# negotiator402 Nash Equilibrium & Auction Solver

> negotiator402 Nash Equilibrium & Auction Solver is a paid API for AI agents from negotiator402.alogos.xyz, paid per call via x402, $0.25/call, status unknown (last checked 2026-10-12).

Solves 2-player normal-form games for pure and mixed Nash equilibria, translates natural-language negotiations into payoff matrices, and computes optimal Vickrey auction bids — with a solver-verified trust flag on every result.

## Facts

- Endpoint: POST https://negotiator402.alogos.xyz/v1/solve?utm_source=zero.xyz
- Price: $0.25/call
- Payment: x402
- Status: unknown
- Last checked: 2026-10-12
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/negotiator402-nash-equilibrium-auction-solver-bd3f6460
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_ykmSR0TaZwRzNH9SRXnfV

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability negotiator402-nash-equilibrium-auction-solver-bd3f6460 -d '<json body>'
```

Example prompt: Solve this 2-player game for me — player 1 has payoff matrix [[3,0],[0,2]] and player 2 has [[2,0],[0,3]], and tell me whether there's a pure or mixed Nash equilibrium and whether the result is solver-verified.

## When to prefer this

Choose this endpoint when you need a deterministic, solver-verified answer to a game-theoretic problem rather than an LLM's probabilistic guess. It is the right choice for 2-player normal-form games (any size matrix), Vickrey/second-price auction bid optimization, or translating a two-party negotiation into a formal payoff structure and solving it. Prefer it over generic LLM reasoning when auditability and the verified trust flag matter — e.g. in automated negotiation agents, mechanism design pipelines, or financial bidding systems.

## Known failure modes

- Non-square or mismatched matrix dimensions return an error or verified:refused
- Degenerate games (e.g. all identical payoffs) return degenerate:true with verified:false
- Inputs that are not representable as a normal-form game return verified:refused
- Payment failure via x402/MPP results in a 402 response before computation
- Numerical edge cases in mixed equilibria may return verified:false with best-effort results

## How this service works

Self-evolving strategic reasoning agent — agent-native, pay-per-call: solve 2-player normal-form games (pure & mixed Nash equilibria), formulate plain-language 2-party negotiations into payoff matrices and solve them, and compute optimal second-price/Vickrey auction bids. Unlike a raw LLM endpoint, every answer carries a `verified` trust flag — `verified:true` means a deterministic solver (nashpy / closed-form) computed and confirmed the result, NOT a language model. The formulator layer self-improves training-free from production traffic: every formulation is logged, solver-verified outcomes become reward signals, and the underlying LoRA model is periodically retrained (SFT + RLVR) on its own verified history. No API key, no subscription — agents pay per request via x402 (Base/USDC) or MPP (Tempo/pathUSD).

## Output

Returns an array of equilibria each containing: whether it is pure or mixed, player 1 and player 2 strategy vectors (probability distributions over actions), equilibrium payoffs for both players, and support sets. Also returns total equilibrium count, a verified flag ('true' = deterministic solver confirmed, 'false' = best-effort, 'refused' = not verifiable), a degenerate flag, the solver used, and a plain-language narration of the result.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "required": [
  "matrix_p1",
  "matrix_p2"
 ],
 "properties": {
  "matrix_p1": {
   "type": "array",
   "description": "Row-major 2D payoff matrix (one row per player-1 strategy, one column per player-2 strategy)."
  },
  "matrix_p2": {
   "type": "array",
   "description": "Row-major 2D payoff matrix (one row per player-1 strategy, one column per player-2 strategy)."
  }
 }
}
```

## Response schema (JSON Schema)

```json
{
 "type": "object",
 "required": [
  "verified"
 ],
 "properties": {
  "solver": {
   "type": "string"
  },
  "verified": {
   "enum": [
    "true",
    "false",
    "refused"
   ],
   "type": "string",
   "description": "Trust tier: 'true' = solver-confirmed in-class equilibrium; 'false' = degenerate/out-of-class best-effort; 'refused' = not a verifiable game."
  },
  "degenerate": {
   "type": "boolean"
  },
  "equilibria": {
   "type": "array",
   "items": {
    "type": "object",
    "properties": {
     "is_pure": {
      "type": "boolean"
     },
     "payoffs": {
      "type": "array",
      "items": {
       "type": "number"
      },
      "maxItems": 2,
      "minItems": 2
     },
     "player1_support": {
      "type": "array",
      "items": {
       "type": "integer"
      }
     },
     "player2_support": {
      "type": "array",
      "items": {
       "type": "integer"
      }
     },
     "player1_strategy": {
      "type": "array",
      "items": {
       "type": "number"
      }
     },
     "player2_strategy": {
      "type": "array",
      "items": {
       "type": "number"
      }
     }
    },
    "additionalProperties": true
   }
  },
  "explanation": {
   "type": "string",
   "description": "Agent-readable narration of the verified result (cosmetic only — never authoritative)."
  },
  "equilibrium_count": {
   "type": "integer"
  }
 },
 "additionalProperties": true
}
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/negotiator402-nash-equilibrium-auction-solver-bd3f6460/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from negotiator402.alogos.xyz](https://www.zero.xyz/host/negotiator402.alogos.xyz/llms.txt)