I don't know the solution to this problem, but it seems interesting (to me).
I am playing an optimal computer in a Jacoby money session (think of XG but ignore
any weaknesses XG may have).
XG doesn't know this but I have a side bet with a friend as to whether or not
I win the current game. Furthermore, this win/lose side bet is the only
thing that concerns me.
Would it ever be correct for me to offer the cube when XG has a take?

Thanks,
Paul

On 7/27/2022 6:46 AM, peps...@gmail.com wrote:
> I don't know the solution to this problem, but it seems interesting (to me).
> I am playing an optimal computer in a Jacoby money session (think of XG but ignore
> any weaknesses XG may have).
> XG doesn't know this but I have a side bet with a friend as to whether or not
> I win the current game. Furthermore, this win/lose side bet is the only
> thing that concerns me.
> Would it ever be correct for me to offer the cube when XG has a take?

Let's say you're playing XG for $1 a point and your side bet is $100.
You don't really care about gammon wins or losses, and you're going to
take any cube offered to you.

Seems to me the optimal strategy would be to wait until XG has a pass
and cash the game. Perhaps there's some example where it would be
correct to offer a cube that's a take, but my hunch is that if there is
one then upping the side bet to $1000 or $10,000 would negate it.

On 7/27/2022 2:41 PM, ah...Clem wrote:
> On 7/27/2022 6:46 AM, peps...@gmail.com wrote:
>> I don't know the solution to this problem, but it seems interesting (to me).
>> I am playing an optimal computer in a Jacoby money session (think of XG but ignore
>> any weaknesses XG may have).
>> XG doesn't know this but I have a side bet with a friend as to whether or not
>> I win the current game. Furthermore, this win/lose side bet is the only
>> thing that concerns me.
>> Would it ever be correct for me to offer the cube when XG has a take?
[...]
> Seems to me the optimal strategy would be to wait until XG has a pass
> and cash the game.

It certainly seems this way, but when I tried to prove it, I discovered
that the problem is more subtle than it seems at first.

First of all, it seems to me that Paul left open the possibility that he
is betting against himself; i.e., he wins the side bet if he loses
against XG. Even in this case, it's not clear to me what the answer is.
But let's assume that Paul is betting that he'll win the game. One
could imagine that D/T could change XG's subsequent checker play, in a
way that increases Paul's winning chances. It's hard to imagine, but it
also seems hard to rule out rigorously.

---
Tim Chow

On Thursday, July 28, 2022 at 2:09:22 AM UTC+1, Tim Chow wrote:
> On 7/27/2022 2:41 PM, ah...Clem wrote:
> > On 7/27/2022 6:46 AM, peps...@gmail.com wrote:
> >> I don't know the solution to this problem, but it seems interesting (to me).
> >> I am playing an optimal computer in a Jacoby money session (think of XG but ignore
> >> any weaknesses XG may have).
> >> XG doesn't know this but I have a side bet with a friend as to whether or not
> >> I win the current game. Furthermore, this win/lose side bet is the only
> >> thing that concerns me.
> >> Would it ever be correct for me to offer the cube when XG has a take?
> [...]
> > Seems to me the optimal strategy would be to wait until XG has a pass
> > and cash the game.
> It certainly seems this way, but when I tried to prove it, I discovered
> that the problem is more subtle than it seems at first.
>
> First of all, it seems to me that Paul left open the possibility that he
> is betting against himself; i.e., he wins the side bet if he loses
> against XG. Even in this case, it's not clear to me what the answer is.
> But let's assume that Paul is betting that he'll win the game. One
> could imagine that D/T could change XG's subsequent checker play, in a
> way that increases Paul's winning chances. It's hard to imagine, but it
> also seems hard to rule out rigorously.

Thanks for your interest.
I'm not betting against myself -- I meant to rule that out.
My intuition is that some clever funfair-style construction tricks can be done
to make D/T optimal but I also can't imagine what this might be.

Paul

On Thursday, July 28, 2022 at 8:11:27 AM UTC+1, peps...@gmail.com wrote:
> On Thursday, July 28, 2022 at 2:09:22 AM UTC+1, Tim Chow wrote:
> > On 7/27/2022 2:41 PM, ah...Clem wrote:
> > > On 7/27/2022 6:46 AM, peps...@gmail.com wrote:
> > >> I don't know the solution to this problem, but it seems interesting (to me).
> > >> I am playing an optimal computer in a Jacoby money session (think of XG but ignore
> > >> any weaknesses XG may have).
> > >> XG doesn't know this but I have a side bet with a friend as to whether or not
> > >> I win the current game. Furthermore, this win/lose side bet is the only
> > >> thing that concerns me.
> > >> Would it ever be correct for me to offer the cube when XG has a take?
> > [...]
> > > Seems to me the optimal strategy would be to wait until XG has a pass
> > > and cash the game.
> > It certainly seems this way, but when I tried to prove it, I discovered
> > that the problem is more subtle than it seems at first.
> >
> > First of all, it seems to me that Paul left open the possibility that he
> > is betting against himself; i.e., he wins the side bet if he loses
> > against XG. Even in this case, it's not clear to me what the answer is.
> > But let's assume that Paul is betting that he'll win the game. One
> > could imagine that D/T could change XG's subsequent checker play, in a
> > way that increases Paul's winning chances. It's hard to imagine, but it
> > also seems hard to rule out rigorously.
> Thanks for your interest.
> I'm not betting against myself -- I meant to rule that out.
> My intuition is that some clever funfair-style construction tricks can be done
> to make D/T optimal but I also can't imagine what this might be.
>
> Paul

I can solve my own problem as below. When you see the basic story, it will
be utterly trivial to construct games that meet the demands of the problem.

I have a good game where I'm solidly ahead but never by so much that XG should pass.
Then we get a sequence that dramatically swings the game so that XG has great gammon chances
and few losing chances.

D/T is very likely to be my best play now (under the side bet hypothesis). When you have good gammon chances, you often need
to sacrifice your winning chances to max your equity.

Paul

On 7/28/2022 6:06 AM, peps...@gmail.com wrote:
> I have a good game where I'm solidly ahead but never by so much that XG should pass.
> Then we get a sequence that dramatically swings the game so that XG has great gammon chances
> and few losing chances.
>
> D/T is very likely to be my best play now (under the side bet hypothesis). When you have good gammon chances, you often need
> to sacrifice your winning chances to max your equity.

I don't follow your reasoning. I guess you're saying that with Jacoby,
XG will double you if you leave the cube in the center, whereas if you
double first, XG will hang on to the cube because it will be TG. I can
see that XG's checker play could differ in the two scenarios, but:

- in both scenarios---i.e., whether or not XG owns the cube---XG will
probably be sacrificing some winning chances for gammon chances; and

- even if XG is sacrificing *more* winning chances when it owns the
cube, it's not clear that this will be enough to compensate for your
inability to cash the game if the equity suddenly swings in your favor
again.

---
Tim Chow