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Daily Doubles and Final Jeopardy! are often the most critical junctures of a Jeopardy! game; the amount wagered can make a big difference in a player's overall chances to win. How does Watson decide on the amount?

Daily Double wagering

In principle, to compute the best Daily Double (DD) bet, a player must answer two basic questions:

1) How likely am I to answer the DD clue correctly?

2)How much will a given bet increase or decrease my winning chances when I get the DD right or wrong?

Humans are at best only able to make crude estimates of these quantities. By contrast, Watson uses advanced mathematical models that can answer both questions with far greater precision than humans can achieve.

To address the first question, Watson uses an "in-category DD confidence" model. Based on thousands of tests on historical Jeopardy! categories containing DDs, the model estimates Watson's DD accuracy, given the number of previously seen clues in the category that Watson got right and wrong.

Watson tackles the second question by using a Game State Evaluator (GSE), a complex regression model that estimates Watson's winning chances at any stage of the game, given the information set that describes the current game state (for example, the scores of the three players, the number of remaining clues, the value of remaining clues, and the number of remaining DDs).

The GSE was trained over the course of millions of simulated Jeopardy! contests pitting Watson vs. two simulated human opponents. The human opponent models in these simulations capture important statistical profiles of human contestants, such as how often contestants attempt to buzz in; how often they are right when they win the buzz; their accuracy on DDs and Final Jeopardy!.

Optimal wagering

By combining the GSE with the in-category DD confidence, Watson can compute an overall expected chance to win the game for any given DD bet. This analysis runs for every legal betting amount - from the \$5 DD minimum, to its entire bankroll for a True Daily Double - to come up with an optimal amount. The calculation also uses risk analytics to trade off expected winning chances against the risk of a particular bet.

Watson's resulting bet might seem unusual, in that it frequently may be far more aggressive, or far more conservative, than typical human bets. The amount may also take on non-round values (i.e., not an exact multiple of \$100). Such values may make the arithmetic a little more challenging for the humans when computing their bets.

Final Jeopardy! wagering

In calculating a Final Jeopardy! (FJ) wager, Watson first needs to know if it is playing a single game or a two-game match [see at bottom of article, Call out box: Match Play]. In the latter case, Watson will use very different strategies for game one and game two. The analysis for game one is similar to Daily Double analysis: Watson uses a statistical model of likely human bets, human FJ accuracy, and Watson's FJ accuracy to calculate its expected winning chances for every legal bet. It then selects the bet giving the best risk-adjusted chance to win the match.

While there are no previously revealed clues in the FJ round, Watson does obtain evidence of its likely FJ accuracy from the category title. Given the title, Watson first computes several salient features via Natural Language Processing analysis. It then consults a "FJ prior accuracy" regression model, based on Watson's performance on thousands of historical FJ categories, to predict Watson's accuracy given the category features.

Wagering in game two of a match is similar to FJ in ordinary games. The predominant consideration is score positioning (first, second or third place). In some cases, the contestants may need to use strategic reasoning as in games like Rock-Paper-Scissors - predict the opponents' bets, while taking into account the fact that the opponents are also trying to predict their bets.

Watson has been programmed with a library of known FJ strategy rules, such as Two-Thirds Betting and Shore's Conjecture. The research team also added novel rules for some special situations which we discovered.[1]

Depending on the situation, Watson will either bet according to a suitable strategy rule, or it will run a real-time simulation to calculate the best bet, among all legal bets. For the match with Ken and Brad, Watson will also take into account the prize values for second place (\$300,000) and third place (\$200,000), leading to a different objective than simply trying to win the match.

Match Play: The Watson-Jeopardy Challenge is spread over two games, with combined totals determining the winner. This style of play requires different strategies than a typical game. Final Jeopardy! of game one is analogous to "half time," so requires different strategies by all competitors, compared to when game two is the last chance to win.

