Underpricing in Market Games with a Distinguished
Player
Matthias Blonski
Ulf von Lilienfeld-Toaly
version 01/2006
preliminary, comments welcome
Abstract
We generalize asset pricing theory by merging it with ideas taken from corporate governance. We do this by introducing a so called distinguished player
de…ned by the ability to enhance the value of a company facing private e¤ort
costs. The current fundamental value of the company hinges then on the distinguished player’s prospective e¤ort which is determined by his ex post holdings and
hence by the outcome of the trading game. In this article we characterize underpricing trade equilibria for a general class of market mechanisms. They display
interesting theoretical properties. Shares of the company are traded below their
correctly anticipated value in order to discourage the distinguished player to sell
and withdraw from the company. In particular, this implies a substantial generalization and reinterpretation of the traditional no-arbitrage condition towards
a game-theoretic understanding. No rational investor can gain by deviating even
though shares of the company are traded below their true value. An intriguing empirically testable prediction of this model are excess-returns for companies with a
publicly visible distinguished player which if con…rmed would falsify models based
on the traditional no-arbitrage condition.
JEL Classi…cation Numbers: G12, G32, C72, D43, D46
Johann Wolfgang Goethe Universität Frankfurt, 60054 Frankfurt am Main, Germany; e-mail correspondence: blonski@wiwi.uni-frankfurt.de
y
Johann Wolfgang Goethe Universität Frankfurt, 60054 Frankfurt am Main, Germany; e-mail correspondence: lilienfeld@wiwi.uni-frankfurt.de
1
1
Introduction
In corporate governance theory a company’s value endogenously depends on executives’
decisions. In particular, the identity of the decision maker is known. In a standard asset
pricing no-arbitrage equilibrium, in contrast, a company’s value re‡ects the aggregated
information of all investors participating anonymously in the market. If executives act as
decision makers with known identity and as anonymous traders of the company’s shares
at the same time this not only involves a typical situation of asymmetric information.
Moreover, and on a more basic level, i.e. even in a framework with complete and symmetric information, it creates a semi-anonymous and genuinely interactive situation of
asymmetric trading incentives. The associated information asymmetry problem and its
implications on capital market valuation have been analyzed up to an impressive level
in the theoretical literature on market microstructure theory (Brunnermeier, 2001, for
example, provides a good starting point). In this article we address the complementary question of capital market valuation in the context of full information but with
asymmetric trading incentives. Within this framework we characterize true value and
underpricing equilibria and investigate their consequences for no-arbitrage. The phenomenon of trade at a price below the correctly anticipated equilibrium value in this
context has escaped theoretical analysis with the exception of Bolton and von Thadden
(1993) and the recent paper of von Lilienfeld-Toal (2005). Since underpricing implies
excess returns and thereby is of interest for investors and is empirically observable our
goal is a systematic theoretical understanding of this phenomenon within a more general
and uni…ed framework.
We proceed by formulating a market game setup that concentrates on incentives and
interactions among completely and symmetrically informed players who are rational
traders of shares of a business project (for example a company). One of our main generalizations with respect to standard capital asset pricing theory is to introduce players
who act as traders and as executives without considering any informational asymmetry
at the same time. We call these players distinguished players. A distinguished player
is de…ned by the ability to raise the value of the business project by working hard and
facing private e¤ort costs. By considering the ex-ante-market before the distinguished
player’s e¤ort decision this simple and seemingly obvious generalization of a classical
setup turns out to have profound and surprising consequences for some of the main
pillars of capital market theory that do not turn up in the theory focussing just on
information asymmetries.
2
As our main result we characterize trade and no-trade equilibria for a very general
class of market mechanisms. In particular, we can show that besides the usual true value
equilibria there can also exist underpricing trade equilibria where shares of the company
are traded in strictly positive quantity for a market price below the correctly anticipated
fundamental value. Surprisingly, underpricing trade equilibria continue to exist even for
positive trading costs. These equilibria, however, cannot occur without a distinguished
player.
The underpricing phenomenon super…cially appears to contradict the traditional interpretation of the no-arbitrage condition in equilibrium. A closer game theoretic inspection reveals, however, that in an underpricing equilibrium "no-arbitrage" is valid in
the sense that no investor can gain by buying or selling more or less. Rational traders are
aware of the fact that the shares are traded below their true value but also that trade
at the true market price would encourage the distinguished player to sell his shares
and save on e¤ort costs instead. Therefore, underpricing equilibria are characterized
by the property that any deviation that drives up the market price towards the true
value triggers the distinguished player to withdraw and not to raise the company value
to the anticipated level which in turn causes even bigger losses to everybody. This latter property of underpricing equilibria is called pivotalness and serves as a disciplining
o¤-equilibrium coordination device where wealth can be transferred between rational investors. Put di¤erently, a failure to coordinate on a su¢ ciently low market price below
the fundamental value could destroy wealth for all shareholders by removing incentives
to the distinguished player to work hard and generate positive externalities.
A crucial consequence of underpricing equilibria is that in contrast to the standard
theory "no-arbitrage" does not imply that the market price and the fundamental value
coincide. In underpricing equilibria those traders who manage to buy below the true
value are indeed able to realize strictly positive gains without any informational advantage whereas sellers su¤er a loss.
This theory yields a striking testable prediction that is clearly at odds with the traditional no-arbitrage condition in classical e¢ cient market theory. The theory predicts
that companies with a publicly perceived distinguished player1 on average should yield
excess returns relative to the whole market. Any evidence supporting this prediction
1
Clearly the perception among investors is the critical criterion. One obvious possibility to measure
this is the presence of a large block stakeholder. Although in many instances large stakeholders are not
executives in the narrow sense their marked in‡uence on crucial decisions is what identi…es them as
distinguished players.
3
would at the same time clearly falsify e¢ cient market models based on the traditional
no-arbitrage condition where every private and public information must be re‡ected in
the market price. In contrast, it would be compatible with our generalized no-arbitrage
condition2 . We are not aware of systematic tests of this particular prediction although
some existing evidence points in the right direction (Fahlenbrach, 2005).
This theory establishes progress in another important direction. It is well known
that results in market microstructure theory hinge critically on the exact speci…cation
of the market microstructure. For example, O’Hara (1995) discusses di¤erent widely
used market microstructure models and states already in her foreword (p. ix) that the
"generality of their results, and hence their applicability, is not well understood". This
property is bothering since any market microstructure is only an approximation of real
world trading systems and it is not known what constitutes a good approximation. In
this paper, however, we are able to derive general characterization results that do not
depend on the speci…cation of the market mechanism for market games with or without
a value increasing large shareholder engaged in a publicly listed …rm.
We expect to challenge skeptical readers wedded with the traditional no-arbitrage
interpretation in e¢ cient markets who might doubt that real world capital market investors are as rational as the present game theoretic explanation supposes. These readers
may doubt in particular our assumption that investors are rational enough to understand
the fact that their trading behavior can have an impact on the distinguished player’s
behavior and thereby on the company’s value. We answer this sort of criticism with the
following counterarguments.
First, our model is general in the sense that it contains the case of a frictionless market with no transaction costs and no externalities with the usual true value equilibria as
a special case. In this sense our results show that models with frictionless markets are
not robust with respect to the introduction of arbitrarily small3 distinguished players.
In other words, we do not see any theoretical reasoning that would allow us to dismiss
2
Conversely, the opposite …nding would not falsify our theory since it is a generalization of the
standard model without distinguished players. It would rather allow for the conclusion that either
distinguished players are not that relevant in reality or that if they are relevant the true value equilibrium
which still exists tends to be the chosen one in reality. Furthermore, von Lilienfeld-Toal (2005) explains
in more detail how a validation of excess returns for companies with publicly perceived distinguished
player can contribute to the unexplained gap in the so called equity premium since returns of those
companies with a distinguished player would raise average returns of the total market.
