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broken image fixed (click 'rendered output' or 'side-by-side' to see the difference); for more info, see https://meta.mathoverflow.net/a/4058/70594
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edit approved Mar 26, 2019 at 8:47
Glorfindel
edit approved Mar 26, 2019 at 8:47
Glorfindel
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For the little it might be worth, here are the results of some simulations:

http://www.landsburg.com/ariel.gif


(source: landsburg.com)

The columns correspond to values of $|A|$ and the rows to values of $|B|$. The four-tuple in each cell is (Probability winner is in $A$, Probability winner is in $B$, Probability a given member of $A$ is the winner, Probability a given member of $B$ is the winner).

I simulated each of these 10,000 times, rounded results to the nearest percent, and retyped them (which has a small chance of having introduced additional errors).

I was struck by the non-monotonicity in the third entry as you go down the third column, so I repeated these trials and got the same result.

For the little it might be worth, here are the results of some simulations:

http://www.landsburg.com/ariel.gif

The columns correspond to values of $|A|$ and the rows to values of $|B|$. The four-tuple in each cell is (Probability winner is in $A$, Probability winner is in $B$, Probability a given member of $A$ is the winner, Probability a given member of $B$ is the winner).

I simulated each of these 10,000 times, rounded results to the nearest percent, and retyped them (which has a small chance of having introduced additional errors).

I was struck by the non-monotonicity in the third entry as you go down the third column, so I repeated these trials and got the same result.

For the little it might be worth, here are the results of some simulations:


(source: landsburg.com)

The columns correspond to values of $|A|$ and the rows to values of $|B|$. The four-tuple in each cell is (Probability winner is in $A$, Probability winner is in $B$, Probability a given member of $A$ is the winner, Probability a given member of $B$ is the winner).

I simulated each of these 10,000 times, rounded results to the nearest percent, and retyped them (which has a small chance of having introduced additional errors).

I was struck by the non-monotonicity in the third entry as you go down the third column, so I repeated these trials and got the same result.

Post Undeleted by Steven Landsburg
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Steven Landsburg
Steven Landsburg
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IfFor the little it might be worth, here are the results of some simulations:

http://www.landsburg.com/ariel.gif

The columns correspond to values of $|A|=1$$|A|$ and the rows to values of $|B|$. The four-tuple in each cell is very large, then with extremely high probability(Probability winner is in $A$, someoneProbability winner is in $B$ will receive, Probability a net transfergiven member of 1. However, the net transfer to the person in $A$ is negative as often as it's positive, and if the cardinalitywinner, Probability a given member of $B$ is eventhe winner).

I simulated each of these 10, it's also sometimes zero000 times, rounded results to the nearest percent, and retyped them (which has a small chance of having introduced additional errors). So 

I think your conjectured result cannot be truewas struck by the non-monotonicity in this casethe third entry as you go down the third column, so I repeated these trials and got the same result.

If $|A|=1$ and $|B|$ is very large, then with extremely high probability, someone in $B$ will receive a net transfer of 1. However, the net transfer to the person in $A$ is negative as often as it's positive, and if the cardinality of $B$ is even, it's also sometimes zero. So I think your conjectured result cannot be true in this case.

For the little it might be worth, here are the results of some simulations:

http://www.landsburg.com/ariel.gif

The columns correspond to values of $|A|$ and the rows to values of $|B|$. The four-tuple in each cell is (Probability winner is in $A$, Probability winner is in $B$, Probability a given member of $A$ is the winner, Probability a given member of $B$ is the winner).

I simulated each of these 10,000 times, rounded results to the nearest percent, and retyped them (which has a small chance of having introduced additional errors). 

I was struck by the non-monotonicity in the third entry as you go down the third column, so I repeated these trials and got the same result.

Post Deleted by Steven Landsburg
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Steven Landsburg
Steven Landsburg
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If $|A|=1$ and $|B|$ is very large, then with extremely high probability, someone in $B$ will receive a net transfer of 1. However, the net transfer to the person in $A$ is negative as often as it's positive, and if the cardinality of $B$ is even, it's also sometimes zero. So I think your conjectured result cannot be true in this case.