The expected payoff to Internet auctions (Q626292)

The independent private values paradigm (IPVP) is used to model the expected value of payoff at Internet auctions (IA). Under this paradigm the bidder's valuations at IA are i.i.d. random varables \(X_1\),\dots, \(X_n\) and the payoff is the second largest order statistic \(M_{n-1:n}\) of these valuations. The problem is to estimate the expected payoff \(E M_{n-1:n}\) when the actual number \(n\) of bidders in the auction is unknown as it is the case at IA. The authors investigate the limit behaviour of \(E M_{n-1:n}\) as \(n\to\infty\) and consider estimates for this value by observed payoffs when \(X_i\) are heavy/light-tailed with the extremal value index \(\gamma\). It is shown that for \(0<\gamma<2\) the observed payoff underestimates the expected payoff. For \(E\log M_{n-1:n}\) an adjusted consistent estimator is proposed but this adjustment doesn't work for \(E M_{n-1:n}\). For \(\gamma<0\), when the bidder's valuation distribution has a finite endpoint, the observed payoff overestimates the expected payoff. For \(\gamma=0\), the observed payoff is a consistent estimate for the expected payoff.