3
The size of the distinguished player is de…ned by his maximal contribution to the company’s fundamental value.
4
underpricing equilibria per se if generically distinguished players matter in the way described by this theory. We neither recognize a theoretical rationale as to why true value
equilibria should be dismissed and only underpricing equilibria should be relevant. Consequently, the question whether underpricing or true value equilibria are more relevant
in reality is an empirical question. Hence, it is necessary to wait additional tests that are
speci…cally tailored towards testing this theory. Notably, so far the empirical support
towards underpricing equilibria is more convincing than one may expect at …rst hand
which is laid out further in von Lilienfeld-Toal (2005). Conversely, it is well known that
standard asset pricing theory is far from …nding indisputable empirical support which is
mirrored in several "asset pricing anomalies" (e.g. equity premium, excess volatility, no
trade theorems, lockup agreements, limited stock market participation, ...).
Next, we do agree that many real traders probably are not as rational as supposed
in our theory. Although we do not further elaborate on this in the present article we are
convinced that a limited degree rather than full rationality is su¢ cient for our theory
in the sense that if su¢ ciently many shareholders recognize the advantages to keep the
market price below the fundamental value, they are able to compensate noise trading
and outright "irrational behavior" to a certain degree and uphold the Pareto superior
underpricing equilibrium with its characteristic excess returns.
Further, the multiplicity and rich equilibrium structure of this theory is not surprising
from the perspective of game theory since our model in fact is a semi-anonymous game.
Semi-anonymity of a game means that players’payo¤s only depend on aggregated actions
of player-types rather than on the individual action pro…le. In our market game we only
consider two types of players, regular investors and the distinguished player. In this
particular context, semi-anonymity is a crucial property since it means that traders do
not care about the composition of bids among regular shareholders. However, being of
a di¤erent type, the distinguished player’s actions distinctly enter the payo¤s of traders.
This circumstance brings about the dichotomic nature of true value versus underpricing
equilibria. Equilibria of semi-anonymous games in general have been characterized in
Blonski (2005). Kalai (2004) has shown that equilibria of semi-anonymous games are
surprisingly robust in various senses when the number of players gets large.
Finally, there may indeed be certain periods in time in which bubbles occurred and
the stock market as a whole has been irrational. However, we are convinced that during
these bubbles, numerous examples of companies exist in which promising executives have
indeed opted for selling their stocks. For example in the most recent dotcom bubble,
5
this was attractive as long as exuberant market prices has removed all their incentives
to proceed. This anecdotal observation if con…rmed by harder evidence would in our
view support the picture of this theory rather than obscuring it because it highlights
pivotalness and shows that selling stocks can be a valuable threat.
The structure of the paper is as follows. Section 2 introduces formally the idea of
a distinguished player and sets up the notation of the general market microstructure
model. Within section 2 we de…ne a deterministic market mechanism and come up with
two salient examples, the Kyle (1989) market mechanism and the Amsterdam market
mechanism, a real world trading system. Any particular market mechanism speci…es a
corresponding market game. In section 3 we present our main results and characterize
true value and underpricing trade equilibria for all deterministic market mechanisms. In
section 4 the main …ndings are generalized to stochastic market mechanisms. These for
example allow for the possibility that a price is picked randomly from several market
clearing prices. In section 5 we show existence of underpricing equilibria for the two
market mechanisms introduced at the end of section 2. Section 6 explains in more
detail the relationship to various branches of the existing literature. Section 7 discusses
extensions like multiple distinguished players, more general forms of e¤ort decisions,
and the introduction of a market maker while section 8 concludes. Appendix A explains
more rigorously the rich and general strategy space of the market game while Appendix
B contains all proofs.
2
Market Microstructure Model
Distinguished player. Denote by i = 0 a distinguished player being interpreted as a
founder or "manager-owner" of a business project. Further, denote by i = 1; :::; N outside
investors with (weakly) positive stakes in this business project. Distinguishedness of a
player-investor is de…ned as the ability to enhance the value of the project. In this paper
we study the simple binary case where the project either has value v or v
v+ v
v
and the realization of this value deterministically depends on the distinguished player’s
e¤ort decision e 2 f0; 1g. If the distinguished player exerts low e¤ort 0 at cost 0 the
project yields value v, if e¤ort is 1 incurring cost c with
v > c > 0 the project’s value
is deterministically raised to v. To sum up, v = v + e(v
0
c<
v), c(e) = ec. Clearly, for
v the e¢ cient or …rst best e¤ort choice is e = 1. To compare with known
models in the literature, we are also interested in the case
6
v =v
v = 0 where no
player is a distinguished player.
Project ownership. Our object of investigation is a market game where stakes of
the project can be traded before the distinguished player decides on his e¤ort decision.
The initial ownership structure of the project before the market game takes place is
exogenously given. It is de…ned by an element
(
=
= ( 0 ; :::;
N)
i
of the simplex
2 Q and
N
X
i
=1
i=0
)
where quantities Q can be either continuous Q = [0; 1] or discrete Q = 0; M1 ; M2 ; :::; M
M
[0; 1] with M indivisible shares. Within the latter interpretation, initially player i owns
iM
shares of the project. The market game to be described subsequently endogenously
results in the …nal ownership denoted by ! = (! 0 ; ! 1 ; :::; ! N ) 2
PN
i=0 ! i = 1.
with ! i 2 Q and
E¤ort choice. In the market game to be de…ned, stakes of the project can be traded
before the distinguished player takes his e¤ort decision. Once the market game is over
the distinguished player chooses e¤ort to maximize the net value
! 0 (v + e v)
c(e) = ! 0 v + e (! 0 v
c)
of his …nal stake ! 0 in the project. If he is indi¤erent between 0 and 1 we assume as tie
breaking rule that e¤ort is 1. Hence,
(
e=
1 for ! 0 v
0
c
otherwise.
Similarly, the payo¤ of any outside investor i = 1; : : : ; N after the market game is given
as ! i (v + e v), i.e. the …nal value of his stake after the distinguished player’s e¤ort
choice.
Prices and Strategies. We want to allow for models with discrete or continuous
prices. Therefore, we introduce P
R[ f 1; 1g as the set of feasible prices. Similarly
as for quantities the typical examples will be a continuous P = R[ f 1; 1g or a discrete
price range P = P := f 1; :::; v; v + ; v + 2 ; :::; v; :::; 1g with some exogenous tick
size . Real world market mechanisms distinguish between buy and sell prices pb ; ps .
7
The di¤erence
:= pb
clearly the bid ask spread
size, i.e.
ps
0 is called bid ask spread 4 . If prices p 2 P are discrete
is supposed to be a non negative integer multiple of the tick
2 N. From here p
ps always denotes sell prices whereas the corresponding
buy price is pb = p + . Since in this complete information context no buyer would ever
pay more than pb = v the relevant sell price range is [v; v
]. Clearly, all buy orders
depend on pb and all sell orders depend on ps .
Strategies or market actions ai 2 Ai of an investor are given by a pair of demand
and supply correspondences ai : P ! (}(Q) r ?)2 where }(Q) is the set of all subsets
of Q and
ai = fDi (p); Si (p)g
are the possible demand and supply correspondences. Together, Zi (p) = Di (p) Si (p)
Q is a set of positive or negative net quantities composed by demand and supply quantities that would be acceptable for investor i at price p.5 In appendix A we describe
in more detail how the rich mathematical object of an excess demand correspondence
is the result of a general and realistic set of possible orders such as buy and sell limit
orders, market orders, stop orders and all or nothing orders (…ll or kill orders).
