Percentage win Omaha vs Texas Holdem when omaha player doesn't have to use two cards from his hand.

compute.php

<?php

$games = 0;
$score = (object) ['o' => 0, 'or' => 0, 'th' => 0];

while (TRUE) {
	// build deck
	$cards = [];
	for ($i = 2; $i <= 14; $i++) {
		foreach (['c', 'd', 'h', 's'] as $c) {
			$cards[] = (object) ['card' => $i, 'color' => $c];
		}
	}

	//require 'tests.php'; die;

	// deal cards
	$keys = array_rand($cards, 2 + 4 + 5);
	$hand_th = [
		$cards[$keys[0]],
		$cards[$keys[1]],
	];
	$hand_o = [
		$cards[$keys[2]],
		$cards[$keys[3]],
		$cards[$keys[4]],
		$cards[$keys[5]],
	];
	$com = [
		$cards[$keys[6]],
		$cards[$keys[7]],
		$cards[$keys[8]],
		$cards[$keys[9]],
		$cards[$keys[10]],
	];

	// find winner
	$o = findOmahaValue($hand_o, $com);
	$or = findOmageRulesValue($hand_o, $com);
	$th = findTHValue($hand_th, $com);

	$games++;
	if ($o > $th) {
		$score->o++;
	} else {
		$score->th++;
	}

	if ($or > $th) {
		$score->or++;
	}

	/*
	echo "\r                              \r";
	var_dump($games, $score);
	if ($score->o > 0)
		echo "Omaha is winning in " . $score->o * 100 / $games .  "%";
	//*/
	if ($games % 1e4 === 0) {
		echo "\r                                                             \r";
		echo "Omaha is winning in " . round($score->o * 100 / $games, 3) .  "% [" . round($score->or * 100 / $games, 3) .  "%] (sample $games hands)";
	}
}




function findOmahaValue($hand, $com)
{
	return max([
		findTHValue([$hand[0], $hand[1]], $com),
		findTHValue([$hand[0], $hand[2]], $com),
		findTHValue([$hand[0], $hand[3]], $com),
		findTHValue([$hand[1], $hand[2]], $com),
		findTHValue([$hand[1], $hand[3]], $com),
		findTHValue([$hand[2], $hand[3]], $com),
	]);
}

function findOmageRulesValue($hand, $com)
{
	return max([
		_omahaRulesHelper((object) [$hand[0], $hand[1]], $com),
		_omahaRulesHelper((object) [$hand[0], $hand[2]], $com),
		_omahaRulesHelper((object) [$hand[0], $hand[3]], $com),
		_omahaRulesHelper((object) [$hand[1], $hand[2]], $com),
		_omahaRulesHelper((object) [$hand[1], $hand[3]], $com),
		_omahaRulesHelper((object) [$hand[2], $hand[3]], $com),
	]);
}

function _omahaRulesHelper($hand, $com)
{
	return max([
		findTHValue($hand, (object) [$com[0], $com[1], $com[2]]),
		findTHValue($hand, (object) [$com[0], $com[1], $com[3]]),
		findTHValue($hand, (object) [$com[0], $com[1], $com[4]]),
		findTHValue($hand, (object) [$com[0], $com[2], $com[3]]),
		findTHValue($hand, (object) [$com[0], $com[2], $com[4]]),
		findTHValue($hand, (object) [$com[0], $com[3], $com[4]]),
		findTHValue($hand, (object) [$com[1], $com[2], $com[3]]),
		findTHValue($hand, (object) [$com[1], $com[2], $com[4]]),
		findTHValue($hand, (object) [$com[1], $com[3], $com[4]]),
		findTHValue($hand, (object) [$com[2], $com[3], $com[4]]),
	]);
}

function findTHValue($hand, $com)
{
	$cards = [];
	$bulk = [];
	foreach ($hand as $c) {
		$bulk[] = $c->card;
		$cards[] = $c;
	}
	foreach ($com as $c) {
		$bulk[] = $c->card;
		$cards[] = $c;
	}

