{"query": "Zero-sum and minimax — the mixed value", "count": 20, "results": [{"id": "card_game_zero_sum", "title": "Zero-sum and minimax — the mixed value", "shelf": "codex", "surface": "secular", "snippet": "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 ", "authority_tier": "engine_derived", "source": "Narrow Highway — game theory", "readable": false, "generated": false}, {"id": "card_theory_fixed_point_theorems__brouwer__kakutani", "title": "Fixed-point theorems (Brouwer, Kakutani)", "shelf": "theories", "surface": "secular", "snippet": "Fixed-point theorems (Brouwer, Kakutani) — an engine domain that can touch it: mathematics. Calibration: map-only — an existence proof, not a computation. Brouwer: any continuous map of a convex compa", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_bridge_theory_quantum_mechanics__game_theory", "title": "Bridge: Quantum mechanics (Schrödinger equation)  ↔  Game theory (Nash equilibrium)", "shelf": "bridges", "surface": "secular", "snippet": "Quantum mechanics (Schrödinger equation) and Game theory (Nash equilibrium) are the same form in different domains. a MIXED STRATEGY and a QUANTUM MIXED STATE are the same object: a probability distri", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_bridge_theory_game_theory__linear_programming_duality", "title": "Bridge: Game theory (Nash equilibrium)  ↔  Linear programming & duality", "shelf": "bridges", "surface": "secular", "snippet": "Game theory (Nash equilibrium) and Linear programming & duality are the same form in different domains. von Neumann's minimax theorem for zero-sum games and LP strong duality are equivalent -- each is", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_floor_game_theory", "title": "Game theory — equilibrium, dilemmas, and strategy", "shelf": "codex", "surface": "secular", "snippet": "The mathematics of strategic choice. A Nash equilibrium is a fixed point where no one gains by deviating alone; the prisoner's dilemma shows rational players reaching a worse outcome than cooperation;", "authority_tier": "engine_derived", "source": "Narrow Highway — game theory", "readable": false, "generated": false}, {"id": "card_n_c0950f401038", "title": "Barnabas VI", "shelf": "patristics", "surface": "witness", "snippet": "announced by the prophets._ When, therefore, He has fulfilled the commandment, what saith He? “Who is he that will contend with me? let him oppose me: or who is he that will enter into judgment with m", "authority_tier": "father", "source": "Epistle of Barnabas (trans. Roberts-Donaldson, 1885)", "readable": false, "generated": false}, {"id": "card_n_e90bcaad8cd2", "title": "Amazing Grace", "shelf": "hymns", "surface": "witness", "snippet": "Amazing grace! How sweet the sound\nThat saved a wretch like me!\nI once was lost, but now am found,\nWas blind, but now I see.\n\n'Twas grace that taught my heart to fear,\nAnd grace my fears relieved;\nHow", "authority_tier": "external_aligned", "source": "John Newton (1779)", "readable": false, "generated": false}, {"id": "card_n_a81af80cfb59", "title": "Ignatius to the Ephesians XIII", "shelf": "patristics", "surface": "witness", "snippet": "Take heed, then, often to come together to give thanks to God, and show forth His praise. For when ye assemble frequently in the same place, the powers of Satan are destroyed, and the destruction at w", "authority_tier": "father", "source": "Ignatius to the Ephesians (shorter recension; trans. Roberts-Donaldson, 1885)", "readable": false, "generated": false}, {"id": "card_n_9fab9883e012", "title": "Martyrdom of Polycarp XVIII", "shelf": "patristics", "surface": "witness", "snippet": "The centurion then, seeing the strife excited by the Jews, placed the body in the midst of the fire, and consumed it. Accordingly, we afterwards took up his bones, as being more precious than the most", "authority_tier": "father", "source": "Martyrdom of Polycarp (trans. Roberts-Donaldson, 1885)", "readable": false, "generated": false}, {"id": "card_n_13870d7e3b21", "title": "Psalm 133 — Behold How Good and Pleasant", "shelf": "codex", "surface": "witness", "snippet": "<<A Song of Ascents. By David.