Deterministic market mechanism. By adding up individual behavior a strategy
pro…le a 2 A induces the market excess demand correspondence
Z(p) =
X
Zi (p):
i=0;:::;N
Again market excess demand can be decomposed into buy and sell o¤ers. Call
D(p) =
X
Di (p) and S(p) =
X
Si (p)
i=0;:::;N
i=0;:::;N
the market demand and market supply correspondences. They de…ne sets of quantities
the market as a whole is willing to buy or to sell at a given sell price p. The limit order
4
We explicitly want to include the case of bid ask spread
= 0 to be able to compare this model
with models that abstract away from bid ask spreads.
5
To allow traders as in reality to choose demand and supply rather than just the sum of both, i.e.
excess demand, allows for "beller strategies" in which a trader might, for example, try to bid up the
stock price by submitting buy orders and simultaneously selling stocks. It turns out that these strategies
complicate existence proofs but we want to consider them since they are allowed in many real world
trading systems.
8
trade volume (p) for p 2 P is de…ned by the quantity
(
)
X
X
(p) = min sup
di (p); sup
si (p)
i=0;:::;N
i=0;:::;N
of the short side of the market restricted to limit orders and stop orders6 but excluding
all-or-nothing orders. Let
(p0 ) 8p0 2 P g
P = fp 2 P j (p)
denote the set of prices maximizing the limit order stop order trade volume. Denote by
= (x; y) = ((x0 ; :::; xN ) ; (y0 ; :::; yN )) a buy-sell-transaction vector and
n
X
=
= (x; y) xi 2 Q; yi 2 Q;
xi yi = 0; + x y 2
o
the set of feasible buy-sell transaction vectors and x y a corresponding net trade vector.
For initial allocation
and net trade vector x y the …nal allocation is ! = +x y 2
De…nition 1 For any initial ownership
ministic market mechanism
2
and any strategy pro…le a 2 A a deter-
is a mapping
:
P
.
A!P
where for initial ownership , bid ask spread
and strategy pro…le a the market mech-
anism ( ; ; a) = (p; x; y) picks a sell price p and buy price p +
and for any player a
subset of submitted orders being executed, i.e.
xi 2 Di (p); yi 2 Si (p):
We call this latter property "voluntary trade". Hence, net trades ! i
by submitted orders, i.e. for i = 0; :::; N holds ! i
total trading volume including all orders is
:
de…ned by
6
( ; ; a) =
PN
i=0
i
i
are composed
2 Zi (p). Accordingly, the resulting
A ! [0; 1]
P
xi ( ; ; a).
The small letters di and si indicate convex valued demand and supply correspondences composed
only by limit orders and stop orders. For details see appendix A.
9
The property xi 2 Di (p); yi 2 Si (p) means that trade is voluntary and only submitted
orders can be executed, i.e. nobody can be forced to trade. Market mechanism
to maximize the trade volume if
is said
picks a price that maximizes the limit order and
stop order trade volume (p), or p 2 P . Clearly, market mechanisms can be speci…ed
that do not maximize the trade volume. For example, price could be determined by
maximizing the total trade volume including …ll or kill orders. Or the distinguished
player or other players could be treated with priority if they submit …ll or kill orders.
However, we are not aware of any real world market mechanism not using maximal trade
volume with top priority. Nevertheless, our main results, theorems 2 and 3 do not rely on
maximal trade volume. However, we will see in theorem 1 that a property like maximal
trade volume may be su¢ cient to guarantee existence of trade equilibria. Moreover, we
call a market mechanism anonymous if permuting players’names does not a¤ect price
and …nal allocation. Our approach is to assume as little as possible about the market
micro structure. The following two examples are chosen to demonstrate that real world
market mechanisms as well as theoretical market mechanisms that fare prominently in
the literature are contained in the present setup. Both examples will be studied in more
detail in section 5.
Example 1 Amsterdam Market Mechanism
A.
Price Setting:
(i) The price is set to maximize trade volume (p).
(ii) Should there be more than one such price, the number of unexecuted orders is
minimized.
(iii) Should there still be more than one potential price, the minimal price will be
taken if there is excess supply. For excess demand, the maximum price is
taken.
(iv) Should there still be more than one price, the price closest to a reference price
will be chosen and we choose v to be the reference price.7 Fill or kill orders
can be submitted. However, they do not have an impact on price setting.
7
The reference price in real world trading systems is the last traded price (XETRA, p. 27). Since our
model only allows for one round of trading, we cannot use the last traded stock price as the reference
price.
However, we think that either of the following two stories deliver a justi…able reference price. In a
…rst situation, the distinguished player and all other stock market participants have learned just before
10
This means that the price (and the corresponding executable trading volume
or excess demand) are calculated as if the …ll-or-kill order was not present.
Only after the price has been set and after all other orders were executed will
the …ll-or-kill order be executed, if possible.8
Allocation:
(i) Orders are executed according to price priority.
(ii) If there are unexecuted orders that use the market price as limit price, orders
will be executed proportionally. If, for example, at an equilibrium price p we
have excess demand, every order will be executed using a factor d(p )=s(p ) with
P
PN
d (p ) = N
i=0 di (p ) and s (p ) =
i=0 si (p ).
(iii) Moreover, if more than one …ll-or-kill order can be executed at the auction
price, it is assumed that the allocation guarantees the maximum executable
trading volume.
(iv) If …ll or kill orders on opposing sides of the market exist, they will only be
matched against each other at the auction price if they cannot be executed
against normal bids in the auction.
Example 2 Kyle Market Mechanism
K.
In contrast to the Amsterdam trading
rules, our reformulation of the Kyle microstructure is as follows:
(i) The price is set to maximize the trading volume and
(ii) the quantity allocation maximizes trading volume.
(iii) Among those prices and quantity allocations obeying (i) and (ii), Kyle’s (1989)
market mechanism picks a price p that minimizes absolute value. If both p and
p
satisfy this property the market mechanism picks the positive price.
period t = 0 that a distinguished player is present and the stock was traded at its stand alone value
before, i.e. at a price p = v. In a second story, we see our game as a subgame in an in…nitely often
repeated game and hence assume that our candidate equilibrium price will persist over time. In this
case the stock price chosen as reference price will be any candidate equilibrium price.
8
A
description
of
the
Amsterdam
stock
exchange
(AON
are
nothing
orders
which
is
another
word
for
…ll
or
kill
http://www.keytradebank.com/form.html?level=form&option=rul&market=aex
orders)
from
all
as
is
or
taken
similar:
" on the segment of the double auction, ... the …xing price is calculated without the AON orders. Just
before the …xing, the AON orders are added to the orderbook."
11
P
(xi )2 + (yi )2 .
(iv) Market transaction vector
minimizes
(v) Market transaction vector
does not execute …ll or kill orders9 .
Market game
i
( ; ; ). Any market mechanism
and bid ask spread
induces a market game
together with an initial ownership
( ; ; ) with strategy space A and
payo¤ functions given by
ui (a) = ! i (v + e v) (p + ) xi + pyi
(
! i v (p + ) xi + pyi for ! 0 v c
=
! i v (p + ) xi + pyi
otherwise
for i = 1; :::; N and
u0 (a) = ! 0 v + e (! 0 v c) (p + ) x0 + py0
(
! 0 v c (p + ) x0 + py0 for ! 0 v c
=
! 0 v (p + ) x0 + py0
otherwise
for the distinguished player i = 0.
