Direct Effects On Probabilities of Old And Young Win in

Another direct effect is on probabilities of Old and Young winning in each of the states int−1 based onpt−1. This effect, though is not monotonous For some middle values the

increment of probability that a Young bidder won will take place (based on an increase inpt−1), since the young bidders’ bids distribution has a different range than distribution

of Old bids. First order Stochastic dominance may not, therefore hold. Another aspect is that above a certain point the probability of old bidder’s win is increasing with pt−1

up to the threshold of Young bidders’ bids. Above that threshold the probability that a Young bidder has won is 0 and the probability that an Old bidder won is 1.

Theorem 12. In State A and B, an observation of the second highest bid (observed price) above the threshold of young bids means that there probability that an old bidder won is equal to 1.

Suppose a price is observed in t−1. Conditional on this price observation the prob- abilities of old and young winning can be updated.

Some definitions that are needed:

The distribution density function of the highest bid coming from young bidder in state A:

f1∩y,A(.) =f1,A(.)∗fy,A(.)∗Cn1

The distribution function of the highest bid coming from old bidder in state A:

f1∩o,A(.) =f1,A(.)∗fo,A(.)∗Cn2

The distribution function of the highest bid coming from young bidder in state B:

f1∩y,B(.) =f1,B(.)∗fy,B(.)∗Cn3

The distribution function of the highest bid coming from old bidder in state B:

f1∩o,B(.) =f1,B(.)∗fo,B(.)∗Cn4

, where Cn41, Cn2, Cn3, Cn4 are normalizing constants. Each of the above density func-

tions have, of course their corresponding cumulative distribution functions: F1∩y,A(.),

F1∩o,A(.),F1∩y,B(.),F1∩o,B(.). It is worth noting that the distribution of young bids has a

threshold which is the bid of the highest valuationb(VH). This means above that thresh-

The probability of young winning in stateA is the probability that the highest bidb1

is from a young bidder and that it is above the observed price. The probability of young winning in state A is updated as below:

P rA(y|pt−1 =x) =P(b1 ∈y|A, pt−1) = N 2N−1(1−F1∩y,A(x)) N 2N−1(1−F1∩y,A(x)) + N−1 2N−1(1−F1∩o,A(x)) (1.5.29)

Above the threshold, of the maximum young bid F1∩y,A(x)) is equal to 0, and therefore

the probability that the highest bid comes from a young bidder is also 0 above the thresh- old.

The probability of old winning in state A is the probability that the highest bid b1 is

from an old bidder and that it is above the observed price. The probability of old winning in state A is updated as below:

P rA(o|pt−1 =x) = P(b1 ∈o|A, pt−1) = N−1 2N−1(1−F1∩o,A(x)) N 2N−1(1−F1∩y,A(x)) + N−1 2N−1(1−F1∩o,A(x)) (1.5.30)

Above the threshold, of the maximum young bid F1∩y,A(x)) is equal to 0, and therefore

the probability that the highest bid comes from an old bidder is 1 above the threshold.

The probability of young winning in state B is the probability that the highest bidb1

is from a young bidder and that it is above the observed price. The probability of young winning in state B is updated as below:

P rB(y|pt−1 =x) =P(b1 ∈y|B, pt−1) = N 2N(1−F1∩y,B(x)) N 2N(1−F1∩y,B(x)) + N 2N(1−F1∩o,B(x)) (1.5.31)

Above the threshold of the maximum young bid F1∩y,B(x)) is equal to 0, and therefore

the probability that the highest bid comes from a young bidder is also 0.

The probability of old winning in state B is the probability that the highest bidb1 is

from an old bidder and that it is above the observed price. The probability of old winning in state B is updated as below:

P rB(o|pt−1 =x) =P(b1 ∈o|B, pt−1) = N 2N(1−F1∩o,B(x)) N 2N(1−F1∩y,B(x)) + N 2N(1−F1∩o,B(x)) (1.5.32)

Above the threshold of the maximum young bidF1∩y,A(x) is equal to 0, and therefore the

probability that the highest bid comes from an old bidder is 1 above the threshold.

Both cumulative functions have the same weights in State B and the distribution of bids from old bidders dominates the distribution of bids from young bidders, but the two distributions have a different support. The distribution of Young bidders bids has a sharp increase not far from the cutoff where it ends. The distribution of old bidders bids is naturally more spread-out and more equally distributed in the middle section. This means that the First-Order Stochastic dominance will most likely not hold. It is not clear that with an increase inpt−1 there will be an increasing probability that the bid was from

an old bidder. The distribution of Young Bids is increasing faster than the distribution of Old Bids, but the sharp increase starts later.

This means that it is likely that for some values an increase inxis not equivalent with an increase in P rA(o|pt−1 = x) or P rB(o|pt−1 = x) . Above the threshold, though the

probability that a young bidder won decreases to 0. Just before the threshold of Young bids the Similarly to the Likelihood Ratio Dominance is likely to be satisfied, which is represented in the theorem below:

price observation in that period for prices in some upper-αrange until the threshold. After