>> See how good and how pleasant it is for brothers to live together in unity! It is like the precious oil on the head, that ran down on the beard, even Aaron's beard; th", "authority_tier": "scripture", "source": "Psalm 133", "readable": false, "generated": false}, {"id": "card_n_5d24a8a62b0b", "title": "Psalm 116 — I Love the LORD", "shelf": "codex", "surface": "witness", "snippet": "I love Yahweh, because he listens to my voice, and my cries for mercy. Because he has turned his ear to me, therefore I will call on him as long as I live. The cords of death surrounded me, the pains ", "authority_tier": "scripture", "source": "Psalm 116", "readable": false, "generated": false}, {"id": "card_n_231b2dc8d0b1", "title": "1689 LBCF ch. 17: Of the Perseverance of the Saints", "shelf": "codex", "surface": "witness", "snippet": "Those whom God has accepted in the Beloved, effectually called, and sanctified by his Spirit, and given the precious faith of his elect unto, can neither totally nor finally fall from the state of gra", "authority_tier": "creed", "source": "1689 London Baptist Confession of Faith", "readable": false, "generated": false}, {"id": "card_n_3d4cb70bba83", "title": "Heidelberg Q1", "shelf": "codex", "surface": "witness", "snippet": "What is your only comfort in life and in death?\n\nThat I, with body and soul, both in life and in death, am not my own, but belong to my faithful Saviour Jesus Christ, who with his precious blood has f", "authority_tier": "catechism", "source": "Heidelberg Catechism (1563)", "readable": false, "generated": false}, {"id": "card_n_19012e3ec760", "title": "Psalm 36 — Psalm 36", "shelf": "codex", "surface": "witness", "snippet": "<<For the Chief Musician. By David, the servant of Yahweh.>> An oracle is within my heart about the disobedience of the wicked: \"There is no fear of God before his eyes.\" For he flatters himself in hi", "authority_tier": "scripture", "source": "Psalm 36", "readable": false, "generated": false}, {"id": "card_water_uses", "title": "Matching water quality to the use", "shelf": "water", "surface": "secular", "snippet": "Treated drinking water is precious — don't spend it where lesser water will do. Reserve the cleanest, treated water for DRINKING, cooking, and medical use. Use untreated or lightly-treated water (rain", "authority_tier": "reference", "source": "FEMA / Ready.gov emergency water guidance (U.S. government, public domain)", "readable": false, "generated": false}, {"id": "card_food_water_mulch", "title": "Watering and mulching the garden", "shelf": "agriculture", "surface": "secular", "snippet": "Water DEEP and less often, not shallow and daily — deep watering drives roots down where they survive dry spells; shallow watering makes weak surface roots. Water in the cool morning to reduce loss an", "authority_tier": "reference", "source": "USDA Farmers' Bulletins & extension guidance (U.S. government, public domain)", "readable": false, "generated": false}, {"id": "card_n_05f7744be9ca", "title": "Martyrdom of Polycarp XV", "shelf": "patristics", "surface": "witness", "snippet": "When he had pronounced this _amen_, and so finished his prayer, those who were appointed for the purpose kindled the fire. And as the flame blazed forth in great fury, we, to whom it was given to witn", "authority_tier": "father", "source": "Martyrdom of Polycarp (trans. Roberts-Donaldson, 1885)", "readable": false, "generated": false}, {"id": "card_comm_adam_clarke_ezr_8_26", "title": "Adam Clarke on Ezra 8:26", "shelf": "commentary", "surface": "witness", "snippet": "Silver vessels a hundred talents - That is, The weight of all the silver vessels amounted to one hundred talents; not that there were one hundred vessels of silver, each a talent in weight.\nReckoning ", "authority_tier": "reference", "source": "Adam Clarke's Commentary on the Bible — Public Domain", "readable": false, "generated": false}, {"id": "card_comm_john_gill_job_28_16", "title": "John Gill on Job 28:16", "shelf": "commentary", "surface": "witness", "snippet": "It cannot be valued with the gold of Ophir,.... Which is often spoken of in Scripture as choice gold, if not the best; See Gill on Job 22:24; the sense is, that the gold of Ophir is not of the value o", "authority_tier": "reference", "source": "John Gill's Exposition of the Old and New Testaments — Public Domain", "readable": false, "generated": false}, {"id": "card_comm_john_gill_ezr_8_27", "title": "John Gill on Ezra 8:27", "shelf": "commentary", "surface": "witness", "snippet": "Also twenty basins of gold, of a thousand drams,.... Which were upwards of 1000 pounds of our money; for Bishop Cumberland says (e), the Persian \"daric\", \"drachma\", or \"drachm\", weighed twenty shillin", "authority_tier": "reference", "source": "John Gill's Exposition of the Old and New Testaments — Public Domain", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_game_zero_sum"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}