Outbid Randomness

This consideration is interesting for auctions where strategic bidding is the best strategy (e.g., in Dutch auction or second price sealed bid auction). We want to know what the optimal bid for a strategic bidder is?

We assume:

• Value of the item 1 EUR (common value auction)
• N participants
• (N-1) participants bid randomly (bids are in the range of 0 and 1).
• Participant N is risk-neutral.

The bid g beats a random player with the probability of 0 <= g <= 1.

To be the highest bidder, our participant has to overbid all other competitors.  The probability for player N to win is:

P_w(g,N) = g^N-1

The expected value for the reward is:

E(R(g)) = P_W(g,N)*R(g) = g^N-1(1-g)

The expected reward is maximal for:

dE(R)/dg = (N-1)g^N-2(1-g)-g^N-1=0

g*=(N-1)/N

The expected maximal reward is:

E(R(g*)) = ((N-1)/N)^N-1*1/N

With growing number of participants, the probability to win converges to:

P_W = lim[N-> ∞]((N-1)/N)^N-1 = e^-1 = 0,368