R help - Apr 2010 - How to work out 3-way probabilities

Hello.

I have a quick question.

I try to use logit regression, to work out probabilities in the sport event.
I have work out probabilities for group of 2 players:

p1 - probability, what player1 will beat player2
p2 - probability, what player2 will beat player1
pt - tie probability, p1 <- 1 - p1 - p2;

Now i want to work out probabilities for group of 3 players, like:

pg1 - probability, what player1 will beat player2 and player3
pg2 - probability, what player2 will beat player1 and player3
pg3 - probability, what player3 will beat player1 and player2

I have probabilities for every pair of players in that group. Is there a
function in R, what can simply convert this 2-way probabilities in to the
3-way (or more)? Or can some body suggest, how to do it manually? I have try
to work it out, buy just multiply corresponding probabilities, but it seems
a bad way.

Cheers,
Dmitrij.

	[[alternative HTML version deleted]]

On Tue, 27 Apr 2010, Dimitrij Kudriavcev wrote:
> Hello.
>
> I have a quick question.
>
> I try to use logit regression, to work out probabilities in the sport
event.
> I have work out probabilities for group of 2 players:
>
> p1 - probability, what player1 will beat player2
> p2 - probability, what player2 will beat player1
> pt - tie probability, p1 <- 1 - p1 - p2;
>
> Now i want to work out probabilities for group of 3 players, like:
>
> pg1 - probability, what player1 will beat player2 and player3
> pg2 - probability, what player2 will beat player1 and player3
> pg3 - probability, what player3 will beat player1 and player2
>
> I have probabilities for every pair of players in that group. Is there a
> function in R, what can simply convert this 2-way probabilities in to the
> 3-way (or more)? Or can some body suggest, how to do it manually? I have
try
> to work it out, buy just multiply corresponding probabilities, but it seems
> a bad way.
So each player gets a score and the highest wins?

And your two-way probabilities are just the probability that player1 
scores higher than player2, etc? Or that player1 won over player2 given 
that player3 did not win?

Or you have a bunch of one-on-one competitions and you want to predict 
what will happen in a three way competition?

Have a Google at 'McFadden conditional logit'. There is a literature
that
should point you in the right direction.

Once you have a sense of what goes in conditional logit models you may 
find survival::clogit() to be of assistance.

HTH,

Chuck

>
> Cheers,
> Dmitrij.
>
> 	[[alternative HTML version deleted]]
>
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Charles C. Berry                            (858) 534-2098
                                             Dept of Family/Preventive Medicine
E mailto:cberry at tajo.ucsd.edu	            UC San Diego
http://famprevmed.ucsd.edu/faculty/cberry/  La Jolla, San Diego 92093-0901

If you're willing to assume independence, multiplication is the way to go:

Pr( a beats b & a beats c) = Pr( a beats b ) * Pr( a beats c )

If you're not willing to assume independence, things can get very weird:
http://en.wikipedia.org/wiki/Arrow's_impossibility_theorem
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