{
  "study_id": "gtogecko-run-it-twice-2026-09-03",
  "study_date": "2026-09-03",
  "source_access_date": "2026-09-03",
  "method": "Exact exhaustive enumeration; no simulation or random seed.",
  "model": {
    "game": "Heads-up Texas Hold'em showdown after all betting is complete",
    "settlement": "Two equal pot fractions, one attached to each board",
    "dealing": "Remaining visible board cards are sampled successively without replacement from one uniformly shuffled stub",
    "burn_card_note": "Unobserved burn cards are marginalized out; projecting a uniform shuffled stub onto visible board positions yields the same uniform disjoint runout distribution.",
    "share_values": "Each run awards Hero 0, 1/2, or 1 of that run's pot fraction.",
    "excluded": [
      "fees that change when the feature is used",
      "known dead or exposed cards not listed in the fixture",
      "action-conditioned folded-card ranges",
      "multiway pots and side-pot eligibility",
      "unequal board weights",
      "strategic action before the board is locked"
    ]
  },
  "sample_space": {
    "flop_single_visible_runouts": "C(45,2) = 990",
    "flop_labeled_disjoint_board_pairs": "C(45,2) * C(43,2) = 893970",
    "turn_single_visible_rivers": "44",
    "turn_labeled_distinct_river_pairs": "44 * 43 = 1892",
    "total_paired_runouts_across_fixtures": 1789832
  },
  "sources": [
    {
      "title": "Variance reduction in Texas hold'em and in video poker",
      "url": "https://arxiv.org/html/2409.03607v2",
      "accessed": "2026-09-03"
    },
    {
      "title": "PokerStars: What is Run It Twice?",
      "url": "https://www.pokerstars.com/help/articles/run-it-twice-new/15582/",
      "accessed": "2026-09-03"
    }
  ],
  "results": [
    {
      "scenario_id": "turn-fifteen-out-draw",
      "label": "Turn: 15-out draw",
      "stage": "turn",
      "hero": [
        "Ah",
        "Kh"
      ],
      "villain": [
        "Qs",
        "Qc"
      ],
      "board": [
        "2h",
        "7h",
        "9c",
        "Jd"
      ],
      "stub_cards": 44,
      "cards_to_come_per_board": 1,
      "single_runout_count": 44,
      "paired_runout_count": 1892,
      "hero_equity_run_once": {
        "fraction": "15/44",
        "decimal": "0.340909090909"
      },
      "hero_equity_run_twice": {
        "fraction": "15/44",
        "decimal": "0.340909090909"
      },
      "run_once_variance": {
        "fraction": "435/1936",
        "decimal": "0.224690082645"
      },
      "run_twice_variance": {
        "fraction": "9135/83248",
        "decimal": "0.109732365943"
      },
      "independent_two_board_variance": {
        "fraction": "435/3872",
        "decimal": "0.112345041322"
      },
      "same_stub_covariance": {
        "fraction": "-435/83248",
        "decimal": "-0.005225350759"
      },
      "variance_retained": {
        "fraction": "21/43",
        "decimal": "0.488372093023"
      },
      "standard_deviation_retained": "0.698836241922",
      "single_run_distribution": {
        "losses": 29,
        "ties": 0,
        "wins": 15
      },
      "run_twice_outcomes": {
        "zero": {
          "pot_share": "0/4",
          "count": 812,
          "probability": {
            "fraction": "203/473",
            "decimal": "0.429175475687"
          }
        },
        "quarter": {
          "pot_share": "1/4",
          "count": 0,
          "probability": {
            "fraction": "0/1",
            "decimal": "0.000000000000"
          }
        },
        "half": {
          "pot_share": "2/4",
          "count": 870,
          "probability": {
            "fraction": "435/946",
            "decimal": "0.459830866808"
          }
        },
        "three_quarters": {
          "pot_share": "3/4",
          "count": 0,
          "probability": {
            "fraction": "0/1",
            "decimal": "0.000000000000"
          }
        },
        "scoop": {
          "pot_share": "4/4",
          "count": 210,
          "probability": {
            "fraction": "105/946",
            "decimal": "0.110993657505"
          }
        }
      }
    },
    {
      "scenario_id": "flop-combo-draw",
      "label": "Flop: combo draw",
      "stage": "flop",
      "hero": [
        "Ah",
        "Kh"
      ],
      "villain": [
        "Qs",
        "Qc"
      ],
      "board": [
        "2h",
        "7h",
        "9c"
      ],
      "stub_cards": 45,
      "cards_to_come_per_board": 2,
      "single_runout_count": 990,
