3 research outputs found
Fixed Price Approximability of the Optimal Gain From Trade
Bilateral trade is a fundamental economic scenario comprising a strategically
acting buyer and seller, each holding valuations for the item, drawn from
publicly known distributions. A mechanism is supposed to facilitate trade
between these agents, if such trade is beneficial. It was recently shown that
the only mechanisms that are simultaneously DSIC, SBB, and ex-post IR, are
fixed price mechanisms, i.e., mechanisms that are parametrised by a price p,
and trade occurs if and only if the valuation of the buyer is at least p and
the valuation of the seller is at most p. The gain from trade is the increase
in welfare that results from applying a mechanism; here we study the gain from
trade achievable by fixed price mechanisms. We explore this question for both
the bilateral trade setting, and a double auction setting where there are
multiple buyers and sellers. We first identify a fixed price mechanism that
achieves a gain from trade of at least 2/r times the optimum, where r is the
probability that the seller's valuation does not exceed the buyer's valuation.
This extends a previous result by McAfee. Subsequently, we improve this
approximation factor in an asymptotic sense, by showing that a more
sophisticated rule for setting the fixed price results in an expected gain from
trade within a factor O(log(1/r)) of the optimal gain from trade. This is
asymptotically the best approximation factor possible. Lastly, we extend our
study of fixed price mechanisms to the double auction setting defined by a set
of multiple i.i.d. unit demand buyers, and i.i.d. unit supply sellers. We
present a fixed price mechanism that achieves a gain from trade that achieves
for all epsilon > 0 a gain from trade of at least (1-epsilon) times the
expected optimal gain from trade with probability 1 - 2/e^{#T epsilon^2 /2},
where #T is the expected number of trades resulting from the double auction
Multi-unit Bilateral Trade
We characterise the set of dominant strategy incentive compatible (DSIC),
strongly budget balanced (SBB), and ex-post individually rational (IR)
mechanisms for the multi-unit bilateral trade setting. In such a setting there
is a single buyer and a single seller who holds a finite number k of identical
items. The mechanism has to decide how many units of the item are transferred
from the seller to the buyer and how much money is transferred from the buyer
to the seller. We consider two classes of valuation functions for the buyer and
seller: Valuations that are increasing in the number of units in possession,
and the more specific class of valuations that are increasing and submodular.
Furthermore, we present some approximation results about the performance of
certain such mechanisms, in terms of social welfare: For increasing submodular
valuation functions, we show the existence of a deterministic 2-approximation
mechanism and a randomised e/(1-e) approximation mechanism, matching the best
known bounds for the single-item setting