
- The utility function for player is something like: where is player ’s contribution, is a convex cost exponent, and is a public good multiplier.
My Current Method: “Local Symmetric” Formula
- In the purely symmetric version of this game (all and each ), I know the standard closed-form Nash solution:
- Because my players differ in and , I adapted that formula by substituting for and for . This yields, for each player :
- I then compute all in one shot (one round), effectively giving me a set of contributions .
Simulation
- With those one‐step values, I run a simulation across different random network structures (e.g., Erdős–Rényi) and random .
- I observe each node’s contribution , sum them up, compare it to some “social optimum” measure, and gather the numerical results.
Questionable “Nash”
- In my code, I have referred to from the above local formula as my “Nash equilibrium.”
- However, I realize that because the formula is directly derived from a symmetric game assumption, then just replaced with local and , it might not be the true equilibrium for a fully heterogeneous network.
- This approach might be more of a heuristic (approximation), rather than a rigorously computed Nash equilibrium.
2. The Second Method: Best-Response (BR) Iteration or Fixed-Point Approach
- After looking into more references (e.g., Bramoullé & Kranton, 2007; Galeotti et al., 2010; Mas-Colell et al., 1995), I see that a standard way to handle heterogeneity is:
- Write down each player’s best-response function: .
- Iterate over all players, updating by numerically solving given the latest contributions of others.
- Continue until convergence, yielding a that should be the genuine Nash equilibrium.
- I understand conceptually how that works, but I have not yet implemented it in my code. I also see it can be more computationally intensive because each iteration might involve a root-finding procedure for each node.
3. My Questions to You
- Is my current approach—using the “symmetry-based closed-form” formula but with local and —acceptable as an approximate or heuristic solution in an academic setting?
- Should I instead implement the Best-Response Iteration / Fixed-Point method to ensure I find the real heterogeneous Nash equilibrium?
- Would it be standard practice to report the heuristic as a rough baseline but then highlight that the true equilibrium would require the iterative approach?
I am somewhat concerned that I’m labeling the outcome of my direct formula as “Nash,” while it might only be correct if and are actually uniform across nodes (which they are not in my simulation). Consequently, I’d truly appreciate your advice on how best to present or justify my current method, or whether you recommend transitioning to the more rigorous BR iteration approach.
Thank you very much for your time and guidance. I look forward to your insights!
Best regards.
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Emine Özge Yurdakurban
Ph.D. Candidate, Economics, ITU
Research Assistant, Economics, YU
Istanbul, Turkey