You have 8 teams in a single elimination football/basketball/baseball (doesn't matter) tournament.



There are two ways to organize this tournament:

-  Random draw

-  Seeding (method of seeding is irrelevant)



The average margin of victory in the 7 games (4+2+1) is the same in both instances, correct?



Can someone explain the math behind this?

No game theory/stats guys?

I believe so...assuming both ways play the same amount of games.

Assign point, 2pt W, 1pt T, 0pt L.  Who ever has the most after 7 games gets ranked accordingly.  2 vs 3, winner plays 1 for championship.

Each individual game is a unique and non-repeatable event. Neither the outcome nor the victory margin can be determined statistically or mathematically with confidence.

If you ran a series of Monte Carlo simulations, you would end up with a whole range of outcomes. Some will have higher probabilities, but no individual game and its victory margin can be statistically determined with reasonable confidence. You can add the probable outcomes under the two scenarios, but each will be an estimated range of total victory margins, not a precise value.

That's a bit of a technical way to say "that's why they play the game."

I'm sorry I think I explained it incorrectly.
Case A (random draw single elimination tournament, no two teams are equal and the better team at the start of the tournament will always beat the lesser team throughout the tournament)






- Round 1






–– Game 1: Random team vs Random team






–– Game 2: Random team vs Random team






–– Game 3: Random team vs Random team






–– Game 4: Random team vs Random team






- Round 2






–– Game 5: Winner of Game 1 vs Winner of Game 2






–– Game 6: Winner of Game 3 vs Winner of Game 4






- Round 3






–– Game 7: Winner of Game 5 vs Winner of Game 6
Case B (seeded single elimination tournament, no two teams are equal and the better team at the start of the tournament will always beat the lesser team throughout the tournament)






(The #1 is the best team and #8 lesser team)






- Round 1






–– Game 1: #1 team vs the #8 team






–– Game 2: #4 team vs the #5 team






–– Game 3: #3 team vs the #6 team






–– Game 4: #2 team vs the #7 team






- Round 2






–– Game 5: Winner of Game 1 vs Winner of Game 2






–– Game 6: Winner of Game 3 vs Winner of Game 4






- Round 3






–– Game 7: Winner of Game 5 vs Winner of Game 6
My question:  If you average the margin of victory of Case A and Case B will they be equivalent?







I'm trying to disprove the statement:  "A random unseeded tournament will have an average margin of victory less than a seeded tournament."
Yes, I know each case is unique; however, if repeated infinitely you
are able to determine which format results in more blowouts, if either.
 
 


 

State the null hypothesis you are trying to disprove. Design a statistical model that will allow you to perform a Monte Carlo simulation that works within the parameters you set out. Analyze the date. Determine if you can reject the null.

Null hypothesis:  Seeded 8 team single elimination tournaments result in larger average margin of victories than an unseeded tournament.



I'll post the Monte Carlo method as I go and someone can correct me if I'm wrong...which I'm sure to be.
 

Your null hypothesis is incorrect. It should be that there is NO DIFFERENCE in average margin of victory between the two scenarios. The starting assumption is always that the hypothesized factor has no effect. If you can prove there is a statistically significant effect, than you can conclude that the alternate hypothesis (larger average margin of victory in one scenario) is likely true, with a given level of confidence.

Before you begin your simulation you need to decide on the acceptable alpha and beta risk levels. This will determine the sample size you will need from your Monte Carlo simulations. State these alpha and beta risk levels clearly up front and then do NOT change them (unless the required sample size is unattainable). This will help avoid the temptation to alter your acceptable risk levels after the fact in order to justify the conclusion you wanted in the first place.

Ok, thanks...give me a day to process.  Another day to research.  And a week to find my mistakes.



Thanks though.





 

Is 8 teams even large enough to generate a statistically valid sample?