Bayesian forgiver strategy

[hodgesmr]

Note: This strategy was ideated and developed with the assistance of Claude Code. I did not see an LLM policy in the repository, but I felt it was important to note that, given open source sensitivities to those tools. I did manual review, edits, and iterations on the implementation and testing.

Summary

This PR introduces BayesianForgiver, a strategy that uses Bayesian inference to model opponent behavior and make adaptive forgiveness decisions based on uncertainty.

Motivation

While the Axelrod library contains many strategies, none use Bayesian opponent modeling to inform decision-making. Existing forgiveness strategies use fixed thresholds or probabilities, regardless of confidence in their assessment of the opponent.

BayesianForgiver introduces uncertainty-aware decision making to the IPD strategy landscape.

Key Features

BayesianForgiver maintains a Beta distribution representing beliefs about the opponent's cooperation probability. It uses both the mean (expected cooperation rate) and variance (uncertainty) to make decisions:

• More cautious when uncertain about the opponent (requires stronger evidence of cooperation to forgive a defection)
• More punishing when certain the opponent is hostile (confident exploitation)
• Adaptive threshold that adjusts forgiveness based on confidence in opponent model

This approach provides noise resilience and adaptive behavior compared to fixed-threshold strategies.

Algorithm

1. Maintain a Beta(α, β) distribution for opponent's cooperation probability
2. Start with Beta(1, 1) - neutral/uniform prior
3. Update after each round:
   • If opponent cooperates: α += 1
   • If opponent defects: β += 1
4. Calculate statistics:
   • Mean cooperation probability: μ = α / (α + β)
   • Uncertainty (std dev): σ = √(αβ / ((α+β)²(α+β+1)))
5. Adaptive forgiveness threshold:
   • threshold = base_threshold + uncertainty_factor × σ
6. Decision:
   • If opponent cooperated last round: Cooperate (reciprocate)
   • If opponent defected last round:
     • If μ ≥ threshold: Cooperate (forgive)
     • Otherwise: Defect (punish)

Implementation Details

• Computational complexity: O(1) per round (constant time Beta updates)
• Memory depth: Infinite (uses full history via Bayesian updates)
• Stochastic: No (deterministic given opponent history)
• Parameters (optimized via grid search over 144 configurations):

prior_alpha=1.0 prior_beta=1.0 base_forgiveness_threshold=0.45 # Base component of forgiveness threshold uncertainty_factor=2.5 # Increases caution under uncertainty (raises threshold early)

## Testing

All tests pass:
```bash
python -m pytest axelrod/tests/strategies/test_bayesian_forgiver.py
# 32 passed in 4.35s

Tests cover:

• Initial strategy behavior
• Behavior against classic strategies (TFT, Grudger, Cooperator, Defector, etc.)
• Response to various opponent patterns (alternating, mixed, single defection)
• Parameter customization
• Clone and reset functionality

[drvinceknight]

Thanks for this.

The CI is failing because line 140 of this file is not hit by a test.

 axelrod/strategies/bayesian_forgiver.py                             34      1    97%   140

[hodgesmr]

Thanks @drvinceknight - added the necessary test and also ran the required black formatter.

[hodgesmr]

I've decided to close this PR, especially as #1479 is being considered by the maintainers. I want to make sure contributions are accurate and productive, and I don't want this to distract or interfere with that. Thank you @drvinceknight and the other maintainers for the thoughtful time spent on this.