Unusual limit theorems for the difference of order statistics from a Pareto

Abstract

In this paper we establish various limit theorems for the difference in order statistics from a sample from the Pareto distribution. The underlying density is f (x) = x −2 I(x ≥ 1).We look at both fixed and slowly increasing samples sizes. For our strong and weak laws of large numbers the first moment will be infinite and for our central limit theorem the second moment will be infinite. These theorems are quite unusual since the usual moment conditions do not hold. In order to achieve these results we must attach weights to these random variables and find these appropriate weights and norming sequences in order to establish our results.

Stackelberg Attacks on Auctions and Blockchain Transaction Fee Mechanisms

“…If the contract works, agent 3 enjoys utility v 3 −2ε, a better result than the first price auction utility of v 3 − v 1 − ε. The desirable outcome is also clear for agent 1: the lottery case yields utility 1 2 (v 1 − ε), which is better than the first price auction utility of 0. Therefore, both 1 and 3 will submit to the contract.…”

“…This models a case where there are three buyers who wish to purchase two identical items -we may imagine these big buyers to be exchanges that control large quantities of user transactions, such as Coinbase or Binance, which are juggernauts in the industry [2]. Note that in this example we have α = 1 2 . Suppose that agents 1, 2, 3 have valuations 0 < v 1 < v 2 < v 3 , respectively.…”

“…If agents 2 and 3 comply with the threat, i.e. bid ε and enter a lottery, they will have expected utilities 1 4 − ε 2 and 3 8 − ε 2 , respectively. These utilities are more desirable than ignoring the threat, and the attack can be executed.…”

“…These utilities are more desirable than ignoring the threat, and the attack can be executed. Agent 1 will enjoy expected utility 1 4 − 2ε. Note that agents 1 and 2 have higher utility than they would have had in the first price auction, but agent 3 is hurt by the attack.…”