Setup and Rule Diagrams

[kmclean]

Hi guys!

Can someone please explain why in the digram S is crossed out initially from ever offering 10 year bonds?

TIA
Kyara

[Robert Carroll]

Kyara,

Remember that "10" on that original diagram has a specific meaning: offering 10 year bonds and only 10 year bonds. "B" on that diagram means offering both 5 and 10. So S is not prevented from offering 10 year bonds, but instead prevented from offering ONLY 10 year bonds. S could offer only 5 year, or both; it can never offer just 10.

The reason why that's the case is outlined by Jeremy earlier in the thread:

Hi Ari,

Those are quite tricky, so you're definitely getting to the heart of the hardest part of this game!

Let's say, for the second rule, that S does not offer both types of bonds. That means that V cannot offer 5-year bonds (and cannot offer both), and must offer 10-year bonds alone. That fills out the diagram (according to Jon's original sketch) like so:

Screen Shot 2020-08-28 at 3.45.29 PM.png

Since S and V now cannot offer both types, and H/L can never offer both types, there are only two remaining companies that can offer both types: R and G.


Moving on to the third rule, the contrapositive begins from R offering 5-year bonds (but not 10-year and not both), which means L must not offer 10-year bonds, and must therefore offer 5-year bonds. Since L and H cannot offer the same type, H now has to offer 10-year bonds, diagrammed per below:

Screen Shot 2020-08-28 at 3.45.36 PM.png

The S inference is a little head-spinning, admittedly. But, think about what would happen if S offered only 10-year bonds. That would mean V would have to offer both types, which would trigger the second rule (V offering 5-year) and force S to offer both types. So, S has to offer both types!

Let us know if that clears it up!

Robert Carroll

[kupwarriors9]

In the stimulus, I read this "Exactly 4 of the corporations offer 5ybs and exactly four offer 10ybs" as meaning:
5-year bonds: __ __ __ __
10-year bonds: __ __ __ __

Where, two corporations would offer both.

However, the line "each offer either or both of exactly 2 bonds types—5yb and 10yb" seems to contradict it?

How do I know which one is right? I am having trouble deciphering exactly how the stimulus wants us to set our base up. I feel like this is extra confusing.

Thanks,
KW9

Setup and Rule Diagram Explanation

This is a Defined-Fixed, Balanced, Grouping Game.


This explanation is still in progress. Please post any questions below!




Main diagram:

g4.png

[Robert Carroll]

kup,

I don't see any difference between the wording of the two, and the way you originally were going to set it up is a perfectly valid way to do it. In fact, it's consistent with both statements you quoted. It's also the way I originally set it up. However, the explanations we give seem, on reflection, better because they basically force a game that's not balanced to become balanced, by having six total slots (two for 5-only, two for 10-only, and two for both) for six variables. Balanced games usually have stronger inferences than Unbalanced games, so anything we can do to diagram in a way that will make inferences more apparent is a good idea to consider.

Robert Carroll

[hope]

Jon in your diagram you have rule 2 as saying V5 or VB ----- SB. Could I ask you where you got the VB from? I thought we were never to assume anything. The rule does not mention VB, it only mentions SB. Could you help a sister out?

JON ALSO ON YOUR DIAGRAM YOU PUT S ON 5. WHY DID YOU DO THAT WHEN S IS B WHEN V IS 5 OR BOTH. WHY COULDN'T S BE 10 INSTEAD OF 5?

[Beatrice Brown]

Hi Hope! This is definitely a tricky game, and happy to help you out with this

To understand why we diagrammed rule 2 as V5 or VB SB, let's take a careful look at how rule 2 is presented to us. Rule 2 says that if V offers 5-year bonds, then S must offer both types. There are two ways for V to offer 5-year bonds: either V offers only 5-year bonds, or V offers both types of bonds. If V offers both bonds, that also triggers the conditional and means that S must offer both, since if V offers both, then they definitely offer 5-year bonds.

Let's turn to how S has been represented on the diagram. Jon placed S as a split option, in either 5 or B. This is just meant to show that S can offer only 5-year bonds, or they can offer both types of bonds, but they can never offer only 10-year bonds (which is then represented by Not Laws).

Jeremy did a great job articulating why S can never offer only 10-year bonds above, but I'll also explain it another way: if S offers only 10-year bonds, then the contrapositive of the second rule is triggered because if S offers only 10-year bonds, then S cannot offer both types of bonds. If S cannot offer both types of bonds, then (according to the contrapositive of rule 2) V offers only 10-year bonds, R offers both, and G offers both. But this gets us into trouble, because if V and S occupy both 10 slots and R and G occupy both B slots, then this leaves H and L to both occupy the 5 slots, violating the first rule of the game (H and L cannot offer the same type of bonds). Since S offering only 10-year bonds leads to a rule violation, we can put S Not Laws under the 10 slots.

To sum up, one of the toughest parts of this game is remembering what only offering one type of bond means and how that gets represented on the diagram. The diagram that Jon drew helps us keep track of the difference between offering only one type of bond and offering both types of bonds.

I hope this helps, and let me know if you have any other questions!

