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May 21, 2014

Guide to Shore’s Conjecture: Part One

In today’s video, we’ll begin to break apart one of the most fascinating subfields of wagering theory: a situation called Shore’s Conjecture.

Basic criteria

The scenario was first identified publicly by two-time champion Bob Shore in 2004 or 2005. (The post has since been lost to history, unfortunately.)

The concept is confusing; Andy Saunders of The Jeopardy! Fan came up with a flowchart to show the possibilities. Using my rules, however, we can come up with a simple definition.

Shore’s Conjecture applies when second place can wager an amount that meets BOTH of the following criteria:

  1. if wrong, he won’t fall below a double-up by third (Rule #1)
  2. if right, he’ll cover a zero wager by the leader (Rule #3)

(For those of you who like algebra, the most-straightforward formula is AB < B – 2C, where A through C represent the scores of first through third.)

Take this situation, for example, from October 21, 2013.

Bill Tolany: 10,600
Nolan Martch: 7,800
Sharon Warner: 1,000

Does Nolan, in second, meet both criteria? He could wager:

  1. up to 5,800 and stay above Sharon (Rule #1)
  2. at least 2,800 and cover a zero by Bill (Rule #3)

Therefore, Nolan can wager between 2,800 and 5,800 to make a basic-strategy “Shoretegic” wager. Let’s say he picks a wager of 4,000, which falls inside this range. We’ll see that it fits both requirements:

Shore's Conjecture requirements example

Forms of Shore’s Conjecture

Those who study wagering theory have named three “forms” of Shore’s Conjecture. These forms describe the relationship between first and third. Assuming first makes the minimum “lockout” wager and responds incorrectly:

  1. Weak: third can’t catch first.
  2. Intermediate: third can pass first only if he gets it right.
  3. Strong: third can wager zero and finish ahead of first.

Our example situation falls into the Weak Form of Shore’s Conjecture. Bill needs to wager 5,000 to cover Nolan (Rule #1); if he’s wrong, he’ll have 5,600. Sharon’s too far away (Rule #2).

Shore's Conjecture weak form example

Following my basic rules will reveal the Form; a quick trick, however, is to compare third’s score with the difference between first and second.

  1. Weak: third has less than the difference between first and second.
  2. Intermediate: third has between one and two times the difference between first and second.
  3. Strong: third has more than twice the difference between first and second.

Note that these tricks help in non-Shore’s games, as well. If you’re in third place and you have less than the difference between first and second, you’d better make some cash!

If you’re the type who likes visuals, here are handy slides with these concepts.

Shore's Conjecture forms summaryShore's Conjecture forms shortcuts

Weak Form

Right off the bat, we can ignore the Weak Form; since third place is out of contention, there’s no point in worrying about him.

To illustrate, imagine what would happen if Nolan wagered 5,800 here and got it wrong. Sure, he’d be at worst in a tie with Sharon, but he’d have no way to win against Bill.

Shore's Conjecture weak form example2

Intermediate Form

In this case, the third-place player can beat first – but only by getting Final correct. If you recall my Corollary to Rule #2, a player who needs to get Final right to have a shot should wager everything. Therefore, we should assume third will do so.

In this game, John went for the lockout, Steven wagered 8,000 (very sub-optimal), and Whitney wagered zero, not even giving herself a chance!

Shore's Conjecture intermediate form example

Strong Form

This is the trickiest of the three. Here, third doesn’t have to wager everything – but, of course, he still might. (Jeopardy! players, as we’ve noted innumerable times on this site, are generally pretty bad at this aspect of the game.)

First place also has other motivations. “If I get it wrong,” he might say to himself, “I’ll run the risk of losing not only to second, but third – all without them having to do anything.” Thus, depending on the options available to second – and we’ll get into that a little later – first might be more tempted to eschew the lockout wager.

In this game, all three players missed; Bill (Tolany, the same Bill as our Weak Form example) would have won with a “Shoretegic” wager.

Shore's Conjecture strong form example

Between Two Forms

Special situations arise when third place has exactly 1 or 2 times the difference between first and second. Both are wager-to-tie scenarios for the leader.

When third place’s score is equal to the difference of first and second, this is, simply enough, “first equals second plus third”. I spoke about this in Part Two of my general tutorial.

When third has exactly twice this difference, first will need to withhold the dollar if he goes for the lockout against second. That’s because if first gets it wrong, he’ll have exactly third’s pre-Final total. Can’t risk losing by a dollar against a zero wager!

How do we “solve” Shore’s Conjecture?

I created a special technique for Shore’s Conjecture situations. Stay tuned for Part Two, in which I’ll unveil this approach!

  1. Kelly permalink

    I need to bring up two points:

    1. You need to add a third criterion for qualifying to be a Shore’s Conjecture game – second must have more than 2/3 of first (under Bob Shore’s qualifiers crush games don’t count, since second can’t win without being right anyway).

    2. Faith Love is not the special case where third has exactly twice the difference of first and second’s score – it’s the “first equals second plus half of third” scenario where since third has a strong incentive to bet exactly zero the leader should offer the tie (and thus second may be justified at going all-in). Faith Love is the “breakpoint” between qualifying and not qualifying for Shore’s conjecture where second can exactly keep third “lock-tied” out and tie a zero bet by the leader (with a true Shore game being where second can go beyond that amount with the same restrictions against third) – thus the leader has a plausible wager of zero like the scenario’s namesake used, and third should bet it all.

    • Kelly permalink

      Sorry for any typos there!

    • Ah, crap – thanks! Spend a week writing the copy for the main part, then decide to add something on the “break points” at the last minute, and that second point is what you get.

      I’m going to leave crushes in there to keep things simple. The strategy I teach in Part Two will bear out the need for first to cover.

  2. jdgalt permalink

    The John/Steven/Whitney game that you use for an example of the intermediate form is really the strong form. John’s 12,100 minus Steven’s 10,800 = 1,300, which is less than Whitney’s 1,800.

  3. jdgalt permalink

    Does “Part Two” of this post exist? If so, the tag “Shore’s Conjecture” doesn’t get me there.

What do you think?