{"id": "card_game_zero_sum", "kind": "reference", "title": "Zero-sum and minimax — the mixed value", "body": "When no pure choice wins (matching pennies), players randomize, and the game still has a value (0 for the fair case, sealed) - von Neumann's minimax theorem. Many moves, one value. Rests on the solve path.", "source": {"label": "Narrow Highway — game theory", "url": "", "authority_tier": "engine_derived"}, "shelf": "codex", "box": "game_theory", "bands": ["game theory", "pillar", "zero_sum", "one map"], "subject": "Zero-sum and minimax — the mixed value", "connections": [{"to_card_id": "card_floor_game_theory", "relationship": "part_of", "evidence": "a pillar of game theory (Zero-sum and minimax)"}, {"to_card_id": "card_floor_the_solve_path", "relationship": "connects_at", "evidence": "the minimax value is an optimization - converge on the saddle point"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.game.theory", "facets": {"subject": ["game theory", "one map", "pillar", "zero_sum"]}, "presentation": {"glyph": "•", "kind_label": "game theory", "by": "Narrow Highway — game theory", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_floor_game_theory", "title": "Game theory — equilibrium, dilemmas, and strategy", "relationship": "part of", "why": "a pillar of game theory (Zero-sum and minimax)", "href": "/card/card_floor_game_theory", "resolved": true}, {"id": "card_floor_the_solve_path", "title": "The solve path — two jaws that close on the answer", "relationship": "connects at", "why": "the minimax value is an optimization - converge on the saddle point", "href": "/card/card_floor_the_solve_path", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_game_zero_sum", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_game_zero_sum"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}