Identity V Weapon Refining Cost Optimizer

English | 简体中文

A dynamic-programming-based stochastic optimization framework to derive cost-optimal locking strategies for the weapon-refining mechanics in Identity V, minimizing the expected number of feathers spent.

Game Mechanics Summary

• Slots: 5 slots total (1 color slot + 4 ordinary slots).
• Independent Re-rolls: Each wash re-rolls all unlocked slots independently.
• Dynamic Probabilities: The per-slot gold probability rises linearly with cumulative wash count $n$ from 1% ($n=0$) to a 10% cap ($n \ge 200$).
• Exponential Resource Scaling: Each wash costs $2^t$ feathers, where $t$ is the number of locked slots.
• Permanent Rollback: The game permits players to restore the slot state to immediately before any prior wash at zero cost, preserving the cumulative wash count.

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Summary of Results & Optimal Policies

Comparing the DP-optimal policies against the empirical baseline (the naive "lock everything you already hold" heuristic, evaluated via 5,000 Monte Carlo trials):

Target	Description	Expected Cost	Strategy	Feathers Saved
A	All 5 slots gold (any series)	388.06	Defer locking until $n \ge 150 \sim 177$; lock all held golds once threshold is reached.	33.4%
B	All 5 slots gold in 1 specific series	864.05	Lock color immediately ($c=1$); defer locking ordinary slots until $n \ge 169 \sim 183$.	51.5%
C	4 ordinary same series + 1 color any gold	627.63	Smart-route locks to current max ordinary series; never lock color slot alone ($t_c=0$ always); defer locking until $n \ge 175$.	43.2%

Key Policy Takeaways

1. Never Lock Early ($n < 150$): Because rollback keeps past progress safe for free, the exponential cost $2^t$ severely penalizes locking during early low-probability stages ($p < 7.7%$).
2. Target B Bottleneck: The color slot ($p_c = p/10$) is the extreme bottleneck; once rolled, it must be locked immediately from $n=0$, while ordinary locks should still be deferred.
3. Target C Color Degeneracy: Holding a color slot before completing the 4 ordinary same-series slots provides zero cost reduction ($V(1, m) = V(0, m)$ for $m < 4$). Hence, locking the color slot alone is strictly suboptimal.

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Solver Architecture

• Target A (1D State): Admits an upper-triangular state transition structure. Solved via closed-form back-substitution at the boundary and reverse DP across $n$.
• Target B (2D State): Bi-directional transitions break the triangular structure. Solved via boundary Value Iteration and reverse DP across $n$.
• Target C (4D State): High-dimensional state space ($2 \times 35 = 70$ states per wash count). Solved via exact Policy Iteration (linear system matrix inversion $(I - P)V = c$) at the boundary to eliminate extreme convergence iterations, followed by reverse DP.

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Quick Start

1. Run Dynamic Programming Solvers

Run any target script to compute the exact expected costs and policy tables:

python target_a.py
python target_b.py
python target_c.py

2. Monte Carlo Baseline Comparison

Run 5,000 simulation trials per target comparing naive play against optimal DP values:

python baseline.py

3. Policy Query Interface (CLI & Library)

Query the optimal locking action $(t_c, t_n)$ for any state in Target C:

# Example: c=1 (color held), na=2, nb=1, nc=0 at wash count n=150
python query.py -c 1 --na 2 --nb 1 --nc 0 -n 150

Programmatic usage in Python:

from query import query

# query(c, na, nb, nc, n) -> (t_c, t_n)
tc, tn = query(1, 0, 0, 0, 100)
print(f"Lock color: {tc}, Lock ordinary: {tn}")
# Output: Lock color: 0, Lock ordinary: 0 (defer locks at low n)

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Repository Layout

├── baseline.py      # Monte Carlo simulator for naive baseline policies
├── common.py        # Shared game constants and dynamic probability function p(n)
├── target_a.py      # Target A DP (closed-form back-substitution)
├── target_b.py      # Target B DP (value iteration)
├── target_c.py      # Target C DP (policy iteration + matrix inversion)
├── query.py         # O(1) policy lookup CLI and module interface
├── report.md        # Comprehensive technical report (mathematical derivations)
└── README_zh.md     # Chinese documentation

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Documentation

For full mathematical derivations, Bellman formulations, boundary condition proofs, and detailed value tables, refer to:

• Technical Report (English)
• 技术分析报告 (简体中文)

License

This project is licensed under the MIT License - see the LICENSE file for details.