Tuesday, August 30, 2016

Shapley's Value; mechanisms to determine to split a fare

Together the cost from Alice to Bob’s place is $12 and if they go Separately, Alice will pay $6 and Bob will pay $11.

$6 + $6 = $12 (together)
$6 + $11 = $17(separate)
The pie is $17 - $12 = $5 the pie is split  =


A = $6 - $2.50 = $3.50
B = $11 - $2.50 = $8.50


A = (trip if A went along) - (the split pie)
B = (trip if  Bwent alone) - (the split pie)




Cost to A + B: $12
Cost to B: $11


Detour = $12 - $11 = $1


Pays $3 +$.50 = $3.50
Pays $12 - $3.50 = $8.50


A = (Alice’s split ride home) + (the split pie)
B= (The price start to A to B) - (Alice’s split ride home + the split pie)




Separate 6+ 11 + 18 =     $32
Together: $6 + $6 +$6 = -$18
---------------------------------------
                                         $14
Take Alice and Bob


Possible orders:   1)  Alice Bob   3) Bob Alice


Alice  ½($6 + $1) = $3.50
Bob    ½($6 + $11) = $8.50


Alice = ½((half the cab ride home) + ((total of start to A to B) - (B’s iindividual ride home)
Bob = ½(half the cab ride to A) + (the trip home if went separately)).

If you notice all three equations of the above are equal to $3.50 and $8.50


We will use the Shapley Value:






ABC


ACB


A = $6  B = ?  C = $11   (What is B? Total cab fare: $18)  B = $18 - ($6 + $11) = $1
A  to C to B is equal to $17 now from A to B to C is equal to $18, now B only has to pay $1.
Now comes the Shapley value


CBA

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