      "paired_runout_count": 893970,
      "hero_equity_run_once": {
        "fraction": "268/495",
        "decimal": "0.541414141414"
      },
      "hero_equity_run_twice": {
        "fraction": "268/495",
        "decimal": "0.541414141414"
      },
      "run_once_variance": {
        "fraction": "60836/245025",
        "decimal": "0.248284868891"
      },
      "run_twice_variance": {
        "fraction": "2522761/21072150",
        "decimal": "0.119720151954"
      },
      "independent_two_board_variance": {
        "fraction": "30418/245025",
        "decimal": "0.124142434445"
      },
      "same_stub_covariance": {
        "fraction": "-93187/10536075",
        "decimal": "-0.008844564983"
      },
      "variance_retained": {
        "fraction": "2522761/5231896",
        "decimal": "0.482188675004"
      },
      "standard_deviation_retained": "0.694398066677",
      "single_run_distribution": {
        "losses": 454,
        "ties": 0,
        "wins": 536
      },
      "run_twice_outcomes": {
        "zero": {
          "pot_share": "0/4",
          "count": 180096,
          "probability": {
            "fraction": "4288/21285",
            "decimal": "0.201456424712"
          }
        },
        "quarter": {
          "pot_share": "1/4",
          "count": 0,
          "probability": {
            "fraction": "0/1",
            "decimal": "0.000000000000"
          }
        },
        "half": {
          "pot_share": "2/4",
          "count": 459732,
          "probability": {
            "fraction": "10946/21285",
            "decimal": "0.514258867747"
          }
        },
        "three_quarters": {
          "pot_share": "3/4",
          "count": 0,
          "probability": {
            "fraction": "0/1",
            "decimal": "0.000000000000"
          }
        },
        "scoop": {
          "pot_share": "4/4",
          "count": 254142,
          "probability": {
            "fraction": "2017/7095",
            "decimal": "0.284284707541"
          }
        }
      }
    },
    {
      "scenario_id": "flop-set-vs-overpair",
      "label": "Flop: set vs overpair",
      "stage": "flop",
      "hero": [
        "7s",
        "7d"
      ],
      "villain": [
        "Ah",
        "Ad"
      ],
      "board": [
        "7c",
        "8c",
        "9h"
      ],
      "stub_cards": 45,
      "cards_to_come_per_board": 2,
      "single_runout_count": 990,
      "paired_runout_count": 893970,
      "hero_equity_run_once": {
        "fraction": "175/198",
        "decimal": "0.883838383838"
      },
      "hero_equity_run_twice": {
        "fraction": "175/198",
        "decimal": "0.883838383838"
      },
      "run_once_variance": {
        "fraction": "17749/196020",
        "decimal": "0.090546882971"
      },
      "run_twice_variance": {
        "fraction": "1285703/29501010",
        "decimal": "0.043581660425"
      },
      "independent_two_board_variance": {
        "fraction": "17749/392040",
        "decimal": "0.045273441486"
      },
      "same_stub_covariance": {
        "fraction": "-199637/59002020",
        "decimal": "-0.003383562122"
      },
      "variance_retained": {
        "fraction": "2571406/5342449",
        "decimal": "0.481315965768"
      },
      "standard_deviation_retained": "0.693769389472",
      "single_run_distribution": {
        "losses": 91,
        "ties": 48,
        "wins": 851
      },
      "run_twice_outcomes": {
        "zero": {
          "pot_share": "0/4",
          "count": 4434,
          "probability": {
            "fraction": "739/148995",
            "decimal": "0.004959897983"
          }
        },
        "quarter": {
          "pot_share": "1/4",
          "count": 8352,
          "probability": {
            "fraction": "464/49665",
            "decimal": "0.009342595389"
          }
        },
        "half": {
          "pot_share": "2/4",
          "count": 148838,
          "probability": {
            "fraction": "74419/446985",
            "decimal": "0.166491045561"
          }
        },
        "three_quarters": {
          "pot_share": "3/4",
          "count": 74912,
          "probability": {
            "fraction": "37456/446985",
            "decimal": "0.083796995425"
          }
        },
        "scoop": {
          "pot_share": "4/4",
          "count": 657434,
          "probability": {
            "fraction": "328717/446985",
            "decimal": "0.735409465642"
          }
        }
      }
    }
  ]
}