Question
A booking platform has six appointment slots and eight users. All eight want exactly one appointment, and every booking uses one slot. Normally, when everyone has equal priority, six users are chosen uniformly at random and the other two get no appointment.
In a trial, the platform randomly assigns four users to a “priority” treatment and leaves four as controls. The four priority users are served first, each getting an appointment; the two remaining slots are assigned uniformly among the four controls. All users learn whether they got an appointment, and one user's booking necessarily removes a slot available to others.
Compute the trial's treated-minus-control booking-rate difference. Compare it with the change in the average booking rate if the platform rolls out priority access to all eight users, with equal priority ties settled by a uniform lottery. How much of the observed trial difference comes from displacing controls rather than creating additional appointments?
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Hint 1
In the trial, priority decides which users fill the fixed six slots. It does not change the number of available slots.
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Solution
In the trial, four treated users take four slots, so a treated user books with probability 1. The remaining two slots are allocated among four controls, so a control books with probability \(2/4=1/2\). The standard individual randomized contrast is
\[
E[Y\mid T=1]-E[Y\mid T=0]=1-\tfrac12=\boxed{\tfrac12}.
\]
But the platform always books six of eight people. With no priority treatment, symmetry gives every user booking probability \(6/8=3/4\), so the average rate is 3/4. If everyone receives priority, everyone again ties and six are selected uniformly; the average remains 3/4. The all-users rollout effect on booking rate is \(3/4-3/4=\boxed0\), and the total appointment count remains six.
Relative to an equal-priority baseline, priority recipients in the mixed trial gain \(1-3/4=+1/4\) booking probability each, while controls lose \(1/2-3/4=-1/4\) each. The contrast \(+1/4-(-1/4)=1/2\) includes both a treated advantage and a displaced-control disadvantage. Their effects cancel in aggregate because there are four users of each type and capacity is conserved:
\[
4(1)+4(1/2)=6=8(3/4).
\]
The trial's individual treated-control contrast is therefore not the effect of a universal rollout. The potential outcome for a user depends on how many other users receive priority, violating the usual no-interference assumption.
Assumptions
Every user requests exactly one appointment; there are exactly six slots in all regimes.
The treated group of four is selected uniformly at random; equal-priority ties are resolved uniformly.
Priority affects allocation order, not the total number of slots or demand.
Final Answer
Trial contrast 0.5 (treated 100%, control 50%). Universal rollout changes the overall booking rate by 0, leaving it at \(6/8=75\%\). Relative to equal-priority baseline, treatment raises each treated user's chance by 0.25 and lowers each control's by 0.25.