Approximating Multiple Opponents

[bigchips]

Yes that explanation makes sense with regards to the ties. I hadn't noticed that so that was probably a poor example by me.

<font class="small">Code:</font><hr /><pre>
equity (%) win (%) / tie (%)

Hand 1: 33.1808 % [ 00.01 00.32 ] { 9c8d }
Hand 2: 32.7686 % [ 00.01 00.32 ] { 9d8h }
Hand 3: 34.0506 % [ 00.02 00.32 ] { 9h8s }
</pre><hr />

And as a matter of practicality, I'm thinking more in terms of ranges of hands. Let's say you have 3 players. Players 1 and 2 only play pairs AA down to 99. Player 3 only plays suited connecters AKs to T9s.

Player 1 vs. Player 2 is a 50/50 proposition.

Player 1 or Player 2 vs. Player 3 is as follows:

<font class="small">Code:</font><hr /><pre> equity (%) win (%) / tie (%)

Hand 1: 68.2245 % [ 00.68 00.00 ] { AA-99 }
Hand 2: 31.7755 % [ 00.31 00.00 ] { AKs, KQs, QJs, JTs, T9s }

</pre><hr />

If we multiply the win % to come up with the projected win% for player 1 we get (.5 x .68) = 34% expected win rate in a three way hand.

When I run the actual 3 way matchup in Pokerstove here is what I get:

<font class="small">Code:</font><hr /><pre> equity (%) win (%) / tie (%)

Hand 1: 39.1855 % [ 00.38 00.01 ] { AA-99 }
Hand 2: 39.1855 % [ 00.38 00.01 ] { AA-99 }
Hand 3: 21.6291 % [ 00.21 00.00 ] { AKs, KQs, QJs, JTs, T9s }
</pre><hr />

So for a range (albeit small) of hands, my simple approximation (.68 + .5)/3 = 39% seems to be giving a superior estimate than (.68 x .5) = 34%.

I'm not sure why this is the case. Can anyone help reconcile this for me? Ties don't seem to be a significant factor in this example. I would run a larger sample of hands 3-way against one another but my computer is so slow it takes forever.