Match Play at 2-away/2-away

 Proof for doubling immediately

 From: Robert Koca Address: koca@orie.cornell.edu Date: 8 May 1994 Subject: 2 away 2 away Forum: rec.games.backgammon Google: 074307Z08051994@anon.penet.fi

```Here is a proof that doubling on one's first turn is a version of optimal
play when at 2 away 2 away match score. It supposes optimal play from
opponent (otherwise one can possibly do better with technically incorect
play).

First of all suppose that there is no market loser possibility before
one's first chance to turn cube.  In other words, it is impossible to have
winning chances less than X or greater than 1-X when you first have option
to turn cube, where X:= equity when at 2 away 1 away crawford.  (It is
generally accepted that X is approximately 30%.)

This is virtually impossible to prove rigorously, but i believe no one
would argue that it is false. If we accept this as a fact then the
following proof works:

By symmetry, one's chance of winning the match assuming perfect play is
50%.  Consider the strategy "double at first opportunity and play checkers
optimally for a 1 point match". Since by our above "fact" the double will
always be accepted, the winner of match is winner of game. Probability of
winning the game is 50%. Thus our strategy attains the equity 50% and is a
version of optimal play.

QED

Note that there are other verions of optimal play.  The theorem "optimal
play never requires a double if no market losing sequence" together with
the above theorem implies that "double if and only if there is a market
losing sequence" is also a version of optimal play.

Robert Koca  (bobk on FIBS)
koca@orie.cornell.edu
```

 Bob Johnson  writes: ```Hmmmm. Suppose no one doubles for awhile, since the game seems about even. Later, my opponent has a miracle roll, or maybe I roll something bad. Suppose my opponent suddenly has more than 70% advantage; I can drop a cube, and have a chance to win the match. If, on the otherhand, the cube was already at 2, then I might feel quite stupid for having doubled early. I don't like feeling stupid. (I wonder if that measures in?) Or -- the opposite happens and I can unexpectedly and suddenly double my opponent out of the game. I now have 70% chance for the match. I am no expert, but I tend to feel something can be gained by waiting so a more informed decision can be made, rather than immediately basing the outcome of the match on one game. ```

 Igor Sheyn  writes: ```I like to double at 2away-2away when I am (slight) favorite with maybe 1-2 market losing SEQUENCES ( since it's hard to lose your market just on your roll in the very beginning ) and with developing solid threats down the road. I will NOT double if I am an underdog, or if I have moderate gammon chances. I don't take the strength of my opponent into account, as I play only good players on FIBS, and in real life majority players in tournaments are good by default. I'll probably wait though if I happen to play a fish. I'd like to hear some more opinions on this, since the only expert one I've heard so far is Kit's in his book, and I am still not quite clear on it. Igor ```

 Kit Woolsey  writes: ```OK, Igor, let me present it in the following proposition: You and I are sitting down to play a 2-point match. Before we start, I will promise that I will turn the cube if there is any 2-roll sequence by which I can lose my market next turn. My claim is that if you adopt any different strategy which involves risking losing your market, then you have the worst of it. Proof: It is clear that I will never lose my market -- i.e. no position can ever arise where I will double and you will (correctly) pass, since I have guaranteed that if this is at all possible I will have doubled on the previous turn. If it ever happens that you fail to double and lose your market then you have done less than optimally, since any time you lose your market you would have done better doubling the turn before. Therefore, if you adopt any strategy which risks losing your market you will be at a disadvantage playing me when I adopt the strategy of never risking losing my market. QED Kit Woolsey ```

 Albert Steg  writes: ```I cannot agree with this argument because it begs the question. That is, in seeking to prove the proposition, it uses the proposition itself as an axiom. I have asked you to show that it is correct to double at the prospect of any market-losing parley, and you have effectively answered "because my opoponent will surely double with any market-losing parlay, unless he plays imperfectly." What you HAVE proved is that against anyone who is certain to employ your advice, it is best to follow your advice. You have also demonstrated, though, that this certainty cannot be assumed. After all, you yourself do not employ the strategy against players whom you suspect will not follow your strategy! As you yourself point out, against an opponent who does not double at the first market-losing opportunity, you may do better to back off from your own advice also. I see no reason for saying then that your opponent is playing "imperfectly." Albert ```

 Darse Billings  writes: ```This discussion seems to be going back and forth... perhaps I can clarify. It is a simple exercise to show that an automatic double is the game theoretic optimal play for 2-away 2-away. The first time I saw this posted, it seemed so obvious that it didn't even warrant discussion. *Optimal* play, in game theory, assumes the opponent will play correctly. The doubling result follows directly from symmetry. *Maximal* play, in practice, often exploits sub-optimal play by the opponent. Hence, in *theory* you can do no better than doubling. In *practice*, you might wait before doubling a weak opponent. Kit has explained this distinction rather well. Cheers, - Darse. ```

### Match Play at 2-away/2-away

Basic strategy  (Darse Billings, Feb 1995)
Counterexample?  (Jim Williams+, Mar 1998)
Do you need an advantage to cube?  (Keene Marin+, Feb 2006)
Double immediately?  (Chuck Bower, Oct 1998)
Ever too good to double?  (Kit Woolsey, July 1995)
Minimum game winning chances to double  (Walter Trice, Mar 1999)
Practical strategy  (Walter Trice, July 1995)
Practical strategy  (Albert Steg+, Feb 1995)
Proof for doubling immediately  (Robert Koca+, May 1994)
Proof of doubling with market losers  (Walter Trice+, July 2001)
Sample game  (Ron Karr, Dec 1996)