Traders’ Behaviors in Frequent Batch Auctions
In class, we talk about Budish, Cramton and Shim’s paper (“The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response”). They identify a problem with continuous time limit order book: sniping, which increases the risk faced by market maker, and, hence decreases the market liquidity and increases the spread. They propose to fix the market by replacing the continuous time limit order book with frequent batch auctions. In this problem, we will try to understand how traders would behave in batch auctions. There is a lot of different ways we could model this auction setting: different ways to think about what information bidders have and what payoffs they face. In this problem, I am going to introduce one particular way. We are going to get away from thinking about strategic behavior by both sides of the market: buyers and sellers, and think of a setting with fixed supply. So, there are k identical objects (e.g., shares) for sale, and there are a bunch of potential bidders, who can submit bids (potentially multiple bids if they want). Formally, we will consider the following model.
Who are the players?
N bidders: i in {1, . . . , N }.
Information and types
Each player’s type, vi, represents his or her value for the good. Each player knows their type, but not the one of their opponent. We will assume that the vi’s are independently and identically distributed: vi i.i.d. ∼ U ([0, 1]).
What is the action space?
Suppose for now that each bidder demand a single good. So, given their type, each player submits one bid: That is, a nonnegative number that depends on their type. So, for all i, Ai = R+.
Outcome
An outcome of the batch auction is an allocation rule, qi, that specifies who gets the good depending on the bids, and a payment, pi, which also depends on the bids submitted.
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For the batch auction, q and p take the following forms:
qi(bi, b−i) = ( 1 if bi is one of the kth highest bids
0 otherwise.
and pi(bi, b−i) = qi(bi, b−i)b(k+1), where b(k+1) is the (k + 1)th highest bid submitted.
So, bidders pay only when they get the good. Note that we ignore the possibility of ties. We do so because it will not affect the analysis and ties never happens in equilibrium.
Payoffs
Given this outcome, player i’s payoff is: ui(bi, b−i, vi) = viqi(bi, b−i) − pi(bi, b−i), i in {1, . . . , N }. Thus, each player’s expected payoff is their value for the good multiplied by their probability of getting it minus the expected payment payment. Crucially, each buyer’s payoff depends only on their value. The other bidder’s value for the item is irrelevant. So, we are interested in the private value sealed bid batch auction.
Show that each bidder bidding his type is a dominant strategy equilibrium. Hint: Think of the solution for the second-price auctions.
Argue that all the other players 2, . . . , N still want to bid their
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