Bettor's Advantage

The math shows that poker is a bettor's game. The passive
defender is at a huge disadvantage.
The showdown game. Clare is dealt one card. It says
winner or loser. Half the cards are marked winner and the
other half are marked loser. She is playing against Dean.
This is a fair game. Clare wins 50% of the time.
Add a street of betting to make this a clairvoyant game.
Only Clare has the opportunity to bet. Clare can bet or
give up. Dean wins if Clare gives up. If Clare bets, Dean
can fold or call. With pot size bets Clare needs 1/3 winners
for this to be an even game. With half pot size bets she
needs 3/8 winners. Either way the bet favors Clare, the
clairvoyant.

Review the entire game. The pot starts as one chip. It
is treated as strange money. Clare is dealt one card
which is marked winner or loser. The winner's frequency
is known to both players. There are three streets of
betting. On each street Clare may check or bet the pot.
Dean can fold or call. Dean cannot benefit from betting
as only Clare knows who holds the winner. Clare always
bets the winner and bets a portion of her losers. Dean
will call 50% on each street.

Clare is dealt 16% winners.

Street one.

Clare should bet 19 losers to 8 winners. Clare bets 16
winners and 38 losers. She gives up on the remaining 46.
Dean calls half the time.

0 * 46 no net change in EV for Clare.
1 * 27 Clare wins one chip
The remaining 27 cases go to street two. The pot is now
3 chips. Clare has 1 chip invested.

Street two.

Clare bets 5 losers to 4 winners. Clare bets her remaining
8 winners and 10 of her losers. She gives up on the
remaining 9. Dean calls half the time.

-1 * 9 Clare gives up on this street.
2 * 9 Dean folds, Clare wins 2 chips.
The remaining 9 cases go to street three. The pot is
now 9 chips. Clare has 4 chips invested.

Street three.

Clare bets 2 losers to 4 winners. She gives up on the
remaining 3. Dean calls half the time.

-4 * 3 Clare gives up on this street.
5 * 3 Dean folds, Clare wins 5 chips.
-13 * 1 Dean calls and wins.
14 * 2 Dean calls and Clare wins.

Summary

Clare was dealt 16% winners. Yet she won 41 pots out of 100.

0 * 46 = 0
1 * 27 = 27
-1 * 9 = -9
2 * 9 = 18
-4 * 3 = -12
5 * 3 = 15
-13 * 1 = -13
14 * 2 = 28

Clare won a gross of 54 chips on 100 plays. She contributed
50 chips in antes. Her net winnings was 4 chips. With
three streets of betting Clare was able to convert 16%
winners to 4% favorite.

Uniform [0,1] game.

Now examine the uniform [0,1] game. Clare and Dean each
ante one chip. Clare and Dean are each dealt one number
from uniform [0,1]. Only Clare is allowed to bet. She
may check or bet the pot, two chips. Dean can fold or call.
Best number wins on showdown. With no betting the game is
obviously equal. With betting Clare bets all her winners.
Clare should also bet bet/(pot + bet) losers to every
winner. With pot size bets Clare is a 1/9 chip favorite.
With half pot size bets Clare is a 1/10 chip favorite.