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         {\bf Game Theory in Finance
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 Eric Rasmusen\\
     \par\noindent
June 25, 1991 draft. Published: {\it  The New Palgrave Dictionary of Money and
Finance,} edited by John
Eatwell, Murray Milgate, and Peter Newman. New York: Stockton Press, 1992.\\



        {\it Abstract}\\
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   \hspace*{.2in} Draft: 4.5.  (Draft 1.1,
 July 1990). 

% Thank the CSES. Brennan, Png, Wlech.
 

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\begin{small}
               \noindent 
\hspace*{20pt} 2000: Eric Rasmusen, 	Professor of Business Economics and Public
Policy and Sanjay Subhedar Faculty Fellow,   Indiana University,
Kelley School of Business, BU 456,   
  1309 E 10th Street,
  Bloomington, Indiana, 47405-1701.
  Office: (812) 855-9219.   Fax: 812-855-3354. Erasmuse@indiana.edu.
Php.indiana.edu/$\sim$erasmuse.
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  Game theory is a modelling approach which drops perfect
competition's assumption that individuals are price-takers and
instead requires them to behave strategically, taking into account
that their actions will alter the behaviour of the rest of the market.
This may be in a context in which two players consciously choose actions
that affect each other, as in duopoly, but in finance it is more
common for one player to try to manipulate the behaviour of  competing
players on the other side of the market.  In either case, game theory
addresses strategic behaviour by defining the players in the game, the
payoff functions they are maximizing, and the strategies available to
them.  It is crucial to delineate carefully the order of actions and
the information available to each player: precommitment and
information transmission are the two pillars of modern game theory.

 Game theory's most important contribution to finance is a very old
one: the theory of expected utility, which was detailed  in the second
edition of Von Neumann and Morgenstern's {\it Theory of Games and
Economic Behavior} in 1947.  Putting that aside, game theory has been
most useful in the context of asymmetric information, which has
increased in research importance as returns have diminished in the
economics of uncertainty. Finance, perhaps even more than other
subjects, is amenable to game theory's approach.  In financial
markets informational advantages matter, events are cleanly defined,
and the important participants are experienced players with enough at
stake to justify careful thought.  Thus, the stylized models and
sophisticated rationality of game theory may apply better to markets
for corporations than for cantaloupes.

  This article will attempt no more than to convey the flavour of game
theory in finance. For particular techniques, see Kreps (1990) or
Rasmusen (1989); for references, see Harris and Raviv (forthcoming).
Rather than survey the literature, I will here convey its flavour
using three typical models, of signalling, commitment, and incentive
design.



{\it Example 1. Signalling in Tender Offers: Why Do Tender Offers
Sometimes Fail? } A major use of game theory is to formalize
intuition, obtain a combination of intuitive and counterintuitive
results, and then refine the intuition by modifying the model until
the results become realistic and their origins understood.  A
sequence of models of tender offers illustrates this nicely.  If a
bidder who can increase a target firm's value by $z$ makes a
conditional tender offer that is $x<z$ above the current stock price,
no shareholders will sell, even though they would jointly benefit.
The shareholders are in a prisoner's dilemma: it is better (by $z-x$)
to be a holdout than a tenderer if the offer succeeds and no worse if
it fails, so every shareholder holds out (Grossman and Hart [1980]).
By this argument, tender offers should never occur, but Shleifer and
Vishny (1986) point out that if the bidder begins with a stake of
$\alpha$ in the company, tender offers can be profitable. The bidder
can profit on his original shares even if he offers $x>z$ and loses
on the tendered shares. 

 Hirshleifer and Titman (1990) explain why offers  succeed sometimes,
but not always.  Nature chooses the bidder's synergy value---his
``type''--- to be $z \in (0, \overline{z}]$. The bidder offers a
premium of $x$ for each of proportion $\omega$ of the shares, and
each of a continuum of shareholders decides whether to accept or
reject the offer. If over $(.5-\alpha)$ accept, the payoffs are $x$
for those that accept and $z$ for those that refuse; otherwise, all
payoffs are zero.

