Research map / Multi-agent systems and games / Games and equilibria
Bounded Rationality Equilibrium Learning in Mean Field Games
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Introduces quantal-response (QRE) and receding-horizon (RH) equilibria for discrete-time mean field games and gives generalized fixed-point iteration and fictitious play algorithms to learn them, evaluated on SIS, RPS and random MFGs.
Problem
Nash equilibria in mean field games assume perfectly rational agents, which is unrealistic; learning bounded-rationality MFG equilibria in discrete time is the target.
Approach
Combine quantal response equilibria (noisy value estimates, Gumbel noise giving softmax policies) and receding-horizon planning with MFGs, define QPiRE, Q*RE and RH variants, relate them to entropy-regularized equilibria, and generalize FPI and fictitious play to learn them.
Key results
- GFP converges quickly to QPiRE, Q*RE and RE on the SIS MFG with alpha=1.0 and beta=0.95, while exploitability stays nonzero, showing bounded rationality.
- On RPS, as alpha goes to infinity all concepts give the uniform policy, and as alpha goes to 0 they converge to the Nash solution; intermediate temperatures give distinct solutions.
- RH QRE with longer lookahead H are closer to the full-horizon QPiRE; parallel RH-GFP needs roughly 1000 iterations total versus about 1000 per MFG sequentially.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use GFP to learn QπRE, Q*RE, or RE instead of NE | SIS MFG · SIS MFG | ΔQπRE, ΔQ*RE, or ΔRE | Helped |
| Use GFP to learn QπRE, Q*RE, or RE instead of NE | SIS MFG · SIS MFG | ΔJ exploitability | Hurt |
| Use GFPI for regularized equilibria instead of GFP | Random MFG and SIS MFG · Random MFG and SIS MFG | distance to equilibrium | Helped |
| Use GFPI for regularized equilibria instead of GFP | Sequential RPS MFG · Sequential RPS MFG | convergence | Hurt |
| Use GFP instead of GFPI | Random MFG, SIS MFG, and sequential RPS MFG · Random MFG, SIS MFG, and sequential RPS MFG | convergence | Helped |
| Use GFP instead of GFPI when GFPI converges | Random MFG and SIS MFG · Random MFG and SIS MFG | convergence speed | Hurt |
| Parallelize RH-GFP across the receding-horizon MFGs instead of solving them sequentially | RH QπRE · RPS MFG, total horizon 10 and receding horizon 5 | iterations to compute all MFGs | Helped |
| Increase the receding horizon (ablation) | RH QπRE · Random MFG | distance to full-horizon QπRE | Helped |
| Shorten the receding horizon (ablation) | RH QπRE · Random MFG | distance to full-horizon QπRE | Hurt |
| Use intermediate temperature rather than the low- or high-temperature limits (ablation) | QπRE, Q*RE, and RE · Sequential RPS MFG | difference between equilibrium policies | Helped |
Surprising findings
- GFPI fails to converge on the sequential RPS MFG, so GFP is needed for generality.
- QPiRE, Q*RE and RE are distinct at intermediate temperatures despite coinciding in the low- and high-temperature limits.
Limitations
- Convergence proof for GFP covers only RE under monotonicity and MF-independent transitions; QRE and RH variants are left to future work.
- GFPI requires sufficiently high temperature and does not converge on the RPS game.
- Experiments are on small tabular toy MFGs.
Experimental setup
Datasets: SIS MFG modified rock-paper-scissors MFG random MFG
Benchmarks: SIS MFG sequential RPS MFG random MFG
Metrics: exploitability distance to equilibrium (Delta QPiRE, Delta Q*RE, Delta RE)
Baselines: Nash equilibrium (MFNE) regularized equilibrium (RE) Boltzmann equilibrium (Q*RE)
Builds on: Nash equilibrium in MFGs quantal response equilibrium entropy-regularized MFG equilibria model predictive control fictitious play
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