Information and Efficiency in Thin
Markets over Random Networks
Michiel van de Leur
Università Ca’ Foscari Venezia & University of Amsterdam
[email protected]
Abstract
More information in a market (presumably) leads to a higher allocative efficiency. We consider a bipartite Erdös-Rényi network where buyers and sellers can
only trade if they are connected. We assume that valuations are private information and that traders use linear markup strategies. We show that in a thin market
the amount of information available to traders has a non-monotonic effect on efficiency. Information on one’s own links is preferable to no information, but adding
information about others’ links reduces efficiency.
In this paper we consider a market in which
transactions can only take place between linked
traders. These links are realised as in a bipartite Erdös-Rényi network[2] where every link is realised with the same probability independently of
each other. Traders use markup and markdown
strategies where the intensity of the markup or
markdown depends on the information set that is
available to the trader.
We calculate expected efficiency as a function of
the probability of a link for three nested sets of information on the network structure. This allows us
to determine whether more information improves
the expected allocative efficiency.
1. The Model
In a bipartite Erdös-Rényi network buyer i and
seller j are connected with probability p independently of other links. Trade is possible only if a
link exists. A buyer desires to obtain one unit of
a good and a seller desires to sell one unit. The
valuations vi of buyers and costs cj of sellers are
private information and are uniformly distributed
on the interval [0, 1].
All traders simultaneously submit their bids and
asks based on their valuation or cost. A buyer
ranks his connected sellers by their asks, and
a seller ranks his connected buyers by their
bids. Trades respect such preferences: preferred
buyer-seller pairs are matched with each other
until no further trades are possible. The trade is
executed at a price that is equal to the average of
bid and ask.
There is an incentive to act strategically and bid
below the valuation and ask above the cost to obtain a higher profit. Similar to Zhan & Friedman[3]
and Cervone, Galavotti & LiCalzi[1], we use convex markup and markdown strategies symmetric
on [0, 1]. These strategies transform the valuation
and cost as follows:
A buyer with valuation vi bids bi = vi (1 − mbi ).
A seller with cost cj asks aj = cj + msj (1 − cj ).
The values mbi and msj denote the intensity of the
markdown of buyer i and the markup of seller j.
The higher these values, the further away bids
and asks are from the valuations and costs. Depending on the information that is available to
traders, the intensity of markup and markdown
depends on the knowledge of the network structure.
2. The Information Sets
We study the Nash equilibrium in markdown and
markup strategies in this market depending on
the information set available to traders. The number of traders and the distribution of valuations
and costs are known. We consider the following nested sets of information about the network
structure:
• Limited
tial and full information strategies are similar albeit the volatility is significantly larger in the latter
case. Volatility of strategies has a negative effect;
lower markups cause a slightly higher efficiency
whereas higher markups may result in absence
of trade. Partial information leads to the highest expected efficiency and the negative effects
of higher markups for limited information and high
volatility for full information are similar.
information: The probability of a link is
known.
• Partial
information: The probability of a link is
known as well as the realisation of the own
links.
• Full
information: The realisation of the entire
network is known.
With limited information only the probabilities of
all networks can be calculated and hence the
equilibrium strategy depends only on the probability of a link. Partial information allows a trader
to base the strategy on the number of own links
and hence the equilibrium strategy depends on
the number of a player’s own links and the probability that other links are realised. With full information the entire network is known and the equilibrium strategies are based on the realisation of
all links.
This allows us to determine the expected efficiency in equilibrium as a function of the probability of a link.
3. Does more information lead to a higher
expected allocative efficiency?
To compare the expected efficiency given different information sets we consider a market with
two buyers and two sellers. We find that the
amount of information available to traders has a
non-monotonic effect on efficiency; irrespective of
the probability of a link, partial information leads
to the highest expected efficiency. For values of
p smaller than the benchmark c (≈ 0.16) limited
information outperforms full information and for
large values of p the opposite holds:
0 < p < c : E(effpartial) > E(efflimited) > E(efffull)
c < p < 1 : E(effpartial) > E(efffull) > E(efflimited)
These results can be explained by the equilibrium strategies for which the average value and
the volatility for every p are displayed in the graph
below. The limited information strategies are the
highest, but are not subject to volatility. The par-
IMW
4. Discussion
In a bipartite Erdös-Rényi market with two buyers and sellers, we compared three ordered sets
of information about the network structure. We
showed that partial information leads to the highest efficiency since markups in limited information and volatility of strategies in full information
are higher. Higher markups and larger volatility
increase the probability of absence of trades and
hence decrease the expected efficiency.
Knowledge of the own links rather than only
the probability distribution improves efficiency,
but adding knowledge of the links of others decreases efficiency. It is optimal, if only the realisation of own links is known and therefore more
information does not necessarily lead to a higher
expected allocative efficiency.
References
[1] R. Cervone, S. Galavotti & M. LiCalzi, Symmetric Equilibria in Double Auctions with Markdown Buyers and Markup Sellers, Artificial
Economics, 2009.
[2] P. Erdös & A. Rényi, On the Evolution of
Random Graphs, Mathematical Institute of the
Hungarian Academy of Sciences, 1960.
[3] W. Zhan & D. Friedman, Markups in Double
Auction Markets, Journal of Economic Dynamics and Control, 2007.
December 3, 2013
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