 Two kinds of equilibria are possible.  One kind is a ``separating
equilibrium'' in ``mixed strategies'': each type of bidder behaves
differently and the shareholders randomize their behaviour. The bidder
offers $x=z$ for $\omega=.5$ of the shares, and the shareholders
randomize their acceptances so that with probability
$(x/\overline{z})^{\omega/\alpha}$ the offer succeeds. The high-$z$
bidder will not offer a lower $x$ because the offer would more likely
fail and he would lose the potential gain on his initial $\alpha$
shares.  A second kind of equilibrium, less plausible here, is a
``pooling equilibrium,'' in which different types of bidders behave
the same. In one of the  pooling  equilibria, $x=0$ for any $z$,
and offers always fail because any positive offer would be
rejected---the shareholders would all hold out under the
``out-of-equilibrium belief'' that if $x>0$, then $z=\overline{z}$.
Pooling equilibria are ruled out here by the reasonable
out-of-equilibrium belief that a higher bid signals a higher value of
$z$, in which case a low-$z$ bidder could profitably deviate from the
pooling equilibrium by offering a low $x$ and the shareholders would
accept his offer.

 This 
is an example of a ``signalling'' model, with a low premium
signalling low synergy.  What makes a signal credible is that it be
more expensive for one type of player than for another, as the low-premium
offer (with its lower probability of success) is for the
low-synergy player.  Signalling models often have multiple
equilibria, and much research effort has been devoted to refining the
equilibrium concept to reduce the number of equilibria (see Chapter
xxx of Kreps [1990]). 


 %---------------------------------------------------------------

{\it  Example 2. Capital Structure as Precommitment: Can High Debt Help
Business?}
  Players often undertake actions to restrict their actions or
information later in a game. An example is the following capital
structure model based on Brander and Lewis (1986).  The players are
two firms  in the same market. In the first move, the firms
simultaneously choose debt levels, and in the second they
simultaneously choose output levels $q_1$ and $q_2$. Nature then
chooses the level of a random demand shock $z$ and profits are
realized.  
It is assumed that firm $i$'s profit, $\Pi_i(q_i,q_j,z)$, is
decreasing in $q_j$ and increasing in $z$, and that the {\it
marginal} profit ($\partial \Pi_i/\partial q_i$) is increasing in the shock
$z$. When $z$ is large, a firm's profits are higher,
especially if it has chosen a high output level.

  If both firms choose zero debt, this is  the Cournot game with
uncertainty: the firms trade off the advantage of high output when
$z$ is large against the disadvantage when $z$ is small.  A firm with
heavy debt, however, would go bankrupt if $z$ were low in any case,
and its shareholders do not care about the disadvantage of high
output in that state, thanks to limited liability.  They do benefit
from high output when $z$ is high, so heavy debt is an incentive for
high output.

 A seller in a Cournot duopoly would like to be able to commit to
high output, because this induces his rival to choose a lower output.
Debt is a form of precommitment. When firm $i$ incurs debt, firm $j$
knows that $i$'s incentive to produce high output has increased, so
$j$ will cut back.  If both firms incur debt, however, which is the
equilibrium here, both of their incentives for high output have
increased, and compared to zero debt both outputs are greater and
both profits lower.

 Unlike Example 1, this is a game of symmetric information, where the
focus is on commitment, not information transmission. Each firm
deliberately risks bankruptcy to create a conflict of interest
between debt and equity that increases its aggressiveness in seeking
market share.  The outcome is worse for the firms than if they
jointly avoid debt, because debt lowers industry profits while
helping the firm that uses it as a commitment tool---another example
of the prisoner's dilemma.


