Consider a two-period game between a monopolist and a buyer. Let's assume
in each period the monopolist can produce an item for sale at zero cost
and its knowledge about the buyer's valuation of the item is described by
a uniform distribution over [0,1]. In each period the monopolist offers a
price to sell the item and the buyer can either accept or reject it.
First let's consider when the monopolist cannot tell that the buyer is the
same person in the two periods. Then of course we simply have a
two-period repetition of the classical monopoly in which the monopolist's
profit is maximized at p=0.5 in each period. The seller's expected profit
is 0.5, the buyer's expected surplus is 0.25.
Now let's consider when the monopolist knows that the same buyer is in
both periods. In this case (we assume) the monopolist knows that the
buyer's valuation for the item is the same in both periods. Therefore he
can use the information he learns about that valuation in the first period
to set the price for the second period. I claim the following is a
perfect baysian equilibrium for this game:
Buyer's strategy: If in period 1 valuation (v) >= twice seller's offer
(2*p1), then accept, otherwise reject. If valuation >= second period
offer, then accept in period 2, otherwise reject.
Seller's beliefs: If buyer accepts in period 1, then buyer's valuation is
distributed uniformly over [2*p1, 1], otherwise it is distributed
uniformly over [0, 2*p1].
Seller's strategy: In period 1 offer p1=3/10. If the buyer accepts, offer
6/10 in period 2, otherwise offer 3/10 in period 2.
To verify this, note that the seller's second period strategy simply
involves maximizing his 2nd period expected profit given his posterior
beliefs. Then it is easy to check that p1=3/10 maximizes his expected
total profit given his second period strategy. The seller's beliefs are
clearly consistent with the buyer's strategy. The buyer's strategy in
period 2 is obvious. In period 1 he can not gain by deviating because if
he accepts, his utility is 2*v-3*p1, otherwise it is max(0, v-p1).
Now we can see some counterintuitive results by looking at the expected
outcome. The seller's expected profit is now 0.45 and the buyer's
expected surplus is 0.325. The total surplus has increased from 0.75 to
0.775 as we expected from the improved information gathering/utilization.
But surprisingly the seller is actually *worse off*, whereas the buyer is
much better off.