Indirect Effects Through Links Between States

The probabilities of states in t−1 can change in one direction or another, and this fur- ther triggers updating of probabilities for states in t. The expectation of future State probabilities in t and t+ 1 are affected through links between states (edges on the state diagram). We can measure the strength of the link between states through the expected probability of that link conditional on state of the origin node.

All of the periods that follow after t will have updated probabilities of states A and B. In addition this updating will be in the same direction as in state t:

Theorem 16. If the probability of StateB in t is increased following a price observation in t−1, then the probability of State B will also increase for all the following periods:

∀i∈N+ dP(Bt|pt−1) dpt−1 >0 =⇒ dP(Bt+i|pt−1) dpt−1 >0 (1.5.35)

The change in probabilities of the two States has the same direction for all future periods.

Proof. Implied by theorem 17.

In state B there are more old bidders than in state A. Looking at the ratio of probabilities of state B following state A, and probability of state B following state B allows to see what impact would an increase in probability of one of the states in t−1 (and decrease in the other) have. If the weight of the edge pointing to State B from

state B is higher than the weight of the edge pointing to State B from StateA, then the following is true:

Theorem 17. Increased probability of state B in any time period t leads to an increased probability of state B in the following period t+ 1.

This is proved below:

Proof. The probability of state B in t conditional on state B in t−1 or A in t −1 is

shown as below: P(Bt|Bt−1) = P rB(o) = Z VH 0 f1B(b)P rB(b=old)db= Z VH 0 f1B(b) 1 2fo,B(b) 1 2fo,B(b) + 1 2fy(b) db P(Bt|At−1) =P rA(o) = Z VH 0 f1,A(b)P rA(b =old)db= Z VH 0 f1,A(b) N−1 2N−1fo,A(b) N−1 2N−1fo,A(b) + N 2N−1fy(b) db

The proof will follow stepwise argumentation. In the first step it needs to be determined which of the statesAorB imply higher probability of old vs young winning. The second step is the fact that an increase in probability of old winning in t leads to an increase in the probability of state B, while an increase of the probability of young winning in t−1 implies an increase in the probability of state A int. Old bidders in both states, A and B bid higher than young bidders (who discount bids). State B has one more of old bidders than state A. The number of young bidders is exactly the same in both states. In addition to the reduced number of old bidders remaining in stateA, their distribution is also changed - they are the remaining bidders who in their young period did not win, even though one young bidder won in t−1. In state B, the remaining bidders have not won as young, but in the case where an old age bidder has won. Comparison of these two distributions of old bidders (or young in t−1, but after the winner has been removed) gives information about which distribution is dominating the other one. As old bidders, their distribution is transformed by a monotonous function, and therefore the same relation is sustained. If distribution of bidders during their young period dominates another distribution of bidders in their young period, then the first distribution of bidders

will also be dominating the second one in the old period. A distribution that dominates is the one that gives a higher probability of old winning. The distributions to compare are below:

The distribution of State A old bidders when they’re young (int−1):

fyt−1,A(z) = P(A)(fR,A(z)fR,y,A(z)∗C) +P(B)(fR,B(z)fR,y,B(z)∗C4) (1.5.36)

The distribution of State B old bidders when they’re young (in t−1):

fyt−1,B(z) = P(A)(fR,A(z)fy(z)∗C1) +P(B)(fR,Bfy(z)∗C2) (1.5.37)

We can compare the pairs of fR,y,A(z) and fy(z) as well as fR,y,B(z) and fy(z) in terms

of likelihood ratio dominance, since this is where the two equations differ. First of all:

fy(z) (1.5.38)

is the same in each state, A or B, since there are the same number of young bidders entering each period, and the valuations are drawn randomly. Next, for the first pair, we need to compare it to:

fR,y,A(z) =

1

N −1(N fy(z)−(f1,y(z)f1,A(z)∗C5))∗C6

Both, f1,y(z) and f1,A(z), are dominating fy(z) in terms of likelihood ratio dominance.

This implies that the distribution after subtraction and rescaling will be lower in terms of likelihood ratio dominance. In summary, this means thatfy(z) dominates fR,y,A(z) in

terms of likelihood ratio dominance. The distribution used in the second pair:

fR,y,B(z) =

1

Analogically, it is also dominated by fy(z) in terms of likelihood ratio dominance.

Remark. Both,fR,y,B(z)andfR,y,A(z)are dominated byfy(z)in terms of likelihood ratio

dominance.

The remark above implies that fyt−1,B(z) dominatesfyt−1,A(z) since a mixture of two

dominating distributions has to be dominating the mixture of two dominated distribu- tions. Since the old bidders bids are just a transformation (by b−1()) of young bidders bids, the fact that the distribution of young bidders bids int−1 of those old bidders who are in state B is dominating the distribution of young bidders who follow to be old in stateA implies that the old bidders distribution in stateB is dominating the old bidders distribution in state A.

The distribution of young bidders is the same in both cases, so the fact that in state B the distribution of old bidders is dominating in terms of likelihood ratio dominance the distribution of old bidders in state A implies that the probability of old winning in state B is higher than in state A.

All two of the edges pointing to state B have weights associated with the probability of old winning. If state B became more likely, then, out of the two edges, the one with origin at stateB will become more important. The fact that this edge has a higher weight than the other one (with origin at state A) means that it would result in an increase in expected probability of state B occurring in period t, following t−1. State A, on the other hand will become less likely in t if the probability of stateB increases int−1.

The above explains that if observed price leads to an update towards an increase in probability of state B in any state t, then in expectation the probability of state B in- creases int+ 1 as well (and the probability of stateA int+ 1 decreases). If the opposite happens, so stateA becomes more likely int, then in periodt+ 1 the probability of state B decreases, while the probability of state A increases.

Theorem 18. The update in probabilities of future states based on pt−1 is diminishing with time. ∀i∈N+,i>1 d di dP(Bi|pt−1) dpt−1 <0 (1.5.39)

Theorem 19. The effect of update of future probabilities is decaying at a diminishing rate: ∀i∈N+,i>1 d dt dP(Bi|pt−1) dpt−1 − dP(Bi+1|pt−1) dpt−1 <0 (1.5.40)

The convergence continues towards the equilibrium of no additional information: Theorem 20.

limi→∞P(Bi) = P∗(B) (1.5.41)

,where i > t

Of course the above theorems aboutP(Bt) relate also toP(At), but it follows directly,

since P(A) = 1−P(B)

The prevailing assumption here is that the observations are made only by an external observer, not by the bidders who take part in the auctions. In the case whereby young bidders could observe price in previous period, after updating their probability of states A and B, they would update their strategies accordingly. In the end the expected price path could not be increasing. The young bidders, by reducing their bids would most probably lead to correction of the price path. The strategies become very complicated, and include infinite dependencies to all the future and past periods.

From the point of view of the hypothetical external observer, the expected price path follow a proportional path to the probability of State B. The expected highest bid in State B is higher than in State A and the expected price is the weighted sum of probabilities and prices in both states.

probabilities of states. The expected price path will follow proportionally and monotonously the same path as the expected probability of StateB: initially the change will be the highest in t and after that it will start converging to equilibrium at a diminishing rate with time.