A strategy pro…le a is a pure strategy mar-
ket game Nash equilibrium or just equilibrium under market mechanism
if every
player plays a best response ai to other players action pro…le a i . Correspondingly,
(p ; x ; y ) = ( ; ; a ) and ! =
y are the respective equilibrium price, equi-
+x
librium trades and equilibrium allocation induced by market mechanism
game equilibrium a . Here, the property ! i
i
and a market
2 Zi (p) guarantees that for any deter-
ministic market mechanism there always exists a no-trade equilibrium where all players
submit no buy and sell orders. In the remainder of this article we concentrate on more
interesting equilibria where we can observe a price such that trade occurs.
3
Trade Equilibrium Characterization
Equilibrium buyers and sellers. An equilibrium a of the market game
( ; ; )
where trade (p ) > 0 occurs at equilibrium price p is called a trade equilibrium of
. A player i who buys a strictly positive quantity in equilibrium is called equilibrium
buyer and conversely someone who sells a strictly positive quantity in equilibrium is
9
Actually Kyle does not allow …ll or kill orders which is equivalent to not being executed. This
assumption is necessary to guarantee convex valued excess demand correspondences. Our results,
however, do not rely on this assumption.
12
called equilibrium seller. The voluntary trade property implies that an equilibrium
buyer submits buy orders di (p ) > 0 and for an equilibrium seller si (p ) > 0. Clearly, in
any trade equilibrium there must exist equilibrium buyers and sellers. Note, however,
that in principle it may happen in equilibrium that the same player i is a buyer and a
seller at the same time10 .
We call player i wealth constrained i¤ i can only submit sell orders11 . For a wealth
constrainedn player i a bid strategy consists only of sellingo bids and therefore takes the
1
2
form ai = ( 1i ; 2i ; :::); ( 1i ; 2i ; :::); ( 1i ; 2i ; :::); ( i ; i ; :::) or zi (p) = Si (p).
True value trade equilibria. A trade equilibrium with high e¤ort e = 1 and equior with low e¤ort e = 0 and equilibrium price p = v is called
librium price p = v
true value trade equilibrium since the project is traded for the fundamental price, i.e.
the same price as the project value enters the payo¤ functions.
Theorem 1 Consider the market game
( ; ; ) de…ned by market mechanism , bid
ask spread , and initial ownership .
(i) For strictly positive bid ask spread
> 0 high true value trade equilibria do not
exist in which no investor k 2 f1; :::; N g is pivotal in the sense that selling less
or buying more than speci…ed by equilibrium strategies triggers the distinguished
player i = 0 to sell more or buy less and to reduce e¤ort subsequently.
(ii) For strictly positive bid ask spread
> 0 low true value trade equilibria do not
exist.
(iii) Assume from here bid ask spread
= 0. For
libria exist if the market mechanism
0
v > c true value trade equi-
maximizes the trade volume and if ei-
ther quantities are continuous or the number of shares M is su¢ ciently large
(M >
v= (
0
v
c)). For
0
v > c any true value equilibrium is a high
price equilibrium with p = v. If conversely
equilibria exist if the market mechanism
0
v < c and v > 0 true value trade
maximizes the trade volume and if ei-
ther quantities are continuous or if M is su¢ ciently large. Any of them yields low
price p = v.
10
11
If this happens we could denote this player as equilibrium "beller ".
For example, this is likely to be the very reason why the distinguished player needs funding by
outsiders. Otherwise he would prefer to run the project himself.
13
Clearly, to be an equilibrium seller in the high true value equilibrium is unpleasant
since equilibrium sellers obtain p = v
for something that is worth v in equilibrium.
The only way to keep them happy in a true value trade equilibrium is pivotalness, i.e. the
o¤ equilibrium threat of the distinguished player to decrease their payo¤ even further.
Traditionally noise traders have to be introduced to initiate trade as long as transaction cost is strictly larger than 0. This observation dates back to the work on no-trade
theorems, as for example in Milgrom and Stokey (1982). Already theorem 1 has shown
that this observation is only partly con…rmed if we consider a market game equilibrium
with a distinguished player. In our framework we propose trade equilibria without noise
traders displaying another explanation for trade. More intriguingly, theorem 1 rules out
trade equilibria with bid ask spread
> 0 unless some traders are pivotal, i.e. they are
forced to sell by the distinguished player. Since in true value equilibria the distinguished
player has no strict incentive to enforce trade at the true price without noise traders no
trade seems to be as realistic.
Underpricing trade equilibria. Theorem 1 does not rule out an equilibrium with
v
p <v
and high e¤ort e = 1. We call such equilibria simply underpricing trade
equilibria. We will see that in contrast to true value equilibria in underpricing equilibria
there may be strict incentives for the distinguished player to support trade at a lower
price. Hence, pivotalness becomes a substantial part of the structure. Now, we turn to
the intriguing questions. Do underpricing trade equilibria exist? If yes, how do they look
like? For equilibrium theorists it is not surprising that existence is the more di¢ cult
task which is postponed to section 5. We continue with the more general results, the
characterization of underpricing trade equilibria.
Theorem 2 Let a be a trade equilibrium with equilibrium price v < p < v
equilibrium net trades !
properties are satis…ed.
6= 0. Then for any market mechanism
and
the following
(i) Equilibrium a is an underpricing trade equilibrium, i.e. the distinguished player
exerts e¤ort e = 1.
(ii) In equilibrium a the distinguished player i = 0 is not a seller ! 0
0.
(iii) Each investor is pivotal in the sense that selling less than speci…ed or buying more
than speci…ed by equilibrium strategies triggers the distinguished player i = 0 to sell
more or buy less and to reduce e¤ort subsequently or it triggers a price increase.
14
(iv) If
0
c= v, the distinguished player i = 0 submits a selling order at least of
quantity s0
0
c.
No distinguished player. This formulation of the model certainly contains the special case
v = 0 with no distinguished player. The following theorem shows that models
without distinguished players have no underpricing equilibria and therefore are not robust with respect to the introduction of arbitrarily small distinguished players
v>0
if underpricing equilibria exist.
Theorem 3 For a model without a distinguished player
v = 0 underpricing trade
equilibria do not exist.
Our interpretation is that the presence of a distinguished player should turn our
attention to both, true price trade equilibria and to underpricing trade equilibria.
4
Stochastic Market Mechanisms
Stochastic trade equilibria. Real world market mechanisms often are speci…ed by a
list of rules with decreasing order of priority. Sometimes there remains some ambiguity
with respect to equilibrium price or allocation if all rules are satis…ed by more than one
price and/or set of executed orders such that a random choice may be implemented. In
this section we consider risk neutral investors facing a stochastic market mechanism.
A stochastic ownership structure ~ 2 ~ is an element of the space of probability
measures ~ on simplex
. Accordingly, de…ne stochastic market prices p~ 2 P~ and
stochastic trade vectors (~
x; y~) 2 ~ .
De…nition 2 For any initial ownership
2
market mechanism ~ is a mapping
~:
P
and strategy pro…le a 2 A a stochastic
A ! P~
where for initial ownership , bid ask spread
~
and strategy pro…le a the market mech-
anism ~ ( ; ; a) = (~
p; x~; y~) picks a stochastic sell price p~ (and buy price p~ + ) and for
any player trade is voluntary. This means that only submitted orders can be executed,
i.e. for any state of nature (~
xi ; y~i ) 2 Di (p)
Si (p) and therefore !
~i
the stochastic trading volume is
~ :
A ! [0; 1]
P
15
i
2 Zi (p). Again,
and ~ ~ ( ; ; a) =
PN
i=0
x~i ( ; ; a).