	$value = 0;

	// 1e16 | straight flush
	$colors = ['c' => 0, 'd' => 0, 'h' => 0, 's' => 0];
	foreach ($cards as $c) {
		$colors[$c->color]++;
	}
	$flush = [];
	foreach ($colors as $col => $v) {
		if ($v >= 5) {
			foreach ($cards as $c) {
				if ($c->color === $col) {
					$flush[] = $c->card;
				}
			}
		}
	}
	rsort($flush);
	if ($flush) {
		$last = NULL;
		$count = 1;
		$top = NULL;
		foreach ($flush as $c) {
			if ($last === $c + 1) { // iterating from top
				if ($count === 1) {
					$top = $last;
				}
				$count++;
			} else if ($count < 5) {
				$count = 1;
			}
			$last = $c;
		}
		if ($count >= 5) {
			return 1e16 * $top;
		}
	}


	// 1e14 | poker
	rsort($bulk);
	$last = NULL;
	$count = 1;
	foreach ($bulk as $c) {
		if ($c === $last) {
			$count++;
			if ($count === 4) {
				return 1e14 * $c;
			}
		} else {
			$count = 1;
		}
		$last = $c;
	}


	// 1e12 | full house
	$last = NULL;
	$last2 = NULL;
	$trips = NULL;
	$b = $bulk; // clone so we may remove cards if we find trips
	foreach ($bulk as $c) {
		if ($last2 === $c) { // highest trips
			$value += 1e6 * $c;
			unset($b[array_search($c, $b)]);
			unset($b[array_search($c, $b)]);
			unset($b[array_search($c, $b)]); // the card is thrice there
			rsort($b); // renew keys
			$trips = $c;
			break;

		} else if ($last === $c) {
			$last2 = $c;
		} else {
			$last = $c;
		}
	}
	// if trips found, we are still looking for a pair
	if ($trips) {
		$last = NULL;
		foreach ($b as $c) {
			if ($last === $c) { // we found highest pair
				return $trips * 1e12 + $c * 1e2;
			}
			$last = $c;
		}
	}
	$value = 0; // mandatory reset from fullhouse trips without additional pair


	// 1e10 | flush
	if ($flush) { // already computed in straight flush resolution
		for ($i = 0; $i < 5; $i++) {
			$value += 1e10 * $flush[$i];
		}
		return $value;
	}


	// 1e8 | straight
	$top = NULL;
	$last = NULL;
	$count = 1;

	$f = $bulk;
	if (in_array(14, $f)) {
		$f[] = 1; // add wheel
	}
	foreach ($f as $c) {
		if ($last === $c + 1) { // iterating from top
			if ($count === 1) {
				$top = $last;
			}
			$count++;
		} else if ($last === $c) {
			// continue

		} else if ($count < 5) {
			$count = 1;
			$top = NULL;
		}
		$last = $c;
	}
	if ($count >= 5) {
		return $top * 1e8;
	}


	// 1e6 | trips
	// already computed in full house resolution
	if ($trips) {
		$value = 1e6 * $trips;
		for ($i = 0; $i < 2; ++$i) {
			$value += $bulk[$i]; // value of the two other cards beside trips
		}
		return $value;
	}


	// 1e2, 1e4 | one or two pairs
	$last = NULL;
	$pair = [];
	foreach ($bulk as $c) {
		if ($c === $last) {
			$pair[] = $c;
			unset($bulk[array_search($c, $bulk)]);
			unset($bulk[array_search($c, $bulk)]); // the card is twice there
			if (count($pair) == 2)
				break; // do not count more than two pair
		}
		$last = $c;
	}
	if ($pair) {
		if (count($pair) === 2) {
			foreach ($pair as $p) {
				$value += 1e4 * $p;
			}
		} else {
			foreach ($pair as $p) {
				$value += 1e2 * $p;
			}
		}
		rsort($bulk); // renew keys
		for ($i = 0; $i < 5 - 2 * count($pair); ++$i) {
			$value += $bulk[$i]; // value of other cards beside pairs
		}
		return $value;
	}