{\it Example 3. Incentive Design: Why Use Financial Intermediaries?} In some
models, the players begin
with symmetric information, but they know that certain players will
later acquire an advantage. In Diamond's 1984 model of financial
intermediaries, $M$ risk-neutral investors wish to finance $N$
risk-neutral entrepreneurs. Each entrepreneur has a project that
requires 1 in capital and yields $Y$, where $Y$ is initially unknown
to anyone.  If $Y <1$, he honestly cannot repay the investors, but
the problem is that only he, not the investors, will observe $Y$,
 so they cannot validate his claim that $Y<1$.  They must
rely on one of two things to elicit the truth: monitoring or an
incentive contract. Under monitoring, each investor pays $K$ to
observe $Y$, which makes it a contractible variable, on which the
repayment can be made contingent.  Under the incentive contract, the
entrepreneur suffers a dissipative punishment $\phi(z)$ if he repays
$z$.  The cost of monitoring is $MK$, while the expected cost of an incentive
contract is  $E\phi$, so in the absence of an intermediary the
incentive contract is preferred if $ E\phi < MK$.

 The purpose of a financial intermediary is to eliminate redundancy
by replacing the $M$ individual monitors with a single central
monitor. The intermediary itself requires an incentive contract, at
cost $E\phi$, and it must spread this cost over several entrepreneurs
to make its existence worthwhile.  Otherwise, if $N=1$, the
intermediary incurs a cost of $K$ for monitoring and $E\phi$ for its
own incentive, whereas a direct investor-entrepreneur contract would
cost only $E\phi$.

The use of this model is to show that (a) an intermediary helps
 only if there are both many investors and   many entrepreneurs,
and (b) incentive contracts have economies of scale compared to
monitoring.  The model is an example of theory-based institutional
economics: the institution takes its particular form to avoid
information problems by contracting while information is still
symmetric.


 Each of the three models discussed above is typical of a literature,
and other literatures in finance (e.g., executive compensation,
market microstructure) also use game theory.  The number and
intricacy of the models can be daunting, and they have been
criticized for the difficulty of empirically verifying the
assumptions and for sensitivity to seemingly minor changes in
assumptions about what information is available or who moves first.
Whether these criticisms apply depends on the model, but they may
apply without being truly objectionable. It is unfortunate if
important variables are hard to measure, but that does not diminish
their importance; rather, case-by-case verification must replace the
regression-running that has dominated economists' empirical work.
Sensitivity to assumptions is not a drawback but a {\it contribution} of
game theory, pointing out the importance of what were once thought to
be insignificant details of the world.  Just as marginalism is more
than the application of calculus to old problems in economics, so
game theory is as important for changing the agenda as for introducing new
techniques.


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\noindent
BIBLIOGRAPHY.\\

Brander, J. and Lewis, T. 1986. Oligopoly and financial structure:
The limited-liability effect. {\it American Economic Review} 76: 956-970.

 Diamond, D. 1984. Financial intermediation and delegated monitoring.
{\it Review of Economic Studies} 51: 393-414.

 Grossman, S. and Hart, O. 1980. Takeover bids, the free-rider
problem, and the theory of the corporation. {\it Bell Journal of
Economics} 11: 42-64.

 Harris, M. and Raviv, A. Forthcoming.  Financial contracting theory.
In {\it Advances in Economic Theory: Sixth World Congress}, edited by
Jean-Jacques Laffont, Cambridge: Cambridge University Press.

 Hirshleifer, D. and Titman, S. 1990. Share tendering strategies and
the success of hostile takeover bids. {\it Journal of Political
Economy} 98: 295-324.

 Kreps, D. 1990. {\it A Course in Microeconomic Theory.} Princeton:
Princeton University Press.

 Rasmusen, E. 1989. {\it Games and Information}. Oxford: Basil
Blackwell.


 Shleifer, A. and Vishny, R. 1986. Large shareholders and corporate
control. {\it Journal of Political Economy} 94: 461-88.

Von Neumann, J. and Morgenstern, O. 1947.  {\it The Theory of Games
and Economic behaviour.} Second Edition. New York: Wiley.

  
 
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