The distinguished player picks his e¤ort decision e after the stochastic market game
is over and the realizations of all random variables are known. Denote by e~ the random
e¤ort decision induced by the realization of (~
x; y~) which determines the …nal stake of the
distinguished player. Similar as before, a stochastic market mechanism ~ together with
an initial ownership and bid ask spread induces a stochastic market game ~ (~ ; ; )
with strategy space A and risk neutral payo¤ functions given by
!0 v
u0 (a) = E [~
! 0 v + e~ (~
ui (a) = E [~
! i (v + e~ v)
c)
(~
p + ) x~0 + p~y~0 ] and
(~
p + ) x~i + p~y~i ] for i = 1; :::N
where E means expectation value.
Stochastic true value and underpricing equilibria. A stochastic true value trade
equilibrium is an equilibrium where E (p
~ ~ ) = E (~
v ~ ~ ) with E (~ ~ ) > 0 and a
stochastic underpricing trade equilibrium is de…ned as an equilibrium where E (p
~~ ) <
E (~
v ~ ~ ) with E (~ ~ ) > 0. With the appropriate modi…cations of the deterministic case
the following theorems 4-6 show that the essence of theorems 1-3 is not a¤ected by
switching from a deterministic to a stochastic market mechanism.
Theorem 4 Consider the stochastic market game ~ (~ ; ; ) de…ned by stochastic market mechanism ~ , initial ownership
spread
and (by assumption) a strictly positive bid ask
> 0. Then, a true value stochastic trade equilibrium does not exist in which
there are no pivotal rational investors.
Theorem 5 Let a be a stochastic underpricing trade equilibrium of the stochastic market game ~ (~ ; ; ). Then, the underpricing equilibrium a satis…es the following properties.
(i) In equilibrium a the distinguished player i = 0 exerts e¤ort with positive probability.
(ii) The probability that the distinguished player i = 0 is not a seller ! 0
0
is strictly
positive.
(iii) Each investor is pivotal in the sense that a deviation from equilibrium strategies
that results in selling less or buying more or decreasing the probability to sell or
16
increasing the probability to buy triggers to increase the probability that the distinguished player i = 0 sells more or buys less and reduces e¤ort subsequently or
triggers an increase in the expected price.
(iv) If
0
c= v, the distinguished player i = 0 submits a selling order at least of
quantity s0
0
c.
As in the deterministic case, in a model with
v = 0 without distinguished player
again any costly e¤ort is useless since it does not yield anything. Hence e¤ort is e = 0
with probability 1 which again rules out the stochastic underpricing equilibrium. This
proves the next theorem.
Theorem 6 For a stochastic market game ~ (~ ; ; ) without a distinguished player
v = 0 underpricing trade equilibria do not exist.
5
Existence of Underpricing Equilibria
So far we learned something about the nature of underpricing equilibria but we don’t
know whether they exist. In this section we explicitly construct underpricing equilibria.
Unsurprisingly, the details of the structure of an underpricing equilibrium depend on the
speci…cation of the market microstructure. Therefore in this part of the paper we have
to be more speci…c and therefore we continue with the two examples given in section 2.
Amsterdam Market Microstructure The Amsterdam Trading Rules are adapted
from the Euronext Amsterdam stock exchange. We picked the Amsterdam Market
Mechanism as an example for real world trading systems.
Proposition 1 Let i = 0 be a wealth constrained distinguished player with initial stake
0
c= v. Let prices be discrete with su¢ ciently small tick size
tinuous variables. Consider the market game
spread
where
A
(
A;
and quantities be con-
; ) with su¢ ciently small bid ask
is de…ned by the rules given in the example of the Amsterdam Market
Mechanism. Then, there are initial ownership structures
equilibrium exists.
17
such that an underpricing
The second set of rules, our reinterpretation of the Kyle (1989) market microstructure12 was chosen because this market microstructure is well known in the literature.
Thereby it is possible to compare our results with the results of other papers in the
literature building on the Kyle market microstructure.
Proposition 2 Let i = 0 be a distinguished player with initial stake
prices be discrete with su¢ ciently small tick size
Consider the market game
(
K;
0
> c= v. Let
and quantities be continuous variables.
; ) with su¢ ciently small bid ask spread
where
K
is de…ned by the rules given in the example of the Kyle Market Microstructure. Then,
there are initial ownership structures
6
such that an underpricing equilibrium exists.
Related Literature
Since there is a long tradition and a huge literature on capital market valuation we only
refer to contributions that are most closely related to the two important dimensions of
progress in this context that we claim for our paper, namely merging ideas of capital asset
pricing and corporate …nance to explain underpricing and expanding and generalizing
the theory of market microstructure. As far as the market microstructure is concerned,
our general results host a wide variety of contributions in the literature.
Most of the literature concerned with large shareholders and trading games are exclusively interested in true value equilibria. Prominent examples include Shleifer and
Vishny (1986), Admati, P‡eider and Zechner (1994), Maug (1998), DeMarzo and Urosevic (2000), Kahn and Whinton (1998) or Magill and Quinzii (2002). These papers all
operate in a similar environment. They are interested in a large and value increasing
shareholder who may increase a …rm’s value while increasing a …rm’s value causes private
e¤ort costs. These papers are more general in other important respects (continuous effort, uncertain environments, repeated interactions, ...) while our paper is more general
with respect to the market microstructure.
As far as the focus on underpricing equilibria is concerned, closest to our paper is
von Lilienfeld-Toal (2005). This papers investigates underpricing equilibria in speci…c
market microstructures and turns attention to the empirical implications of underpricing
or excess returns. In particular, it argues that underpricing equilibria are consistent with
12
Actually, Kyle (1989) did not mention trade volume maximization. However, without this latent
assumption and market transactions minimization the market mechanism would always implement no
trade. It is obvious from Kyle’s paper and its successors that this is not what they investigated.
18
i) negative abnormal returns around unlock days and ii) positive abnormal returns for
founder-CEO …rms.
We mentioned in the introduction that underpricing equilibria may occur in the
model of Bolton and von Thadden (1993) which is also concerned with corporate control issues. This article does not focus on (asset) pricing implications of underpricing
equilibria and in particular does not relate underpricing to no-arbitrage in asset pricing.
Rather, they are interested in the question when blocks of shares remain, vanish or are
newly created. The reason as to why underpricing equilibria may exist in the model of
Bolton and von Thadden (1993) is similar to our notion of pivotalness. It has also been
employed by Bagnoli and Lipman (1985) and Holmstrom and Nalebu¤ (1992). The
latter papers analyze potential solutions to the free rider problem …rst mentioned by
Grossman and Hart (1980).13
Since all above mentioned papers derive the results using a particular market microstructure, we are not aware of characterization results that are valid for a broad class
of market microstructures. On the other hand, our assumptions on complete symmetric
information in a static market game are clearly more speci…c in other respects than a
large number of the above mentioned contributions.
Our setting can be viewed as double sided auctions with strategic trading and the
paper relates to this branch of market microstructure theory. Papers falling within our
framework are for example Kyle (1985), Kyle (1989) or Rochet and Vila (1994). While
their exact speci…cation of price setting and quantity allocation rules is within our class
of market mechanisms, the economic environment we are interested in is distinct as
compared to these papers. Moreover, market microstructure theory is also interested
in the price impact of individual trades which is aptly pointed out by O’Hara (2003):
"... asset pricing ignores the central fact that market microstructure focuses on: Asset
prices evolve in markets".
Our paper is also related to the literature on no trade theorems, for example Milgrom
and Stokey (1982) or Tirole (1982). The driving force behind no trade theorems is the
fact that there are no gains from trade or negative gains from trade in the presence of
transaction costs. In the class of models we are interested in, gains from trade are zero
for true value equilibria and consequently, true value equilibria in the traditional sense
fail to exist for positive bid ask spreads. In underpricing equilibria, in contrast, gains
13
In fact, Bagnoli and Lipman (1985) and Holmstrom and Nalebu¤ (1992) can also be seen as a
special case of our model if the strategy space of the distinguished player is limited to a takeover bid
and other shareholders can only submit sell orders.
