# The Final Wager – Friday, October 17, 2014

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All men so far this week. Let’s see how our lone Brooklyn girl does.

She’s in the lead thanks to the penultimate clue. Will she hold on?

John Campbell: 10,000

Alexander Persaud: 13,000

Sarah Horvitz: 14,400

The Final Jeopardy! category: **COATS OF ARMS**

Let’s check out this thorny situation.

## Basic strategy – first vs. second

Sarah should wager 11,600 to cover Alexander.

Alexander can wager up to 10,200 to stay above Sarah if they’re both wrong.

John can wager up to 7,200 to stay above Sarah should they both miss.

## Basic strategy – second vs. third

Alexander should wager 7,000 to cover John.

John should wager at most 4,000.

## Basic strategy – Rule #3

Alexander should wager at least 1,400 to cover a zero wager by Sarah, and at least 2,800 to cover an “unsafe” wager by Sarah, but he’s already looking at a 7,000 wager, so we’ll ignore this.

John should wager 3,000 to cover a zero wager by Alexander, 4,400 to cover a zero wager by Sarah, or 6,000 to cover an “unsafe” wager by Alexander.

## Mind games – first vs. second

If Alexander wagers 10,200, his total will be 23,200. In response, Sarah might wager 8,800.

An incorrect response with that wager would leave Sarah with 5,600. Alexander, therefore, might cap his wager at 7,400.

John might cap his wager at 4,400 – exactly what he needs to cover a zero wager by Sarah.

## What actually happened

Whoops – John picked the wrong range, and it cost him the game. New (male) champ on Monday!

## The Final Jeopardy! clue

**COATS OF ARMS**

THIS COUNTRY’S COAT OF ARMS FEATURES A PALM TREE & A 19th CENTURY AMERICAN SAILING SHIP

Correct response: | Show |
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My intuition was that Coats of Arms is not an easy category, and that turned out to be correct. Looking through the last 5 years in J-Archives finds that contestants have netted -$1600 over $9200 worth of Coats of Arms clues, counting Daily Doubles at the default clue value.

It made me look at this wagering situation a little differently.

There are eight possible outcomes for FJ responses: RRR, RRW, RWR, RWW, WRR, WRW, WWR and WWW. Each of those outcomes is of varying likelihood, and for the sake of analogy let’s pretend they’re wedges on a wheel of chance. Through wagering, contestants determine which of those outcomes they’ll win on once the “Wheel of Trivia” is spun. The largest single wedge is WWW (23%), followed by RRR(20%).

The leader always has the opportunity to make a lockout wager, which guarantees wins on RRR, RRW, RWR, or RWW (47%), but they also have the option to make an unsafe wager which guarantees a win on RWW and WWW (36%). Securing that 47% is normally the most important wagering consideration, which is why it’s wagering rule #1. But in a difficult category, those percentages can change, to where Rule #1 becomes “Don’t let your opponents win with a wrong answer.”

This doesn’t apply to 2nd versus 3rd if the leader is presumed to be making an unsafe wager, since on a triple-stumper they’d only be competing for second place.

Following your Basic Wagering rules with reversed priorities for 1st v 2nd and 1st v 3rd:

1st v 2nd: Sarah should wager no more than 1400 to force her opponents to answer correctly. Alexander should wager at least 2800 to cover Sarah, should they both get it right.

1st v 3rd: John should wager at least 5800 to cover Sarah if they both get it right.

2nd v 3rd: Alexander should wager at least 7000 to lock out John. If he misses, he’ll be left with 6000, but John can’t stay above that total and still cover Sarah’s unsafe wager.

Improvements: If John misses with his minimum wager of 5800, he’ll be left with 4200. To stay ahead of that if they both miss, Alexander can cap his wager at 8800.

Final ranges:

Sarah: 0-1400

Alexander: 7000-8800

John: 5800-10000

This is what I was actually thinking the contestants would actually do:

Alexander wagering at least 7,000 seems like the most certain action. He locks up WRW and WRR, still has a very good chance of winning on the all-important WWW. While he could try to catch Sarah making a less-than-lockout wager by going all-in, but the 6,000 he’ll be left with if he’s wrong seems far more likely to be relevant. I would’t be surprised to see him bet 7,000, but I actually thought that 7,400 was the better wager. There aren’t that many scenarios where having 6,000 left over is better than 5,600, but there is a real chance that Sarah ends up at 20,400.

Sarah could have thought about it longer, and put Alexander squarely on the 7,000-7,400 range. Being able to put him on such a narrow range lets her crush it with a wager of 6,000. John has a possible reason to wager 1,600, so there’s no reason to go any higher than 6,000.

John probably has the easiest wager of all. He’s very likely to be locked out by both other players, or at a minimum by one of them. He should wager something very close to zero, to maximize his chances if the other two miss. Sarah could end up at 8,400 if she misses with one of her better possible wagers. Alexander could end up anywhere from 10,000-11,600 (after missing with his 1,400-3000 range) if he’s uncomfortable with the category, but still wants to cover a zero wager from Sarah. Wagering exactly 1,600 lets John maximize against all of those possibilities except a 3,000 wager from Alexander, which would be best answered with a zero wager.

I’m interested enough in this now that I think I’m going to put some payoff matrices together and see if I can find the Nash Equilibrium strategies. I’d guess it would be a mix of the following, in order of frequency.

John: 1,600/10,000

Alexander: 7,400/13,000/1,400

Sarah: 11,600/6,000/0