The same card, going first or going second
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Nothing here is tied to one game: any game that pays the side without the initiative in cards rather than in anything else works out the same way.
Results
| Chance the one going first holds it, in per cent | — |
| Chance the one going second holds it, in per cent | — |
| The gap between them, in points | — |
| Turns of head start that gap is worth | — |
| What the gap was on the very first turn, in points | — |
Turn by turn, both sides at once
| Turn | Cards seen going first | Chance going first, in per cent | Cards seen going second | Chance going second, in per cent | Gap, in points |
|---|
The last column is the whole of it, and the useful thing about it is that it shrinks. The head start is worth most on the very first turn and less every turn after, because the same extra card is a smaller and smaller share of everything already seen.
Which is why the honest way to describe it is not as a better chance but as an earlier one. Whatever the one going second has now, the one going first has a fixed number of turns later, and that number does not change as the game goes on even though the gap in points keeps falling.
What this does not weigh is the other half of the trade, because the side with the initiative gets to act first and that is worth something the card count cannot see. This page prices one half exactly, which is more than the argument usually gets.
How much is the extra card actually worth?
About four and a half points on the very first turn, with sixty cards and four copies of what you want, and less every turn after that.
By turn ten the same extra card is worth about two and a third points, and by turn thirty it is worth a third of a point. The card does not change; the pile it is added to does.
| Turn | Chance going first, in per cent | Chance going second, in per cent | Gap, in points |
|---|---|---|---|
| 0 | 39,9500 | 44,4820 | 4,5321 |
| 3 | 52,7721 | 56,5503 | 3,7782 |
| 6 | 63,4224 | 66,5354 | 3,1130 |
| 10 | 74,6921 | 77,0464 | 2,3542 |
So going second gives a better chance?
Not really, and this is the honest way to put it: it gives the same chance earlier. Whatever the second player holds now, the first player holds a fixed number of turns later.
That number is the extra cards divided by the cards drawn each turn, and unlike the gap in points it does not shrink as the game goes on.
Why does the gap shrink if the extra card never changes?
Because it is a share of everything already seen. One more card on top of eight matters more than one more card on top of thirty, even though it is the same card.
That is also why a head start is at its most valuable before anything has happened, which is the opposite of how people usually talk about card advantage.
Does this settle whether to go first?
No, because the side with the initiative acts first, and nothing on this page can see what that is worth. It prices the card half of the trade exactly and leaves the other half open.
What it does rule out is arguing about the size of the card advantage, which is a fixed and countable thing.
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