19
from trade are no longer zero sum since the owner manager’s threat to sell is viable and
trade prevents the owner manager from selling.
7
Extensions
It is straightforward to set up a similar theory with more than one distinguished player
j = 0; :::; M each of them characterized by his stake, e¤ort cost and potential value
j ; cj ;
vj . In an e¢ cient equilibrium those for which cj <
vj should exert e¤ort.
Clearly, since distinguished players can behave as regular investors as well the presence of
a single distinguished player is su¢ cient to raise the possibility of underpricing equilibria.
For empirical testing we expect that the signi…cance of the most important distinguished
player is a better predictor for excess returns than the number of distinguished players
since by triggering this player to withdraw causes the biggest payo¤ losses. If the market
price o¤ers the right incentives for this distinguished player it should as well do for minor
distinguished players.
Another minor and straightforward extension is to extend the e¤ort choice model to
a more realistic model with a more general set of e¤ort actions and a stochastic value
production function. As long as the e¤ort choices that maximize the preferences of the
distinguished player and the e¢ cient e¤ort choice from the perspective of all traders
both are unique and do not coincide the reduced form of the problem shrinks exactly to
the binary problem with high and low e¤ort proposed in this article.
So far, we have not discussed the potential role of market makers in our model even
though our market microstructure allows for an analysis of market makers. Analyzing
the role of market makers can be done as follows. Suppose agent N + 1 is the market
maker. Moreover, agent N + 1 submits buy and sell orders (1; 1) and (1; 1). In other
words, the agent N + 1 submits to buy and sell the entire …rm. Hence, any market
imbalance can be bought by agent N + 1 or sold to agent N + 1 without violating our
"voluntary trade" assumption. Moreover, the market maker is programmed such that it
maximizes uN +1 , potentially subject to institutional rules such as market clearing, i.e.
o¤ering a clean up price. The resulting allocations and prices will be the same as in a
market microstructure in which all agents i = 0; 1; :::; N submit their shares, a market
maker observes these orders and sets a price to maximize his utility, subject to obeying
institutional rules.
20
8
Conclusions
We characterized equilibria for market games with a value increasing large shareholder
for a large class of market microstructures. Several lessons can be learned from our
characterization results. No trade theorems remain valid for true value equilibria while
positive trade volumes might occur even if all agents behave rational if we are interested in underpricing equilibria. There exists little theoretical foundation to dismiss
underpricing equilibria per se as they can only be ruled out if a …rm does not have a
distinguished player at all.
As far as underpricing equilibria are concerned, pivotalness plays a crucial role. In
our interpretation, pivotalness, i.e. the price impact of players in conjunction with
the supposition that players are aware of it, can be seen as an extension of the usual
property of arbitrage free stock markets if one considers underpricing equilibria. Hence, if
underpricing equilibria turn out to be an empirically important aspect, pivotalness must
be seen as an important characteristic of the equilibrium. As a consequence, observing
abnormal returns due to a certain investment strategy, as documented for example in
Fahlenbrach (2005), need not be a sign of irrational behavior but might be the result of
underpricing equilibria and (even stronger) rational behavior due to pivotalness.
Since our model exhibits multiple equilibria, empirical observations might help to
judge which equilibria are relevant in reality. Building on this observation, the following
questions can be addressed in future empirical research: Can we observe underpricing
equilibria in reality? Can we detect the importance and magnitude of private costs of
control from these underpricing equilibria or can we rule out underpricing equilibria.
Are there any di¤erences in trading volume for …rms with and without a distinguished
player, especially if we control for transaction costs? Moreover, it might be possible to
test more general aspects of game theory using stock market data and the methodology of
games in aggregated form developed in Blonski (2005) which allows robust predictions on
the structure of large semianonymous games without speci…c knowledge about individual
preferences. The numerous implications of underpricing equilibria promise an interesting
combination of theoretical and empirical work on incentives, game theory and asset
pricing.
Our model is a static model which does not incorporate any dynamic aspects of
real world trading. Therefore, it is important to see how our model fares in a dynamic
version. Finally, as emphasized in the introduction information is asymmetric if distinguished players trade in stock markets. It is an interesting and challenging project to
21
study the interplay between the structure of underpricing equilibria with informational
asymmetries. As the notion of no-arbitrage equilibrium in traditional capital market valuation theory generalizes from the complete information case to the case of information
asymmetries we conjecture that the notion of generalized no-arbitrage as formulated in
this article generalizes as well to the case with informational asymmetries.
9
Appendix A
Orders. The following description of possible market actions denoted as orders is
chosen to model real world market mechanisms as closely as possible. First, denote
by B = Q
pb = p +
quantity q
P the space of buy limit orders with typical element
= (b; p) where
2 P denotes the limit price up to which a player is willing to buy any
b 2 Q. Conversely, for a sell limit order
= (s; p) 2 S = Q
P the price p
is the minimal price from which the submitting trader is willing to sell q up to quantity
s. Buy and sell orders can be interpreted as upward sloping step correspondences. For
= (b0 ; p0 ) is precisely de…ned by the correspondence
example, the buy limit order
: P ! Q where
(p) =
A trader who submits
(
b0 g for p
fq 2 Q jq
otherwise
0
= (b; 1) or
p0 +
:
= (s; 1) is said to submit a market order
since a certain quantity is ordered for buy or sell independently of price. Market order
correspondences are bounded by vertical lines. Similarly, the graphs of buy stop orders
denoted by
= (bst ; p) and sell stop orders denoted by
= (sst ; p) are upward-sloping
step correspondences. The interpretation is that any quantity up to bst is bought above
price p or quantity sst is sold below price p. We also allow so called …ll or kill orders or
all or nothing orders that specify that a certain quantity is to be bought or sold entirely
or not at all. A …ll or kill order is denoted by
for
= (b0 ; p0 ), the related correspondence
de…ned as
(p) =
(
= (b; p), , , or . More precisely, say
: P ! Q has a non-convex graph and is
f0; b0 g for p
p0 +
otherwise
0
.
Strategies. A market game strategy ai of player i = 0; :::; N is a collection of orders
ai = f(
(
1
i;
1
i;
1
2
i :::); ( i ;
1
2
i :::); ( i ;
2
1
i ; :::); ( i ;
2
i ; :::); (
22
1
i;
2
1
2
i ; :::); ( i ; i ; :::);
2
1
2
i ; :::); ( i ; i ; :::)g:
Denote by Ai the corresponding strategy space of player i and by A = A0
AN the
strategy pro…les. Adding up buy and sell orders for some player i yields the individual
excess demand correspondence
Zi (p) = zi (p) + zi (p) composed by
X
(p) + (p)
(p)
zi (p) =
(p) and
zi (p) =
(p)
; ; ; 2ai
X
(p) + (p)
(p)
; ; ; 2ai
adding up buy and sell orders14 of player i. A market game strategy can be decomposed
into buy orders and sell orders. Denote by
X
X
X
X
Di (p) = di (p) + di (p) =
(p) +
(p) +
(p) +
(p)
2ai
2ai
Si (p) = si (p) + si (p) =
X
(p) +
2ai
X
2ai
2ai
(p) +
2ai
X
2ai
(p) +
X
(p)
2ai
player i’s individual demand and supply correspondences given as quantities player i is
willing to buy or to sell at a given price p. In particular di (p) and si (p) specify individual
demand and supply excluding …ll or kill orders.
