Ask The Wizard #220

Backgammon ranks among my top favorite gambling games. I tend to avoid writing about it due to the complexity of player-versus-player games, which makes thorough analysis quite challenging. Moreover, I feel as though I've exhausted my exploration in the game. Therefore, I think it’s best to leave the expert advice to others. Here are a few resources I highly recommend:

Backgammon by Paul Magriel: This book is considered the definitive guide to backgammon. I own a vintage hardcover version, and it serves as an excellent starting point. Despite being published in 1976, the insights and strategies it offers remain relevant today.

501 Essential Backgammon Problems by Bill Robertie: I've been attempting to make my way through this book for several years, and I'm still not quite done with it. It's a bit disheartening to find myself struggling with many of the problems, which makes me feel rather inept at backgammon—much like how I feel about golf. Nevertheless, each mistake teaches a valuable lesson, making it a great resource for intermediate to advanced players who are eager to learn.

Snowie backgammon software : I engage in around 1000 games each year of this exciting game. Snowie doesn’t just play with incredible precision; it also highlights the costs of your mistakes as you make them. There are still many features within the software that I haven’t yet explored. If there’s one lesson I’ve learned using Snowie, it's that the primary flaw in my gameplay often arises from basic blunders—missing the most obvious moves, much like in chess where one slip can negate the value of a hundred good moves.

Motif website : Prior to acquiring Snowie, I had played numerous matches against Motif. I found Motif's strategies to be quite well-grounded. Playing against a stronger opponent is one of the best ways to elevate your own skills.

Let's set a few premises. First, we’ll assume that these three players lack any insights into betting strategies typical for Final Jeopardy, except for the probabilities shown in the table. Second, we’ll assume that the category doesn’t provide any advantage. Lastly, we’ll suppose all three players are determined to win outright rather than tie.

Starting with player C, he must consider that player A might wager $6001 to ensure he stays ahead of player B if B is correct. However, if A happens to be wrong, he would fall to $3999. Therefore, for C to surpass A, he needs to wager at least $500 and be accurate in his answer. Still, in my view, if winning depends on being correct, C might as well wager everything he has.

Player B is caught in a dilemma between making a sizable wager or a smaller one. A minimal wager should be capped at $999 to keep ahead of C if C answers correctly. The advantage of this smaller bet lies in maintaining a lead over C regardless of the outcome, while hoping player A decides to make a large bet and misses. On the other hand, a larger wager doesn’t always require betting it all, but could potentially yield higher rewards. The downside of a big bet is that it relies on B being right while hoping player A falls into a trap of either wagering small or going big and missing.

Player A essentially seeks to mirror player B's approach. A small wager could range from $0 to $1000, ensuring that A remains above player B’s potential $999. On the contrary, for a larger bet, A might consider $6001 to secure a win if he is correct, while still leaving open the chance of winning if B swings for a big bet and all three contestants end up wrong.

To better understand the probabilities of the eight possible outcomes based on correct and incorrect answers, I analyzed Final Jeopardy results obtained from seasons 20 to 24 using j-archive.com, which is unfortunately no longer in service. The results illustrate the standings where player A leads, followed by player B, with player C in third place.

Utilizing the game theory principles I outline in problem 192 on my website, mathproblems.info , I've determined that players A and B should incorporate some degree of randomness in their strategies as follows.

Player A should opt for a large wager 73.6% of the time and a small one 26.4% of the time.
Player B should aim for a big bet 67.3% of the time, while keeping a smaller bet option 32.7% of the time.
Player C ought to confidently bet big 100.0% of the time.

If these strategies are adhered to, the probability of winning for each player would break down like this:

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

As a side note, according to the aforementioned table, the likelihood of the leader answering Final Jeopardy correctly stands at 54.4%, with the second-place contestant at 49.8% and the third-place player at 48.7%. Overall, the average probability sits at 51.0%.

Practically speaking, players are generally aware of betting behaviors. In my experience, players typically opt for larger wagers more frequently than what mathematical models would suggest. Interestingly, I observe that betting in Daily Doubles tends to be more conservative than the math would imply. I believe one factor that contributed to Ken Jennings’ remarkable success was his aggressive betting on Double Doubles. However, if I were competing in the show, I would predict that the other players would also bet aggressively. Thus, my actual wagers would consist of $6000 if I were player A (to be considerate of player B), $0 as player B, and $3495 as player C (leaving a small amount unbet to account for the possibility that player A might make an ill-advised all-in bet and be incorrect).

Before anyone raises concerns regarding how to generate a random number in a live setting, I’d recommend utilizing the second hand of your watch to pick a random number between 1 to 60, according to the Stanford Wong approach.

Setting aside the impracticality of such a system, I would consider charging around 50 million dollars for it. If no one was interested, I would have no problem moving forward and generating significantly more on my own.

One minor reason is to disrupt card counters. However, instead of burning x cards, the dealer could simply advance the cut card x positions and achieve the same effect. The primary motive for burning cards is to protect the integrity of the game. For example, if a player were to catch a glimpse of the top card, it could unfairly influence their betting and strategy choices. Furthermore, there are multiple cheating techniques that can target the top card. It might be marked, or the dealer could secretly view it, or possibly manipulate a specific card to the top. If the dealer knows what the top card is, they might convey that information to a partner in the game, giving them a significant edge.