	// 1 | board, possible kicker
	for ($i = 0; $i < 5; ++$i) {
		$value += $bulk[$i];
	}
	return $value;
}

tests.php

<?php

// board, kicker
hope($t1 = findTHValue([h('11h'), h('3h')], [h('5s'), h('6s'), h('7s'), h('9d'), h('10d')]), 11+10+9+7+6);
hope(findTHValue([h('2h'), h('3h')], [h('5s'), h('6s'), h('7s'), h('9d'), h('10d')]), 10+9+7+6+5);

// one pair
hope($t2 = findTHValue([h('2h'), h('2c')], [h('5s'), h('6s'), h('7s'), h('9d'), h('10d')]), 2*1e2 + 10+9+7);

// two pair
hope($t3 = findTHValue([h('2h'), h('2c')], [h('3s'), h('3d'), h('7s'), h('9d'), h('10d')]), 2*1e4 + 3*1e4 + 10);
hope(findTHValue([h('14h'), h('14c')], [h('13s'), h('13d'), h('12s'), h('12d'), h('11d')]), 14*1e4 + 13*1e4 + 12); // three pairs, count only top two

// trips
hope($t4 = findTHValue([h('2h'), h('2c')], [h('2s'), h('3d'), h('7s'), h('9d'), h('10d')]), 2*1e6 + 9+10);

// straight
hope($t5 = findTHValue([h('2h'), h('3c')], [h('4s'), h('5d'), h('6s'), h('10d'), h('11d')]), 6*1e8);
hope(findTHValue([h('2h'), h('3c')], [h('4s'), h('5d'), h('14s'), h('10d'), h('11d')]), 5*1e8); // wheel
hope(findTHValue([h('14h'), h('13c')], [h('12d'), h('11s'), h('10s'), h('14d'), h('13d')]), 14*1e8);

// flush
hope($t6 = findTHValue([h('2h'), h('3h')], [h('8h'), h('9h'), h('12h'), h('13d'), h('14d')]), 1e10*(2+3+8+9+12));

// full house
hope($t7 = findTHValue([h('2h'), h('2c')], [h('2h'), h('3c'), h('3h'), h('13d'), h('14d')]), 1e12 * 2 + 100 * 3);

// poker
hope($t8 = findTHValue([h('2h'), h('2c')], [h('2h'), h('2h'), h('12h'), h('13d'), h('14d')]), 1e14 * 2);

// straight flush
hope($t9 = findTHValue([h('6h'), h('7h')], [h('2c'), h('2c'), h('8h'), h('9h'), h('10h')]), 10 * 1e16);
hope(findTHValue([h('6h'), h('7h')], [h('2h'), h('2c'), h('8h'), h('9h'), h('10h')]), 10 * 1e16);

// various
hope(findTHValue([h('2c'), h('3c')], [h('4h'), h('5h'), h('6h'), h('13h'), h('14h')]), 1e10*(4+5+6+13+14)); // flush over straigh
hope(findTHValue([h('14c'), h('14d')], [h('14h'), h('13c'), h('13d'), h('13h'), h('12h')]), 14*1e12 + 13*100); // two trips
hope(findTHValue([h('10c'), h('14d')], [h('10h'), h('14c'), h('10d'), h('14h'), h('12h')]), 14*1e12 + 10*100); // ordering
hope(findTHValue([h('14c'), h('14d')], [h('8h'), h('7d'), h('6c'), h('2d'), h('2s')]), 14*1e4 + 2*1e4 + 8); // ordering


