Toy Game Question

[FatTony]

Toy game:
- Heads up, 1/2 blind structure
- SB can fold, call, or raise.
- BB can call or fold.
- NO further betting.

Questions:
1) BB holds TT. Is he hoping SB calls or raises?
2) What is smallest pocket pair BB can hold while hoping for a raise from the SB?
3) What are all the hands the BB can hold where he hopes the SB raises?

[edit: Question #3 added]

[mwette]

I would think this is a very simple game. Since
SB gets no reads whatsoever, his decision should
be based purely on EV w/ 2 players: if > 50% raise,
if < 50% fold. SB should know exactly what BB will
do, if BB's strategy is based this way.

[creedofhubris]

[ QUOTE ]
I would think this is a very simple game. Since
SB gets no reads whatsoever, his decision should
be based purely on EV w/ 2 players: if > 50% raise,
if < 50% fold. SB should know exactly what BB will
do, if BB's strategy is based this way.

[/ QUOTE ]

Perhaps you missed that SB can call.

[lurkling]

1)hopes for a raise
2)22
3)anything with 33% equity against a random hand

embrace all 5 cards with no more action!

[FatTony]

I'm working my way through Chen's book, and the toy game is something I contrived to determine if I'm understanding the material.

Here's my solution to the Toy Game:
1) Call
2) JJ
3) AA to QQ, AKs to AJs, AKo to AJo

Here's how I arrived at it:
1) Determine SB's action for each possible hand he can hold. From his perspective, BB holds a random hand. For each possible hand the SB can hold, calc EV for call & raise. (Assume SB calls all raises, which is reasonable, as getting 5-to-1 odds. This precludes the SB bluff-raising with weak hands.) The analysis shows the SB raises with the following hands:
a. Any pocket pair
b. AXs, KXs, QXs, JTs to J6s, T9s to T7s, 98s
c. AXo, KXo, QJo to Q5o, JTo to J8o, T9o

2) Calculate the Fold or Call EV for the BB. The BB knows the hands that the SB will call or raise with, so use this distribution as part of the calcs. Results confirm that BB will call a raise with any hand.

3) For each hand the BB can hold, compare the BB’s EV for when the SB calls or raises. Identify those hands the BB holds where his EV increases when the SB raises (as opposed to SB calling). Those hands are the answer to #3 above.

These results seem modestly counter-intuitive to me, though not obviously incorrect. I've been careful working through this, though certainly do not preclude the fact that I have erred. Flame away if I have!

[img]/images/graemlins/smile.gif[/img]

[creedofhubris]

1. If SB raises, BB is getting 3:1 to call the raise, not 5:1. (There's $6 in the pot, he needs to call $2.) This means that he folds his worst hands, somewhere around 10%. This in turn allows the SB to raise hands that aren't 50% when called, because he gets fold equity.

2. Small blind will be availing himself of that calling option fairly often.

3. I did some preliminary math, and I came up with SB folding 32o and 42o, calling about 20%, and raising about 70-80%. BB calls ~90% of hands and folds about 10%.

4. Given all that, BB likes a raise when he's holding 33+ for sure. I forget what other hands he likes a raise with, it depends on the raise proportions.

[FatTony]

Yes, the BB is getting 3:1 to call a raise. I stated this incorrectly as 5:1, but my analysis correctly used 3:1. However, there is no hand the BB should fold to a raise. Using the hand distribution the SB raises with:

a. Any pocket pair
b. AXs, KXs, QXs, JTs to J6s, T9s to T7s, 98s
c. AXo, KXo, QJo to Q5o, JTo to J8o, T9o

the BB's calling EV is still roughly 0.38 with the worst hand he can hold, 72o (using Pokerstove to verify my calcs: 72o against the above hand range). Hence, the SB will never bluff raise with weak holdings.

[creedofhubris]

[ QUOTE ]
Yes, the BB is getting 3:1 to call a raise. I stated this incorrectly as 5:1, but my analysis correctly used 3:1. However, there is no hand the BB should fold to a raise. Using the hand distribution the SB raises with:

a. Any pocket pair
b. AXs, KXs, QXs, JTs to J6s, T9s to T7s, 98s
c. AXo, KXo, QJo to Q5o, JTo to J8o, T9o

the BB's calling EV is still roughly 0.38 with the worst hand he can hold, 72o (using Pokerstove to verify my calcs: 72o against the above hand range). Hence, the SB will never bluff raise with weak holdings.

[/ QUOTE ]

Oops. For some idiotic reason I was looking for .33 equity as the cutoff point for BB's calls rather than .25 equity. Ditto for SB's fold vs. call frequency.

So, yeah, SB never folds, BB never folds, SB raises hands that have + EV vs. a random hand and limps in with the rest, game solved.

[FatTony]

In closing, I find it surprising that the only hands the BB holds in which he's hoping the SB raises are:
- AA to QQ
- AKs to AJs
- AKo to AJo

way counter-intutive....

[jogsxyz]

Your talking about a real game. MOP toy games are usually uniform [0,1](uniform zero-one) or one player is clairvoyant.

In real games the player in the lead on one street may not remain in the lead the next street. And the distribution is not uniform.