Ask The Wizard #220

Backgammon is one of my favorite gambling games. I don’t write about it because player vs. player games are extremely hard to analyze. I also can’t seem to find any new ground to break in the game. So, I’ll leave the advice to others. Here are my suggested resources:

Backgammon by Paul Magriel: If there were a Bible to backgammon, this would be it. I’m a proud owner of an old hard-cover edition. This book would be a great place to start. Although it was written in 1976, the advice still holds up well.

501 Essential Backgammon Problems by Bill Robertie: I’ve been trying to get through this book for years, and I’m still only half way there. It is disheartening to get half the problems wrong, enough to make me think I’m as bad at backgammon as I am at golf. However, with every problem missed, there is a valuable lesson to be learned. For the intermediate to advanced player, this book is a valuable, and humbling, learning tool.

Snowie backgammon software: I play about 1000 games a year against this game. Snowie not only plays a near-perfect game, but tells you exactly how costly your errors are, when you make them. There are lots of other features that I have never explored. If there is one thing I’ve learned from Snowie, it’s that the biggest problem with my game is bone-headed mistakes of not seeing perfectly obvious plays sometimes. Much like chess, one bad move can wipe out 100 good ones.

Motif website: Before I purchased Snowie, I played countless games against Motif. The strategy employed by Motif is very solid, in my opinion. There is nothing like playing against a stronger opponent to improve your own game.

Let me start by making some assumptions. First, I’m going to assume that the three players have no prior knowledge of betting behavior in Final Jeopardy, except the probabilities of being correct in the table presented. Second, I’m going to assume that knowing the category is of no help. Third, I’m also going to assume that all three contestants want to go for the win, not wishing to take another player along in a tie.

Let’s start with player C. He should anticipate that A might bet $6001, to stay above B if B is right. However, if A is wrong, that would lower him to $3999. C would need to bet at least $500, and be right, to beat A in such a scenario. However, in my opinion, if you must be right to win, you may as well bet big. So if I were C I would bet everything.

B is torn between betting big or small. A small bet should be $999 or less, to stay above C if C is correct. The benefit of a small bet is staying above C no matter what, hoping that A will go big, and be wrong. A big bet does not necessarily have to go the whole way, but it may as well. The benefit of a big bet is hoping that either A goes small, or goes big and is wrong, but both require B to be right.

A basically wants to go the same way as B. A small bet for A can be anything from $0 to $1000, which will stay above B if B bets $999. A big bet should be $6001, to guarantee a win if A is right, and still retain hope if B goes big, and all three players are wrong.

To help with the probabilities of the eight possible outcomes of right and wrong answers, I looked at the Final Jeopardy results for seasons 20 to 24, from j-archive.com(no longer available). Here is what the results look like, where player A is the leader, followed by player B, and C in last.

Using the kind of game theory logic I explain in problem 192 at my site mathproblems.info, I find that A and B should randomize their strategy as follows.

Player A should bet big with probability 73.6% and small with probability 26.4%.
Player B should bet big with probability 67.3% and small with probability 32.7%.
Player C should bet big with probability 100.0%.

If this strategy is followed, the probability of each player’s winning will be as follows:

Player A: 66.48%
Player B: 27.27%
Player C: 6.25%

As an aside, based on the table above, the probability of the leader getting Final Jeopardy correct is 54.4%, for the second-place player, 49.8%, and 48.7% for the third-place player. The overall probability is 51.0%.

As a practical note, players do have knowledge of betting behavior. In my judgment, players tend to bet big more often than mathematically justified. Interestingly, I find wagering in Daily Double to be too conservative than mathematically justified. One of the reasons I believe Ken Jennings did so well was aggressive wagering on the Double Doubles. Anyway, in reality if I were actually on the show, I would assume the other two players would bet aggressively. So my actual wagers would be $6000 as A (being nice to B), $0 as B, and $3495 as C (leaving a little un-bet, in case A foolishly bets everything or all but $1, and is wrong).

Before somebody challenges me about how one could draw a random number in the actual venue, let me suggest the Stanford Wong strategy of using the second hand of your watch to draw a random number from 1 to 60.

Putting aside the issue that such a system would be impossible, I would charge about 50 million dollars. If I had no buyers, fine, I would just go out and make much more than that on my own.

A minor reason is to foil card counters. However, instead of burning x cards, the dealer could move the cut card x cards forward, and achieve the same purpose. The major reason is game protection. For one, the player might catch a glimpse of the top card, and alter his bet and strategy, based on this information. Such a tactic would not be cheating, I might add. The top card is also vulnerable to lots of cheating schemes. It could be marked, the dealer could peek at it, or force a desired card to the top. If for any reason the dealer knew what the top card was, he could signal that information to a confederate player, giving him a huge advantage.