10
Appendix B: Proofs
Theorem 1. Proof.
(i) Consider the converse case of a high true value equilibrium with p = v
with bid ask spread
and
> 0 where no equilibrium seller k 2 f1; :::; N g is pivotal
in the sense that selling less than speci…ed by equilibrium strategies triggers the
distinguished player i = 0 to sell and to reduce e¤ort subsequently. In this case
any equilibrium seller can improve by selling less since by not being pivotal he can
be sure that this does not trigger a distinguished player’s e¤ort.
(ii) In a low true value trade equilibrium with p = v with bid ask spread
> 0 a
buyer could improve by not buying since he pays pb = v + . In contrast to the
high true value equilibrium pivotalness does not play a role since by de…nition
players payo¤ cannot decrease below v per share.
14
It is necessary to di¤erentiate in our notation the cases including and excluding …ll or kill orders
since in most real world market mechanisms kill or …ll orders are treated di¤erently. For example, they
are not written in the order book and thereby have no direct in‡uence on the market price.
23
(iii) Construct a strategy pro…le with resulting net trade x
y 6= 0 such that no
player can strictly improve by deviating. To do this let a player j with positive
initial stake
j
> 0 just submit a single limit sell order aj = f j g = f(sj ; v)g.
Moreover, the constraints sj
j
and sj
0
c
v
must be satis…ed. Since
either quantities are continuous or M is large enough, we can …nd quantities that
satisfy the latter inequality. Some other player
a
j
=
j
j submits a single limit buy order
= f(sj ; v)g. And all other players submit nothing. If
maximizes the
trade volume at equilibrium price p = v then quantity sj is traded between player
j and player
j and therefore we observe a non-zero net transaction x
y 6= 0. No
player can improve since for any other price players can only trade with themselves
and at true value equilibrium price p = v they are indi¤erent between trading
and not trading. Especially, the distinguished player i = 0 cannot sell enough to
choose e = 0 subsequently. This proves existence for a trade volume maximizing
market mechanism .
Consider now
0
v < c and v > 0. This implies ! 0 v < c and a low e¤ort level
e = 0. This implies that if a true value equilibrium exists then it yields low price
p = v. Existence is constructed similarly as before by picking a pair of players
submitting one buy order and one sell order at this price but nothing else. Again,
a trade volume maximizing market mechanism guarantees that the pair of orders
is executed at this price and nobody can improve. If quantities are continuous or
M is su¢ ciently large, we can again …nd traded quantities that do not allow the
distinguished player to buy shares and exert e¤ort subsequently.
Theorem 2. Proof.
(i) By de…nition in a trade equilibrium there exist equilibrium buyers and sellers.
Assume in contrast to (i) that in equilibrium the distinguished player i = 0 does
not exert e¤ort. Then any buyer can improve by not buying.
(ii) For the distinguished player i = 0 to exert e¤ort e = 1 and to be a seller ! 0 <
0
is not optimal, since by not selling and exerting e¤ort i = 0 can improve. This
proves (ii) and shows that there must be other players being equilibrium sellers.
Otherwise the trade volume would be 0.
24
(iii) Suppose not. Then, equilibrium sellers could bene…t from not selling their shares
since they are worth more than the equilibrium price. All investors could bene…t
from increasing demand if they were not pivotal and if they could buy at the
equilibrium price.
(iv) Sellers only sell in equilibrium if they are pivotal with respect to the distinguished
player i = 0 e¤ort choice.
If
0
s0
c= v, a seller can only be pivotal if the distinguished player sells at least
0
c shares. Since the market microstructure obeys the voluntary trade
rule, this is only possible if the distinguished player o¤ers s0
0
c shares for
sale.
Theorem 3. Proof. Let
v = 0. This means that any costly e¤ort is useless since it
does not yield anything. This rules out the only remaining possibility of an underpricing
equilibrium for p < v and high e¤ort e = 1 since we know already by theorem 1 that
there exists no trade equilibrium with p < v.
Theorem 4. Proof.
Suppose to the contrary that there exists a stochastic true
value trade equilibrium. Then, there exists a rational equilibrium buyer j with strictly
positive probability. The buyer can improve by not submitting any buy order and not
paying
because the buyer is not pivotal.
Theorem 5. Proof.
(i) By de…nition in a stochastic trade equilibrium there exist equilibrium buyers and
sellers. Assume in contrast to (i) that in equilibrium the distinguished player i = 0
exerts e¤ort e = 0 with probability 1. Then, the …rm value will always be low and
any seller can improve by not selling.
(ii) As in the deterministic case, for the distinguished player i = 0 to exert e¤ort e = 1
and to be a seller ! 0 <
0
is not optimal, since by not selling and exerting e¤ort
i = 0 can improve. However, we have seen in (i) that i = 0 exerts e¤ort with
strictly positive probability. This proves (ii) and shows that there must be other
players being equilibrium sellers with strictly positive probability. Otherwise the
equilibrium would not be a strong stochastic trade equilibrium.
25
(iii) We have seen that in equilibrium the distinguished player exerts e¤ort e = 1 with
strictly positive probability. Since the stocks are on average traded below their
expected value, every equilibrium seller would prefer not to sell. The only way to
guarantee that they sell less is pivotalness. If they sell less, the stocks they keep
are worth even less because selling less triggers the distinguished player i = 0 to
reduce e¤ort. Similarly, all investors want to buy stocks at the expect stock prices.
There are now two ways to prevent them from increasing demand. Pivotalness with
respect to the distinguished players e¤ort decision which is similar to the seller’s
pivotalness. Additionally, outside investors may refrain from buying additional
stocks because they would bid up the stock price up or above its expected value
which then makes buying additional stock unattractive.
(iv) If
c= =v, the distinguished player i = 0 must sell at least quantity s0
0
0
c in order to reduce e¤ort subsequently. Since the market mechanism obeys
voluntary trade, this can only happen if the distinguished player submits a selling
order at least of s0
Proposition 1. Proof.
0
c.
Specify
<
c
0
; <
1
2
c
and
0
0
c= v. To prove
existence of an underpricing equilibrium we have to specify a strategy pro…le a such
that the price and net trade (p ; x; y) =
mechanism satisfy (1) p < v
A(
, (2) ! 0 v
; a ) induced by the Amsterdam market
c high e¤ort of the distinguished player
i = 0; and (3) which is a best response for all players i = h0; :::; N . Before proving
c
c
existence under A , note that there exists a unique price p0 2 v
;v
+ \ P.
0
0
The crucial property of this price is that p0 is the maximal price in P below which the
distinguished player prefers not to sell and exerting e¤ort e = 1 to selling all his shares
and exerting low e¤ort e = 0 since
p0 < v
c
0
+ ,
0v
c>
0
(p0
(*)
)
We call this property (*). Now, consider strategy pro…le a speci…ed by the following
conditions.
(i) a0 =
0
with
0
(p0 ) = (
0 ; p0 ):
The distinguished player i = 0 submits a single
…ll-or-kill order using p0 as the limit price to sell his entire
26
0
stocks.
(ii) Further at least 2 investors i 6= 0 submit a …ll-or-kill order to buy
a market order, i.e. for these investors
i
=(
0 ; 1)
2 ai .
(iii) Buying shareholders submit market buy orders (bi ; 1) with
stocks using
0
PN
i=1 bi
= q, hence
the entire market quantity is bought by investors who use market orders.
