1-table Tournament Expectation Question

[parappa]

Standard NLHE One-Table Tournament.

10 players, all of equal ability, start a tournament with 1000 chips each. Prize Money is 50/30/20 for first 3 places.

Our hero, in seat 1, doesn't find a hand he likes during the first 4 hands and folds them all, as does the player in seat 2. The first 4 hands are played as follows.

Hand 1: seat 10 goes all-in and loses all his chips to seat 3.
Hand 2: seat 9 goes all-in and loses all his chips to seat 4.
Hand 3: seat 8 goes all-in and loses all his chips to seat 5
Hand 4: seat 7 goes all-in and loses all his chips to seat 6.

So after the first 4 hands, the stacks are like this:
Hero (1000)
Seat 2 (1000)
Seat 3 (2000)
Seat 4 (2000)
Seat 5 (2000)
Seat 6 (2000).

Hero started the tournament with a (because the players abilities are all the same) 10% chance of winning and a 30% chance of being in the top 3.

I assume that Hero still has his 10% chance of winning, because he has 10% of the chips. But has his probability of finishing in the top 3 changed?

Thanks.

[pzhon]

There are many different models. One simple one is the independent chip model. One way of looking at it is that a random chip is chosen to be thrown away, then another, etc. Another way of looking at it is that the winner is decided proportionately, then among the losers, second place is decided proportionately, etc.

According to this model, the two players with 1000 chips have probability

.100 = 1/10 of placing 1st,
.111 = 1/9 of placing 2nd,
.127 = 8/63 of placing 3rd,
.152 = 16/105 of placing 4th,
.203 = 64/315 of placing 5th, and
.306 = 193/630 of placing 6th.

The four players with 2000 chips have probability

.200 = 1/5 of placing 1st,
.194 = 7/36 of placing 2nd,
.187 = 47/252 of placing 3rd,
.174 = 73/420 of placing 4th,
.148 = 187/1260 of placing 5th, and
.097 = 61/630 of placing 6th.

With a 50/30/20 prize distribution, the players with 1000 chips expect to win $10.87 each. The players with 2000 expect to win $19.56 each.

There are other reasonable models, but the results are similar: The players with 1000 chips win slightly more than $10, and the players with 2000 chips win slightly less than $20.