Undefined Equity - Part 1

Very interesting as usual Douglas!

What is wrong with the following logic in the Paradox Position: Suppose Red redoubles. You have already shown that Redoubling gains if White fails to redouble if Red dances. So only if White's recube after Red's dance is correct mat Red lose out by redoubling the original position. However, since Red's and White's decisions are identical, White's recube can only be correct if Red's recube is also correct. So the maniac strategy is correct.

Suppose I assume that my opponent is a maniac, but I am forbidden to use the maniac strategy. In that case, it is wrong to redouble. In some ways this resembles real life, since I won't redouble to stakes I can't cover.

I think real life may be modelled better by part of the n- brave versus m-brave table. You (as first roller) get to choose the row, while your opponent chooses the column. If a player will not double to stakes that can't be covered, this limits the choices to the first few rows or columns.

If you are willing to redouble up to 3 times (to 4, 16, and 64), and your opponent is willing to redouble up to 3 times (up to 128), then the restricted table appears as follows:

Second Roller 0 1 2 3 First 0 .361 .361 .361 .361 1 .721 .220 .220 .220 2 .721 .916 -.050 -.050 3 .721 .916 1.292 -.572

In this restricted game, the last column dominates the others. The entries in the last column are the lowest in each row. So, the second player should be willing to redouble 3 times, and the last column is the only relevant one. The highest entry is in the first row, so you should not redouble.

However, If your opponent is not able to redouble the third time, this is the payoff table:

Second Roller 0 1 2 First 0 .361 .361 .361 1 .721 .220 .220 2 .721 .916 -.050 3 .721 .916 1.292

Now the bottom row dominates the other rows, so you should be willing to redouble 3 times, while your opponent should not redouble.

Before you test your opponent, you might not know whether you are in the former or latter restricted game, and this information determines the best move.

You might say that rather than the first and second player's choices being identical, they are exactly opposite. If the first player is willing to redouble more than the second, the first player should be willing to redouble, but the second player should refrain from redoubling. After redouble/take and a miss, the position is not the same, because the roller has changed, and who is willing to redouble more has switched from roller to nonroller or from nonroller to roller.

Douglas Zare

I like the presentation. I'm looking forward to your second part (table stakes and match play).

If I remember right, a sufficient condition for the 1- animal (beavers allowed, raccoons not allowed) table to be valid is (1) the cubeless equity (of the player on roll) is greater than 0 and less than 0.5, (2) correct checker play is idential to money-cubeless checker play, and (3) the cube has no value to either player once someone rolls 6-x from the bar. [Equivalent to (1) is "the value of the first 6 is 61/22 = 2.77 times the cube"]. These are very robust assumptions. "Clearly" (1) is true by a comfortable margin and (2) and (3) are almost true, so overall the 1- animal table is almost surely valid, which is a "nice" result in my opinion. I'm looking forward to using this knowledge, but unfortunately in my chouettes I've yet to reach this Paradox position... Still, it's fun to think about the results...