(iv) Selling shareholders use limit orders (si ; p0
keeps at least a proportion
) to sell their shares. Each seller
of his initial stake, i.e. si
PN
sum up to the traded quantity i=1 si (p0
) = q.
v+
(v) Finally, some shareholder j 6= 0 submits a limit sell order
limit p0 and a quantity sj satisfying
0
> sj
v
v+
i.
j (sj ; p0 )
Selling bids
using a price
maxi si , i.e. between
0
and the
largest equilibrium seller quantity maxi si .
6
q+2
q+
0
0
q
p = p0
-
p
q
q
sj
q
sj +
0
Figure 1: Underpricing equilibrium aggregated excess demand correspondence for the
Amsterdam rules
27
First, we observe that applying the Amsterdam market mechanism
pro…le a implements market price p = p0
A
and trade volume (p ) = q. To see this
note that the trade volume is maximized and the market clears at all p 2 [p0
Further, our assumption
0
to strategy
c= v () v
c
0
v implies v
;v
].
p and hence p is
the price closest to the reference price v. Inspection of …gure 1 yields that the trade
volume at p = p0
is (p ) = q. It remains to show that strategy pro…le a satis…es
conditions (1), (2), and (3).
Since by speci…cation the bid ask spread satis…es
v
+ and hence p = p0
<v
c
<
0
we have p0 < v
c
0
+ <
which shows that condition (1) holds. Since the
distinguished player does not trade at p and therefore ! 0 v =
0
v
c and condition
(2) holds. Unsurprisingly, the most work is condition (3) to show that nobody can
gain by deviating. Instead of tediously checking the universe of all possible deviations
we proceed by discussing only optimal deviations. Moreover, any particular deviation
will lead to a certain price and allocation given the behavior of all other players. We
organize possible deviations …rst by the resulting price level denoted by p^. Second,
among all deviations implementing price p^ only consider an optimal deviation for any
player i in the sense that no other deviation of this player can yield more.
(i) p^ = p : To increase demand or supply at p can be achieved by all players but is
not pro…table. Increasing demand at p is not bene…cial since the order will not
be executed due to price priority. Increasing supply is not bene…cial since either
the supply bid is not executed or the supply bid is served and the agent looses
v
p per share sold. The distinguished player i = 0 cannot gain from deviating
by property (*) because p < p0 . The 2 or more shareholders who o¤er to buy a
block of stocks cannot gain from cancelling the …ll or kill order since their orders
are not executed in equilibrium. Finally, reducing supply does not result in p^ = p .
Reducing demand or not trading at all is not bene…cial neither. To see this note
that not buying yields 0 instead of v
speci…cation
<
1
2
c
0
implies v
pb = v
p0
p0
per share. But our
> 0.
(ii) p^ < p : This can only be achieved by increasing supply. Since all shares not
sold are worth 1 > p in equilibrium, this is not a pro…table deviation and the
distinguished player does not bene…t by construction of p .
(iii) p^ > p : An equilibrium seller i could deviate by using a beller strategy, buying
and selling at a price p^ > p . Although buying and selling some amount at the
28
same time cancels out seller i gains si (p ) (^
p
p ) for his original equilibrium sell
quantity by raising the sell price from p to p^. The downside is that this deviation
of an equilibrium seller only works if he buys the additional supply sj . This
deviation triggers the distinguished player to sell and to reduce e¤ort. Thereby
player i looses sj (^
p
v) since p^ is what he has to pay for any such share and v is
what it is worth later on. Comparing gains and losses yields
gains = si (p ) (^
p
since sj
p)
max si (p ) (^
p
i
maxi si by speci…cation of a and
p^ p
p^ v
p)
sj (^
p
v) = losses
< 1 since p > v. Alternatively
an equilibrium seller i could increase the stock price just by
by reducing supply
to s0i < si and not buying stocks. By this deviation seller i gains less than si . It
is not pro…table, however, since sellers loose
stock
si
i
s0i
v
v+
i
>
i
v on each share of their remaining
si . In our speci…cation of a for every equilibrium seller holds
, si <
v(
si ). Thus, comparing gains and losses yields
i
gains < si <
v(
si ) <
i
v(
i
s0i ) = losses.
Furthermore, a price p^ > p could be induced by other investors who increase
demand. This is not bene…cial since this triggers the …ll or kill order to be executed
and decrease e¤ort of the distinguished player. This is also true for the k players
who submit …ll or kill orders. The distinguished player’s threat to sell his shares
will still be viable because more than one player submits a …ll or kill order.
Proposition 2. Proof.
Specify
<
1
2
c
and
0
c= v. Since the strategy of this proof is similar to
0
the existence proof of underpricing equilibria for the Amsterdam market rules we provide
a less detailed proof. Consider strategy pro…le a speci…ed by the following conditions.
(i) a0 =
0
with
0
(p0 ) = (
0 ; p0 ):
The distinguished player i = 0 submits an order
using p0 de…ned as in the existence proof of the Amsterdam trading rule as the
limit price to sell his entire
0
stocks.
(ii) An investor j 6= 0 submits a limit order to sell less than half of what he owns, i.e.
quantity q <
j
2
at limit price p0
.
29
(iii) Each investor i = 0; 1; :::; N submits a market buy order with quantity bi = N q+1 .
P
Together these orders add up to N
i=0 bi = q, hence the entire market quantity is
bought by investors who use market orders.
(iv) Finally, it holds that
to buy (at least) bk
0
q
N +1
c= v < q. Investor k submits an additional buy order
using a price limit p0
.
The aggregated excess demand correspondence induced by a is shown in …gure 2.
aggregated demand
bk +
6
k
k
p0
p0
-
price
k
(
k
+
0)
Figure 2: Underpricing equilibrium aggregated excess demand correspondence for the
Kyle market rules
Applying the Kyle market rules to strategy pro…le a yields price p = p0
!0 v =
0
v
and
c and in turn high e¤ort e = 1. It remains to show that there
is no strictly improving deviation for some investor. As before we organize possible
deviations …rst by the resulting price level denoted by p^ and second, among all deviations
implementing price p^ only consider an optimal deviation for any player i in the sense
that no other deviation of this player can yield more.
(i) p^ = p : Deviations leading to the same price p^ = p are not bene…cial. All N + 1
investors submit buy orders to buy stocks at p . Since the Kyle trading mechanism
P
picks the allocation that minimizes i (xi )2 + (yi )2 , no investor can increase the
executed order size beyond
q
N +1
by increasing demand at p . To reduce supply
at p is not possible without triggering an increase in the stock price. Finally,
30
the distinguished player i = 0 cannot gain from selling at p by construction of p0 .
Again, as for the Amsterdam rule not buying yields 0 instead of v pb = v p0
per share. But our speci…cation
<
1
2
c
0
implies v
p0
> 0.
(ii) p^ < p : Picking a stock price at p^ < p is not possible for buying shareholders and
not pro…table for the selling shareholder.
(iii) p^ > p : Deviations leading to stock prices p^ > p are clearly not bene…cial for
shareholders buying stocks but could potentially be bene…cial for the distinguished
player i = 0 or the selling shareholder i = j and achievable using a beller strategy.
Since D (p) = q = (p ) for all p^ > p , the distinguished player cannot increase
! 0 above ! 0 and hence to drive up the stock price p^ > p only makes buying
stocks more expensive. The selling shareholder j cannot gain from picking a stock
price p^ > p since j can at most sell q stocks at a price p^ > p . This price
increase, however, triggers the distinguished player to reduce e¤ort to e = 0 and
thus comparing j’s deviation payo¤ with his equilibrium payo¤ yields
p^ q + 0 (q
j)
1 q + 0 (q
j)
<p
q+1 (
j
q)
while the last inequality holds by speci…cation of a , in particular by q <
j
2
.
The low e¤ort is triggered even if the selling shareholder withdraws his buy order
because then investor’s k buy order will be served.
31
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