[g_lawyered]

Hi P.S.,
After thoroughly reading all explanation, I still have some remaining questions.
While I now understand the implications of rule 2. I don't understand this bolded part:

"To understand why we diagrammed rule 2 as V5 or VB SB, let's take a careful look at how rule 2 is presented to us. Rule 2 says that if V offers 5-year bonds, then S must offer both types. There are two ways for V to offer 5-year bonds: either V offers only 5-year bonds, or V offers both types of bonds. If V offers both bonds, that also triggers the conditional and means that S must offer both, since if V offers both, then they definitely offer 5-year bonds."

Rule 2 specifically states a CONDITIONAL statements IF V is 5yr. How are we suppose to assume that this rule means 2 things. More specifically, based on what are we suppose to assume that this rule means that: V can go in both 5 and 10 yr? I thought we weren't suppose to assume from conditional rules and only make inferences on what the rule specifically states.
Question #2: Additionally, in rule 3 I was able to correctly infer the contrapositive means; R in 5 L5 and H in 10. However, I didn't further infer about S being in both 5 and 10 because this rule doesn't trigger V at all. I understand that we could infer about S being in both IF V is in 5. But I don't see where SB (S being in both 5 and 10) can added to the inference of rule 3? Can someone please explain this to me. Any guidance would help understanding this confusing game.

Thanks in advance!

[Adam Tyson]

You don't have to assume anything about that rule, GGIBA003@FIU.EDU . What you need to do is avoid making an unwarranted assumption, which is what it looks like you might be doing.

The rule does not say that if V offers ONLY 5 year bonds, then S offers both. It says if V offers 5, S offers both. If V offers both, that means V offers 5 and 10 - that's what "both" means in this context! So offering both includes offering 5, which is all it takes to trigger that rule!

Now be careful about the contrapositive regarding R and L. R offering 5 doesn't by itself trigger the contrapositive, because R might offer both 5 and 10. The rule itself neither says nor should be interpreted to mean that R does NOT offer 5-year bonds. If L offers 10, R has to offer 10, but there's nothing to say that R couldn't also offer 5. After all, two of these corporations do have to offer both! So to trigger the contrapositive you would need to have R offering only 5-year bonds and not offering 10.

As far as that inference goes, let's break it down. If R offers ONLY 5-year bonds, then L also offers only those bonds, and H would offer only 10-year bonds. That's three of our 6 variables offering just one or the other. But we still need two corporations to offer both. Who could they be? It could be V and S - that's not a problem. And it could be G and S, because that is also not a problem. But why not V and G as the pair that offers both? Because if V offers both (5 and 10) then S has to offer both, because V is offering 5. So regardless of who else offers both in this scenario, S will have to!

4 corporations offer 5-year bonds; 4 of them offer 10-year bonds; but there are only 6 corporations. To make this happen, two of them offer both. But "both" doesn't mean not 5 and not 10 - it means 5 AND 10! Once you see that, the rest should all make sense.

[g_lawyered]

Hi Adam,
Thank you so much for taking the time to explain that further. I now understand your point about making incorrect assumption about rule 2 & rule 3. However, I'm wrapped on the inference of why S CAN'T be in 10 only. I went back to review the explanation and the part that V must be in both isn't clicking for me:

" The S inference is a little head-spinning, admittedly. But, think about what would happen if S offered only 10-year bonds. That would mean V would have to offer both types, which would trigger the second rule (V offering 5-year) and force S to offer both types. So, S has to offer both types!"

.

After reading the explanation about the correct way to translate rule 2. This is what I re-wrote to understand it for myself:
V5 OR V Both S Both
Contrapositive: S NOT in BOTH (S either in 1 of only 5 or 10) V NOT IN 5 AND V NOT IN BOTH. Which forced V to be only in 10 (V10).
Doesn't this contrapositive contradict what the explanation quoted above states??

Based on this I tried to draw out & find WHY S can't be only in 10. Here's how I diagrammed:

5 yr: L/H, ___ (NOT LAW: S CAN'T BE IN THIS SECTION)
10: L/H, S, V
Both: __, __ (NOT LAWS: CAN'T BE IN THIS SECTION: H,L, S,V)

To test out the inference of S NOT being allowed in 10 only. I used the contrapositive of Rule 2. Which states S CAN'T be in BOTH (hence why I placed the Not-Law in the "Both Section) which means S CAN only be in one of either 5 OR 10. In this case, I chose S to be in 10. It further states that V CAN'T be in BOTH NOR V CAN'T be only in 5. Hence, why I placed V only in 10 year. Observing this diagram, I see that forces 3 slots into the 10 year section (as opposed to 2 which is what the original set up inferred about distributing the numbers of corps into the years). Is this the reason why S CAN'T be ONLY in 10 year? Because rule 2 contrapositive would FORCE V to be in 10 only, as well? Which messes up the 10 year slot because 1 of either H or L must be in (Split block- H/L)?

Can someone confirm my diagram or explain where my reasoning might be wrong? I think I'm almost done understanding the complexity of this game but I'm stuck here.

Thanks in advance