// compare hand values
hope( // pair is more than high card
	findTHValue([h('2h'), h('2c')], [h('3c'), h('4s'), h('6s'), h('8d'), h('10d')]) > 
	findTHValue([h('14h'), h('13c')], [h('2c'), h('4s'), h('6s'), h('8d'), h('10d')])
, TRUE);
hope( // two pair are more than one pair
	findTHValue([h('2h'), h('2d')], [h('3c'), h('3s'), h('8h'), h('9h'), h('10h')]) >
	findTHValue([h('14h'), h('14c')], [h('2c'), h('4s'), h('6s'), h('8d'), h('10d')])
, TRUE);
hope( // trips over two pair
	findTHValue([h('2h'), h('2d')], [h('2c'), h('3s'), h('8h'), h('9h'), h('10h')]) >
	findTHValue([h('14h'), h('14c')], [h('13c'), h('13s'), h('6s'), h('8d'), h('10d')])
, TRUE);
hope( // trips kicker
	findTHValue([h('2h'), h('2d')], [h('2c'), h('14s'), h('13h'), h('8h'), h('6h')]) >
	findTHValue([h('2h'), h('2d')], [h('2c'), h('14s'), h('12h'), h('8h'), h('6h')])
, TRUE);
hope( // straight over trips
	findTHValue([h('2h'), h('3d')], [h('4c'), h('5s'), h('6h'), h('9h'), h('10h')]) >
	findTHValue([h('14h'), h('14c')], [h('14d'), h('13s'), h('6s'), h('8d'), h('10d')])
, TRUE);
hope( // ace straight over wheel
	findTHValue([h('14h'), h('13c')], [h('12d'), h('11s'), h('10s'), h('8d'), h('7d')]) >
	findTHValue([h('14h'), h('2d')], [h('3c'), h('4s'), h('5h'), h('9h'), h('10h')])
, TRUE);
hope( // same straights
	findTHValue([h('14h'), h('13c')], [h('12d'), h('11s'), h('10s'), h('14d'), h('13d')]) ===
	findTHValue([h('14h'), h('13c')], [h('12d'), h('11s'), h('10s'), h('2d'), h('2s')])
, TRUE);
// @TODO add flush
hope( // full house over straight
	findTHValue([h('2h'), h('2c')], [h('2d'), h('3s'), h('3d'), h('4d'), h('6h')]) >
	findTHValue([h('14h'), h('13c')], [h('12d'), h('11s'), h('10s'), h('2d'), h('2s')])
, TRUE);
hope( // full houses
	findTHValue([h('2h'), h('2c')], [h('3d'), h('3s'), h('3d'), h('4d'), h('6h')]) >
	findTHValue([h('2h'), h('2c')], [h('2d'), h('3s'), h('3d'), h('4d'), h('6h')])
, TRUE);
hope( // poker over full house
	findTHValue([h('2h'), h('2c')], [h('2d'), h('2s'), h('3d'), h('4d'), h('6h')]) >
	findTHValue([h('2h'), h('2c')], [h('2d'), h('3s'), h('3d'), h('4d'), h('6h')])
, TRUE);
hope( // straight flush over poker
	findTHValue([h('2h'), h('3h')], [h('4h'), h('5h'), h('6h'), h('8s'), h('9c')]) >
	findTHValue([h('14h'), h('14c')], [h('14d'), h('14s'), h('13d'), h('4d'), h('6h')])
, TRUE);

hope($t1 < $t2, TRUE);
hope($t2 < $t3, TRUE);
hope($t3 < $t4, TRUE);
hope($t4 < $t5, TRUE);
hope($t5 < $t6, TRUE);
hope($t6 < $t7, TRUE);
hope($t7 < $t8, TRUE);
hope($t8 < $t9, TRUE);

echo "\n";

$test = 0;
function hope($v, $expected) {
	global $test;
	$test++;
	if ($v !== $expected) {
		echo "FAIL test #$test: $v (expected $expected)\n";
	} else {
		echo ".";
	}
}

function h($s) {
	$m = [];
	preg_match('~^(?P<v>\d+)(?P<c>.)$~', $s, $m);
	return (object) ['card' => (int) $m['v'], 'color' => $m['c']];
}

Omaha is winning in 68.929% (sample of 240000 hands) at revision 432222388d4b12f93007626abd8a60adeb6d408f