## Perfect Poker Coin Tosses

12 Oct 2011

We often speak of coin tosses in poker, where two players are all in with hands that have a roughly equal chance of winning. A classic example is AKs v QQ, here the queens actually have a slight edge against the over-cards, with almost 54% equity.

I was wondering if there are any *perfect* coin tosses in Hold'em, where
the players are *exactly* 50-50. Trivial examples exist when both players
have the same hand, such as AA v AA
but these consist mostly of tied outcomes, and are not particularly interesting,
what of the non-trivial solutions?

Well, after running an analysis on all possible pre-flop match-ups it turns out there aren't any, the closest is As10s v 3s3c (or any equivalent suit permutation) with 49.99% v 50.01% equity, close but no cigar.

### Coin tosses after the flop

It's not very surprising that there are no perfect pre-flop coin tosses, as
there are *48 choose 5*
possible flop/turn/river combos, that's over 1.7 million outcomes that need to
be perfectly divided into two groups (barring ties), highly unlikely. After the
flop however there are only *45 choose 2* = 990 possible turn/river combos,
much more likely that there are hands that can divide this perfectly, and in fact
there are:

On the flop: 5s3h v 9h2d, Flop: 4d 2s Kh

On the turn: Qc8s v 2d2s, Flop: 5s 5c Js, Turn: 9s

3 players on the flop: 3sAs v 4dQc v 10c8d, Flop: 6c 3c 7h

3 players on the turn: 5h6d v Qh10c v 2c4h, Flop: 4d 7c Jh, Turn: Kd

4 players on the turn: 2d2s v AhKs v 5s4s v 6c7h, Flop: Jd 8h Js, Turn: 9s

It's neat that none of these involve any ties in order to achieve the equal
equities, it turns out that won't be possible on the flop or turn for five players,
that is because there are *39 choose 2* = 741 possible turn/river combos on
the flop, and 38 possible rivers on the turn, neither of these numbers are multiples
of five. For six, seven and eight players these numbers do divide evenly, though
I'd be quite surprised if coin tosses did exist for any of these higher number
of players (then again the four player one surprised me, so who knows?).

My search algorithm that found the hands above is naive, dealing completely random hands until the criteria is met, if you have any clever ideas on how to improve the search, or if you can manually construct some other coin toss hands let me know.

Update! I was able to construct a rather contrived 5 way example on the turn where a tie occurs only ~21% of the time, and a similar 8 way example on the turn where a tie occurs 25% of the time.