[1] One such rule in ordinary FJ applies when the leader's score exactly equals the sum of the other two players' scores, for example, if Watson has \$20,000 and the two humans have \$13,000 and \$7,000. Watson would normally bet \$6,001, to win by \$1 when the second place player doubles her score. However, in this case Watson will bet \$6,000 to tie for first place. The reason is that if Watson bets \$6,001 and is wrong, it gives the third place player a chance to win by \$1 (\$14,000 to \$13,999) if the second place player is wrong.

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07:37 AM on 02/24/2011
The rules of the contest were favoring the computer in one way. You can't press the buzzer until Alex finishes reading the question. Sometimes the humans and the computer all know the answer before that, but they have to wait and then race to press the button. Nerves and muscle can't beat electronics in that kind of race. By the end of the third day, Jennings was trying to tell the audience this by holding his buzzer up where it was visible like he was trying to show, his fastest click isn't good enough.
04:32 PM on 02/23/2011
For his next trick I would love to see Watson calculate coaching decisions for one NFL team for an entire season.
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obtusegoose
aka David in the O.C.
02:44 PM on 02/23/2011
The fact that Watson didn't round his wagers to the nearest \$100, made it look like it didn't know how humans played the game. It was a sloppy mistake that the programmers should have taken into account.
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jabailo
(Participant) Texeme.Construct()
04:39 PM on 02/23/2011
Yes, the cosmetics would have been nice.

And also, are Watson's rules good enough that every digit is significant so if not, it would be just as good to bet \$100 instead of \$99 or \$98.
02:27 PM on 02/23/2011
The entire exercise left me wondering whether Watson was fed the question information in computer ready format (e.g. as a text message) or whether it had to scan and interpret an image of the screen with the letters on it, or whether Watson had to interpret the question from listening to Alex Trebek's voice.

Another thing that left me wondering is whether Watson might have had the advantage of electronic reaction time in pressing its answer "button" whereas the humans had the usual small fraction of a second delay called "reaction time".

Since humans are limited by the speed of nerve messages to the brain from the eyes as they read, it would have been nice to know whether Watson's programming compensated for such potential advantages as electronic (rather than human) reaction times, and for having to use brain function to interpret the audio visual question inputs.

Maybe this was explained in the course of the programs, and I missed it, or I missed reading the answers in some of the print commentary. Sorry if my question has been answered elsewhere.
02:02 PM on 02/23/2011
Did Watson use John Nash's Game Theory which won him a Nobel Prize ? (As in the movie with Russell Crowe - A Beautiful Mind)
01:07 PM on 02/23/2011
Thanks -- the family (a bunch of card counters) was discussing this last night, so your description of the algorithm is timely.
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BlueZoo
Independent voter, Independent thinker!
11:58 AM on 02/23/2011
Because Watson bet such unusual amounts (one even ending in 46 cents!), I suspected it had been programmed to calculate percentages. It's good to read the thought process behind the wagering. I'd still like to know how Watson managed to get the Final Jeopardy question wrong re US Cities. I suspect it's because the category was read out by Trebek as "US Cities" instead of United States Cities. Here is the example of humans having the advantage. Computers don't have eyes to read the category heading, whereas humans do. Both humans gave the correct response but Watson failed miserably.
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Realbluesky
11:39 AM on 02/23/2011
Watson's performance reflected the brilliance of the IBM staff who developed him. I am glad I had the chance to first view the NOVA program that took a look behind the scenes before watching the Jeopardy program. Looks easy on the show, not so much in the background. Congratulation guys, you did a masterful job.
01:31 PM on 02/21/2011
Dr. Tasauro is absolutely correct in his analysis of what Watson's needed computation to select
the correct amount to bet - essentially the probability of the correct answer times the amount to be won. The human mind runs such an expected value on every decision it makes; and, until the Artificial Intelligence scientists understand this and set up a program to increase the possibility of the machines own survival, they will not program the human mind in a computer. See: RevolutionOfReason.com and YouTube: RobertLBlackburn
01:37 PM on 02/17/2011
Very fascinating discussion. thanks for posting!