Question

Write down the optimal strategies for n participants regarding the following mechanisms:
(a) First price sealed bid auction. (b) Second price sealed bid auction.

Answer 1

• First price sealed bid auction

Optimal strategy n participants -

The optimal strategy for the bidder is to bid the expected highest value for all remaining potential buyers. This takes into account the probability of various numbers of bidders. When the number of potential buyers has a geometric distribution, Bn = (1-n) 8n. Then, B*n = (1-8)8n-1 and Q(v) = (1-n)/(1-nG(v)). When n is small, Q(0) is large, B(v) will be near zero.

Consider G(v) = v for 0 < v < 1, which in the case of fixed n leads to B(v) = v(n-1)/n.

B(v) = v + log(1-nv)/n

• Second price sealed bid auction

Dominant strategy for each bidder is to opt for b_{i =} v_i, for the bidders the payoffs are never more than v_i. The value that i would pay if i wins remains the same but the change in the value of i only decides on whether i wins or loses.The payoff is determined by what other bidders are willing to pay. Truthful bidding is the dominant strategy. Second price bidder decides on the amount the winner pays.

The optimal strategy is to decide on the equilibrium value of expected payoff. If v_i is the payoff, the probability of this value being the highest is v_i^n-1

Expected payoff for the bidder is G(vi) = v_iⁿ⁻¹(vi − s(vi))

from the revelation principle, s(v_i) = 0.5