[0,1]- all-in or fold two player tournament

[well]

I'll keep it short but will post more numbers if anyone seems to be interested...

This is about a well-known poker game simplification where each player is dealt a

sample of the uniform [0,1] distribution. In a show-down, the highest number wins.

There have been loads of posts about this, with different betting structions, but to

my knowledge never in a tournament set-up. I have done some calculations for this

situation and the main result is this:

The probability of winning the tournament is not equal to the player's chip share.

The correct strategy assuming the opposite is exploitable.

These are the betting rules:

- the SB posts 1
- the BB posts 2
- both players recieve a "hand"
- the SB decides to fold or go all-in
- if the SB folds, he loses his 1 to the BB
- if the SB raises, the BB either calls or folds
- if the BB folds, he loses his 2 to the SB
- if the BB calles, the highest "hand" wins the pot

For the tournament version:

- all-in means raising to the amount of chips the shortstacked has
- the SB and BB roles are switched after each hand
- the player who loses all his chips loses the tournament

At the beginning of the tournament, each player recieves t tournament chips; so

there are t big blinds in play. A coin is tossed to decide which player starts

in the BB.

I calculated the cash equilibria ($), where EV's are in terms of chip amounts,

tournament equilibria (T), where EV's are in terms of winning probabilities and

the best strategies against the cash optimal strategy (^); for the values

t=4, 5, ..., 20.

Furthermore, I ran the tournament strategies against each other (TT), the

tournament strategie against the cash strategy (T$) and the cash

counter-strategty against the cash strategy (^$).

For t=20,

- the probability of player I winning is:

TT: .50
T$: .52
^$: .57

- with a 1:3 chip disadvantage, in the SB:

TT: .24
T$: .30
^$: .38

- in the BB:

TT: .25
T$: .30
^$: .39

- 3:1 chip advantage, SB:

TT: .75
T$: .76
^$: .79

- BB:

TT: .76
T$: .77
^$: .80

Are these numbers too close to .50, .25 and .75 to matter? Not in my opinion. I don't

know (yet) how these results compare to a HE version but hey - it's someting!

All comments and questions are appreciated (by me),

Regards,

Well.